The Geometry of
Quantization (II)
Ennio Gozzi
Dept. of Physics
Theoretical Section
University of Trieste
Strada Costiera 11
Miramare-Trieste
Phys. Rev.Lett. 105, 150606 (2010)
E.G. and C.Pagani
MOTIVATIONS
● Superposition of states is one of the main
feature of Quantum Mechanics (QM). It is at
the heart of typical quantum mechanics effects
like interpherence, entanglement etc.
● In Classical Mechanics (CM) instead we do not
have superpositions and this creates all the
troubles we encounter when we try to couple a
quantum mechanical system with a classical one.
● In this work we shall try to understand what
kills the superposition in Classical Mechanics.
This will shed a new light not only on CM but
also on QM.
Contents
● Koopman and von Neumann Hilbert Space
formulation of classical mechanics (KvN) 1931;
● Path integral counterpart of the operatorial formalism
mentioned above (CPI), which is different from the
quantum one (QPI);
● Geometrical analysis of the space where the path
integral formulation lives. Introduction of two
Grassmann partners θ and θ¯ to time t and of
supervariables;
● Superposition principle in KvN?;
● Local invariances;
● Observables;
● Solution of the problem.
-1-
Classical Mechanics
(Review of Hamiltonian formalism)
a
● Phase space: ϕ ≡ (q1 · · · qn ; p1 · · · pn ), a = 1, · · · , 2n
● Hamiltonian: H(ϕ)
● Equations of motion:
● Poisson brackets:
ϕ˙ a = ω ab ∂b H
{A1 (ϕ), A2 (ϕ)}pb = ∂a A1 ω ab ∂b A2
● Probability density distribution in phase space:
a
microcanonical ρ(ϕ , t) ∝ δ[H − E]
a
canonical ρ(ϕ , t) ∝ exp[−βH]
● Evolution of observables:
∂O
dO
=
+ {O, H}pb
dt
∂t
● Evolution of probability density:
∂ρ
dρ
= 0 =⇒
= −{ρ, H}pb = −∂a ρ ω ab ∂b H
dt
∂t
-2-
Classical Mechanics
in Operatorial Form
Koopman, von Neumann (1931)
●
∂ρ
ˆ
= −{ρ, H} = ∂a Hω ab ∂b ρ ≡ −iLρ
∂t
ˆ = i∂a Hω ab ∂b = i∂q H∂p − i∂p H∂q (Liouvillian)
L
●
⇓
ρ(ϕ , ∆t) = ρ(ϕ − ω ∂b H∆t; 0)
a
a
ab
Same functional form but different arguments.
●
Propagator:
ρ(ϕ, t) =
!
dϕi K(ϕ; t|ϕi ; ti )ρ(ϕi , ti )
K(ϕf ; tf |ϕi ; ti ) = δ[ϕf − φcl (tf ; ϕi , ti )]
φcl (t; ϕi , ti ) is a solution of Hamilton equations of
motion.
-3-
Koopman-von Neumann
(KvN) Postulates
1)
KvN introduced a Hilbert space made up of
complex and square integrable functions on the
phase space ψ(ϕ) ∈ L2 and imposed the following
two postulates:
2): [ˆq, pˆ] = 0
3): |ψ(ϕ)|2 = ρ(ϕ)
Probability density in the phase space.
4): The evolution of ψ is via the Liouvi llian Lˆ
i
∂ψ
ˆ
= Lψ
∂t
ˆ = i∂a Hω ab ∂b = i∂q H∂p − i∂p H∂q
L
● As Lˆ is linear in the derivatives one can derive:
i
5): The observables?
∂ρ
ˆ
= Lρ
∂t
-4-
Classical Dynamics
!
∂ Dynamics
Classical
ˆ
t) = Lψ(ϕ,
t)
i ψ(ϕ,
Same equation !
∂t
 ∂
ˆ
t)
i ∂ ρ(ϕ, t) = Lρ(ϕ,

%
i ∂tψ(q, p, t) = Hψ(q,
p, t)
∂t
same equation!

