Mathematical Foundations of QFT. Quantum Field Theory
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1 Mathematical Foundations of Quantum Field Theory Wojciech Dybalski TU München Zentrum Mathematik LMU,
2 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
3 Spacetime symmetries Lorentz group Minkowski spacetime: (R 4, η) with η := diag(1, 1, 1, 1). 1 Lorentz group: L := O(1, 3) := { Λ GL(4, R) ΛηΛ T = η } 2 Proper ortochronous Lorentz group: L + - connected component of unity in L. L = L + T L + PL + TPL +, where T (x 0, x) = ( x 0, x) and P(x 0, x) = (x 0, x). 3 Covering group: L + = SL(2, C) = { Λ GL(2, C) det Λ = 1}
4 Spacetime symmetries Lorentz group Minkowski spacetime: (R 4, η) with η := diag(1, 1, 1, 1). 1 Lorentz group: L := O(1, 3) := { Λ GL(4, R) ΛηΛ T = η } 2 Proper ortochronous Lorentz group: L + - connected component of unity in L. L = L + T L + PL + TPL +, where T (x 0, x) = ( x 0, x) and P(x 0, x) = (x 0, x). 3 Covering group: L + = SL(2, C) = { Λ GL(2, C) det Λ = 1}
5 Spacetime symmetries Lorentz group Minkowski spacetime: (R 4, η) with η := diag(1, 1, 1, 1). 1 Lorentz group: L := O(1, 3) := { Λ GL(4, R) ΛηΛ T = η } 2 Proper ortochronous Lorentz group: L + - connected component of unity in L. L = L + T L + PL + TPL +, where T (x 0, x) = ( x 0, x) and P(x 0, x) = (x 0, x). 3 Covering group: L + = SL(2, C) = { Λ GL(2, C) det Λ = 1}
6 Spacetime symmetries Poincaré group 1 Poincaré group: P := R 4 L. 2 Proper ortochronous Poincaré group: P + := R4 L +. 3 Covering group: P + = R4 L + = R4 SL(2, C)
7 Relativistic Quantum Mechanics Symmetries of a quantum theory 1 H - Hilbert space of physical states. 2 For Ψ H, Ψ = 1 define the ray ˆΨ := { e iφ Ψ φ R }. 3 Ĥ - set of rays with the ray product [ˆΦ ˆΨ] := Φ, Ψ 2. Definition A symmetry of a quantum system is an invertible map Ŝ : Ĥ Ĥ s.t. [Ŝ ˆΦ Ŝ ˆΨ] = [ˆΦ ˆΨ].
8 Relativistic Quantum Mechanics Theorem (Wigner 31) For any symmetry transformation Ŝ : Ĥ Ĥ we can find a unitary or anti-unitary operator S : H H s.t. Ŝ ˆΨ = ŜΨ. S is unique up to phase. Application: 1 P + is a symmetry of our theory i.e., P + (a, Λ) Ŝ(a, Λ). 2 Thus we obtain a projective unitary representation S of P + S(a 1, Λ 1 )S(a 2, Λ 2 ) = e iϕ 1,2 S((a 1, Λ 1 )(a 2, Λ 2 )). 3 Fact: A projective unitary representation of P + corresponds to an ordinary unitary representation of the covering group P + (a, Λ) U(a, Λ) B(H).
9 Relativistic Quantum Mechanics Theorem (Wigner 31) For any symmetry transformation Ŝ : Ĥ Ĥ we can find a unitary or anti-unitary operator S : H H s.t. Ŝ ˆΨ = ŜΨ. S is unique up to phase. Application: 1 P + is a symmetry of our theory i.e., P + (a, Λ) Ŝ(a, Λ). 2 Thus we obtain a projective unitary representation S of P + S(a 1, Λ 1 )S(a 2, Λ 2 ) = e iϕ 1,2 S((a 1, Λ 1 )(a 2, Λ 2 )). 3 Fact: A projective unitary representation of P + corresponds to an ordinary unitary representation of the covering group P + (a, Λ) U(a, Λ) B(H).
10 Relativistic Quantum Mechanics Theorem (Wigner 31) For any symmetry transformation Ŝ : Ĥ Ĥ we can find a unitary or anti-unitary operator S : H H s.t. Ŝ ˆΨ = ŜΨ. S is unique up to phase. Application: 1 P + is a symmetry of our theory i.e., P + (a, Λ) Ŝ(a, Λ). 2 Thus we obtain a projective unitary representation S of P + S(a 1, Λ 1 )S(a 2, Λ 2 ) = e iϕ 1,2 S((a 1, Λ 1 )(a 2, Λ 2 )). 3 Fact: A projective unitary representation of P + corresponds to an ordinary unitary representation of the covering group P + (a, Λ) U(a, Λ) B(H).
