MATHEMATICAL INTRODUCTION
TO QUANTUM ELECTRODYNAMICS
JAN DEREZIŃSKI
Slides of the minicourse at IHP 21-25.06.2010
Contents
3
0 Introduction
5
1 General scattering theory
29
2 Axioms of quantum field theory
65
3 Neutral scalar bosons
89
4 Massive photons
137
5 Massless photons
175
6 Linearly perturbed quadratic Hamiltonians
209
7 Charged scalar bosons
221
8 Dirac fermions
267
Chapter 0
Introduction
5
Quantum Electrodynamics (QED) is an extremely successful theory. It depends on very few
adjustable parameters – the fine structure constant and the masses of charged particles – and it
can be used to compute with great precision many physical quantities, explaining a wide range
of phenomena. It has a rich and complex formal structure, even though it can be developed
from a very small number of postulates without any arbitrariness,
The agreement of the QED computations with the experimental data is astounding. Its most
spectacular successes are the computation of the anomalous magnetic moment of the electron
and of the energy levels of light atoms (the Lamb shift). Its range of applicability is however
much wider: one can argue that a large part of modern physics can be derived from QED.
QED has serious problems. It has been noticed already by F. Dyson that the perturbation
expansion computed in QED is probably divergent. Even worse: as noted by L. D. Landau, if
we try to sum up this series we encounter a singularity, the so-called Landau pole. The common
wisdom says that QED does not exist as a consistent satisfactory nonperturbative theory.
QED is an example of a quantum field theory (QFT). It is a relatively sophisticated example
of a QFT because of the gauge invariance.
It is natural to ask whether QED, or more generally, other QFT’s, fit into the framework
of usual quantum mechanics and of some common-sense postulates: whether they can be
described on a Hilbert space, with a bounded from below Hamiltonian, satisfy the Poincaré
covariance, and the Einstein causality. In the literature two basic sets of such postulates can
be found: the Wightman axioms and the Haag-Kastler axioms. They are viewed by many
physicists as an irrelevant pedantry of narrow-minded mathematicians. In my opinion, this is
a wrong assessment – in reality, most of these postulates express basic intuitions coming from
quantum mechanics. One can view these axioms not as an unshakable foundation, but as a
set guiding principles.
Unfortunately, so far the only known nontrivial models of Wightman or Haag-Kastler axioms
in 1+3 dimensions are free theories. Interacting quantum field theories, such as eg. the λ:φ4 :
theory, satisfy these axioms at best in the sense of formal power series. There is probably no
hope that a model of interacting QFT will be rigorously and non-perturbatively constructed in
a near future. In particular, it is commonly believed that it is impossible to construct the λ:φ4 :
theory satisfying the Wightman axioms (because of the Landau pole). The situation with QED
is, to my understanding, even worse, since it does not even satisfy Wightman axioms on the
formal level, because of the gauge invariance.
According to the Wightman axioms, the full informations about a QFT can be encoded in the
so-called Wightman functions. Actually, in usual physics textbooks, QFT is studied following
a somewhat different philosophy. According to them, the objective of QFT is computation of
various quantities related to the scattering operator (of course, as formal, probably divergent
power series). These quantities are easier to describe in massive theories. In theories containing
massless particles, the situation is messed-up by the so-called the infra-red problem.
Basic quantities computed in QFT
massive QFT
QFT possessing massless particles
N -point time-ordered
N -point time-ordered
Green’s functions
Green’s functions
(with a physical normalization); (without a physical normalization);
scattering amplitudes;
————-
scattering cross-sections;
inclusive scattering cross-sections;
(complex) energy levels;
(complex) energy levels.
In standard modern textbooks, N -point time-ordered Green’s functions are treated as the basic objects of a given QFT. Other quatities, such as the energy levels and scattering amplitudes
and cross-sections can be in principle derived from Green’s functions.
Scattering amplitudes (that is matrix elements of the scattering operator) do not, however,
exist in the massless case (even as formal power series). Physically measurable are inclusive
cross-sections and energy levels.
In QED (both massive and massless), there is a problem with Green’s functions, because
they depend on the gauge (both in the massless and massive case). Neither the (inclusive)
scattering amplitudes nor energy levels depend on the gauge.
One can name 4 major issues in QED that need a special attention:
1. the ultra-violet problem;
2. the infra-red problem;
3. the gauge-invariance;
4. bound states of many particle systems.
The ultra-violet problem is common to all kinds of interacting relativistic quantum field
theories. It is the main reason why we seem to be limited to a perturbative treatment and why
the renormalization is necessary. Historically, it posed the biggest obstacle in the development of
QED, and was mastered only in the late 40’s, with major contributions of Schwinger, Feynman,
Tomonaga and Dyson.
The infra-red problem is common to all QFT’s possessing massless particles. In QED it is
due to the presence of massless photons and is absent in the massive QED. It appears already
in non-relativistic quantum mechanics – in scattering theory with Coulomb forces. These forces
are long-range, which makes the usual definition of scattering operators impossible. Its another
manifestation is the appearance of inequivalent representations of canonical commutation relations when we consider scattering of photons against a classical current that has a different
direction in the past and in the future. Thus, even in these toy non-relativistic situations
the usual scattering operator is ill-defined. Therefore, it is not surprising that (much bigger)
problems are present in the full relativistic case. One can cope with the infra-red problem
by approximating massless QED with the massive one and restricting computations only to
inclusive cross-sections justified by an imperfect resolution of the measuring device.
The expression gauge invariance has in the context of QED several meanings.
1. The most common, discussed in classical electrodynamics, is the fact that if we add to a
4-potential solving the Maxwell equation a total derivative, then it still solves the Maxwell
equations. Of course, this no longer holds for the Proca equations – the massive generalization of the Maxwell equations. Therefore, it is often stressed that gauge-invariance
implies that the photons are massless.
2. There exists another meaning of the gauge-invariance: we can multiply charged fields by a
space-time dependent phase factor and compensate it by changing the external potentials.
1. and 2. go together in the full Lagrangian of QED, which is invariant with respect to
these two gauge transformations applied simultaneously.
3. In QED one often uses the term “gauge invariance” in yet another meaning: we have the
freedom of the choice of the (free) photon propagator. This meaning applies both to the
massive and massless QED. Some of these propagators are distinguished. In particular, the
Feynman propagator plays a special role in the renormalization in the ultraviolet regime.
There are several distinct approaches to treat the problem of the gauge invariance QED, all
of them equivalent what concerns the physical quantities. These approaches are quite different
in the massive and massless case. Nevertheless, all physical quantities depend continuously on
the mass, down to zero.
Bound states energies are defined as the positions of singularities of Green’s functions. They
usually have a nonzero imaginary part.
The understanding of bound states seems to be one of the major problems of QFT. It is in
fact dificult to study bound states by purely perturbative methods, since the basic quantities
that we compute involve elementary particles as asymptotic states. The same problem would
appear if we wanted to compute the scattering matrix of an N -body Schrödinger operator in
a purely perturbative way.
QED is a quantized theory that describes charged particles interacting with photons. Charged
particles are either fermions described by the Dirac equation or bosons described by the KleinGordon equation. From the point of view of applications, the case of charged fermions is
much more important, nevertheless, from the theoretical point of view, charged bosons can be
considered on the same footing as charged fermions.
Photons are massless vector bosons, described by the Maxwell equation. The zero mass of
photons leads to the infra-red divergencies, which are difficult from the theoretical point of
view. The usual way to cope with them is to assume that photons are massive, described by
the Proca equation. Then we can treat the (physical) massless QED as the limit of the (nonphysical) massive one. Therefore, it is useful to discuss in parallel the massless and massive
QED.
In full QED both photons and charged particles are quantized. It is however useful to
consider the formalism, where only one kind of particles is quantized. In fact, the following
three theories can be viewed as the main pillars of the full QED:
1. charged scalar bosons interacting with an external potential;
2. Dirac fermions interacting with an external potential;
3. (massless or massive) photons interacting with an external current.
All the above three theories are (or at least can be) well understood in the non-perturbative
sense. They have a well-dynamics on a Hilbert space, or at least in an appropriate C ∗ -algebra.
They can be used to compute quite a number of physically relevant quantities and to explain
tricky theoretical questions.
One can distinguish three, increasingly complex, basic varieties of the above theories:
(i) theory without external potentials, resp. currents;
(ii) theory with external potentials, resp. currents in the vacuum representation;
(iii) theory with external potentials, resp. currents in the interaction (Furry) representation.
Only the first theory is Poincaré invariant. All of them are in some sense very simple-minded.
Their Hamiltonians are quadratic, and hence their equations of motions are linear. After solving
the classical equations, we can say everything about their quantizations.
Full QED is much more difficult. It involves quantizing simultaneously charged particles and
photons.
One way to formulate full QED goes as follows. First we formulate the theory of charged
particles in an external potential and the theory of photons in an external current as perturbative
theories for many-point Green’s functions and scattering amplitudes. An elegant and convenient
way to do this a representations of terms in the perturbation expansions as diagrams. Rules to
evaluate them are often called the Feynman rules. Then we put the Feynman rules for charged
particles and photons together obtaining the Feynman rules for the full QED.
These rules (in a somewhat different language) were known already in the 30’s. If one
applies them to the so-called tree diagram, one obtains physically meaningful and well defined
answers (such as the Klein-Nishina formula, Bhabha formula, etc.). When one tries to evaluate
diagrams involving loops, one usually encounters divergencies.
Only at the end of 40’s physicists found methods that enabled them to make sense of these
diagrams.
As a result, the quantities of interest can be expressed in terms of formal power series. These
power series, though probably divergent, involve a very small coupling constant. Therefore, for
practical purposes, if we take a few first terms, they seem to behave as convergent series.
To my experience, in most textbooks, QED is not presented as a consistent theory. Essentially all commonly available textbooks on QED present it as a collection of “cookbook recipes”,
somewhat chaotic and not quite honest. The reader is supposed to excuse the inconsistencies
and use his or her “physical intuition” to cope with its problems.
To my understanding, in reality, properly interpreted, QED is a rigorous theory with unambiguous rules. There are several methods of renormalizing Feynman diagrams. All of them, to
my understanding, are equivalent and lead to a uniquely defined formal power series for various
quantities of interest depending on very few parameters.
The parameters of QED do not appear explicitly in the inital Feynman rules. They enter the
theory in the so-called normalization conditions, usually imposed on the values of certain Green’s
functions at some chosen points in the momentum space. There is a considerable freedom in
choosing these conditions. For instance, we can demand that the 2-point Green’s functions
have an appropriate singularity at the value of the physical mass (the on-shell normalization
condition). However, other conditions can often be more convenient.
The value of the electric charge (and hence of the fine structure constant) can be fixed by
demanding that the appropriate Ward(-Takahashi) identity holds.
One can distinguish at least 3 basic varieties of the full QED:
(i) with no external potentials;
(ii) with weak external potentials;
(iii) in the interaction (Furry) picture.
QED with no external potentials is fully Poincaré covariant. It can be viewed as the basic
theory. The normalization conditions for the masses and the charge are formulated within its
framework.
QED with external potentials can in principle be derived as an effective approximation of
the QED with no external potentials. We need to assume that some infinitely heavy charged
particles generate a potential, which can be treated classically (in addition to quantized photons
already present in the theory). One possibility is to keep the vacuum representation and expand
the external potential in the perturbation series. Another possibility is to use the interaction
(Furry) picture, where we assume that the external potential (or at least its major part) is
time-invariant and its effect can be included in the propagator of charged particles. (Of course,
there can be an additional time-dependent external potential treated perturbatively).
QED in an external potential is very important from the point of view of practical computations. The definition of the anomalous magnetic moment involves QED in a weak external
potential. A natural setup for the Lamb shift is the QED in the Furry picture.
To sum up, one can view QED as a family of closely related theories. Beside the full QED,
this family contains the theories of one kind of particles interacting with external potentials
or currents. These partial theories can be understood non-perturbatively. Even though they
are in some sense “trivial”, they are actually quite interesting, both physically and mathematically. A number of nontrivial and often pathological problems can be partly understood
within their setup, such as the infra-red problem, the infinite renormalization of the charge,
non-implementability of the dynamics, gauge invariance.
The full QED is much less trivial. It seems to be necessarily perturbative.
Chapter 1
General scattering theory
29
Heisenberg picture
Let H be a Hilbert space with a self-adjoint operator (called the Hamiltonian) H. Let B be
an operator and t ∈ R. The operator B in the Heisenberg picture at time t is defined as
BH (t) := eitH Be−itH .
Wightman functions
Let Φ be a normalized vector satisfying HΦ = EΦ.
Given operators BN , . . . , B1 and times tn , . . . , t1 (not necessarily ordered), the corresponding
Wightman function is defined as
W (BN , tN , . . . , B1 , t1 )
:= (Φ|BN,H (tN ) · · · B1,H (t1 )Φ)
i(−tN +tN −1 )(H−E)
i(−t2 +t1 )(H−E)
···e
B1 Φ .
= Φ|BN e
Dyson’s time ordered product
Let tn . . . , t1 ∈ R be pairwise distinct. We define Dyson’s time-ordered product of Bn (tn ),...,
Bi (t1 ) by
T (Bn (tn ) · · · B1 (t1 )) := Bin (tin ) · · · Bi1 (ti1 ),
where i1 , . . . , in is a permutation such that tin ≥ · · · ≥ ti1 .
Green’s functions
The (time-ordered) Green’s functions are defined for pairwise distinct tN , . . . , t1 as
G (BN , tN , . . . , B1 , t1 )
:= (Φ|T (BN,H (tN ) · · · B1,H (t1 )) Φ) .
Clearly, we can pass from the Wightman to Green’s functions:
G(BN , tN , . . . , B1 , t1 )
X
=
θ tσ(N ) − tσ(N −1) · · · θ tσ(2) − tσ(1) W (Bσ(N ) , tσ(N ) , . . . , Bσ(1) , tσ(1) ).
σ∈SN
Analytic continuation of Wightman functions
Assume in addition, that Φ is a ground state of H, that is H ≥ E. Then it is easy to see
that Wightman functions can be extended to a function defined for Im(−zN + zN −1 ) ≥ 0,
Im(−z2 + z1 ) ≥ 0:
W (BN , zN , . . . , B1 , z1 )
i(−zN +zN −1 )(H−E)
i(−z2 +z1 )(H−E)
= Φ|BN e
···e
B1 Φ .
The Wightman function is holomorphic in the interior of its domain and continuous up to
the boundary. For any real tN > · · · > t1 we have
G(BN , tN , . . . , B1 , t1 ) = W (BN , tN , . . . , B1 , t1 ).
(1.1)
This allows us in principle to determine the Wightman functions from Green’s functions. This
follows from the Edge of the Wedge Theorem, which says that in certain situations a holomorphic function is determined by its values on the boundary of its domain.
Note that the holomorphic extension of thw Wightman function at its “euclidean points” is
often called the Schwinger function:
S(BN , sN , . . . , B1 , s1 ) := W (BN , isN , . . . , B1 , is1 ).
Standard Møller and scattering operators
Assume now that H0 and V are self-adjoint operators and H := H0 + λV .
The standard Møller and scattering operators (if they exist) equal
S ± := s− lim eitH e−itH0 ,
t→±∞
+∗ −
S := S S .
Theorem 1.1 1. If the standard Møller operators exist, they are isometric and satisfy S ± H0 =
HS ± .
2. If the standard scattering operator exists, it satisfies H0 S = SH0 .
Problem with eigenvalues
It is easy to show the following fact:
Theorem 1.2 If the standard Møller operators exist and H0 Ψ = EΨ, then HΨ = EΨ.
This is the reason why in practice the standard formalism of scattering theory is usually
applied in a suituations where the unperturbed Hamiltonian H0 has only absolutely continuous
spectrum.
In quantum field theory, even when one can define (even formally) H0 and H, typically both
have ground states, and these ground states are different. Thus standard scattering theory is
not applicable. Instead one can sometimes try other approaches to scattering theory.
Time-ordered exponentials
Consider a family of operators [t− , t+ ] 3 t 7→ B(t). We define the time-ordered exponential
Z t+
Z
Z
∞
X
Texp
B(t)dt :=
···
B(tn ) · · · B(t1 )dtn · · · dt1
t−
n=0 t ≥t ≥···≥t ≥t
+
n
1
−
Z
Z
∞
t+
X t+
1
T (B(tn ) · · · B(t1 )) dtn · · · dt1 ,
n!
t
t
−
−
n=0
Z t+
−1
Z t−
Texp
B(t)dt := Texp
B(t)dt
=
···
t−
t+ ≥ t1 ;
t+
=
∞
X
Z
Z
···
n=0 t ≤t ≤···≤t ≤t
+
1
n
−
(−1)n B(t1 ) · · · B(tn )dt1 · · · dtn , t+ ≤ t− .
For shortness, let us write
Z
t+
U (t+ , t− ) := Texp
B(t)dt .
t−
Note that
d
U (t+ , t− ) = B(t+ )U (t+ , t− ),
dt+
d
U (t+ , t− ) = U (t+ , t− )B(t− ),
dt−
U (t, t) = 1l,
U (t2 , t1 )U (t1 , t0 )
If B(t) = B, then U (t+ , t− ) = e(t+ −t− )B .
=
U (t2 , t0 ).
Time dependent Hamiltonian
Let R 3 t 7→ H(t) be a family of self-adjoint operators. H(t) is called the Hamiltonian at
time t. We introduce the dynamics (evolution)
Z
U (t+ , t− ) := Texp −i
t+
H(s)ds .
t−
The dynamics U (t+ , t− ) is unitary.
Heisenberg picture for time-dependent Hamiltonians
If R 3 t 7→ A(t) is an operator-valued function, the operator A(t) in the Heisenberg picture
is defined as
AH (t) := U (0, t)A(t)U (t, 0).
Note that if A(t) does not depend on t, then AH (t) is the solution of
d
AH (t) = i [HH (t), AH (t)] ,
dt
AH (0) = A.
Interaction picture
Let H0 be a self-adjoint operator. Let R 3 t 7→ V (t) be a family of self-adjoint operators.
Set H(t) := H0 + λV (t). Let R 3 t 7→ A(t) be an operator-valued function. The operator
A(t) in the interaction picture is defined as
AI (t) := eitH0 A(t)e−itH0 .
The evolution in the interaction picture is defined as
UInt (t+ , t− ) := eit+ H0 U (t+ , t− )e−it− H0 .
The interaction Hamiltonian is defined as
HInt := VI (t).
Note that
Z
UInt (t+ , t− ) = Texp −iλ
t+
t−
HInt (t)dt .
Scattering operator for time-dependent perturbations
Quite often, the dynamics in the interaction picture has a limit t− → −∞ or t+ → ∞. This
is in particular the case if V (t) decays in time sufficiently fast.
Definition 1.3 The Møller or wave operators, resp. the scattering operator (if they exist) are
defined as
S ± := s− lim UInt (0, t),
t→±∞
+∗ −
S := S S .
Theorem 1.4 1. If S ± exist, then they are isometric.
2. If both S + and S − exist, then so does S and
S = w−
lim
t+ ,−t− →∞
UInt (t+ , t− ).
3. If RanS + = RanS − , then S is unitary.
Note that all the operators UInt (t+ , t− ), resp. S ± can be viewed as special cases of the
scattering operator, if we multiply V (t) by 1l[t− ,t+ ] (t), resp. 1l[0,+∞[ (±t).
