Renormalisation Circumvents Haag's Theorem

Some no-go results & canonical quantum ﬁelds Axiomatics & Haag’s theorem Renormalisation bypasses Haag’s theorem

Renormalisation circumvents Haag’s theorem

Lutz Klaczynski, HU Berlin

Theory seminar DESY Zeuthen, 31.03.2016

Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag’s theorem Some no-go results & canonical quantum ﬁelds Axiomatics & Haag’s theorem Renormalisation bypasses Haag’s theorem

Outline

1 Some no-go results & canonical quantum ﬁelds Sharp-spacetime quantum ﬁelds Interaction picture Theorems by Powers & Baumann (CCR/CAR) Schrader’s result 2 Axiomatics & Haag’s theorem Wightman axioms Haag vs. Gell-Mann & Low Haag’s theorem for nontrivial spin 3 Renormalisation bypasses Haag’s theorem Renormalisation Mass shift wrecks unitary equivalence Conclusion

Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag’s theorem Sharp-spacetime quantum fields Some no-go results & canonical quantum fields Interaction picture Axiomatics & Haag’s theorem Theorems by Powers & Baumann (CCR/CAR) Renormalisation bypasses Haag’s theorem Schrader’s result The quest for understanding canonical quantum fields

1 1 Constructive approaches by Glimm & Jaffe : d ≤ 3 2 2 Axiomatic quantum field theory by Wightman & G˚arding : axioms to construe quantum fields in terms of operator theory hard work, some results (eg PCT, spin-statistics theorem) interesting: triviality results (no-go theorems) 3 other axiomatic approaches, eg algebraic quantum field theory

General form of no-go theorems φ quantum ﬁeld with properties so-and-so ⇒ φ is trivial.

3 forms of triviality (1) φ free, (2) φ = cI or (3) φ = 0 1J.Glimm, A.Jaffe, Springer (1981) 2A. S. Wightman, L.G˚arding,Arkiv förFysik, 28, 129 - 184 (1964) Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag’s theorem Sharp-spacetime quantum fields Some no-go results & canonical quantum fields Interaction picture Axiomatics & Haag’s theorem Theorems by Powers & Baumann (CCR/CAR) Renormalisation bypasses Haag’s theorem Schrader’s result Triviality of sharp-spacetime fields

Theorem (Wightman, 1964) Let ϕ(x) be a Poincar´e-covariant Hermitian scalar ﬁeld, that is,

U(a, Λ)ϕ(x)U(a, Λ)† = ϕ(Λx + a)

and suppose it is a well-deﬁned operator with vacuum Ω0 in its domain. Then there is a constant c ∈ C such that

√ n/2 ϕ(x)Ω0 = cΩ0 thus hΩ0|ϕ(x1)...ϕ(xn)Ω0i = c .

Free canonical ﬁeld Z 3 d p 1 −ip·x ip·x † ϕ(x) = 3 p [e a(p) + e a (p)]. (2π) 2Ep

Question: ||ϕ(x)Ω0|| = ?

Answer: unphysical question, result:

Z 3 2 1 d p 1 ||ϕ(x)Ω0|| = hΩ0|ϕ(x)ϕ(x)Ω0i = = ∞ 2 (2π)3 pp2 + m2

1 Implement Heisenberg’s uncertainty: Z ϕ(t, f ) = d3x f (x)ϕ(t, x),

free ﬁelds smoothed out in space, then

Z 3 2 2 1 d p |f (p)| ||ϕ(t, f )Ω0|| = < ∞. 2 (2π)3 pp2 + m2

2 Too strong an assumption: x 7→ hΨ|U(x, I)Φi continuous for all state vectors Ψ, Φ ∈ H, must be weakened.

1 Smearing in both space and time such that

f 7→ hΨ|ϕ(f )Φi Ψ, Φ ∈ H,

is distribution on Schwartz space S(M). 2 Poincar´ecovariance: for all f Schwartz

U(a, Λ)ϕ(f )U(a, Λ)† = ϕ({a, Λ}f ),

where ({a, Λ}f )(x) = f (Λ−1(x − a)). 3 Sharp-time ﬁeld

ϕ(t, f ) := lim ϕ(δt ⊗ f ) →0 if existent, {δt }>0 Dirac family centred at t ∈ R.

