Reflection positivity. Abstracts from the workshop held November 26 – December 2, 2017. (English) Zbl 1409.00095
Summary: The main theme of the workshop was reflection positivity and its occurences in various areas of mathematics and physics, such as Representation Theory, Quantum Field Theory, Noncommutative Geometry, Dynamical Systems, Analysis and Statistical Mechanics. Accordingly, the program was intrinsically interdisciplinary and included talks covering different aspects of reflection positivity.
MSC:
| 00B05 | Collections of abstracts of lectures |
| 00B25 | Proceedings of conferences of miscellaneous specific interest |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
| 22E65 | Infinite-dimensional Lie groups and their Lie algebras: general properties |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 81T08 | Constructive quantum field theory |
| 22-06 | Proceedings, conferences, collections, etc. pertaining to topological groups |
| 17-06 | Proceedings, conferences, collections, etc. pertaining to nonassociative rings and algebras |
Software:
GMRFLibReferences:
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| [53] | Arthur Wightman, Quantum field theory in terms of vacuum expectation values,https:// journals.aps.org/pr/abstract/10.1103/PhysRev.101.860{\it P}{\it hysical Review}, 101 (1956), 860-866. Direct construction of pointlike observables in the Ising model Daniela Cadamuro Relativistic quantum field theories are described by their set of local observables. These are linear bounded or unbounded operators associated with regions of Minkowski space. They form∗-algebras that are expected to satisfy, e.g., the Haag-Kastler axioms, which are relevant to their interpretation as physical “measurements”. The problem of constructing models of quantum field theory, i.e., exhibiting algebras of local observables with such properties, is a notoriously hard task due to the complicated structure of local observables in the presence of interaction. Quantum integrable models in 1+1-dimensional Minkowski space are simplified models of interaction, rendering the mathematical structure of quantum field theory more accessible. In these models, the scattering of n particles is the product of two particle scattering processes, namely the S-matrix is said to be “factorizing”, a property connected to integrability. Examples include the Ising model, the O(N ) nonlinear sigma models and the Sine-Gordon model. 3282Oberwolfach Report 55/2017 We are interested in studying the content of local observables in these theories. This can be investigated in various mathematical frameworks: as Wightman fields [1], as algebras of bounded operators [2], or as closed operators affiliated with those algebras. For example, the task of constructing the Wightman n-point functions in integrable models from a given S-matrix has been widely studied, see, e.g., [3], but convergence of the associated series expansions has not been established so far, despite some progress [4]. An alternative approach considers fields localized in unbounded wedge-shaped regions as an intermediate step to the construction of sharply localized objects, which is handled indirectly [5,6,7,8], thus avoiding explicit computation of pointlike fields. The existence proof of local observables is reduced to an abstract condition on the underlying wedge algebras. While the generators of the wedge algebras are explicitly known, the passage to the von Neumann algebras includes the weak limit points of this set. These limit points include the elements of local algebras, but of these much less is known. Our task is to gain more information on the structure of these local observables. For that, we characterize the local observables in terms of a family of coefficient functions fm,n[A]in the following series expansion [9]: X∞dmθθθdnηηη (1)A =fm,n[A](θθθ, ηηη)z†(θ1)· · · z†(θm)z(η1)· · · z(ηn), m,n=0m!n! where z†, z are “interacting” creators and annihilators fulfilling a deformed version of the CCR relations which involves the scattering function. Due to the form of this expansion, local observables are defined as quadratic forms in a suitable class. We denoteHω,fthe dense space of finite particle number states Ψ fulfilling the conditionkeω(H/µ)Ψk < ∞, where H is the Hamiltonian, µ > 0 is the mass, and ω : [0,∞) → [0, ∞) is a function with the properties of [10, Definition 2.1]. In (1), A is a quadratic form onHω,f× Hω,fsuch that kQkAe−ω(H/µ)Qkk + kQke−ω(H/µ)AQkk < ∞ for any k∈ N0, where Qkis the projector onto the space of k or fewer particles. We denote this class of quadratic forms byQω. In order to characterize the coefficients fm,n[A]in terms of the localization of A in spacetime, we need a notion of locality which is adapted to quadratic forms in the classQω: We say that A∈ Qωis ω-local in the double coneOx,y:=Wx∩ Wy′ (whereWxdenotes the right wedge with edge at x andWy′the left wedge with edge at y, with x to the left of y) if and only if [A, ϕ(f )] = [A, ϕ′(g)] = 0 for all f∈ Dω(Wy′) and all g∈ Dω(Wx), as a relation inQω. Here ϕ, ϕ′are the left and right wedge-local fields, respectively,Dω(Wx) is the space of smooth functions compactly supported inWxwith the property that θ7→ eω(cosh θ)f±(θ) is bounded and square integrable (f±is positive and negative frequency part of the Fourier transform, respectively.) The notion of ω-locality is weaker than the usual notion of locality in the net of C∗-algebrasA(Ox,y). It does not imply that A commutes with unitary operators Reflection Positivity3283 eiϕ(f )−, or with an element B∈ A(Wx): if A is just a quadratic form, it would not be possible to write down these commutators in a meaningful way. We therefore clarify how ω-locality is related to the usual locality: Proposition 1. (i) Let A be a bounded operator; then A is ω-local inOx,yfor some x, y∈ R2 if and only if A∈ A(Ox,y). (ii) Let A be a closed operator with coreHω,f, andHω,f⊂ dom A∗. Suppose that ∀g ∈ DωR(R2) : exp(iϕ(g)−)Hω,f⊂ dom A.