i ∂ ρ(q, p, t) = Hρ(q,
%
p, t)
ˆ is linear in
Why? L
the
∂t derivatives.
% is first order in ∂q and ∂p :
Why? H
% = −i∂p H(q, p)∂q + i∂q H(q, p)∂p
H
Quantum Dynamics
Quantum Dynamics
!
 i ∂ ψ(x, t) = Hψ(x,
ˆ
t)
∂
 ∂t
%
Different
different i ∂t ψ(x, t) = Hψ(x, t)
Equations
equations 
i ∂ ρ(x, t) = − ! div(ψ ∗ ∇ψ
$ − ψ ∇ψ
$ ∗)
∂t
2m
!2 ∂ 22
∂ V (x) is second order in ∂
% =H
ˆ = − ¯h +
H
−
+ V (x) is second order in x∂x
Reason:
2m ∂x2 2
2
2m ∂x
3
-5-
Classical transition amplitude via path
integral
As the evolution of the ψ and the ρ is the same, the
propagator will be the same:
K(ϕf ; tf |ϕi ; ti ) = δ[ϕf − φcl (tf , ϕi )]
K(ϕf ; tf |ϕi ; ti )
= lim

−1 %
N$
N →∞  j=1


dϕj δ[ϕj − φcl (tj , ϕi )] δ[ϕf

− φcl (tf , ϕi )]
δ[ϕj − φcl (tj , ϕi )] = δ[ϕ˙ a − ω ab ∂b H]tj
!
dλa eiλa [ϕ˙
a
−ω ab ∂b H]
!
det[δba ∂t − ∂b (ω ac ∂c H)]tj
a
d¯
ca dca e−¯ca [δb ∂t −∂b (ω
ac
∂c H)]cb
So in the continuum limit we get
!
! ϕf
D!! ϕDλDcD¯
c exp i dt L"
K(ϕf ; tf |ϕi ; ti ) =
ϕi
L! = λa [ϕ˙ a − ω ab ∂b H] + i¯
ca [δba ∂t − ∂b (ω ac ∂c H)]cb
-6-
Z ≡ "ϕ ; t |ϕ ; t # =
!
D!! ϕ δ(ϕa − φacl (t; ϕi ))
f f i i
Propriet`
aa delle
Propriet`
delle delta
delta di
diDirac:
Dirac:
a
a
a
ab
a
ab
δ(ϕ
c H)
δ(ϕa −
−φφclacl(t;
(t;ϕϕii))
))==δ(
δ(ϕ˙ϕ˙ a−−ωω ab∂∂b bH)
H)det(∂
det(∂t δt cδca−−ωω ab∂∂
b∂
c H)
b∂
Propriet`a delle delta di Dirac:
Path Integral and
!! t
Operatorial
Formalism
Z
cc exp
dt
Z=
= D
D!! ϕDλDcD¯
ϕDλDcD¯
expii
dtL"L"
a
δ(ϕEsponenziazione:
− φacl (t; ϕi )) = δ(ϕ˙ a − ω ab ∂b H) det(∂t δca − ω ab ∂b ∂c H)
Esponenziazione:
!!
ftf
!!
titi
Esponenziazione:
!
! tf
""=
Z● =
iH where:
dtL"
L
=D
+i¯
i¯
−H
L
λλ!!aaϕDλDcD¯
ϕϕ
˙˙aa +
ccaac˙c˙aaexp
−
ti
ab ∂ H + i¯
H=
= λλaaωωab
Hcdd
H
∂bbH + i¯
ccaaωωabab∂∂bb∂∂ddHc
a + i¯
L" =
λa ϕ˙can
caturn
c˙a − H into an operator?
How
we
H = λa ω ab ∂b H + i¯
ca ω ab ∂b ∂d Hcd
44
● Commutators:
![ϕa (t), λb (t)]" ≡ lim !ϕa (t + #)λb (t) − λb (t + #)ϕ(t)"