Wightman, 1967 R 3 Hamiltonian H = H0 + Hint, where Hint = d x Hint(x). Then Z iH0t −iH0t 3 iH0t −iH0t HI (t) = e Hinte = d x e Hint(x)e

entails ||HI (t)Ω0|| = 0 for all t, thus HI (t)Ω0 = 0.

−iH t Proof. By e 0 Ω0 = Ω0 and translational invariance one has Z Z 2 3 3 ||HI (t)Ω0|| = d x d y hΩ0|Hint(x)Hint(y)Ω0i Z Z 3 3 = d x d y hΩ0|Hint(0)Hint(y)Ω0i

||HI (t)Ω0|| not of interest, yet in Vacuum expectation of Dyson’s series

R −i dt HI (t) hΩ0|SΩ0i = hΩ0|T {e }Ω0i X (−1)n Z Z = dt ... dt hΩ |T{H (t ) ... H (t )}Ω i n! 1 n 0 I 1 I n 0 n≥0

there one ﬁnds Z Z dt ds hΩ0|T{HI (t)HI (s)}Ω0i = vacuum diagrams

but cancel out in Gell-Mann-Low formula.

Triviality of ﬁelds obeying CCR/CAR

Theorem (Baumann): scalar ﬁeld ϕ(t, f ) in d ≥ 5 Nontrivial commutator given by

[ϕ(t, f ), ∂t ϕ(t, g)] = i(f , g) ⇒ ϕ(t, f ) free

Weyl form of CCR: d ≥ 4 (Sinha).

Theorem (Powers): Dirac ﬁeld ψ(t, f ) in d ≥ 3 Nontrivial anticommutator given by

{ψ(t, f ), ψ†(t, g)} = i(f , g) ⇒ ψ(t, f ) free. Response: so be it.

Theorem (Schrader, 1974) 0 2 dµ0(ϕ) Gaussian measure on distributions D (R ), interaction term Z 2 V`(λ, ϕ) = λ d x : P(ϕ(x)) : B`

2 P bounded normalised polynomial, B` = [−`/2, `/2] and

−[V`(λ,ϕ)−E`(λ)] dµ`,λ(ϕ) = e dµ0(ϕ)

−E (λ) R −V (λ,ϕ) with e ` = e ` dµ0(ϕ). Then dµ∞,λ exists but

dµ0 and dµ∞,λ have mutually disjoint support.

Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag’s theorem Some no-go results & canonical quantum ﬁelds Wightman axioms Axiomatics & Haag’s theorem Haag vs. Gell-Mann & Low Renormalisation bypasses Haag’s theorem Haag’s theorem for nontrivial spin

Wightman framework: axioms (scalar ﬁeld)

Axiom 0 (Relativistic Hilbert space) Hilbert space H, strongly continuous unitary representation (a, Λ) 7→ U(a, Λ) of the connected Poincar´egroup, unique vacuum Ω0 ∈ H

U(a, Λ)Ω0 = Ω0 ∀(a, Λ).

Axiom I (Spectral condition) generator of translations Pµ has spectrum in forward lightcone σ(P) ⊂ V +. Axiom II (Quantum field) Poincaré-covariant operator family {ϕ(f )}f ∈S(M), common dense and Poincaré-stabledomain D ⊂ H, generate dense subspace D0 = C[ϕ(f ): f ∈ S(M)]Ω0 ⊂ H. Axiom III (Locality) [ϕ(f ), ϕ(g)] = 0 if f , g have spacelike-separated support.

Haag’s theorem

Haag’s theorem (Wightman, Jost, Schroer, Reeh, Schlieder)

1 ϕ(t, f ) and ϕ0(t, f ) sharp-time Wightman quantum ﬁelds of mass m > 0 with Hilbert spaces H and H0, respectively. In any spacetime dimension d ≥ 2.

2 Suppose at time t there is a unitary intertwiner V : H0 → H such that −1 ϕ(t, f ) = V ϕ0(t, f )V .

Then if ϕ0(t, f ) is free, so is ϕ(t, f ).