Wick’s time-ordered product
For further applications, it is convenient to modify the definition of Dyson’s time-order
product. Assume that I is a unitary involution on H. We call an operator B even, resp. odd,
if B = ±IBI. Let t 7→ Bk (t), . . . , B1 (t) be time dependent operators of pure parity.
Definition 1.5 Let tn . . . , t1 ∈ [t+ , t− ] be pairwise distinct. We define Wick’s time-ordered
product of Bn (tn ),..., Bi (t1 ) by
T (Bn (tn ) · · · B1 (t1 )) := sgna σBσn (tσn ) · · · Bσ1 (tσ1 ),
where σ1 , . . . , σn is a permutation such that tσn ≥ · · · ≥ tσ1 and sgna σ is the sign of the
permutation restricted to the odd elements among Bi .
Note that Dyson’s time-ordered product is a special case of Wick’s time-ordered product for
I = 1l. Therefore, we will keep the same symbol for Dyson’s and Wick’s time-ordered product.
Note also that if B(t) is even for all t, then all the formulas of (1) remain true for Wick’s
time-ordered product.
Green’s functions
Assume that H0 and V (t) are even. Let Φ0 be a fixed vector with H0 Φ0 = 0. Suppose that
there exist
Φ± := lim UInt (0, t)Φ0 = lim U (0, t)Φ0 .
t→±∞
t→±∞
(1.2)
The Green’s function for the dynamics U is defined as
G(Ak , tk , . . . , A1 , t1 )
:=
Φ+ |T Ak,H (tk ) · · · A1,H (t1 ) Φ− .
(1.3)
Let G0 (. . . ) denote Green’s functions for the free dynamics. We can express interacting
Green’s functions by the free ones:
G(Ak , tk , . . . , A1 , t1 )
Z
Z ∞
∞
X
(−iλ)n ∞
dsn · · ·
ds1 G0 (V, sn , · · · , V, s1 , Ak , tk , · · · , A1 , t1 ) .
=
n!
−∞
−∞
n=0
In fact, it suffices to assume that tk ≥ · · · ≥ t1 . Then the left hand side of (1.3) equals
lim lim U (0, t+ )e−it+ H0 Φ0 |U (0, tk )Ak U (tk , 0) · · ·
t+ →∞ t− →−∞
−it− H0
×U (t2 , t1 )A1 U (t1 , 0)U (0, t− )e
Φ0
= Φ0 |UInt (∞, tk )Ak,I (tk )UInt (tk , tk−1 ) · · · UInt (t2 , t1 )A1,I (t1 )UInt (t1 , −∞)Φ0 ,
which equals the right hand side of (1.3).
Abstract LSZ approach
Setting some of ti = ±∞ in Green’s functions (1.3) has the obvious meaning – we take
the corresponding limit limti →±∞ . Green’s functions can be used to compute the scattering
operator. In fact, if B1 , . . . , Bm , Ak , . . . , A1 are some operators, then
(Bm · · · B1 Φ0 |SAk · · · A1 Φ0 )
∗
= G B1∗ , ∞, · · · , Bm
, ∞, Ak , −∞, · · · A1 , −∞ .
(1.4)
Adiabatic dynamics.
Let V be a self-adjoint operator and > 0. We define V (t) := e−|t| V . Then we will write
H (t), U (t+ , t− ), S (t+ , t− ), S± , S for the corresponding time-dependent Hamiltonian, etc.
Proposition 1.6 We have

H (t+ )U (t+ , t− ) − U (t+ , t− )H (t− )
0 ≥ t+ ≥ t− ;
iλ∂λ U (t+ , t− ) =
−H (t )U (t , t ) + U (t , t )H (t ) t ≥ t ≥ 0;
+
+ −
+ −
−
+
−

HInt (t+ )S (t+ , t− ) − S (t+ , t− )HInt (t− )
0 ≥ t+ ≥ t− ;
iλ∂λ UInt (t+ , t− ) =
−H (t )S (t , t ) + S (t , t )H (t ) t ≥ t ≥ 0;
Int + + −
+ −
Int −
+
−
iλ∂λ S± = ±HS± ∓ S± H0 ,
iλ∂λ S = −H0 S + S H0 .
Proof. Display the dependence on λ by writing U,λ (t+ , t− ). For t+ ≥ t− ≥ 0 we have
U,λ (t+ , t− ) = U,λe−θ (t+ + θ, t− + θ).
Hence,
d
U −θ (t+ + θ, t− + θ)
θ=0
dθ ,λe
d
d
= −λ∂λ U,λ (t+ , t− ) +
U,λ (t+ , t− ) +
U,λ (t+ , t− )
dt+
dt−
= −λ∂λ U,λ (t+ , t− ) − iH,λ (t+ )U,λ (t+ , t− ) + iU,λ (t+ , t− )H,λ (t− ).
0 =
This proves the first identity, from which the remaing follow. 2
Assume that Φ0 is an eigenvector of H0 with H0 Φ0 = E0 Φ0 . Set
S± Φ0
,
(Φ0 |S± Φ0 )
(Φ0 |HS± Φ0 ) (Φ0 |HΦ±
)
:=
=
.
±
±
(Φ0 |S Φ0 )
(Φ0 |Φ )
Φ±
:=
E±
Proposition 1.7
(H − E0 ∓ iλ∂λ )S± Φ0 = 0,
±iλ∂λ log(Φ0 |S± Φ0 ) = E± − E0 ,
H − E± ∓ iλ∂λ Φ±
= 0.
The Gell-Mann and Low formula for eigenvectors.
Theorem 1.8 Assume that there exist
±
±
Φ±
GL := lim |(Φ0 |S Φ0 )|Φ .
&0
and
lim λ∂λ Φ±
= 0.
&0
Then there exist
±
EGL
:= lim E±
&0
and
±
±
HΦ±
GL = EGL ΦGL .
Suppose that E0 is a discrete nondegenerate eigenvalue of H0 . Then for small enough λ0 > 0
and |λ| < λ0 , H has a unique nondegenerate eigenvalue close to E0 , which we denote E. If
±
+
−
EGL
depends continuously on λ, then we see that EGL
= EGL
= E and Φ+
GL is proportional
to Φ−
GL .
Suppose now that the we have time reversal invariance. More precisely, assume that T is
an antiunitary operator such that T H0 T −1 = H0 , T V T −1 = V , T Φ0 = Φ0 . Then, obviously,
−
+
−
T U (t+ , t− )T −1 = U (−t+ , −t− ), T S+ T −1 = S− . Therefore, T Φ+
GL = ΦGL , EGL = EGL and
−
(Φ0 |Φ+
GL ) = (Φ0 |ΦGL ).
If we assume both the discreteness of E0 and the time reversal invariance, then we can
−
conclude the Φ+
GL is equal ΦGL .
−
We will assume in the following that Φ+
GL = ΦGL =: Φ.
Lemma 1.9 Let B be an operator. Then
(S+ Φ0 |BS− Φ0 )
.
&0
(Φ0 |S Φ0 )
(Φ|BΦ) = lim
Proof. The right hand side of (1.5) equals
−
−
(Φ+
(Φ+
(S+ Φ0 |BS− Φ0 )
|BΦ )
GL |BΦGL )
=
lim
=
.
+
−
−)
&0 (Φ+
&0 (S+ Φ0 |S− Φ0 )
|Φ
(Φ
|Φ
)
GL GL
lim
But Φ±
GL = Φ and (Φ|Φ) = 1. 2
(1.5)
The Sucher formula
Theorem 1.10
iλ
∂λ log(Φ0 |S Φ0 ).
&0 2
E − E0 = lim
Proof.
iλ∂λ S = H0 S + S H0 − 2S+∗ HS− ,
We sandwich it with Φ0 and divide with (Φ0 |S Φ0 ) obtaining
(S+ Φ0 |HS− Φ0 )
iλ∂λ log(Φ0 S Φ0 ) = 2E0 − 2
.
(S+ Φ0 |S− Φ0 )
The last term, by Lemma 1.9, converges to 2E. 2
(1.6)
The Gell-Mann and Low formula for the N -point function
Theorem 1.11
G (Ak , tk , · · · , A1 , t1 )
Z ∞
∞
n Z ∞
X
(iλ)
dsn · · ·
ds1
= lim Z−1
&0
n!
−∞
−∞
n=0
×G0 (V , sn , · · · V , s1 , Ak , tk , · · · A1 , t1 ) ,
where
Z =
Z
∞
X
(iλ)n
n=0
n!
∞
Z
∞
dsn · · ·
−∞
ds1
−∞
×G0 (V , sn , · · · , V , s1 ) .
(1.7)
Proof. The left-hand side of (1.7) equals
lim (Φ|Ak,H (tk ) · · · A1,H (t1 )Φ)
&0
(S+ Φ0 |Ak,H (tk ) · · · A1,H (t1 )S− Φ0 )
&0
(Φ0 |S Φ0 )
= lim
But Z = (Φ0 |S Φ0 ). 2
Adiabatic Møller wave operators
Theorem 1.12 1. Assume that there exists the adiabatic or Gell-Mann–Low wave operator
|(Φ0 |S± Φ0 )| ±
S ,
:= w− lim
&0 (Φ0 |S± Φ0 )
(1.8)
|(Φ0 |S± Φ0 )| ±
w− lim λ∂λ
S = 0.
&0
(Φ0 |S± Φ0 ) (1.9)
±
±
SGL
(H0 − E0 ) = (H − E)SGL
,
(1.10)
±
SGL
Φ0 = Φ ±
GL .
(1.11)
±
SGL
and
Then
2. Define the wave function renormalization operators
±∗ ±
Z ± := SGL
SGL .
Clearly Z ± ≥ 0,
Z ± H0 = H0 Z ± ,
Z ± Φ0 = Φ0 .
3. Assume that KerZ ± = {0}.Define the renormalized Møller operators
±
±
Sren
(Z ± )−1/2 .
:= SGL
±
Then Sren
are isometric and
±
±
Sren
(H0 − E0 ) = (H − E)SGL
.,
±
Sren
Φ0 = Φ±
GL .
(1.12)
(1.13)
±
4. Assume that (1.8) is true in the sense of strong limits. Then SGL
are isometric and Z ± = 1l,
so that there is no need to renormalize Gell-Mann – Low wave operators.
q
|f |
|f |
|f |
f
i∂λ f
Proof of 1. Using f =
f , we obtain i∂λ f = − f Re f . Therefore, setting f :=
(Φ0 |S± Φ0 ) we compute
|(Φ0 |S± Φ0 )|
|(Φ0 |S± Φ0 )| (Φ0 |(HS± − S± H0 )Φ0 )
±iλ∂λ
= −
Re
(Φ0 |S± Φ0 )
(Φ0 |S± Φ0 )
(Φ0 |S± Φ0 )
|(Φ0 |S± Φ0 )|
±
Re
E
−
E
= −
0 .
(Φ0 |S± Φ0 )
Therefore,
|(Φ0 |S± Φ0 )| ±
|(Φ0 |S± Φ0 )|
±
±
±
±iλ∂λ
S
=
−
HS
−
S
H
+
Re(E
−
E
)
.
0
0
(Φ0 |S± Φ0 )
(Φ0 |S± Φ0 )
±
±
Using ReEGL
= EGL
, we obtain (1.10). (1.11) follows by definition. 2
Adiabatic scattering operator
±
Assume that SGL
exist. Define the adiabatic or Gell-Mann–Low scattering operator
+∗ −
SGL = SGL
SGL .
Set
Sren := (Z + )−1/2 SGL (Z − )1/2
+∗ −
= Sren
Sren .
−
Assume that Φ+
GL = ΦGL .
(1.14)
Theorem 1.13 1. We have
H0 SGL = SGL H0 ,
SGL Φ0 = Φ0 .
2. We have
H0 Sren = Sren H0 ,
Sren Φ0 = Φ0 .
+
−
= RanSren
, then Sren is unitary.
3. If RanSren
±
exist as strong limits, then
4. If SGL
S
.
&0 (Φ0 |S Φ0 )
SGL := w− lim
Chapter 2
Axioms of quantum field theory
65
Consider the Minkowski space R1,3 with variables xµ , µ = 0, 1, . . . , 3. By definition, it is the
vector space R4 of signature (− + ++). In particular, it is equipped with the pseudo-Euclidean
quadratic form for x = (xµ ) ∈ R1,3 equal to
0 2
−(x ) +
3
X
i=1
(xi )2 .
Writing R3 we will mean the spatial part of the Minkowski space obtained by setting x0 = 0.
If x ∈ R1,3 , then ~x will denote the projection of x onto R3 . Latin letters i, j, k will sometimes
denote the spatial indices of a vector.
On R1,3 we have the standard Lebesgue measure denoted dx. The notation d~x will be used
for the Lebesgue measure on R3 ⊂ R1,3 .
A nonzero vector x ∈ R1,3 is called
timelike if x2 < 0,
causal if x2 ≤ 0,
lightlike if x2 = 0,
spacelike if x2 > 0.
A causal vector x is called
future oriented if x0 > 0,
past oriented if x0 < 0.
The set of future/past oriented causal vectors is denoted J ± . We set J := J + ∪ J − .
If O ⊂ R1,3 , its causal shadow is defined as J(O) := O + J. We also define its future/past
shadow J ± (O) := O + J ± .
Let Oi ⊂ R1,3 . We will write O1 × O2 iff J(O1 ) ∩ O2 = ∅, or equivalently, O1 ∩ J(O2 ) = ∅.
We then say that O1 and O2 are causally separated.
A function on R1,3 is called space-compact iff there exists a compact K ⊂ R1,3 such that
suppf ⊂ J(K). It is called future/past space-compact iff there exists a compact K ⊂ R1,3
such that suppf ⊂ J ± (K).
∞
(R1,3 ). The set of fuThe set of space-compact smooth functions will be denoted Csc
∞
(R1,3 ).
ture/past space-compact smooth functions will be denoted C±sc
Lorentz and Poincaré group
The pseudo-Euclidean group O(1, 3) called the Lorentz group. Its connected component of
unity is denoted SO↑ (1, 3). The affine extension of the Lorentz group R1,3 o O(1, 3) is called
the Poincaré group.
These groups have double coverings P in(1, 3), Spin↑ (1, 3) and R1,3 o P in(1, 3).
Note that Spin↑ (1, 3) happens to be isomorphic to Sl(2, C).
An element of R1,3 o P in(1, 3). will be often written as (a, Λ̃) and then the corresponding
element of R1,3 o O(1, 3) will be written as (a, Λ).
Representations of Spin↑ (1, 3) can be divided into two categories. The integer spin representations induce a representation of SO↑ (1, 3) and the half-integer representations do not.
Basic rules of relativistic quantum mechanics
Physical states are described by a Hilbert space H equipped with a strongly continuous
unitary representation of the double cover of the Poincaré group (or one of its subgroups)
R1,3 o Spin↑ (1, 3) 3 (x, Λ) 7→ U (x, Λ) ∈ U (H).
The self-adjoint generator of space-time translations is denoted P = (H, P~ ). H is called the
Hamiltonian and P~ – the momentum. Thus
~
U ((t, ~x), 1) = e−itH+i~xP .
It is natural to impose the following conditions:
1. Existence of a Poincaré covariant vacuum: There exists a (normalized) vector Ω invariant
with respect to R1,3 × Spin↑ (1, 3).
2. Spectral condition: sp(P ) ⊂ J + .
3. Uniqueness of the vacuum: The vector Ω is unique up to a phase factor.
Note that (2) implies H ≥ 0. Conversely, the Poincaré invariance and H ≥ C implies (2).
(2) implies also that Ω is the ground state of H. (3) implies that this groundstate is
nondegenerate.
In the mathematical literature one can find two basic sets of axioms for a quantum theory,
which try to express the concept of causality: Haag-Kastler and Wightman axioms. Even
though the Wightman axioms were formulated earlier, it is in my opinion more natural to start
with the Haag-Kastler axioms.
Note that we keep the “basic rules of relativistic quantum mechanics” as a part of both sets
of axioms.
Haag-Kastler axioms
To each open bounded set O ⊂ R1,3 we associate a distinguished von Neumann algebra
A(O) ⊂ B(H). The family A(O), O ⊂ R1,3 , satisfies the following conditions
1. Isotony: O1 ⊂ O2 imples A(O1 ) ⊂ A(O1 ).
2. Covariance: for (a, Λ̃) ∈ R1,3 ×Spin↑ (1, 3) we have U (a, Λ̃)A(O)U (a, Λ̃)∗ = A ((a, Λ)O)).
3. Einstein causality: O1 × O2 and Ai ∈ A(Oi ), i = 1, 2 imply A1 A2 = A2 A1 . This means
that measurements in spatially separated regions are independent.
S
4. Irreducibility: ( O A(O))0 = C1l.
Local algebras
The algebras A(O) describe algebras of observables in O. This means that in principle an
observer contained in O can perturb the dynamics by a self-adjoint operator from A(O), and
only from A(O).
The least obvious axiom is that of irreduciblity.
Quantum fields
In practical computations of quatum field theory the information is encoded in a family of
functions R1,3 3 x 7→ φa (x), where a = 1, . . . , n is a finite index set enumerating the “internal
degrees of freedom”, typically, species of particles and the value of their spin projected on a
distinguished axis. These functions can be viewed as “operator valued distributions”, which
become (possibly unbounded) self-adjoint operators when smeared out with test functions in
Cc∞ (R1,3 ). Some of the particles are bosons, some are fermions. They commute or anticommute for causally separated points, which is expressed with the commutation/anticommutation
relations
[φa (x), φb (y)]± = 0, (x − y)2 > 0.
Smeared quantum fields
We can organize the internal degrees of freedom into a finite dimensional vector space
V = Rn . Thus for any f = (fa ) ∈ Cc∞ (R1,3 , Rn ) we obtain an operator
XZ
fa (x)φa (x)dx.
φ[f ] :=
a
From the mathematical point of view it is natural to treat φ[f ] as the basic objects. This
leads to the following set of axioms.
Wightman axioms
We assume that V = Vs ⊕ Va is a finite dimensional real vector space. Vs , resp. Va is called
the space of bosonic, resp. fermionic particles. We set := 1l ⊕ (−1l) on V. We will write
|s| = 0, |a| = 1. The space V is equipped with a representation
Spin↑ (1, 3) 3 Λ̃ 7→ S(Λ̃),
(2.1)
which preserves the decomposition V = Vs ⊕ Va .
We assume that H = Hs ⊕ Ha . We set I := 1l ⊕ (−1l) on H. Hs , resp. Ha is called even
fermionic, resp. odd fermionic.
We suppose that D is a dense subspace of H containing Ω. We have a map
Cc∞ (R1,3 , V) 3 f 7→ φ[f ]
(2.2)
into linear operators on D satisfying the following conditions:
1. Continuity: For any Φ, Ψ ∈ D,
Cc∞ (R1,3 , V) 3 f 7→ (Φ|φ[f ]Ψ)
(2.3)
is continuous.
h
i
−1
2. Covariance: for (x, Λ̃) ∈ R ×Spin (1, 3) we have U (x, Λ̃)φ[f ]U (x, Λ̃) = φ S(Λ̃)f ◦ (x, Λ) .
1,3
↑
∗
3. Einstein causality: O1 ×O2 and fi ∈ Cc∞ (Oi , Vji ), i = 1, 2, imply φ[f1 ]φ[f2 ] = (−1)|j1 ||j2 | φ[f2 ]φ[f1 ].
nS
o
4. Irreducibility: Span
f φ[f ]Ω is dense in H.
