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Mathematical Foundationsof

Quantum Field Theory

Wojciech Dybalski

TU MünchenZentrum Mathematik

LMU, 02.11.2016

Wojciech Dybalski (TU München) Mathematical Foundations of QFT

Outline

1 Spacetime symmetries

2 Relativistic Quantum Mechanics

3 Relativistic (Haag-Kastler) QFT

4 Relativistic (Wightman) QFT

5 Relativistic (perturbative) QFT

6 Status of Quantum Electrodynamics

Spacetime symmetries

Lorentz group

Minkowski spacetime: (R4, η) with η := diag(1,−1,−1,−1).

1 Lorentz group: L := O(1, 3) := Λ ∈ GL(4,R) |ΛηΛT = η

2 Proper ortochronous Lorentz group: L↑+ - connectedcomponent of unity in L.

L = L↑+ ∪ TL↑+ ∪ PL↑+ ∪ TPL↑+,

where T (x0, ~x) = (−x0, ~x) and P(x0, ~x) = (x0,−~x).

3 Covering group: L↑+ = SL(2,C) = Λ ∈ GL(2,C) | det Λ = 1

Lorentz group

L = L↑+ ∪ TL↑+ ∪ PL↑+ ∪ TPL↑+,

Lorentz group

L = L↑+ ∪ TL↑+ ∪ PL↑+ ∪ TPL↑+,

Poincaré group

1 Poincaré group: P := R4 o L.

2 Proper ortochronous Poincaré group: P↑+ := R4 o L↑+.

3 Covering group: P↑+ = R4 o L↑+ = R4 o SL(2,C)

Relativistic Quantum Mechanics

Symmetries of a quantum theory

1 H - Hilbert space of physical states.

2 For Ψ ∈ H, ‖Ψ‖ = 1 define the ray Ψ := eiφΨ |φ ∈ R .

3 H - set of rays with the ray product [Φ|Ψ] := |〈Φ,Ψ〉|2.

Definition

A symmetry of a quantum system is an invertible map S : H → Hs.t. [SΦ|SΨ] = [Φ|Ψ].

Theorem (Wigner 31)

For any symmetry transformation S : H → H we can find a unitaryor anti-unitary operator S : H → H s.t. SΨ = SΨ. S is unique upto phase.

Application:1 P↑+ is a symmetry of our theory i.e., P↑+ 3 (a,Λ) 7→ S(a,Λ).

2 Thus we obtain a projective unitary representation S of P↑+S(a1,Λ1)S(a2,Λ2) = eiϕ1,2S((a1,Λ1)(a2,Λ2)).

3 Fact: A projective unitary representation of P↑+ corresponds toan ordinary unitary representation of the covering group

P↑+ 3 (a, Λ) 7→ U(a, Λ) ∈ B(H).

Theorem (Wigner 31)

P↑+ 3 (a, Λ) 7→ U(a, Λ) ∈ B(H).

Theorem (Wigner 31)

P↑+ 3 (a, Λ) 7→ U(a, Λ) ∈ B(H).

Theorem (Wigner 31)

P↑+ 3 (a, Λ) 7→ U(a, Λ) ∈ B(H).

Positivity of energy

Consider a unitary representation P↑+ 3 (a, Λ) 7→ U(a, Λ) ∈ B(H).

1 Pµ := i−1∂aµU(a, I )|a=0 - energy momentum operators.

2 If SpP ⊂ V+ then we say that U has positive energy.

Distinguished states

1 Def: Ω ∈ H is the vacuum state if U(a, Λ)Ω = Ω for all(a, Λ) ∈ P↑+.

2 Def: H1 ⊂ H is the subspace of single-particle states of massm and spin s if U H1 is the irreducible representation [m, s].

DefinitionA relativistic quantum mechanical theory is given by:

1 H - Hilbert space.

2 P↑+ 3 (a, Λ) 7→ U(a, Λ) ∈ B(H) - a positive energy unitary rep.

3 B(H) - possible observables.

H may contain a vacuum state Ω and/or subspaces ofsingle-particle states H[m,s].

Relativistic (Haag-Kastler) QFT

DefinitionA local relativistic QFT is a relativistic QM (U,H) with a net

R4 ⊃ O 7→ A(O) ⊂ B(H)

of algebras of observables A(O) localized in open bounded regionsof spacetime O, which satisfies:

1 (Isotony) O1 ⊂ O2 ⇒ A(O1) ⊂ A(O2),

2 (Locality) O1 ∼ O2 ⇒ [A(O1),A(O2)] = 0,

3 (Covariance) U(a, Λ)A(O)U(a, Λ)∗ = A(ΛO + a).

