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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
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Spacetime symmetries
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)
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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].
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic Quantum Mechanics
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).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic Quantum Mechanics
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).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic Quantum Mechanics
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).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic Quantum Mechanics
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).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic Quantum Mechanics
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.
P
0P
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic Quantum Mechanics
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].
P
m
Ω
P0
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic Quantum Mechanics
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].
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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...?
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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 .
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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 .
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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 .
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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 .
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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.
x
I I
I
I
O OO’ ’
x0
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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 .
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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( ·).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Haag-Kastler) QFT
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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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↑+.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
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
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
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
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
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)).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
Scattering theory
1 Consider a massive Wightman theory of a scalar field ϕ.
P
m
Ω
P0
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
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).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
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).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
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).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
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).
Infrared problems in scattering theory1 Scattering states of massless particles [Buchholz 77].
Ω
0
P
P
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
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).
Infrared problems in scattering theory1 Scattering states of massive particles in presence of massless
particles.[W.D. 05, Herdegen 13, Duell 16]
Ω
0
P
m
P
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
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).
The problem of asymptotic completeness
1 Hout = H? [Gérard-W.D. 13, W.D. 16]
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
Scattering matrix and Green functions (LSZ)
.
p1
p2
p3
p4
pn
...
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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
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.
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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
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.
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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (Wightman) QFT
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.
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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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
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
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
Relativistic (perturbative) QFT
Path-integral for the free theory
1 Free Euclidean action
SE ,f [φ] :=12
∫d4x
(∂µφ(x)∂µφ(x) + m2φ(x)2).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (perturbative) QFT
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) =
δn
δJ(x1) . . . δJ(xn)GΛ0E ,f [J]
∣∣J=0.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (perturbative) QFT
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) =
δn
δJ(x1) . . . δJ(xn)GΛ0E ,f [J]
∣∣J=0.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (perturbative) QFT
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) =
δn
δJ(x1) . . . δJ(xn)GΛ0E ,f [J]
∣∣J=0.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (perturbative) QFT
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) =
δn
δJ(x1) . . . δJ(xn)GΛ0E ,f [J]
∣∣J=0.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (perturbative) QFT
Path-integral for the free theory
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
Relativistic (perturbative) QFT
Path-integral for the free theory
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
Relativistic (perturbative) 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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (perturbative) 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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Relativistic (perturbative) 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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Status of QED
Perturbative QED
1 Formally given by the action
S =
∫d4x
ψ(iγµ(∂µ + ieAµ)−m)ψ − 1
4FµνF
µν,
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]
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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.
x
I I
I
I
O OO’ ’
x0
1 Indeed, due to the Gauss Law one can determine the electriccharge in O by operations in O′.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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.
x
I I
I
I
O OO’ ’
x0
1 Indeed, due to the Gauss Law one can determine the electriccharge in O by operations in O′.
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Status of QED
Status of ψ:An alternative criterion proposed in [Buchholz-Roberts 13]
π A(C c) ' π0 A(C c)
C
C
CC
= CCC
:
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]
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Status of QED
Status of ψ:An alternative criterion proposed in [Buchholz-Roberts 13]
π A(C c) ' π0 A(C c)
C
C
CC
= CCC
:
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]
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Status of QED
Status of ψ:An alternative criterion proposed in [Buchholz-Roberts 13]
π A(C c) ' π0 A(C c)
C
C
CC
= CCC
:
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]
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
Status of QED
Status of ψ:An alternative criterion proposed in [Buchholz-Roberts 13]
π A(C c) ' π0 A(C c)
C
C
CC
= CCC
:
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]
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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).
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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?
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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?
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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?
Wojciech Dybalski (TU München) Mathematical Foundations of QFT
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
Wojciech Dybalski (TU München) Mathematical Foundations of QFT