Relativistic Quantum Field in Theoretical Physics
by
fEJSA HU, ETTI !NSTITUTE
OF TECHNOLOLGY
Trung Van Phan
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
AUG 102015
LIBRAR IES
Bachelor of Science in Physics
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2015
@ Trung Van Phan, MMXV. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute
publicly paper and electronic copies of this thesis document in whole or in part in
any medium now known or hereafter created.
Author ..........................................
Signature redacted
Department of Physics
May 8, 2015
C ertified by ......................................
Signature redacted
Jeise D. Thaler
Assistant Professor
Thesis Supervisor
Accepted by ...............................
Signature redacted
Nergis Mavalvala
Associate Department Head for Education
2
Relativistic Quantum Field in Theoretical Physics
by
Trung Van Phan
Submitted to the Department of Physics
on May 8, 2015, in partial fulfillment of the
requirements for the degree of
Bachelor of Science in Physics
Abstract
Quantum field theory is the most well-developed tool in theoretical physics to study about the
dynamics at microscopic scales, with particles and quantum behaviors. In this thesis I'll review
about the construction of quantum field theory from the S-matrix point of view (building up
particles and formulating interactions), then poke at interesting topics that are usually briefly
mentioned or even ignored in standard quantum theory of field textbooks. In more details, chapter
1 will be about how a quanta is defined from the analysis of group theory, chapter 2 will be focus on
how the language of field to describe the quantum behaviors of quanta is desirable for interactions
and arised quite naturally (it should be noted that, string theory can be viewed as a totally
different theory in quantum interactions, with conformal symmetric and topological natures, but
at low energy scale it can always be reduced back to quantum field theories), and chapter 3 will
be about several angles of quantum field theory.
Thesis Supervisor: Jesse D. Thaler
Title: Assistant Professor
3
Acknowledgments
I'd like to express my sincere gratitude to Massachusetts Institute of Technology (MIT) for letting
me pursuing my dream about studying theoretical physics here. I would also like to thank Prof.
Jesse Thaler and the Physics Department for guiding me with this thesis from start to finish. I'm
indebted to Prof. Jesse Thaler, my thesis supervisor, for his understanding, patience, enthusiasm
and encouragement during all these years in MIT. To Prof.
Leonid Levitov, Prof.
Edmund
Bertschinger, Prof. lain Stewart, Prof. Hong Liu and Prof. Washington Taylor, I'm extremely
grateful for the past discussions and perspectives from you in different topics in theoretical physics,
which some of them are mentioned in this thesis. To all my friends for supporting me during
hardship, especially Thai Pham, Duy Ha, Dan Doan, Nhat Cao, Tru Dang and Truong Cai. To
my family for always having faith and not giving up on me.
4
Contents
Quanta
.
. . . . . . . .
1.1.2
Poincare Group and Poincare Algebra
.
Under Spacetime Transformation
. . . . . . . . . . .
8
. . . . . . . . . . . 11
1.2.1
Casimir operators . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 11
1.2.2
Quanta definition . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 12
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 14
1.3.1
4-momentum Label . . . . . . . . . . . . . . . .
. . . . . . . . . . . 14
1.3.2
Details Dynamical Information of Quanta
. . .
. . . . . . . . . . . 16
1.3.3
More on Spin and Helicity . . . . . . . . . . . .
. . . . . . . . . . . 18
1.3.4
Quanta in Flat Spacetime and Curved Spacetime
. . . . . . . . . . . 21
.
.
.
.
Fixed Characteristics . . . . . . . . . . . . . . . . . . .
Labeling a Quanta
22
. . . . . . . . . . . 22
2.1.1
Scattering Setting . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 22
2.1.2
Creation and Annihilation Operator . . . . . . .
. . . . . . . . . . . 24
2.1.3
Causality Condition
. . . . . . . . . . . . . . .
. . . . . . . . . . . 26
2.1.4
The General Form of S-Matrix . . . . . . . . . .
. . . . . . . . . . . 28
2.1.5
Internal Symmetry and Conserved Charge
. . .
. . . . . . . . . . . 28
2.1.6
Cluster Decomposition . . . . . . . . . . . . . .
. . . . . . . . . . . 30
. . . . . . . . . . . . . . . . .
31
. . . . . . . .
31
Macroscopic Description
.
.
.
.
.
.
.
.
Construction of Multi-particles Dynamics . . . . . . . .
2.2.1
Creation and Annihilation Fields
2.2.2
Finite-Dimensional Representations of the Lorentz Group.
5
.
2.2
8
. . . . . . . . . . . 10
2 Field
2.1
. . . . . . . . . . .
. . . . .
.
1.3
1.1.1
.
1.2
Spacetime Symmetry . . . . . . . . . . . . . . . . . . .
.
1.1
7
.
1
32
. 36
2.2.4
Spacetime Rotation and Particle Self-Rotation . . . . . . . . . . . . . . .
41
.
.
Physical Requirements for the Theory . . . . . . . . . . . . . . . . . . . .
Quantum Field Theory
.
Massive Quantum Field
. . . . . . . . .
. . . . . . . . . . . . . . . 45
3.1.2
Massless Quantum Field . . . . . . . . .
. . . . . . . . . . . . . . . 47
. . . . . . . .
. . . . . . . . . . . . . . . 49
3.2.1
Gauge Freedom . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 49
3.2.2
Renormalization . . . . . . . . . . . . . .
. . . . . . . . 51
3.2.3
Long Distance Physics . . . . . . . . . .
3.2.4
Maxwell's Electromagnetism and Einstein's General Relativity . . . . . . . . 62
3.2.5
Classical Field Configuration . . . . . . .
.
. . . . . . . . . .
. . . . . . . . . . . . . .
.
Functional Integration
. . . . . . . . 57
. . . . . . . . 64
.
.
.
.
Collective Behaviors of Quantum
.
.
.
3.1.1
. . . . . . . . . . . . . . . . . . . 65
.
. . . . . . . . . . . . . . . . . . . 66
3.3.1
Connection to Canonical Quantization
3.3.2
Is Functional Integration always Superior? . . . . . . . . . . . . . . . . . . . 70
3.3.3
Mathematical Rigors . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 74
3.3.4
Effective Field Theory
. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 75
3.3.5
Quantum Anomaly . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 93
3.3.6
W avefunctional
.
. . . . . . . . . . . . . . . . . . .
.
3.3
. . . . . . . . . . . . . . . 44
.
3.2
The Free Theory Lagrangian . . . . . . . . . . .
.
3.1
44
.
3
2.2.3
6
. . . . . . . . . . . . . . 100
Chapter 1
Quanta
Unchanged properties under the change of observers and reference frames are of great interests in
understanding nature, since they are very useful to formulate an universal theory and also easy
to keep track of, experimentally. Since it is expected that the law of physics is the same in every
inertial frame, which is a postulation in special relativity (with a depth philosophical reason),
hence one can partially define the building blocks of the universe to be objects associated with
the set of most the fundamental and fixed characteristics under the transformations of spacetime
that relate different inertial frames together (to get the full definition, one has to take into account
the interactions). The transformations are described by Lorentz group, which is the result from
the other postulation of special relativity (unsurprisingly, based on a constant - the speed of
light), and the spacetime translation group, which is seen in flat spacetime.
Combining these
two gives the Poincare group. From the experiments, it is known that nature exhibits quantum
(discretization) behavior, therefore the fundamental building blocks should be one-particle states
and multi-particles states.
A quanta is the physical realization of an one-particle state.
From
the quanta point of view, the physical transformation that unchanges the presenting physics is a
classical interpretation for symmetry.
7
1.1
1.1.1
Spacetime Symmetry
Under Spacetime Transformation
Transformation of Spacetime Coordinates
The spacetime symmetry is a global symmetry described by the Poincare transformation, with
Lorentz rotation Al and spacetime shift a":
T(A, a):x:-+" a' = AI",xv +a , T(A', a')T(A, a) = T(A'A, A'a +a')
(1.1)
Since Lorentz transformation preserves the 4-vector length, the Lorentz parameter satisfies:
, --+ A =
To,,AvA , =
-1
0
0
0
0
+1
0
0)
0
0
+1
0
0
0
0
+1
(1.2)
(1.3)
uir
1 ", = AAnan , (Ado)t , = A" = lsoP te hAPt:
To have the connection with unitary, the determinant det A should be 1. Also note that:
n/A "oAvo= qoo -+ -(A
o)2 + (A'o) = -I,
(1.4)
(A00) 2 >
There are disjoint regions of the real axis for possible A 0 0 , and for linking requirement with identity
transformation, one needs to have A 0 0 > 1. From the Lorentz proper orthochronous subgroup,
with det A = 1 and A 0 0 > 1, to go to the other part of the Lorentz group, spatial inversion P and
time reversal T are employed:
po
1
,
p=
P
'P2P>
_
2
7
3
1
(1.5)
From now on, the proper orthochronous Lorentz group is refered as simply Lorentz group, unless
mention explicitly. Also, Poincare group is with the the proper orthochronous Lorentz group, not
8
the full original Lorentz group. The infinitesimal transformation of the Lorentz group:
At', -+ P' + w", , wA =
p,
w",
=
r""w ,p, ;
ap
-+
16
(1.6)
The Lorentz transformation of the metric indicates that w.,, is anti-symmetric:
AU
77t = 7PAP
7
=
= TipcG(P + w"i)(5" + w 0 ,)
(1.7)
w,,
(1.8)
+ wI, + WVA +
... --
Transformation in the Hilbert space
In quantum terminology, any collection of properties can be represented by a ray of normalized
vectors in the physical Hilbert space. The transformation in spacetime coordinates is realized by
a unitary linear transformation, with similar composition rule [36]:
1IT) -+ U(A, a) IT) , U(A', a')U(A, a) = U(A'A,A'a+ a')
(1.9)
Note that, there's a caveat here. The associatity of the symmetry group representation on the
physical states is not always exactly equal, because the transformation T (a general symmetry
transformation, not just restrict to the spacetime Poincare transformation) take a ray to another
ray but don't really put a restriction at the possible phase arise:
U(T')U(T) = eOT1,T) U(T'T)
(1.10)
Because of linearity (also, anti-linearity), the phase should be independence of the state which
the transformation acts on.
The name for a representation with a general
#
is the projective
representation [36]. The symmetry group cannot tell by itself whether the physical states furnish
an ordinary or a projective representation, but its topology can tell us whether the group has any
intrinsically projective representations (supporting non-zero phase
#
in what way). But since any
symmetry group with projective representations can always be enlarged in a way without changing
its physical impications (for example, go from single-covered to double-covered or even multicovered) so that its representations can all be defined as non-projective, hence set 0 = 0. From
now on, unless specify, the choice of U is always that of the representations after the enlargement
9
was done:
U(T')U(T) = U(T'T)
(1.11)
The spatial inversion and time reversal are the extensions of Lorentz rotation, described by:
=
U(P, 0)
,
7 = U(T, 0)
(1.12)
If a state is the eigenstate of the Poincare transformation, then the same still holds after spatial
inversion and time reversal (in more details, there's only a phase diffent between these).
1.1.2
Poincare Group and Poincare Algebra
If there is no transformation, it is expected that U(1, 0) is the same as the identity. Under an
infinitesimal Poincare transformation, the corresponded transformation of quantum state can be
written with independence operators J" and P", which should be Hermittian so U is unitary:
U(1 + W,
=
I) -iJ"
1+
2
- iP
+
,
...
J"= -J"
(1.13)
The J" for Lorentz rotation and P" for translation shift are called the generators of the Poincare
group. One can associated physical meaning to these operators. From quantum mechanics, since
the energy operator, also known as the Halmintonian H, is related to the translation generator
in the time direction, therefore H = P'. Similar arguments can be applied to the 3-momentum
P = {P1 , P2 , P3 } and the angular momentum f= {J 23 ,
31
, 712}. The remaining opertors don't
have a classical analogy since they come from special relativity, thus we will borrow the name from
there, the boost 3-vector K
{J0 1 , J 2 , J0 3 } [36]. The algebra of these generators can be read-off
by considering a following transformation:
U(A, a)U(1 + w, e)U'(A, a)
U (A(1+ w)A- 1 , Ac - AwA-1a)
U(A, a)(1 + -iwitV
=
+
AP'J)U
(1.14)
1+ 2i(AwA-1),,Jv - (Ae - AwA-'a),P1'
(1.15)
Hence, the operators JI" and PP are transformed as:
U(A,a)JIlvU-(A,a)= A"A,"(JP'
-
alPv + avPl) , U(A, a)PPU-l(A, a)
10
A,"PV
(1.16)
-=H,
H,-1 = -H
Take A -+ 1
,
P
=
f-I
-1,
-
p
=
1
TJ_
- 1 = -k
f0,
=-
,9K,-7
=K
(1.17)
(1.18)
+ w' and a -+ E' infinitesimally, then one arrives at:
2
The commutation algebra of the generators of the Poincare group is found to be:
i[JW", JPa] -
qVPjA01 - 7J
- 7
i[P", J"'] = r 1VPP - nr P"
,
/J"
+ 7O"
,
[Pt, P"] = 0
(1.20)
(1.21)
Lorentz generators J", in general, obey the 0(1, 3) algebra, but the restriction to the proper
orthochronous subgroup make it SO+(1, 3) [17, 361. The spacetime translation algebra, from the
.
generators PY, is trivially T 4 . Combining these, one arrives at the Poincare algebra SO(1, 3)+ 0T4
Operators that constructed from the Lie algebra generators which commutes with all of these
are called Casimir operators, and the eigenvalues associated with Casimir operators are fixed
characteristic of the transformation with generators in that algebra. In other words, the eigenvalues
of the Poincare algebra's Casimir operators are what define the fundamental building blocks of
physics, and the argument should applies to a quanta.
1.2
1.2.1
Fixed Characteristics
Casimir operators
The Casimir operators C1, 02 of SO(1, 3)+ 0 T4 (which commute with all generators of the symmetry group) are the length-squared of the 4-momentum P-0 and the Pauli - Lubanski 4-vector
WA [17, 26, 32, 36]:
C1 = -P
2
=
(P)
2
__ P 2 ,
C2 = W2
,
11
W
=
2
A"U"PJPU ,
WVV P, = 0
(1.22)
It's straight-forward to see that C1 is nothing but the mass-squared M2 in special relativity. The
mass-squared, however, can be zero, positive or negative.
The negativity of the mass-squared
seems to violated the causality of special relativity (as from the energy-momentum dispersion
relation, a quanta with negative mass - a tachyon - should move faster than light), however,
when constructing a field theory out of the individual particles and interactions (perturbative
description), one realizes that a tachyon is not only non-localized when moving faster than light [7]
but also indicating an unstability of the configuration where the full (nonperturbative) theory is
formulated around. Therefore, we will only consider the cases where the mass-squared is positive
and zero. Also note that, the mass is always chosen to be non-negative, so that the positive energy
branch is associated with particle creation, and the negative branch is for particle annihilation.
1.2.2
Quanta definition
Massive Quanta and Spin
Consider a massive quanta C1
=
M 2 > 0 and go to its rest frame, then the meaning of C2 in
quantum mechanics can be seen:
PP = (m, 0)
+
W = Mj
,
C2 =1W2 = -M2P =M
2
s(s + 1)
(1.23)
Hence, C2 encodes the information of the spin s, a well-known concept in quantum mechanics
describing the spatial rotational nature of the quanta with s E 1N [32].
Massless Quanta and Helicity
For massless quanta, both C1 and C2 vanishes (not quite, it's actually that non-zero C2 is never
seen in nature, so for physical purposes, let's just consider C2
=
0 only), and from the vanishing
of W"P,, since the only null 4-vectors whose products with a given non-zero null vector vanish
should be those propotional to that vector, therefore:
Wh = hP
,
P" = (PO, 0 ,,0 PO) -> h =
(1.24)
|PI
12
The ratio h is the helicity, which can be viewed as the angular momentum in the direction of
motion. To see the possible value for the helicity, note that in the chosen frame above:
Wo
J=P
3 ,
W 1 =P0 (
+K 2 )
,
W2 = P"(J2 - K
1
)
(1.25)
The algebra of Wo, W 1 , W2 is closed and same as E2 , isomorphic to SO(2) 0 T 2 [32]:
[W1, W 21= 0 , [W2 , J3] = iW1
,
[W1, J3 ]=-iW2
Although the helicity comes in pairs (with opposite sign
-+
1
2
h -
(1.26)
), it changes sign under spatial inversion,
and since that transformation is not connected to unitary, therefore the value of h is a fixed
characteristic. The helicity for massless quanta plays the similar role as the spin for massive quanta,
therefore, Jhi can also be called spin. The different is that each helicity h is one-dimensional while
each spin s is (2s + 1)-dimensional, and we will see this later.
Particle Type
In summary, there are always two fixed characteristics of the Hilbert physical states, which can
be used in the particle definition (different characteristics for different species): the mass-squared
C1 = M
2
and the spin s c 1N (for massive quanta) or the helicity E
Z (for massless quanta).
For interacting theories, there should be more unchanged characteristics (doesn't have anything to
do with spacetime symmetry), such as the charges, which will be discussed later. After take in to
account of all those characteristics, one has the particle's definition IF (for example, an electron,
a proton, or a neutron). The subset of the complete characteristics can be used to define families
(such as baryons, mesons, leptons).
It is not enough, however, to study the dynamics by just
knowing the particles. It is important to know different dynamical states of the very same quanta.
Therefore, one needs labels to further specify more information of the fundamental degrees of
freedom (the particles) in the theory.
13
Labeling a Quanta
1.3
1.3.1
4-momentum Label
A quanta is partially defined by the fixed characteristics under Poincare transformation (the collection of these are colled IF; we will add more to it later from the information of the interactions),
and can be labeled by the eigenvalues of a complete set of mutually commute operators. The natural choice for the label is the eigenvalue of the 4-momentum p", since the spacetime translation
group T 4 is trivial, which means this choice of label is good. However, the label p
(which will
be shown later) is not a complete description for the dynamical states of a quanta, so let's denote
others be a general label a:
P"I;p,u -) =p";p,)
,
U
e Ip)
, U)(,a)4, Im UTE; p, -) =
(1.27)
Since all information about Pl is known, let's consider a Lorentz transformation, with:
P4U(A) I'T; p,
IT; p, a) =
U(A) (U-1(A)PPU(A))
U)
= A'p"U(A)I ; p,a)
-
U(A); p, a) =
U(A) (A %P") [IT; p, a)
C,(A, p)IT; Ap, a') , U(A, 0) = U(A)
(1.28)
(1.29)
Since the combination is linear, in principle the a label can be choose in such a way that the
matrix C,(A, p) is block-diagonal, as the state
a representation of the Poincare group [36].
14T; p, a)
with a within any block by itself furnish
It is then natural to identify the states with the
components of a irreducible representation of the Poincare group.
Wigner's Little Group Representation
For each value of -P
2
(a specific value of C1, the mass-squared M 2 ),
choose a standard 4-
momentum kP and express all other 4-momentum of the same length by a Lorentz transformation,
then the states can be related by a normalization facter N(p):
p
J=LP,,(p)k' , I4;p,a) = N(p)U(L(p))14;k,a)
14
(1.30)
Applied an arbitrary Lorentz transformation on the above state:
U(A) 'IF;p, o-) = N(p)U(AL(p)) W; k, o) = N(p)U (AL(Ap)) U (L-1(Ap)AL(p))II; k, o)
(L-1(Ap)AL(p))k
=
L 1 '(Ap)Ap = k
-+
W(A,p)
=
(L-1(Ap)AL(p))
,
W,kv = kL
(1.31)
(1.32)
The Lorentz subgroup that leaves the 4-momentum invariant is called the little group, and one
can see already see the relation between this and the Pauli - Lubanski vector, in the sense that
they both respect the very same E2 algebra, which we will discuss in a moment. The little group
transformation can be written as:
U(W)I; k, a) =
Dr,(W)I|;k, o-)
(1.33)
It can be seen that the coefficients D(W) furnish a representation of the little group:
S Data(W'W)I T; k, a') = U(W'W)|IT; k, o-) = U(W')U(W)|IJT; k, o-)
(1.34)
a'
=
>11Da(W)Daa,,,(W') IIF; k, a') -+ Data(W'W) =3
Dai"(W')Da/a(W)
(1.35)
Use this in equation (1.30), one arrives at:
U(A)jT;p, o-) = N(p)
D,',(W(Ap))U(L(Ap)) IT; k, a')
ND
(1.36)
(1.37)
l,(W (A, p)) I ; Ap, a')
This is known as the Wigner's little group representation [24, 36].