5. Self-adjointness: φ[f ]∗ = φ[f ].
6. Fermionic fields change the fermionic parity: φ[f ] = Iφ[f ]I
One can show the theorem about the connection of spin and statistics saying that the
representation (2.1) has integer spin in Vs and half-integer in Va .
Wightman axioms are satisfied by free fields.
Neutral vs charged fields
In practice, only the so-called neutral fields are assumed to be self-adjoint. One has operator
valued distributions R1,3 3 x 7→ ψa (x), ψa∗ (x), a = 1, . . . , m. They are the so-called charged
fields. After smearing with complex test functions we obtain operators
XZ
ψ[g] :=
g a (x)ψa (x)dx,
a
ψ ∗ [g] :=
XZ
ga (x)ψa∗ (x)dx,
a
such that
ψ[g]∗ = ψ ∗ [g].
Clearly, by setting
1
φa,R (x) := √ (ψa (x) + ψa∗ (x)),
2
1
φa,I (x) := √ (ψa (x) − ψa∗ (x)),
i 2
for any charged field we obtain a pair of neutral fields.
Thus one can organize the space describing the species of particles into two finite dimensional
spaces: a real space Vn describing neutral fields and a complex space Vc describing charged
fields.
Relationship between Haag-Kastler and Wightman axioms
”Morally”, Wightman axioms are stronger than the Haag-Kastler axioms. In fact, let Falg (O)
be the ∗-algebra in L(D) (linear operators on D) generated by φ[f ] with suppf ⊂ O. It is
equipped with an involution α given by α(φ[f ]) = φ[f ], so that Falg (O) as a vector space is
alg
alg
a direct sum of its even and odd part, Falg
0 (O) and F1 (O). Then the family F0 (O) almost
satisfies the Haag-Kastler axioms, except that elements of Falg
0 (O) are defined only on D and
not on the whole H, and often do not extend to bounded operators on H.
We know that the fields φ[f ] are hermitian on D. Suppose they are essentially self-adjoint.
Then we could consider the von-Neumann algebra F(O) generated by bounded functions of
φ[f ], suppf ⊂ O. Then the family F0 (O) satisfies the the Haag-Kastler axioms.
In practice, observable algebras A(O) can be even smaller than F0 (O). This is so in particular
if we have a group acting G on the space V responsible for a (global) symmetry of the fields.
For example, in the case of charged fields the group equals U (1) and this symmetry is obtained
by multiplying fields with a phase factor. The action of G on fields induces an action of G on
F0 (O). One can argue that the subalgebra of fixed points of this action should be taken as
the observable algebra.
(Note that what we described is the case of a global symmetry and not of a local gauge
invariance).
N -point Wightman functions
Under the Wightman axioms we obtain a multilinear map
Cc (R1,3 , V) × · · · × Cc (R1,3 , V)
3 (fN , . . . , f1 ) 7→ (Ω|φ[fN ] · · · φ[f1 ]Ω) ∈ C,
which is separately continuous in its arguments.
(2.4)
By the Schwartz Kernel Theorem, it can be extended to a linear map
Z
1,3 N
⊗N
3 F 7→ W (xN , . . . , x1 )F (xN , . . . , x1 )dxN · · · dx1 ,
Cc (R ) , V
where R(1,3)N 3 (xN , . . . , x1 ) 7→ W (xN , . . . , x1 ) is a distribution on R(1,3)N with values in the
space dual to V ⊗N , called the N -point Wightman function, so that (2.4) equals
Z
W (xN , . . . , x1 )fN (x1 ) · · · f1 (x1 )dxN . . . dx1 .
From the point of view of the Wightman axioms, the collection of Wightman functions WN ,
N = 0, 1, . . . , contains all the information about a given quantum field theory. In particular,
(φ[fN ] · · · φ[f1 ]Ω|φ[gM ] · · · φ[g1 ]Ω)
Z
=
W (y1 , . . . , yN , xM , . . . , x1 )
×f1 (x1 ) · · · fN (xN )gM (yM ) · · · g1 (y1 )dx1 · · · dxN dyM · · · dy1 .
N -point Green’s functions
In practical computations, the functions that play an important role are not Wightman
functions but the so-called (time-orderded) Green’s functions. Their formal definition is as
follows:
G(xN , . . . , x1 )
(2.5)
X
:=
sgn (σ)θ x0σ(N ) − x0σ(N −1) · · · θ x0σ(2) − x0σ(1) W (xσ(N ) , . . . , xσ(1) ),
σ∈SN
where sgna (σ) is the sign of the permutation of the fermionic elements among N, . . . , 1.
Note that in (2.5), we multiply a distribution with a discontinuous function, which strictly
speaking is illegal. Disregarding this problem, the Green’s function is a covariant, due to the
(anti-)commutativity of fields at spacelike separations.
Chapter 3
Neutral scalar bosons
89
In this chapter we consider the Klein-Gordon equation with an external source
(−2 + m2 )φ(x) = j(x).
We will quantize the space of its real solutions.
(3.1)
Special solutions of the homogeneous equation
Every function ζ that solves the Klein-Gordon equation
(−2 + m2 )ζ(x) = 0.
can be written as
1
ζ(x) =
(2π)3
Z
eikx g(k)δ(k 2 + m2 )dk
q
Z
√
1 X
d~k
±ix0 ~k 2 +m2 ∓i~x~k
2
2
~
~
p
g(± k + m , k)e
=
,
(2π)3 ±
~k 2 + m2
where g is a function on the hyperboloid (mass shell) k 2 + m2 = 0.
(3.2)
A special role is played by the following 3 solutions of the homogeneous Klein-Gordon equation:
1. The standard positive, resp. negative frequency solution.
Z
i
(±)
eikx θ(±k 0 )δ(k 2 + m2 )dk
D (x) = ∓
3
(2π)
Z
√
d~k
i
±ix0 ~k 2 +m2 ∓i~x~k
p
e
= ∓
(2π)3
2
2
~
2 k +m
√
miθ(x2 )
1
mθ(−x2 ) ±sgnx0 p 2
√ K1 (m x2 ).
H1
(m −x ) ∓
=
sgnx0 δ(x2 ) ∓ √
4π
8π −x2
4π 2 x2
where H1± are the Hankel functions and K1 is the MacDonald function of the 1st order.
2. The Pauli-Jordan function (or the commutator function)
Z
i
D(x) =
e−ikx sgn(k 0 )δ(k 2 + m2 )dk
3
(2π)
q
Z
d~k
1
i~x~k
0 ~2
p
=
e sin x k + m2
(2π)3
2
2
~k + m
p
1
mθ(−x2 )
0
2
=
sgnx δ(x ) − √
J1 (m −x2 ).
2π
4π −x2
D(x) is the unique solution of the homogeneous Klein-Gordon equation satisfying
D(0, ~x) = 0, Ḋ(0, ~x) = δ(~x).
We have, suppD ⊂ J.
Green’s functions
Solutions of
(−2 + m2 )ζ(x) = δ(x),
(3.3)
are called Green’s functions or fundamental solutions of the Klein-Gordon equation. In particular, let us introduce the following important Green’s functions:
1. advanced, resp. retarded Green’s function.
Z
1
e−ikx
±
dk
D (x) =
(2π)4
k 2 + m2 ∓ i0sgnk 0
p
1
mθ(−x2 )θ(±x0 )
0
2
√
=
θ(±x )δ(x ) −
J1 (m −x2 ).
2
2π
4π −x
We have suppD± ⊂ J ± . In the literature, D+ (x) is usually denoted Dret (x) and D− (x)
is usually denoted Dadv (x).
2. causal or Feynman(-Stueckelberg) Green’s function.
Z
1
e−ikx
c
D (x) =
dk
(2π)4
k 2 + m2 − i0
√
mθ(−x2 ) − p 2
miθ(x2 )
1
2
√ K1 (m x2 ).
δ(x ) − √
H1 (m −x ) +
=
2
4π
8π −x
4π x2
Relationships between special solutions and Green’s functions
D(x) = −D(−x) = D(+) (x) + D(−) (x)
= D+ (x) − D− (x),
D(+) (x) = −D(−) (−x),
D+ (x) = D− (−x) = θ(x0 )D(x)
D− (x) = θ(−x0 )D(x)
Dc (x) = Dc (−x) = θ(x0 )D(−) (x) − θ(−x0 )D(+) (x).
Let us prove the last identity.
Z
1
e−ikx
dk
D (x) =
(2π)4
k 2 + m2 − i0
Z
1
e−ikx
dk
p
p
=
(2π)4
2
2
2
2
~
~
2 k +m
k + m − |k0 | − i0
Z
e−ikx
1
p
dk
p
+
(2π)4
2
2
2
2
~
~
2 k +m
k + m + |k0 | + i0
Z
~
eik0 x0 −ik~x
1
dk
p
p
=
(2π)4
2
2
2
2
~
~
2 k +m
k + m − k0 − i0
Z
~
1
eik0 x0 +ik~x
p
dk
p
+
(2π)4
2
2
2
2
~
~
2 k +m
k + m + k0 + i0
Z
~
1
eik0 x0 −ik~x
p
dk
p
=
(2π)4
2
2
2
2
~
~
2 k +m
k + m − k0 − i0
Z
~
eik0 x0 +ik~x
1
p
dk
p
+
(2π)4
2
2
2
2
~
~
2 k +m
k + m + k0 + i0
c
This equals
i
θ(x0 )
=
(2π)3
Z
~
eik0 x0 −ik~x ~
i
p
dk +
θ(x0 )
3
(2π)
2
2
~
2 k +m
Z
= θ(x0 )D(−) (x) + θ(−x0 )D(−) (−x),
where we used the identity
Z
eits
ds = 2πie−ita θ(t).
a − s − i0
~
e−ik0 x0 +ik~x ~
p
dk
2
2
~
2 k +m
Retarded and advanced solutions of the inhomogeneous problem
Among solutions of the inhomogeneous Klein-Gordon equation one can distinguish the advanced and the retarded solution. They can be obtained by the convolution with the advanced/retarded Green’s function.
∞
(R1,3 ), solutions
Theorem 3.1 For any f ∈ Cc∞ (R1,3 ) there exist unique functions ζ ± ∈ C±sc
of
(−2 + m2 )ζ ± = f, suppζ ± ⊂ J ± (suppf ).
Moreover, ζ ± (x) =
R
D± (x − y)f (y)dy.
Note that by duality D± can be applied to distributions of compact support. Therefore, in
the above theorem we can assume that f is a distribution of a compact support, and then ζ ±
is a distribution.
The Cauchy problem
Solutions of the Cauchy problem are uniquely parametrized by their Cauchy data (the value
and the normal derivative on a Cauchy surphace). They can be expressed by the Cauchy data
with help of the Jordan-Pauli function.
∞
(R1,3 ) that solves
Theorem 3.2 Let ς, ϑ ∈ Cc∞ (R3 ). Then there exists a unique ζ ∈ Csc
(−2 + m2 )ζ = 0
(3.4)
with initial conditions ζ(0, ~x) = ς(~x), ζ̇(0, ~x) = ϑ(~x). It satisfies suppζ ⊂ J(suppς ∪ suppϑ)
and is given by
Z
ζ(t, ~x) = −
Z
Ḋ(t, ~x − ~y )ς(~y )d~y +
R3
D(t, ~x − ~y )ϑ(~y )d~y .
R3
Conserved current
Let YKG,R , resp. YKG denote the space of real, resp. complex, space-compact solutions of
the Klein Gordon equation. Clearly, YKG = CYKG,R .
Let ζ1 , ζ2 ∈ C ∞ (R1,3 , R). We define
j µ (x) = j µ (ζ1 , ζ2 , x) := ∂ µ ζ1 (x)ζ2 (x) − ζ1 (x)∂ µ ζ2 (x).
We easily check that
∂µ j µ (x) = −(−2 + m2 )ζ1 (x)ζ2 (x) + ζ1 (x)(−2 + m2 )ζ2 (x),
Therefore, if ζ1 , ζ2 ∈ YKG , then
∂µ j µ (x) = 0.
One says that j µ (x) is a conserved current.
Symplectic form
The flux of j µ across a space-like subspace S of codimension 1 does not depend on its
choice. It defines a symplectic form on YKG,R
Z
ζ1 ωζ2 =
j µ (ζ1 , ζ2 , x)dsµ (x)
ZS =
ζ̇1 (t, ~x)ζ2 (t, ~x) − ζ1 (t, ~x)ζ̇2 (t, ~x) d~x.
(3.5)
Clearly, the form (3.5) is well defined if only ζ2 ∈ YKG,R , and ζ1 is a distributional solution
of the Klein-Gordon equation.
Poincaré covariance
The Poincaré group acts on YKG,R by
r(a,Λ) ζ(x) := ζ (a, Λ)−1 x .
r(a,Λ) are symplectic for Λ in the orthochronous Poincaré group, otherwise they are antisymplectic.
We will write for shortness ra for r(a,1l) , where a ∈ R1,3 .
Solutions parametrized by space-time functions
The Jordan-Pauli function D can be used to construct solutions of the Klein-Gordon equation, which are expecially useful in the axiomatic formulation of QFT. We will write
Z
Df (x) := D(x − y)f (y)dy.
Theorem 3.3 1. For any f ∈ Cc∞ (R1,3 , R), Df ∈ YKG,R .
2. Every element of YKG,R is of this form.
R
3. (Df1 )ωDf2 = f1 (x)D(x − y)f2 (y)dxdy.
4. If suppf1 × suppf2 , then
(Df1 )ωDf2 = 0.
Classical fields
We will also consider the space dual to YKG,R . More precisely, we can endow the space
YKG,R with the standard topology of Cc∞ (R3 ) ⊕ Cc∞ (R3 ) given by the initial conditions. The
#
space of real continuous functionals on YKG,R . will be denoted by YKG,R
, complex valued by
#
#
CYKG,R
. The action of T ∈ CYKG,R
on ζ ∈ YKG,R will be denoted by hT |ζi.
For x ∈ R1,3 , φ(x), π(x) will denote the functionals on YKG,R given by
hφ(x)|ζi := ζ(x),
hπ(x)|ζi := ζ̇(x).
They are called classical fields. Clearly, for any ζ ∈ YKG,R we have
(−2 + m2 )hφ(x)|ζi = 0.
Thus the equation
(−2 + m2 )φ(x) = 0
is a tautology.
#
#−1
On YKG,R
we have the action of the Poincaré group (a, Λ) 7→ r(a,Λ)
. Note that
#−1
r(a,Λ)
φ(x) = φ(Λx + a).
Clearly, φ̇(x) = π(x) and
Z
Z
φ(t, ~x) = Ḋ(t, ~x − ~y )φ(0, ~y )d~y + D(t, ~x − ~y )π(0, ~y )d~y .
The symplectic form can be written as
Z
ζ1 ωζ2 =
(hπ(t, ~x)|ζ1 ihφ(t, ~x)|ζ2 i − hφ(t, ~x)|ζ1 ihπ(t, ~x)|ζ2 i) d~x,
or more simply,
Z
ω=
π(t, ~x) ∧ φ(t, ~x)d~x.
Spatially smeared fields
We can also consider smeared out fields. One way to smear them out is to use the symplectic
#
form to pair YKG,R
and YKG,R .
More precisely, for ζ ∈ YKG,R we introduce the functional on YKG,R given by
hφ((ζ))|ρi := ζωρ,
Clearly,
φ((ζ)) =
Z ρ ∈ YKG,R .
ζ̇(t, ~x)φ(t, ~x) − ζ(t, ~x)π(t, ~x) d~x.
(3.6)
Note that
{φ((ζ1 )), φ((ζ2 ))} = ζ1 ωζ2 .
(3.7)
Space-time smeared fields
We can also smear fields with space-time functions f ∈ Cc∞ (R1,3 ):
Z
φ[f ] := f (x)φ(x)dx.
Clearly,
φ[f ] = φ((Df )),
Z Z
{φ[f1 ], φ[f2 ]} =
f1 (x)D(x − y)f2 (y)dxdy.
(3.8)
Poisson structure
The symplectic structure on the space YKG,R leads to a Poisson bracket on functions on
YKG,R :
{φ(t, ~x), φ(t, ~y )} = {π(t, ~x), π(t, ~y )} = 0,
{φ(t, ~x), π(t, ~y )} = δ(~x − ~y )
(3.9)
The relations (3.9) can be viewed as mnemotechnic identities that yield the correct Poisson
bracket for more regular functions, eg. the smeared out fields in (3.7) or (3.8). Note that
formally φ(t, ~x) and π(t, ~x) generate all functions on YKG,R .
Using (3.6) we obtain {φ(x), φ(y)} = D(x − y). Therefore, the Jordan-Pauli solution is
often called the commutator function.
The Hamiltonian density and momentum density are quadratic functionals on YKG,R :
1
2
2
2
2
~
π(x) + ∇φ(x) + m φ(x) ,
H(x) =
2
~
~ x φ(x).
P(x)
= π(x)∇
They give the total Hamiltonian and momentum
Z
Z
~ ~x)d~x,
H = H(0, ~x)d~x, P~ = P(0,
the generators of the time and space translations. The Poisson bracket of H and P~ vanishes.
Plane waves
We would like to diagonalize simultaneously the Hamiltonian, the momentum and the symp
~
~
~
plectic form. To this end, for any k = (ω(k), k) with ω(k) = ~k 2 + m2 , we introduce the
corresponding plane wave
Φ(k) =
1
q
eixk .
(2π)3/2 2ω(~k)
Note that
iΦ(k)ωΦ(k 0 ) = 0,
iΦ(k)ωΦ(k 0 ) = δ(~k − ~k 0 ).
(3.10)
Plane waves in a box
Sometimes we use the box normalization:
1
ΦV (k) = q
eixk .
V 2ω(~k)
Then
iΦV (k)ωΦV (k 0 ) = 0,
iΦV (k)ωΦV (k 0 ) = δ~k,~k0 .
Plane wave functionals
Define the following functionals on YKG,R :
a(k) = φ((iΦ(k)))
s
3
= (2π)− 2
Z



ω(~k) −i~x~k
i
~

e
φ(0, ~x) − q
e−ik~x π(0, ~x) d~x,
2
2ω(~k)
a(k) = φ(( − iΦ(k)))
s

Z
3
i
~
 ω(~k) i~x~k

= (2π)− 2 
e φ(0, ~x) + q
eik~x π(0, ~x) d~x.
2
2ω(~k)
Diagonalization
Note that
{a(k), a(k 0 )} = {a(k), a(k 0 )} = 0,
{a(k), a(k 0 )} = iδ(~k − ~k 0 ),
Z
H =
d~kω(~k)a(k)a(k),
Z
P~ =
d~k~ka(k)a(k).
If ζ1 , ζ2 ∈ YKG,R , then
Z
iζ1 ωζ2 =
(ha(k)|ζ1 iha(k)|ζ2 i − ha(k)|ζ1 iha(k)|ζ2 i) d~k.
The fields can be written as
d~k
q
eikx a(k) + e−ikx a(k)
2ω(~k)
Z =
Φ(k)a(k) + Φ(k)a(k) d~k.
− 32
Z
φ(x) = (2π)
Thus, every ζ ∈ YKG,R can be written as
− 32
ζ(x) = (2π)
d~k
Z
q
2ω(~k)
eikx ha(k)|ζi + e−ikx ha(k)|ζi .