(∗) Then A is ω-local inOx,yif and only if it is affiliated withA(Ox,y). (iii) In the case S =−1, statement (ii) is true even without the condition (∗). This proposition gives criteria for affiliation of closed operators to local algebras, but in examples, closability of a quadratic form A is difficult to characterize in terms of the coefficients in the expansion (1). Moreover, not much is known about the domain of the closed operator. We therefore look for sufficient (but not necessary) conditions that allow to apply Proposition1. We will understand (1) as an absolutely convergent sum on a certain domain, using summability conditions on the norms of the coefficients fm,n[A]. The following proposition provides a sufficient criterion for closability of A as an operator: Proposition 2. Let A∈ Qω. Suppose that for each fixed n, X∞ 2m/2 √kfm,n[A]kωm×n+kfn,m[A]kωn×m<∞. m=0m! Then, A extends to a closed operator A−with coreHω,f, andHω,f⊂ dom(A−)∗. To apply Proposition1we therefore need to fulfill the condition in Proposition2 and to show ω-locality of A. Hence, we formulate the ω-locality condition in terms [A] of properties of the functions fm,n. This is the content of [10, Theorem 5.4], which we summarize briefly: A is localized in the standard double coneOrof [A] radius r if and only if the coefficients fm,nare boundary values of meromorphic functions (Fk)∞k=0on Ck(with k = m + n) with a certain pole structure, which are S-symmetric, S-periodic, and fulfill certain bounds in the real and imaginary directions, depending on ω and r, and which fulfill the recursion relations 1YnYk resζn−ζm=iπFk(ζ) =−S(ζj− ζm)1−S(ζm− ζp)Fk−2(ˆζ). 2πi j=mp=1 The problem is now to find examples of functions (Fk)∞k=0fulfilling the above conditions of ω-locality and closability via Proposition2. In the case S =−1 (Ising model) this is possible, and we aim at constructing a large enough set of observables so that they have the Reeh-Schlieder property. 3284Oberwolfach Report 55/2017 To that end, let k≥ 0, let g ∈ D(Or) with some r > 0, and let P be a symmetric Laurent polynomial of 2k variables. We define the analytic functions XYk (2)F2k[2k,P,g](ζζζ) := ˜g(p(ζζζ))P (eζζζ)sign σsinhζσ(2j−1)− ζσ(2j), 2 σ∈S2kj=1 and Fj[2k,P,g]= 0 for j6= 2k. For these the properties above hold with respect to this r and for example with ω(p) := ℓ log(1 + p) for some ℓ > 0. Another example, involving the Fjfor odd j, is the non-terminating sequence (3)F2j+1[1,P,g](ζζζ) :=1g(p(ζζζ))P˜Y (2πi)k2j+1(eζζζ)tanhζℓ− ζ2r, 1≤ℓ<r≤2j+1 where g∈ Dω(Or), and P = (P2j+1)∞j=0are symmetric Laurent polynomials in 2j + 1 variables such that P2j+1(p,−p, q) = P2j−1(q). We set F2j[1,P,g]= 0. Also for these Fj, the properties above hold with respect to r and ω(p) = pαwith α∈ (0, 1). Hence in both examples the associated quadratic form A given by (1) is ω-local in the double coneOr. Additionally, the families of functions fulfill the condition of Proposition2, which implies that A extends to a closed operator affiliated with the local algebrasA(Or). This is in fact trivial for (Fj[2k,P,g])∞j=0as the sequence terminates; but for (Fj[1,P,g])∞j=0, it involves careful norm estimates of a sequence of singular integral operators, as one is concerned precisely with the boundary values of the function at the poles of the hyperbolic tangent. Further, by choosing different polynomials P we can generate a large set of observables which has the Reeh-Schlieder property. References |
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| [71] | Ruijsenaars S. N. M., On Bogoliubov transforms II. The general case., Ann. Phys. 116, 105-134 (1978). 3288Oberwolfach Report 55/2017 Ward identities as a tool to study functional integrals. Margherita Disertori (joint work with T. Spencer, M. Zirnbauer) In the context of field theory Ward identities are functional identites generated by internal symmmetries of the model. They generally appear as relations between Feynman diagrams, allowing to simplify the pertubative expansion, and in some cases even to close the Schwinger-Dyson equation. Such applications require nevertheless the model to be described by a small perturbation of a free (quadratic) action. A natural question is whether symmetry-generated identities may also help studying models where standard renormalization group techniques do not apply. In this context, in collaboration with M. Zirnbauer and T. Spencer [2], we considered the so-called H2|2supersymmetric nonlinear sigma model, introduced in [1] as a toymodel for quantum diffusion. It is also a key ingredient in the construction and study of certain stochastic processes with memory (cf. [4] [5] [6]). For this model we constructed a multiscale procedure whose key ingredient is a infinite family of Ward identities generated by supersymmetry. We hope a similar strategy may extend to other models with and without supersymmetry. References |
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| [73] | M. Disertori, T. Spencer, and M. R. Zirnbauer. Quasi-diffusion in a 3D supersymmetric hyperbolic sigma model. {\it Comm. Math. Phys.}, 300(2):435-486, 2010. · Zbl 1203.82018 |
| [74] | M. Disertori, T. Spencer. Anderson localization for a supersymmetric sigma model. {\it Comm.} {\it Math. Phys.}, 300:659-671, 2010. · Zbl 1203.82017 |
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| [76] | M. Disertori, C. Sabot, and P. Tarr‘es. Transience of Edge-Reinforced Random Walk. {\it Comm.} {\it Math. Phys.}, 339(1):121-148, 2015. · Zbl 1329.60116 |