!→0
4
! = λa ϕ˙ a + i¯
With our L
ca c˙a + · · · the only non-zero
commutators are:
![ϕa (t), λb (t)]" = iδba ,
●
![ca (t), c¯b (t)]" = δba
A particular realization of the commutator above is:
λa −→ −i
∂
∂ϕa
c¯a −→
∂
∂ca
ˆa , cˆa as multiplicative operators.
and ϕ
●
In particular ![ϕa (t), ϕb (t)]" = 0 which confirms
that we are doing classical mechanics.
-7-
Path Integral Classico
Path
Integral
ϕa ≡ (q,
p) variabili diClassico
spazio delle fasi
Path
φacl (t; ϕi ) cammino classico.
a ≡ (q, p) variabili di spazio delle fasi
ϕ
ϕ˙ a = ω ab ∂ H(ϕ)
Integrals --> Risolve
Operators
b
φacl (t; ϕi ) cammino classico.
∂ ω ab ∂b H(ϕ)
Risolve ϕ˙ a =
!
λa (t) −→ −i a
∂ϕ
Z ≡ "ϕf ; tf |ϕi ; ti # = D!! ϕ δ(ϕa − φacl (t; ϕi ))
⇓
ˆ = −iω ab ∂b H∂a = L
ˆ!
H = λa ω ab ∂b H −→ H
Z ≡ "ϕf ; tf |ϕi ; ti # = D!! ϕ δ(ϕa − φacl (t; ϕi ))
(Liouville operator)
Propriet`a delle delta di Dirac:
!
ϕf
ϕi
D!! ϕDλ exp i
↓
!
a
ab
a
ab
δ(ϕa ˆ− φacl (t; ϕi )) =
ˆ δ(ϕ˙ − ω ∂b H) det(∂t δc − ω ∂b ∂c H)
dtL" −→ exp −iHt
= exp −iLt
Propriet`a delle delta di Dirac:
↓
δ(ϕa − φacl (t; ϕi )) = δ(ϕ˙ a − ω ab ∂b H) det(∂t δca − ω ab ∂b ∂c H)
Esponenziazione:
Operator
! formulation
! tf
(Koopman-von
Z! = Neumann
D!! ϕDλDcD¯
dtL"
1931) c exp i
ti
Esponenziazione:
!
! tf
!!
Z
=
D
ϕDλDcD¯
c
exp
i
dtL"
a
a
"
Question: If the bosonic part of L =
is λ
related
i¯cthe
a ϕ˙ +to
a c˙ − H
Path integral
ti
ab ∂ H + i¯
Liouville operator, what
is the
H=
λa ωmeaning
ca ω ab ∂b ∂d Hcd
b
a
a
"
ca c˙ − H
of the full L?= λa ϕ˙ + i¯
H = λa ω ab ∂b H + i¯ca ω ab ∂b ∂d Hcd
L! = λa [ϕ˙ a − ω ab ∂b H] + i¯
ca [δba ∂t − ∂b (ω ac ∂c H)]cb
-8-
4
4
Campi di Jacobi
c- variables and Jacobi fields
•
δϕa (t)
a
ab
!
L = λa [ϕ˙ − ω ∂b H] + i¯
ca [δba ∂t − ∂b (ω ac ∂c H)]cb
δϕa (0)of motion:
Equations
●• Variation with respect to λ
gives
ϕ˙ a − ω ab ∂b H = 0
↓
standard equations of
 d
ϕ δϕa
a
ab
d !
motion fordi