Positive/negative frequency solutions
(±)
YKG will denote the subspace of YKG consisting of positive, resp. negative frequency solutions, that is,
(+)
YKG := {ζ ∈ YKG : ha(k)ζi = 0, k = (ω(~k), ~k)},
(−)
YKG := {ζ ∈ YKG : ha(k)ζi = 0, k = (ω(~k), ~k)}.
Hilbert space of positive energy solutions
(+)
Every ζ ∈ YKG can be written as
− 32
ζ(x) = (2π)
Z
d~k
q
eikx ha(k)|ζi
2ω(~k)
(+)
For ζ1 , ζ2 ∈ YKG we define the scalar product
Z
(ζ1 |ζ2 ) := iζ1 ωζ2 = ha(k)|ζ1 iha(k)|ζ2 id~k
(+)
(+)
We set ZKG to be the completion of YKG in this scalar product. The ortochronous Poincaré
(+)
group leaves ZKG invariant.
Quantization of free fields
The Klein-Gordon equation defines a symplectic space equipped with a linear symplectic
dynamics generated by a strictly positive Hamiltonian. For such systems, we can apply the
prcedure of the positive energy quantization.
Let us describe the quantization of the Klein-Gordon equation. We will use the “hat” to
denote the quantized objects.
We want to construct a Hilbert space H, a positive self-adjoint operator Ĥ called the
Hamiltonian, a normalized vector Ω, which is a ground state of Ĥ and a distribution
R1,3 3 x 7→ φ̂(x),
(3.11)
which smeared with Cc∞ (R1,3 , R) functions has values in self-adjoint operators satisfying
1. (−2 + m2 )φ̂(x) = 0, [φ̂(x), φ̂(y)] = iD(x − y).
2. eitĤ φ̂(x0 , ~x)e−itĤ = φ̂(x0 + t, ~x).
3. Ω belongs to the domain of polynomials in smeared out φ̂(x) and is cyclic for these
operators.
There exists an alternative equivalent formulation of the quantization program, which uses
the smeared fields instead of point fields. Instead of (3.11) we look for a linear function
YKG,R 3 ζ 7→ φ̂((ζ))
with values in self-adjoint operators such that
1. ’
[φ̂((ζ1 )), φ̂((ζ2 ))] = iζ1 ωζ2 .
(3.12)
2. ’
φ̂((r(t,~0) ζ)) = eitĤ φ̂((ζ))e−itĤ .
3. ’ Ω belongs to the domain of polynomials in φ̂((ζ)) and is cyclic for these operators.
One can pass between these two versions of the quantization problem by
Z φ̂((ζ)) =
ζ̇(t, ~x)φ̂(t, ~x) − ζ(t, ~x)π̂(t, ~x) d~x,
˙
where π̂(x) := φ̂(x).
(3.13)
The above problem has a solution, which is unique up to a unitary equivalence. We set H =
(+)
Γs (ZKG ). The creation/annihilation operators will be denoted by â∗ and â. For k = (ω(~k), ~k)
we set
â∗ (k) := â∗ (−iΦ(k)).
Ω will be the Fock vacuum. We set
− 23
φ̂(x) = (2π)
Z
d~k
ikx
q
e
2ω(~k)
−ikx ∗ ~
~
â(k) + e
â (k) .
The Hamiltonian and the momentum are
Z
Ĥ =
â∗ (~k)â(~k)ω(~k)d~k,
Z
~
P̂ =
â∗ (~k)â(~k)~kd~k.
the whole orthochronous Poincaré group is unitarily implemented on H by U (a, Λ) :=
Note that Γ r(a,Λ) (+) This is true even though in our construction we only required that time transZKG
lations are implemented. We have
U (a, Λ)φ̂(x)U (a, Λ)∗ = φ̂ (a, Λ)x .
Note the identities
(Ω|φ̂(x)φ̂(y)Ω) = −iD(+) (x − y),
(Ω|T(φ̂(x)φ̂(y))Ω) = −iDc (x − y).
In fact,
−3
Z Z
(Ω|φ̂(x)φ̂(y)Ω) = (2π)
d~kd~k 0 ikx−ik0 y
√ √ e
(Ω|â(~k)â∗ (~k 0 )Ω)
2ω 2ω 0
d~k ik(x−y)
e
2ω
= −iD(−) (x − y);
= (2π)−3
Z
(Ω|T(φ̂(x)φ̂(y))Ω) = θ(x0 − y 0 )(Ω|φ̂(x)φ̂(y)Ω) + θ(y 0 − x0 )(Ω|φ̂(y)φ̂(x)Ω)
= −iθ(x0 − y 0 )D(−) (x − y) − iθ(y 0 − x0 )D(−) (y − x)
= −iDc (x − y).
Axioms
The family Cc∞ (R1,3 , R) 3 f 7→ φ̂[f ] :=
R
φ̂(x)f (x)dx satisfies the Wightman axioms with
(+)
D := Γfin
s (ZKG ).
For an open set O ⊂ Rd we set
A(O) := {exp(iφ̂[f ]) : f ∈ Cc∞ (O, R)}00 .
The algebras A(O) satisfy the Haag-Kastler axioms.
External source
We go back to the classical theory. In the presence of an external source R1,3 3 x 7→ j(x) ∈
R, the Klein-Gordon equation becomes
(−2 + m2 )ζ(x) = −j(x).
(3.14)
Introducing the fields R1,3 3 x 7→ φ(x) we can rewrite (3.14) as
(−2 + m2 )φ(x) = −j(x).
(3.15)
Lagrangian formalism
(3.15) can be obtained from the Lagrangian formalism. The Lagrangian density can be taken
as
L(x) = − 12 ∂µ φ(x)∂ µ φ(x) − 12 m2 φ(x)2 − j(x)φ(x).
The Euler-Lagrange equations
∂φ L − ∂µ
yield (3.15).
∂L
=0
∂(∂µ φ)
(3.16)
Hamiltonian formalism
The variable conjugate to φ(x) is
π(x) :=
∂L
= ∂0 φ(x).
∂0 φ(x)
We have the equal-time Poisson-brackets (3.9). The Hamiltonian density equals
1
m2 2
1 2
φ (x) + j(x)φ(x).
H(x) = π (x) + ∂i φ(x)∂i φ(x) +
2
2
2
The Hamiltonian
Z
H(t) =
d~xH(t, ~x)
can be used to generate the dynamics
φ̇(t, ~x) = {H(t), φ(t, ~x)},
π̇(t, ~x) = {H(t), π(t, ~x)}.
If j does not depend on time, then the Hamiltonian does not depend on time as well and we
can write H instead of H(t).
We can also introduce the stress-energy tensor density
1
Tµν (x) := ∂µ φ(x)∂ν φ(x) − gµν L(x),
2
which is conserved
∂µ Tµν (x) = 0.
We have
H(x) = T00 (x), Pi (x) = T0i (x).
Quantization in the presence of external source
We are looking for quantum fields satisfying
(−2 + m2 )φ̂(x) = −j(x)
(3.17)
coinciding with the free fields for t = 0.
Such fields are given by
Z 0
Z t
Ĥ(s)ds .
φ̂(t, ~x) := Texp −i
Ĥ(s)ds φ̂(0, ~x)Texp −i
t
0
where the Hamiltonian Ĥ(t), and the corresponding Hamiltonian in the interaction picture
equal
Z
Ĥ(t) =
1 2
1
m2 2
d~x : π̂ (~x) + ∂i φ̂(~x)∂i φ̂(~x) +
φ̂ (~x) + j(t, ~x)φ̂(~x) :,
2
2
2
Z
ĤInt (t) =
j(t, ~x)φ̂I (t, ~x)d~x.
In Ĥ(t), the fields are at time zero: φ̂(~x) = φ̂(0, ~x), π̂(~x) = π̂(0, ~x).
In ĤInt (t), the fields φ̂I (x) are free, that is defined with H0 .
N -point Green’s functions
For xN , . . . , x1 , the N -point Green’s function are defined as follows:
G(φ̂, xN , . . . , φ̂, x1 )
+
:= Ω |T φ̂(xN ) · · · · · · φ̂(x1 ) Ω− ,
where
Z t
Ĥ(s)ds Ω.
Ω := lim Texp −i
±
t→±∞
0
Note that the fields φ(t, ~x) according to the notation from first chapter would be denoted
φH (t, ~x).
Feynman rules in momentum space for Green’s functions.
We have 1 kind of lines and 1 kind of vertices. At each vertex just one line ends.
1. In the nth order we draw all possible topologically distinct Feynman diagrams with n
vertices and N external lines. This means, we put N dots (for external legs) and n dots
(for vertices). Then we connect them in all possible ways.
2. To each line we associate a propagator
−i
.
k 2 + m2 − i0
For internal lines we integrate over the variables with the measure
3. To each vertex we associate the factor −ij(k).
d4 k
(2π)4 .
Scattering operator
Z
∞
S = Texp −i
ĤInt (t)dt .
−∞
A special role is played by matrix elements of the scattering operator between plane waves,
called scattering amplitudes
+
−
−
Φ(k1+ ) · · · Φ(kN
+ )| S Φ(kN − ) · · · Φ(k1 ) .
For simplicity, we assume that the momenta are distinct.
Feynman rules in momentum space for scattering amplitudes.
To compute scattering amplitudes with N − incoming and N + outgoing particles we draw
the same diagrams as for N − + N + -point Green’s functions, where we adopt the convention
that the incoming lines are drawn on the right and outgoing lines on the left. The rules are
changed only concerning the external lines.
1. With each incoming external line we associate √
2. With each outgoing external line we associate √
1
.
V 2ω(~k)
1
.
V 2ω(~k)
Chapter 4
Massive photons
137
Let Aν (x) be a vector field. Set
Fµν := ∂µ Aν − ∂ν Aµ .
(4.1)
In this chapter we discuss the quantization of the so-called Proca equation
−∂µ F µν (x) + m2 Aν (x) = 0.
(4.2)
Let J ν (x) be a given vector function called current. We will assume that the current is
conserved, that is
∂ν J ν (x) = 0.
(4.3)
We will also consider the Proca equation in the presence of the current J µ :
−∂µ F µν (x) + m2 Aν (x) = J ν (x).
(4.4)
Space of solutions
Let YPr,R , resp. YPr denote the set of real, resp. complex smooth space-compact solutions
of
−∂ µ (∂µ ζν − ∂ν ζµ ) + m2 ζν (x) = 0,
(4.5)
(−2 + m2 )ζ µ (x) = 0,
(4.6)
∂ν ζ ν (x) = 0.
(4.7)
or what is equivalent
Only ζ~ is dynamical and satisfies the obvious analog of Theorem 3.2. ζ0 can be calculated by
~˙
ζ0 = (−∆ + m2 )−1 divζ.
Symplectic form
Clearly, the following expression defines a conserved current:
j µ (ζ1 , ζ2 , x) := ∂ µ ζ1,ν (x)ζ2ν (x) − ζ1,ν (x)∂ µ ζ2ν (x).
YPr,R is a symplectic space with the symplectic form
Z
ζ1 ωζ2 =
j µ (ζ1 , ζ2 , x)dsµ (x)
ZS ν
ν
=
ζ̇1,ν (t, ~x)ζ2 (t, ~x) − ζ1,ν (t, ~x)ζ̇2 (t, ~x) d~x.
Alternatively, we could have used another conserved current to define the symplectic form:
j0µ (ζ1 , ζ2 , x)
:= (∂ µ ζ1,ν (x) − ∂ν ζ1µ (x)) ζ2ν (x) − ζ1,ν (x) (∂ µ ζ2ν (x) − ∂ ν ζ2µ (x)) .
Using the integration by parts and the Lorentz condition we see that it leads to the same
symplectic form on YPr,R .
Classical potentials
We introduce the functionals Aµ (x)
hAµ (x)|ζi := ζµ (x).
Moreover, we introduce the functionals Ei (x)
hEi (x)|ζi := ζ̇i (x) − ∇i ζ0 (x).
~˙ − ∇A0 .
Clearly, E = A
Poisson structure
The symplectic structure on the space YPr,R leads to a Poisson bracket on functions on
YPr,R :
{Ai (t, ~x), Aj (t, ~y )} = {Ei (t, ~x), Ej (t, ~y )} = 0,
{Ai (t, ~x), Ej (t, ~y )} = δij δ(~x − ~y ).
Using
~˙
A0 = (−∆ + m2 )−1 divA,
we deduce
{Aµ (x), Aν (y)} =
gµν
∂µ ∂ν
− 2
m
D(x − y).
(4.8)
Spatially smeared fields
We can use the symplectic form to pair distributions and solutions. For ζ ∈ YPr,R we
introduce also the functional on YPr,R given by
hA((ζ))|ρi := ρωζ.
Note that
{A((ζ1 )), A((ζ2 ))} = ζ1 ωζ2 .
Z µ
µ
A((ζ)) =
ζ̇µ (t, ~x)A (t, ~x) − ζµ (t, ~x)E (t, ~x) d~x.
(4.9)
Space-time smeared fields
We can also smear the potentials with space-time vector valued functions f ∈ Cc∞ (R1,3 , R1,3 ):
Z
A[f ] := fµ (x)Aµ (x)dx.
Note that A[f ] = A((ζ)), where
∂µ ∂ ν
Dfν .
ζµ = Dfµ −
m2
Let ijk be the standard fully antisymmetric symbol. We introduce
1 ~2
−2
2
2
2
2 ~2
~
~
~
~
E (x) + m (divE) (x) + (∇A) (x) − (divA) (x) + m A (x)
H(x) =
2
~
~ i (x),
P(x)
= Ei (x)∇A
S(x) = Ei (x)ijk ∇k Aj (x).
They give the total Hamiltonian, momentum and polarization
Z
Z
Z
~ ~x)d~x, S = S(0, ~x)d~x.
H = H(0, ~x)d~x, P~ = P(0,
The Poisson bracket of any pair from H, P~ , S is zero.
Plane waves
We would like to diagonalize simultaneously the Hamiltonian H, the momentum P~ , the
polarization S and the symplectic form ω.
p
1,3
~
~
~
Let k ∈ R with k = (ω(k), k), ω(k) = ~k 2 + m2 . We fix two spatial vectors e1 (~k), e2 (~k)
that form an oriented orthonormal basis of the plane orthogonal to ~k. Define
!
~
~
~
|k| ω(k)k
u(k, 0) :=
,
,
(4.10)
m m|~k|
1
(4.11)
u(k, ±1) := 0, √ (e1 ± ie2 ) .
2
Note that
uµ (k, σ)k µ = 0,
uµ (k, σ)uµ (k, σ 0 ) = δσ,σ0 ,
X
uµ (k, σ)uν (k, σ) = gµν +
σ=0,±1
kµ kν
.
m2
A plane wave is defined as
Φ(k, σ) =
1
q
uµ (k, σ)eikx .
(2π)3/2 2ω(~k)
Note that iΦ(k, σ)ωΦ(k 0 , σ 0 ) = δ(~k − ~k 0 )δσ,σ0 .
(4.12)
Plane wave functionals
Define the following functionals on YPr,R :
a(k, σ) = A((iΦ(k, σ)))
s
Z ω(~k) −i~x~k
− 32
= (2π)
e
uµ (k, σ)Aµ (0, ~x)
2
i
µ
−i~k~x
uµ (k, σ)E (0, ~x) d~x.
−q
e
~
2ω(k)
Diagonalization
{a(k, σ), a(k 0 , σ 0 )} = {a(k, σ), a(k 0 , σ 0 )} = 0,
{a(k, σ), a(k 0 , σ 0 )} = iδ(~k − ~k 0 )δσ,σ0 ,
Z
X
H =
d~k
ω(~k)a(k, σ)a(k, σ),
σ=0,±1
P~ =
Z
d~k
X
~ka(k, σ)a(k, σ),
σ=0,±1
Z
S =
d~k
X
σ|~k|a(k, σ)a(k, σ).
σ=0,±1
If ζ1 , ζ2 ∈ YPr,R , then
iζ1 ωζ2 =
X Z
σ=0,±1
(ha(k, σ)|ζ1 iha(k, σ)|ζ2 i − ha(k, σ)|ζ1 iha(k, σ)|ζ2 i) d~k.
The fieds can be written as
− 32
Aµ (x) = (2π)
X Z
σ=0,±1
=
X Z σ=0,±1
d~k
q
2ω(~k)
ikx
−ikx
uµ (x, σ)e a(k, σ) + uµ (x, σ)e
a(k, σ)
Φ(k, σ)a(k, σ) + Φ(k, σ)a(k, σ) d~k.
Positive/negative frequency solutions
(±)
YPr will denote the subspace of YPr consisting of positive, resp. negative frequency solu(+)
tions. Every ζ ∈ YPr can be written as
− 32
ζµ (x) = (2π)
X Z
σ=0,±1
d~k
q
eikx uµ (k, σ)ha(k, σ)|ζi
2ω(~k)
(+)
For ζ1 , ζ2 ∈ YPr we define the scalar product
X Z
ha(k, σ)|ζ1 iha(k, σ)|ζ2 id~k
(ζ1 |ζ2 ) := iζ1 ωζ2 =
σ=0,±1
(+)
(+)
We set ZPr to be the completion of YPr in this scalar product. The ortochronous Poincaré
(+)
group leaves ZPr invariant.
Quantization of free fields
We want to construct a Hilbert space H, a positive self-adjoint operator Ĥ, a normalized
vector Ω, which is a ground state for Ĥ and a self-adjoint operator-valued distribution R1,3 3
x 7→ µ (x) satisfying
1.
−∂ µ (∂µ Âν − ∂ν µ ) + m2 Âν (x) = 0,
∂µ ∂ν
[µ (x), Âν (y)] = i gµν − 2 D(x − y).
m
2.
eitĤ µ (x0 , ~x)e−itĤ = µ (x0 + t, ~x).
3. Ω belongs to the domain of polynomials in smeared out µ (x) and is cyclic for these
operators.
An alternative set of conditions:
We look for a linear function YPr,R 3 ζ 7→ Â((ζ)) satisfying
1. ’
h
i
Â((ζ1 )), Â((ζ2 )) = iζ1 ωζ2 .
2. ’
Â((r(t,~0) ζ)) = eitĤ Â((ζ))e−itĤ .
3. ’ Ω belongs to the domain of polynomials in Â((ζ)) and is cyclic for these operators.
(+)
The above problem has a solution unique up to a unitary equivalence. We set H = Γs (ZPr ).
The creation/annihilation operators will be denoted by â∗ and â. In particular, we set
â∗ (k, σ) := â∗ (−iΦ(k, σ)).
Ω will be the Fock vacuum. We set
Z
X 3
d~k
−ikx
−2
ikx ∗
q
uµ (k, σ)e
â(k, σ) + uµ (k, σ)e â (k, σ) ,
µ (x) = (2π)
2ω(~k) σ=0,±1
The quantum Hamiltonian, momentum and polarization are
X Z
Ĥ =
â∗ (k, σ)â(k, σ)ω(~k)d~k,
σ=0,±1
~
P̂ =
X Z
â∗ (k, σ)â(k, σ)~kd~k,
σ=0,±1
Z
Ŝ =
d~k
X
σ=0,±1
σ|~k|â∗ (k, σ)â(k, σ).
The whole orthochronous Poincaré group is unitarily implemented on H by U (a, Λ) :=
Γ r(a,Λ) (+) We have
ZPr
U (a, Λ)µ (x)U (a, Λ)∗ = Λνµ Âν (a, Λ)x .