11 Relativistic Quantum Mechanics Theorem (Wigner 31) For any symmetry transformation Ŝ : Ĥ Ĥ we can find a unitary or anti-unitary operator S : H H s.t. Ŝ ˆΨ = ŜΨ. S is unique up to phase. Application: 1 P + is a symmetry of our theory i.e., P + (a, Λ) Ŝ(a, Λ). 2 Thus we obtain a projective unitary representation S of P + S(a 1, Λ 1 )S(a 2, Λ 2 ) = e iϕ 1,2 S((a 1, Λ 1 )(a 2, Λ 2 )). 3 Fact: A projective unitary representation of P + corresponds to an ordinary unitary representation of the covering group P + (a, Λ) U(a, Λ) B(H).
12 Relativistic Quantum Mechanics Positivity of energy Consider a unitary representation P + (a, Λ) U(a, Λ) B(H). 1 P µ := i 1 aµ U(a, I ) a=0 - energy momentum operators. 2 If Sp P V + then we say that U has positive energy. P 0 P
13 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: H 1 H is the subspace of single-particle states of mass m and spin s if U H 1 is the irreducible representation [m, s]. P 0 m Ω P
14 Relativistic Quantum Mechanics Definition A relativistic quantum mechanical theory is given by: 1 H - Hilbert space. 2 P + (a, Λ) 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 of single-particle states H [m,s].
15 Relativistic (Haag-Kastler) QFT Definition A local relativistic QFT is a relativistic QM (U, H) with a net R 4 O A(O) B(H) of algebras of observables A(O) localized in open bounded regions of spacetime O, which satisfies: 1 (Isotony) O 1 O 2 A(O 1 ) A(O 2 ), 2 (Locality) O 1 O 2 [A(O 1 ), A(O 2 )] = {0}, 3 (Covariance) U(a, Λ)A(O)U(a, Λ) = A(ΛO + a).
16 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...?
17 Relativistic (Haag-Kastler) QFT Charges and gauge groups 1 A := O R 4 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 global gauge group. Charge conjugation C := taking inverse. 4 Def. π : A B(H π ) is an admissible representation if π(α (a, Λ) (A)) = U π(a, Λ)π(A)U π (a, Λ), A A, for some relativistic QM (U π, H π ). 5 Def: Vacuum rep. π 0 : (U π0, H π0, Ω), [π 0 (A)Ω] = H π0.
18 Relativistic (Haag-Kastler) QFT Charges and gauge groups 1 A := O R 4 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 global gauge group. Charge conjugation C := taking inverse. 4 Def. π : A B(H π ) is an admissible representation if π(α (a, Λ) (A)) = U π (a, Λ)π(A)U π (a, Λ), A A, for some relativistic QM (U π, H π ). 5 Def: Vacuum rep. π 0 : (U π0, H π0, Ω), [π 0 (A)Ω] = H π0.
19 Relativistic (Haag-Kastler) QFT Charges and gauge groups 1 A := O R 4 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 global gauge group. Charge conjugation C := taking inverse. 4 Def. π : A B(H π ) is an admissible representation if π(α (a, Λ) (A)) = U π (a, Λ)π(A)U π (a, Λ), A A, for some relativistic QM (U π, H π ). 5 Def: Vacuum rep. π 0 : (U π0, H π0, Ω), [π 0 (A)Ω] = H π0.
20 Relativistic (Haag-Kastler) QFT Charges and gauge groups 1 A := O R 4 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 global gauge group. Charge conjugation C := taking inverse. 4 Def. π : A B(H π ) is an admissible representation if π(α (a, Λ) (A)) = U π (a, Λ)π(A)U π (a, Λ), A A, for some relativistic QM (U π, H π ). 5 Def: Vacuum rep. π 0 : (U π0, H π0, Ω), [π 0 (A)Ω] = H π0.
21 Relativistic (Haag-Kastler) QFT Charges and gauge groups 1 Doplicher-Haag-Roberts (DHR) criterion: For any O π A(O ) π 0 A(O ), where O := { x R 4 x O}. x 0 I O O O I I x I
22 Relativistic (Haag-Kastler) QFT Charges and gauge groups 1 Doplicher-Haag-Roberts (DHR) criterion: For any O π A(O ) π 0 A(O ), where O := { x R 4 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 π 0. ρ is called the charge conjugate representation.
