Fedosov Quantization and Quantum Field Theory
Giovanni Collini
Joint work with Prof. Stefan Hollands
Universit¨
at Leipzig
Vienna, May 2014
1 / 20
Fedosov Quantization and QFT
Outline
Outline
Introduction
Free QFT
Interacting QFT
Fedosov quantization scheme
Some details for the φ4 -interaction
Fedosov quantization vs ”ordinary” perturbation theory
Conclusion
2 / 20
Fedosov Quantization and QFT
Introduction
Let consider a phase space S with Poisson bracket
t, u.
“Traditional” quantization
Promote observables, sub-algebra of C 8 pS q, to operators on Hilbert space.
Dynamics: the Hamiltonian Ñ self-adjoint operator Ñ time evolution
(Ex) Consider S tpq, p q P R2 u and tq, p u 1 standard Poisson braket.
Quantization on L2 pR, dq q is
q Ñ qˆ,
p Ñ pˆ : i~Bq
F pq, p q Ñ Fˆ ? (choice of ordering)
Ñ “deformation” of classical mechanics
Consider C 8 pS q, define a new associative product () such that
F G FG o p~q
pF G G F q i~tF , G u o p~2q
Deformation quantization
(+ technical properties)
Change product, not observables (weaken to formal power series in ~).
No Hilbert space, no ordering choice, but non-equivalent star-products.
3 / 20
Fedosov Quantization and QFT
Introduction
Does any (finite-dim) Poisson manifold admit a deformation quantization?
1
Simplest case:
S : R2n with x µ
pqj , pj q, the symplectic form and the Poisson bracket are
¸ BF BG BF BG µ
ν
j
σ σµν dx ^ dx : dq ^ dpj
tF , G u :
Bqj Bpj Bpj Bqj
j
A well-defined star-product is the Moyal product
F
M G :
¸ pi~qk
k
k!
σ µ1 ν1 σ µk νk Bµ1 ...µk F Bν1 ...νk G
2
Symplectic case:
pS, σq is a symplectic manifold, tF , G u : σ1 pdF , dG q Poisson bracket.
D deformation quantization [De Wilde & Lecomte, Omori & Maeda &
Yoshioka, Fedosov].
3
General Poisson manifold case:
pS, t, uq is a Poisson manifold (not induced by any symplectic form).
D deformation quantization [Kontsevich].
4 / 20
Fedosov Quantization and QFT
Free QFT
Free QFT
We would like to apply the philosophy of deformation quantization to field
theories arising from a classical Lagrangian.
ñ ‘Symplectic case’ but infinite dimensional phase space S
Simplest example is the linear KG-theory (on Minkowski space)
p m 2 q u 0
In this case the phase space is linear
S : tu smooth sol, compact supp on Σu
tpq, pq P C08pΣq C08pΣq where q u|Σ, p B0u|Σu
there is a natural symplectc form σ : S S Ñ R
σ pu1 , u2 q »
Σ
pq1p2 p1q2qdx
5 / 20
Fedosov Quantization and QFT
Free QFT
Deformation quantization in this framework [D¨
utsch & Fredenhagen]:
1
Choose ‘observables’ (i.e. replacing C 8 pS q) as
F
2
»
f px1 , . . . , xn qϕpx1 q ϕpxn q
where ϕpx qru s : u px q evaluation functional on S,
n
f is a compactly supp distribution having WFpf q X pV
~-product
F
~ G :
¸ ~k »
k
k!
Y V nq H
W px1 , y1 q W pxk , yk q
k
k
δϕpx q .δ. . δϕpx q F δϕpy q .δ. . δϕpy q G
1
k
1
k
W = Wightman function (or 2-point function of an Hadamard state).
6 / 20
Fedosov Quantization and QFT
Free QFT
Relation to “traditional quantization”:
Corresponds to the standard approach to QFT if
F
Ñ Fˆ »
f px1 , . . . , xn q : ϕˆpx1 q ϕˆpxn q :
where ϕˆ standard field operator (creation + annihilation op),
: : normal ordering.
Then ~ is equivalent to the Wick’s theorem.
Relation to ‘simplest case’:
The finite dimensional version of
~ is the Wick star-product, obtained
iσ µν Ñ W µν g µν iσ µν
in the formula of of the .
Note:
g µν is the inverse of a metric on S (linearity of S);
Jνµ g µλ σλν is a complex structure.
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Fedosov Quantization and QFT
Interacting QFT
Interacting QFT
How can we generalize this methods to interacting theories?