Normalization
To determine the normalization, choose the usual orthonormalization in quantum mechanics (the
6-function arises and the completeness relation follows because the states are eigenstates of a
complete set of mutually commute Hermittian operators) [361:
(';
k', 'I T; k, o-)
=
- )6a'a6i-'
15
-('
-+ Dt(W) =D(W)
,
(1.38)
Consider another inner product, with p = L(p)k and p' = L(p)k' (same Lorentz transformation):
(T; p', o-'j T; p, a-) = N(p)N(p)t (I; p', o'lU- (L(p)) U (L(p)) IiT; k, o-)
=
N(p)2D (W (L-1(p), p) )6(3)(
k)
(1.39)
(1.40)
Since at p' -+ p, the transformation W(L-lp,p') -÷ 1, so:
(';p',o'l I; p, u-) = IN(p) 2 (3 ('
O),
-
(3) p
(1.41)
The trick is, under Lorentz transformation, the 4-vector length stay as a scalar, thus [22, 36]:
'-
-+
U(A)I'; p, o-)
p
k0 6( 3)(k'
=
(Ap)0
-
L)
D,,
-+
N(p)
(W(A, p))II;Ap,o')
(1.42)
(1.43)
The only remaining problem now is finding the representations of the little group. The exact form
of D(W) should depend on the definition of the quanta, therefore, in general:
Do, (W(A, p)) = D(, (W(A, p))
1.3.2
(1.44)
Details Dynamical Information of Quanta
Massive Quanta Labeling
For mass positive-definite -p
2
=
M2 > 0, the little group is SO(3), isomorphic to SU(2)/Z 2 so
the representation theory of SU(2) naturally including that of SO(3), with unitary representations
can be broken into the direct sum of irreducible representations spin-s D() of (2s + 1)-dimensional
SU(2), corresponds to (2s + 1) different spin states, with s E -N. The specific value of the a can
be associated with the eigenvalue of one of the little group's generators. Under an infinitesimal
rotation Rij =
Do(1
6
ij + E9i [36]:
9)-
6
+
(
)
,
= -S5,
16
,
a= -s,-s+1,...,s- 1, s
(1.45)
(J()
iJs),
=
Jls)) =
(JlS)
',c,
T o)(j
1/(s
i
a
+ 1)
,
= OUalc
= (JlS)),
(j (s)
(1.46)
The rotational generators in the spin-s representation is given in details, above. To calculate the
element of the little group - Wigner's rotation, one choose a standard transformation L(p) to
carries the 4-momentum kA = (11, 0, 0, 0) to p", and this can be found straight-forwardly:
L
=(p)
(1.47)
, Lo(p) ==a b read-off:
L7 t o or
The Lorentz transformation rule for a massless particle of spin-s can be read-off:
-91) (W(A, p)
0'
T;A,
(1.48)
/
Ap 0
U(A)IT; p, a) =
o
Massless Quanta Labeling
For massless quanta -p 2 = 0, the little group is ISO(2), also known as the Euclidean group
E2, which is composed of rotation and translation. It's a little bit different from SO(2), where
one has only rotation. To see how the translation can arise, choose the standard 4-momentum
k
(k 0 , 0, 0, k 0 ), and note that the little group transformation satisfies [21, 361:
WL k" = k= -+ W = S(a, 3)R(0)
1+
S(a,/3)
(a2+#2)
a
a
13
-(a2+32)
1
0
-a
0 1
-3
R(O)
=,
13
(a2 +
2)
a /
(1.49)
1
0
0
0
0
cos0
sinO
0
0
-sin0
cos0
0
0
1
0
1-0 (a 2 + 0 2 )/
0
(1.50)
From equation (1.12), the corresponding transformation on the physical Hilbert space:
U(S(a, O)R(O)) = I+iaNiiN
2
+i0J
3
, Ni =
+ k,
N2 = -JI
+K$2
(1.51)
The product of any group element can be work out, and one can already see that it's ISO(2)
instead of SO(2) [32, 36]. It can be seen directly from the algebra:
[J3 , N1] = iN 2
[13 ,N 2 ]= -iN1
17
,
[N 1 ,N 2 ] = 0
(1.52)
These results are very similar with what was found by studying the Pauli - Lubanski vector, given
in equation (1.24) and (1.25). The unitary irreducible representations of the group E2 are labeled
by the eigenvalue n'
=
(ni, n2 ) of generator N = (N1 , N2 ) and h of generator J3 . For n2 > 0 one
gets an infinite dimensional irreducible representations, which can be interpreted as particles of
infinity spin, but since there is no such particles ever observed in nature, these representations
are of no physical interests [261. For n2
=
0, the group E2 effectively acts as SO(2) rotation, and
its representations are one-dimensional, labeled by real, integer of half-integer eigenvalue h E !Z,
which is called helicity. The transformation of massless particle is now finished:
Dal, (W) = e
60,
U(A),h;p) =
(Ap) 0 e
h; Ap)
(1.53)
The transformation is independence of a and 13, and the freedom of these parameters is related to
the problem of gauge ambiguity, which will be discussed later in details.
1.3.3
More on Spin and Helicity
Multi-valued Problem of Representations
There's a "sneaky" feature of the continuous group that we ignored when nailing down the possible
values for helicity [36]. Indeed, every continuous group has the possibility of having multi-valued
representations. But it turns out that the existence of multi-valued representations is not without
restriction - it is intimately tied to connectedness, the global topological property of the group
parameter [32j. For example, in the case of SO(2), the group parameter space (which is the unit
circle) is multiply-connected, which implies the allowance of multi-valued representations. Thus, it
is possible to determine the existence and the nature of multi-valued from an intrinsic property of
the group. The occurrence of double-valued representations can be traced to the connectedness of
the group manifolds of symmetry associated with the physical four-dimensional spaces, and that's
the topological reason why the helicity can only be an integer or a half-integer, instead of any
possible real number. For the group SO(3) and SU(2), there's a clear algebraic reason restricting
the spin to be of jN [32, 36].
18
Topology of Spacetime Symmetry
Let's settle the point with helicity of massless particles, one and for all. Any real 4-vector VP can
be used to constructed a Hermittian 2 x 2 matrix:
2
V3V
=-det v
V=
(1.54)
VO-
=
V+V3V 1 -iV
V1i + iV
v
2
,
0oy
In this language, a Lorentz transformation A can be realized as a Hermittian 2 x 2 matrix A [361:
(A,4(A)V)a-, = AvAt , A(A'A) = A(A')A(A)
, det A = 1
(1.55)
The required properties of A means it should be a special linear complex matrix SL(2, C). It is
important to note that although the algebra of A is the same as A, the group SL(2, C) is not
the same as SO(1, 3)+, and this can be seen from the fact that
A represent the same Lorentz
transformation. Mod that out (identifying +A together), one has SL(2, C)/Z2 , which is now the
same as the Lorentz group. Now that the setting is done, we can study the topology of the Lorentz
group (the Poincare group topology is just a trivial extension of this, since the T4 group is nothing
more that adding R' fiber bundle). By polar decomposition theorem, any complex non-singular
matrix can be written as:
A = AeB
A t A=1 , Bt=B ;
dctA=1
-+
detA=1 , TrB=O
(1.56)
In the components form, one can see the topology of A is S 3 and B is R3 :
S
d + ie
-+ig
f + ig
d2 + e2 + f 2 +g 2 =1 ; h=
d-ic
C
a - ib
a+ib
-c
The topology of the group SL(2, C), therefore, is S3 0 R3, which is simply-connected.
SL(2, C)/Z 2 , identifying A with -A
is the same as merging the unitary factors A and -A
(1.57)
For
(since
eB is always positive), then the topology of the Lorentz group SO(1, 3)+ can be understood as
S 3 /Z
2
D R 3 = RP(3) D R 3 , and the geometry is not simply-connected anymore. The manifold
is doubly-connected since the 1st homotopy group r 1 (RP(3)) = Z2 , so a double loop can be
19
continuously deformed to a point. Consider a transformation trajectory 1 -+ A'
(U(A')U(A)U-1(A'A))
-+
A'A
= 1 -+ U(A')U(A) = +U(A'A)
-
1 twice:
(1.58)
Here, we use the caveat mentioned earlier, as the representation is before the enlargement is
done and the phase is set to 0. The sign
can be understood from the phase of the Lorentz
transformation when acts on a physical state, integer-spin always see +, while half-integer can see
- in some cases. It is best to see this from equation (1.52), since after applying a full 2wr rotation,
a phase appears:
A 2 ,p
=
p ,
0(A 2 ,,, p)
= 27r
U(A 2 ,)IT, h; p) =
-+
ei 2 7rhO(A,p) IT,
h; p)
(1.59)
If h is an integer, then the state transform back to itself, bit if h is an half-integer, then the
state gets an opposite sign. Because of the doubly-connected requirement, then ei4 7h should be 1,
therefore the helicity cannot be more fractional than half. Even for higher spacetime dimension,
the arguments hold for both massive spin
this, note that SO(1, D)+ ~ SO(D) 0
(s E
HD.
IN) and massless helicity (h E 1Z). and to see
The hyperbolic D-space
HD
has a trivial topology
of RD, therefore the 1st homotopy can be calculated entirely from SO(D). But it is known that
7r
(SO(D)) = Z2 for D > 3 [11, thus the Lorentz group at higher dimension theory is also doubly-
connected, which only supports spin and helicity no more fractional than half. About the spin states
and the helicity states of a quanta, these can be understood from the projective representations of
the special orthogonal groups. For SO(N), for N odd, it has one spinor representation, categorized
by
N-i
2
half-integers pi
>
-2'1,
and for N even, there are mirror conjugated
spinor representations,
categorized by z2 half-integers pi >-_ - [141.
Still,did we left out something?
The answer is yes, actually we did. To see this, let's take a look at the massless particle again, and to
be precise, its helicity h = -.
This operator is not boost-invariant, therefore Lorentz-invarance
of helicity is not generic [371. The realization leads to the discovery o a new Lorentz-covariant
massless particle type known as the continuous-spin particle (which is labeled by a spin-scale p
with units of mass in 1 + 3 dimensional spacetime). This type of particle is not (yet) found in
nature, but still of physical interests [11, 23].
20
1.3.4
Quanta in Flat Spacetime and Curved Spacetime
Let's conclude this section by summarize the important points, and a remark. From spacetime
symmetry, it's possible to define a quanta (can be a fundamental particle or a composite particle)
to a certain degree, with fixed characteristics and labels for the particle's dynamical information.
To represent massive quanta, the following physical state can be used:
I : M > 0, s ; p
)
-÷
I;p, -)
(1.60)
With massless quanta, the physical state is of the form:
IX : M
= 0, h ; p)
--+
I, h; p)
(1.61)
Note that, all arguments in this section are done in flat spacetime. In a general curved spacetime,
quanta, or particle, is usually a concept of a locally flat region, which is meaningful for the observer
only if its characteristic wavelength (an estimation for the size of the quanta in quantum mechanics,
which is the Compton wavelength for massive particle and the normal wavelength for massless
paritcle) is less than the curvature scale at the position of the observer. It should be noted that
in some special cases, such as in AdS space, particles can be defined similarly by finding the
irreducible representations of the AdS algebra, but that's different from flat space algebra. Indeed,
although the local Lorentz symmetry is fine as the spin is still good for labeling, the 4-momentum
and also the mass are not well-defined and position-dependence since the spacetime translation
is not a symmetry of the theory of general relativity anymore.
This is one of the reasons why
quantum field theory on a curved spacetime, even fixed and non-dynamical, is extremely hard
(but very thought-provoking).
21
Chapter 2
Field
A single quanta is just a perturbative description of the universe. To go beyond that picture, one
should try to study the collective behaviors of many quantum with interactions (multi-particles
dynamics), and realize that the language of field comes out naturally. It should be noted that, field
formulation of the quantum world is so desirable, not only because it's naturally arise, but also
it's a lot easier to dealt with, compare with the language of creation and annihilation operators. It
should be mentioned that, while most of the time theoretical particle studies tend to use the former
language, the later one is still very popular in condense matter physics of interacting system, when
the interactions are not quite complicated.
2.1
2.1.1
Construction of Multi-particles Dynamics
Scattering Setting
To simplify the notation, for a single massless quanta state:
II: MAp= 0, h; p) -+
F: AMi= 0, s =Ihl;p, - = h)
(2.1)
If the particles in the theory are not interacted, then one can always build a multi-particles state
as a direct product of single quanta states. To short-hand the exhausting labels, let's just use H
to keep track of the multi-particles state (the sum integral only include configurations that do not
22
simply differ by just permutation of particles):
Jd3f;id39P
(J;d=
(2.2)
2 ...
The Poincare transformation, the normalization and the completeness relation naturally follow.
The (perm.) takes into account the possibility that a permutation of one of these multi-particles
is totally depend on the quantum statistical nature
states is the same as the other, and the sign
of the particles [36]:
U(A, a) (;p,
-).=
(A)..ea(+
D1 (W (A, p,).. (IF;p,
p1..
o1..
(2.3)
K
'(p', 1'I)... (1;pio)...) =(3)(
(
) ='6=
(perm.)
,
1=
(2.4)
f
d='')(E'I -+ = fd-'I') (E'I E)
(2.5)
To make a connection between theoretical predictions and experimental data, it is important to
have the same setting. Interactions between particles are usually understood through scattering
experiments, which can be described by S-matrix from the theory side. The preparation always
requires a definite particle contents, which means an effectively non-interacting multi-particle state,
at a far past and let the particles interact inside a scattering region of the size much much smaller
than the distance between the particles initially, then at a far future, when what come out are
away from others, the measurement takes place. The S-matrix encodes the information of the
very same process, telling the probability amplitude for the transition between the initial and final
states, which is nothing but multi-particles states that are approximately the direct products of
the effectively non-interacting degrees of freedom (can be fundamental particles and composite
particles) in the theory at long distance scale. It is important to realize that some interactions
cannot be seen directly, at least at low energy, for example, the communication between single
quarks, because at the usual energy of experiments, these are confined and cannot exist alone.
The interactions have to be observed indirectly from the scattering of composite particles with
quarks inside, such as baryons and mesons, which can appear as a single quanta.
In general,
S-matrix cannot be used to describe short-distance quantum process (might be useful for weakly
interacting theories, but usually fails for strongly interacting ones).
23
2.1.2
Creation and Annihilation Operator
Creation Operator
Let's define creation operators, which arises in the context of non-interacting multi-particles very
naturally. The creation operator, for a given quanta of specific labels, is define to be an operator
that simply add a new particle to a state:
aT;p,, (1;
pi, G1)(4 2 ;P2 , 72 )...
(;
=
p
)('F1; pi, U)(' 2; p 2 , 0 2 )...
(2.6)
With this, under the Poincare transformation of the non-interacting theory:
U0 (A,
a)aF.,,U6-1(A, a) = ei(AP)a
Define a vacuum state
(A0
D(,
W (A,
a Q))pfAp,
(2.7)
IQ), which has no particle and is usually assumed to be invariant under
Poincare transformation and also the spatial inversion and the time reversal. Any multi-particles
states can be built by acting many creation operators on IQ), and due to the quantum statistical
nature of the particles, the adding order might be important [361:
XFP,0
'2a2*..)
= (T 1;pi,9
1 )(T2;p2,9 2)...
(2.8)
There's only two possible quantum statistic for particles of the same definition (identical particles),
because if one switch them twice, the state should be back to the original, hence the effect of each
switching can only gives a factor of +1 or -1.
the sign
Sperm.(n)
Boson statistic is
+, and fermion statistic is -, and
for the permutation of n identical particles, label ordering 1, 2, ... , n, can always
be kept in track by finding the integer r < n which label the particle switch place with the first
one, and the sign
Sperm.(n-1)
for the permutation of the rest:
Sperm.(n)
(
)r*
24
Sperm.(n-1)
(2.9)
Annihilation Operator
The annihilation operators can be defined as the Hermittian adjoint of the creation operators, and
from the normalization of physical Hilbert states:
(
i
; P', c)...(
; p',, u,4) ap ;p,, ('P ; Pi, o-l)...(i ; Pr, -n)
p,
=
(e+; u))(9 '1' , o)...(";P
, -,m) ('P ; P1, a1)-...(
Pr) H
6
5iperm
(2.11)
m
ni
E per.(n-1)r _ -
; Pn, -n))
(2.10)
( - Pperm.(i))
m+1,n (2.12)
-
r=1 perm.(n-1)
j=1
r6
r-1perm.(n-1)
Y
Jm+1,n
-
Pr)
(2.13)
r=1 perm.(n-1)
x (+;p1
I..(+
('T
.',.
Pn, an) (2.14)
( ;;,o1..W;p_, r1(iP+,J+)-(
Therefore, one arrives at the realization of annihilation operators in the Hilbert space that it takes
particles out of a state, like the description in its own name:
awp;p,,j9) = 0
,
ap,;p,, ('i;P1,u)..(
;Pauo1 )
(2.15)
N
Z()'
1
0ua,(
4
)(p _ Pr) (P';Pi,.).--('+;Pr-,ao-)(J
;Pr+1,) r+)--('F ; Pn, on)
(2.16)
T-1
From this, under the Poincare transformation of the non-interacting theory:
Uo(A, a)aq;p,U -1(A, a) = eiAP)
A 0
D(', (W1 (A,(p))2a;Ap,
(.17)
Interaction from Creation and Annihilation Operator
It's trivial to see that creation and annihilation operators of different quanta must be commute.
To see the commutation relation between these of the same quanta, apply an annihilation operator
on equation (2.6) and an creation operator on equation (2.16):
an'' p ~- p) + P1, 9~1)... ( ; pn, n)
(2.18) ;''
;p,
(
1,11) ..
; Pn,
n
25
N
+Z()r 2
-Pr) ('I ;p ,c)('I Pr-,u-1)(
((
;Pr+,
r+)...(I
;Pn, an))
r= 1
(2.19)
ap,;ap;y,-, (T; pi, UI)...(J'; pn, Un)
a
(2.20)
N
SZ(
)r+1jo,
3(4 )(P/
_
; Pn,on)
;Pr-i,0r-1)(F ;Pr+l, r+) ...
P+r) I('F ;1Pui)... ('
(2.21)
r=1
[alp ;p,,-', aXP;Pu] , j34)p-
T- aq,;P,,,aQ;,,aI
-+;PO
(2.22)
In addition to the above relation, one can also find [361:
{A, B}
= AB 4 BA , {at
7
,,, at
} = {aW ;p,&, af ;,,}=
0
(2.23)
Since any state in the physical Hilbert space can be reached by adding in or snatching out particles
and the connection (the matrix elements) between any two states can always be assigned to any
value, therefore, with creation and annihilation operators, any operator on the physical Hilbert
space can be construct, so the interactions in the theory can be described. With the algebra found
in equation (2.22) and (2.23), in any given operators one can have all creation operators are moved
to the left and all annihilation operators to the right. This arrangement is called normal ordering.
2.1.3
Causality Condition
The evolution of the quatum world depends on the Halmintonian H = HO + V, with HO is the
free theory Halmintonian that including all the particles (can be both fundamental and composite
particles) and V describe the interactions, all of these can always be written in terms of creation
and annihilation operators of normal ordering:
H,
HO, V
(J
d
nmn
*1, 42,.
'
'..o
...
..=
j=1
k=1
x hmrn, homw, Vmn ((T'1; P'" Ul)...( qf' P;
dp
a
1
j api
;P)(,
pjo)
(2.24)
1=1
1'm), (y1; P1, a,)---(..
P77,7
)
,
(2.25)
nm
?n=0
n=O
The requirement of having real energy means the Hamiltonian should be Hermittian, both the free
theory and the interactions. For S-matrix calculation, one can always take that V is effectively
turn-off everywhere except inside the scattering region, or having the interactions die out slowly
26
when approach the past and future infinity, where the initial in-coming and final out-going state
are set [22, 36]. Use the later approach, S-matrix can be expressed as:
S
= (E
;-) =(ESi)
,E;~)
=
) = IE7f,t = -+oo)
|Eit =-oC) I
(2.26)
The S-matrix operator is nothing but the renormalized evolution operator:
S
=
E(+oo, -oo) = eiHotf e-iH(tf-ti)e
(2.27)
-iHoti
tf=+oo,ti=-oO
Since we want to build a relativistic theory, the Poincare invariance should be preserved, hence Smatrix should be unchanged under Poincare transformation of the in-coming and out-going states.
In other words, S-matrix commutes with all generators of the Poincare group in a free theory:
Uo(A, a)SU- 1 (A, a) = S
,
[P0 = Ho, S] = [Po, S] = [Jo, S] = [KO, S] = 0
(2.28)
In the interacting picture, S-matrix can be expressed using the Dynson series [22, 36]:
V(t)
=
,=
eiHotVe-iHot
S
-Te
fj
(2.29)
dtV(t)
dt1...dt(EIT (V(t1)...V(t)
jdn
(
,
I7i)
(2.30)
n=O
To make S-matrix Poincare invariance, the interaction density should be a Poincare scalar:
Uo(A, a) V(x)U- (A, a) = V(Ax + a) , V(t) =
dxl...dex(BfIT(V(xi)...V(xn))IEi)
(2.32)
=
0=
f d 5V(5,t)
3
-
S = Te-ifdaxV(x)
(2.31)
4
n=O
However, even with this, the S-matrix still doesn't fit the bill yet, because of the time-ordering
operator is not Lorentz invariant, in the sense that it loses its meaning for space-like separation.
Therefore, a condition for interaction density should be impose, for the sake of consistency, and it
is called the causality condition:
(X'
-
X)2 > 0 -+ [V(x'), V(x)j = 0
27
(2.33)
2.1.4
The General Form of S-Matrix
The Poincare invariant feature of S-matrix theory indicates that it should vanish unless the total
4-momentum of all particles disappears (in-coming momentum and out-going momentum are of
opposite sign in the summation), since under an arbitrary Poincare transformation U(A, a) on both
the initial and final states, there's always a single phase with a-dependence pops out and it should
disappear so that S-matrix isn't depend on the choice of a. This is nothing but the momentum
conservation law, which is also realized in classical physics:
eiapPA
P,= p,
= 0 ,
P
(2.34)
Therefore, a general S-matrix can be written as:
i2w
-
-n,=os
)(ps,
-
pE1)Ma
(2.35)
,
If the theory has no interaction V = 0, then S-matrix is simply
, so the information of
interactions are all encoded inside Ma+sf. In a general interacting theory, S-matrix should be
unitary since it connects two complete sets of orthogonal states, therefore:
J
dE'S~~S&+~
-~
i(~
~
-
Af f4)-
2-71
J
4 dE54(P=, _ P=f
'~,MI.
(2.36)
By taking Ej = Ef = 7, one arrives at the optical theorem:
Im MIa
2.1.5
-IrfdE'5(Ps,
-
p6) M
2
.1+s|
(2.37)
Internal Symmetry and Conserved Charge
Because the particle zoo in nature is very crowded, there are too many possibilities for the dynamics
can be encoded inside the S-matrix, then to put more constraints on the possible interactions
(and make the theory easier to keep track of), one can employs internal symmetry, the kind of
symmetry that has nothing to do with the Poincare group.