Note the identities
∂µ ∂ν
(Ω|µ (x)µ (y)Ω) = −i gµν − 2 D(+) (x − y),
m
∂µ ∂ν
(Ω|T(µ (x)Âν (y))Ω) = −i gµν − 2 Dc (x − y).
m
Axioms
For f ∈ Cc∞ (R1,3 , R1,3 ) set
Z
Â[f ] :=
µ (x)f µ (x)dx.
We obtain a family that satisfies the Wightman axioms with D := Γfin
s (ZPr ).
For an open set O ⊂ Rd we set
n
o
∞
1,3
1,3
A(O) := exp(iÂ[f ]) : f ∈ Cc (R , R ) .
The algebras A(O) satisfy the Haag-Kastler axioms.
External current
We return to the classical Proca equation. Recall that in the presence of the external current
the Proca equations take the form
∂ν J ν (x) = 0,
−∂µ (∂µ Aν − ∂ν Aµ ) + m2 Aν (x) = J ν (x).
(4.13)
(4.14)
Note that (4.14) and (4.13) imply
∂ν Aν (x) = 0.
(4.15)
(−2 + m2 )Aµ (x) = J µ (x).
(4.16)
We have therefore
The temporal component of (4.14) has no time derivative:
~ − m2 A0 = J 0 .
∆A0 − ∂ 0 divA
(4.17)
~ at the same time:
Therefore, we can compute A0 in terms of A
~ + J 0 ).
A0 = (∆ − m2 )−1 (∂ 0 divA
(4.18)
The only dynamical variables are the spatial components, satisfying the equation
~ = J.
~
(−∂02 + ∆ − m2 )A
(4.19)
The Lagrangian
The Lagrangian density equals
L :=
=
=
=
1
m2
µν
− Fµν F −
Aµ Aµ + Jµ Aµ
4
2
1
m2
1
Aµ Aµ + Jµ Aµ
− ∂µ Aν ∂ µ Aν + ∂µ Aν ∂ ν Aµ −
2
2
2
1
1
1
1
− ∂i Aj ∂i Aj + ∂i A0 ∂i A0 + ∂0 Ai ∂0 Ai + ∂i Aj ∂j Ai − ∂0 Ai ∂i A0
2
2
2
2
2
2
m
m 2
+ A20 −
A + Ji Ai − J0 A0
2
2 i
2
2
1
~ 2 + 1 (∇A
~ 0 )2 + 1 A
~ 2 + J~A
~˙ 2 − A
~ − J0 A0 .
~˙ ∇A
~ 0 + m A20 − m A
− (rotA)
4
2
2
2
2
As noted before, only spatial components Ai are dynamical. We have the conjugate variables
Ei =
∂L
= F0i
∂ Ȧi
~ =A
~˙ − ∇A
~ 0 . In terms of E,
~ we have
or E
A0 =
1
~
(J0 − divE).
m2
(4.20)
The Hamiltonian
The Hamiltonian density
~ ·A
~−L
H = E
1 ~2 ~ ~
1
m2 2 m2 ~ 2 ~ ~
2
~
= E + E ∇A0 + (rotA) −
A +
A − J A + J0 A0
2
4
2 0
2
1 ~2
1
1 ~ ~ 2 1
m2 ~ 2
2
2
~
~
= E
+
(J
−
div
E)
+
(
∇
A)
−
(div
A)
+
A
0
2
2m2
2
2
2
~ + J0 (−∆)−1 J0 + total derivative.
−J~A
~ ∇A
~ 0 with −divEA
~ 0 and 1 (rotA)
~ 2 with 1 (∇
~ A)
~ 2 − 1 (divA)
~ 2 , the
In the last step we replaced E
4
2
2
difference being a total derivative in spatial variables. We also inserted A0 as in (4.20).
Quantization in the presence of external source
We are looking for quantum potentials R1,3 3 x 7→ µ (x) satisfying
−∂µ (∂ µ Âν − ∂ ν µ ) + m2 Âν (x) = J ν (x).
They are given by
Z t
Z 0
Ĥ(s)ds ,
Ĥ(s)ds µ (~x)Texp −i
µ (t, ~x) := Texp −i
t
0
where the Hamiltonian Ĥ(t), and the corresponding Hamiltonian in the interaction picture
equal
1
2
1
~2
~ 2 1 ~ ~ 2 1
~ 2 m ~2
Ĥ(t) =
d~x : Ê +
(J0 − divÊ) + (∇Â) − (divÂ) +
Â
2
2m2
2
2
2
~
−1
~
−J Â + J0 (−∆) J0 :
Z
1
1 2
~
~
−1
~
J + J0 (−∆) J0 :.
ĤInt (t) =
d~x : − 2 divÊI J0 − J ÂI +
m
2m2 0
Z
N -point Green’s functions
For xN , . . . , x1 , the N -point Green’s function are defined as follows:
G(µN , xN , . . . , µ1 , x1 )
+
−
:= Ω |T(µN (xN ), . . . , µ1 (x1 ))Ω ,
where
Z t
Ĥ(s)ds Ω.
Ω± := lim Texp −i
t→±∞
0
Feynman rules for Green’s functions.
We have 1 kind of lines and 1 kind of vertices. At each vertex just one leg ends.
1. In the nth order we draw all possible topologically distinct Feynman diagrams with n
vertices and external lines.
2.
−i
To each line we associate a propagator m2 +k
2 −i0
4
d k
grate over the variables with the measure (2π)
4.
3. To each vertex we associate the factor −iJ µ (k).
gµν −
kµ kν
m2
. For internal lines we inte-
Scattering operator
Z
∞
S = Texp −i
ĤInt (t)dt .
−∞
As usual, matrix elements of the scattering operator between plane waves are called scattering
amplitudes
+
+
−
−
−
−
Φ(k1+ , σ1+ ) · · · Φ(kN
+ , σN + )| S Φ(kN − , σN − ) · · · Φ(k1 , σ1 ) .
For simplicity, we assume that the momenta are distinct.
Feynman rules for scattering amplitudes.
To compute scattering amplitudes with N − incoming and N + outgoing particles we draw
the same diagrams as for N − + N + -point Green’s functions, where we adopt the convention
that the incoming lines are drawn on the right and outgoing lines on the left. The rules are
changed only concerning the external lines.
1. With each incoming external line we associate √
2. With each outgoing external line we associate √
1
u(k, σ).
V 2ω(~k)
1
u(k, σ).
V 2ω(~k)
Propagators for massive photons
The causal propagator used to compute Green’s functions and scattering amplitudes that
follows directly from the interaction Hamiltonian is
kµ kν
1
c
Dµν = 2
gµν + 2 .
m + k 2 − i0
m
(4.21)
If we compute scattering amplitudes, we can pass from this propagator to another by adding
kµ fν (k) + fµ (k)kν for an arbitrary function fµ (k).
To see this note that after adding kµ fν (k) + fµ (k)kν the contribution of each line changes
by
J µ (k) (kµ fν (k) + fµ (k)kν ) J ν (k),
which is zero, because kµ J µ (k) = 0.
For scattering amplitudes the external line does not involve the propagator. Therefore,
scattering amplitudes do not change. However, Green’s functions change, because of external
lines.
We will list a number of useful propagators. We will drop the superscript c (for causal).
For any α ∈ R, we can pass to the following propagators
1
k
k
µ
ν
α,I
Dµν
=
gµν − (α − 1) 2 ,
m2 + k 2 − i0
m
kµ kν
1
α,II
gµν − (α − 1) 2
Dµν =
.
m2 + k 2 − i0
k
The above propagators coincide for α = 1, and will be called the propagator in the Feynman
gauge, that is
Feyn
Dµν
(k) =
1
.
m2 + k 2 − i0
We can introduce the propagator in the Yukawa gauge:
Yuk
D00
=−
1
m2 + ~k 2
,
Yuk
D0j
= 0,
DijYuk
ki kj
1
δij −
= 2
,
m + k 2 − i0
m2 + ~k 2
Feyn
Yuk
+ kµ fνYuk (k) + fµYuk (k)kν , where
= Dµν
We have Dµν
f0Yuk (k) =
k0
(k 2
−
i0)2(m2
+ ~k 2 )
, fiYuk (k) = −
ki
(k 2
−
i0)2(m2
+ ~k 2 )
.
The propagator in the Yukawa gauge is the massive analog of the propagator in the Coulomb
gauge.
The propagator in the temporal gauge equals
tem
D00
= 0,
tem
D0j
= 0,
Dijtem
ki kj
1
δij − 2 .
= 2
k − i0
k0
tem
Feyn
We have Dµν
= Dµν
+ kµ fνtem (k) + fµtem (k)kν , where
f0tem (k) =
ki
1
tem
,
f
.
(k)
=
−
i
(m2 + k 2 − i0)2k0
(m2 + k 2 − i0)2k02
Chapter 5
Massless photons
175
For a vector potential Aν (x) we define
Fµν := ∂µ Aν − ∂ν Aµ ,
as in (4.1). In this chapter we discuss the quantization of the Maxwell equation
−∂µ F µν (x) = 0.
(5.1)
We will also consider an external conserved current, that is a given vector function J ν (x)
satisfying
∂ν J ν (x) = 0.
(5.2)
The Maxwell equation in the presence of the current J reads
−∂µ F µν (x) = J ν (x),
(5.3)
Gauge invariance
It is well known that the Maxwell equation
−∂µ (∂ µ ζ ν (x) − ∂ ν ζ µ (x)) = 0
(5.4)
is invariant wrt the replacement of ζµ with ζµ + ∂µ χ, where χ is an arbitrary smooth function
on the space-time. In particular, there is no uniqueness of the Cauchy problem for ζµ .
This poses problems both for the classical and quantum theory. One could avoid the problem
of the gauge invariance by considering fields and not potentials as basic objects. However,
when one quantizes the Maxwell equation including the currents, it is more convenient to use
potentials. Therefore, we will stick to potentials.
Space of solutions of the Maxwell equation
There exist several ways to cope with gauge invariance. In the first approach, we start with
the space of real/complex smooth space-compact solutions of the Maxwell equation (5.4),
which we denote by ỸMax,R /ỸMax We check that the following expression defines a conserved
current
j̃ µ (ζ1 , ζ2 , x)
:= (∂ µ ζ1,ν (x) − ∂ν ζ1µ (x)) ζ2ν (x) − ζ1,ν (x) (∂ µ ζ2ν (x) − ∂ ν ζ2µ (x)) .
It defines in the usual way the antisymmetric form
Z
ζ1 ω̃ζ2 =
j̃ 0 (ζ1 , ζ2 , (t, ~x))d~x.
(5.5)
(5.5) does not depend on the gauge. The ortochronous Poincare group acts on ỸMax by
transformations preserving ω̃:
r̃(a,Λ) ζ µ (x) := Λµν ζµ (a, Λ)−1 x .
The Coulomb gauge
We say that a solution ζ of the Maxwell equation is in the Coulomb gauge if ζ0 = 0 and
divζ~ = 0.
Note that every ζ ∈ ỸMax is gauge-equivalent to a unique solution of the Maxwell equation
in the Coulomb gauge, which however does not have to be space-compact. If for ζ ∈ ỸMax ,
we denote by ζ Coul the corresponding Coulomb gauge solution, then
ζ1 ω̃ζ2 = ζ1Coul ω̃ζ2Coul .
Symplectic reduction
The space ỸMax,R is presymplectic. The kernel of the symplectic form coincides with potentials that are gauge equivalent to zero. We define YMax,R to be the symplectic reduction
of ỸMax,R . In other words, YMax,R is ỸMax,R divided by the gauge equivalence. YMax,R is
equipped with a natural symplectic form ω. The ortochronous Poincare group acts on YMax,R
by symplectic transformations.
Analogously we define the space YMax of gauge classes of complex smooth space-compact
solutions of (5.4).
We introduce the functionals ACoul
(x) on YMax,R , called the classical potentials in the
µ
Coulomb gauge,
hACoul
(x)|ζi := ζµCoul (x),
µ
where ζ Coul on the right hand side is the representative of the class ζ in the Coulomb gauge.
Clearly,
ACoul
(x) = 0,
0
~ Coul (x) = 0.
divA
Moreover, we introduce the functionals Fµν (x) called the fields:
hFµν (x)|ζi := ∂µ ζν (x) − ∂ν ζµ (x).
We will write Ei (x) = F0i (x). Clearly,
~ = ∂t A
~ Coul , divA
~ Coul (x) = 0.
E
Poisson structure
The symplectic structure on the space YMax,R leads to a Poisson bracket on the level of
functions on YMax,R :
{ACoul
(t, ~x), ACoul
(t, ~y )} = {Ei (t, ~x), Ej (t, ~y )} = 0,
i
j
∂i ∂j
Coul
δ(~x − ~y )
{Ai (t, ~x), Ej (t, ~y )} = δij −
∆
From the above relations we deduce
{ACoul
(x), ACoul
(y)}
i
j
∂i ∂j
= δij −
D(x − y).
∆
Spatially smeared potentials
Let us study various constructions related to YMax . In what follows we drop the superscript
Coul on ACoul
(x).
µ
We can use the symplectic form to pair distributions and solutions. For ζ ∈ YMax,R we
introduce also the functional on YMax,R given by
hA((ζ))|ρi := ρωζ.
Note that
{A((ζ1 )), A((ζ2 ))} = ζ1 ωζ2 .
Z µ
µ
A((ζ)) =
ζ̇µ (t, ~x)A (t, ~x) − ζµ (t, ~x)E (t, ~x) d~x.
Let us stress that A((ζ)) depends on ζ only modulo gauge transformations.
(5.6)
Space-time smeared potentials
We can also smear the potentials with space-time vector valued functions f ∈ Cc∞ (R1,3 , R1,3 ):
Z
A[f ] := fµ (x)Aµ (x)dx.
(5.7)
Note that A[f ] = A((ζ)), where
∂i ∂ j
ζi = D fi −
fj , ζ0 = 0.
∆
As for massive photons, we introduce
1 ~ 2 ~ ~ 2
E(x) + ∇A(x) ,
H(x) =
2
~
~ i (x),
P(x)
= Ei (x)∇A
S(x) = Ei (x)ijk ∇k Aj (x).
They give the total Hamiltonian, momentum and polarization
Z
Z
Z
~ ~x)d~x, S = S(0, ~x)d~x.
H = H(0, ~x)d~x, P~ = P(0,
The Poisson bracket of any pair from H, P~ , S is zero.
Plane waves
Let k ∈ R
1,3
with k = (ω, ~k), ω(~k) :=
p
~k 2 .
The vectors u(k, ±1) are defined as in (4.11). u(k, 0) are not defined at all. Note that
~u(k, σ) · ~k = 0, u0 (k, σ) = 0,
uµ (k, σ)uµ (k, σ 0 ) = δσ,σ0 ,
X
ki kj
ui (k, σ)uj (k, σ) = δij −
.
~k 2
σ=±1
A plane wave is defined as
Φ(k, σ) =
1
q
uµ (k, σ)eikx .
(2π)3/2 2ω(~k)
Note that iΦ(k, σ)ωΦ(k 0 , σ 0 ) = δ(~k − ~k 0 )δσ,σ0 .
(5.8)
Plane wave functionals
Define the following functionals on YMax,R :
a(k, σ) = A((iΦ(k, σ)))
s
Z 3
ω(~k) −i~x~k
= (2π)− 2
e
uµ (k, σ)Aµ (0, ~x)
2
i
µ
−i~k~x
uµ (k, σ)E (0, ~x) d~x.
−q
e
2ω(~k)
Diagonalization
{a(k, σ), a(k 0 , σ 0 )} = {a(k, σ), a(k 0 , σ 0 )} = 0,
{a(k, σ), a(k 0 , σ 0 )} = iδ(~k − ~k 0 )δσ,σ0 ,
Z
X
~
H =
dk
ω(~k)a(k, σ)a(k, σ),
σ=±1
P~ =
Z
d~k
X
~ka(k, σ)a(k, σ),
σ=±1
Z
S =
d~k
X
σ|~k|a(k, σ)a(k, σ).
σ=±1
If ζ1 , ζ2 ∈ YMax,R , then
XZ
iζ1 ωζ2 =
(ha(k, σ)|ζ1 iha(k, σ)|ζ2 i − ha(k, σ)|ζ1 iha(k, σ)|ζ2 i) d~k.
σ=±1
The potentials can be written as
XZ
3
d~k ikx
−ikx
−2
q
uµ (x, σ)e a(k, σ) + uµ (x, σ)e
a(k, σ)
Aµ (x) = (2π)
σ=±1
2ω(~k)
Z
X
=
Φ(k, σ)a(k, σ) + Φ(k, σ)a(k, σ) d~k.
σ=±1
Positive/negative frequency solutions
(±)
YMax will denote the subspace of YMax consisting of positive, resp. negative frequency
solutions.
(+)
Every ζ ∈ YMax can be written as
− 32
ζ(x) = (2π)
XZ
σ=±1
d~k
q
eikx u(k, σ)ha(k, σ)|ζi
2ω(~k)
(+)
For ζ1 , ζ2 ∈ YMax we define the scalar product
XZ
(ζ1 |ζ2 ) := iζ1 ωζ2 =
ha(k, σ)|ζ1 iha(k, σ)|ζ2 id~k.
σ=±1
(+)
(+)
We set ZMax to be the completion of YMax in this scalar product. The ortochronous Poincaré
(+)
group leaves ZMax invariant.
Spin averaging
For a given k ∈ R1,3 with k 2 = 0, let M, N be vectors with
M µ kµ = N ν kν = 0.
Then
X
σ=±1
M µ uµ (k, σ)uν (k, σ)N ν = M µ Nν .
(5.9)
In fact,
X
uµ (k, σ)uν (k, σ) = gµν + δµ0 δν0 −
σ=±1
Therefore, the left hand side of (5.9) equals
M µ gµν N ν + M 0 N 0 −
But
M0 =
~ ~k)(N
~ ~k)
(M
.
ω2
~k N
~k M
~
~
, N0 =
.
ω
ω
~kµ~kν
.
ω2
Quantization of free fields
There are several approaches to the quantization of Maxwell equation. In particular, one
can either reduce first and then quantize, or first quantize and then reduce. We will treat the
former choice as the basic one. This means, we will use the space YMax,R as the input for
quantization. The other possibilities will be discussed later.
The quantization of YMax,R is very similar to the quantization of YPr,R . The main difference
is that Condition 1. is replaced with
(−2 + m2 )Âi (x) = 0,
∂i Âi (x) = 0,
Â0 (x) = 0
∂i ∂j
[Âj (x), Âi (y)] = i δij −
D(x − y).
∆
(+)
The above problem has a solution unique up to a unitary equivalence. We set H = Γs (ZMax ).
The creation/annihilation operators will be denoted by â∗ and â. In particular, we set
â∗ (k, σ) := â∗ (−iΦ(k, σ)).
Ω will be the Fock vacuum. We set
Z
3
d~k X −ikx
−2
ikx ∗
√
ui (k, σ)e
â(k, σ) + ui (k, σ)e â (k, σ) ,
Âi (x) = (2π)
2ω σ=±1
The Hamiltonian, the momentum and the polarization are
XZ
Ĥ =
â∗ (k, σ)a(k, σ)ω(~k)d~k,
σ=±1
XZ
~
P̂ =
â∗ (k, σ)â(k, σ)~kd~k,
σ=±1
Ŝ =
XZ
σ=±1
d~kσ|~k|â∗ (k, σ)â(k, σ).