Consider the V pφq-interacting massive scalar field theory, i.e.
Pφ : φ m2 φ V 1 pφq 0
Let S be the set of non-linear solutions.
Fix φ P S the linearized equation around φ is
Pφ u : u m2 u V 2 pφqu
0
Linearized solution Ø element in the tangent space Tφ S.
For each φ P S define an Hadamard pure quasi-free state for the
linearized theory, denote Wφ px, y q its 2-point function.
Consider the *-algebra
Wφ : t non-lin functions F : Tφ S
Ñ Cu with ~
Note that this defines a bundle of *-algebras W
φ PS Wφ.
8 / 20
Fedosov Quantization and QFT
Interacting QFT
Further structure on S:
1
2
Decompose Wφ Gφ i {2Eφ .
Eφ is the causal propagator for the linear operator Pφ .
The causal propagator defines a non-degenerate skew-symm tensor
field on S
»
φÑ
Eφ px, y q
δ
δφpx q
b δφδpy q P Tφ S ^ Tφ S
(= inverse symplectic form).
3
(Quasi-free) ñ Gφ defines a non-degenerate symm tensor field on S.
It is the inverse of a Riemannian metric gφ on S.
4
(Pure) ñ Gφ σ gives an almost complex structure on S,
i.e. p1, 1q tensor field J such that J 2 id.
5
non-lin solutions
6
almost K¨ahler manifold.
Define the Yano connection ∇, i.e. unique ∇g 0 , T 1{4N
(ñ ∇ respects symplectic form and almost complex structure);
9 / 20
Fedosov Quantization and QFT
Interacting QFT
Aim:
Define a -product on a suitable set of observables F
should (at least) contain
F pφq »
f px q
¹
pBqk φpx q
i
with f
P F pS q which
P C08 pM q
i
Note that SectpW q is an algebra under the “fibrewise” product
pF ~ G qpφq : Fφ ~ Gφ
since Fφ P Wφ for any φ P S and Wφ is an algebra (the “free field Wick
polynomial algebra” of the linearized theory around φ).
However this is not the desired deformation quantization of F pS q since
SectpW q is a different (much larger) space!
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Fedosov Quantization and QFT
Fedosov quantization scheme
Fedosov quantization scheme
How to relate F pS q with SectpW q?
The idea is perform this procedure
Construction of -product
Define a flat connection D on SectpW q such that the Leibniz rule
wrt ~ holds.
Construct a bjection Q : F pS q Ñ Sect0 pW q between observables and
flat sections.
Define pF
G qpφq : Q 1pQ pF q ~ Q pG qq.
In finite dimensions this procedure works:
Symplectic case [Fedosov], almost K¨ahler case [Karabegov & Schlichenmaier]
Many issues due to
8-dim nature of field theory!
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Fedosov Quantization and QFT
Fedosov quantization scheme
Consider the Fedosov operator δ : SectpS q Ñ W-valued 1-form
pn, 0q-homogeneous Ñ pn 1, 1q-homogeneous
(as element of the bigraded algebra of W-valued forms).
Theorem (Fedosov, Karambegov & Schlichenmacher)
D!r rp0q
rp1q ~
rp2q ~2
. . . W-valued 1-form on S, such that
D : ∇ δ
i
rr , s
~
is flat and satisfies the Leibniz rule (+ technical requirements on rp0q ).
Moreover for each F P F pS q there is a unique Q pF q P Sect0 pW q such that
Q pF q Fp0q
Fp1q ~
Fp2q ~2
. . . with Fp0q
F idW
12 / 20
Fedosov Quantization and QFT
Some details for the φ4 -interaction
Some details for the φ4 -interaction
Consider an ultra-static space-time M
metric ds 2 dt 2 hij px qdx i dx j .
Consider the V pφq φ4 , non-lin eq
S : smooth sol non-lin eq
R Σ with Σ compact with
ñ Pφ φ m2φ 4φ3.
Ø E : C 8 pΣq C 8 pΣq Cauchy data
8-dim (Fr´echet) manifold and S Ñ E is a global chart.
The corresponding linearized equation is Pφ u p m2 12φqu 0.
ñ there exists a 1-to-1 correspondence TφS Ø E ;
ñ natural symplectic form
»
»
Ð
Ñ
σ pu, v q :
u pt, x q Bt v pt, x q u px qσ px, y qv py q
Moreover S is an
Σ
M2
where σ px, y q is a compactly supported distribution which
“regularizes” the restriction on the Cauchy surface.