23 Relativistic (Haag-Kastler) QFT Charges and gauge groups 1 Doplicher-Haag-Roberts (DHR) criterion: For any O π A(O ) π 0 A(O ), where O := { x R 4 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 π 0. ρ is called the charge conjugate representation.
24 Relativistic (Haag-Kastler) QFT Charges and gauge groups 1 Doplicher-Haag-Roberts (DHR) criterion: For any O π A(O ) π 0 A(O ), where O := { x R 4 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 π 0. ρ is called the charge conjugate representation.
25 Relativistic (Haag-Kastler) QFT Charges and gauge groups 1 Doplicher-Haag-Roberts (DHR) criterion: For any O π A(O ) π 0 A(O ), where O := { x R 4 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 π 0. ρ is called the charge conjugate representation.
26 Relativistic (Haag-Kastler) QFT Charge-carrying fields Definition A twisted-local relativistic QFT is a relativistic QM (U, H, Ω) 1 With algebras of charge-carrying fields O F(O) B(H). 2 With a unitary k B(H) s.t. k 2 = 1 and kf(o)k F(O) which gives F ± (O) := { F F(O) kfk = ±F }. which satisfies: 1 (Isotony) O 1 O 2 F(O 1 ) F(O 2 ), 2 (Twisted locality) O 1 O 2 [F ± (O 1 ), F ± (O 2 )] ± = {0}, 3 (Covariance) U(a, Λ)F(O)U(a, Λ) = F(ΛO + a).
27 Relativistic (Haag-Kastler) QFT Charge-carrying fields and gauge group Theorem DHR 74, DR90 Given a local relativistic QFT (U, H, Ω, A) one obtains: 1 A representation π ph : A B(H ph ) containing all DHR representations. 2 A twisted local relativistic QFT (U ph, H ph, Ω, F, k), 3 A compact gauge group G of unitary operators on H ph containing k in its center. 4 π ph (A(O)) = { F F(O) gfg = F, g G }.
28 Relativistic (Haag-Kastler) QFT Spin-statistics connection Theorem (Fierz 39, Pauli 40, Dell Antonio 61...DHR 74) 1 Suppose [F + Ω] H ph,[m,s+]. Then s + is integer. 2 Suppose [F Ω] H ph,[m,s ]. Then s is half-integer.
29 Relativistic (Haag-Kastler) QFT CPT theorem Theorem (Lüders 54, Pauli 55, Jost 57,...Guido-Longo 95) Under certain additional assumptions there exists an anti-unitary operator θ on H ph which has the expected properties of the CPT operator i.e. 1 θf(o)θ = F( O), 2 θu ph (a, Λ)θ = U ph ( a, Λ), 3 θh ph,ρ = H ph, ρ and θρ( )θ = ρ( ).
30 Relativistic (Haag-Kastler) QFT Pointlike localized fields Definition (Fredenhagen-Hertel 81, Bostelmann 04) A quadratic form φ j is a pointlike field of a relativistic QFT, if there exists F j,r F(O r ), where O r is the ball of radius r centered at zero, s.t. (1 + P 0 ) l (φ j F j,r )(1 + P 0 ) l r 0 0 for some l 0. Theorem (Bostelmann 04) Under certain technical assumptions one obtains that φ j (x) := U(x, I )φ j U(x, I ) are relativistic quantum fields in the sense of Wightman.
31 Relativistic (Haag-Kastler) QFT Pointlike localized fields Definition (Fredenhagen-Hertel 81, Bostelmann 04) A quadratic form φ j is a pointlike field of a relativistic QFT, if there exists F j,r F(O r ), where O r is the ball of radius r centered at zero, s.t. (1 + P 0 ) l (φ j F j,r )(1 + P 0 ) l r 0 0 for some l 0. Theorem (Bostelmann 04) Under certain technical assumptions one obtains that φ j (x) := U(x, I )φ j U(x, I ) are relativistic quantum fields in the sense of Wightman.
32 Relativistic (Wightman) QFT Definition A Wightman QFT is a relativistic QM (U, H, Ω) with distributions S(R 4 ) f φ j (f ) =: d 4 x φ j (x)f (x) [operators on H] defining quantum fields. They satisfy 1 (Twisted locality) [φ j (x), φ k (y)] ± = 0 for x y spacelike, 2 (Covariance) U(a, Λ)φ j (x)u(a, Λ) = D( Λ 1 ) j,k φ k (Λx + a), where D is a finite-dimensional representation of L +.