Unlike Poincare symmetry which
acts on the spacetime and is realized by each quanta individually, the internal symmetry mixes
different particles together and leaves the S-matrix unchanged. It is also desirable that the internal
28
symmetry can still be seen when the interactions are smoothly goes to vanish at far distance past
and future, for the initial and final states, as the symmetry transformation in free theory works:
Uo(I)SU-1 (I) = S
(2.38)
Let's focus on the symmetry that can be realized as linear transformation between particles (nternal
symmetry realized in nature is mostly of this kind, for example, in nuclear physics under the
interchange of neutrons and protons):
Uo(I')Uo(I) = Uo(I'I) -+ D(I')D(I) = D(I'I)
,
Dt (T)D(T) = 1
Dq, q(I)Dpj.. (I';piFi)(P';p2 ,cr2 )...
UO(I) (F1; p1, Oi)(4 2 ;P 2 , 0 2 )...) =
1'
(2.39)
(2.40)
2'**
Consider the cases when the symmetry group is a Lie group, which can be described by a continuous
0' parameter, then it can be written in an exponential form, with the symmetry generators,
Hermittian
Q'.
This generator should commute with all Poincare generators of the free theory,
and, through the commutation realtion, put the restriction on the interactions:
UO (I(oa)) = eiQaoa , [P = HO, Q] = [PoQ] = [,Q"] = [KO, Q"] = 0
Hence,
Qa
unchanged.
,
[V, Qa] =0 (2.41)
should be the conserved charge in the theory, and any process should make the charge
To see this, consider the symmetry with one-dimensional parameter 0 that gives
different particle species independence charges, then the S-matrix invariant under that symmetry
indicates charge conservation as the total charge of the in-coming state is the same as the outgoing state, follows in the same way as momentum conservation arises from translation invariant
in equation (2.34):
DTx10)=
6tp
q
-> q-
=
q~f
(2.42)
And indeed, charge conservation can be seen in nature and in many different physical processes,
such as the conservation of electrical charges in electromagnetism interactions.
29
Cluster Decomposition
2.1.6
Another requirement for the interactions is, quite a philosophical one, if the events are sufficiently
far apart, then these should be uncorrelated. This is a fundamental principle of science indeed,
because it's allow us to make predictions and formulate theories without the need of knowing
everything in the universe [361. It's called cluster decomposition (or, sometimes, locality), and it
allows S-matrix to be factorizable into different scattering processes that happened at scattering
regions far away from others:
S i
=EE, S-+,;S=
s..Sa
,
E.(E')(E')--(') (2.43)
=(7 1)(B 2)...(En) ,
This is indeed a very strict condition on the possible Halmintonian of the interacting theory,
hence also for the Halmintonian of the free theory and the interactions. For realization of cluster
decomposition, the coefficients for the direct product of creation and annihilation operators of
these three in equation (2.25) should be proportional to a single 6-function of 3-momentum
hmn, homn, Vmn ((O;
--
u $ ), (1; P1,
o1)...' nPn, 0'n)
6(3)
m-Pi)rn, 1Omn,
__
i=1
P' ,)
"Drnn
o ), (P1 P1, o1)...(I'n; Pn, 9-n))
N
(e";
,
1361:
(2.44)
(
i=1
(2.45)
In other words, there's no 3-function in h, ho and f. The need for a 6-function in cluster decomposition can be easily proved by using equation (2.30) then going from the momentum space to
the position space, and the restriction for that 6-function of the 4-momentum to be the only one
can be seen from the topological nature of the perturbative Feynman diagram (which replaces the
algebraic language of creation and annihilation operators by a geometric language of lines and
vertices) at all order of expansion [361.
30
'4' a
2.2
2.2.1
Macroscopic Description
Creation and Annihilation Fields
Since T already specifies the quanta definition, so the on-shell condition is known, therefore, one
can always drop p0 out of the notation. Define the creation <bj and annihilation fields <bt:
d p a aui(xI|';
i, -), Ir(x) =f dpa ,.vi(x IT; , o)
'11-(x) =f
(2.46)
a
01
Basically, we group the creation and annihilation operators in spacetime symmetry irreducible
representations, which are specified by the label 1. The function tensor coefficients ut and v, can
always be chosen so that the fields have nice properties [36]:
Uo(A, a)'Uo-1 (A, a) = ED(A-1)Tj(Ax+a)
(2.47)
, D(A')D(A) = D(A'A)
1'
From equation (2.7) and (2.17), one arrives at:
Ulf (Ax + a I T; Ap, o-')D ( (W(A, p)) =
P
Di(A)ei(AP)aui(xI1p; , c-)
,
(2.48)
a1
D (A)e-(AP)avi(x14; p5 -)
Iv/ (Ax + a IT; Ap, -')Ds (W(A,p)) =
(2.49)
Under transformation alone, the two function tensor coefficients change to tensor coefficient:
ui(01';j ,ar) = u(AI; ,-)
-*
ui(x1';
a,) = (27r)-2eiPu(T; , -)
e
= (27r)- ZZld3Pata
-f+(x)
-+
, o-)
=
vz(T;
, o-)
qf-(x) = (27)-2
1
-
(2.51)
,
DPO
EDri(A)u(P;i7,-)
ui,(I; Ap, aD')D, (W(A, p)
vi(0IW;
7t, (IF; ,a-)
v(x|T; , o-)
1
f
d1pa
31
=
(27r) -eAiPxv(IF;
e- Pxvl(P;7,U)
(2.50)
,
', a)
,
;
,
(2.52)
(2.53)
(2.54)
viF(P;Aip, -')D/, (W (A, p)) =
2.2.2
P
Dii1 (A) vi(I; p5 o-)
(2.55)
Finite-Dimensional Representations of the Lorentz Group
While D(s)(W) represents only the little group, D(A) represents the full Lorentz group with generators J and K. The non-compact Lie algebra SO(1, 3) of the Lorentz group can be reduced to
the direct product of two non-mixing compact subalgebras SU(2) 0 SU(2) by separating in the
following way [32, 36]:
A = -(J+iK)
2
[A 3 A] = iEjklAl ;
,
=
-(J-iK) ,
2
,5 k] = iEkBk , [Ai, B]
0 (2.56)
=
Thus, any finite dimensional irreducible representation of the non-compact Lorentz group can be
written as a pair of integers or half-integers (A, B) with A and B specifying the spins representing
the operator A and B. The simplest non-trivial irreducible representations of the group are the twodimensional
(1, 0)
left-handed and (0, 1) right-handed, and these are fundamental representations
so that any finite dimensional representation of the group can be obtained by the decomposition
of repeated direct products of these [32], and the results in components [36]:
A1+A
0
(A 1 , B 1)®(A 2 , B 2 )
2
Bi+B 2
0
(A, B) ;
AaIbI,ab= Jaab'b ,
Ba'b',ab= a'aJ,
;
(2.57)
A=IA1-A 21 B=IBI-B21
(J)))a/a
= a'a
,
(J1)
2 A),
= 6a,a i /(A
T a)(A
a + 1)
(2.58)
The (A, B) representation is irreducible with respect to the Lorentz group as a whole, but reducible
under the rotation subgroup [24, 32]. To see this, note that J = A+ B, and the tensor product of
spin A and spin B, in general, can be reduced to objects of rotation spin s [361:
IA - BI < s < A + B
(2.59)
Although irreducible representations of SO(1, 3) are also irreducible representations of SU(2) 0
SU(2), some important properties of the presentation, such as unitarity, are not presered, since the
generators A, B and K cannot simultaneously be Hermittian [32]. In this particular representation,
the generator K is anti-Hermittian, therefore the finite dimensional representations of the Lorentz
32
group are non-unitary, which is expected from the fact that non-compact groups do not have nontrivial finite dimensional unitary representation, therefore cannot be directly realized as physical
states since all symmetry operations should be realized as unitary operators on the physical Hilbert
space[32. This is not a lethal problem, however, since here we are dealing with fields instead of
physical states. Also note that, for any spin s, there's infinite many way to choose (A, B) to gets
to the final results, so one has the freedom to choose the simplest one, for example, to have D(A)
choice is (1, 1), for half-integer spin s it's can be
right-handed or
+
,
-
, +
-
A +
,
-
left-handed or
-
,
+
of the spacetime transformation form AAA... as much as possible. For integer spin s then a good
Dirac.
Massive Field
For the field of massive quantum, choose the standard 4-vector k" has vanishing spatial components
k
=
0, then:
D t (L(q))u(;Oo) , v),(t;, ,I-)
up (T; , a) =
i(L(q))v(J;, c-)
=IF;D
(2.60)
R, so that the standard 4-vector is
Now, let's the Lorentz transformation to be a rotation W(A, p)
left unchanged, then with
j(s
and j to be the angular-momentum matrices in the representations
D(W)(R) and D(R):
(Jls)
=
(-)7'-+
1j(S),,,
;
(2.61)
Dpi(R)u(I;( , c-) -+ up (4; 0, -') J = Su (T; 6, a-)
u11(4; 0, a-')D, (R)
1
0~
1
to,
(2.62)
V3 (T ; G,
D-')D t(R)
Dz'(R)v,('I; , o-)
=
-+
vv(T; d, o-')(J
)f
=
-
3J
1
(T; U, a-)
(2.63)
The eigenstates of J are called the polarization tensor, and from the analysis of the little group
we know that these correspond to 2s + 1 degrees of freedom. In (A, B) representation, one arrives
at the famous equations for Clebsh - Gordan coefficients [36]:
>a3'b(W; 0,
'
)
>
a
tab'(X;
ab
33
J, c-) +
Ua'b( 4 ; G, O-) ,
(2.64)
alb
Uab(';
-+
( V
G, o-)
'
0
a
1
ab)
=CAB(so-;
a'a) Vab',(P,,u
1
-
2
Va'b(T&;
.. d
ba~;
Vab(T; ', 0)
"nab(T;
(2.65)
G, cr)
(2.66)
, -o-)
For a general finite momentum, the two tensor coefficients (with our choice of convention) are:
Uab
>1(e+'jjA), (C
-
;T ,-)a
jB)O)bCAB(su;
a'b'
a'b')
Vabe'
U0,,) /
(-;0
-)
(2.67)
Massless Field
For massless field, equation (2.52) and (2.55) can be simplified with (1.52):
ul (T; Ap, a-)e''O
vi(P;
4,
(V;
(Ap) 0 >Dii(A)ui(';,
-e--ioo(,A)
Choose the standard 4-momentum k
U1
-
p 0
(2.69)
, o-)
= (k 0 , 0, 0, k 0 ), so:
PO kO Dii ( L( p) iiN; , 0-)
, or)
Dij(A)vi(T;
(2.68)
)
,
U) = - Uab(T;
k
vi/(TI; , C-)
D
0 > D1,1(L(p))vi(r'; k-, 0-)
(2.70)
The polarization ternsors are assigned as the eigenstates of D(R), which should be two in total, as
seen from the SO(2) part of little group. The full little group transformation is E 2 , with rotations
and translations in a two-dimensional spatial plane, which is described in equation (1.48):
u1 (4';
Ak, o)e"
>3 E Jq ( R())ui(P; yj, c-)
V(X;AkO-)e-i
=
,j(R(O) vj((;
,2-)
(2.71)
'ul(T; Ak, o-)
=
3([(S~c,
3))ui(P; Y, a)
vI;Ak,-) = >
i'i(S(a,3))vI(IF; -,ao)
From the polarization tensors, one can get the tensor coefficients, with a prefactor
malization.
34
1
m2
(2.72)
for nor-
Restriction on Massless Field Construction
However, it's not possible to find the tensor coefficients satisfy these equations for any general
representation of the homogeneous Lorentz group. To see this, consider (A, B) representation with
an infinitesimal transformation on Uab [361:
Jia'b',ab =
-+ (J1A)
-
ii (Jl,
k
ab+
k
jA)aa ua'b(C; k,
o)
= (J'B)
(2.73)
aa(I))V
+ oa'a ka'aubb
Z
b(I
B))a
k, a)
0
(2.74)
Same goes for v, these requires the two tensor coefficients to disappear unless a = -A and b = B.
Thus, a massless field of type (A, B) can only be used to described particles of helicity a(A - B). For later interests, let's consider massless fields of helicity
1 and t2. Instead of using
a vector (1, -), one needs an anti-symmetric tensor (1, 0) D (0, 1) to get massless a =
1, and
instead of using a symmetric tensor (1, 1), one needs an 4-rank tensor (2, 0) D (0, 2) to get massless
a
t2 [361. However, if the objects isn't exactly a Lorentz 4-vector and a Lorentz symmetric
tensor, then these are still possible to be massless, and we will see that later.
The Building Blocks of Polarization Tensors
Since there's only two polarization tensors for each massless quanta, corresponding to the highest
and lowest possible value for the angular momentum along the propagating direction, therefore
the polarization tensors of a field with many spacetime indices can always be built trivially by a
sum of direct products of the polarization tensors from fields with fewer spacetime indices. The
most fundamental polarization spinors and vectors are:
=
(1, 0)
,
_ =(0,1) ; e t (kt)
(0,1,ti,0)
(2.75)
For example, if we want to have the polarization tensors for a symmetric tensor of spin 2, then:
E (k, ) = d'(k,
)eL(f, t) =
0
0
0
1
0
S
35
0
i 0
i
0
0
-1
0
0
0
(2.76)
However, like mentioned before, one should note that in general they didn't have the right transformation properties to be massless, under a little group transformation in equation (1.48) [211:
) = (A-)E"(Ak, +) +
e TiYE(,
(A-)",(A)",eP(Ak,
e*2i6 PE(k)
Va(a
2.2.3
1
,,F2k 0
) + kl(A v )v +
i)c4((,k(a i3) 2(Ak)"
=
(a
i )k
;
(2.77)
kv(A-1v )
(2.78)
,
1
(2.79)
4(kO)2
Physical Requirements for the Theory
The four Requirements
In order to put quantum mechanics and special relativity together, with some philosophical questions in consideration, the dynamics of the theory, which is encoded in the interacting Halmintonian
(the free theory Halmintonian and the interactions) should satisfies four main requirements
124,
361:
1. Hermicity
2. Causality
3. Conserved charge
4. Cluster decomposition
Quantum Field Theory pops out
All requirements can be satisfied with the language of field, and what is known as quantum field
theory does pop out from here. Note that, Hermicity is the easiest requirement since it can can
always be employed when writing down the field dynamics, so we won't discuss any about it.
Checking Cluster Decomposition
Construct the Lorentz scalar interaction density out of the creation and annihilation fields:
VWx
mZ
''
rn
S.l S--1
..
n
36
(X
)
,j
(2.80)
D j.(A~1))g.,
(JJD
1 T,,(A-1)
I...I 11...In
1
= g .,
..
1
T
(2.81)
j=1
i=1
~n)
(21r)-
=
-3
-P-i_
)
3 !m -
'1 01)--WM; Pm, Omr), OF1; P1, O~1)---.(Fn; Pn,7
va (W';
)
The interactions in the full Halmintonian, from equation (2.31), (2.51) and (2.54):
(2.82)
..4,1.. (V'; -5, a'.) vl, (XF ;pA, 9-k))
(2.83)
X
H
.
I,...,l'
1
i,..,1m
k=1
3=1
This expression automatically satisfies cluster decomposition.
The non-interacting free theory
Halmintonian gets this requirements trivially.
Quantum Field
In general, the creation and annihilation fields are not commute, even separated space-like. Therefore, instead of just always treat them separately, one can try to combine them into a new field,
called quantum field, and make the dynamics depend on it and its Hermittian adjoint. For example:
'I'(x) = T- + AI'-
-- > 'Jab(X)
=
AT-
KI'+
(2.84)
The causality requirement in equation (2.33) can be satisfied, if one can show that it's possible to
have the commutation relation, with considerations of the quantum statistics:
(x -
X') > 0 -*
[P 1 (x), JTI/,(x')]
= 0
(2.85)
At first glance it might seems impossible for such a simple combination can solve the problem
with causality, but by some extensions with a new type of quanta known as anti-particle, then it's
actually the case. We will discuss more about this later. Also note that, causality can be seen
from the free theory Halmintonian easily, because of the on-shell condition for each quanta.
37
Checking Charge Conjugation
The conservation charge
Q, which is Hermittian,
should be commute with the whole Hamintonian,
from equation (2.41). If the quanta carries a non-zero charge qT:
[Q, A ] = qa
,
[Q, TI (x)] = -Fq,'F'(x)
[Q, a;g,] = -qap;,-+
For the whole Halmintonian to be written in
(2.86)
i, it's necessary for the field to have a simple
commutation relation with the conserve charge:
[Q, TI(x)]
=
q,'I(x) ,
[Q,
It(x)] =
qqI'(x)
(2.87)
And one can keep track of the free theory and the interactions with operators of no total charge:
2.-+
q
+q
2 +.-
-
q..
- ... = 0
-qp
(2.88)
Equation (2.82) can be taken on by introducing the concept of anti-particle, and like mentioned
before, it is crucial for causality to be realized in the theory. The idea is to have a doubling of
particle species (which means same mass and spin) carries the charges with opposite signs:
[Q, at
)]
-qafcg
,
[Q, ac;g,,] = qapc;, -+ [Q, 'I+] =
q'Ic+
,
[Q, IV-]
-
- (2.89)
Generalize by redefining the quantum field in equation (2.79), and this is the form we will use from
now on:
Ii(x) = Ki+ + A'- -+ 'Fab(x) --
ab
+ AT'ab-
(2.90)
If the particle is its own anti-particle ap = apc, then it should carry no charge and we get back
to equation (2.79). The reason why we want the anti-particle to have the same mass and spin
with the normal particle is, since we want to put the particle annihilation (creation) field and
anti-particle creation (annihilation) field in the same quantum field, the on-shell condition should
be the same, therefore the mass and spin should be in agreement.
38
Checking Causality
We will only consider massive particles since the extension to massless particles is straight-forward
but usually needs lengthy case-by-case analysis. Let's look at the quantum field in more details,
with creation and annihilation opertors
J
Tab(x) = (27r)-
d3a
; , ) + AjcU(e-ab
;veab(
(P;p~, a))
(2.91)
The commutation relation can be written in terms of the spin sum E(4I; p):
x
[
Eab ,
,
i(X
; i)
cd
3
2p
2
01
3
-2i-
E
J
1
Uab(;
,, o- (; j, cr)
3
2p Vab(; P, ab,(W;P, )
>3CAB(SY; cd)CAB(sA JA
''
IA12ei(x-x')
,
(2.92)
(2.93)
(2.94)
;
c'd'
P0 , A
P) = Eaab'(;
Eab,a'(P;
E-a~''1
2eip(x-x')
(P
-P
-
(- 2(A+I) E1ab,a'b1(q;P1
-E1
22a,a',61
,
(aba'b'(;
Af + 2P0 E2a,alb, ,
-A -- (-)2(A+B)+
For a space-like separation, choose a Lorentz frame so that x0
E 2 ab, a'b,(O; )
(2.95)
(2.96)
x' 0 , then the commutation relation
becomes much simpler [361:
[qPab(X), q'"
_
1
-ilk
-A
[W(,
+(112
(2.98)
F ()2(A+B)
This should be vanished when F 4
IK1 2
If A
/- 2
(\tiD
\EhI) I ab a'b'(' 1V)20 (l_ 3i
f d3jT :\b'(X)J
(_)2(A+B)
(2.97)
l
i
12
2)
2
ab,ab _j(3)
-
, therefore:
= -(-)2(A+B)A
1
2
T
_)2(A+B)
-1
12 _A
2
(2.99)
+ B is an integer then the sign should be +, and if A + B is an half-integer then the sign
should be -.
This is nothing but the relation between the quantum statistical of the field with
boson (+) full integer and fermion (-) half-integer nature. With this, the causality requirement
is solved, if the full interacting Halmintonian are made of T, and Pl (cannot be only Ti or Tt,
39
because of Hermicity). Note that, the relative phase arise from the relation Ir,12 -
A 2 can always
be eliminated by the redefinition of the creation and annihilation operators, and also the overall
scale of the field, hence, in the end [361:
'j(x)
I'(x)
=
(2 7r)-
Ti(x) = (27rTI'ab(X)
=(27r)-
Z
= '' +(-)21c- ,
d3(a
2
'ai(X) =T+ +()Tca- ;
(2.100)
pePxui(T; , or) + (-)2aac;g6e-xvi(4;5,cx))
d a qiug,;
ziYf
a) +
creg- ~Vs(; 0, ))
', )2
,
(2.101)
(2.102)
(2.102)
dPfA Ilr;l&Pxab(aIVz7, )a+)
Other interesting properties of the field that should be mentioned:
{'i(x), I''(x')}+= 0
{'ab(X), Ja'b'(X')}
{J/(x), J),(x')}= 0
,
= 0 ,
{4Ia(X), ',,(')}
I =0
,
(2.103)
(2.104)
Anti-Particle and CPT theorem
Consider the massive case only, for the same reasons as the previous subsubsection. We start by
constructing a quanta out of spacetime symmetry, then arguing the dynamics of multi-particles
and arriving at the collective behaviors encoded in quantum field with the four physical requirements. The concept of anti-particle arise natrurally when considering about charge conservation,
which is theoretically used to restrict the possible interactions and experimentally observed in
nature. There's a relation between the normal quanta and the anti-quanta known as the charge
conjugation, or charge inversion, Co which reverses all charges of the the internal symmetry. Thus,
the charge conjugation should be anti-commute with the charge operator, but commute with the
Halmintonian, the momentum, the angular momentum and the boost 3-vector in the free theory.