Questions:

1 Where are charges and gauge groups?

2 Where are charge-carrying (possibly anti-commuting) fields?

3 How about spin-statistics connection and CPT theorem?

4 Where are pointlike-localized fields, Green functions,path-integrals...?

Charges and gauge groups1 A :=

⋃O⊂R4 A(O) ⊂ B(H), α(a,Λ)( · ) := U(a, Λ) · U(a, Λ)∗.

2 Idea: Charges label ‘reasonable’ irreducible reps. of A.

3 ‘Reasonable’ reps. form a group whose dual is the globalgauge group. Charge conjugation ’C’:= taking inverse.

4 Def. π : A → B(Hπ) is an admissible representation if

π(α(a,Λ)(A)) = Uπ(a, Λ)π(A)Uπ(a, Λ)∗, A ∈ A,

for some relativistic QM (Uπ,Hπ).

5 Def: Vacuum rep. π0: (Uπ0 ,Hπ0 ,Ω), [π0(A)Ω] = Hπ0 .

Charges and gauge groups

1 Doplicher-Haag-Roberts (DHR) criterion: For any O

π A(O′) ' π0 A(O′),

where O′ := x ∈ R4 |x ∼ O.

O OO’ ’

Charges and gauge groups1 Doplicher-Haag-Roberts (DHR) criterion: For any O

π A(O′) ' π0 A(O′),

where O′ := x ∈ R4 |x ∼ O.

2 That is, π(A) = WAW ∗ for A ∈ A(O′) and a unitary W .

3 Clearly, ρ(A) := W ∗π(A)W for A ∈ A is unitarily equiv. to π.

4 Fact: ρ : A → B(H) is an endomorphism ρ : A → A.Endomorphisms, in contrast to reps., can be composed!

5 Fact: For any ρ there is a unique ρ s.t. ρ ρ contains π0.ρ is called the charge conjugate representation.

π A(O′) ' π0 A(O′),

where O′ := x ∈ R4 |x ∼ O.

π A(O′) ' π0 A(O′),

where O′ := x ∈ R4 |x ∼ O.

π A(O′) ' π0 A(O′),

where O′ := x ∈ R4 |x ∼ O.

Charge-carrying fields

DefinitionA twisted-local relativistic QFT is a relativistic QM (U,H,Ω)

1 With algebras of charge-carrying fields O 7→ F(O) ⊂ B(H).

2 With a unitary k ∈ B(H) s.t. k2 = 1 and kF(O)k∗ ⊂ F(O)which gives F±(O) := F ∈ F(O) | kFk∗ = ±F .

which satisfies:

1 (Isotony) O1 ⊂ O2 ⇒ F(O1) ⊂ F(O2),

2 (Twisted locality) O1 ∼ O2 ⇒ [F±(O1),F±(O2)]± = 0,

3 (Covariance) U(a, Λ)F(O)U(a, Λ)∗ = F(ΛO + a).

Charge-carrying fields and gauge group

Theorem DHR 74, DR90Given a local relativistic QFT (U,H,Ω,A) one obtains:

1 A representation πph : A → B(Hph) containing ‘all’ DHRrepresentations.

2 A twisted local relativistic QFT (Uph,Hph,Ω,F , k),

3 A compact gauge group G of unitary operators on Hphcontaining k in its center.

4 πph(A(O)) = F ∈ F(O) | gFg∗ = F , g ∈ G .

Spin-statistics connection

Theorem (Fierz 39, Pauli 40, Dell’Antonio 61...DHR 74)1 Suppose [F+Ω] ⊃ Hph,[m,s+]. Then s+ is integer.

2 Suppose [F−Ω] ⊃ Hph,[m,s−]. Then s− is half-integer.

CPT theorem

Theorem (Lüders 54, Pauli 55, Jost 57,...Guido-Longo 95)

Under certain additional assumptions there exists an anti-unitaryoperator θ on Hph which has the expected properties of the CPToperator i.e.

1 θF(O)θ∗ = F(−O),

2 θUph(a, Λ)θ∗ = Uph(−a, Λ),

3 θHph,ρ = Hph,ρ and θρ( · )θ∗ = ρ( ·).

Pointlike localized fields

Definition (Fredenhagen-Hertel 81, Bostelmann 04)

A quadratic form φj is a pointlike field of a relativistic QFT, if thereexists Fj ,r ∈ F(Or ), where Or is the ball of radius r centered atzero, s.t.