33 Relativistic (Wightman) QFT Irreducible representations of L + = SL(2, C) 1 Representation space: H ( j 2, k 2 ) := Sym( j C 2 ) Sym( k C 2 ) 2 Representation: D ( j 2, k 2 ) ( Λ) = ( j Λ) ( k Λ) Example 1. Some familiar fields 1 D = D (0,0) - scalar field ϕ 2 D = D ( 1 2, 1 2 ) - vector field j µ 3 D = D ( 1 2,0) D (0, 1 2 ) - Dirac field ψ 4 D = D (1,0) D (0,1) - Faraday tensor F µν
34 Relativistic (Wightman) QFT Irreducible representations of L + = SL(2, C) 1 Representation space: H ( j 2, k 2 ) := Sym( j C 2 ) Sym( k C 2 ) 2 Representation: D ( j 2, k 2 ) ( Λ) = ( j Λ) ( k Λ) Example 1. Some familiar fields 1 D = D (0,0) - scalar field ϕ 2 D = D ( 1 2, 1 2 ) - vector field j µ 3 D = D ( 1 2,0) D (0, 1 2 ) - Dirac field ψ 4 D = D (1,0) D (0,1) - Faraday tensor F µν
35 Relativistic (Wightman) QFT Example 2. Free scalar field ϕ f 1 Consider a scalar field which satisfies ( + m 2 )ϕ f (x) = 0, := µ µ. 2 Fact: This is the usual free scalar field on H = Γ(L 2 (R 3 )) ϕ f (x) = 1 (2π) 3/2 d 3 p 2ω( p) (e iω( p)x0 i p x a ( p) + e iω( p)x0 +i p x a( p)).
36 Relativistic (Wightman) QFT Example 3. Interacting scalar field ϕ 1 Consider a scalar field which satisfies ( + m 2 )ϕ(x) = λ 3! ϕ(x)3. Theorem (Glimm-Jaffe 68...) 1 This theory, called ϕ 4, exists in 2 and 3 dimensional spacetime and satisfies the Haag-Kastler and Wightman postulates. 2 Furthermore, the theory is non-trivial.
37 Relativistic (Wightman) QFT Scattering theory 1 Consider a massive Wightman theory of a scalar field ϕ. P 0 m Ω P
38 Relativistic (Wightman) QFT Scattering theory 1 Consider a massive Wightman theory of a scalar field ϕ. 2 Define a non-local field ϕ ε by ϕ ε (p) := χ [m 2 ε,m 2 +ε](p 2 ) ϕ(p). 3 Set a t (g t ) := d 3 x ϕ ε (t, x) 0 g(t, x) where g is a positive energy Klein-Gordon solution. Theorem (Haag 58, Ruelle 62) The following limits exist Ψ out/in := lim t +/ a t (g 1,t )... at (g n,t )Ω and span subspaces H out, H in H naturally isomorphic to Γ(H 1 ).
39 Relativistic (Wightman) QFT Scattering theory 1 Consider a massive Wightman theory of a scalar field ϕ. 2 Define a non-local field ϕ ε by ϕ ε (p) := χ [m 2 ε,m 2 +ε](p 2 ) ϕ(p). 3 Set a t (g t ) := d 3 x ϕ ε (t, x) 0 g(t, x) where g is a positive energy Klein-Gordon solution. Theorem (Haag 58, Ruelle 62) The following limits exist Ψ out/in := lim t +/ a t (g 1,t )... at (g n,t )Ω and span subspaces H out, H in H naturally isomorphic to Γ(H 1 ).
40 Relativistic (Wightman) QFT Scattering theory 1 Consider a massive Wightman theory of a scalar field ϕ. 2 Define a non-local field ϕ ε by ϕ ε (p) := χ [m 2 ε,m 2 +ε](p 2 ) ϕ(p). 3 Set a t (g t ) := d 3 x ϕ ε (t, x) 0 g(t, x) where g is a positive energy Klein-Gordon solution. Theorem (Haag 58, Ruelle 62) The following limits exist Ψ out/in := lim t +/ a t (g 1,t )... at (g n,t )Ω and span subspaces H out, H in H naturally isomorphic to Γ(H 1 ).