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Fedosov Quantization and QFT
Some details for the φ4 -interaction
Choice of functionals F pS q
Consider F : S Q φ Ñ C having an off-shell extension F˜ : C 8 pM q Ñ C
1 pM k q dist. with compact supp s.t.
smooth, with F˜ pk q pφq P EW
!
WFpF˜ pk q pφqX px1 , . . . , xk ; ξ1 , . . . , ξk q|ξi
PV
or ξj
R V & ξi j P V
or
)
Ø
1 pM n q .
A n-cov tensor is defined as a distribution pσ Eφ qbn α
˜ with α
˜ P EW
ñ cotangent space “symplectical cotangent space”.
We provide a notion of cov tensor field compatible with F pS q:
A n-cov tensor fields is a map A : S Q φ Ñ bn Tφ S such that there exists
˜ : C 8 pM q Ñ bn T C 8 pM q smooth with A
˜ pk q pφq P E 1 pM n k q and
A
W
b
n
pσ Eφq A˜ pφq Apφq.
(Ex) the kernel σ px, y q doesn’t depend on the background
S
Q φ Ñ pσ Eφ qb2 σ pσ Eφ σq
is a 2-cov tensor field.
14 / 20
Fedosov Quantization and QFT
Some details for the φ4 -interaction
Given a 2-point function Wφ of an Hadamard pure quasi-free state for the
linearized theory, is
S
Q φ Ñ pσEφ qb2 Wφ pσWφ σq “symplectically smeared” 2-point function
a cov tensor field?
In general ’no’!
However there exists a state for which the answer is ‘yes’.
ð [Fulling, Narcowich, Wald] deformation argument
For this specific choice of the state the formal analogy between finite-dim
almost K¨ahler manifold and QFT can be made rigorous.
In particular this framework allows a infinite dimensional FKS theorem.
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Fedosov Quantization and QFT
Fedosov quantization vs “ordinary” perturbation theory
Fedosov quantization vs “ordinary” perturbation theory
Let F
P F pS q be a polynomial, local function of the field φ P S
F pφpx qq F pφpx q, B φpx q, . . . , B n φpx qq
Perturbation theory formula [Haag], [Fredenhagen & D¨
utsch]
Fock-space formula for interacting field for F px q
:
¸
N
pi {~q
N
»
¡ ¡¡yN0
x 0 y10
r. . . r: F pϕˆpx qq :, : V pϕˆpy1 qs, . . . , : V pϕˆpyN qq :s
It corresponds to the “Gell-Mann-Low” formula.
This formula, like others in perturbation theory, requires renormalization.
16 / 20
Fedosov Quantization and QFT
Fedosov quantization vs “ordinary” perturbation theory
In order to compare the “ordinary” perturbation theory and the Fedosov
construction, we re-interprete the perturbative formula as an element of
the algebra bundle W.
For each φ P S, one can define a retarded product as a map
Rφ : F p S q b
N
ª
F pC 8 pM qq Ñ Wφ
Retarded products Rφ rF , G bN s are constructed using Epstein-Glaser
renormalization [Fredenhagen & D¨
utsch], [Hollands & Wald].
Remark
F pS q has to be replaced by an “extension” F~ pS q,
ð imposing the equation of motion and the axiomatic approach to
time-ordered products (which define retarded products) are incompatible
[Hollands & Wald].
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Fedosov Quantization and QFT
Fedosov quantization vs “ordinary” perturbation theory
Using retarded products, define the map ρ : F~ pS q Ñ SectpW q
ρpF q ¸ pi {~qN
N
N!
»
Rφ F φ ;
M
V pqd 4 x
bN Then it is natural asking
Are the Fedosov quantization and the “ordinary” perturbation theory
equivalent?
i.e. we ask if there exists a unitary section U P SectpW q (U ~ U id)
such that
ρpF q U ~ Q pF q ~ U where Q : F~ pS q Ñ Sect0 pW q is the quantization map obtained by
Fedosov procedure
18 / 20
Fedosov Quantization and QFT
Conclusions
Conclusion
In this talk I have outlined the general idea for an approach to quantization
of interacting fields based on the Fedosov quantization scheme. The
construction gives an algebra of observables which has the structure of a
space of flat sections in a bundle of algebras over the space of classical
solutions. Each fibre is a canonical CCR type algebra, corresponding to the
linearized theory around the given classical solution. The non-trivial
dynamical content is encoded in the flatness condition for the sections.
There are still open questions...
19 / 20
Fedosov Quantization and QFT
Conclusions
Thanks a lot for your attention!
20 / 20
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