‖(1 + P0)−`(φj − Fj ,r )(1 + P0)−`‖ →r→0

0 for some ` ≥ 0.

Theorem (Bostelmann 04)

Under certain technical assumptions one obtains that

φj(x) := U(x , I )φjU(x , I )∗

are relativistic quantum fields in the sense of Wightman.

Pointlike localized fields

Definition (Fredenhagen-Hertel 81, Bostelmann 04)

A quadratic form φj is a pointlike field of a relativistic QFT, if thereexists Fj ,r ∈ F(Or ), where Or is the ball of radius r centered atzero, s.t.

‖(1 + P0)−`(φj − Fj ,r )(1 + P0)−`‖ →r→0

0 for some ` ≥ 0.

Theorem (Bostelmann 04)

Under certain technical assumptions one obtains that

φj(x) := U(x , I )φjU(x , I )∗

are relativistic quantum fields in the sense of Wightman.

Relativistic (Wightman) QFT

DefinitionA Wightman QFT is a relativistic QM (U,H,Ω) with distributions

S(R4) 3 f 7→ φj(f ) =:

∫d4x φj(x)f (x) ∈ [operators on H]

defining quantum fields. They satisfy

1 (Twisted locality) [φj(x), φk(y)]± = 0 for x − y spacelike,

2 (Covariance) U(a, Λ)φj(x)U(a, Λ)∗ = D(Λ−1)j ,kφk(Λx + a),

where D is a finite-dimensional representation of L↑+.

Irreducible representations of L↑+ = SL(2,C)

1 Representation space: H( j2 ,

k2 ) := Sym(⊗jC2)⊗ Sym(⊗kC2)

2 Representation: D( j2 ,

k2 )(Λ) = (⊗j Λ)⊗ (⊗k Λ)

Example 1. Some familiar fields

1 D = D(0,0) - scalar field ϕ

2 D = D( 12 ,

12 ) - vector field jµ

3 D = D( 12 ,0) ⊕ D(0, 12 ) - Dirac field ψ

4 D = D(1,0) ⊕ D(0,1) - Faraday tensor Fµν

Irreducible representations of L↑+ = SL(2,C)

1 Representation space: H( j2 ,

k2 ) := Sym(⊗jC2)⊗ Sym(⊗kC2)

2 Representation: D( j2 ,

k2 )(Λ) = (⊗j Λ)⊗ (⊗k Λ)

Example 1. Some familiar fields

1 D = D(0,0) - scalar field ϕ

2 D = D( 12 ,

12 ) - vector field jµ

3 D = D( 12 ,0) ⊕ D(0, 12 ) - Dirac field ψ

4 D = D(1,0) ⊕ D(0,1) - Faraday tensor Fµν

Example 2. Free scalar field ϕf

1 Consider a scalar field which satisfies

( + m2)ϕf(x) = 0, := ∂µ∂µ.

2 Fact: This is the usual free scalar field on H = Γ(L2(R3))

ϕf(x) =1

(2π)3/2

∫d3~p√2ω(~p)

(eiω(~p)x0−i~p~xa∗(~p) + e−iω(~p)x0+i~p~xa(~p)).

Example 3. Interacting scalar field ϕ

1 Consider a scalar field which satisfies

( + m2)ϕ(x) = − λ3!

”ϕ(x)3 ”.

Theorem (Glimm-Jaffe 68...)

1 This theory, called ϕ4, exists in 2 and 3 dimensional spacetimeand satisfies the Haag-Kastler and Wightman postulates.

2 Furthermore, the theory is non-trivial.

Scattering theory

1 Consider a massive Wightman theory of a scalar field ϕ.

Scattering theory1 Consider a massive Wightman theory of a scalar field ϕ.

2 Define a non-local field ϕε by

ϕε(p) := χ[m2−ε,m2+ε](p2)ϕ(p).

3 Set a∗t (gt) :=∫d3~x ϕε(t, ~x)

↔∂ 0g(t, ~x) where g is a positive

energy Klein-Gordon solution.

Theorem (Haag 58, Ruelle 62)

The following limits exist

Ψout/in := limt→+/−∞

a∗t (g1,t) . . . a∗t (gn,t)Ω

and span subspaces Hout,Hin ⊂ H naturally isomorphic to Γ(H1).