41 Relativistic (Wightman) QFT Theorem (Haag 58, Ruelle 62) The following limits exist Ψ out/in := lim t +/ a t (g 1,t )... at (g n,t )Ω and span subspaces H out, H in H naturally isomorphic to Γ(H 1 ). Infrared problems in scattering theory 1 Scattering states of massless particles [Buchholz 77]. P0 Ω P
42 Relativistic (Wightman) QFT Theorem (Haag 58, Ruelle 62) The following limits exist Ψ out/in := lim t +/ a t (g 1,t )... at (g n,t )Ω and span subspaces H out, H in H naturally isomorphic to Γ(H 1 ). Infrared problems in scattering theory 1 Scattering states of massive particles in presence of massless particles.[w.d. 05, Herdegen 13, Duell 16] P0 m Ω P
43 Relativistic (Wightman) QFT Theorem (Haag 58, Ruelle 62) The following limits exist Ψ out/in := lim t +/ a t (g 1,t )... at (g n,t )Ω and span subspaces H out, H in H naturally isomorphic to Γ(H 1 ). The problem of asymptotic completeness 1 H out = H? [Gérard-W.D. 13, W.D. 16]
44 Relativistic (Wightman) QFT Scattering matrix and Green functions (LSZ) p 1 p 3 p 2 p n p Theorem (Lehmann-Symanzik-Zimmermann 55, Hepp 66) out p 3, p 4,..., p n p 1, p 2 in = ( i) n G a,c ( p 1, p 2, p 3,..., p n ), where G a,c denotes connected, amputated Green functions.
45 Relativistic (Wightman) QFT Theorem (Lehmann-Symanzik-Zimmermann 55, Hepp 66) out p 3, p 4,..., p n p 1, p 2 in = ( i) n G a,c ( p 1, p 2, p 3,..., p n ) where G a,c denotes connected, amputated Green functions. Green functions 1 G(x 1,..., x n ) := Ω, T ϕ(x 1 )... ϕ(x n )Ω, where T is time ordering. 2 G(x 1,..., x n ) = π P R π G(x i R 1,..., x i R R ) c, for example G(x 1, x 2 ) c := G(x 1, x 2 ) G(x 1 )G(x 2 ). 3 G a,c (x 1,..., x n ) := ( 1 + m 2 )... ( n + m 2 )G(x 1,... x n ) c.
46 Relativistic (Wightman) QFT Theorem (Lehmann-Symanzik-Zimmermann 55, Hepp 66) out p 3, p 4,..., p n p 1, p 2 in = ( i) n G a,c ( p 1, p 2, p 3,..., p n ) where G a,c denotes connected, amputated Green functions. Green functions 1 G(x 1,..., x n ) := Ω, T ϕ(x 1 )... ϕ(x n )Ω, where T is time ordering. 2 G(x 1,..., x n ) = π P R π G(x i R 1,..., x i R R ) c, for example G(x 1, x 2 ) c := G(x 1, x 2 ) G(x 1 )G(x 2 ). 3 G a,c (x 1,..., x n ) := ( 1 + m 2 )... ( n + m 2 )G(x 1,... x n ) c.
47 Relativistic (Wightman) QFT Theorem (Lehmann-Symanzik-Zimmermann 55, Hepp 66) out p 3, p 4,..., p n p 1, p 2 in = ( i) n G a,c ( p 1, p 2, p 3,..., p n ) where G a,c denotes connected, amputated Green functions. Green functions 1 G(x 1,..., x n ) := Ω, T ϕ(x 1 )... ϕ(x n )Ω, where T is time ordering. 2 G(x 1,..., x n ) = π P R π G(x i R 1,..., x i R R ) c, for example G(x 1, x 2 ) c := G(x 1, x 2 ) G(x 1 )G(x 2 ). 3 G a,c (x 1,..., x n ) := ( 1 + m 2 )... ( n + m 2 )G(x 1,... x n ) c.
48 Relativistic (perturbative) QFT Path-integral formula G(x 1,..., x n ) = 1 Dφ φ(x 1 )... φ(x n ) e is[φ], Dφ := dφ(x), N x R ( 4 1 S[φ] := d 4 x 2 µφ(x) µ φ(x) m2 2 φ(x)2 λ ) 4! φ(x)4. 1 Wick rotation G E (x 1,..., x n ) = 1 Dφ φ(x 1 )... φ(x n ) e S E [φ], N G E (x 1,..., x n ) := G( ix1 0, x 1,..., ixn, 0 x n ), S E [φ] 0 2 We want to determine G E as formal power series in λ: G E (x 1,..., x n ) = λ r G E,r (x 1,..., x n ). r=0 No control over convergence of the series.
49 Relativistic (perturbative) QFT Path-integral formula G(x 1,..., x n ) = 1 Dφ φ(x 1 )... φ(x n ) e is[φ], Dφ := dφ(x), N x R ( 4 1 S[φ] := d 4 x 2 µφ(x) µ φ(x) m2 2 φ(x)2 λ ) 4! φ(x)4. 1 Wick rotation G E (x 1,..., x n ) = 1 Dφ φ(x 1 )... φ(x n ) e S E [φ], N G E (x 1,..., x n ) := G( ix1 0, x 1,..., ixn, 0 x n ), S E [φ] 0 2 We want to determine G E as formal power series in λ: G E (x 1,..., x n ) = λ r G E,r (x 1,..., x n ). r=0 No control over convergence of the series.