Proof of Haag’s theorem part I (sketch)

1. First prove intertwiner relates vacua: V Ω0 = Ω. Then follows

−1 −1 hΩ0|ϕ0(t, f )ϕ0(t, h)Ω0i = hΩ0|V ϕ(t, f )VV ϕ(t, h)V Ω0i = hΩ|ϕ(t, f )ϕ(t, h)Ωi.

2. This implies

hΩ0|ϕ0(t, x)ϕ0(t, y)Ω0i = hΩ|ϕ(t, x)ϕ(t, y)Ωi

in the sense of distributions, ie

1 Z d3p eip·(x−y) hΩ|ϕ(t, x)ϕ(t, y)Ωi = . 2 (2π)3 pp2 + m2

Proof of Haag’s theorem part II (sketch)

3. Use Poincar´ecovariance to show

1 Z d3p eip·(x−y) hΩ|ϕ(x)ϕ(y)Ωi = . 2 (2π)3 pp2 + m2

for spacelike distances (x − y)2 < 0. 4. Technical (edge of the wedge theorem, ...): then for all x − y ∈ M. 5. Employ Jost-Schroer theorem : ϕ has free-ﬁeld 2-point function =⇒ ϕ has free-ﬁeld n-point correlators.

Countering canonical narrative

Heisenberg picture ﬁeld (interacting ﬁeld)

ϕ(t, x) = eiHt ϕ(x)e−iHt

Interaction picture ﬁeld (free ﬁeld)

iH0t −iH0t ϕ0(t, x) = e ϕ(x)e

Intertwining relation

iHt −iH0t iH0t −iHt † ϕ(t, x) = e e ϕ0(t, x)e e = Vt ϕ0(t, x)Vt

Interaction picture S-matrix & Gell-Mann-Low formula

Interaction picture S-matrix

iH0t −iH(t−s) −iH0s † S = lim lim e e e = lim lim Vt V t→∞ s→−∞ t→∞ s→−∞ s as time-ordered exponentials:

R R t −i HI −i HI S = T {e } Vt = T {e 0 }

Gell-Mann-Low formula

hΩ0|T{Sϕ0(x1)...ϕ0(xn)}Ω0i hΩ|T{ϕ(x1)...ϕ(xn)}Ωi = hΩ0|SΩ0i

6= hΩ0|T{ϕ0(x1)...ϕ0(xn)}Ω0i Contradiction to Haag’s theorem? No.

Nontrivial spin case

Axioms modiﬁed Poincar´ecovariance (Axiom II) n X −1 U(a, Λ)φs (x)U(a, Λ) = Ssr (Λ )φr (Λx + a) r=1

Ssr ﬁnite-dimensional representation of Poincar´egroup Anticommutator (Axiom III) For half-integer spin

† [φs (f ), φr (g) ]+ = 0

if f , g have mutually spacelike-separated support. Then: Higher spin fields also affected by Haag’s theorem. Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag’s theorem Some no-go results & canonical quantum fields Wightman axioms Axiomatics & Haag’s theorem Haag vs. Gell-Mann & Low Renormalisation bypasses Haag’s theorem Haag’s theorem for nontrivial spin

Special species: gauge ﬁelds

Theorem (Strocchi, 1967)

Aµ Wightman ﬁeld with Poincar´ecovariance (Axiom II)

† σ U(a, Λ)Aµ(f )U(a, Λ) = Λµ Aσ({a, Λ})f )

or commuting at spacelike distances (Axiom III). µν Assume Fµν = ∂µAν − ∂νAµ satisﬁes ∂µF = 0, then

hΩ, Fµν(f )Fρσ(g)Ωi = 0.

If all axioms are fulﬁlled, then Fµν = 0 and thus Aµ = ∂µg(x). Expedient (free case): Gupta-Bleuler quantisation in Krein space.

Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag’s theorem Some no-go results & canonical quantum ﬁelds Renormalisation Axiomatics & Haag’s theorem Mass shift wrecks unitary equivalence Renormalisation bypasses Haag’s theorem Conclusion

Change of story: renormalisation

Haag’s theorem no big surprise: UV divergences in PT backpedal: so far, wrong Lagrangian 1 1 g L = (∂ϕ)2 − m2ϕ2 − ϕ4 2 2 4!

new, renormalised Lagrangian 1 1 g L = (∂ϕ )2 − m2ϕ2 − r ϕ4 + L r 2 r 2 r r 4! r ct with counterterm Lagrangian 1 1 g L = (Z − 1)(∂ϕ )2 − m2(Z − 1)ϕ2 − r (Z − 1)ϕ4 ct 2 r 2 r m r 4! g r

Counterterms induce mass shift

Canonical narrative continued New renormalised interaction part replace old by new interaction part g 1 1 Lr = − r Z ϕ4 − (Z − 1)(∂ϕ )2 − m2(Z − 1)ϕ2 . int 4! g r 2 r 2 r m r | {z } induce mass shift and put it through ’Gell-Mann-Low machine’ with new renormalised S-matrix

i R Lr Sr = T{e int } ⇒ ﬁnite results in perturbation theory, physically sensible.

Mass shift in free theory

2 2 2 2 3 Consider mass shift m0 → m = m0 + δm in free theory : Theorem (Haag’s theorem for free ﬁelds)

Let ϕ and ϕ0 be two free ﬁelds of masses m and m0, respectively. If at time t there is a unitary map V such that

−1 ϕ(t, x) = V ϕ0(t, x)V ,

then m = m0. Interacting theory unitarily equivalent to free theory?

3Reed, Simon (Academic Press, 1975) Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag’s theorem Some no-go results & canonical quantum ﬁelds Renormalisation Axiomatics & Haag’s theorem Mass shift wrecks unitary equivalence Renormalisation bypasses Haag’s theorem Conclusion

Mass shift, canonical treatment

Mass shift as interaction 2 2 2 mass shift m = m0 + δm implemented by 1 1 1 L = (∂ϕ)2 − m2ϕ2 − δm2ϕ2 m 2 2 0 2 perturbative Gell-Mann-Low procedure

hΩ |T{Sϕ (x)ϕ (y)}Ω i hΩ|T{ϕ(x)ϕ(y)}Ωi = 0 0 0 0 , hΩ0|SΩ0i

−i 1 δm2 R ϕ2 with S-matrix S = T{e 2 } and formal intertwiner

−i 1 δm2 R t ϕ2 Vt = T {e 2 0 }

Mass shift intertwiner not unitary

Claim

ϕ0(t, x) free ﬁeld of mass m0, for

i 1 δm2 R t ϕ2 −i 1 δm2 R t ϕ2 ϕ(t, x) = T {e 2 0 0 }ϕ0(t, x)T {e 2 0 0 }

subjected to Gell-Mann-Low procedure with mass shift interaction one gets

hΩ0|T{Sϕ0(x)ϕ0(y)}Ω0i 2 hΩ|T{ϕ(x)ϕ(y)}Ωi = = i∆F (x −y; m ), hΩ0|SΩ0i

1 2 R 2 2 2 2 −i δm ϕ0 with mass m = m0 + δm and S-matrix S = T {e 2 }. −i 1 δm2 R t ϕ2 Therefore Vt = T {e 2 0 0 } not unitary.

Mass shift destroys unitary equivalence

Renormalisation leads to mass shift

Full renormalised theory: self-energy Σr (gr , p, mr ) i Gr (gr , p) = 2 2 + p − mr − Σr (gr , p, mr ) + i0 incurs momentum (and scale) and coupling-dependent mass shift! Theory cannot possibly be unitary equivalent to theory with i G0,r (p) = 2 2 + . p − mr + i0

Conclusion

1 Axiomatic framework interesting but not the answer. 2 Some aspects of import, especially Haag’s theorem to be taken seriously. 3 Haag’s theorem most likely averted by renormalisation. 4 Thus renormalised interacting quantum ﬁelds not unitarily equivalent to free ﬁelds. Further problems: renormalised S-matrix still unitary ... ?

Lutz Klaczynski, HU Berlin Renormalisation circumvents Haag’s theorem