| [77] | M. Disertori, F. Merkl, and S.W.W. Rolles. A supersymmetric approach to martingales related to the vertex-reinforced jump process. {\it ALEA}, 14:529-555, 2017. Symmetric R-spaces, reflection positivity and the Berezin form Jan Frahm (joint work with Gestur ´Olafsson, Bent Ørsted) We give a brief introduction to the Berezin form on symmetric R-spaces and explain its relation to reflection positivity. The main new result is a complete answer to the question for which parameters the Berezin form is positive semidefinite, or in other words, for which parameters the reflection positivity condition holds. Reflection Positivity3289 1. Symmetric R-spaces Roughly speaking, a symmetric R-space is a compact symmetric space K/L which is, at the same time, a homogeneous space for a larger non-compact semisimple group G. Typical examples are the Grassmannians X = Grp(Rp+q) consisting of all p-dimensional R-linear subspaces of Rp+q, where p, q≥ 1. The group G = SL(p + q, R) acts transitively on X by g· b = gb. Fixing the base point b0= Re1+· · · + Repwe can identify X≃ G/P , where P is the stabilizer of b0, a maximal parabolic subgroup of G. Since the maximal compact subgroup K = SO(p + q)⊆ G already acts transitively on X, we further have X ≃ K/L with L = P∩ K = S(O(p) × O(q)), which expresses X as a compact symmetric space. For simplicity, we focus on the example X = Grn(R2n), i.e. p = q = n, for the rest of this note and refer the interested reader to [5] for the general statements. 2. The Berezin form For λ∈ C we define a representation πλof G onE = C∞(X) by πλ(g)f (b) = j(g−1, b)−λ+n2f (g−1b),g∈ G, b ∈ X, where j(g, b) = det(prb◦ gtg◦ ib) with ib: b ֒→ R2nthe natural embedding and prb: R2n։b the orthogonal projection with respect to the standard inner product on R2n. This normalization is chosen, so that πλextends to an irreducible unitary representation on L2(X), the unitary principal series, if and only if λ∈ iR. For λ∈ R there is a πλ(G)-invariant Hermitian form onE given by ZZ hf1, f2iλ=|Cos(b1, b⊥2)|λ−nf1(b1)f2(b2) db1db2, XX where b⊥∈ X is the orthogonal complement of b ∈ X and the kernel function is Cos(b1, b2) = Vol(prb2(Eb1)) for any convex subset Eb1⊆ b1of volume 1 containing the origin (see e.g. [7]). We remark that this form depends meromorphically on λ and has to be regularized at all simple poles of the integral kernel. The form h·, ·iλ(or its regularization) is positive definite if and only if λ∈ (−1, 1). The corresponding irreducible unitary representations of G are called complementary series representations. Now consider the involutive automorphism τ of G given by τ (g) = In,ng−⊤In,n, where In,n= diag(In,−In). Its fixed point group is given by H = Gτ= SO(n, n) and the Lie algebra g decomposes as g = h + q into±1 eigenspaces of τ. The real form gc= h + iq of gCis given by gc= su(n, n). On the representation spaceE we also have an involution τ∗:E → E, τ∗f (b) = f (In,nb⊥) which is compatible with τ in the sense that πλ(τ (g)) = τ∗◦ πλ(g)◦ τ∗. 3290Oberwolfach Report 55/2017 Therefore, twisting the πλ(G)-invariant Hermitian formh·, ·iλby τ∗gives a πλ(H)and πλ(gc)-invariant Hermitian form ZZ hf1, f2iτ,λ:=hf1, τ∗f2iλ=|Cos(b1, In,nb2)|λ−nf1(b1)f2(b2) db1db2, XX which is called the Berezin form and was previously studied in [1,2,3]. 3. Reflection positivity Since the Berezin form is H- and gc-invariant, we can restrict it toE+= Cc∞(O) for any open H-orbitO ⊆ X and obtain an H- and gc-invariant Hermitian form onE+. We ask the following natural question: Question. For which open H-orbitsO in X and for which parameters λ ∈ R is the Berezin formh·, ·iτ,λpositive semidefinite onE+= Cc∞(O)? The positivity of the Berezin form is nothing else than reflection positivity for the involution τ∗with respect to the subspaceE+. In this case, we hope that the Lie algebra representation πλcof gconE+yields a unitary representation of the 1-connected group Gcwith Lie algebra gcon the Hilbert space completion ofE+ with respect to the Berezin formh·, ·iτ,λ. The open H-orbits in X are given by Oj={b ∈ X : ω|b×bhas signature (n− j, j)}(0≤ j ≤ n). Every open H-orbit is a symmetric space, more preciselyOj≃ SO(n, n)/S(O(n − j, j)× O(j, n − j)). In particular, every Ojhas an H-invariant pseudo-Riemannian metric which is Riemannian if and only if j = 0 or j = n. Theorem (see [5]). The restriction of the Berezin formh·, ·iτ,λtoE+= Cc∞(Oj) (0≤ j ≤ n) is positive semidefinite if and only if j ∈ {0, n} and λ ∈ (−∞, 1) ∪ {1, 2, . . . , n} or if j ∈ {1, . . . , n − 1} and λ = n. The first case leads to (scalar type) unitary highest weight representations of the group Gc= fSU(n, n). This was first observed by Enright [4] and Schrader [8] in special cases and later generalized by Jørgensen– ´Olafsson [6]. In the second case, the Berezin kernel is trivial, and the corresponding unitary representation πcλof Gc= SU(n, n) is the trivial representation. References |
| [78] | F. A. Berezin: Quantization in complex symmetric spaces. {\it Izv. Akad. Nauk SSSR Ser. Mat.} 39 (1975), no. 2, 363-402, 472. · Zbl 0312.53050 |
| [79] | G. van Dijk and S. C. Hille: Maximal degenerate representations, Berezin kernels and canonical representations. In: Komrakov, B., Krasil’shchik, J., Litvinov, G., Sossinky, A. (eds.) {\it Lie groups and Lie algebras, their representations, generalizations, and applications.} Dordrecht: Kluwer Academic, 1997, pp. 1-15. · Zbl 0895.22008 |
| [80] | G. van Dijk and V. F. Molchanov: The Berezin form for rank one para-Hermitian symmetric spaces. {\it J. Math. Pures Appl. }77 (1998), no. 8, 747-799. · Zbl 0919.43007 |
| [81] | T. J. Enright, Unitary representations for two real forms of a semi simple Lie algebra: A theory of comparison, {\it Lie Group Representations, I (College Park, Maryland, 1982/1983)}, LNM 1024, Springer 1983, pp. 1-29. Reflection Positivity3291 · Zbl 0531.22012 |
| [82] | J. Frahm, G. ´Olafsson and B. Ørsted, {\it The Berezin form on symmetric }R{\it -spaces and re-} {\it flection positivity}, to appear in the proceedings of the 50th Seminar Sophus Lie in Bedlewo, Banach Center Publications. · Zbl 1395.22005 |
| [83] | P. E. T. Jørgensen and G. ´Olafsson, {\it Unitary representations of Lie groups with reflection} {\it symmetry}, J. Funct. Anal. 158 (1998), no. 1, 26-88. · Zbl 0911.22012 |
| [84] | G. ´Olafsson and A. Pasquale, {\it The }Cosλ{\it and }Sinλ{\it transforms as intertwining operators} {\it between generalized principal series representations of }SL(n + 1, K), Adv. Math. 229 (2012), no. 1, 267-293. · Zbl 1244.22007 |