50 Relativistic (perturbative) QFT Path-integral formula G(x 1,..., x n ) = 1 Dφ φ(x 1 )... φ(x n ) e is[φ], Dφ := dφ(x), N x R ( 4 1 S[φ] := d 4 x 2 µφ(x) µ φ(x) m2 2 φ(x)2 λ ) 4! φ(x)4. 1 Wick rotation G E (x 1,..., x n ) = 1 Dφ φ(x 1 )... φ(x n ) e S E [φ], N G E (x 1,..., x n ) := G( ix1 0, x 1,..., ixn, 0 x n ), S E [φ] 0 2 We want to determine G E as formal power series in λ: G E (x 1,..., x n ) = λ r G E,r (x 1,..., x n ). r=0 No control over convergence of the series.
51 Relativistic (perturbative) QFT Path-integral for the free theory 1 Free Euclidean action S E,f [φ] := 1 d 4 x ( µ φ(x) µ φ(x) + m 2 φ(x) 2). 2
52 Relativistic (perturbative) QFT Path-integral for the free theory 1 Free Euclidean action S E,f [φ] := 1 2 d 4 p ( φ(p)(p 2 (2π) 4 + m 2 ) φ( p) ) 2 S E,f [φ] = 1 2 φ, C 1 φ, where C(p) := 1 covariance. p 2 +m 2 is called 3 S Λ 0 E,f [φ] := 1 2 φ, (C Λ 0 ) 1 φ, where C Λ 0 (p) := C(p)e p 2 +m 2 Λ 0. 4 G Λ 0 E,f [ij] := e 1 2 J,C Λ0(p)J = 1 N Dφ e i φ,j e SΛ 0 E,f [φ]. G Λ 0 E,f (x 1,..., x n ) = δ n δj(x 1 )... δj(x n ) G Λ 0 E,f [J] J=0.
53 Relativistic (perturbative) QFT Path-integral for the free theory 1 Free Euclidean action S E,f [φ] := 1 2 d 4 p ( φ(p)(p 2 (2π) 4 + m 2 ) φ( p) ) 2 S E,f [φ] = 1 2 φ, C 1 φ, where C(p) := 1 covariance. p 2 +m 2 is called 3 S Λ 0 E,f [φ] := 1 2 φ, (C Λ 0 ) 1 φ, where C Λ 0 (p) := C(p)e p 2 +m 2 Λ 0. 4 G Λ 0 E,f [ij] := e 1 2 J,C Λ0(p)J = 1 N Dφ e i φ,j e SΛ 0 E,f [φ]. G Λ 0 E,f (x 1,..., x n ) = δ n δj(x 1 )... δj(x n ) G Λ 0 E,f [J] J=0.
54 Relativistic (perturbative) QFT Path-integral for the free theory 1 Free Euclidean action S E,f [φ] := 1 2 d 4 p ( φ(p)(p 2 (2π) 4 + m 2 ) φ( p) ) 2 S E,f [φ] = 1 2 φ, C 1 φ, where C(p) := 1 covariance. p 2 +m 2 is called 3 S Λ 0 E,f [φ] := 1 2 φ, (C Λ 0 ) 1 φ, where C Λ 0 (p) := C(p)e p 2 +m 2 Λ 0. 4 G Λ 0 E,f [ij] := e 1 2 J,C Λ0(p)J = 1 N Dφ e i φ,j e SΛ 0 E,f [φ]. G Λ 0 E,f (x 1,..., x n ) = δ n δj(x 1 )... δj(x n ) G Λ 0 E,f [J] J=0.
55 Relativistic (perturbative) QFT Path-integral for the free theory 1 Free Euclidean action S E,f [φ] := 1 2 d 4 p ( φ(p)(p 2 (2π) 4 + m 2 ) φ( p) ) 2 S E,f [φ] = 1 2 φ, C 1 φ, where C(p) := 1 covariance. p 2 +m 2 is called 3 S Λ 0 E,f [φ] := 1 2 φ, (C Λ 0 ) 1 φ, where C Λ 0 (p) := C(p)e p 2 +m 2 Λ 0. 4 G Λ 0 E,f [ij] := e 1 2 J,C Λ0(p)J = 1 N Dφ e i φ,j e SΛ 0 E,f [φ]. G Λ 0 E,f (x 1,..., x n ) = δ n δj(x 1 )... δj(x n ) G Λ 0 E,f [J] J=0.