(δϕ
)
=
ω
∂
∂
H(δϕ
)
→
equazione

b
d
 dt
a
ac
b
● Variation with respect to c¯ gives c˙ − ω ∂c ∂b Hc = 0

 d a

ab
d
a
c = ω ∂b ∂d Hc → equazione di c
dt
a
a
c have the same evolution equation as the first variations of ϕ
a ≡ δϕac
a
ab
c
Jacobi
(δ ϕ˙ ) − ω ∂c ∂b H(δϕ ) = 0
fields.
• H genera
l’evoluzione
siaofdei
punti
dello of
spazio
generates
the evolution
both
the points
the phase space
and their
delle
fasi first
che variations.
delle loro prime variazioni.
• H: le sue propriet`a spettrali forniscono
informazioni
!c(t)c(0)"
∼ exp λt
sulle popriet`a dinamiche del sistema (ergodicit`a,
caos, esponenti di Lyapunov).
a
c (0)
Lyapunov exponents.
[Fig. 1]
ca (t)
-914
1
November 4, 2003 17:17 WSPC/139-IJMPA
01598
c and differential forms
Forme Differenziali
●
Infinitesimal
evolution:
Evoluzionetime
temporale
infinitesima
Cartan Calculus via Pauli Matrices 5233

a! = ϕa + "ω ab ∂ H

ϕ
b

where the Lagrangian is given by
∂ϕa! b
ca! = ca + "ω ab ∂b ∂d Hcd =
c


b
L = λa ϕ˙ a + i¯
ca c˙a − H , with H = λa ω ab ∂b H +∂ϕ
i¯
ca ω ac ∂c ∂b Hcb .
(2.5)
From the Lagrangian
(2.5)
one can derive, besides the standard
equaa Hamilton’s
b a
a
as abase
basisper
forleforms;
dϕa ∧ dϕb
ca transform
si trasforma
una
forme
⇒cca ·≡c dϕ↔
tions (2.1) for ϕ,c the
following come
equations
of motion
for
the Grassmann
variables:
⇒
a
∈ basis
cab = dϕ
of Tϕ∗ M
bc
a
c˙ = ω ∂c ∂a Hc ,
(2.6)
c¯˙b = −¯
ca ω ac ∂c ∂b H .
a + ψ dϕa ∧ dϕb + . . .
ψ(ϕ, dϕ)(p)= ψ01(ϕ) + ψa (ϕ)dϕ
ab
1 H are given
So infinitesimal transformations
by Fˆ ≡ 1 Fa ...a ca1 · · · cap
∧ · · · ∧ dϕap −→
Fa1 ...ap dϕaby
● F = p! generated
↓
p! 1 p
!
ϕa = ϕa + %ω ab ∂b H ,
a
a
b
ψ(ϕ, c) =
ψ0 (ϕ)
ψa (ϕ)c + ψabfunctions
c c + . .of
. ϕ and c
forms
on+M
!
!
∂ϕa b
`e ac
una formab differenziale
ca =ψ(ϕ,
ca +c)%ω
∂c ∂b Hc =
c ,
(2.7)
∂ϕb
←→
∂ϕb
c¯!a = c¯a − %¯
cb ω bc ∂c ∂a H =
c¯b .
∂ϕa!
Hψ(ϕ, c) = Lh ψ(ϕ, dϕ)
a
We notice from these equations that c transforms, under the diffeomorphism
L : c¯derivata
di Lieaslungo
il flusso
hamiltoniano
a basis
for vector
fields
a transform
generated by H, as a hbasis
for the differential forms dϕa , while c¯a transforms as
∂
(Cartan 1900).
a basis for the vector fields ∂ϕ
a , see Refs. 1 and 7. From the kinetic part of the
Lagrangian (2.5) we can derive, 1as Feynman did for quantum mechanics,
the graded
1
a1 ...ap
∂a1 ∧ · · · ∧ ∂ap −→ Vˆ ≡ V a1 ...ap c¯a1 · · · c¯ap
V (p) They
= Vare:
commutators of the ●
theory.
p!
p!
●
13
a
"[ϕa , λantisymmetric
"[¯
cb , ca ]+ # = δba . functions of ϕ and
(2.8)
c¯
tensors
b ]− # = iδb ,
←→
All other commutators are zero. In order to satisfy the above commutators we can
realize ϕa and ca as multiplication operators and λa and c¯a as derivative ones:
λa = −i
∂
,
∂ϕa
c¯a =
∂
.
∂ca
(2.9)
Substituting the previous relations into the Hamiltonian H of Eq. (2.5) we obtain
the following operator (from now on we will omit the hat signs ! on the symbols of
the abstract operators. Instead we will use them on the symbols which indicate the
matrices associated to these operators):
H = −iω ab ∂b H∂a − iω ac ∂c ∂b Hcb
∂
.
∂ca
(2.10)
The first term of (2.10) is just the Liouville operator that appears in the operatorial
formulation of classical mechanics due to Koopman and von Neumann.2,3 This
confirms that (2.4) is just the correct functional counterpart of the operatorial
formulation for classical mechanics. Moreover, thanks to the presence of Grassmann
variables, the CPI provides also the evolution of more generalized objects.1 In fact
the following kernel:
# " tf
$
"
K(ϕf , cf , tf |ϕi , ci , ti ) = D!! ϕ Dλ D!! c D¯
c exp i
dt L
(2.11)
ti
-10-
Simmetrie
e Geometria
del adCPI (1)
a
a
ab
b
!
L = λa ϕ˙ + i¯
ca c˙ − λa ω ∂b H − i¯
ca ω ∂d ∂b Hc
L! = λa ϕ˙ a + i¯
ca c˙a − λa ω ab ∂b H − i¯
ca ω ad ∂d ∂b Hcb
Trasformata di Legendre
Trasformata di Legendre
! = λa ω ab ∂b H + i¯
H
ca ω ad ∂d ∂b Hcb
! = λa ω ab ∂b H + i¯
H
ca ω ad ∂d ∂b Hcb
! i cui
Esistono alcune simmetrie globali e universali di H
! i cui
Esistono alcune
simmetrie sono
globalii eseguenti:
universali di H
generatori
generatori sono i seguenti:


QBRS =a ica λa




(conserved
Universal
Symmetries
QBRS =charges
ic λa for any system)

Simmetrie
puramente



¯ BRS = i¯
Simmetrie puramente 


Q
ca ω ab λb



QBRS = i¯
ca ω ab λb
a
ab

¯

QBRSGeometriche
≡
ic
λ
;
Q
≡
i¯
c
ω
Qg ≡ ca c¯a
BRS
a 1 λb ;a b
a
Geometriche
K
=
ω
2 caab
K = 12 ωab
cbc c
(Non
H(ϕ)) 



1
(Nondipendono
dipendono1da
da H(ϕ))
1 abab

¯=
1=ab
ω¯aωc¯bc¯ac¯c¯abc¯b


K
K = ωab ca cb ; 
KK
=


2 ω2 c


Algebra
2

Algebra Isp(2)
Isp(2)

Q =a 2ca c¯

Qgh gh
= c c¯a a
Global symmetries


NN
==
ca ∂caaH(ϕ)

H H
∂a H(ϕ)









N
=
c¯a ωc¯abω∂ab


¯
H
b H(ϕ)

N
=
∂b H(ϕ)
Dynamical
Symmetries
Simmetrie Dinamiche
Dinamiche
H
a
Simmetrie
 QQ
H = QBRS − NH
(they
depend on
)
(Dipendono
da H(ϕ))