Along the discrete symmetry of spacetime given in equation (1.16) and (1.17), spatial inversion P0
and time reversal 7To, since these are all commute with the Halmintonian, so there're only phases
different in the quanta state between before and after these unitary change of basis transformations:
C0p;P1 or) =,q,, y f; , or) , PoKT; k, 0) =ntl4Fc; -, or) , TOX; , a) =It C; --, -o) (2.105)
apc;1.a, Poa P-- = ta.
,Ta
(2.106) =
Coal3_,.XCo7
1= (-)-aq.ta
n
40
Brute-force calculations from equation (2.102), the combined transformation COPoT is an antiunitary transformation (To is anti-unitary, which is known from quantum mechanics, while PO and
CO should be unitary) change the field in the representation (A, B):
Po
T4
,~c ()-2A-a-b-spf(B
)'x) 61
'(AB)O-1
(CoPo))_1=)(x),(C1Po_))
- ,
=
t
CP =c
(2.107)
t()
),(
A+B-s 4 B,A)(_,
,(-)A+B+a+b-2sq(A,B)
7(-)2B,,(A,B)
,A)
(y
)
(2.108)
,
0
)
CO 'I(,B) (X)C
(2.110)
Set qf't = 1 for any particle [36] (for a charged field, the charge operator can be employed to
construct an unitrary transformation which modifies the phase of the field and also the discrete
symmetry operator, hence
41
A,,B',
F,
jct). Since any Poincare scalar written in terms of the quantum field
,,'B1) (might be with derivatives) must has EZ
Ai and >js Bi are integers, therefore
the S-matrix, which is related to the Hermittian scalar interaction density in equation (2.31), is
invariant under CPT transformation:
(CoPoTo)V(x)(CoPoTo)-
1
= V(-x) -+ (CoPo7T)S(CoPTo)-
1
= S
(2.111)
It should be noted that, instead of setting q't =1 for any particle, one can keep it and show that
for every scattering process with the same incoming and ongoing particles, the S-matrix is only
shifted by a constant phase, and it's not affected the physics of theory, therefore can be ignored
effectively. The invariant of S-matrix under CPT transformation is known as the CPT theorem,
and the theorem holds for any order of applying the discrete symmetry transformations, since only
an overall phase is produced as the order is changing (T is anti-Hermittian, so any on the right of
it should be complex conjugated). A more rigorous mathematical proof in the context of axiomatic
quantum field theory yields the very same result [15].
2.2.4
Spacetime Rotation and Particle Self-Rotation
There's an interesting relation between the physics at macroscopic scale and microscopic scale
that cannot be seen from classical physics, know as the relation between spacetime rotation and
41
particle rotation.
The idea is that if we interchange the position of the two identical fermion
particles, the new configuration will be different from the original, and it's actually the same as
rotating one of the fermion around only a single time.
To see that, note that one can always
characterize the transformation into two different type, the active transformation (which acts on
the field, generating the self-rotation) and the passive transformation (which acts on the spacetime,
generating the change of particle's positions). We will show this interesting realization, at least
schematically. The active transformation can be expressed as:
IF (x) -+ Uw(A)'P(x)Ujj(A)
(2.112)
Also, write down the passive transformation:
qJ(x) -+ T(A)T(A-lx)
(2.113)
Equate the passive and active transformations, one arrives at the relation between the two:
TT(A)U,(A)T(x)Ugjj(A) = T(Ax)
(2.114)
Consider a half-integer spin fermionic particle, then desire relation to be shown is the equality of
switching place the two particles (which correspond to the passive spacetime rotation 0 = -r) and
rotating one of these around once (the active particle rotation 0 = 7r). Note that, for a fermion:
T 1(0) = A-'(0)
, A- 1 (27r) = -1
, A- 1 (47r) = 1
(2.115)
Illustratively, two particles can be represented by field insertions at two different position, say,
For a clearer description, use the test functions f+localizes at xo and
f_
x.
localizes at -xO so that
f+(-x) = f- (x), and the spatial rotation simply exchange these 2 functions. Then, let's passively
rotate them an angle
7r:
I(Xo)XF(-xo)
Jdxjif+(x1)P(x) Jdx 2 f-(x 2 )(x
-dx'f'(x1)V(x1)
42
fdx'2f'(X2)qj'(X2)
2
)
(2.116)
(2.117)
=
dxifi(x)A-'(7r)1(-x1) fdX 2 f+(x 2 )A-'(7r)P(-X 2 )
~ Aq 1 (27r)P(xo)'(-xo) = -I (xo)4(-xo)
(2.118)
(2.119)
The result is exactly equal, if we actively exchange these two by a rotation of angle 27r:
41(X)X2(-X)
- UT,(27r)AP(x)U--1(27r)w(-x) = A-1(27r)qI(x)T(-x)
=
-I(x)T(-x)
(2.120)
Indeed, there's a very nice way to keep track of the spacetime rotation and the particle self-rotation
of fermions, by the topology of a rubber band [101.
43
Chapter 3
Quantum Field Theory
Quantum field theory, undoubtedly, is the best tool for physicists nowadays to understanding the
nature at microscopic scales. Start from scratches with a philosophical question of what it means
by a particle, and then let the particles interact with others through some guiding principles with
great depth of how nature works at quantum level, a field description with quantum nature arise.
In this chapter, we will learn how quantum field theory is formulated by a field Halmintonian and
Lagrangian, and consider some exciting ideas about the theory and interesting applications. We
will not go into many details, only briefly discuss the core ideas and intriguing examples. This
part can be enjoyable if the readers have some certain understandings about quantum field theory
priory.
3.1
The Free Theory Lagrangian
The quantum field is constructed by summing over all particles of the same specific definition with
different labels. Each quanta is a physical Hilbert state of the free theory on its own, an on-shell
state. From the mass and spin given in the quanta definition, also restriction from the possible
polarization states, one can derive a suitable equation of motion with constraints for the field that
gives rise to all possible labels of the particle. Then, the field Halmintonian and Lagrangian can
be arrived at that describe the found equation of motion with expected constraints. These, in
the free theory, should be quadratic in field (due to propagation and the addictive nature of the
Halmintonian when adding more particles in the theory), and should has no more or less than
second derivative for bosonic field and first derivative for fermionic field (because of the position of
44
the propagation pole and its residue). The specific form of the Halmintonian and the Langrangian
is fixed after taking the four requirements into account along with some physical considerations.
We will not give the details treatment since it can be done easily case-by-case. Rather, we will only
give the results from some general cases that are of many interests in theoretical particle physics.
Also, it should also be noted that, in principle, by using the optical theorem given in equation
(2.37) (in details, for a process with a single particle of interests propagates in the middle of it) and
the polarization tensors of the particle, then propagation properties of the field can be read-off,
and one can get out the free theory Halmintonian and Lagrangian. It's not always straight-forward
to use, however.
3.1.1
Massive Quantum Field
Massive Bosonic Field
Consider the massive bosonic field T, . . . of mass m and spin s E N of the Lorentz representation
( E), then the field must be a totally symmetric traceless rank-s tensor:
rfAVA3 ...,
=0
,
(3.1)
.
The value of the Casimir operator C1 is encoded in the equation of motion, and the Casimir C2
requires that all the lower spin values be eliminated, which impossing the transversality condition
[9, 29]:
I
=-8N
,(L
2- ,l... 11(x) 0, OPT
2... 4L(x) =0
(3.3)
The number of degrees of freedom can be counted to be 2s + 1, which is expected. The free theory
Lagrangian for quantum field T(s) can be written down, with some symmetric traceless tensor
auxiliary fields 4 [27]:
LO =
s(s -
1)2
2s-1
(s-2)I
d
()(
2
-
(S2)(L -
45
+ 2(a2(s))2
M)
a2
(3.4)
1)1(s- 2 ) + b2 (4 (
2(35)
2
E
c(. 11
q=3
]O-q) - aq2j)s-q ( -q))2 -- M2,(s-q) (s---1))
(3.6)
1
q(2s-q+1)(s-q+2)
2(2s - 2q+ 3)(s - q+ 1)
aq
,
k=2
b
(s-q+2)2
2s-2q+7
q(s-q+2)2 (s-q+4)(2s-q+4)
2(s - q+ 3)(2s - 2q+ 5)
(3.7)
q
This is known as the Singh - Hagen Lagrangian for bosonic fields. It can be checked that the
Lagrangian satisfy equation (3.3).
Massive Fermionic Field
For the massive fermionic field T,1.../, of mass m and spin s E N + 1 of the Lorentz representation
+
,
then the field must be a totally symmetric rank-(s
-
Dirac
-
tensor-spinor:
d3P a
(27)-
~ U,1... P(;C,
ue
c-)
(3.8)
i
+ a
Pxvpi...n(I; yo))
-+
)
,41...,
1
2'
n=s-
7AI-2-... n =0 ,
(3.9)
The equation of motion and the transversality condition can be read-off [9, 291:
948, ,
0 =
1A(i
-0)111...
/(n)
=0 ,
0" TP'-2... P.(x)
=0 (3.10)
The number of degrees of freedom can be counted to be 2n + 2 = 2s + 1, which is expected. The
free theory Lagrangian for quantum field T(n) can be written down, with some symmetric traceless
tensor-spinor auxiliary fields <) and <' [281:
ALL)dx'n' -+ 2c49- 1 (&((+))
LO-
2n2
2n + 1
+ a1Mk)<DC"--0 +
fll)(if
2n
2n
120'(n-2)(O<D(n-))
_ 1'(fl-) (if
+ aM)b(-
+
+
(3.11)
(3.12)
b2+
n
+(-)q
q=3
- b2
24(-n-)
(
+ a2 AMq)<(-
(3.13)
q-1
(
Ckbk) ( 2
"-q) (0(n-+1)) _ g (n-q)
k=2
46
+ aq_
TMIM -)
(3.14)
1(,$'(n-q),(n-q)
-bq
S
+
n+ 1
n- q+1
+ aqA1,)<D(n-))
("~q4)/(n-q)) - bqI(n-q) (i
(q - 1)(2n - q + 3)
2n - 2q+3
'
bq-
_
,
(3.15)
2(n - q + 4) 2
2n - 2q + 7
(3.16)
This is known as the Singh-Hagen Lagrangian for fermionic fields. It can be checked that the
Lagrangian satisfy equation (3.10).
3.1.2
Massless Quantum Field
Massless Bosonic Field
Similar to the massive bosonic field, but the free theory Langriangian is significantly simplify, since
all of the auxiliary fields decouple except for the one with rank-(s - 2). Combine the symmetric
traceless rank-s and rank-(s - 2) together into a single symmetric double-traceless (not singletraceless) tensor A ... A [12, 291:
rfr[IV 77..OT=0
Lo
f
dX(
s(s
-
(3.17)
T1.'.-T...s +)(N1t2...11)2
1)
+
+
V32prh,3...2
2
, TPVlJ . PO'W1'P...
s(s - 1)(s - 2) (aPqW
Ps
(3.18)
.
)2
(3.19)
This is known as the Fronsdal - Fang Lagrangian for massless bosonic fields. Counting the number
of the degrees of freedom, one gets 0 for s = 0 and 2 for all s > 0. There's an ambiguity in the
theory, which can already be seen at quanta level as the freedom in the transformation S(a, 3)
given in equation (1.49) and demonstrated for massless spin-1 and spin-2 quanta in equation (2.77)
and (2.78) as "strange" pieces proportional to the propagation direction (we will discuss about it
more later). That's the gauge freedom that leaves the Lagrangian unchanged:
TA1-... A(X) -+ TP,...,(+D..(
,
)
<Ds(
(,L1AA 2...As)()
,
rf"Apjj...t,_1 = 0 (3.20)
The gauge parameter A is a symmetric single-traceless tensor of rank-(s - 1). For later interests,
let's consider the cases of massless spin-1 and spin-2 field, which is later known as photon electromagnetism field and graviton gravitational field, for later interests. After some integration by
47
part, the massless spin-1 and spin-2 field free theory Halmintonian and Lagrangian can found:
4 -+ A
Tpy, -
hjv ,
1 FtvF
4L
-
,
Fp, = OpAv -
vA,
,
1=)
O"h"POhvp + &"hPV0phPv + 21hOlh + hO'Ovhj,
L
h
0
2
Vhv, 1j
=
#
(3.21)
SAY = o,A(x) ;
2~V&h~
rhg
0 , 6h,,(x) = 0,,A,(x) + &A,(x)
,
(3.22)
(3.23)
Massless Fermionic Field
By taking the massless limit of the fermionic Singh - Hagen Langrangian, all auxiliary fields
decouple from the theory except for the ones of rank-(n - 1) and (n - 2). Combine the symmetric
traceless rank-n, (n - 1) and (n - 2) tensor-spinor field together into a single symmetric triple..
,
then one arrives at the Fronsdal
-
traceless (not single- or double-traceless) tensor-spinor 'I'
Fang Lagrangian for massless fermionic fields [8, 291:
P
'-I"'I/
A
L
3=d(P-1
...+i 12../n .
4... A
-
(3.24)
0
(3.25)
-
4
2
(
VJ
-An)
3 ~.
PP3. An3_.(9PW
i3/
(3.27)
The number of degrees of freedom is always 4, for all spin. The ambiguity gauge freedom has a
symmetric single-traceless tensor-spinor gauge parameter:
An (x) -F1...
-- qP, 1 .... ,(x) + 6'fP 1 . . y
J
A1 ...
a(x) =
0(/1AP2 ...
in)c(x)
48
(x)
, 7AAI/ 2 ...
(3.28)
,
pn-I
= 0
(3.29)
3.2
Collective Behaviors of Quantum
3.2.1
Gauge Freedom
The degrees of freedom in each quanta can be understood through the polarization tensors, which
is encoded in the tensor coefficients of the field, a collection of all possible physical particle modes.
As it is mentioned before, one cannot build a massless particle out of a general representation of
the Lorentz group, and at quanta level, it can be understood that the polarization tensors are not
transformed under the little group as it should be as Lorentz tensors in some cases. Nevertherless,
the particle, although not really a Lorentz tensor are indeed realized in nature. In order to make
the theory with these massless particles consistent, one has to associated the strange "pieces" that
comes out after little group transformation of the polarization tensors to be unphysical, hence
must be ruled out of the theory. Thus, the ambiguity in defining that massless particles becomes
the ambiguity of the whole theory.
Gauge symmetry
Consider the cases of massless spin-1 and spin-2, the transformation in the polarization tensors
given in equation (2.77) and (2.78) can be seen from the field, photon A,, (A = Ac) and graviton
hl,
(h = hc), as:
U(A)A" (x)U-1(A) = AAv(Alx) + O"A(x)
U(A)h"(x)U- 1 (A) = A,"A,"hPv(A-'x) +
,
Ak(x) + "IA"(x)
(3.30)
(3.31)
Note that, from the quanta point of view, the unphysical degree of freedom is just the longitudinal
part induced to the polarization tensors. However, from the field point of view, the strange "pieces"
are 0,IA(x) for the photon field and O(PM<(x), and in general, the specific forms of these are
horrendous. For example, the shift in the photon field is both Lorentz transformation dependence
and field dependence:
/A(X) =
-i(2r)-2
E
~
/
d75
vcr
f V~O
1((a
-
(ae - ioui3aW> ei);
0()a+A;4 .aeipx
(3.32)
'AAXI
Therefore, instead of having the unphysical degrees of freedom to be a specific tensor function A
and A', make them a generic tensor function, and then agree with the results found in equation
49
(3.21) and (3.23), which come from the massless limit of massive particles free theory.
Thus,
these equations are used, legitimately, to describe the free theory of massless spin-1 and spin-2
particles. The ambiguity of the theory, which is the unphysical degrees of freedom, is known as the
gauge freedom. The freedom is local, since it can be different at distinct positions. By couple the
field with this gauge freedom (the gauge field) to other field in the theory, a symmetry associated
with that freedom can be realized, hence the theory has gauge local symmetry. This is indeed
desirable, since it can be viewed as the internal symmetry of theory that helps to put a constraint
on possible interactions. Note that, gauge symmetry is the symmetry realize at the level of field,
as the collective behaviors of quantum.
Gauge fixing
Gauge fixing arises naturally as the technique to get rid of the gauge freedom and take out only
the physical part, so that meaningful dynamics can be read-off from the theory. It is found that
the Hilbert space of a gauge theory is defined by BRST symmetry (a symmetry that replacing the
gauge symmetry by adding ghosts in the theory and let these absorb all the unphysical information,
such as the longitudinal polarization tensor of the gauge field) or to be precise, BRST cohomology
[2]. The method of using BRST cohomology to do gauge fixing is known as BRST quantization,
and it has interesting topological origins [381. Note that, this complication comes from the fact
that we want to use the language of field to described the theory. In operators and states, one
can always specify the physical Hilbert space from the very beginning, and build out consistent
interacting theory with creation and annihilation operators. However, the good thing with field
formulation is that one can imposed internal symmetry easily, and that symmetry is not there
for the only purpose to cancel the unphysical degrees of freedom of the gauge field. Another way
of thinking, the internal symmetry is already in the theory in the first place, and massless gauge
fields, with all unphysical polarization tensors ruled out, is what come along as the indication of
the gauge symmetry.
Is Gauge Symmetry Physical?
The gauge freedom is associated with unphysical degrees of freedom. One can always artificially add
a gauge symmetry in the theory, since it's just a redundancy of the description for the dynamics.
Although one can thought of gauge symmetry as a way to constraint the possible interactions,
50
which has a very physical implication, but it should be noted that the very reason why we need
restriction in the first place is that the mathematical origin of the formulation has more than what
the physical theory needs. So, the final verdict is, gauge symmetry is not physical.
3.2.2
Renormalization
Renormalization associated physical meanings to mathematical concepts can be read-off from a
theory. There are many uses of renormalization in theoretical physics, such as in statistical field
theory [16], but here we're only interested in the context of quantum field theory.
Process of Renormalizing
In the language of perturbative quantum field theory, which encodes the dynamics of particles
and fields in a diagrammatic form known as the set of Feynman diagrams, divergence usually
arise because of loops since the integration over the momentum can go up to gigantic scale with
no limitation. In order to make the calculation well-define, one usually let the divergence at loop
levels to be cancel by the physical quantities at tree level (in technical details, bare terms and
counter terms), and that process of cancellation is known as regularization. In other words,
regularization is how the theory can be made finite, and all the calculation for the quantum
contributions at any tree or loop orders can be given with finiteness.
The validity of the perturbative description of the calculation with quantities after regularization
is that as one goes higher in loop order, the contribution decreases, and these all converge to a
specific result. But usally that's not the case, since perturbative series in quantum field theory
are usually not convergent but rather asymptotic, hence after a certain loop order, the
contributions grow larger and larger. Most of the time, the super-complicated and
time-consuming loop calculations don't allow us to reach up to that point (indeed that's the case
with quantum electrodynamics), but in a strong interacting theory (such as quantum
chromodynamics), some calculation goes on that side track at 3-loop order [30]. If the calculation
is known at all orders, one can use analytic continuation, such as Borel transformation, to make
the final result finite, since indeed the divergent is usually an artifact of the perturbative
description. Usually, quantum field theory isn't that nice, and we have to make predictions base
on the calculation that regulated all higher loop orders after where the divergent starts (it's not
51
mathematical rigorous at all, but still pretty much the best we can do, and as expected, the
predictions can be very wrong, such as the quark-quark potential in quantum chromodynamics
predicted from perturbative side of the theory).
It's important to understand that regularization removes the physics at UV region, not IR
region. There are two different type of divergences, one at IR region and the other at UV region,
and both of them can be physical. While UV divergence can sometime be unreal due to the lack
of knowledge in the degrees of freedom in the theory at high energy scale, IR divergence is always
physically meaningful, and it's indeed play a very important role in effective field theories since
the IR behaviors should agrees with that of the original full theory. Thus, infinity isn't always
bad, it's just that when one needs to make a connection to observables, the results should better
be finite.
There are cases when the physical quantities actually goes off to infinity or becomes badly
defined, but that's usually when the nature of that quantities cannot be understood normally any
more as the physical systems has a change in phase, such as the quark mass, which is can be
read-off physically as the pole of the quark field propagator, lost its meaning as the quark cannot
exist as a single quanta normally under the confinement energy scale.
After regularization, the calculation is finite, but there's still one more step to come out of the
hard-core maths to the reign of the physics world. Renormalization scheme is the way we choose
to define physical meanings to specific calculated object, and with the physical quantities are
renormalized, we conclude the process of renormalization. It should be noted the observables are
not scheme-dependence, at all, and even if the choice of renormalization is bad with huge
divergences appear, usually these are all cancelled out the others hence the observables are finite.
Indeed the divergences are artifacts from splitting up the physics at different energy scales.
If the number of physical quantities need to absorb the divergent is finite, then in principle, it
can make good predictions as we climb up in the energy scale, at least as long as we reach where
one of the quantities blown up, such as in quantum electrodynamics for the gauge couplings at
the Landau pole energy scale. If the number of physical quantities is infinite, it's lost the
predictive power, but still work in a certain range, and indeed, this is the key principles of the
effective field theories. The theory in the later case is said to be non-renormalizable.
52
Keeping track of Regularization
For an integration Jy of a graph 'y with external momentum p and internal momentum k which
are integrated over, define the Taylor expansion operator t of order N in the external momentum:
.1(pi,
J
(3.33)
)
...
O,
;k i,...
(
(2,7r)4...I
N
n!
lN
DPnI
'n
n=O
The counter term of a subgroup can be described by the operator tY, then regulated integration
is defined up to a polynomial in the external momentum of degree D-, which is the freedom of
regularization choice [5]:
D~y
i-t = (1 - tY) J_ , J-Y -+ j-Y +
!fd..,p
..