| [85] | R. Schrader, {\it Reflection positivity for the complementary series of }SL(2n, C), Publ. Res. Inst. Math. Sci. 22 (1986), 119-141. Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality Rupert L. Frank (joint work with Elliott H. Lieb) For 0 < λ < N and functions f and g on RNwe abbreviate Z Zf (x) g(y) Iλ[f, g] :=dx dy . RN×RN|x − y|λ According to the Hardy-Littlewood-Sobolev (HLS) inequality there is a constant HN,λsuch that for all f, g∈ Lp(RN) with p = 2N/(2N− λ), (1) Iλ[f, g] ≤ HN,λkfkpkgkp. In [7] Lieb computed the sharp (that is, smallest possible) constantHN,λin (1) and characterized the cases of equality. An alternative proof was given in [1]. The precise statement is Theorem 1. Let 0 < λ < N and p = 2N/(2N− λ). Then (1) holds with 1−λ/N Γ((N− λ)/2)Γ(N ) Γ(N− λ/2)Γ(N/2). Equality holds if and only if −(2N−λ)/2−(2N−λ)/2 f (x) = α β +|x − γ|2andg(x) = α′β +|x − γ|2, for some α, α′∈ C, β > 0 and γ ∈ RN. Our goal here is to sketch our proof [2] of Theorem1under the additional assumption N− 2 ≤ λ < N if N ≥ 3. In contrast to the proofs in [7] and [1], which rely on the technique of Schwarz symmetrization, our proof in [2] relies on reflection positivity. We also refer to [3] for a variation of the ideas in [2] and the extension of our method to the so-called logarithmic HLS inequality. Yet another rearrangement-free proof of Theorem1was given in [5], this time in the whole range 0 < λ < N . The work [5] was motivated by our proof of the sharp HLS inequality on the Heisenberg group [4], which is a situation where one cannot expect rearrangement techniques to work. It is hoped that the methods from 3292Oberwolfach Report 55/2017 [2,3,4,5] will be relevant in other situations, where rearrangement techniques cannot be used effectively. Inequality (1) is clearly invariant under translations and dilations. It is less obvious that it is invariant under the whole conformal group [7,1]. This fact will play a crucial role in our proof. Reflection positivity. Let B ={x ∈ RN:|x − a| < r}, a ∈ RN, r > 0, be a ball and denote by r2(x− a) |x − a|2+ a the inversion of a point x through the boundary of B. This map on RNis lifted to an operator acting on functions f on RNaccording to r2N−λ (ΘBf )(x) :=f (ΘB(x)) . |x − a| One easily finds that with p = 2N/(2N− λ) Iλ[f ] = Iλ[ΘBf ]andkfkp=kΘBfkp, where we abbreviated Iλ[f ] := Iλ[f, f ]. Similarly, let H ={x ∈ RN: x· e > t}, e∈ SN−1, t∈ R, be a half-space and denote by ΘH(x) := x + 2(t− x · e) the reflection of a point x on the boundary of H. The corresponding operator is defined by (ΘHf )(x) := f (ΘH(x)) and it again satisfies Iλ[f ] = Iλ[ΘHf ]andkfkp=kΘHfkp. Our first ingredient in the proof of Theorem1is the following Theorem 2 (Reflection and inversion positivity). Let 0 < λ < N if N = 1, 2, N− 2 ≤ λ < N if N ≥ 3, and let B ⊂ RNbe either a ball or a half-space. If f∈ L2N/(2N−λ)(RN) and (( fi(x) :=f (x)if x∈ B ,fo(x) :=ΘBf (x)if x∈ B , ΘBf (x)if x∈ RNB ,f (x)if x∈ RNB , then 1 Iλ[fi] + Iλ[fo]≥ Iλ[f ] . 2 If λ > N− 2 then the inequality is strict unless f = ΘBf . For half-spaces and λ = N− 2 this theorem is well known. The half-space case with N− 2 < λ < N was apparently first proved by Lopes and Mari¸s [8]. The case of balls seems to be new for all λ. Our original proof [3] was simplified by E. Carlen, to whom we are grateful, using the conformal invariance [2]. Reflection Positivity3293 The Li-Zhu lemma. Our second ingredient in the proof of Theorem1is a geometric characterization of the optimizing functions α β +|x − γ|2−(2N−λ)/2, extending a result of Li and Zhu [6]. Theorem 3 (Characterization of inversion invariant measures). Let µ be a finite, non-negative measure on RN. Assume that (A) for any a∈ RNthere is an open ball B centered at a such that µ(Θ−1B(A)) = µ(A)for any Borel set A⊂ RN. and for any e∈ SN−1there is an open half-space H with interior unit normal e such that µ(Θ−1H(A)) = µ(A)for any Borel set A⊂ RN. Then µ is absolutely continuous with respect to Lebesgue measure and −N dµ(x) = α β +|x − y|2dx for some α≥ 0, β > 0 and y ∈ RN. For absolutely continuous measures dµ = v dx assumption (A) is equivalent to the fact that for any a∈ RNthere is an ra> 0 such that for any x r2N v(x) =avr2a(x− a)+ a, |x − a||x − a|2 and similarly for reflections. Finally, let us deduce Theorem1from Theorems2and3. We make use of the fact that there is an optimizer f for the inequality in TheoremR1. Given aR∈ RN, we apply Theorem2to the ball B centered at a withB|f|pdx = (1/2)RN|f|pdx (or, given eR∈ SN−1Rto the half-space H with interior unit normal e such that H|f|pdx = (1/2)RN|f|pdx). We infer that both fiand foare optimizers. Moreover, if λ > N− 2, then the strictness statement in Theorem2implies that f = ΘBf . The same conclusion holds for λ = N− 2, but in this case we need an additional argument based on unique continuation. Therefore, we obtain in any case that assumption (A) in Theorem3is satisfied for the measure dµ =|f|pdx. We conclude from that theorem that f has the claimed form. References |
| [86] | E. A. Carlen, M. Loss, {\it Extremals of functionals with competing symmetries}. J. Funct. Anal. 88 (1990), no. 2, 437-456. · Zbl 0705.46016 |
| [87] | R. L. Frank, E. H. Lieb, {\it Inversion positivity and the sharp Hardy-Littlewood-Sobolev in-} {\it equality}. Calc. Var. Partial Differential Equations 39 (2010), no. 1-2, 85-99. · Zbl 1204.39024 |
| [88] | R. L. Frank, E. H. Lieb, {\it Spherical reflection positivity and the Hardy-Littlewood-Sobolev} {\it inequality}. In: Concentration, Functional Inequalities and Isoperimetry, C. Houdr´e et al. (eds.), Contemp. Math. 545, Amer. Math. Soc., Providence, RI, 2011. · Zbl 1378.39013 |
| [89] | R. L. Frank, E. H. Lieb, {\it Sharp constants in several inequalities on the Heisenberg group}. Ann. of Math. 176 (2012), no. 1, 349-381. 3294Oberwolfach Report 55/2017 · Zbl 1252.42023 |
| [90] | R. L. Frank, E. H. Lieb, {\it A new, rearrangement-free proof of the sharp Hardy-Littlewood-} {\it Sobolev inequality}. In: Spectral Theory, Function Spaces and Inequalities, B. M. Brown et al. (eds.), 55-67, Oper. Theory Adv. Appl. 219, Birkh¨auser, Basel, 2012. · Zbl 1297.39023 |
| [91] | Y. Y. Li, M. Zhu, {\it Uniqueness theorems through the method of moving spheres}. Duke Math. J. 80 (1995), no. 2, 383-417. · Zbl 0846.35050 |