The whole orthochronous Poincaré group is unitarily implemented on H by U (a, Λ) :=
Γ r(a,Λ) (+) We have
ZPr
0
0
U (a, Λ)F̂µ,ν (x)U (a, Λ)∗ = Λµµ Λνν F̂µ0 ,ν 0 (a, Λ)x .
Note the identities
∂i ∂j
(Ω|Âi (x)Âj (y)Ω) = −i δij −
D(+) (x − y),
∆
∂i ∂j
(Ω|T(Âi (x)Âj (y))Ω) = −i δij −
Dc (x − y).
∆
Axioms
The family Cc∞ (R1,3 , R3 ) 3 f 7→ Â[f ] :=
R
Âi (x)fi (x)dx with D := Γfin
s (ZMax ) does not
satisfy the Wightman axioms because of two problems: the nonlocality of the commutator and
the absence of the Lorentz covariance.
If we replace the fields µ with F̂µν , we restore the Poincare covariance.
For an open set O ⊂ R1,3 we set
A(O) := {exp(iF̂ [f ]) : f ∈ Cc∞ (R1,3 , R3 )}.
The algebras A(O) satisfy the Haag-Kastler axioms.
External current
We return to the classical Maxwell equation. We consider an external conserved current Jµ .
The Maxwell equation with an external current reads
−∂µ ∂ µ ζ ν + ∂ ν ∂µ ζ µ = J ν .
Given an arbitrary solution ζµ , by setting
ζµCoul := ζµ + ∂µ χ,
~˙ ~x) we obtain a soluton satisfying
where χ(t, ~x) := −∆−1 divA(t,
divζ~Coul = 0,
ζ0Coul = −∆−1 J 0 .
We call ζ Coul the representative of ζ in the Coulomb gauge.
(5.10)
Consider the Maxwell equation written in terms of Aµ :
−∂µ ∂ µ Aν + ∂ ν ∂µ Aµ = J ν .
We write separately the temporal and spatial equations:
~˙ = J 0 ,
−∆A0 + divA
~−∇
~ Ȧ0 + ∇div
~
~ = J.
~
(∂0 )2 − ∆ A
A
(5.11)
~ at the same time:
We can compute A0 in terms of A
~ − J 0 ).
A0 = ∆−1 (∂ 0 divA
(5.12)
~ obtaining
We can insert this into spatial equations, using J˙0 = divJ,
~ tr = J~tr .
(−∂02 + ∆)A
(5.13)
where
~ tr := A
~ − ∇∆
~ −1 divA,
~
A
~ −1 divJ.
~
Jtr := J~ − ∇∆
~ =: Θ is an arbitrary
Thus the only dynamical variables are the transversal spatial fields. divA
space-time function.
The Lagrangian and Hamiltonian
The Lagrangian density
1
L := − Fµν F µν + Jµ Aµ
4
1
~˙ ∇A
~ 0 + J~A
~ − J0 A0
~˙ 2 − A
~ 2 + 1 (∇A
~ 0 )2 + 1 A
= − (rotA)
4
2
2
1 ~ ~ 2 1 ~˙ 2 ~ ~
1
~
= − (∇
Atr ) + (Atr ) + Jtr Atr + J 0 ∆−1 J 0 − ∂ 0 (J 0 ∆−1 divA)
2
2
2
+total spatial derivative.
~ = A
~˙ − ∇A0 , where clearly divE
~ = J0 . The Hamiltonian
The conjugate variables are E
density is
~ tr · A
~ tr − L
H = E
1 ~2 1 ~ ~ 2 ~ ~
−1
= E
tr + (∇Atr ) − Jtr Atr + J0 (−∆) J0 + total derivative.
2
2
Quantization in the presence of external source
We are looking for quantum potentials R1,3 3 x 7→ µ (x) satisfying
−∂µ (∂ µ Âν − ∂ ν µ ) + m2 Âν (x) = J ν (x),
~
divÂ(x) = 0,
~
Â0 = ∆−1 (∂ 0 div − J 0 ).
(5.14)
(5.15)
(5.16)
To construct them note that it is enough to impose the condition (5.15) at t = 0. Then
this condition propagates for all times. (5.16) can be used as a definiton of Â0 (x). One can
use the Hamiltonian Ĥ(t) and the corresponding Hamiltonian in the interaction picture equal
Z
1
~2 1 ~ ~ 2 ~~
−1
Ĥ(t) =
d~x : Ê + (∇Â) − J Â + J0 (−∆) J0 :
2
2
Z
~
−1
~
ĤInt (t) =
d~x −J ÂI + J0 (−∆) J0 .
Propagators for massless photons
The causal propagator used to compute Green’s functions and scattering amplitudes that
follows directly from the interaction Hamiltonian is the propagator in the Coulomb gauge
k
k
1
1
i
j
Coul
Coul
D00
δij −
,
= − , D0j
= 0, DijCoul = 2
~k 2
~k 2
k − i0
If we compute scattering amplitudes, we can pass from this propagator to another by adding
kµ fν (k) + fµ (k)kν for an arbitrary function fµ (k).
Let us list a number of useful propagators in other gauges. In particular, we distinguish the
family of propagators
kµ kν
1
gµν + (α − 1) 2
.
k 2 − i0
k
They have the following names
Lan
Dµν
Feyn
Dµν
FY
Dµν
k k
1
gµν − µk2 ν
:= k2 −i0
1
gµν
:= k2 −i0
kµ kν
1
:= k2 −i0 gµν + 2 k2
Landau or Lorentz gauge,
Feynman gauge,
Fried and Yennie gauge.
Coul
Feyn
We have Dµν
= Dµν
+ kµ fνCoul (k) + fµCoul (k)kν , where
f0Coul (k) =
k0
(k 2 − i0)2~k 2
, fiCoul (k) = −
ki
(k 2 − i0)2~k 2
.
The propagator in the temporal gauge
tem
D00
= 0,
tem
D0j
= 0,
Dijtem
ki kj
1
δij − 2 .
= 2
k − i0
k0
tem
Feyn
We have Dµν
= Dµν
+ kµ fνtem (k) + fµtem (k)kν , where
f0tem (k) =
1
ki
tem
,
f
(k)
=
−
.
i
(k 2 − i0)2k0
(k 2 − i0)2k02
Chapter 6
Linearly perturbed quadratic Hamiltonians
209
Baker-Campbell-Hausdorff formula
The BCH formula says that if [A, [A, B]] = [B, [A, B]] = 0, then
1
eA+B = eA eB e− 2 [A,B] .
Time-dependent Baker-Campbell-Hausdorff formula
Let us describe a time-dependent version of the BCH formula.
Let [t− , t+ ] 3 t 7→ A(t), B(t) be operator valed functions such that
[A(s), A(s0 )] = [B(s], B(s0 )] = 0,
[[A(s), B(s0 )], A(s00 )] = [[A(s), B(s0 )], B(s00 )] = 0.
Then
!
t+
Z
Texp
ds(A(s) + B(s))
t−
Z
!
t+
= exp
Z
t+
dsA(s) exp
t−
t+
× exp
Z
t+
!
t+
ds2 θ(s1 − s2 )[A(s2 ), B(s1 )]
ds1
t−
= exp
dsB(s)
t−
Z
Z
t−
!
Z
t+
dsA(s) exp
t−
!
!
dsB(s) exp
t−
1
2
Z
t+
Z
ds1
t−
!
t+
ds2 G(s2 , s1 ) ,
t−
where
G(s2 , s1 ) = θ(s1 − s2 )[A(s2 ), B(s1 )] + θ(s2 − s1 )[A(s1 ), B(s2 )].
Time-dependent Van Hove Hamiltonians
Cosider a Hilbert space L2 (Ξ, dξ) and the corresponding Fock space Γs (L2 (Ξ, dξ)). Consider
the self-adjoint operator
Z
H0 =
ω(ξ)a∗ (ξ)a(ξ)dξ
We perturb it by a time-dependent operator.
Z
Z
V (t) = v(t, ξ)a∗ (ξ)dξ + v(t, ξ)a(ξ)dξ.
Clearly, the Hamiltonian in the interaction picture equals
Z
Z
itω(ξ)
∗
v(t, ξ)a (ξ)dξ + e−itω(ξ) v(t, ξ)a(ξ)dξ.
HInt (t) = e
Scattering for van Hove Hamiltonians
The scattering operator is then given by
Z
Z
∗
S = exp i v(ω(ξ), ξ)a (ξ)dξ exp i v(ω(ξ), ξ)a(ξ)dξ
!
Z
i
v(τ, ξ)v(τ, ξ)ω(ξ)
× exp
dτ dξ ,
2π
ω(ξ)2 − τ 2 − i0
where v(τ, ξ) =
R
v(t, ξ)e−itτ dt.
Time-independent Van Hove Hamiltonians
Consider now a time-independent operator
Z
Z
∗
V = v(ξ)a (ξ)dξ + v(ξ)a(ξ)dξ.
Clearly, the Fock vacuum Φ0 = Ω is the eigenstate of H0 vacuum with the eigenvalue E0 = 0.
The eigenstate and eigenvalue of H := H0 + V are
Z
|v(ξ)|2
E = −
dξ,
ω(ξ)
!
Z
Z
v(ξ) ∗
v(ξ)
a (ξ)dξ −
a(ξ)dξ Ω.
Φ = exp
ω(ξ)
ω(ξ)
Adiabatic scattering theory for van Hove Hamiltonians
To compute the Gell-Mann and Low wave operators for the pair H0 , H, consider v+ :=
θ(t)e−|t| v(ξ). Then v+ (τ, ξ) =
Z
−iv(ξ)
τ −i .
Thus
Z
v(ξ)
v(ξ)
a∗ (ξ)dξ exp −
a(ξ)dξ
ω(ξ) − i
ω(ξ) + i
Z
|v(ξ)|2 ω(ξ)dτ dξ
i
.
× exp
2π
(τ 2 + 2 )(ω(ξ)2 − τ 2 − i0)
S+ = exp
!
Using the residue calculus we see that
Z
i
i
ω(ξ)dτ dξ
=
2π
(τ 2 + 2 )(ω(ξ)2 − τ 2 − i0)
2(ω(ξ) − i)
iω(ξ)
1
=
−
.
2(ω(ξ)2 + 2 ) 2(ω(ξ)2 + 2 )
(6.1)
(6.2)
Now the real part of (6.2) equals − 2(ω(ξ)12 +2 ) . Therefore
|(Ω|S+ Ω)| +
S
(Ω|S+ Ω) Z
v(ξ)
a∗ (ξ)dξ exp −
ω(ξ) − i
Z
1
|v(ξ)|2
× exp −
dξ .
2 ω(ξ)2 + 2
= exp
Z
v(ξ)
a(ξ)dξ
ω(ξ) + i
!
Therefore, if
R
|v(ξ)|2
ω(ξ)2 dξ
< ∞, then
If
R
|v(ξ)|2
ω(ξ)2 dξ
v(ξ) ∗
a (ξ)dξ exp −
ω(ξ)
Z
|v(ξ)|2
× exp −
dξ .
ω(ξ)2
+
SGL
= s− lim S+ = exp
&0
Z
+
= ∞, then SGL
does not exist.
Z
v(ξ)
a(ξ)dξ
ω(ξ)
!
−
+
An analogous computation yields SGL
= SGL
. Therefore,
+∗ −
SGL := SGL
SGL = 1l.
One can also set v(t, ξ) := e−|t| v(ξ). Then v(τ, ξ) =
2
τ 2 +2 v(ξ).
Therefore,
!
Z
Z
S
v(ξ)2
v(ξ)2 ∗
= exp i
a
(ξ)dξ
exp
−i
a(ξ)dξ .
(Ω|S Ω)
2 + τ 2
2 + τ 2
This converges strongly to 1l.
Infrared problem
A classical particle travels with trajectory t 7→ ~y (t), and a constant profil f (~x). Then its
current equals
J 0 (t, ~x) = f (~x − ~y (t)), J i (t, ~x) = f (~x − ~y (t))
~y (t)
.
dt
Assume that ~y (t) = t~v ± for ±t > 0. Then
Z
~
µ
J (k) =
J µ (t, ~x)e−ik~x+ik0 t dxdt
i(1, ~v+ )µ
i(1, ~v− )µ
= −
+
f (~k)
~k~v+ − k0 − i0 ~k~v− − k0 + i0
ipµ+
ipµ−
= −
+
f (~k),
kp+ − i0 kp− + i0
where p = √
m
(1, ~v ± ).
1−(~v ± )2
Consider photons of mass m ≥ 0 coupled to the current J µ . Then the scattering operator
is well defined if
Z
Jµ (k)J µ (k)
~k 2 + m2 − k 2 − i0
0
dk
(6.3)
is finite. This is the case if m > 0.
If m = 0, then (6.3) is infinite. If we want to define the scattering operator in this case, we
need to consider different representations of the CCR for the incoming and outgoing photons.
Chapter 7
Charged scalar bosons
221
In this chapter we consider again the Klein-Gordon equation
(−2 + m2 )ψ(x) = 0.
This time we will quantize the space of its complex solutions.
The analysis of this equation is essentially a trivial generalization of its real case. Instead
of smooth space-compact real solutions we consider smooth space-compact complex solutions.
However, the formalism used to quantize in the complex case is different from the real case,
therefore we devote to it a separate chapter.
Another difference between the real and complex case is the possibility to include an external
4-potential Aµ (x) and to consider the equation
− (∂µ − iAµ (x)) (∂ µ − iAµ (x)) + m2 ψ(x) = 0.
Space of solutions
Recall from the previous chapter that YKG denotes the space of smooth space-compact
complex solutions of the Klein-Gordon equation
(−2 + m2 )ζ = 0
(7.1)
Clearly, the space YKG is equipped with the complex conjugation ζ 7→ ζ and a U (1) symmetry
ζ 7→ eiθ ζ.
Complex classical fields
#
We will also consider the space dual to YKG , denoted YKG
. In particular, for x ∈ R1,3 , we
introduce the functionals on YKG given by
1
hψ(x)|ζi := √ ζ(x),
2
1
hη(x)|ζi := √ ζ̇(x),
2
Clearly,
1
hψ(x)|ζi := √ ζ(x),
2
1
hη(x)|ζi := √ ζ̇(x).
2
Z
ψ(t, ~x) =
Z
Ḋ(t, ~x − ~y )ψ(0, ~y )d~y +
D(t, ~x − ~y )η(0, ~y )d~y .
Hermitian form
Just as for the real Klein-Gordon equation, we can define the symplectic (bilinear) form
ζ1 ωζ2 . As the basic structure it is however more natural to choose the hermitian form iζ 1 ωζ2 .
In fact, on YKG we have a conserved current
µ
µ
µ
µ
J (x) = ij (ζ 1 , ζ2 , x) := i ∂ ζ1 (x)ζ2 (x) − ζ1 (x)∂ ζ2 (x) .
The flux of J µ across an arbitrary space-like subspace S of codimension 1 defines a hermitian
form on YKG :
Z
J µ (ζ 1 , ζ2 , x)dsµ (x)
ZS = i
ζ̇1 (t, ~x)ζ2 (t, ~x) − ζ1 (t, ~x)ζ̇2 (t, ~x) d~x.
iζ 1 ωζ2 =
(7.2)
Poisson structure
The imaginary part of the hermitian form (7.2) is a symplectic form and leads to a Poisson
bracket on functions on YKG . To describe it, it is sufficient to restrict oneself to operators for
equal times, since they (in some sense) generate all functions on YKG . The only non-vanishing
equal-time Poisson brackets are
{ψ(t, ~x), η(t, ~y )} = {ψ(t, ~x), η(t, ~y )} = δ(~x − ~y ).
Using (7.3) we obtain
{ψ(x), ψ(y)} = {ψ(x), ψ(y)} = 0,
{ψ(x), ψ(y)} = D(x − y).
(7.3)
Spatially smeared fields
We can use the charged symplectic form to pair distributions and solutions. For ζ ∈ YKG
we introduce also the functional on YKG given by
hψ((ζ))|ρi := −iζωρ,
hψ((ζ))|ρi := iζωρ,
ρ ∈ YKG .
Note that
{ψ((ζ1 )), ψ((ζ2 ))} = {ψ((ζ1 )), ψ((ζ2 ))} = 0,
{ψ((ζ1 )), ψ((ζ2 ))} = ζ 1 ωζ2 .
Space-time smeared fields
We can also smear fields with space-time functions f ∈ Cc∞ (R1,3 , C):
Z
ψ[f ] :=
f (x)ψ(x)dx,
Z
ψ[f ] :=
f (x)ψ(x)dx.
Clearly, ψ[f ] = ψ((Df )), ψ[f ] = ψ((Df )),
{ψ[f1 ], ψ[f2 ]} = {ψ[f1 ], ψ[f2 ]} = 0,
Z Z
{ψ[f1 ], ψ[f2 ]} =
f1 (x)D(x − y)f2 (y)dxdy
Charge and current
We can introduce the current of a single ζ ∈ YKG :
J µ (ζ, x) := J µ (ζ, ζ, x)
= i∂ µ ζ(x)ζ(x) − iζ(x)∂ µ ζ(x).
(Note that the imaginary unit makes it real). It can be written as
J µ (x) = i∂ µ ψ(x)ψ(x) − iψ(x)∂ µ ψ(x).
It satisfies
∂µ J µ (x) = 0.
The total charge is
Z
Q=
J 0 (0, ~x)d~x.
We have the Hamiltonian and momentum density
~
~
· ∇ψ(x)
+ m2 ψ(x)ψ(x),
H(x) = η(x)η(x) + ∇ψ(x)
~
~
~
+ ∇ψ(x)η(x).
P(x)
= η(x)∇ψ(x)
Acting on ζ ∈ YKG , the Hamiltonian density equals
2
~
H(ζ, x) = |ζ̇(x)|2 + |∇ζ(x)|
+ m2 |ζ(x)|2 .
The total Hamiltonian and momentum are
Z
H = H(0, ~x)d~x,
P~ =
Z
~ ~x)d~x.
P(0,
The Poisson brackets between the Hamiltonian, momentum and charge vanish.
Plane waves
We would like to diagonalize simultaneously of the Hamiltonian, the momentum, the charge,
and the symplectic form.
It will be convenient to use different letters for the generic notation of the energy-momentum
in the neutral and charged case. In the charged case, the momentum will be denoted generically
p
by p and the energy by E = E(~p) = p~2 + m2 (which were denoted k and ω(~k) in the neutral
case.)
For p ∈ R1,3 satisfying p2 + m2 = 0 (which is equivalent to |p0 | = E(~p)), we define the
corresponding plane waves
Φ(p) =
1
p
eixp .
(2π)3/2 2E(~p)
Note that
iΦ(p)ωΦ(p0 ) = 0,
iΦ(p)ωΦ(p0 ) = sgnp0 δ(~p − p~0 ), sgnp0 p00 > 0,
iΦ(p)ωΦ(p0 ) = 0,
sgnp0 p00 < 0.
For plane waves with negative frequency, we will also use the alternative notation
Ξ(p) := Φ(−p).
(7.4)
Plane wave functionals
Define the following functionals on YKG :
a(p) = ψ(( − iΦ(p)))
!
Z r
3
E(~p) −i~xp~
i
= (2π)− 2
e
ψ(0, ~x) − p
e−i~p~x η(0, ~x) d~x,
2
2E(~p)
a(p) = ψ((iΦ(p)))
!