(335
n=O
In dimensional regularization, the divergent can be followed with the dimensional variation E -* 0:
00
00
J
S
j(y) an
a.," , j(7) = (1 - ty)Jy =,
n=-D-f
j(y) +d
(3.36)
n=O
Now consider an arbitrary Feynman diagram G, then a subgraph -Y C G (can be the full G or the
empty
0) is a renormalization part if the degrees of divergent is non-negative D ;> 0, which means
superficially divergent. Two subgraphs are disjoint if:
71
(3.37)
n72 = 0
If one is contain totally inside another, then these subgraphs are nested:
71 C
(3.38)
72 C 71
2
The subgraphs are overlapping if these share lines and vertices:
!(
1
n 2 = 0) (
!(_Y1 'C
2) ( !(2 c -Y
1)
53
(3.39)
#
A set of non-full renormalization parts y
G is a forest, a set with all non-full renormalization
parts is a full forest F and sets with only disjoint and nested non-full renormalization parts is nonoverlapping forests Ta. Weinberg's power counting theorem states that by eliminating potentially
divergent contributions from a full graph, the leftover should be absolutely convergent. Therefore,
after regularization, one arrives with a finite answer with the Bogoliubov's R-operator [6]:
RGIG
fl(I
-
t,) IG , RGIG =(1 - tG)RGIG
(3.40)
7EF
The order of carrying out the regularization for subgraphs should be inside out, but it's not applicable to overlapping subgraphs. However, one can get rid of that problem, since any divergent of
the Feynman diagram cannot be non-local, and one arrives at the Zimmermann's non-overlapping
forests formula [39]:
(
RGIG = (1 - t G)
a
(
))IG
(3.41)
tE.Fc,
With this formula, one can keep track of the regularization, for any Feynman diagram. This is
a very crucial part of the standard Bogoliubov - Parasiuk - Hepp - Zimmermann renormalization
method.
An example: Quark Mass Ambiguity
As mentioned before, renormalization scheme is the way we interpret the mathematical results
physically. A bad choice of scheme may lead to a bizarre answer, and we will demonstrate it
through the axample with quark mass, by considering a set of diagrams that's can be calculated
at all loop orders, and analytical continuation can be imposed to get a non-perturbative result.
Quarks, at quantum chromodynamics scale, are at confinement phase, which is a collective
behavior that's not allow a single quark to be alone in the theory, therefore, it can only be found
in bound states, such as baryons and mesons. Indeed, if one try to define a quark as a particle in
a conventional way, and read-off the pole mass from its relation with the MS mass (which comes
out from just regularization of the Langrangian in the MS scheme), then an ambiguity of the
size of the quantum chromodynamics scale arises if the selected energy modes for quantum
contributions is not treated with cares [30].
Consider a particle subset of Feynman diagram in quantum chromodynamics, the bubble chain
54
(a gauge vector field chain with fermionic bubbles) correction to the propagation of quark.
Indeed, the bubbles sum diagram is unique in any order of perturbation theory that gives gauge
invariant contributions to the flavor or color structure of the theory, and it has the most power of
nf (the number of active fermions running in each bubble). The fundamental ingredient after
regularization, the bubble (in Landau gauge) is [30]:
Dab(p,a.) =
Z
g 2 4167r
30 a,
92 = 47a, =
4- )00 =-n+-CA (3.43)
-
(
11
2
5
-
Taking a Borel transformation of the sum of all bubble chains (ignore the
0X)
f(a)
n+
fna" 1
=
(3.42)
e
PPV) 6 ab OOCS)l1-
6
ab):
00
n
due-uF(u) , F(u) = f_ 1 6(u) +
=
fnu
(3.44)
;
n=O
n=-1
as) (=2
GpV (p, GtwPas
n(
(3.45) nn
E P (/0o
= 47
P2 nn" 41r
- l4v)
(p)
_ 00 4
(45
167r 2
(9 2(G
(3.46)
ec)
(
PPv)
("2
2
(p, U) =
The mass pole nmz can be read-off, after sticking the bubble chain sum to the quark propagator
with the MS mass fi [301:
+i4V'u(p)
2G ab
(3.47)
11 (p2
-iCF
ffi, U)
-
-+
M,('u) =
CF
67ro
k4 i(p)'t(fi +
(
2 ecu6(1 -
(n6(u)-
2
(p + k) 2 +
J (27r)4
u)F(u)r(1
-
CF
A2 c
/o(j
67r,30 AM2
U6(1
)
2u)+
+
F(3 - u)
2
(3.47)
-u)
(u)F(1
7(
)+
17(3 - u)
(3.48)
...
-2u)(
... )
(3.49)
The 6(u) part comes from transforming the factor 1, which corresponds to mr = f- at lowest
order, to Borel space. The omitted terms contain both the terms where there is a pole structure
of the form 1 rendering it regular at u = 0 and terms that regular in u, which are not needed for
the analysis of interests. Prom the F-function structure, the strongest pole, which is closest to 0,
is at u = ., corresponds to a renormalon with 2"n! growth. The mass pole can be further
55
simplified to express the effect of that pole in more details, and inverse Borel transform it:
CF
MA(u) = 11 6 7
3
pei
(M
)
2
0\
+
MP(OS)
42_
due -as MA(u)
(3.50)
Because of the u = j renormalon, there's an ambiguity which is realized by analytical
continuation above or below the pole, hence half of the residue around u =1:
AmP(a) =
Res (MP(u =
)
=
1(2i17u)i
()=
=
CF
C
C
(3.51)
Since the pole mass has this ambiguity, one should avoid this choice of renormalization scheme for
mass. Note that, the mass pole uncertainty doesn't depend of the use of in and it's independent
of the regulated energy scale p, which is the evidence for the fact that it comes from the bad
choice of separating physics at different energy scales, and to be precise, IR region [30]. To cure
the ambiguity, one has to introduce a new energy scale R, and in general a scheme change gives:
mp = m(R) + Am , Am = R
alnk
n=1
( ) (7
4
(3.52)
k
The mass m(R) can be chosen to be free of renormalons, if Am properly substracts the pole
mass renormalons. The physical picture of the new renormalization scheme is that the energy
scale R can be considered a floating cut-off which sets the scale for absorbing the IR fluctuations
(that causes the instability by dressing up the pole mass) together with the pole mass to yield a
well-defined mass m(R).
It's not trivial why the origin of renormalons is in the IR region. To understand this, one has to
note that the MS scheme separates short and long distance physics for logs correctly, but for
powers it relies on setting the scale of integration to zero, which is forced from the very definition
of dimensional regularization and the scheme itself. This treatment leaves residual sensitivity to
power divergences from including the wrong regions of momentum space in the quantum field
theory integrals, which results in renormalons [30]. The problem wasn't in Wilsonian picture of
renormalization, however, since Wilsonian has a hard cut-off so in general the calculations are
very difficult (not to mention that the symmetries will not be preserved order-by-order). One can
always start with the MS-scheme and perturbatively go toward the Wilsonian picture in a
56
Lorentz and gauge symmetries preserving way, and indeed, it is the case with the so-called
MSR-scheme [30].
3.2.3
Long Distance Physics
The long distance physics in quantum field theory is dominated by the exchange of massless
particles. For massive particles of mass m, the estimated interaction range from the uncertainty
principle in quantum mechanics is nothing but ~ -, hence dies off at long distance. It can also be
seen from the tree-level Born approximated potential, which the potential is decayed exponentially
~ mr (along with a negative power of r) with the separation r. This should be the case, since
the physics of quantum field theory always decouples the massive degrees of freedom (dynamically
inactive) as one goes to an arbitrary small energy scale, passing the mass thresholds. Here, for
integer spin field, we will discussed about the decouple of spin-> 3 as a possible soft-exchange
from the theory at IR region. A generalization to massless half-integer spin field can be done, also
yields the decouple of spin-> '2 at long distance scale [3]. The implication is that particles with
spin more than 2 cannot have couplings that survive at low energy limit.
S-matrix with Massless External Quanta
Consider a scattering process in which a massless -y particle is emitted with momentum q and
integer helicity states h, then under a Lorentz transformation, the S-matrix should be unchanged
(in general, the S-matrix for emission and absorption of several massless particles can be treated
similarly). The S-matrix is simply refer as Sh(q, p), with p stand for all the momentum and spin
states of other particles, well, schematically [35]. From equation (1.53):
Sh(,p) =
ite*O(AS(Aq, Ap)
(3.53)
Use the standard 4-momentum trick with qO and po, so that a Lorentz transformation AO can carry
these to the momentum q and p of interests. Since in a general physical process, the only Lorentz
transformation that leaves both q and p invariant is the identity, so the carrying transformation
Ao is uniquely determined by q and p [35]. Let's try to write down the S-matrix in the following
form, with the polarization tensor of any massless quanta with integer spin can be built purely
57
from the known polarization vector given in equation (2.75):
1hI
qa(,
=t
1)
t(
Aa..,,(q, p)
,E'(ql)
= Rl,(q)cv
(3.54)
j=1
Some useful properties of the polarization vector can be lists [35]:
(0qheI(q) =
,)
1,
Ef(gjIE, 1 (q) = 0
q12q'q"
EZ(qE'I(q)
h h = 'rf" + qIqV I +
(3.55)
,
CP1t(q) = CA
, qI = (Iq1, -q')
(3.56)
Also, equation (2.77) can be rewritten in a more usable form as:
A
iAv
(q)
ih(A
=
q-)
(3.57)
With these, one can construct the standard "M-object" from the standard S-matrix:
E(q
o))Sh(_, PO)
2Ig- (
(q , po) =
1h
(3.58)
"
j=1
Define the general "M-object" by Lorentz transformation, hence it's a Lorentz tensor:
hi
(, P)
Mlh
jh
-(Fl
=
=1
(Ao)"j, (q,T P)) A"hI'
( qO
, , PO)
Ihi
(3.59)
IhI
(Ao)PkVk (q,
(Ao)lj V (q' P)) Q
H
Ap)) Mh7.~P~h (q, Ap)
k=1
A,PP) MhI (IhI A36)
j=1
(3.60)
This expression for the M-tensor satisfies equation (3.54) for all possible q and p. To see that,
rewrite equation (3.57), then rewrite equation (3.59):
-ihO
AO
q,
(A1(, p) -
C'(q) = e
qI(Ao OV~*P
E (q)
(3.61)
Ihi
--
(
h
j=1
58
1 ...ijhl(qj
p)
(3.62)
ihe
-(,
A-(-)
e
qh
q
|J
(q0) -- 0hh
(o, po)
t
( )(A )O
(3.63)
M
k=1
Using equation (3.55) and equation (3.54):
A
-+
(jj
M
!()
0
= e(A
...
)
j=1
1t J(q)
q, ) =
A
SI(qIo po) (3.66)
(qic
(3.64)
(AI, po)
t(ql.))M.
(3.65)
k=1
-- +
_2-I
E
qJj1
(JJh
~
MhB. A~hI qP)=
Setting A = A- 1 , then one arrives back at equation (3.54). Now, after knowing that the "M-object"
is just a Lorentz tensor (even symmetric), so the S-matrix Lorentz transformation infinitesimally
can be read-off:
A
)=
Sq,
=
ihO(Aq)
'viI eiO(ASh (
,1
+ W", ,(3.67)
- (Aq) PiAvOEh (Aq)
IT'
q, Ap) -
-I/21
WE
(
q,
t
IE
j=2
)A'i...
,,, (-, p)
(3.69)
To get the right transformation for S-matrix, then [35]:
ihi
)(AJhMl)... Afhp (Aqp) =0
qh
(3.70)
j=2
This imples the on-shell gauge invariance, since from the equation above, S-matrix should be
invariant under a regauging of the polarization vector with an arbitrary parameter:
EI(q --+ El(q-) + Ah(q)qP
(3.71)
For a massless vector field with spin-1, then one arrives at the Ward - Takahashi identity:
q'Ai,
1
59
=0
(3.72)
For a massless tensor field with spin-2, then:
qlEI iA2,v =
0 -+ q"A'i 2
(3.73)
11 q"
But because of equation (3.55), a shift in M,, proportional to q,, will not change the S-matrix,
so one can always get the choice of M-tensor so that:
(3.74)
q"Ma22MV= 0
Soft-Exchange of Massless Quanta
To describe the long distance physics, one needs to go to the IR region of the quantum field theory.
For a high-energy scattering process Ej -+ E, the emission of a soft-massless quanta of integer
helicity h and momentum q- -+ 0 from a particle of mass mj and spin sj (use rg to specify incoming
+1
and outgoing -1)
in the process, as the internal line is off-shell (very close to on-shell) with
mementum pj - q and spin state 0' and the external line is truely on-shell with momentum p and
state o-, contributes to the S-matrix:
(pj+q)2 +m
2
pjq
2pjq
(3.75)
Note that, there's also possible contrubition to the scattering S-matrix from emission of photon
from the internal lines, but these aren't singular at the soft-limit, therefore we neglect [35]. Indeed,
external particles are on-shell, so when q --+ 0, the propagator diverges as (pj +
q)
2
-+
n2 ,
while
for virtual particles, since these aren't on-shell in the first place, so in general divergent of that
form doesn't arise [21]. From equation (3.70):
>
, ..)j8e(=0 (3.76)
q,)
> 3( ...
1
k=2
j=1
Thus, the only possible form for the M-tensor so that the above can be true for any input:
q"'M,1...Lh(sj;pj, o-j, o') Vq
--
-)
(
k7j,
k=1
60
)
(3.77)
The constraint now becomes:
|h|
j
Z)f
0
,h"((s;pjo-,
(3.78)
k=2
,
As the soft-momentum goes to 0, one expects a smooth change so that the scattering process
should know nothing about the interaction between the high-energy particles and the massless
soft-particle, hence there must be a factorization, which can only arise if [351:
|hj
ff hI)(s.;P. p, a',,) = f (h|)(S'; P_)
) f( (s; pj) =0 (3.79)
j
The value of
k=2
f(IhI)(sj; pj) can be defined to be the renormalized charge under the coupling with
the massless particles. With M-tensor is now known, the soft-interaction vertex can be read-off
[35]:
V(h)
)
-
(2r)%,,CT (pjEc~t)jhi
h __f
_2i
(p)
=1p fy (sj; pj)
-(
(3.80)
/'2Ij 2p 0
(27r)
Let's look at the interaction ,with the exchange of spin-i and spin-2 particles, of two particles with
mass ma and mb, momentum Pa and
the momentum transfer t = -(Pa -
PA,
spin-1 coupling ea and eb, spin-2 coupling ,/87Gg. Let
Pb)2 -+
0, then the S-matrix is dominated by a single spin-1
and spin-2 exchange, and can be calculated at tree-level (ignore 6C,,) [351:
S(Pa, Pb) = 41 2 t
___1
eaCb(PaPb) + 87rGg2 ((PaPb)2
_
mamb)
2 2
(3.81)
Choose the frame where particle b is at rest, then:
S(Pa,Pb)=
eaeb + 87rGg(2po -
4ra
'7rt
a gmb
0)
(3.82)
As the massless spin-1 particle is photon, and the massless spin-2 particle is graviton, hence ea is
the electrical charge and the effective gravitational mass f
fila = g(2p
-
, p
(ma, i)
61
-+
is:
ma = gm7a ,
mb = gmb
(3.83)
With the physical meaning of the charges is related to classical physics, one can take a look at
equation (3.79) and interpret the found constraint. For spin-1, it's the conservation of electrical
charges (the whole arguments can be generalize to non-Abelian gauge group):
7 jf (1)(s;p)
=
0
1 )(sj;p,)
Zf
-+
(3.84)
out
in
ff2 )
For spin-2, the only possible constraint for
f )(s; p)
=
is that it should be universal, so momentum
conservation leads to satisfaction trivially. This is the interactions point of view of the equivalent
principle in general relativity:
njtpff 2 (sj; p) = 0
For spin-> 2, in general there's no possible
f('),
-*
f} (s2;)p.) = f (2)
or in other words
(3.85)
f(>2 )
= 0, therefore these
massless particles should be decoupled from the theory at long distance scale, low energy limit.
3.2.4
Maxwell's Electromagnetism and Einstein's General Relativity
Maxwell's Electromagnetism and Einstein's General Relativity can be seen from the part of the
Lagrangian contributed by the dynamics of the massless spin-1 photon field and the massless spin-2
graviton field, given in equation (3.21) and equation (3.22).
Also, long distance physics, which
closely relates to the reign of classical physics, does give-off the nature of electromagnetics (charge
conservation) and gravity (equivalent principle), which is described in equation (3.84) and (3.85).
The Maxwell Lagrangian is exactly given, while the Einstein-Hilbert Lagrangian can only be seen
perturbatively.
However, the important underlying nature of these two classical theory can be
understood through the symmetry (in which the classical field equations are read-off from), and
indeed, the gauge symmetry of the electromagnetism vector potential is the same as the gauge
freedom of the photon field, while the diffeomorphism of spacetime, which can be encoded in the
metric, agrees with the freedom in defining the graviton field. Therefore, quantum field theory
gives rise to electromagnetism and gravity. It should be noted that the quantum field theory for
perturbative gravity is non-renormalizable.
It's still work well as an effective field theory of low
energy scale, but to look more into the UV region, one needs a new method. Superstring theory
is the way to put quantum field theory and gravity consistently at perturbative level.
62
An Example: Geodesics from Field Theory in Curved Spacetime
The particle-wave duality in quantum mechanics comes from the fact that a single particle
behaviors are described from a wave equation. In classical field theory, a particle can be
described by a localized field configuration, and the classical field can be realized from the
quantum field theory point of view, even with the discreteness of quantization. In this example,
we will use this realization of quanta in a classical field language to verify that quantum field
theory in a curved spacetime background indeed does have the features of general relativity, and
in details, we will verify that a free particle should move a long the geodesics. We will consider
massive particle only, and skip the generalization to massless one.
In general relativity, the reasons why a particle is moving along the geodesics can be understood
from the Lagrangian point of view. Since the action is proportional to spacetime interval, by
extremize it one gets the geodesics:
ds = -m
-m
S+
d 2 Xp
dr 2
rp
dr ~ ds
-
JdrL
= -m
dX" dXO
aO dr dT
d 2Xp
2
dr2
f
dT V/gpTOX9XV
1 dL dXA
=0
L dr dT
+ F'3
"
dX"dXO
di
dT
(3.86)
(3.87)
'
S[s]
= 0
(3.88)
Now, let's look at the problem of classical field theory in a curved spacetime. Consider a real
scalar field action, the equation of motion can be read-off:
SA[
+
O4"( + m2 2)
- _'fd4(g,"
(09 - M 2)4)
0
,
=
E]4
1 &(
9g"J4)
(3.89)
(3.90)
Schematically, a localized wave-packet can be used to describe a particle. With A is the size of
the and r is the curvature length scale of the spacetime background, then a good description for
a particle from the classical field point of view is:
A
so
A (x)4 ei =1 > - ~-+ 0 , kA= -01,0 , Vk, = Vk,
(3.91)
E
63
The equation of motion is now change to:
1(,A -&O#
+... =0
A
"
At the leading order
-,
-
m2 +
+
(3.92)
one arrives at the on-shell condition of the particle:
(3.93)
At the next leading order , the size of the wave packet A(x) has the evolution:
k ,, ln A
1
-
(
k)
Vc,--
(3.94)
From the metricity condition, one arrives at the equation of geodesics flow:
V,,8 = 0
V, (k2 = -m 2 ) = 0 = 2k"V,k = 2kV ak = 2VkkA
(3.95)
The position of the wave packet can be seen from the flow with respect to the proper time T,
then once again, the geodesics equation arises:
d2X
dX"
dT
3.2.5
dT
1
2
dXa dXl3
aO dT
dT
(3.96)
Classical Field Configuration
Since the physical Hilbert space is completed, therefore any physical field configuration can be
made, in general, by superposition of multi-particle states.
To make a connection the classical
world of physics, let's construct a classical field configuration, and for simplicity, consider a free
theory of massive scalar field:
H =
2]
d X_((O)2
+ (G)2+
~'
m242)
(3.97)
We want to build a classical field state:
(1)1) = ID( 0)1 ))
64
(3.98)
One can always try to construct the Halmintonian and Lagrangian from the creation and annihilation operators, but since it's a lot simpler to deal with functions, usually one choose to start with
the Halmintonian and Langrangian first, find the degrees of freedom and the associated canonical
momentum, and then use canonical momentum to quantize the theory, thus get the operators and
states language back again. The field and the canonical momentum, represented as creation and
annihilation operators:
P
<bz)=
2(27)
rl(y) = -i
3
(ag + at -)epsf
,
(3.99)
(a# - at g)eigx
(3.100)
/pl+m2
23x2(2
da
In a functional form, the canonical momentum can be written as the derivatives with respect the
the field, and vice versa:
Z') -+ _(x)#) =-
)
, b(X)1)
=
(3.101)
= [<(5),
-i,(,)i|) f(i')]
-
((l
Hence, it's straight-forward to build up a classical field state from the vacuum
I|) =
eif d3
d(I
d
(x)r(x)I) = exp
)
d
2(2w)
IQ):
(ap.- aLe)
(3.102)
To see that, let's act 4i(Y)-operator on it:
<b C~i
=
JDr
) (7r -I_z,(:
=
3.3
f
Dw
f
f d3
(
)7(i) IQ)
Kr(i()ef
Xs~rs)
if d
JD17)
-D(Y) 11(S))
(3.103)
(wj.(x)eifd33Y )(Y) 7(Y)
(3.104)
"(i)
|I) = 5(Y)eid 3 I(')H()IQ)
(3.105)
=
Functional Integration
As mentioned before, the S-matrix language from particle construction of quantum field theory
isn't good for short distance study, but with field formalism one can always theorize the theory
65
first (usually with the gauge symmetry, since it puts the constraint on possible fields and
interactions one can write down) and read-off particles out from it then try to match with
observables in experiments, instead of start with the particles and then guess the underlying
degrees of freedom. Canonical quantization is the method to get the quanta out of the theory,
and generally very useful for understanding physical interpretation at microscopic scale. The
calculations with quantum fields are fairly simple for weak interacting theory, such as quantum
electrodynamics, and the predictions come out are generally good, but the same cannot be
applied for interacting theory, such as quantum chromodynamics, since the perturbative
description is ill-defined. Hence the analytical results, even just approximated ones, are hard to
see, and it's desirable to have a numerical method for quantum field theory.
