TheUsesofReflectionPositivity - ETH Z · TheUsesofReflectionPositivity Erhard Seiler...

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The Uses of Reflection Positivity Erhard Seiler Max-Planck-Institut f ¨ ur Physik, M¨ unchen (Werner-Heisenberg-Institut) Schrader memorial, ETH 2106 – p.1/25

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The Uses of Reflection Positivity

Erhard Seiler

Max-Planck-Institut fur Physik, Munchen

(Werner-Heisenberg-Institut)

Schrader memorial, ETH 2106 – p.1/25

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from Robert’s home page at FU Berlin

Schrader memorial, ETH 2106 – p.2/25

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1. Prelude: Res Jost“The General Theory of Quantized Fields”, 1964

∃ vector valued functions

Ψ(z1, . . . , zn) = ” Φ(z1) . . . Φ(zn)Ω ”,

holomorphic for z1, . . . , zn ∈ T+

n ,

T +n ≡z1, . . . , zn ∈ C

4|Im z1 ∈ V+,

Im(zk − zk−1) ∈ V+, k = 2, . . . n .

V+ forward light cone,T +

n can be extended by permutation symmetry to T pn .

T +n , T p

n contain Euclidean points (~zk real, z(4)k = ix

(4)k ).

Schrader memorial, ETH 2106 – p.3/25

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Imaginary time reflection

ϑ(~xk, ix(4)k ) ≡ (~xk,−ix

(4)k ) ,

ΘΨ(z1, . . . , zn) ≡ Ψ(ϑz1, . . . , ϑzn) .

||Ψ(z1, . . . , zn)||2 = (Ω,Θ[Φ(z1) . . . Φ(zn)]∗Φ(z1) . . . Φ(zn)) ≥ 0 .

“Reflection positivity in a nutshell”Extends to polynomials in Φ.

RP basis for probabilistic interpretation of EQFT

Remark: ∃ many situations where Euclidean correlationfunctions neither positive, nor moments of a measure.RP paramount.

Schrader memorial, ETH 2106 – p.4/25

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2. Osterwalder&SchraderAxioms for Euclidean Green’s functions:Two landmark papers (OS 1973, 1975):

• Euclidean Covariance

• Reflection positivity

• (Anti-)symmetry

• Cluster property

• “Linear growth condition”

=⇒Wightman axioms for Minkowski QFT

In a nutshell: P0 ≥ 0 =⇒

contractive semigroup e−P0t ↔ unitary group eiP0t

Schrader memorial, ETH 2106 – p.5/25

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3. Duality: Lie groups / algebrasSymmetric spaceLie group G, closed connected subgroup H, involutiveautomorphism σ s.t.

σ H = id .

⇒ “symmetric Lie algebra” g, involutive automorphism σ,

g = h⊕m ,

σ h = id , σ m = −m .

[h, h] ⊂ h , [h,m] ⊂ m , [m,m] ⊂ h , g∗ = h⊕ im ,

g∗ different real Lie algebra

g←→ g∗ “c-duality” (P. E. T. Jørgensen et al)

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g∗ generates simply connected Lie group G∗:“c-dual Lie group”

g and g∗ different real forms of complex Lie algebra gC

Examples:

g = o(n), h = o(n− 1), g∗ = o(n− 1, 1) ,

g = e(n), h = e(n− 1), g∗ = p(n)

e(n) euclidean Lie algebrap(n) Poincaré Lie algebraSometimes: g compact, g∗ noncompact

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4. Lüscher-Mack“Global Conformal Invariance in Quantum Field Theory”Lüscher, Mack CMP 1975(Resolved some confusion over ‘weak’ vs ‘strong’conformal invariance.)

Use: ∃ contractive semigroup S ⊂ G of full dimension=⇒ unitary rep. of simply connected dual Lie group.Here:G∗ = ˜SO(4, 2)e, g = so(5, 1), g∗ = so(4, 2), m generates S.m : generated by so(4, 1) , P0 and commutators.

Schrader memorial, ETH 2106 – p.8/25

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Result:Assume

(1) Schwinger functions SO(5, 1)e invariant on S4

“weak conformal invariance”(2) RP

Then ∃ analytic continuation to Wightman-like QFT on M

(infinite cover of Minkowski space = Einstein universe)

carrying unitary rep. of G∗ = ˜SO(4, 2)e.

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Crucial toolTheorem:Assume: ∃ open convex cone V ⊂ m s. t.i) ∀X ∈ V, u ∈ Ge uXu−1 ∈ V

ii) V and h span g

S0 ≡ g ∈ G| ∃X1, . . . , Xk ∈ V, u ∈ Ge : g = eX1 . . . eXku

S ≡ S0 ∪Ge

– π contractive rep. of S, strongly cont. at e

– π(g)∗ = π(σ(g)−1)

Then π continues analytically to cont. unitary rep. of G∗.

Schrader memorial, ETH 2106 – p.10/25

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5. Statistical mechanicsRP u Existence of positive transfer matrixStructure:Net of commutative or Grassmann algebrasZd 3 O 7→ A(O). Global algebra A, A+,A− ⊂ A.Antilinear antimorphism Θ(A− → A+

Multiplicative convex cone C generated by elementsΘ(A)A, A ∈ A+; Boltzmann factor eS with S ∈ C.A priori expectation functional 〈·〉0 RP⇔ 〈A〉0 ≥ 0 forA ∈ C

Consequence (RP): 〈AΘA〉 ≡ 〈A (ΘA)eS〉0〈eS〉0

≥ 0 .