Z r
3
E(~p) i~xp~
i
e ψ(0, ~x) + p
ei~p~x η(0, ~x) d~x.
= (2π)− 2
2
2E(~p)
Diagonalization
{a(p), a(p0 )} = {a(p), a(p0 )} = 0,
{a(p), a(p0 )} = isgnp0 δ(~p − p~0 ), sgnp0 p00 > 0,
{a(p), a(p0 )} = 0,
Z
H =
d~p
sgnp0 p00 < 0,
X
E(~p)a(p)a(p),
p0 =±E(~
p)
P~ =
Z
d~p
X
±~pa(p)a(p),
p0 =±E(~
p)
Z
Q =
d~p
X
±a(p)a(p).
p0 =±E(~
p)
Z
iζ 1 ωζ2 =
d~p
X
p0 =±E(~
p)
±ha(p)|ζ1 iha(p)|ζ2 i.
The fields can be written as
− 32
Z
X
ψ(x) = (2π)
p0 =±E(~
p)
Z
X
=
d~p
p
eipx a(p)
2E(~p)
Φ(p)a(p)d~p
p =±E(~
p)
=
Z0 Φ(p)a(p) + Ξ(p)a(−p) d~p.
Thus, every ζ ∈ YKG can be written as
− 32
ζ(x) = (2π)
X
p0 =±E(~
p)
Z
d~p
p
eipx ha(p)|ζi.
2E(~p)
Positive/negative frequency solutions
(±)
YKG were defined in Section 3. Every ζ ∈ YKG can be uniquely decomposed as ζ =
(±)
ζ (+) + ζ (−) with ζ (±) ∈ YKG . For ζ1 , ζ2 ∈ YKG we define the scalar product
(+)
(+)
(−)
(−)
(ζ1 |ζ2 ) := iζ1 ωζ2 − iζ1 ωζ2
Z
Z
ha(p)|ζ1 iha(p)|ζ2 id~p + ha(−p)|ζ1 iha(−p)|ζ2 id~p,
=
(+)
(−)
where p = (E(~p), p~). We set ZKG = ZKG ⊕ ZKG to be the completion of YKG in this scalar
(+)
(−)
product. The ortochronous Poincaré group leaves ZKG and ZKG invariant.
Quantization of free fields
In this section we describe the quantization of the complex Klein-Gordon equation. As usual,
we will use the “hat” to denote the quantized objects.
In principle, we could quantize it as a pair of real Klein-Gordon fields. However, we will use
a different formalism, adapted to the complex (charged) case.
We want to construct a Hilbert space H, a positive self-adjoint operator H called the
Hamiltonian, a normalized vector Ω, which is a ground state of Ĥ and a distribution
R1,3 3 x 7→ ψ̂(x),
(7.5)
which smeared with Cc∞ (R1,3 , C) functions has values in closed operators satisfying
1.
(−2 + m2 )ψ̂(x) = 0,
[ψ̂(x), ψ̂ ∗ (y)] = iD(x − y).
[ψ̂(x), ψ̂(y)] = 0.
2. eitĤ ψ̂(x0 , ~x)e−itĤ = ψ̂(x0 + t, ~x).
3. Ω belongs to the domain of polynomials in smeared out ψ̂(x), ψ̂ ∗ (x) and is cyclic for these
operators.
An alternative equivalent formulation of the quantization program uses the smeared fields
instead of point fields. Instead of (3.11) we look for an antilinear function
YKG 3 ζ 7→ ψ̂((ζ))
with values in closed operators such that
1. ’
[ψ̂((ζ1 )), ψ̂ ∗ ((ζ2 ))] = iζ 1 ωζ2 .
(7.6)
[ψ̂((ζ1 )), ψ̂((ζ2 ))] = 0.
2. ’ ψ̂((r(t,~0) ζ)) = eitĤ ψ̂((ζ))e−itĤ .
3. ’ Ω belongs to the domain of polynomials in ψ((ζ)), ψ ∗ ((ζ)) and is cyclic for these operators.
One can pass between these two versions of the quantization problem by
Z ζ̇(t, ~x)ψ̂(t, ~x) − ζ(t, ~x)η̂(t, ~x) d~x,
ψ̂((ζ)) =
˙
where η̂(x) := ψ̂(x).
(7.7)
(+)
The above problem has a solution unique up to a unitary equivalence. We set H = Γs (ZKG ⊕
(−)
(+)
ZKG ). Creation/annihilation operators on ZKG will be denoted â∗ /â; creation/annihilation
(−)
operators on ZKG will be denoted b̂∗ /b̂. For plane waves we use the notation
â∗ (p) = â∗ (−iΦ(p)),
p0 = E(~p);
b̂∗ (p) = b̂∗ (iΦ(−p)) = b̂∗ (iΞ(p)),
Ω will be the Fock vacuum.
p0 = −E(~p).
We set
− 32
Z
d~p
ipx
p
ψ̂(x) = (2π)
e â(p) + e
2E(~p)
Z ∗
=
Φ(p)a(p) + Ξ(p)b (p) d~p.
b̂ (−p)
−ipx ∗
The Hamiltonian, the momentum and the total charge are
Z ∗
∗
Ĥ :=
â (p)â(p) + b̂ (p)b̂(p) E(~p)d~p,
Z ~
∗
∗
P̂ :=
â (p)â(p) + b̂ (p)b̂(p) p~d~p,
Z ∗
∗
Q̂ :=
â (p)â(p) − b̂ (p)b̂(p) d~p.
Note that the whole orthochronous Poincaré group acts unitarily on H by U (a, Λ) :=
Γ r(a,Λ)
, with
ZKG
U (a, Λ)ψ̂(x)U (a, Λ)∗ = ψ̂ (a, Λ)x .
Note the identities
(Ω|ψ̂(x)ψ̂ ∗ (y)Ω) = −iD(+) (x − y),
(Ω|T(ψ̂(x)ψ̂ ∗ (y))Ω) = −iDc (x − y).
Quantized current
The quantized current density is given by
µ
µ
∗
∗
µ
Jˆ (x) = : i∂ ψ̂ (x)ψ̂(x) − iψ̂ (x)∂ ψ̂(x) :.
It satisfies
∂µ Jˆµ (x) = 0,
(Ω|Jˆµ (x)Ω) = 0,
[Jˆµ (t, ~x), ψ̂(t, ~y )] = −ψ̂(t, ~y )δ(~x − ~y ),
[Jˆµ (t, ~x), ψ̂ ∗ (t, ~y )] = ψ̂ ∗ (t, ~y )δ(~x − ~y ),
[Jˆµ (t, ~x), Jˆν (t, ~y )] = 0
The total charge equals
Z
Q̂ =
Jˆ0 (0, ~x)d~x.
The current at point x can be defined without the use of the Wick ordering, by the so-called
point-splitting method.
If g ∈ Cc∞ (R3 ) with
R
g(~x)d~x = 1, then
Z
µ
µ ∗
∗
µ
ˆ
J (x) = lim g(~y /) i∂ ψ̂ (x + ~y )ψ̂(x) − iψ̂ (x + ~y )∂ ψ̂(x) d~y − −3 .
&0
Axioms
For f ∈ Cc∞ (R1,3 , C) we set
Z
ψ̂[f ] :=
∗
ψ̂ [f ] :=
ψ̂(x)f (x)dx,
Z
ψ̂ ∗ (x)f (x)dx.
We obtain an operator valued distribution satisfying the Wightman axioms with D :=
(+)
(−)
Γfin
s (ZKG ⊕ ZKG ).
For an open set O ⊂ Rd we set
n
o00
∗
∞
F(O) := exp iψ̂ [f ] + iψ̂[f ] : f ∈ Cc (O, C) .
We define A(O) to be the subalgebra of F(O) fixed by the automorphism eiθQ̂ · e−iθQ̂ . The
algebras A(O) satisfy the Haag-Kastler axioms.
External potential
Let us go back to the classical theory. Let R1,3 3 x 7→ A(x) ∈ R1,3 be a given function.
The (complex) Klein-Gordon in an external potential A is
−(∂µ − ieAµ (x))(∂ µ − ieAµ (x)) + m2 ζ(x) = 0.
(7.8)
If ζ satisfies (7.8) and R1,3 3 x 7→ χ(x) ∈ R is an arbitrary function, then eiχ ζ satisfies (7.8)
with A replaced with A + ∇χ. If we introduce the fields R1,3 3 x 7→ ψ(x), ψ(x), then we can
rewrite (7.8) as
−(∂µ − ieAµ (x))(∂ µ − ieAµ (x)) + m2 ψ(x) = 0.
(7.9)
The Lagrangian
(7.9) can be obtained as the Euler-Lagrange of a variational problem. The Lagrangian
density can be taken as
L(x) = −(∂µ − ieAµ (x)) ψ(x) (∂ µ − ieAµ (x)) ψ(x) − m2 ψ(x)ψ(x),
The Euler-Lagrange equations
∂ψ L − ∂µ
yield (7.9).
∂L
=0
∂(∂µ ψ)
(7.10)
The Hamiltonian
Let us introduce the variable conjugate to ψ(x):
η(x) :=
∂L
= ∂0 ψ(x) − ieA0 (x)ψ(x).
∂0 ψ(x)
We have the equal-time Poisson-brackets (7.3). The Hamiltonian density
H(x) = η(x)η(x) − ieA0 (x) ψ(x)η(x) − η(x)ψ(x)
+(∂i − ieAi (x))ψ(x)(∂i − ieAi (x))ψ(x) + m2 ψ(x)ψ(x).
The Hamiltonian
Z
H(t) =
d~xH(t, ~x)
can be used to generate the dynamics
ψ̇(t, ~x) = {H(t), ψ(t, ~x)},
η̇(t, ~x) = {H(t), η(t, ~x)}.
We can also introduce the stress-energy tensor density
Tµν (x) := ∂µ ψ(x)∂ν ψ(x) − gµν L(x),
which is conserved
∂µ Tµν (x) = 0.
We have
H(x) = T00 (x), Pi (x) = T0i (x).
Quantization in the presence of external potentials
We are looking for quantum fields satisfying
−(∂µ − ieAµ (x))(∂ µ − ieAµ (x)) + m2 ψ̂(x) = 0.
coinciding with the free fields for t = 0. Such fields are given by
Z t
Z 0
Ĥ(s)ds ,
Ĥ(s)ds ψ̂(0, ~x)Texp −i
ψ̂(t, ~x) := Texp −i
t
0
(7.11)
where the Hamiltonian Ĥ(t), and the corresponding Hamiltonian in the interaction picture
equal
Z
Ĥ(t) =
d~x: η̂ ∗ (~x)η̂(~x) − ieA0 (t, ~x) ψ̂ ∗ (~x)η̂(~x) − η̂ ∗ (~x)ψ̂(~x)
+(∂i − ieAi (t, ~x))ψ̂ ∗ (~x)(∂i − ieAi (t, ~x))ψ̂(~x)
2 ∗
+m ψ̂ (~x)ψ̂(~x) :,
Z
µ
2~
2 ∗
ˆ
ĤInt (t) =
d~x: ieAµ (t, ~x)JInt (t, ~x) + e A(t, ~x) ψ̂Int (t, ~x)ψ̂Int (t, ~x) :
Z
µ
∗
=
d~x: ieAµ (t, ~x)JˆInt
(t, ~x) + e2 A(t, ~x)2 ψ̂Int
(t, ~x)ψ̂Int (t, ~x)
2
2 ∗
+e A0 (t, ~x) ψ̂Int (t, ~x)ψ̂Int (t, ~x) :.
(7.12)
(7.13)
Note that the above Hamiltonians should be understood as formal expressions. They need not
correspond to (time-dependent) self-adjoint operators, and if they do, it is not guaranteed that
they generate a unitary dynamics. Some of the terms in the following perturbation expansions
may turn out to be infinite – however, they lead to well defined scattering cross-sections.
2N -point Green’s functions.
For yN , . . . y1 , xN , . . . , x1 , the 2N point Green’s function are defined as follows:
∗
∗
G ψ̂ , y1 , · · · ψ̂ , yN , ψ̂, xN , · · · ψ̂, x1
+
∗
∗
:= Ω |T ψ̂ (y1 ) · · · ψ̂ (yN )ψ̂(xN ) · · · ψ̂(x1 ) Ω− .
Feynman rules for Green’s functions.
We have 1 kind of lines and 2 kind of vertices. Each line has an arrow. At each vertex
two lines meet, one with an arrow pointing towards, one with an arrow pointing away from
hthe vertex. The 1-photon vertex is denoted by an attached wavy line, the 2-photon vertex,
by attached two wavy lines. The names of vertices are motivated by the full QED.
1. In the nth order we draw all possible topologically distinct Feynman diagrams with n
vertices and 2N external lines.
2. To each line we associate a propagator
Dc (p) =
−i
.
p2 + m2 − i0
For internal lines we integrate over the variables with the measure
d4 p
(2π)4 .
3.
(i) To each 1-photon vertex we associate a factor
ν +
−
ν +
− −
e(p+
ν Â (p − p ) + Â (p − p )pν ).
(ii) To each 2-photon vertex we associate a factor
−2ie2 (Aν Aν )(p+ − p− ).
The derivation of the Feynman rules for charged scalar bosons is more complicated than for
neutral bosons, since it involves not just expectation values of “configuration space fields”, but
also the “momentum space fields”:
(Ω|T(ψ̂I (x)ψ̂I∗ (y))Ω) = −iDc (x − y),
(Ω|T(η̂I (x)ψ̂I∗ (y))Ω) = −i∂x0 Dc (x − y),
(Ω|T(ψ̂I (x)η̂I∗ (y))Ω) = −i∂y0 Dc (x − y),
(Ω|T(η̂I (x)η̂I∗ (y))Ω) = −i∂x0 ∂y0 Dc (x − y) − iδ(x − y).
Scattering operator
Z
∞
S = Texp −i
ĤInt (t)dt .
−∞
Scattering amplitudes: matrix elements of the scattering operator between plane waves
+
+
−
−
+0
+0
−0
−0
Φ(p1 ) · · · Φ(pN + )Ξ(p1 ) · · · Ξ(pN +0 )| S Ξ(pN −0 ) · · · Ξ(p1 )Φ(pN − ) · · · Φ(p1 ) .
For simplicity, we assume that the momenta are distinct.
Feynman rules for scattering amplitudes.
To compute scattering amplitudes with N − incoming and N + outgoing particles we draw
the same diagrams as for N − + N + -point Green’s functions, where we adopt the convention
that the incoming lines are drawn on the right and outgoing lines on the left. The rules are
changed only concerning the external lines.
1. With each incoming external line we associate
1
.
V 2E(~
p)
anti-boson: √ 1 0 .
V 2E(~
p)
(i) charged boson: √
(ii) charged
2. With each outgoing external line we associate
1
.
V 2E(~
p)
anti-boson: √ 1 0 .
V 2E(~
p)
(i) charged boson: √
(ii) charged
Abstract gauge covariance
Let us adopt for a moment an abstract setting. Let R 3 t 7→ H(t) be a time-dependent
Hamiltonian generating the dynamics
Z
t+
U (t+ , t− ) := Texp −i
H(s)ds .
t−
Let t 7→ R(t) be a time dependent family of self-adjoint operators. Introduce the timedependent gauge transformation
Z
t
W (t) := Texp −i
R(s)ds .
−∞
Assume that W (t) converges to identity as t → −∞. Then
Z t
∗
W (t+ )U (t+ , t− )W (t− ) = Texp −i
HR (s)ds .
−∞
where the gauge-transformed Hamiltonian equals
HR (t) := W (t)H(t)W ∗ (t) + R(t).
(7.14)
Gauge covariance
Let us go back to the setting of quantized charged scalar fields. Let R1,3 3 x 7→ χ(x) ∈ R
be a smooth function that decays in all directions. We define the gauge transformation
Z
W (χ, t) := exp −i d~x χ(t, ~x)Jˆ0 (~x) .
Z t
Z
= exp −i
ds d~x χ̇(s, ~x)Jˆ0 (~x)
(7.15)
−∞
Z t
Z
0
= Texp −i
ds d~x χ̇(s, ~x)Jˆ (~x) .
−∞
To see the second above identity it is enough to note that [Jˆ0 (~x), Jˆ0 (~y )] = 0, hence we can
replace Texp with exp in (7.15). In the interaction picture we have
WInt (χ, t) := eitĤ0 W (χ, t)e−itĤ0
Z
= exp −i d~x χ(t, ~x)JˆI0 (t, ~x) .
(7.16)
Let Ĥ(A, t), resp. ĤInt (A, t) denote (7.12), resp. (7.13). Let U (A, t+ , t− ), UInt (A, t+ , t− )
and S(A) be the corresponding dynamics, the dynamics in the interaction picture and the
scattering matrix. We have the following identities, which express the gauge covariance:
W (χ, t+ )U (A, t+ , t− )W ∗ (χ, t− ) = U (A + ∇χ, t+ , t− ),
∗
WInt (χ, t+ )UInt (A, t+ , t− )WInt
(χ, t− ) = UInt (A + ∇χ, t+ , t− ).
To see (7.17), we use (7.14) and
∗
W (χ, t)Ĥ(A, t)W (χ, t) +
= Ĥ(A + ∇χ, t).
The second identity follows from the first.
Z
d~xχ̇(t, ~x)Jˆ0 (~x)
(7.17)
(7.18)
Ward(-Takahashi) identities for the scattering operator
Using that WInt (χ, ±∞) = 1l, we see that (7.18) implies
S(A) = S(A + ∇χ).
Differentiating this identity wrt χ and setting χ = 0 we obtain one of versions of the identities
for the scattering operator
∂yµ
∂
∂Aµ (y)
S(A) = 0.
In the momentum representation these identities read
pµ
∂
∂Aµ (p)
S(A) = 0.
Ward(-Takahashi) identities
for Green’s functions
∗ 0
We will write G A; ψ̂ , x1 , · · · , ψ̂, x1 to express the dependence of Green’s functions on
the external potential A. The following property follows from (7.17):
∗ 0
G A; ψ̂ , x1 , · · · , ψ̂, x1
iχ(x01 ) ∗ 0
−iχ(x1 )
= G A + ∇χ; e
ψ̂ , x1 , · · · , e
ψ̂, x1 .
By differentiating with respect to χ(y) and setting χ = 0 we obtain the Ward identities for
Green’s functions in the position representation:
∂
∗ 0
∂yµ
G A; ψ̂ , x1 , · · · , ψ̂, x1
∂Aµ (y)
!
N
N
X
X
0
∗ 0
= i
δ(y − xj ) − i
δ(y − xj ) G A; ψ̂ , x1 , · · · , ψ̂, x1 .
j=1
j=1
In the momentum representation these identities read
∂
∗ 0
kµ
G A; ψ̂ , p1 , , · · · , ψ̂, p1
∂Aµ (k)
∗ 0
∗ 0
= iG A; ψ̂ , p1 − k, · · · , ψ̂, p1 + · · · − iG A; ψ̂ , p1 , · · · , ψ̂, p1 + k .
Chapter 8
Dirac fermions
267
Let γ µ be matrices satisfying
[γ µ , γ ν ]+ = −2g µ,ν ,
γ0 γ µ γ0−1 = γ µ∗ .