=
Q

BRS − NH
(Dipendono da H(ϕ)) 
 H



QH¯ = QBRS
+ NH ¯


¯

Q
=
Q
H

&
'( BRS +)NH
Algebra N = 2 SUSY
Algebra
N =N=2
2 SUSY
algebra
susy:
4
&
'(
!
[QH , QH ] = 2iH
)
¯ H ] = 2iH
!
¯
H, Q
[QH[Q
,Q
H ] = 2iH
4
Square roots of the Hamiltonian:
¯H ,
Q(1) ≡ QBRS − N
¯ BRS + NH
Q(2) ≡ Q
Q2(1) = Q2(2) = −iH
If we apply twice the transformations generated by Q(i)
we get a time translation.
-11-
CARTAN CALCULUS
Differential Geometry
Operations
dF (p)
exterior derivative
ιV F (p)
interior contraction
Lh = dιh + ιh d
Lie derivative
←→
Path Integral
Corrispondent
←→
[QBRS , Fˆ (p) ]
←→
[Vˆ , Fˆ (p) ]
←→
[d, Lh ] = 0
←→
deq = d − ιh
←→
equivariant exterior
!
derivative
H
[QBRS , H] = 0
Q(1)
-12-
OBSERVABLES
in KvN theory
●
●
KvN did not specify which are the
observables in their formulation
In standard classical mechanics the
observables are only:
!
O(ϕ)
So all the observables commutes and
there is no interpherence effects.
●
If the observables were hermitian
operators but functions also of λ , c, c¯
!
O(ϕ,
λ, c, c¯)
then they would not commute and
interpherence effects would appear
because superposition is admitted in KvN
●
The Way Out ?.
λ
-13-
Superspace and Superfields
●
t
⇓
θ
θ¯
t
base space
¯
t −→ (t, θ, θ)
superspace
●
(ϕ, c, c¯, λ)
target space
↓
Superfield :
¯ = ϕa (t) + θca (t) + θω
¯ ab c¯b (t) + iθθω
¯ ab λb (t)
Φa (t, θ, θ)
-14-
Confronto Tra Path Integral
Ampiezza di transizione classica:
!
!
= D!! Φq DΦp exp i idtdθdθ¯ L[Φ]
Comparison of
Path Integrals
!Φqf ; tf |Φqi ; ti "
Quantum transition amplitude (QPI):
Ampiezza di transizione quantistica:
!
!
i
!qf ; tf |qi ; ti " = D!! qDp exp
dtL[ϕ]
!
Confronto Tra Path Integral
Classical transition amplitude (CPI) :
Ampiezza di transizione classica:
!
Il path
integral
classico
ha la stessa! Lagrangiana e la
q
q
!Φf ; tmisura
= path
D!!integral
Φq DΦp exp
i idtdθd
θ¯ L[Φ]
f |Φi ; ti "del
stessa
quantistico
con
i super-
campi al posto dei campi e il supertempo al posto del
tempo.
Ampiezza di transizione quantistica:
!
!
8
i
The classical
Lagrangian
!qf ; tf path
|qi ; ti "integral
= D!!has
qDpthe
expsame dtL[ϕ]
! path integral
and the same measure as the quantum
but with the fields replaced by the superfields and the
time integration replaced by the supertime integration.
Il path integral classico ha la stessa Lagrangiana e la
stessa misura del path integral quantistico con i supercampi al posto dei campi e il supertempo al posto del
tempo.
8
-15-
Local universal symmetries
●
●
!
!
¯ ab c¯b
ϕa → ϕa + ε(t)θca
ϕa → ϕa + ε(t)θω
ca → ca − ε(t)ca
c¯b → c¯b − ε(t)¯
cb
! a
¯ ab λb
ϕ → ϕa + iε(t)θθω
λa → λa − ε(t)ωab ϕb
The only quantity invariant under
a
all three
! transformations above is Φ .
So L[Φ]dθdθ¯ is invariant and the
transformations turn out to be local
symmetries of our systems
● As they are like gauge transformations,
the only admissible observables are
the following ones:
ˆ a)
O(Φ
which are gauge invariant.
-16-
Heisenberg picture
for θ and θ¯
●
Let us go from the Heisenberg
picture in θ, θ¯ to the
Schroedinger one
● e−θQ
BRS
¯ BRS θ¯ ! a
−Q
¯ θQ
Φ (t, θ, θ)e
BRS
¯ BRS θ¯
+Q
=ϕ
!a (t)
¯ is the Heisenberg picture,
! a (t, θ, θ)
● So Φ
in ( θ , θ¯ ), of
●
e−θQ
●
BRS
ϕ
!a (t)
Let us then turn the physical
! a ) into the
observables O(Φ
Schroedinger picture form:
¯ BRS
−θ¯Q
!H (Φa )eθQBRS +θ¯Q¯ BRS ≡ OS = O(ϕa )
O
We get in this way the standard
! a ) of CM
observables O(ϕ
-17-
Super-selection rule
●
●
●
a
We have an operator ϕ! which