(Fröhlich-Israel-Lieb-Simon 1978; Osterwalder-S. 1978)

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Fröhlich-Simon-Spencer 1976:Infrared Bounds =⇒ first (only?) proof of sponaneousbreaking of a continuous symmetry

Fröhlich-Israel-Lieb-Simon 1978, 1980:Numerous results on existence of phase transitions:

• Long range interactions,

• antiferromagents in external field

• classical&quantum Coulomb gases

• six-vertex models

• · · ·

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Many further applications• Liquid crystals: Heilmann& Lieb 1979

• Lattice dipole gases: Fröhlich& Spencer 1981

• · · ·

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6. Lattice gauge theoryRP⇒ particle masses from Euclidean correlations

(BMW collaboration 2012)

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Applying IR bounds in LGTC. Borgs, E. S. 1983: Deconfinement transition inquenched QCD (T > 0)

M. Salmhofer, E. S. 1991: Chiral symmetry breaking atstrong coupling

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7. ‘Virtual Representations’(Fröhlich-Osterwalder-S. 1983; cf. Klein-Landau 1983)

Abstracts group theoretical essence from O-S

Motivation: O-S/Wightman axioms problematic for gaugetheories

F-O-S: minimal assumptions; no Euclidean functionalmeasure, no unitary rep. of Euclidean group.

(Relevant for:QCD with positive density,QFT with topological term)

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Setup• Rn Eucl. space, En univ. cover of Eucl. group

• Rn 3 O 7→ S(O) isotonic net of complex top. vectorspaces over C; S ≡

O∈Rn separable

• cont. representation τ of En on S

• cont. time reflection ϑ on S

• antilinear involution J on S with JS(O) = S(O);Jτg = τgJ ∀ g ∈ En; Θf ≡ Jϑ(f)

• RP: cont. bilinear functional S : S× S→ C withS(Θf, f) ≥ 0 ∀f ∈ S+ (S+ =

O∈Rn+, Rn

+ : xn > 0)

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Results:The structure determines

• Hilbert space H

• continuous linear map φ : S+ →H

• continuous unitary representation π∗ of univ. cover ofPoincaré group P on H

Schrader memorial, ETH 2106 – p.18/25

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Bisognano-Wichmann 1975Wedge: W ≡ x ∈M | x1 > |x0|

B1 ≡M01

boost leaving invariant (s.a.)exp(2πB1) = ∆W modular operatorfor vacuum representation of wedge algebra

Borchers 1995: Generalization to double cones forconformal QFTs

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8. Schrader: duality of repsRobert Schrader 1986:Group theoretic setup: ≈ as before (symmetric spaceG/H, symmetric Lie algebra g = h⊕m, invol. autom. σ)Furthermore:– Hilbert space H,– unitary rep. π of G on H,– closed subspace H+ ⊂ H,– unitary involution Θ

RP: 〈f, f〉Θ ≡ 〈f,Θf〉 ≥ 0 ∀f ∈ H+ =⇒ HΘ ≡ H+/NΘ

Semigroup S ≡ g ∈ G|π(g)H+ ⊂ H+ of full dim.!π(S) contractive on H+, HΘ (slight abuse)=⇒ Lüscher-Mack theorem applies!

Schrader memorial, ETH 2106 – p.20/25

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Concretely:G = SL(2n, C), G∗ = SU(n, n)× SU(n, n),π ∈ complementary series.

σX ≡ −JX∗J ∀X ∈ sl(2n, C); J =

(

0 1I

−1I 0

)

h = sub-Lie algebra leaving J invariant =⇒

h ∼= su(n, n); g∗ ∼= su(n, n)⊕ su(n, n)

Result:π(S) analytically continues tostrongly cont. unitary rep. π∗ of G∗ on HΘ.

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9. Further workP. E. T. Jørgensen et al ≥ 1986: Generalize virtualrepresentations (dubbed “local reps”) and duality ofunitary reps.

P. E. T. Jørgensen and G. Ólafsson 1999: “UnitaryRepresentations and O-S Duality”“Successfully glues various topics together” (A. I. Shtern)

Examples:– π complementary series unitary irrep.; using L-Mtheorem⇒ unitary highest weight rep. of G∗

– π Generalized principal series unitary irrep.; usingvirtual rep. strategy⇒ unitary highest weight rep. of G∗.

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RP in various contexts“Radial Quantization” (see e.g. Rychkov 2012, EPFLlectures )

A.Jaffe et al 2008-2016≥ 10 articles with RP in title:

• free fields on static Riemannian manifolds

• parafermions

• Majoranas

• stochastic quantization

• complex measures

• . . .

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10. Schrader: Regge calculusRobert Schrader 1981-2015: Long term favorite subject(lattice) quantum gravity

Robert’s last paper:Schrader 2015: Reflection Positivity in simplicial gravity– Piecewise linear manifold– Reflection symmetric– Einstein-Hilbert action with cosmological term=⇒ Reflection positivity

Schrader memorial, ETH 2106 – p.24/25

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11. Robert Schrader

1939 – 2015

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