| [92] | E. H. Lieb, {\it Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities}. Ann. of Math. (2) 118 (1983), no. 2, 349-374. · Zbl 0527.42011 |
| [93] | O. Lopes, M. Mari¸s, {\it Symmetry of minimizers for some nonlocal variational problems}. J. Funct. Anal. 254 (2008), no. 2, 535-592. RP, PCT, KMS, Etc. J¨urg Fr¨ohlich (joint work with L. Birke, K. Osterwalder, E. Seiler, and others) Dedicated to the memory of R. Haag, R. Jost and R. Schrader In my lecture I gave a brief survey of results on general quantum statistical mechanics, all related to “Osterwalder-Schrader Positivity”, or “Reflection Positivity” (RP), and to the “KMS condition” for so-called “temperature-ordered Green functions”, and I discussed some consequences of these results for relativistic quantum field theory and representation theory. My lecture was originally prepared in 2004 but was presented at the Oberwolfach meeting, in November 2017, for the first time. Earlier results, which my research and the lecture presented in Oberwolfach have been based on, are alluded to below, and some relevant papers are listed in the bibliography. Time did not permit to discuss some of the rather spectacular applications of Reflection Positivity to concrete models of statistical mechanics, in particular to the theory of phase transitions and spontaneous symmetry breaking in certain classes of classical and quantum lattice systems and lattice gauge theories; but see [21] and references given there. A. Remarks on Early History of Reflection Positivity and the KMS condition: The first seed of Reflection Positivity can be found in the book [3]. Subsequently, it was formulated explicitly and used to reconstruct (real-time) vacuum expectation values of local quantum fields from Euclidean (imaginary-time) Green functions by K. Osterwalder and R. Schrader in their celebrated work [10]. Inspired by their proposal, related results appeared in [11]. The first shadow of the KMS condition and an example of a “modular conjugation” – namely the anti-unitary PCT symmetry operation of local relativistic quantum field theory – appeared in the context of a proof of the PCT theorem in the framework of the Wightman axioms in Jost’s famous paper [1]. The KMS condition was formulated explicitly in [2]. An understanding of this condition as a general characterization of thermal equilibrium states in quantum statistical mechanics, as well as very interesting mathematical consequences thereof appeared in seminal work of Haag, Hugenholtz and Winnink [4]. An analysis of thermal Green functions and, in particular, of their analyticity properties in the time variables was presented in [5]. Results on analyticity properties of thermal Green functions Reflection Positivity3295 of concrete systems of quantum statistical mechanics (dilute quantum gases described in terms of “reduced density matrices”) were sketched in [8]. The findings in [4] apparently played a role in the genesis of the deep discoveries, due to Tomita, that gave rise to the modular theory of von Neumann algebras, details of which are described in [7]; (see also references given there). B. More recent developments related to RP and the KMS condition: Cousins of the Osterwalder-Schrader reconstruction theorem for “temperatureordered” (imaginary-time) Green functions of thermal equilibrium states of simple quantum field models at positive temperatures appeared in [12,13]. An attempt to formulate and prove a general reconstruction theorem of real-time thermal Green functions from imaginary-time, temperature-ordered Green functions was undertaken in a course on equilibrium statistical mechanics I taught at Princeton in 1977. Although various technical details were not straightened out in my course, yet, it did lead to the rather important result on selfadjoint extensions of locally densely defined symmetric semigroups published in [15]. A crucial idea used in my proof of this result was generously contributed by Edward Nelson. A related result was subsequently proven in [16]. It has been pointed out to me by Arthur Jaffe at the Oberwolfach meeting that an earlier (seemingly slightly weaker) result on selfadjoint extensions of symmetric semigroups was proven by A. E. Nussbaum in [6], almost ten years earlier! A first general reconstruction theorem (in the context of stochastic processes) appeared in [17]. A very general theorem on the reconstruction of real-time thermal Green functions from temperature-ordered Green functions appeared in [20], (the genesis of which was independent of [17]). The reason these results are of interest is that, in many examples of quantum many-body systems, the temperature-ordered Green functions are more directly accessible to construction than the real-time Green functions. In [18], Reflection Positivity, the KMS condition and the semigroup theorem alluded to above were first applied to problems in the representation theory of Lie groups: “virtual representations of symmetric spaces and their analytic continuation”. The notion of “virtual representations” was coined in this paper. This line of research caught the attention of a number of mathematicians, including P. E. T. Jorgensen, K.