56 Relativistic (perturbative) QFT Path-integral for the free theory 1 G Λ 0 E,f [ij] := e 1 2 J,C Λ 0(p)J is called generating functional. It is: (a) continuous on S(R 4 ) R, (b) of positive type i.e. G E,f [i(j k J l )] is a positive matrix, (c) normalized i.e. G E,f [0] = 1. Theorem (Bochner-Minlos) A functional on S(R 4 ) R satisfying (a), (b), (c) is the Fourier transform of a probabilistic Borel measure on S (R 4 ) R i.e. G Λ 0 E,f [ij] = e i φ,j dµ(c Λ 0, φ) Remark: Due to Λ 0 the measure is supported on smooth functions.
57 Relativistic (perturbative) QFT Path-integral for the free theory 1 G Λ 0 E,f [ij] := e 1 2 J,C Λ 0(p)J is called generating functional. It is: (a) continuous on S(R 4 ) R, (b) of positive type i.e. G E,f [i(j k J l )] is a positive matrix, (c) normalized i.e. G E,f [0] = 1. Theorem (Bochner-Minlos) A functional on S(R 4 ) R satisfying (a), (b), (c) is the Fourier transform of a probabilistic Borel measure on S (R 4 ) R i.e. G Λ 0 E,f [ij] = e i φ,j dµ(c Λ 0, φ) Remark: Due to Λ 0 the measure is supported on smooth functions.
58 Relativistic (perturbative) QFT Path-integral for the interacting theory 1 Interaction S Λ 0 E,int [φ] = (V ) d 4 x ( a Λ 0 1 φ(x)2 + a Λ 0 2 µφ(x) µ φ(x) + a Λ 0 3 φ(x)4). 2 The generating functional of interacting Green functions G Λ 0 E [ij] = e i φ,j e SΛ 0 E,int [φ] dµ(c Λ 0, φ). 3 Perturbative renormalizability: Find a Λ 0 i = r 1 aλ 0 i,r λr s.t. G E,r (x 1,..., x n ) = lim Λ 0 G Λ 0 E,r (x 1,..., x n ) exist and renormalization conditions are satisfied.
59 Relativistic (perturbative) QFT Path-integral for the interacting theory 1 Interaction S Λ 0 E,int [φ] = (V ) d 4 x ( a Λ 0 1 φ(x)2 + a Λ 0 2 µφ(x) µ φ(x) + a Λ 0 3 φ(x)4). 2 The generating functional of interacting Green functions G Λ 0 E [ij] = e i φ,j e SΛ 0 E,int [φ] dµ(c Λ 0, φ). 3 Perturbative renormalizability: Find a Λ 0 i = r 1 aλ 0 i,r λr s.t. G E,r (x 1,..., x n ) = lim Λ 0 G Λ 0 E,r (x 1,..., x n ) exist and renormalization conditions are satisfied.
60 Relativistic (perturbative) QFT Path-integral for the interacting theory 1 Interaction S Λ 0 E,int [φ] = (V ) d 4 x ( a Λ 0 1 φ(x)2 + a Λ 0 2 µφ(x) µ φ(x) + a Λ 0 3 φ(x)4). 2 The generating functional of interacting Green functions G Λ 0 E [ij] = e i φ,j e SΛ 0 E,int [φ] dµ(c Λ 0, φ). 3 Perturbative renormalizability: Find a Λ 0 i = r 1 aλ 0 i,r λr s.t. G E,r (x 1,..., x n ) = lim Λ 0 G Λ 0 E,r (x 1,..., x n ) exist and renormalization conditions are satisfied.
61 Status of QED Perturbative QED 1 Formally given by the action S = d 4 x { ψ(iγ µ ( µ + iea µ ) m)ψ 1 4 F µνf µν}, where 1 ψ - Dirac field, 2 A µ - electromagnetic potential, 3 F µν := µ A ν ν A µ - the Faraday tensor. 2 QED is a perturbatively renormalizable theory. [Feldman et al 88, Keller-Kopper 96]
62 Status of QED Axiomatic QED: Consider a Haag-Kastler theory (U, H, Ω, A) whose pointlike localized fields include F µν and j µ = e ψγ µ ψ s.t. µ F µν = j ν, α F µν + µ F να + ν F αµ = 0.
63 I I I I Status of QED Status of ψ: The DHR criterion not suitable for electrically charged representations π of QED i.e. π A(O ) π 0 A(O ) fails. x 0 O O O x 1 Indeed, due to the Gauss Law one can determine the electric charge in O by operations in O.