Functional integration is a different way to define a quantum theory, where instead of quantum
fields as operators and physical Hilbert states one has to deal with the quantum fields as
functions, which is more familiar and easier to keep track of. It's not only good for analytical
purposes, but also can be formulated - to be precise, reformulated - numerically in a very
well-defined way, with some guidances from constructive quantum field theory as the theory is
Euclidean. Short distance physics can now be seen with correlation functions in functional
integration, where insertion of fields can be anywhere, as the need of the non-interacting picture
for initial and final states in S-matrix is not needed. In other words, functional integration is
usually a more convenient choice of describing a quantum field theory than the canonical method.
It should be noted that, in most cases of physical interests, functional integration quantization
and canonical quantization are equivalent. The quantized nature of functional integration is
encoded in the the quantum field itself, and the counting of quanta in non-interacting theory can
be track down by the number of quantum field insertions inside the integration of functional. For
interacting theory, one has to connect it back to canonical quantization to see the quanta more
clearly.
3.3.1
Connection to Canonical Quantization
Let's derive the functional integration from canonical quantization, and then see the most important aspect of canonical method, the commutation relation, again from functional integration point
of view. In a general physical system, start with the the choice for degrees of freedom, the action
66
S in a spacetime volume V"3 can be written down since the behavior of the system, by definition,
extremize the action. Hence, the Lagrangian density L can be read-off, and the Halmintonian
density W at a time t in a space-like hypersurface slice Vi comes right after by introducing an unit
timelike vector N,, = (1, 0), then thus canonical quantization.
The Importance of Space-like Hypersurface
Indeed, only a space-like hypersurface can provide initial conditions for the time evolution. This
statement is true in a general curved spacetime. If the initial hypersurface is not space-like, then
some events on the surface will be in causal contact with others, implying that field values
cannot be chosen independently. This violates the requirements of phase space that the variables
might be chosen freely on an initial hypersurface. It's not only a bad idea to do quantization on a
non-space-like surface, but also impossible to formulate a well-posed initial value problem, which
is true in both classical and quantum physics.
Another way to put this, is that cannonical quantization can only make sense on a space-like
hypersurface. Typically, a slice of constant time is already a space-like hypersurface, but
sometimes it's not, such as the choice of time in Schwarzchild metric spacetime. Strictly
speaking, there's no need to use a slice of constant time, and just a space-like hypersurface is
enough (of course, it can always be defined as a space of constant time in appropriate
coordinates). Quantizing on a time-like hypersurface means trying to quantize on a surface of
constant position in space there, which has to be bad. There's also a technical issue with
non-space-like initial hypersurface, that the Laplace operator (for the fields equation of motion in
usual cases of physical interests) is no longer elliptic, meaning that it gives a spectrum with
singularities, thus cannot be inverted.
Derivation of Functional Integration from Canonical Quantization
For simplicity, let's just consider a theory in flat Minkowski spacetime with the degrees of freedom
are N scalar fields <D, and its canonical momentum H,, can be read-off from the Langrangian:
N
Svl,3 =
d12
= ill-tI,. - r
,4 I =oaE =N, (PI -+W
(3.106)
nr=1
67
[<Dn(x), <bm(X')
[Hn(x), Ilm(X')] = 0
=
fln(x)J#)
=
,
Canonical quantization means imposing the commutation relation operator-wise:
[<'n(x), JIm(X')] =
jnmo()(X
, Dn(X)brKn) =
ns|T)
-- C00.(E)I#n)
-~
(3.107)
X')
-
(3.108)
From the Halmintonian density, which is the operator form for the evolution of the quantum theory
with respect to time, one gets the change with time in the definition of the fields and its canonical
momentum, and also the eigenstates:
H(t)
=
f
d 3 N(t, X)
(3.109)
Jv?
Sei
fln.(t,
f dTH(T)
-i fot drH(r)
-r)Hnef>
9) = eifi
-+
H(r ) -+
idrH; t)
-i
drH(r)
(3.110)
fo d-H(rTiw)
(3.111)
Equation (3.108) can be used to proof that:
N; tj 71, ...
7n t,
7I1,
-i,
7--) =
f
ei27)f d-2n(
7(
)
(#1, ...
)n1(
(3.112)
The inequal time inner product can be read-off, with the time ordering is implemented inside the
function integration measure Do and D7r for each history of configuration [181:
(3.113)
im
M-1
N
f
j=1 72=1
M-1 N
=
im
M-1 N
D
j1
,
tj+1 - tj
=
Jt =
t'
-
t
Al
(3.114)
M-1
D-.
+1; t+1 7rj ; tj ) (7rj; tj| # j; tj
(3.115)
)
j=1 71=1
j=1 n=1
M-1
-
)
( fj (#+; tj+1 I
j=0
D~)
)
M-1
N
j=0
M-1 N
(27r)-N lim
fv3 dajrN jxr(#i
D7r ) e
j=1
j=1
n=1
-#iV)-H(tj)jt
' 1
(3.116)
-f
=
(27r)-NJ
N -
(m
n=1
#';t'
N
4
e'7fV1,3 d
n)(1[
n=1
68
(ZEQ ir2 Ot-7-
(3.117)
The most popular form of the Halmintonian density is quadratic in the canonical momentum:
1
W = 17rnCnrnm [01.,qN17m +O0(71
2
170 )
(3.118)
Hence, the Gaussian functional integration can be evaluated, with stationary constraint [181:
,1
0,; t)
(#'1, ... , #'n; t'| 1 ,1 ...
=
f
10n)
N
n=1
i fy1,3
O/
d4X(E N 17rato._7j).,
13dX[
4,0
(2 7r)N v'e(4)(O) f,3
dOTr
IC[1,.,N](3.119)
nC[i,. NI(
9
#,
one get back the
K)S = L
(3.120)
But at stationary solution of the canonical momentum 7r in terms of the fields
Lagrangian, which is written with the fields only:
N
at#O = O,
,
-=
-1an
a
7rt#n -
-+
n=1
If the operator C is independent of the field, then one arrives at the building block of functional
integration, which states that the oscillatory weight of a history of field configuration should be
the action:
4' ;t'
N
(0' ...
n; t'i #01,.,
D
o; t) ~
(3.121)
) es'
n=1
4;t
There will be a disagreement between the canonical method and the conventional functional integration if the operator C is field-dependent, as the weight associated with each history should
be history-dependent. However, this is just a problem of definition (can always modify the original formulation of functional integration so that it has history-dependent), thus for any canonical
method there will be a corresponded functional integration.
Commutation Relation in Functional Integration
The order of operator in the equal-time commutation relation of a single field type of interests
(inequities only yield 0) of canonical method corresponds to the order in time of the insertions in
the functional integration:
[<Xt
[ ), D 1(t,
t) I:')]
4t-+o - lim (<b(t + dt, x)pt, i') -6~,s)<~
69
t, Y))
(3.122)
The correlation value of the insertions can be calculated:
m D
o5Di(#f; t + t|Ib(t + St, )HI(t, ) - f(t, F'>1(t
lim f17(3.123)
st-+0
f Df Doi (of; t + &| #1; t - 6t)
f Df D#4D7re
f
vr
(
ot,)I#; t - 6t)
-
(Y) - #i(Z))
)f
(of W)(3.124)
DfDjDwe
Using the method of discretization with P x P x P grids of size
13,
with label
= (ji, J2, j3) runs
from 0 to P =(P, P, P), on V 3 , then:
f ]
d dd7rleNr
7'
f
-+
-(3.125)
f Hl; 6 d} d#7 dirieT-
f
d dfdwid# d# dirte'94eT ( le ( - r'(#
o d
(
dd( Wd 7(i 'f i eN ' -e
dddd7r/dd
-
(3.2)
-
(3 .12
)
_fi
(3.127)
o*, d#/5,' (dora &-+4) (i i'el
f d#d#d'
f di
f
d..
(##))
d.5-S(
'
-
(os(#
-
)36(
=-f o
-.
-
-
-
f d
/ f fen9(4- ) de+
-
f#d/4de de5
-+
-)(
i6N()(- f)
(3.129)
7')
This is precisely in agreement with the canonical method.
3.3.2
Is Functional Integration always Superior?
It should be noted that, although functional integration is easy to keep track of, sometime it's
not as convenient as the canonical method, such as in understanding the polology nature of the
S-matrix, since the completeness of the physical Hilbert space gives on-shell particles interpretation
natural, while the correspondence in the functional integration is not that straight-forward to see.
One should be flexible with the two methods to read-off interesting physics from quantum field
70
theory.
An example: Polology
The study of pololology in S-matrix (or, correlation function of insertions with non-vanishing
factorization) gives nonperturbative physics of the local theory, scanning the mass spectrum as
the energy-momentum scale goes up. Let's start by looking at a n-point insertions and factorize
it (by on-shell unitarity complete insertion) [36]:
G(xi, X2, ...
,X) = (Q|T (01(Xl)02(X2)...O(X.)) IQ) ,
J
1
(27r) 3 2p,+M2
e1ir(QIT
f2 VpT+ mq,
01(x)...O,(x,)) I')(PI (o,+(
IF =g,2 )(IF I
+ 1)...O(X.)) I) + ...
(3.130)
(3.131)
The assumption is that the theory has spacetime translation symmetry, so that the 4-momentum
P is a good quantum labeling. Note that, choosing x' =
j - x 1 for j < r and
'= xj - x,+1 for
j > r:
eir
=(ti -
=0(min{ti,
... ,
tr})
- max{t,+1, .--,
tn})
tr+1 + min{0, t', --, t'} - max{0, t+ 2, ... , t})
(3.132)
(3.133)
For now, let's just assume that the ground state is Lorentz has no 4-momentum, as PQ) = 0
[36], then:
(Q T (01 (Xi) ... O,(X,)) I)
= (QIT (ePx1-i
01(x1)e-iPX ... e
Or (x,) e-iPXl e-iPX))
(3.134)
-
e-iPqx (Q
IT (ePaO1(x 1 )e-iPx .. .eiPX O,(xr)eiP1)IXF)
= e-iPqx1 (QT (01 (0)o2(X'2)...Or(X')) )IT
(3.135)
(3.136)
Using the 0-function integral representation:
f()' dw i _-iWX
d
0(x) =1 -e
071- 2iw
71
(3.137)
d
Define At = min{0, t'
0n(PliP2, ---
1,} - max{0, t'
2
, ...
df
Ryg
, P.) =
,
LJ, then
1-
d4Xieipii...
+ m4 f
S(27r)3 2 V/p
x e-iP'(x1 xr+1)e i(p2+-+r)xl
work in the momentum space:
fd
Pj _=P
x(27r)()
- j=e
(3
i
d
27r
Ej - Vp
dPi
+ Mi +
f
E
Ex(
(2ffr)j
(3
j (Ej - Ej
(A)
i
p,+ m2,
+ n4 - w)
4 3(4)(p +
2
2
p - Mr,
q)
(3.142)
(3.143)
(3.144)
j=r+
r~E - E ,)At
=
j I Ej -E
j=r+1
j=1
i=r+1
(QIT (01(0)O 2 (X'2)...Or (4') T)(1 IT (Or+1(0)Or+2((Xr+2 )...On(X'n))IQ) +
_i(2wr)
=
(3.141)
(3.146)
j=1
x
(3.140)
j+1r+=
E
1
n
=
( 27r)3 j 1rif2
ET (3.145)
r'e-
...
i
j=1
d 14
+
w
j=11
(T IT (Or+1(0)0r+2(r+2)..0n(X'n))IQ) +( . ; pj , q= P
/
(3.139)
T (01(0)02(2)-...E0,(',))IT)
j=r+)
x
2 )...On(X'))IQ)
e
j=
(3.138)
27r w
n1
r
x
Jff
3
1 (2-r3 )
f
e-i(t-tr+i)e-WALt
ei(Pr+2+-+Pn)xr+1
x (QIT (01(0)02 (X'2)...Or(X')) IT)(4 IT(Or+1(0)r+
2 ('+
2
1
2/pE + m2
dd4X'
f
...
(3.147)
I
(QIT 0 1 (pI)...Or(pr)JI)(T'IT (Or+(pr+1)...On(Pn))I)
+
...
(3.148)
Unitarity of the theory means the completeness of (on-shell states) Hilbert space, and the used
notation in the calculations above is short-handed, and the better one should be:
( 3
)(W
-
(2
3
2E1q |T;p)( I; p
(3.149)
The poles of the n-points correlation in momentum space are at the mass of a particles in the
theory. This is a nonperturbative physics, since the mass is not only for fundamental particles
(appears in the Lagrangian) but for all the possible ones (such as bound states and
multiple-particles, even unstable particles). For a single fundamental particle insertion
72
factorization, the pole arise when the particle goes onshell, and this is nothing but the Lehmann
- Symanzik - Zimmermann reduction, with a nonperturbative derivation. In that description,
S-matrix can be read-off as the residual (up to a prefactor cj = (QJDj(O)Jp)) near the pole of
onshell insertion. For details:
Gn(P1, P2,..., P
=
n
pj)
-Z
-
= J7G 2 (p,-pj)Sn(pI,
j=1
p 2,..., pn)
G 2 (pj, -pj)
,
=
j=1
2
2
z
+
n-1
m3
(3.150)
The momentum of the theory can be constructed from the Lagrangian as the conserved charge of
spacetime translation. For a free scalar field 4D theory, it's simply:
-fd3H()V()
H = Jd35Q
2(
)+
(V()) +
(3.151)
,
m2
X2))
,
GX)
=
& ()
(3.152)
Note that, the assumption that the vacuum state is annihilated by the momentum operator P, in
general, isn't always true. Essentially, in quantum field theories, the kinematic structure is fixed
by the canonical commutation relations between fields and their canonical conjugate momentum.
In non-relativistic quantum mechanics there is a finite number of degrees of freedom, and so the
Stone von-Neumann theorem tells you there is a unique Hilbert space and choice of operators on
the Hilbert space (let's just take it as the mathematical fact). The theorem, however, does not
hold for infinite number of degrees of freedom, and actually there are infinitely many inequivalent
choices for the Hilbert space for quantum fields [34]. In flat spacetime, the requirement that
there exists a state such that PIG)
=
0 picks out a unique Hilbert space, and one can check it, in
principle, with lattice quantum field theory (to avoid horrendous analytical calculations):
(QIPIQ)
=
JD2N eisP
(3.153)
In curved spacetime (which in general have no killing vectors), it is unclear how to single out a
unique Hilbert space, and this is a big difficulty [341.
It's important to note that, already in a free scalar quantum field theory, the ground state has a
bad divergence for P'JQ) (PIQ) = 0, well-define). However, it is possible to redefine the
73
Halmintonian and Lagrangian in the way that these contribution goes away, and the theory
realize Lorentz symmetry as expected from observations and experiments. Supersymmetry
actually kills-off this ambiguity, as there's no freedom to shift the Halmintonian around. One can
think of it as a trick to some how regulate unwanted physics of the theory, and then if
supersymmetry cannot be seen, give it a breaking mechanism so that the effect of superpartners
become unobservable.
3.3.3
Mathematical Rigors
From the hopes of having a more rigorous framework of formulating quantum field theory, axiomatic
quantum field theory and constructive quantum field theory were born. As at the moment, we
cannot say that the two theories are very rigor and successful, but still, it's important to note
that, even quantum field theory is usually "sloppy" with maths, it still give us more valuable
understandings of the nature.
Axiomatic Quantum Field Theory
Since until now there are several times we use the axioms in axiomatic quantum field theory, it is
necessary to mention the full list of Wightman axioms for real scalar field theory (generalization
to more complicated theory can be done similarly), to do the theory more justices [311:
1. There is a separable Hilbert space V, the states of the theory are described by unit rays.
2. There is a unitary, positive-energy representation U(A, a) of the Poincare group on /.
3. There exists a Poincare-invariant vacuum states
IQ).
.
4. The quantum field 4(x) is an operator-valued distribution, for x E Rl 3
5. The states are of the form F(x1)...(x2)IQ) E 71.
6. The field '1(x) transforms covariantly under U(A, a).
7. If two fields are space-like separated, then the fields either commute or anticommute.
8. The space of invariant vectors
IQ) is one-dimensional (uniqueness of the vacuum).
Constructive Quantum Field Theory
In general, the Lorentzian signature is problematic, so it's tempting to change the Lorentzian time
to the Euclidean time by an imaginary factor, and the spacetime change from R'
3
to a more
familiar R 4 . In constructive quantum field theory (there are many other interesting things in this
74
topics, but not of our interests), from the point of view of the functional integration, the Euclidean
field theory is nothing but the analytical continuation of the field theory in Minkowski spacetime
[19].
However, there are mismatches between these, for example, there's no Hilbert space (in
the conventional sense) in the theory with Euclidean signature, and instead of the S-matrix in
Lorentzian quantum field theory, one has correlation functions in Euclidean quantum field theory.
Osterwalder-Schrader quantization gives a set of axioms (we will not mention them in details here)
for constructive quantum field theory, so that for an Euclidean theory there's a corresponding
well-defined theory in Minkowski spacetime. Unfortunately, although rigorous constructions for
two and three-dimensional 04 -theory, Yukawa theory and Gross-Neveu theory were found, there
are still many open problems in four-dimensional theory and beyond. Nevertheless, indeed, this
trick is used, even overused, in theoretical studies, such as in lattice quantum field theory for
well-defined behavior of the far away contributions (oscillatory becomes decay) and string theory
to define a Riemann surface (for perturbative calculations of stringy worldsheets). Even without
mathematical rigors, the found answers are very interesting, and some even fit with reality, for
example, the spectrum of baryons and mesons found with lattice quantum chromodynamics. The
method of Wick's rotation to go from Lorentzian time to Euclidean time will be used in later
discussions, so be prepared for the "sloppiness".
3.3.4
Effective Field Theory
Functional integration provided an intuitive way to think about the process of taking out the
degrees of freedom in the quantum field theory, by explicitly doing the integration.
One can
integrated out a field totally from a theory, or get rid of the contributions from quantum fluctuation
at the energy scale outside of the interval of interests.
An example: Effective Theory of Quantum Gravity
Let's consider an effective theory that can be obtained by simply integrate out the field, from
quantum gravity coupled with a real scalar field. For simplicity, deal with it in a general
Euclidean curved spacetime (Wick's rotation the Lorentzian time t
S[1,
gPV
f
d2lvy/-(gjvO
+ V4))
lA
75
d -t
-+
f
-it) [201:
g-4P<D
,
(3.154)
V = V(x)
, F = -Og
+ V
,
094 = ---
,(Vg""uO4)
(3.155)
The function operator F has eigenfunctions <D,, (normalized) and eigenvalues A.:
<b=
An
)\n
,
(4n, (Dm) = 6m,n
,
d2 w
(F, G) =
x/FG
(3.156)
The wishful thinking is that those gives a basics for any arbitrary function 1b (that's not always
the case, though; for example, the whole spacetime manifold of the Schwarzchild blackhole):
00
100
=(<YD,<D)
<D=c,,
->
S[4,9g""v= 2E
n=O
c2A,
(3.157)
n
The (vacuum) functional integral in this basis can be calculated as:
f=
-1/2
/, 00
7
Z =-Db-S
<
e-n"4 =
f
A)
(3.158)
Let's have a remark here. In the Euclidean manifold, it's natural to impose zero boundary
conditions on the basis functions 4D,, (the choice of regularization). To calculate anything, such as
S-matrix, put insertions in points in the bulk of the regularized manifold, using this very
definition of functional integral. The logic is, if the regularization is of any good, the correlation
comes from applying the described rule should be insensitive to the boundary conditions [20].
There are cases where this trick brings new insight to the table, such as quantum anomaly.
The effective action of the metric, which comes from 1-loop physics (integrating out the scalar
field degrees of freedom), can be read-off:
Seff[g"v=
In Z = 2
A
2ln det F
(3.159)
This functional determinant should be regularized so the final answer is the renormalized one
(well-define and get rid of divergence problems). To do that, one srart with an auxiliary Hilbert
space,which has a complete basis of generalized vectors jx), normalized as:
(Xj X') = 6(x - X') , Jd2z2v)(XI = 1
76
(3.160)
The task is to determine a Hermittian operator 0 that has precisely the same eigenvalues An as
F, that is:
01JFn) = AnIJn) , (lml IXn)
J
=
d2
(3.161)
= jmn
WX4m(X)1n(X)
Hilbert space, in general, is different than quantum mechanics. The appearance of a Hilber space
and of the Dirac notation does not mean that the vector IT) are states of some quantum system.