ϕε(p) := χ[m2−ε,m2+ε](p2)ϕ(p).

a∗t (g1,t) . . . a∗t (gn,t)Ω

ϕε(p) := χ[m2−ε,m2+ε](p2)ϕ(p).

a∗t (g1,t) . . . a∗t (gn,t)Ω

Infrared problems in scattering theory1 Scattering states of massless particles [Buchholz 77].

a∗t (g1,t) . . . a∗t (gn,t)Ω

Infrared problems in scattering theory1 Scattering states of massive particles in presence of massless

particles.[W.D. 05, Herdegen 13, Duell 16]

a∗t (g1,t) . . . a∗t (gn,t)Ω

The problem of asymptotic completeness

1 Hout = H? [Gérard-W.D. 13, W.D. 16]

Scattering matrix and Green functions (LSZ)

Theorem (Lehmann-Symanzik-Zimmermann 55, Hepp 66)out〈p3, p4, . . . , pn|p1, p2〉in = (−i)nG a,c(−p1,−p2, p3, . . . , pn),

where G a,c denotes connected, amputated Green functions.

Theorem (Lehmann-Symanzik-Zimmermann 55, Hepp 66)out〈p3, p4, . . . , pn|p1, p2〉in = (−i)nG a,c(−p1,−p2, p3, . . . , pn)

Green functions1 G (x1, . . . , xn) := 〈Ω, T ϕ(x1) . . . ϕ(xn)Ω〉, where T is time

ordering.

2 G (x1, . . . , xn) =∑

π∈P∏

R∈π G (xiR1, . . . , xiR|R|

)c, for example

G (x1, x2)c := G (x1, x2)− G (x1)G (x2).

3 G a,c(x1, . . . , xn) := (1 + m2) . . . (n + m2)G (x1, . . . xn)c.

ordering.

2 G (x1, . . . , xn) =∑

π∈P∏

R∈π G (xiR1, . . . , xiR|R|

)c, for example

G (x1, x2)c := G (x1, x2)− G (x1)G (x2).

3 G a,c(x1, . . . , xn) := (1 + m2) . . . (n + m2)G (x1, . . . xn)c.

ordering.

2 G (x1, . . . , xn) =∑

π∈P∏

R∈π G (xiR1, . . . , xiR|R|

)c, for example

G (x1, x2)c := G (x1, x2)− G (x1)G (x2).

3 G a,c(x1, . . . , xn) := (1 + m2) . . . (n + m2)G (x1, . . . xn)c.

Relativistic (perturbative) QFT

Path-integral formula

G (x1, . . . , xn) = “1N

∫Dφφ(x1) . . . φ(xn) eiS[φ] ”, Dφ := ”

∏x∈R4

dφ(x)”,

S [φ] :=

∫d4x

(12∂µφ(x)∂µφ(x)− m2

2φ(x)2 − λ

4!φ(x)4

1 Wick rotation

GE (x1, . . . , xn) = “1N

∫Dφφ(x1) . . . φ(xn) e−SE [φ], ”

GE (x1, . . . , xn) := G (−ix01 , ~x1, . . . ,−ix0

n , ~xn), SE [φ] ≥ 0

2 We want to determine GE as formal power series in λ:

GE (x1, . . . , xn) =∞∑r=0

λrGE ,r (x1, . . . , xn).

No control over convergence of the series.Wojciech Dybalski (TU München) Mathematical Foundations of QFT

G (x1, . . . , xn) = “1N

∫Dφφ(x1) . . . φ(xn) eiS[φ] ”, Dφ := ”

∏x∈R4

dφ(x)”,

S [φ] :=

∫d4x

(12∂µφ(x)∂µφ(x)− m2

2φ(x)2 − λ

4!φ(x)4

1 Wick rotation

GE (x1, . . . , xn) = “1N

∫Dφφ(x1) . . . φ(xn) e−SE [φ], ”

GE (x1, . . . , xn) := G (−ix01 , ~x1, . . . ,−ix0

n , ~xn), SE [φ] ≥ 0

GE (x1, . . . , xn) =∞∑r=0

λrGE ,r (x1, . . . , xn).

G (x1, . . . , xn) = “1N

∫Dφφ(x1) . . . φ(xn) eiS[φ] ”, Dφ := ”

∏x∈R4

dφ(x)”,

S [φ] :=

∫d4x

(12∂µφ(x)∂µφ(x)− m2

2φ(x)2 − λ

4!φ(x)4

1 Wick rotation

GE (x1, . . . , xn) = “1N

∫Dφφ(x1) . . . φ(xn) e−SE [φ], ”

GE (x1, . . . , xn) := G (−ix01 , ~x1, . . . ,−ix0

n , ~xn), SE [φ] ≥ 0

GE (x1, . . . , xn) =∞∑r=0

λrGE ,r (x1, . . . , xn).