64 I I I I Status of QED Status of ψ: The DHR criterion not suitable for electrically charged representations π of QED i.e. π A(O ) π 0 A(O ) fails. x 0 O O O x 1 Indeed, due to the Gauss Law one can determine the electric charge in O by operations in O.
65 Status of QED Status of ψ: An alternative criterion proposed in [Buchholz-Roberts 13] π A(C c ) π 0 A(C c ) C C C : = C CC 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 can be constructed. [Alazzawi-W.D. 15]
66 Status of QED Status of ψ: An alternative criterion proposed in [Buchholz-Roberts 13] π A(C c ) π 0 A(C c ) C C C : = C CC 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 can be constructed. [Alazzawi-W.D. 15]
67 Status of QED Status of ψ: An alternative criterion proposed in [Buchholz-Roberts 13] π A(C c ) π 0 A(C c ) C C C : = C CC 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 can be constructed. [Alazzawi-W.D. 15]
68 Status of QED Status of ψ: An alternative criterion proposed in [Buchholz-Roberts 13] π A(C c ) π 0 A(C c ) C C C : = C CC 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 can be constructed. [Alazzawi-W.D. 15]
69 Status of QED Status of A µ. 1 Suppose j µ = 0 and F µν 0. Then A µ is not a Wightman field. [Strocchi]. 2 Standard way out: A µ as a Wightman field on indefinite metric "Hilbert space" [Gupta-Bleuler]. Can one avoid it? 3 Free A µ in the axial gauge (i.e. e µ A µ = 0) is a string-like localized field. [Schroer, Mund, Yngvason 06]. 4 In fact, A µ (x, e) = 0 dt F µν (x + te)e ν, U( Λ)A µ (x, e)u( Λ) 1 = (Λ 1 ) µ ν Aν (Λx, Λe).
70 Status of QED Status of A µ. 1 Suppose j µ = 0 and F µν 0. Then A µ is not a Wightman field. [Strocchi]. 2 Standard way out: A µ as a Wightman field on indefinite metric "Hilbert space" [Gupta-Bleuler]. Can one avoid it? 3 Free A µ in the axial gauge (i.e. e µ A µ = 0) is a string-like localized field. [Schroer, Mund, Yngvason 06]. 4 In fact, A µ (x, e) = 0 dt F µν (x + te)e ν, U( Λ)A µ (x, e)u( Λ) 1 = (Λ 1 ) µ ν Aν (Λx, Λe).
71 Status of QED Status of A µ. 1 Suppose j µ = 0 and F µν 0. Then A µ is not a Wightman field. [Strocchi]. 2 Standard way out: A µ as a Wightman field on indefinite metric "Hilbert space" [Gupta-Bleuler]. Can one avoid it? 3 Free A µ in the axial gauge (i.e. e µ A µ = 0) is a string-like localized field. [Schroer, Mund, Yngvason 06]. 4 In fact, A µ (x, e) = 0 dt F µν (x + te)e ν, U( Λ)A µ (x, e)u( Λ) 1 = (Λ 1 ) µ ν Aν (Λx, Λe).
72 Status of QED Status of A µ. 1 Suppose j µ = 0 and F µν 0. Then A µ is not a Wightman field. [Strocchi]. 2 Standard way out: A µ as a Wightman field on indefinite metric "Hilbert space" [Gupta-Bleuler]. Can one avoid it? 3 Free A µ in the axial gauge (i.e. e µ A µ = 0) is a string-like localized field. [Schroer, Mund, Yngvason 06]. 4 In fact, A µ (x, e) = 0 dt F µν (x + te)e ν, U( Λ)A µ (x, e)u( Λ) 1 = (Λ 1 ) µ ν Aν (Λx, Λe).
73 Status of QED Status of A µ. Open questions: 1 What is the role of A µ from the DHR perspective? Is it a charge-carrying field of some charge? 2 How to construct the local gauge group starting from observables? 3 What is the intrinsic meaning of local gauge invariance?
74 Status of QED Status of A µ. Open questions: 1 What is the role of A µ from the DHR perspective? Is it a charge-carrying field of some charge? 2 How to construct the local gauge group starting from observables? 3 What is the intrinsic meaning of local gauge invariance?
75 Status of QED Status of A µ. Open questions: 1 What is the role of A µ from the DHR perspective? Is it a charge-carrying field of some charge? 2 How to construct the local gauge group starting from observables? 3 What is the intrinsic meaning of local gauge invariance?
76 Physics Spacetime symmetries Charges, global gauge symmetries CPT symmetry Quantum fields Path integrals Renormalizability Mathematics Representations of groups Representations of C -algebras Tomita-Takesaki theory Theory of distributions Measure theory in infinite dim. Combinatorics/ODE
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