-H. Neeb, G. Olafsson, and their followers, working in the field of group representation theory. New results have been described at the Oberwolfach meeting by Jan Frahm whose notes I refer the reader to. A very general understanding of a construction of local quantum theories with infinitely many degrees of freedom from generalized “imaginary-time Green functionals” emerged from the line of work begun in [18]. It has been described in detail in [18,20]. This work has given rise to – among other results – very simple proofs of the celebrated PCT theorem and of the connection between spin and statistics, (which were sketched in my lecture). The PCT theorem says that the product of space reflection (P), charge conjugation (C) and time reversal (T) is a symmetry of every local relativistic quantum field theory on space-times of even dimension; this implies that to every species of particles there corresponds a species of anti-particles with the same mass and the same quantum numbers, but 3296Oberwolfach Report 55/2017 of opposite charge. The spin-statistics theorem says that local quantum fields of half-integer spin located at space-like separated points in space-time anti-commute (Fermi-Dirac statistics), while local quantum fields of integer spin located at spacelike separated points commute (Bose-Einstein statistics), implying what one calls “Einstein causality”. The proofs of these results described in [20] are somewhat inspired by results of Bisognano and Wichmann [14] concerning the KMS condition satisfied by the vacuum state of a local relativistic quantum field theory with respect to one-parameter subgroups of Lorentz boosts; (results that, in turn, continue the thought processes initiated in [1,4]). A variant of Reflection Positivity – connected to Markov traces on Hecke algebras and to “planar algebras” and related objects – has first appeared and been used in subfactor theory and then in the general theory of tensor categories. This line of work was initiated by Vaughan Jones [19] and is still going on. It has important implications in the theory of knots and links, (low-dimensional) local quantum field theory and quantum information theory. In his lecture at Oberwolfach, Arthur Jaffe has sketched some of the recent research in this direction that he and his collaborators have undertaken. I refer the reader to Jaffe’s notes.1 C. Applications of Reflection Positivity to Statistical Mechanics: Some of the most spectacular uses of Reflection Positivity and of some of its consequences have occurred in the theory of phase transitions and of spontaneous breaking of (continuous) symmetries in classical and quantum systems – a theory that belongs to equilibrium statistical mechanics; see [21]. The first such use appeared in a proof of existence of a phase transition accompanied by the spontaneous breaking of the (ϕ7→ −ϕ) - symmetry in the λϕ4- theory in two space-time dimensions, due to Glimm, Jaffe and Spencer. They used a more powerful version of estimates in [9], which they derived directly from Reflection Positivity, to carry out a so-called Peierls argument patterned on a method first discovered by Sir Rudolf Peierls for the example of the two-dimensional Ising model, which proves the existence of a phase transition. In work of Simon, Spencer and myself, a new method to exhibit phase transitions in lattice models of magnets and in λ| ϕ|4theories with continuous symmetries in three or more (space-time) dimensions was discovered. It draws inspiration from the so-called K¨allen-Lehmann representation of vacuum expectation values of two fields in local relativistic quantum field theory. A variant of this representation, called “infrared bounds”, can be established using Reflection Positivity. Infrared bounds have since been very widely used. Many interesting results can be found in [21]. Files of my lecture at Oberwolfach and of my Vienna lectures [21], IV., are available on request. 1 In the past, I have contributed various results to the theory of braided tensor categories and their applications in the theory of superselection sectors in quantum field theory in collaborations led by Thomas Kerler, Ingo Runkel and Christoph Schweigert. Reflection Positivity3297 Acknowledgements: I am indebted to Arthur Jaffe for a useful discussion and for having drawn my attention to a paper of A. E. Nussbaum [6] on selfadjoint extensions of symmetric semigroups related to results of mine (see [15]), but found several years before mine. I thank our colleagues Joachim Hilgert and Karl-Hermann Neeb for their encouraging interest in my results, and Christian G´erard, Roberto Longo and Jakob Yngvason for useful comments. References |
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| [110] | A. Klein, L. J. Landau, {\it Stochastic Processes Associated With KMS States}, J. Funct. Analysis 42 (1981), 368-428 · Zbl 0498.60098 |
| [111] | J. Fr¨ohlich, K. Osterwalder, E. Seiler, {\it On Virtual Representations of Symmetric Spaces and} {\it Their Analytic Continuation}, Annals of Math. 118 (1983), 461-489 · Zbl 0537.22017 |
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| [114] | I. J. Glimm, A. Jaffe, T. Spencer, {\it Phase Transition for }ϕ42{\it Quantum Fields}, Commun. Math. Phys. 45 (1975), 203-216 II. J. Fr¨ohlich, B. Simon, T. Spencer, {\it Infrared Bounds, Phase Transitions and Continuous} {\it Symmetry Breaking}, Commun. Math. Phys. 50 (1976), 79-85 III. C. Borgs, E. Seiler, {\it Lattice Yang-Mills Theory at Non-Zero Temperature and the} {\it Confinement Problem}, Commun. Math. Phys. 91 (1983), 329-380 IV. J. Fr¨ohlich, {\it Phase Transitions and Continuous Symmetry Breaking}, Lectures, Vienna, August 2011 J. Fr¨ohlich, ETH Zurich, December 10, 2017 Email: juerg@phys.ethz.ch Periodic striped ground states in Ising models with competing interactions Alessandro Giuliani (joint work with J. Lebowitz, E. Lieb, R. Seiringer) In this talk, I will review some selected results obtained in the last few years on the existence of periodic minimizers in two- and three-dimensional spin systems with competing interactions. The model that we consider is an Ising model in dimension d (the most interesting cases being d = 2 and d = 3), with short range ferromagnetic and long range, power-law decaying, anti-ferromagnetic interactions. The Hamiltonian describing the energy of the system is XX(σxσy− 1) |x − y|p, hx,yi{x,y}: x6=y where J > 0 is the ratio between the strengths of the ferromagnetic and of the anti-ferromagnetic interaction, and p > d is the decay exponent of the long-range interaction. The first sum ranges over pairs of nearest-neighbor sites in the discrete torus TdL:= Zd/LZd, while the second over pairs of distinct sites in TdL. The spins σx, x∈ TdL, take values in{±1}, and the constant −1 appearing in the two terms is chosen in such a way that the energy of the homogeneous configuration σx≡ +1, is equal to zero. A physically relevant case is d = 2 and p = 3, in which