One uses the Hilbert space formalism because it simplifies shits when calculating the
renormalized functional determinants. It is possible but more cumbersome to work directly with
the partial differential equations. Using the decomposition of unit operator, one can rewrite the
eigenvalue problems in the coordinate basis in the following way, then do the matching [201:
Of d 2wXIIXI)
+
= AnII.) Tn (x1O d2wXI Ix') (x'I X)
(x'I I)
dJ2w (xO|x')4fn(x') = An (x| IF)
A 7,
,
(3.162)
(x) , XI' =
(xIOx')4 = g114 (x)(-Eig(x) + V(x)) (g-1/4(x)6(x
-
/4
,
x')4)
(3.163)
(3.164)
Before defining (rigorously) the regularization for the quest of finding the functional determinant
of 0, let's study right about the trick (hand-waving) that helps physicists to find the finite part
of the functional determinant using the (-function:
00
(6(s) = Tr(6~) -
A-
(3.165)
n=O
The trick is similar to how the Riemann's (-function is used for getting a finite result (except
poles), by analytical continuation (the relation between analytical continuation and
renormalization will be mentioned later):
00
ds(6(s) = d
e-lf" Ad = -
C0
=>
In det O = in fA
n
es In A, In An
(3.166)
00
= ZA
n
-d,(6(s)
(3.167)
8=0
The function (6(s) is usually regular at s = 0, so the derivative should be finite, which is good.
This equation can be thought as the choice of regularization for the functional determinant of
77
operator 0 (this trick can also be employed for finite dimensional system).
The calculation of (6(s) can be reduced to the problem of solving the partial differential equation
for the heat Kernel. Given a Hermittian operator 0 and a complete set of eigencevetor
IJT), the
heat Kernel operator is defined as:
00
K(7) = e-OT
Ee~AIn)(nl
Z
,
k(0) = 1
(3.168)
71
The proper time
T
(> 0) is auxiliary and will disappear from all physical results. The trace of the
hear Kernel gives the (-function of interests:
00
Tr k(T) = E
nk(T)IT) =
J
(6(s
0wA
1
[00
0)
6T5-1dT
1-A
( - T00
F(O)
_=
)=
(3.169)
,
eTAr- dr
= A'
Re(s) > 0)
(3.170)
,
T
(
F(s)
En
n1
J-
"-idrT = F(s)]
From the definition of the heat Kernel, given above, the evolution along the proper time
dK(T)
-
=
-OK(r)
,
T:
(xIk(T)IX') = K(x, X'; r)
d2XII(x61 xd")K(x", x; T)
dTK(x, X'; T) =
(3.171)
TrTk(T)T-dT
(3.172)
(3.173)
At the intitial time-slice:
K(-r
0)
=
1
-
K(x, x'; 0) = 3(x -
')
(3.174)
Since the trace of the operator doesn't depend on the choice of the orthornomal basis, hence:
TrK(T)
J d2,xK(x,X; T)
The steps to find the functional determinant of 0 with the matrix element
(3.175)
t0
x') can be
summarize as follow: (i) solve the heat Kernel equation with the initial condition, then determine
K(x, X';
T);
(ii) subtitute the expression the the trace of hear Kernel into zetag(s) and calculate
78
the convergences; (iii) analytically continue the obtained function (6(s) to get out in det 0, and
the determinant functional is found straight-forwardly after that. Let's try to solve the heat
Kernel equation, in flat Euclidean spacetime, with V(x)
(xO = F = -- ")4
= 91/ 4(-)
(7-1/46(x -
=
0:
X")4) = 2(6(x - x")4)
(3.176)
The heat Kernel equation is simple enough:
dTK(x, X'; T) = & K(x, x'; T) , K(x, x'; r)
-+ dk(k, x'; r) = -k 2 K(k, X'; T)
=
f
-eik k(k, X'; T)
k(k, x'; 0)
,
(3.177)
(3.178)
e-ikx'
The solution for the heat Kernel, unsurprisingly, is just a Gaussian:
k(k, X'; T) = e_,k _ikx'
2
-
K(x, X'; T)
exp
=
-
)
(3.179)
For the general curved spacetime, this is a very hard problem. Therefore, instead of attack it
nonperturbatively, consider a perturbative description around a flat spacetime configuration:
171W = 6pL
, gpv = 7711 + h,
+ g'L" =
, + hl"" ,
l, I < 1
(3.180)
The operator can be rewritten [20]:
(XIOx)4
=
=
-g-
1 4
0,
X')4)) + V(x)6(x -
(g 1/2gv"(g1/46(X -
EZ (6(x' - x)4) +
s[V
, V(x)]4
,
s[h"", V(x)]
=
X')4
h+F+ P
(3.181)
(3.182)
The operator h and F are linear, even nonperturbatively order [20]:
hA
=
hV",0, (6(x - x')4) , P4 = Oh"'&,1 (6(x - x')4)
(3.183)
The operator P has a quite horrendous full form [201:
P(x) =
4~)
4
O ,h", 3 ,h
- 1 g1gg"0,,,hao
79
- 4
h&"ga6Ophc1
(3.184)
1
gg
01, haOvhzA - V(x)
,
P4 = P(x)S(x
- x')46
(3.185)
The heat Kernel can be solved order-by-order of perturbation (in .):
Ko(T) + Ki(T) +K 2(T)
=
+ ..
drK = (E] +
,
K(T)
s)(3.186)
Well, basically (drop the 6-function, notationally):
KO
,
dKo(T) =
dTKI(T) =
Ko = 1 , K>1 = 0
Ko(r)
Ki(T) +
(3.187)
The solution can be read-off:
Ko(T) = e
-+
0(7)
dT'k-1 (r')sko(T')
=
(3.188)
ki(() =ko(r)O(r)
ki(T) =
/
d'r'Ko(T - T')ko(T)
(3.189)
To find the trace of the heat Kernel, one first need to calculate the matrix elements of ko and ki:
(XIKO(T)y) = (xIeEIly) = eTl6(x
-
y)
2
2
_4+&2-)
d Wkw
(2w)
(3.190)
Hence, one arrive at the boring Gaussian:
xJO(r)Jy)
(3.191)
4r
=
-
-(4lrT
Calculation for higher orders of K., is very cumbersome, in general. For the next leading order:
k 1 = kit, + k1r + kip
(3.192)
Let's just do the last term, since it's easiest (and others are similar):
(xlKiplx)
=
dTd2w yd 2u(XI
dr'(xIko (T - T') PJo(T')
IX)
o(T - T')Iy)(yIPIZ)(ZIko(T')IX)
80
(3.193)
(3.194)
dT'
0
I
2e
(47r(T -
),-7'Els(r
1(
(4lT)w
(47rT')WP
r'))
r
dr'e+7e
J
(3.195)
xP(x)
The diagonal element of k1 p will need more work [20]:
(XIKlh IY)
dT
4r
I
d2 wk
d 2 wk eikxPk
P(k)
X'+r-r)
11(r- T)k+ik
7P(X>J=
(2i_)w
.)
(3.196)
(2(w
For the other parts of k 1 , after tiresome brute-force calculations [201:
(xIKlhIl) = -
/
-(
1
j
O6( hl"
dT'e
x
d'eI
(3.197)
-
h +
T ) 2 o 0h "
(3.198)
(
(3.199)
)
(xIKirjx) =
The trace of the heat Kernel, at linear order:
J d2wX(xlko+Ak1x) =
irk(r)
1
(47r-r)w
d2 wx
1- 1 ,-rV+
From dimensional analysis and diffeomorphism invariant of the action, for a general curved
metric:
TrK = (47rT)w
J
R
d2wX
(1
(3.200)
+'r
At the next next leading order (as one can see, controlled by the proper time r) [20]:
1
(47rT)w
I
+T 2 Rf 3 (-TL g)R+
f2(/z) =
fi(z)
6
(R-v
d2w XV4I+
fi(z)-1
2z
2R,,f
R-
4 (-TEg)R"
)
1k(T) =
if1 (-Tg)V
, fi(z) =
, fi(z) -1+I
h (Z)
f3(Z)
z2
=
+T2V 'f 2 (-rEg)R (3.201)
(3.202)
eu(1u)zdu
f1(z)2+ fi(z) - 1
3
8z
f
(z)
8
(3.203)
The f-function contains a glimpse of nonperturbative physics, at all orders of r (take T go to 0
81
inside f-function to get the T2 -order result). In conclusion, the trace of the heat Kernel:
TrK(
( 4 1)=
)
d2wxv
1
VR
2RV)+T(V2
fQ2
(7T
6
120
0(73)
2+
60iv~y~k)
f
(3.204)
The function a,(x) and a 2 (x) are the famous Seeley - deWitt coefficients. Since one will need to
integrate T from 0 to oo, so the Seeley - deWitt expansion (the result above) in general cannot be
used to calculate the corresponded (-function (at least, normally, but analytical continuation
actually might works) since it's only valid for small
T.
The behavior of the heat Kernel at small
T
gives UV region physics, as an effective field theory. The fastest way to see this is to note that r
has the dimension of x 2 and thus small T means short distance (or high energy scale), describes
things like vacuum polarization [201. On the other hand, large value of T tells about IR region
physics, which is related to particles production [201. Note that, for a slightly curved (almost
flat) metric, the expansion should be written with the power counting of R and V instead of -:
1 fd2xv 1+ R-V T+ (V1
TrK(T)
VR+
IR 2+ RRPv
(47FT)'w
)T2+(R3,V3)
d
6
J\\
/\
6
2
120
601
(3.206)
The (-function of interests can be used to read-off the effective action:
1
(s=(s)
1
1
8Tr K(T)dT
,
(3.207)
Se55[g"] = -- dSP(s)
s=O
For s = 0, the integral diverges at
T -+
0, hence the definition above is application only for those
s for which the integral converges. The value of (p(s = 0) is then obtained by analytical
continuation procedure, removes all the divergences in the effective action, without any
justification, hence does not reveal the physical meaning of those divergences.
Define the (-function for small s by regularizing the integral through an explicit cut-off instead of
perferming the analytical continuation, for the sake of studying the divergence origins. Consider
massless field V = 0 (note that, UV physics at an energy scale a lot higher than the mass does
perceive the particles effectively massless) in d = 4 (or w = 2). For small
82
T,
it's best to use the
Seeley - deWitt expansion, for cut-off Tc:
d4 x/
(S) = (
(4wr) 2 F(S)
+(
R2 +
Ts- 3 dT
f
f
RAvR
(3.208)
RTs- 2 dT
+
foc
)]
T5-
1
(3.209)
dr + O(Ts)>0).
Change the cut-off scale to T' < -rc, and define:
A') = f
T-d
~
,
f
B(T')
2
dT- ~'-1
dT
,
C (T')
T
dT~
In(
(3.210)
The for r' -+ 0, the leading divergences is of the structure:
(Ps) = (41r)
(s)
d4x /(A(7') + RB(') +
1 R 2 + 1RVR")C(T')
(3.211)
For small s, the F-function has the structure:
1
S + O(s2)
F(s)
(3.212)
The contribution of the part of UV physics (between r, and Tc) to the Euclidean effective action,
the regularized effective action as the cut-off scale approaches 0:
Seff [g'
] ~
d4X
(a-2
+ bRr'-1 + c(1 R2 + I RWRt" ) lfl(Tc')
(3.213)
The backreaction of the quantum field on the gravitational background causes a modification in
the Einstein's field equation. The total action for the gravitational background is the sum of the
free gravitational action, Einstein-Hilbert SEH [9v], and the effective contribution Seff [gill] as
quantum corrections. The classical action of general relativity SEH [gv] contains the cosmological
term A and the spacetime curvature R, and the renormalization procedure for those is
implemented as follows: assume that the free gravitational action (without the backreaction from
83
quantum fields) has terms quadratic in curvature [20]:
dd4xV
SEH,bare[1v]
R-+2Ab
ay2
R2 +
(3.214)
RR")
With the bare constants Ab, Gb, ab, and these values are not observable since the quatum fields
always present and cannot be decoupled from the theory of gravity. The modified action for
gravity is the sum of the free action and the effective quantum contributions:
S[g[v"] = SEH,bare[9gW
l
f d4X
+ Seff g"I
+(-I + B (Tc))R+ (CT)
167rGb 1927r2 R+ 327r2
r b
(3.215)
+
ab) (~R2 +I
,Rv
120 R2+60RyR"(.2)
(3.216)
The renormalization procedure assume that the bare constants are chosen so that they cancel the
divergences in the effective action:
A
_
8G
Ab
8TG
A(r')
32W 2
'
1
1
B(')
167G
167rGb
1927r 2
O
,
C(T')
327r 2
a
(3.217)
Set the cut-off to T' = 0, then the renormalized constants are interpreted as the observable values.
Now, let's go down to the IR region, and since it's not a good idea to use Seeley - deWitt
expansion anymore, one should try that with R and V as plate-holder of order expansion (which
is only good at slightly curved spacetime). For simplicity, still choose V = 0 as the matter field is
massless, and then the (-function:
jIis
J
d2wx17/j
dTTs1 I(1
+
R+ 2Rf 3(-Tg)R + T2R wf4 (-QTEg)R"
(3.218)
For d
2 (w = 1), there's a simple relation for the metric:
Ry= IgZVR
Cd(s)
I-
]))R
0 dTr'R(Rf3(--Eg)R+ f4((-r4
2xx/
84
(3.219)
The corresponded effective action (to do the integration, change the variable from
Sef[g"] =
--
Jd x
2
jR
j
dr (Rf, (- T,)R +
2f4 (-rig))
967fd
2
T
to
-Trg):
x/RGi--R
(3.220)
With the Green's function of the Laplacian Elg, then one arrives at the gravity Polyakov action:
1
2
dxd2 y Vg(x)g(y)R(x)R(y)GE(XY)
Seff[g"] = 96
(3.221)
For 4D result (d = 4, w = 2), the answer is:
(3.222)
Sejj[gI"] ~ [d'x1 -R n )R+...R,,InR"
The mass scale A 2 is only introduced for a dimensional reasons. The logarithm of the Laplacian
operator:
In
(
)
j
d(r(2
\ p2
Note that a change in the mass scale p
'
(A 2 +M2
)
-9
+M2)
(3.223)
would add a term ~ R In b to the action, thus
changing the constant in front of the R 2 term. This constant hence is scale dependent and
runable. This is the manifestation of the renormalization group of the theory (which can be
explained in more details from statistical field theory [161). The coupling value at a given energy
(normalization point) must be determined experimentally and then the dependence of this
constant from the energy is obtained by solving the renormalization group equation.
Top-Down and Bottom-Up
As we move toward more and more general theories, things become harder to compute, for
example, hydrogen energy levels with quantum field theory rather than nonrelativistic quantum
mechanics and elliptic orbits of planets with general relativity rather than Newtonian gravity.
Generalization is very interesting, but the same can also be said about the other direction,
toward finding the simplest framework that captures the essential physics in a manner that can
be corrected to arbitrary precision, such as an expansion in v/c < 1 in a non-relativistic
quantum field theory. This is the guiding principal of the effective field theory. For the
construction of any effective field theory, one should take into consideration of [30]:
85
1. Fields -+ Relevant degrees of freedom.
2. Symmetries -+ Interactions and possibility of broken symmetry.
3. Power counting -+ Expansion parameters, leading order description.
The power counting in an effective field theory is just as important as gauge symmetry.
Effective field theory is based on the realization, with both physical and philosophical reasons,
that the description of physics at some energy scale m should not be dependence on the the
dynamics of much higher energy scales A >> m. By integrate out all the degrees of freedom but
the relevent ones in the physical situations of interests, the calculations can be simplified.
However, this real insensitiveness of nature to high energy physics indicates that to probe short
distance physics at low energies, usually the only way is to increase the precision.
There are two main ways to build up effective field theories, the top-down method and the
bottom-up method.
In top-down, the high energy theory is understood, and the questis to look for a simpler effective
theory at low energies. Usually, we integrate out (the phrase corresponds to explicitly doing the
integration of the high energy field modes in the path integral formulation) and remove the
heavier particles, and this procedure yields new operators and new low energy couplings. More
specifically, the full Lagrangian is expanded as a sum of terms of decreasing relevance
LhighZ~
. The Lagrangians
Chigh
and
4,,
will agree in the IR. region, but will differ in
the reign of UV. Some examples of top-down effective field theories are [30]:
1. Integrate out heavy particles (top quark, W, Z, and Higgs bosons) from the Standard Model.
2. Heavy quark effective theory for charm and bottom quarks at energies below their masses.
3. Non-relativistic quantum chromodynamics or quantum electrodynamics for bound states of
two heavy particles.
4. Soft-collinear effective theory (I and II) for quantum chromodynamics processes with energetic hadrons or jets.
There are some usual aspects of top-down effective field theories one should aware of. For
effective theories that built out of quantum chromodynamics, a separation of scales is strictly
needed to distinguish physics that are perturbative in the coupling c, (p) (evaluated at the scale
86
p=
Q)
AQCD
from effects that are non-perturbative in the coupling (evaluated at a scale close to
< Q)
[30]. Also note that, the summation E, L() is a power expansion of the counting
parameter, but there are also logarithms which appear with the arguments that are of the ratio
between mass scales or the power counting parameters. In a perturbative effective field theory
with a coupling like a,, the renormalization of L(")
low allows us to sum the large logs a, ln (MI)
M
~ 1
when m 2 < m 1 [30]. Indeed, any logarithms that appear in qyantum field theory should be
related to renormalization in some effective field theories [301. About the precision, the desired
precision will be tracked by where to stop the expansion (how far we go with the sum on n).
In bottom-up, the underlying theory is unknown, and the construction of the effective field
theories is mainly without reference to any other theory. Even if the underlying theory is known,
construction of the effective field theory from the bottom-up is still desirable whenever the
matching is difficult, such as in cases where the matching would have to be nonperturbative in a
coupling and hence is not feasible analytically. One can construct K1 L(") by writing down the
most general set of possible interactions that consistent with all symmetries, using fields for the
relevant degrees of freedom. Couplings are unknown but can be fit to experimental or numerical
data, such as output from lattice quantum chromodynamics. Some examples of bottom-up
effective field theories are [30]:
1. Chiral perturbation theory for low energy pion and kaon interactions.
2. The Standard Model of particles and nuclear physics (the argument for this is that Majorana
neutrino masses can come from a dimension-5 operator).
3. Einstein gravity made quantum with graviton loops.
In bottom-up effective field theories, usually the E, expansion is in powers, but there are also
logs. Renormalization of L(" allows us to sum large logs ln (m)
(M 2
<
i)
together [30]. Even
when mi and mi2 are not masses particles, the same is still true, as that logs in quantum field
theory are summed up with some effective field theories [30]. About the precision of this effective
description, the desired precision tells us how high do we go in the sum over n before stopping.
General Representation Independence Theorem
The change in the degrees of freedom in an effective field theory is very important, and the
general representation independence theorem tells us how to keep track of the modification, so
87
that the rewritten effective field theory is the same as the original one, at least on-shell.
Make the degrees of freedom satisfies 0 = XF(X) with F(O) = 1, so that one can Taylor expand
the field around
of quantized
#
#
= 0 with the leading term being q = x, hence the one-particle representation
and quantized x are the same. The statement of reparamentrization independence
invariant is that calculations of observables with L(O) with quantized field
#
should be no
different to the theory L'(x) = L(XF(X)) with quantized field x. To see how that works, consider
a scalar field theory, with q < 1 keeps track of the order [301:
1
L=
2
2
-
12
m 2 2 - Aq
2
_ 1gg 6 - r0 2 q 3 Lk +
4
The last term can be dropped by making a field redefinition
#
-+
q
+
(q2)
Wg2q3,
(3.224)
or using the
equation of motion to replace Ei0:
]Oq- m 20 - 4A#3 +
(77)
=
(3.225)
0
The new Lagrangian in both case of substitution are the same:
L'
=
1
1
01'00,0 - -m02
2
2
-
-
6
77'g,0
+
0(9772)
(3.226)
Explicit computations of the 4-point and 6-point tree-level and loop-level Feynman diagrams up
to the order 0(;) from L(O) with quantized
#
or L'(x) with quantized k arrive at the same
results.
The general representation independence theory states that, the field redefinition that preserves
symmetries and have the same one-particle states with the original theory allows classical
equations of motion to be used for simplification of a local effective field theory Lagrangian
without changing the predictions to the observables. A schematic illustration for this theorem in
a field theory with complex scalar
LEFT = E
"
7 7,,(n)
#
can be shown as follows. From the effective Lagrangian
(the expansion parameter 7 < 1), let's try removing a general first order term
1T[O]D24 from C') that preserves symmetries of the theory, with T[4'] is generally a local
function of various fields
'7.
In other words, removing linear terms D 24 in the effective field
theory.
88
The Green's function with sources J can be obtained by functional derivatives of the partition
function with respect to sources:
Z[J]
f J
i
Do exp
(L( 0) +
d
()-
2TD)+ 2 7 TD2q+
Jk
+0(92)))
(3.227)
k
i
It's straight-forward to see that removing the term IjT[V)]D 2 0 is relevant to redefining the field
#*
=
#*- rT[/]
in the path integral:
i fD0exp
d
Z[J(=
-
Jy4
+n(L) - ITD21) + 1 qTD21+
-
YT
-
)
(3.228)
(3.229)
JoqT + 0(772)
One can see that there are three changes: the Lagrangian, the Jacobian and the source term JP.
Without changing the S-matrix, we can remove the change in Jacobian and the source and only
left with the change of variable in the Lagrangian L (this is precisely the statement of the
generalize representation independence theorem). The piece 5L needs
#*
+
77T
to transform like
0* so that the symmetries of the theory is preserved:
L
1
2
I(D#)*(Dq) 22
2(D,#')*(D,#') -
12
M2#'*#'
2
m2*
-
1
2
+ (...)
T(D 2 #'
The -! 7 7 TD 2 #' term from L(0) after redefining the field cancels
-
(3.230)
m 2 0') + (...)'
j71TD #', which is good.
2
(3.231)
Since
the effective field theory Lagrangian at all orders q contains all terms allowed by symmetries, all
operators in (...)' are already present in (...) as the field redefinition also respects the symmetries.
Thus couplings are simply redefined, and this poses no problem, since the values of couplings of
an effective field theory aren't fixed. In other words, we still have the same effective field theory.