commutes with all the observables
and which is not a multiple of the
identity.
! ϕ
O(
!a )
This triggers a superselection
mechanism which says that the
Hilbert space of the system is given by
the eigenvariety of the superselection
operator ϕ! :
ϕ
!a |ϕa0 ! = ϕa0 |ϕa0 !
Another eigenvariety will be:
ϕ
!a |ϕa1 ! = ϕa1 |ϕa1 !
-18-
No-superposition in CM
●
As |ϕa0 ! and |ϕa1 !belongs to different
Hilbert spaces we cannot superimpose
them:
|ϕ
!a ! ≡ |ϕa0 ! + |ϕa1 !
|ϕ
!a ! is
●
not a physical state.
Not all hermitian operator are
observables . The non-observables
ones are all those hermitian operators
which connects different superselected
sectors.Example: the Liouvillian
operator:
! = λa ω ab ∂b H
L
which moves from one point to another.
-19-
Schoredinger picture
in θ , θ¯ of the KvN
waves
● Besides the obsevables we have to
bring also the KvN-waves in the
Schoeredinger picture in θ¯, θ
¯
ψ!S (t, θ, θ)
BRS
¯ BRS θ¯
+Q
=
eθQ
=
¯ ab c¯b , c)
ψ(ϕa + θca + θω
ψ(t) =
● Note that cˆa commutes with all the
observables so the system lives also in
the eigenvariety:
cˆa |ca0 ! = ca0 |ca0 !
●
The physical states are then:
ψ
=
=
!ϕ|ϕa0 "!c|ca0 " =
δ(ϕ − ϕ0 )δ(c − c0 )
-20-
Schroedinger picture
● Is
ψ! = δ(ϕ − ϕ0 )δ(c − c0 )
of the form
¯ ab c¯b , ca ) ?
ψ!S = ψ(ϕa + θca + θω
● The only manner to achieve that is to
put c = c¯ = 0 so:
ψ!S = δ(ϕ − ϕ0 )δ(c)
and this is isomorphic to:
ψ!S = δ(ϕ − ϕ0 )
which are the physical states.
-21-
Role of c , c¯
● If we had neglected the differential
forms in the KvN procedure, we
would have had only functions of
the ϕ!, λ! . These would make up non! and
! λ)
commuting observables O(ϕ,
there would be no way to get rid of
them via some principle but only via
a postulate:
! to the O(
! ϕ)
! λ)
restrict the O(ϕ,
"
¯ instead allow us to go to the
● The c, c
superfield and have a gauge
invariance which automatically selects
the correct observables.
-22-
Quantization and superposition
● In this framework quantization is
achieved by sending θ, θ¯ → 0
(identical to “geometric quantization”).
●
The “gauge invariace” disappears if
θ, θ¯ → 0 :
!
●
ϕa → ϕa + ε(t)θca
ca → ca − ε(t)ca
Having no gauge invariace the
superselection mechanism is not
triggered anymore and we can have
superposition.
-23-
Physical Meaning of the Superfield
● We would like to get CM in the CPI formulation
by starting from the path integral of QM and
performing a “block spin” transformation to “block
spin” of phase space of size >> ¯h?
●
points in
phase space
QPI
●
block spin in
phase space
CPI
We already passed from the QPI to the CPI
via:
a
ϕa
Φ
so maybe the Φa represent the block spin:
Φa (ϕ, c, . . .) =
c = side
ϕ = center of
the block spin
Crucial Physical Freedom that we have
If the new basic variables in CM are
the blocks of phase space, then we can
reparametrize its internal points as we
like provided the block remain the
same.
The local-transformation on the
components of the superfields that we have
used for the superselection rule are,
“maybe”, the above Internal Point
Reparametrization
RENORMALIZATION GROUP
APPROACH
We proved that the Lagrangians in the
QPI and the CPI are the same, except
for the exchange field-superfield, it
means there is only a field
“renormalization” and not a coupling
and mass renormalization. This is
typical of supersymmetric field theory .
Can we find an “analog” of the Gammafunction of the Callan-Symanzik
renormalization group equation and its
associated zeros ?