case (1) models the low-temperature equilibrium properties of thin magnetic films, embedded in the three-dimensional space, with the easy-axis of magnetization coinciding with the axis orthogonal to the film; in this case, the long range term models the dipolar interaction among the localized magnetic moments, while the short-range term models a ferromagnetic exchange interaction. The goal is to characterize the structure of the ground states of the system, for any (even, sufficiently large) L∈ N. Ideally, one would also like to characterize the low-temperature infinite volume Gibbs states, but this is beyond our current abilities. Note that the short-range interaction favors a homogeneous state, that is σx≡ +1 or σx≡ −1, while the long-range term favors an anti-ferromagnetic ‘N‘eel’ state, that is σx= (−1)x1+···+xdor σx= (−1)x1+···+xd+1. The fact that Reflection Positivity3299 the long-range contribution to the energy is minimized by the N‘eel state is not obvious, and was proved in [2] by Reflection Positivity (RP) methods. In the presence of both terms, the competition between the short-range ferromagnetic and the long-range anti-ferromagnetic interaction induces the system to form domains of minus spins in a background of plus spins, or vice versa. This happens in an intermediate range of values of J: in fact, if J is sufficiently small, the ground state is the same as for J = 0, that is, it is the N‘eel state [2]; if J is sufficiently large and p > d + 11, the ground state is the same as for J = +∞, that is, it is the homogeneous state. For intermediate values of J the ground state is characterized by non-trivial structures, whose typical length scale diverges as J→ Jc(p) from the left; here Jc(p) is the critical value of J, beyond which the ground state is homogeneous. It coincides with the value of J at which the surface tension of an infinite straight domain wall, separating a half space of minuses from a half space of pluses, vanishes [5]. It is expected that, for values of J close to Jc(p) and slightly smaller than it, all the ground states are quasi-one-dimensional (i.e., they are translationally invariant in d− 1 directions), and periodic, provided the box size L is an integer multiple of an ‘optimal period’ 2h∗, which can be explicitly computed. We shall refer to these expected ground states as the ‘optimal periodic striped states’: they consist of ‘stripes’ (in d = 2, or ‘slabs’, in d = 3) of spins all of the same sign, arranged in an alternating way (that is, neighbouring stripes have opposite magnetization), and all of the same width h∗. The conjecture that optimal periodic striped states are ground states of (1) has been first proved in [3,4], via a generalization of the standard RP technique, which we named ‘block reflection positivity’, because the reflections are performed across the bonds that separate a block of plus spins from a block of minus spins. The same proof shows that in any dimension, optimal periodic striped states are the states of minimal energy, among all the possible quasi-one-dimensional states. More recently, in a work in collaboration with R. Seiringer, we succeeded in proving this conjecture [6], for all dimensions d≥ 1 and sufficiently large decay exponents, namely p > 2d. The result has been recently extended to the continuum setting and p > d + 2 [1]. The proof is based on the following main steps: (1) We re-express the energy of the spin configuration as the energy of an equivalent droplet configuration. Here the droplets are the connected regions of minus spins, in a background of plus spins. The energy, if expressed in terms of droplets, consists of (i) a sum of droplet self-energies, which include the ferromagnetic contribution to the surface tension, plus the long range interaction of the minus spins in each droplet δ with a ‘sea’ of plus spins in the complement of the droplet δc= TdLδ, and (ii) a droplet-droplet pair interaction, which is repulsive. Remarkably, the long 1for p ≤ d + 1, the homogeneous state is {\it not }the ground state, for any finite value of J. In these cases, which include the case d = 2, p = 3 mentioned above, it would be interesting to characterize the ground states for J sufficiently large; unfortunately, we do not have rigorous results to report on this case yet, with the only exception of the one-dimensional case d = 1. 3300Oberwolfach Report 55/2017 range contribution to the self-energy of a droplet δ behaves (for the purpose of a lower bound) as−2Jc(p)|∂δ|, where |∂δ| is the length (if d = 2, or area, if d = 3) of the boundary of the droplet, plus a positive constant times the number of corners, that is, the points where the domain walls bend by 90o. In this respect, the corners look like the elementary excitations of the system. (2) We localize the droplet energy in bad boxes, characterized by a local ‘atypical’ configuration (which either has corners, or too large uniformly magnetized regions – called ‘holes’), and good regions, which are the connected components of the complement of the union of the bad boxes. By ‘localizing’, we mean here that the original energy is bounded from below in terms of a sum of local energy functionals, each depending only on the local droplet configuration (supported either in a bad box or in a good region). By construction, the configuration in a good region is quasione-dimensional, and consists of stripes all in the same direction, but not necessarily all of the same width. (3) We use our lower bound on the self-energy of the droplets, to infer that the localized energy in a bad box is much larger than the energy of an optimal striped configuration in the same box. The energy difference scales like the number of corners contained in the bad box, plus the volume of the holes. We shall refer to this energy difference as the energy gain associated with each bad box. (4) We use a slicing procedure, combined with block RP and an optimal control of the boundary errors, to derive an optimal lower bound on the localized energy in a good region. Such a lower bound scales like the energy