In this chapter we study the Dirac equation
(−iγ µ ∂µ + m)ψ(x) = 0
and its quantization. For further reference, let us note the Dirac equation in the momentum
representation:
(γ µ pµ + m)ψ(p) = 0.
We will also consider the Dirac equation in the presence of an external potential Aµ (x):
(−iγ µ (∂µ − ieAµ (x)) + m) ψ(x) = 0.
Representations of Dirac matrices
To find a representation of Dirac matrices, that is to find γ µ , µ = 0, . . . , 3, such that
(γ 0 )2 = 1l, (γ i )2 = −1l,
γ µ γ ν + γ ν γ µ = 0,
(γ 0 )∗ = γ 0 , (γ i )∗ = −γ i ,
we need the space C4 .
i = 1, 2, 3;
0 ≤ µ < ν ≤ 3;
i = 1, 2, 3.
One of the most common representations is the so-called Dirac representation
"
#
"
#
1
0
0
~
σ
γ0 =
, ~γ =
.
0 −1
−~σ 0
Here is the Majorana representation:
"
#
"
#
"
#
"
#
0
−1
0
σ
−1
0
0
σ
1
3
γ0 = i
, γ1 = i
, γ2 = i
, γ3 = i
,
1 0
σ1 0
0 1
σ3 0
and the spinor representation:
"
γ0 =
0 1
1 0
#
"
, ~γ =
0 −~σ
~σ
0
#
.
We also introduce the spin operator
1
σ µν := [γ µ , γ ν ].
2
In the Dirac representation
"
σ 0i =
1 ij
σ ijk =
2
0 σi
σi 0
"
σk 0
0 σk
#
,
#
.
Special solutions and Green’s functions
Note the identity
−(−iγ∂ + m)(iγ∂ − m) = −2 + m2 .
Therefore, if
(−2 + m2 )ζ(x) = 0,
then (iγ µ ∂µ + m)ζ(x) is a solution of the homogeneous Dirac equation:
(−iγ µ ∂µ + m)(iγ µ ∂µ + m)ζ(x) = 0.
In particular, we have special solutions of the homogeneous Dirac equation
S (±) (x) = −(iγ∂ + m)D(±) (x),
S(x) = −(iγ∂ + m)D(x).
We have suppS ⊂ J.
If
(−2 + m2 )ζ(x) = δ(x),
then −(iγ µ ∂µ − m)ζ(x) is a Green’s function of the Dirac equation, that is
(−iγ∂ + m)(iγ µ ∂µ + m)ζ(x) = δ(x).
In particular, a special role is played by the Green functions
S ± (x) = −(iγ∂ + m)D± (x),
S c (x) = −(iγ∂ + m)Dc (x).
We have suppS ± ⊂ J ± .
Note the identities
S(x) = −S(−x) = S (+) (x) + S (−) (x)
= S + (x) − S − (x),
S (+) (x) = S (−) (−x),
S + (x) = S − (−x) = θ(x0 )S(x)
S − (x) = θ(−x0 )S(x)
S c (x) = S c (−x) = θ(x0 )S (−) (x) − θ(−x0 )S (+) (x).
Solution of the inhomogeneous equation
∞
(R1,d , C4 )
Theorem 8.1 For any f ∈ Cc∞ (R1,3 , C4 ) there exist unique functions ζ ± ∈ C±sc
solutions of
(−iγ∂ + m)ζ ± = f.
They are given by
±
±
Z
ζ (x) = (S f )(x) =
R1,3
S ± (x, y)f (y)dy.
(8.1)
The Cauchy problem
We set αi = iγ 0 γ i , i = 1, . . . , 3, and β := γ 0 , obtaining
β 2 = 1l, (αi )2 = 1l,
βαi + αi β = 0, αi αj + αj αi = 0,
β ∗ = β, αi∗ = αi ,
In the Dirac representation we have
"
β=
1
0
0 −1
#
"
, α
~=
i = 1, . . . , 3;
1 ≤ i < j ≤ 3;
i = 1, . . . , 3.
0 ~σ
~σ 0
#
.
We can rewrite the Dirac equation in the hamiltonian form
i∂t ζ(t, ~x) = Dζ, D := αi pi + mβ.
∞
Theorem 8.2 Let ϑ ∈ Cc∞ (R3 , C4 ). Then there exists a unique ζ ∈ Csc
(R1,3 ) that solves
the Dirac equation with initial conditions ζ(0, ~x) = ϑ(~x). It satisfies suppζ ⊂ J(suppϑ) and
is given by
Z
S(t, ~x − ~y )βϑ(~y )d~y .
ζ(t, ~x) =
R3
Conserved current
∞
Let YD be the space of space-compact solutions of the Dirac equation, that is ζ ∈ Csc
(R1,3 , C4 )
satisfying (−iγ µ ∂µ + m)ζ = 0. Let ζ1 , ζ2 ∈ C ∞ (R1,3 , C4 ). Set
j µ (ζ1 , ζ2 , x) := ζ1 (x)βγ µ ζ2 (x).
We easily check that
∂µ j µ (x) = (−iγ∂ + m)ζ1 (x)βζ2 (x) − ζ1 (x)β(−iγ∂ + m)ζ2 (x),
Therefore, if ζ1 , ζ2 ∈ YD , then
∂µ j µ (x) = 0,
Scalar product
For ζ1 , ζ2 ∈ YD , the flux of j µ across a space-like hypersurface does not depend on its choice.
It defines a scalar product on YD
Z
ζ 1 · ζ2 =
j µ (ζ1 , ζ2 , x)dsµ (x).
S
In terms of the Cauchy data we have
Z
ζ 1 · ζ2 =
ζ1 (t, ~x)ζ2 (t, ~x)d~x.
Solutions parametrized by test functions
For any f ∈ Cc∞ (R1,3 , C4 ), we will write
Z
Sf (x) := S(x − y)f (y)dx.
Theorem 8.3 1. For any f ∈ Cc∞ (R1,3 , C4 ), Sf ∈ YD .
2. Every element of YD is of this form.
RR
f1 (x)βS(x, y)f2 (y)dxdy.
3. Sf1 · Sf2 =
4. If suppf2 × suppf2 , then
Sf1 · Sf2 = 0.
Classical Dirac field
We will also consider the space dual to YD , denoted YD# . In particular, for x ∈ R1,3 ,
ψ(x), ψ(x) will denote the functionals on YD with values in C4 given by
hψ(x)|ζi := ζ(x),
Clearly,
hψ(x)|ζi := ζ(x).
Z
S(t, ~x − ~y )βψ(0, ~y )d~y .
ψ(t, ~x) =
It is convenient to introduce the symbol
ψ̃(x) := βψ(x).
(In a large part of the physics literature, ψ̃ is denoted ψ.)
Spatially smeared fields
We can use the scalar product to pair distributions and solutions. For ζ ∈ YD we introduce
also the functional on YD given by
hψ((ζ))|ρi := ζ · ρ,
hψ((ζ))|ρi := ζ · ρ,
ρ ∈ YD .
Clearly,
Z
ψ((ζ)) =
ψ(t, ~x)ζ(t, ~x)d~x,
Z
ψ((ζ)) =
ψ(t, ~x)ζ(t, ~x)d~x.
Space-time smeared fields
We can also introduce
Z
f (x)ψ(x)dx = ψ((Sf )),
ψ[f ] :=
Z
ψ[f ] :=
f (x)ψ(x)dx = ψ((Sf )).
Current and charge
We can introduce the current
J µ (ζ, x) := j µ (ζ, ζ, x)
= ζ(x)βγ µ ζ(x).
We can write it as
J µ (x) = ψ(x)βγ µ ψ(x).
It satisfies
∂µ J µ (x) = 0.
The total charge is
Z
Q=
J 0 (0, ~x)d~x.
Plane waves
We would like to diagonalize simultaneously the following commuting operators on the space
~ and the polarization −iijk σij ∇k .
YD : Dirac Hamiltonian D, the momentum −i∇
To this end we introduce for p = (E(~p), p~) with E = E(~p) =
" #
" #
1
0
χ+ :=
, χ− :=
,
0
1
u(p, s) =
u(p, s) =
√
√
"
E+m
−~σ p~
E+m χs
χs
p~2 + m2 ,
#
, p0 = E > 0;
~σ p~
E+m χs
"
E+m
χs
p
#
, p0 = −E < 0.
We also introduce the corresponding plane waves
Φ(p, s) =
u(p, s)
p
eipx .
3/2
(2π)
2E(~p)
We have
Φ(p) · Φ(p0 ) = δ(~p − p~0 ), sgnp0 p00 > 0,
iΦ(p) · Φ(p0 ) = 0,
sgnp0 p00 < 0.
For negative frequencies, that is p0 = −E(~p), it is customary to have an alternative notation
#
"
~σ p~
√
χ
s
,
v(p, s) = u(−p, s) = E + m E+m
χs
v(p, s)
p
Ξ(p, s) = Φ(−p, s) =
e−ipx .
(2π)3/2 2E(~p)
We introduce
a(p, s) := ψ((Φ(p, s)))
Z
− 32
= (2π)
d~pu(p, s)ei~p~x ψ(0, ~x)
We have
ψ(x) =
=
X
X
p0 =±E(~
p)
s
X
s
− 23
(2π)
− 23
(2π)
Z
Z
d~pu(p, s)eipx a(p, s)
ipx
−ipx
d~p u(p, s)e a(p, s) + v(p, s)e
a(−p, s)
We have
ha(p, s)|Dζi = p0 ha(p, s)|ζi,
~
ha(p, s)|(−i∇)ζi
= p~ha(p, s)|ζi,
ha(p, s)|(−iijk σij ∇k )ζi = |~p|sha(p, s)|ζi.
ζ 1 · ζ2 =
XZ
X
p0 =±E(~
p)
ha(p, s)|ζ1 iha(p, s)|ζ2 id~p.
s
Clearly, every ζ ∈ YD can be written as
Z
X X
3
ζ(x) =
(2π)− 2 d~pu(p, s)eipx ha(p, s)|ζi.
p0 =±E(~
p)
s
(±)
YD will denote the subspace of YD consisting of positive, resp. negative frequency solutions.
(±)
Every ζ ∈ YD can be uniquely decomposed as ζ = ζ (+) + ζ (−) with ζ (±) ∈ YD .
(−)
We change the complex structure on YD . For ζ1 , ζ2 ∈ YD we define the scalar product
(+)
(+)
(−)
(−)
(ζ1 |ζ2 ) := ζ1 · ζ2 + ζ1 · ζ2
Z
Z
ha(p)|ζ1 iha(p)|ζ2 id~p + ha(−p)|ζ1 iha(−p)|ζ2 id~p,
=
(+)
(−)
where p = (E(~p), p~). We set ZD = ZD ⊕ ZD
to be the completion of YD in this scalar
(+)
(−)
product. The ortochronous Poincaré group leaves ZD and ZD invariant.
Quantization of free fields
We would like to describe the quantization of the Dirac equation. As usual, we will use the
“hat” to denote the quantized objects.
We need to use the formalism of quantization of charged fermionic systems.
We want to construct a Hilbert space H, a positive self-adjoint operator Ĥ called the
Hamiltonian, a normalized vector Ω, which is a ground state of Ĥ and a distribution
R1,3 3 x 7→ ψ̂(x),
(8.2)
which smeared with Cc∞ (R1,3 , C) functions has values in bounded operators tensored with C4
satisfying
˜
1. Setting ψ̂(x) := γ0 ψ̂ ∗ (x), we have
(−iγ∂ + m)ψ̂(x) = 0,
˜
[ψ̂(x), ψ̂(y)]+ = S(x − y),
2. eitĤ ψ̂(x0 , ~x)e−itĤ = ψ̂(x0 + t, ~x).
3. Ω is cyclic for smeared ψ̂(x), ψ̂ ∗ (x).
[ψ̂(x), ψ̂(y)]+ = 0
There exists an alternative equivalent formulation of the quantization program, which uses
the smeared fields instead of point fields. Instead of (3.11) we look for an antilinear function
CYD 3 ζ 7→ ψ̂((ζ))
with values in bounded operators such that
1. ’
[ψ̂((ζ1 )), ψ̂ ∗ ((ζ2 ))]+ = ζ 1 · ζ2 .
[ψ̂((ζ1 )), ψ̂((ζ2 ))]+ = 0.
2. ’
ψ̂((r(t,~0) ζ)) = eitĤ ψ̂((ζ))e−itĤ .
3. ’ Ω is cyclic for ψ̂((ζ)), ψ̂ ∗ ((ζ)).
(8.3)
One can pass between these two versions of the quantization problem by
Z
ψ̂((ζ)) = ζ(t, ~x)ψ̂(t, ~x)d~x.
(8.4)
(+)
The above problem has a solution unique up to a unitary equivalence. We set H = Γs (ZD ⊕
(−)
(+)
(−)
ZD ). For ZD the standard creation/annihilation operators are denoted â∗ , â. For ZD the
standard creation/annihilation operators are denoted b̂∗ , b̂.
â∗ (p) = â∗ (Φ(p)),
p0 = E(~p);
b̂∗ (p) = b̂∗ (Φ(−p)) = b̂∗ (Ξ(p)),
Ω will be the Fock vacuum.
p0 = −E(~p).
− 32
ψ̂(x) = (2π)
XZ
p0 =E(~
p)
s
− 32
+(2π)
d~pu(p, s)eipx â(p, s)
XZ
s
v(p, −s)eipx b̂∗ (p, s).
p0 =−E(~
p)
The Hamiltonian and the momentum are
Z X
∗
∗
Ĥ =
â (p, s)â(p, s) + b̂ (p, s)b̂(p, s) E(~p)d~p,
~
P̂ =
Ŝ =
s
Z X
s
Z X
â (p, s)â(p, s) + b̂ (p, s)b̂(p, s) p~d~p,
∗
∗
∗
∗
|~p|s â (p, s)â(p, s) + b̂ (p, s)b̂(p, s) p~d~p.
s
The whole orthochronous Poincaré group acts unitarily on H.
We have the quantized current density
Jˆµ (x) = :ψ̂ ∗ (x)βγ µ ψ̂(x):.
It satisfies
∂µ Jˆµ (x) = 0,
(Ω|Jˆµ (x)Ω) = 0.
Besides, we obtain the charge operator
Z
XZ ∗
∗
0
ˆ
â (~p, s)â(~p, s) − b̂ (~p, s)b̂(~p, s) d~p.
Q̂ = J (0, ~x)d~x =
s
We have
˜
(Ω|ψ̂(x)ψ̂(y)Ω) = S (+) (x − y),
˜
(Ω|T(ψ̂(x)ψ̂(y))Ω) = S c (x − y).
Axioms
For f ∈ Cc∞ (O, C4 ) we set
Z
ψ̂[f ] :=
∗
ψ̂ [f ] :=
ψ̂(x)f (x)dx,
Z
ψ̂ ∗ (x)f (x)dx.
(+)
We obtain an operator valued distribution satisfying the Wightman axioms with D := Γfin
a (ZD ⊕
(−)
ZD ).
For an open set O ⊂ R1,3 we set
F(O) := {ψ̂ ∗ [f ], ψ̂[f ] : f ∈ Cc∞ (O, C4 )}00 .
We define A(O) to be the subalgebra of F(O) fixed by the automorphism eiθQ̂ · e−iθQ̂ . The
algebras A(O) satisfy the Haag-Kastler axioms.
External potential
Let R1,3 3 x 7→ A(x) ∈ R1,3 be a given function. The Dirac equation in an external
potential A is
(−iγ µ (∂µ − ieAµ (x)) + m) ζ(x) = 0
(8.5)
If ζ satisfies (8.5) and R1,3 3 x 7→ χ(x) ∈ R is an arbitrary function, then eiχ ζ satisfies
(7.8) with A replaced with A + ∇χ. If we introduce the fields R1,3 3 x 7→ ψ(x) ∈ C4 , then
we can rewrite (8.5) as
(−iγ µ (∂µ − ieAµ (x)) + m) ψ(x) = 0
(8.6)
(7.9) can be obtained as the Euler-Lagrange of a variational problem. The Lagrangian
density can be taken as
i
µ
µ
ψ̃(x)γ (∂µ − ieAµ (x))ψ(x) − (∂µ + ieAµ (x))ψ̃(x)γ ψ(x)
L(x) =
2
−mψ̃(x)γ µ ψ(x).
The Euler-Lagrange equations
∂ψ L − ∂µ
yield (8.6).
∂L
=0
∂(∂µ ψ)
(8.7)
Quantization in the presence of external potentials
We are looking for quantum fields satisfying
(−iγ µ (∂µ − ieAµ (x)) + m) ψ̂(x) = 0.
(8.8)
coinciding with the free fields for t = 0. Such fields are given by
Z t
Z 0
Ĥ(s)ds ,
Ĥ(s)ds ψ̂(0, ~x)Texp −i
ψ̂(t, ~x) := Texp −i
t
0
where the Hamiltonian Ĥ(t), and the corresponding Hamiltonian in the interaction picture
equal
Z
Ĥ(t) =
Z
ĤInt (t) =
d~x: ψ̂ ∗ (~x)((αi (i∂i + eAi (t, ~x)) + mβ + A0 (t, ~x))ψ̂(~x) :,
µ
d~xeAµ (t, ~x)JˆInt
(t, ~x).
2N -point Green’s functions.
For yN , . . . y1 , xN , . . . , x1 , the 2N point Green’s function are defined as follows:
∗
∗
G ψ̂ , y1 , · · · ψ̂ , yN , ψ̂, xN , · · · ψ̂, x1
+
∗
∗
:= Ω |T ψ̂ (y1 ) · · · ψ̂ (yN )ψ̂(xN ) · · · ψ̂(x1 ) Ω− .
Feynman rules for Green’s functions.
1. In the nth order we draw all possible topologically distinct Feynman diagrams with n
vertices and external lines. All the charged lines have a natural arrow.
2. To each line we associate a propagator S(p) =
over the variables with the measure
−ipγ+m
p2 +m2 −i0
For internal lines we integrate
4
d p
(2π)4 .
3. To each vertex we associate a factor −ieγµ Aµ (p+ − p− ).
4. If two graphs differ only by an exchange of two fermionic lines, there is an additional factor
(−1) for one of them. This implies, in particular, that loops have an additional factor
(−1).
Scattering operator
Z
∞
S = Texp −i
ĤInt (t)dt .
−∞
Scattering amplitudes: matrix elements of the scattering operator between plane waves
+ +
− −
+0 +0
−0 −0
Φ(p1 , s1 ) · · · Ξ(p1 , s1 ) · · · | S · · · Ξ(q1− , s1 ) · · · Φ(p1 , s1 ) .
For simplicity, we assume that the momenta are distinct.
Feynman rules for scattering amplitudes.
To compute unrenormalized scattering amplitudes with N − incoming and N + outgoing
particles we draw the same diagrams as for N − + N + -point Green’s functions, where we adopt
the convention that the incoming lines are drawn on the right and outgoing lines on the left.
The arrow on photon lines can be now chosen in the natural way. The rules are changed only
concerning the external lines.
1. With each incoming external line we associate
q
m
(i) fermion:
E(~
p)V u(p, s).
q
m
(ii) anti-fermion:
E(~
p)V ṽ(p, s).
2. With each outgoing external line we associate
q
m
(i) fermion:
E(~
p)V ũ(p, s).
q
m
(ii) anti-fermion:
E(~
p)V v(p, s).
Each incoming and outgoing antifermion has an additional factor (−1). (This follows from
the rule (4) above).