Path-integral for the free theory

1 Free Euclidean action

SE ,f [φ] :=12

∫d4x

(∂µφ(x)∂µφ(x) + m2φ(x)2).

Path-integral for the free theory1 Free Euclidean action

SE ,f [φ] :=12

∫d4p

(2π)4

(φ(p)(p2 + m2)φ(−p)

)2 SE ,f [φ] = 1

2〈φ,C−1φ〉, where C (p) := 1

p2+m2 is calledcovariance.

3 SΛ0E ,f [φ] := 1

2〈φ, (CΛ0)−1φ〉, where CΛ0(p) := C (p)e−

p2+m2Λ0 .

4 GΛ0E ,f [iJ] := e−

12 〈J,C

Λ0 (p)J〉 = ” 1N

∫Dφ ei〈φ,J〉e−S

Λ0E ,f [φ]”.

GΛ0E ,f(x1, . . . , xn) =

δJ(x1) . . . δJ(xn)GΛ0E ,f [J]

∣∣J=0.

SE ,f [φ] :=12

∫d4p

(2π)4

(φ(p)(p2 + m2)φ(−p)

)2 SE ,f [φ] = 1

2〈φ,C−1φ〉, where C (p) := 1

3 SΛ0E ,f [φ] := 1

p2+m2Λ0 .

4 GΛ0E ,f [iJ] := e−

12 〈J,C

Λ0 (p)J〉 = ” 1N

Λ0E ,f [φ]”.

GΛ0E ,f(x1, . . . , xn) =

∣∣J=0.

SE ,f [φ] :=12

∫d4p

(2π)4

(φ(p)(p2 + m2)φ(−p)

)2 SE ,f [φ] = 1

2〈φ,C−1φ〉, where C (p) := 1

3 SΛ0E ,f [φ] := 1

p2+m2Λ0 .

4 GΛ0E ,f [iJ] := e−

12 〈J,C

Λ0 (p)J〉 = ” 1N

Λ0E ,f [φ]”.

GΛ0E ,f(x1, . . . , xn) =

∣∣J=0.

SE ,f [φ] :=12

∫d4p

(2π)4

(φ(p)(p2 + m2)φ(−p)

)2 SE ,f [φ] = 1

2〈φ,C−1φ〉, where C (p) := 1

3 SΛ0E ,f [φ] := 1

p2+m2Λ0 .

4 GΛ0E ,f [iJ] := e−

12 〈J,C

Λ0 (p)J〉 = ” 1N

Λ0E ,f [φ]”.

GΛ0E ,f(x1, . . . , xn) =

∣∣J=0.

1 GΛ0E ,f [iJ] := e−

12 〈J,C

Λ0 (p)J〉 is called generating functional. It is:

(a) continuous on S(R4)R,

(b) of positive type i.e. GE ,f [i(Jk − J`)] is a positive matrix,

(c) normalized i.e. GE ,f [0] = 1.

Theorem (Bochner-Minlos)

A functional on S(R4)R satisfying (a), (b), (c) is the Fouriertransform of a probabilistic Borel measure on S ′(R4)R i.e.

GΛ0E ,f [iJ] =

∫e i〈φ,J〉dµ(CΛ0 , φ)

Remark: Due to Λ0 the measure is supported on smooth functions.Wojciech Dybalski (TU München) Mathematical Foundations of QFT

1 GΛ0E ,f [iJ] := e−

12 〈J,C

Λ0 (p)J〉 is called generating functional. It is:

(a) continuous on S(R4)R,

(b) of positive type i.e. GE ,f [i(Jk − J`)] is a positive matrix,

(c) normalized i.e. GE ,f [0] = 1.

Theorem (Bochner-Minlos)

A functional on S(R4)R satisfying (a), (b), (c) is the Fouriertransform of a probabilistic Borel measure on S ′(R4)R i.e.

GΛ0E ,f [iJ] =

∫e i〈φ,J〉dµ(CΛ0 , φ)

Remark: Due to Λ0 the measure is supported on smooth functions.Wojciech Dybalski (TU München) Mathematical Foundations of QFT

Path-integral for the interacting theory

1 Interaction

SΛ0E ,int[φ] =

∫(V )

d4x(aΛ01 φ(x)2 + aΛ0

2 ∂µφ(x)∂µφ(x) + aΛ03 φ(x)4).