The redefinition of
#
differs from the original one at the lowest order, so lowest order corrections
of L(0) (which are also symmetry-preserving) can be all absorbed into L() couplings. L()
corrections go to higher orders in the Lagrangian, and terms linear in D 2 #0can all be taken out
from L(). Using the same idea, it's possible to cancel D 2 0 to an arbitrary power out of LM by
89
just replacing it using the equation of motion (because of the kinetic term, D 2 4 should always be
there in the theory), which is also relevant to redefining the fields.
About the change in the Jacobian,let's recall that the Fadeev - Popov method:
det (0'D,) =
DcDcexp (if
ddX6i-a1D,)c)
Since the effective field theory is valid for the energy scale A
mass
-
-+ 6 1 - q')c
,
<
-
(3.232)
the ghosts will have
Ae, and hence decouple, just like other particles at this mass scale that were left out.
Note that dropping ghosts can change the couplings.
To see this, consider a field redefinition:
T=0'* - A'*('*)
,
cq*
-+
(1 - 77E + 2TIAO'*O)c
(3.233)
Rescale c -+ crf-2 to have the correctly normalized kinetic term. It then becomes:
c(1-1 - l + 2A#'*#')c ,
-=
Anew
(3.234)
The mass term of the ghost showing that it has the expected mass Anew, which is huge as q -+ 0.
Since ghosts always appear in loops, they can be removed like heavy particles and only
contribute some corrections to the couplings.
For a Green's function of n-points scalar fields, after the redefinition:
G(n)
=
T QIT(#(X)...5(Xn)...)IQ)
= (QT((O'(xi) +qT(xl))...(#(Xn) + qT(Xn))...)IQ) (3.235)
Here the ... can include insertions of other fields in the theory, and for now let's use real q for
notational simplicity. The change of sources can be shown to drop out of the S-matrix from
Lehmann - Symanzik - Zimmermann reduction (field rescaling and field renormalization
cancellation, no pole, no contribution to the scattering) [301:
JJdd
iePi-(OT((x1)...#(xr)...o) 0)(7JY~r
) (P1P2...ISIPjP+1...)
+
90
,.2 I_
(3.236)
Renormalization in Effective Field Theory
Let's take a look at renormalization in more technical details.
Regularization is the technique to cut-off UV divergences in order to pull out an finite results.
Different regularization methods introduce different cut-off parameters (such as hard cut-off A'Uy)
dimensional regularization d --+ d - 2E and lattice spacing). In principle, any regulator can be
acceptable, but for the matter of convenient, it's better choose the regulator to preserve
symmetries (for example, gauge invariance, Lorentz symmetry, chiral symmetry) and also
preserve power counting by not yielding a mixing of terms of different orders in the expansion,
the calculations become easier [301. It is because in general, operators mix with other operators
of the same dimension and same quantum numbers (with a matrix of counter-terms) due to
quantum effects. Good choice of regularizations are usually with mass-independent regulators
(strictly speaking, a new mass scale may still appear but in a way that doesn't directly change
the power counting factor, and it's mass-independent in the sense that it doesn't see the
thresholds of particles' masses in the theory)[301. Still, it should be note that even if the
regulator doesn't have these desired properties (for example, supersymmetry is broken by
dimensional regularization), one can still use counter-terms to restore symmetries and power
counting, therefore making calculations more simple [30j.
Renormalization, with a specific scheme, gives definite physical meaning to each coefficient and
operator of the quantum field theory, and usually introduce some renormalization parameters
along with it (such as ft in MIS, p 2 =-
for off-shell subtraction scheme, A for Wilsonian). The
relation between the bare value abare, renormalized aren and counter-term 6a coefficients a in
different schemes of renormalization (UV cut-off with integrated momenta p, ATV <; IpI and MS
dimensional regularization) are related, in the following way schematically:
abare(Auv) = are"(A) + Sa(Auv, A) , abare ()
aent) + 6a(c, [t)
The definition of renormalizability, in the traditional sense, is that if a theory has divergences
from loop integrations which can be absorbed into a finite set of parameters at any order of
perturbation, then it's renormalizable. However, in the context of an effective field theory,
renormalizability becomes more relax, as a theory can still be good and make nice predictions if
91
it's renormalizable order by order in its expansion parameters [301. This allows for an infinite
number of parameters, but only a finite number at any order of the expansion. Also, it's
interesting to note that, if a theory is traditionally renormalizable, it does not contain any direct
information on the new physics possibly arises at higher energy scale, usually around some mass
thresholds [301.
Decoupling Theorem
Consider making an effective field theory by integrating out the massive degrees of freedom, if
the remaining low energy effective theory is renormalizable and a physical renormalization
scheme is used, such as off-shell momentum subtraction, then all effects due to heavy particles of
mass scale M appear as a change in the couplings or are suppressed as
y [301.
Since the MS
scheme (which is very convenient for calculations) is not physical as it is mass-independent,
physically doesn't see the mass threshold, one must implement the decoupling argument of the
theory by hand, removing particles of mass M for [t < M.
A good example is that, in the MS scheme of quantum chromodynamics [30]:
g3
5
193
3 bo + O(g ) < 0
_
13(g) = p d g(-p) =
dp167
,
bo =
11
CA
3
-
4
4nFTF
3
(3.237)
The quantum chromodynamics fine structure constant behaves assymptotically free as the run
can be read-off from the lowest order solution:
as
=A)
-
-
4x
as(pt) =
+ as(p) In
1 + a, (11)h
27
Define an intrinsic mass scale:
As27
ba8 (eu
ACD
(3.238)
A
)
(3.239)
From equation (3.237), this scale can be seen to be independent of the choice for p. The fine
structure constant is now can be written in a very nice form, , which specifies the energy scale
when quantum chromodynamics becomes non-perturbative
as(p) =
2R
bo In (p/Amsfe
92
(~
200 MeV):
(3.240)
Note that AgD depends on bo, thus the number of light fermionic flavors nF, on the order of
loop expansion for 3(g), and also on the renormalization scheme as one goes beyond 2 loops.
There's a problem that comes from heavy quarks contribution to bo for any A from the point of
view of the unphysical MS renormalization scheme and that contradicts the decoupling theorem
at low energy scale compare to the masses of these quarks, therefore decoupling should be
implemented by hand by together integrating out and changing the fermion number nF which is
allowed in the loop effectively as [z passes through a quark mass threshold. Specifically, nF = 6
for p > mt, nF = 5 for mlb < [t < mt, and so on.
The matching condition, of perturbative diagrams and couplings, between effective theories after
removing the massive particles can be read-off by studying closely the physics around the
transition mass scale m(~ pm = p() = A)), from the fact that the S-matrix elements with light
external particles should be in agreement between these two theories. The leading order
condition for couplings, which is nothing but the continuous condition of the run at the mass
threshold, can be shown to be a,,(Im) =
The general procedure for matching effective field theories L(1) -
L(2)
top-down method for mass thresholds going from higher to lower scale rn
...
L(n) from
-
>
m
2
>
...
> Mn can
be summed up as follows [301:
1. Match the theory Cl) at the scale mi onto C(') by considering the S-matrix.
2. Compute the /-function and anomalous dimension in theory 2, which does not have the
particle 1, to run the couplings down from the evolution equations, then run them.
3. Match the theory
C2)
at the scale mi onto 0 ) by considering the S-matrix.
4. Compute the /3-function and anomalous dimension in theory 3, which does not have the
particle 2, to run the couplings down from the evolution equations, then run them.
5. Follow this procedure for any number of additional steps required.
3.3.5
Quantum Anomaly
Another interesting topic in quantum field theory is quantum anomaly, the different between
classical symmetry and quantum symmetry with topological origins, that cannot be seen
93
perturbatively (S-matrix argument) but pops out naturally nonperturbatively (regularization,
renormalization and functional integration measure).
Violation of global Noether current's conservation is OK, while for violation of gauge current
(caused by quantum anomaly) it means the theory should be inconsistent. To see this, consider a
simple gauge theory and turn on an expectation value for the gauge field A,. This is supposed to
be a spontaneous breaking of the gauge symmetry (as the gauge field gets a vacuum expectation
value), and the effective action after integrating out the matter fields should be invariant under
gauge transformation:
eis[',D,A
sD
eiseff[A]
(3.241)
Change the background by gauge transformation A, -+ A. + iDtE:
isff[A']
-
_
D4(f
,,Sfi[A]
f
D
(,is[,A]
d4xiDe,,J,) eis[D,A]
f
_
_
Df (
f
fis
(s[,A]
_ eiS[,A]+fd4xiDeJP"'
d4xiEDAJP) eis[,,A] =
e
d4 x(D,1J)
4
(3.243)
Thus, if (DJ) # 0, then the effective action should be gauge dependence, hence the gauge
symmetry is explicitly broken. The broken of gauge symmetry means the longitudinal freedom of
the massless vector is not the redundancy of the theory anymore, hence violating unitarity and
Lorentz invariant (which indicates the ambiguity should be there). The breaking of global
symmetry, however, doesn't mean any consistency of the theory, therefore should be fine. Indeed,
the violation of global symmetry gives observable realizations.
What are the implications of quantum anomaly? First, the symmetry at the level of the action
(classical point of view) is accidental, and in reality (which is quantum), no such symmetry exist.
Second, even if the action doesn't seem to be symmetric, if quantum effects give corrections so
that there is a new symmetry arise, one should conclude that the symmetry must be there.
Finally, the role of regularization and renormalization (the origin of quantum anomaly) is
undoubtedly critical in the righteousness of quantum field theories, restricts possible models in
phenomenology. It is important to remember that, quantum anomaly cannot be seen purely from
the "powerful" S-matrix theory arguments (it can still be calculated perturbatively from the
Feynman diagrams, though).
94
As a matter of fact, anomaly is not a failure of bad choice for regularization scheme. It's just
that there's no regularization so that the symmetry can still be preserved when studying
quantum effects, and that's a physical interpretation. Anomaly admits a topological
intepretation, given in term of Chern character, by the celebrated Atiyah - Singer idex theorem
[41, and we will show that.
Use Wick's rotation to get to the simple Euclidean signature:
iY0 =Y
ix =
4
17,'Y6
Y ,t= =7172737Y 4 , T
,
-T
(3.244)
,
4,
(3.245)
The effective action for the gauge fields can be read-off from the functional determinant:
J
e-Seff[V]
is[*,A = det(i
)
(3.246)
Since the determinant is nothing but the product of eigenvalues (the quantized nature), therefore
it's crucial to have a non-ill description of an eigenproblem. Consider an Abelian gauge theory,
the Dirac operator in Euclidean space is Hermittian [41:
, P'y5 0. (x) = -A7nY
Xi(f)
= Antn(x)
5
(x)
,
{'Y, 75}
=
0
(3.247)
One gets the orthogonal complete normalized basis {4'n}. The orthogonality in the non-zero
modes V',, _L y 5 , is due to Pt = P. For the zero-modes, this is not true, and one can use
chirality projection to separate them:
An (750n.1 4) = (754n| 1A'n) = (73tnJ#i n)= -An (-N'V2 0n) ;
(3.248)
1
-=(175)
, +7V1. (3.249)
P
Expand the spinor field with respect to the found eigensolutions:
P(x)
=7a,(x)
=
a1 (xl 4')
,
'(x) =
nl (XIVn
Z
:d1Ot()
95
n/x)
=
a (4'| x)
d
0
(3.250)
The prefactor denote the (infinite dimensional) transformation matrices between the
eigenfunction space and position space, and it's play no roles in the later calculations or the
physics of the theory. The functional determinant can be evaluated simply with a Grassmann
integral:
d4X+il9F
dn ,
=
D'I'D'
f
det(ilg) =
n
(3.251)
IAn)
djnjd esEnAn"nan
T
A local chiral transformation with an infinitesimal parameter
T(x) -+ V'(x) = IF(x)e~1(x)5
a'C=
Cnnam
=
='i
,
Dnmdn ,
Dnm =nm - if
(3.252)
7Fn
a3T'
'I(x) -* T'(x)
,
if
Cnr = 6nm -
d4 X
[4]:
T(x)e~f x)5
=
d4 X'V (X),3(X)Y 5
(3.253)
,
m(X)
,
(3.254)
(3.255)
t(X)/3(X)75 Om1(X) = Cnm
Grassmannian measure transforms with the inverse determinant. To see this:
(i = jij)(
f(x)
,
... i Jil5l... Jinin =
i ... in
dt J
f(+)
,
= ... +
1
...
+ n,
(.i .. (i
+
...
(3.256)
J'Ei...in(iI...(
+
...
(3.257)
-
n
1
..
(
+- ...
...
=
+ -dt
n
But for Grassmannian numbers, derivation means integration, hence:
d(j ... d(i = dot Jdxj1 .. .dXi
-+
(dot J)-dC< ...d(, = dxjI ...dxi
(3.258)
Therefore, the change in the measure pull out a contribution to the action:
da',db'
dct C dot D
dandei
(3.259)
71.U
(SlnmJn2id4XX)()O()Y,
=
5
'prn(X))
1
i
dada
(3.260)
71
e~if~x~'i/1fr5(k
Jda,,dbn = J(O3) JJdand&
1
n
96
,
J (0) - e-2i fdxIJ)
(3.261)
The Jacobian density can be written in a more compact form [25]:
3 =-Tr
(3.262)
6( 4)(0)
=
n
Keep the transform parameter to be a constant, for global transformation, then one get the index
[4]:
(740'1
I
d4X
(X) - 5
(X)
d4X Y
=
n
n
= /P+
()")= 0 , D
,
(3.263)
4
(X) Y5 On',t(X) 75V7
(3.264)
(X)
n
(3.265)
f d4 x~tn+(X)On+(x) -- Zf d4x?4_ (x)4,0 _ (x)
n1
(3.266)
= dim ker D+ - dim ker D_ = index D+
The definition of the index of an operator:
0
index
=
index0 = f d4x lndex(O)
dimker(O) - dim ker(Ot)
(3.267)
Regularization, with regulator function f(s) can be used the pull out the anomaly [25]:
f(0) = 1 , f(oo)= 0 , sf'(s)
(3.268)
=0
s=O COo
Consider global transformation, and anomaly can now be seen from the Jacobian density:
d 4kf"(k 2 ) = igr2
i"][-Y, fo])
f
dssf"(s)
=
= -16ie""Po
-igr2
(3.269)
,
Tr (7 5 [v4,
Jo
dsf'(s) = ir2
(3.270)
00
lrmf
2
k Tr (Y5f (-
d4 k
j(27r)4 T (Y5f
)e)ik(x)
M2)
A/I~]2)
(
(3.271)
)X--+y
, k" --+ MkV
(3.272)
)
M-+00
J
M+0I(2jr) 4
-
3(x) = Jim
-
lim
M-+00
m4 d
(2r
4
Tr
(y5f
- i+
97
]
~ k2 _21kADP
2
(3.273)
if
2
(k
i(-)4 f"(k)Tr (4 91 +
d
1
2
,]F,))
327r2 4vpFvFpc
(3.274)
This non-zero change (known as the Chern-Pontryagin density x - . [251) corresponds to of the
non-conservation of the gauge current. Note that, from Atiyah - Singer index theorem, one also
agrees [4]:
index D+
1
3i
dxel'"
2
Tr(F,,Fp,)
-
2
Tr(F2) =
87r2
ki
Ch(F)
(3.275)
This give rise to the chiral anomaly (also known as singlet anomaly) in this simple Abelian gauge
theory, with the index density related to the topology of the gauge group (the Chern number):
index D+
1
-
-
12 Ch(F) , d * j5 = 2i Index D+
(3.276)
Hence, the anomalous axial current, also can be generalize straight-forwardly to non-Abelian
gauge theory [4]:
(DPJ )
-272
F jttvP;,
Tr (Ta{Tb, Tc})
Al = (A/)
(3.277)
The prefactor i disappear when one goes back to Lorentzian signature. In general, anomaly can
be read-off from the Atiyah - Singer index theorem. If the Dirac operator containing the
Yang-Mills gauge potential A = APT &P, then the characteristic class is given by the Chern
character [4]:
Ch(F) = Tr eF
dimF +
27r
Tr F +
2!
( )Tr(F
2
)+...
,
F=dA+A 2
(3.278)
Tr(Fri)
,
OM = 0
(3.279)
(27r
The Atiyah - Singer index theorem states that [4]:
index D+ =
Ch(F) =
It's important to remind that all calculations of quantum anomaly can be seen, from heat Kernel
regularization or perturbatively from a Feynman diagram, perturbatively, in Pauli-Villar
regularization scheme (might be a good story for another time). Anomalies can be understood as
both UV physics (UV region regularization) and IR physics (zero-modes mismatch), which is
98
another evidence for the fact that it's an universal (all scales) problem in quantum field theory.
For a more horrendous gauge transformation, the topological interpretation of the theory still
exists, and the generalization was done in the 1980s. One of the most highlight is the use of it to
restrict the possible super Yang - Mills gauge group allowed in ten-dimensional supergravity, and
(to some certain surprises) it agrees with string theory (with the conformal consistency
requirements for particles and interactions, already) [13].
The Standard Model is a chiral gauge theory where the gauge fields couple to the right-hand and
the left-hand fermions differently (experimental facts). This means the gauge group of the
standard model should be constraint, so the theory is anomaly-free and no left-right mixing arise.
The Standard Model gauge group U(1) x SU(2) x SU(3) is indeed fit the bill. Along with it,
there's other consistent choice (anomaly-free) of phenomenological interests - only for N < 3 [331:
U(1) x U(1) x U(1) , U(1) x SU(2) x SU(2) ,
U(1) x SU(3) x SU(3)
, SU(2) x SU(2) x SU(2)
(3.280)
(3.281)
For Minimal Supersymmetric Model, a single Higgsino should be anomalous, therefore another
Higgsino with the same helicity is required. Incidentally, the mixed gravity-gauge anomaly and
Witten's global anomaly (a theory with an odd number of SU(2) doublets is anomalous under
large gauge transformations) are also cancelled with this choice [33].
In theories whose spectrum changes with the energy scale, the matching of anomalies between
the high and low energy theories provided nontrivial nonperturbative information. The condition
made by Gerard 't Hooft states that the calculation of any chiral anomaly by using the degrees of
freedom of the theory at some energy scale, must not depend on what scale is chosen for the
calculation [33]. It is also known as 't Hooft UV-IR anomaly matching condition, where UV
stand for the high-energy limit (generator Ta), and IR for the low-energy limit (generator Ta) of
the theory:
>ITr. (Tj{TTj})
ETr (Ta{TI,Tj})
-
R
(3.282)
L
=Z
Tr({Tj,7 }
-
R
Tr (
L
99
L"{JLTf})
(3.283)
Consider calculating the anomaly of a global gauge symmetry current by considering which
fermionic degrees of freedom the theory has. One also may use either the degrees of freedom at
the far IR (low energy limit) or the degrees of freedom at the far UV (high energy limit) in order
to calculate the anomaly. In the former case one should only consider massless fermions which
may be composite particles, while in the latter case one should only consider the elementary
fermions of the underlying short-distance theory. In both cases, the answer must be the same.
Hence, one can think of this as the constraint on expected degrees of freedom at different energy
scales of the theory.
To prove the anomaly matching condition, artificially add to the theory a gauge field which
couples to the current related with this symmetry, as well as chiral fermions which couple only to
this gauge field, and cancel the anomaly (so that the gauge symmetry will remain
non-anomalous, as needed for consistency). In the limit where the coupling constants go to zero,
one gets back to the original theory, plus the added fermions (which remain good degrees of
freedom at every energy scale, as they are free fermions at this limit). The gauge symmetry
anomaly can be computed at any energy scale, and must always vanishes, so that the theory is
consistent. One may now get the anomaly of the symmetry in the original theory by subtracting
the free fermions we have added, and the result is independent of the energy scale.
3.3.6
Wavefunctional
Let's make the connection back to quantum mechanics, by studying the very interesting object
known as the wavefunctional with by using functional integration and canonical method together.
Consider a massive free scalar field quantum theory, described by the action:
S[D]
dtdKZ(
-
-
(6t@) 2 + (0A) 2 + m2
(3.284)
Let's change it to Euclidean signature by a Wick's rotation for time t -+ -it, then the Euclidean
action and the equation of motion for the field:
SE[
3
((
2
+
2
100
+
2
2 2)42>- mfJ
(3.285)
Spatial Fourier expanded the scalar field, one can rewrite the action:
I
<D(t,)=
SE[
-*
(3.286)
t
d
J
=
d3 k
3
(27r)
(3.287)
2
2
k2
2
(27)3
Since the oscillatory behavior in Lorentzian time is replaced by the decay one, therefore the longestlived state is the ground state, and by evolving from past infinity one can pull it out:
4O f ;tf
![#5]
=
(#1 Q) ~ (#f = #; tf = 01 #i;ti = -oo) ~_
Do'e-sE[O'
dO k
(3.288)
k
Of kj~b
dt(I0t(PI 2 +wU Ik
f4e~J
I 2)
e- bwIk
=
~NH
'-e2
sinh (wptf-ti))
H e-2
2_ei=
X
(3.289)
(3.290)
k 1 k.
k
k
Therefore, the ground state wavefunctional:
(3.291)
To find the excited states, one canjustapplies the creation operator on the ground state:
(wi,<bft)
iH(i)) -+
-
I
d 3 z- -i(Y)
-
a = dg6
00(g))
(3.292)
For example,for the first excited state:
J
- ao(2 ) 1Q[#]
=aF[(] = e- W4
)
-f[#
d3 Feik#(-)e-
f
(3.293)
(3.294)
Note that, another way to find the ground state is to find the solution of:
(+) - ifl(-))
-+
101
Id
3
-ee(00)
+ 00(y)
)
d3= &FeWkj(w
(3.295)
a-'JQ [#]
= d xeik (()
+ 0(y))
102
'fQ[]
0 Vk
(3.296)
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