2 The generating functional of interacting Green functions

GΛ0E [iJ] =

∫e i〈φ,J〉e−S

Λ0E ,int[φ]dµ(CΛ0 , φ).

3 Perturbative renormalizability: Find aΛ0i =

∑r≥1 a

Λ0i ,rλ

r s.t.

GE ,r (x1, . . . , xn) = limΛ0→∞

GΛ0E ,r (x1, . . . , xn)

exist and ‘renormalization conditions’ are satisfied.

1 Interaction

SΛ0E ,int[φ] =

∫(V )

2 ∂µφ(x)∂µφ(x) + aΛ03 φ(x)4).

GΛ0E [iJ] =

∑r≥1 a

Λ0i ,rλ

r s.t.

GE ,r (x1, . . . , xn) = limΛ0→∞

GΛ0E ,r (x1, . . . , xn)

1 Interaction

SΛ0E ,int[φ] =

∫(V )

2 ∂µφ(x)∂µφ(x) + aΛ03 φ(x)4).

GΛ0E [iJ] =

∑r≥1 a

Λ0i ,rλ

r s.t.

GE ,r (x1, . . . , xn) = limΛ0→∞

GΛ0E ,r (x1, . . . , xn)

Status of QED

Perturbative QED

1 Formally given by the action

∫d4x

ψ(iγµ(∂µ + ieAµ)−m)ψ − 1

4FµνF

where1 ψ - Dirac field,

2 Aµ - electromagnetic potential,

3 Fµν := ∂µAν − ∂νAµ - the Faraday tensor.

2 QED is a perturbatively renormalizable theory.[Feldman et al 88, Keller-Kopper 96]

Status of QED

Axiomatic QED:Consider a Haag-Kastler theory (U,H,Ω,A) whose pointlikelocalized fields include Fµν and jµ = ”eψγµψ” s.t.

∂µFµν = jν , ∂αFµν + ∂µFνα + ∂νFαµ = 0.

Status of QED

Status of ψ:The DHR criterion not suitable for electrically chargedrepresentations π of QED i.e.

π A(O′) ' π0 A(O′) fails.

O OO’ ’

1 Indeed, due to the Gauss Law one can determine the electriccharge in O by operations in O′.

Status of QED

Status of ψ:The DHR criterion not suitable for electrically chargedrepresentations π of QED i.e.

π A(O′) ' π0 A(O′) fails.

O OO’ ’

1 Indeed, due to the Gauss Law one can determine the electriccharge in O by operations in O′.

Status of QED

Status of ψ:An alternative criterion proposed in [Buchholz-Roberts 13]

π A(C c) ' π0 A(C c)

1 Composition/conjugation of reps. Global gauge group.

2 A promising direction for constructing ψ.

3 If electron is a Wigner particle, Compton scattering states canbe constructed. [Alazzawi-W.D. 15]

Status of QED

Status of Aµ.1 Suppose jµ = 0 and Fµν 6= 0. Then Aµ is not a Wightman

field. [Strocchi].

2 Standard way out: Aµ as a Wightman field on indefinitemetric "Hilbert space" [Gupta-Bleuler]. Can one avoid it?

3 Free Aµ in the axial gauge (i.e. eµAµ = 0) is a string-likelocalized field. [Schroer, Mund, Yngvason 06].

4 In fact,

Aµ(x , e) =

∫ ∞0

dt Fµν(x + te)eν ,

U(Λ)Aµ(x , e)U(Λ)−1 = (Λ−1)µνAν(Λx ,Λe).

Status of QED

field. [Strocchi].

4 In fact,

Aµ(x , e) =

∫ ∞0

Status of QED

field. [Strocchi].

4 In fact,

Aµ(x , e) =

∫ ∞0

Status of QED

field. [Strocchi].

4 In fact,

Aµ(x , e) =

∫ ∞0

Status of QED

Status of Aµ. Open questions:

1 What is the role of Aµ from the DHR perspective?Is it a charge-carrying field of some ‘charge’?

2 How to construct the local gauge group starting fromobservables?

3 What is the intrinsic meaning of local gauge invariance?

Status of QED

Mathematical Foundations of QFT

Physics MathematicsSpacetime symmetries Representations of groupsCharges, global gauge symmetries Representations of C ∗-algebrasCPT symmetry Tomita-Takesaki theoryQuantum fields Theory of distributionsPath integrals Measure theory in infinite dim.Renormalizability Combinatorics/ODE

mathematical foundations of quantum field theory · mathematical foundations of quantum field...

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