of the optimal striped configuration in the same region, minus a boundary error, which is so small that it can be over- compensated by the energy gains of the bad boxes at the boundary of the good region (note that every boundary portion of a good region borders on a bad box). Our result provides the first rigorous proof of the formation of mesoscopic periodic structures in d≥ 2 systems with competing interactions. It leaves a number of important problems open: (1) Extend the result of [6] to smaller decay exponents. In particular, prove that the ground states of (1) with d = 2 and p = 3 are periodic and striped, for all sufficiently large J. (2) Prove that there are at least d infinite volume Gibbs states at low temperatures, which are translationally invariant in d− 1 coordinate directions. Depending on the dimension, prove the existence of Long-Range Striped Order (LRSO), or of quasi-LRSO a’la Kosterlitz-Thouless, in the last coordinate direction. (3) Extend these results to the continuum setting, for an effective free energy functional that is rotationally invariant. In particular, prove the onset of continuous symmetry breaking, both in the ground state and in the low-temperature Gibbs states. Reflection Positivity3301 References |
| [115] | S. Daneri, E. Runa: {\it Exact periodic stripes for a minimizers of a local/non-local interaction} {\it functional in general dimension}, arXiv:1702.07334. · Zbl 1410.82005 |
| [116] | J. Fr¨ohlich, R. B. Israel, E. H. Lieb, and B. Simon: {\it Phase Transitions and Reflection} {\it Positivity. II. Lattice Systems with Short-Range and Coulomb Interactions}, J. Stat. Phys. 22, 297 (1980). |
| [117] | A. Giuliani, J. Lebowitz, E. Lieb: {\it Ising models with long-range dipolar and short range} {\it ferromagnetic interactions}, Phys. Rev. B 74, 064420 (2006). |
| [118] | A. Giuliani, J. Lebowitz, E. Lieb: {\it Striped phases in two-dimensional dipole systems}, Phys. Rev. B 76, 184426 (2007). |
| [119] | A. Giuliani, J. Lebowitz, E. Lieb: {\it Checkerboards, stripes and corner energies in spin models} {\it with competing interactions}, Phys. Rev. B 84, 064205 (2011). |
| [120] | A. Giuliani, R. Seiringer: {\it Periodic Striped Ground States in Ising Models with Competing} {\it Interactions}, Comm. Math. Phys. 347, 983-1007 (2016). Reflection positivity: an operator algebraic approach to the representation theory of the Lorentz group Christian J¨akel (joint work with Jens Mund) The space-time symmetry group of the two-dimensional de Sitter space . dS=x∈ R1+2| x20− x21− x22=−r2,r > 0 , is the Lorentz group SO0(1, 2). A wedge . W = ΛW1⊂ dS ,W1=x∈ dS | x2>|x0|,Λ∈ SO0(1, 2) , is a space-time region, which is invariant under the action of the Lorentz boosts cosh t0sinh t ΛW(t) = ΛΛ1(t)Λ−1,Λ1(t)=.010 . sinh t0 cosh t The reflection at the edge of the wedge W , ΘΛW1= Λ(P1T )Λ−1,Λ∈ SO0(1, 2) ,P1T = Λ1(iπ) , maps W to its space-like complement, the opposite wedge W′. Now, let Λ7→ u(Λ) be a (anti-)unitary irreducible representation of the Lorentz group O(1, 2) on some Hilbert spaceH. Let ℓWbe the self-adjoint generator of the one-parameter subgroup t7→ u ΛWtr. Set δW= e−2πrℓW,jW= u(Θ.W). δWis a densely defined, closed, positive non-singular linear operator onH; jWis an anti-unitary operator onH. These properties allow one to introduce the operator . (1)sW= jWδ1/2W,W = ΛW1. sWis a densely defined, antilinear, closed operator onH with range R(sW) = D(sW) and s2W⊂ 1. Moreover, u(Λ)sWu(Λ)−1= sΛW,Λ∈ SO0(1, 2). 3302Oberwolfach Report 55/2017 The modular localisation map W7→ H(W ), introduced by Brunetti, Guido and Longo [4], associates a closed R-linear subspace . H(W )={h ∈ D(sW)| sWh = h} ofH to a wedge W . Each H(W ) is a standard subspace in H, i.e., H(W ) ∩ iH(W ) = {0} ,H(W ) + iH(W ) = H . The operator sWintroduced in (1) is the Tomita operator ofH(W ), i.e., sW:H(W ) + iH(W ) → H(W ) + iH(W ) h + ik7→ h − ik. In particular, δWitH(W ) = H(W ) and jWH(W ) = H(W )′, withH(W )′the symplectic complement ofH(W ) in H. By construction, u(Λ)H(W ) = H(ΛW ) ,Λ∈ SO0(1, 2) . We now turn from the one-particle picture to quantum field theory. It is convenient to use the coherent vectors Γ(h) =⊕∞n=0√1h⊗s· · · ⊗sh n!|{z} n−times . to define operator algebras on the Fock spaceF ≡ Γ(H)=⊕∞n=0H⊗ns: for h, g∈ H, the relations V (h)V (g) = e−iℑhh,giV (h + g) ,V (h)Ω◦= e−12||h||2Γ(ih) , define unitary operators, called the Weyl operators. They satisfy V∗(h) = V (−h) and V (0) = 1. The one-parameter group Λ7→ u(Λ) induces a group of automorphisms α◦Λ(V (h))= V u(Λ)h.,h∈ H , Λ ∈ SO0(1, 2) , representing the free dynamics. The modular localization available on the one-particle Hilbert space can now be used to associate von Neumann algebras to space-time regions in dS: i.) for the wedge W1, we setA◦(W1)=.{V (h) | h ∈ H(W1)}′′; ii.) for an arbitrary wedge W = ΛW1, we setA◦(W )= α.◦ΛA◦W1; iii.) for an arbitrary bounded, causally complete, convex regionTO ⊂ dS, set A◦(O) =O⊂WA◦W. The mapO 7→ A◦(O) preserves inclusions, the algebras A◦(O) are hyperfinite type III1factors, and α◦Λ(A◦(O)) = A◦(ΛO). It remains to justify that the Fock zero-particle vector Ω◦induces the physically relevant de Sitter vacuum state. This is not completely obvious: there is no global time evolution on de Sitter space and hence no natural notion of energy. However, stability of matter against spontaneous collapse has to be ensured somehow, and also the energy-momentum currents should not fluctuate in an uncontrollable manner. The geodesic KMS condition (proposed by Borchers and Buchholz [2]) ensures such stability properties. It requires that the restriction of the de Sitter vacuum state to the wedge W1is a thermal state with respect to the dynamics Reflection Positivity3303 provided by the one-parameter group t7→ exp(itL◦) of boosts which leave the wedge W1invariant. The unique state satisfying this condition is the one induced by the Fock vacuum vector Ω◦. Somewhat surprisingly theP(ϕ)2model can be formulated on Fock space too. In fact, it can be reconstructed from the vector representing the interacting de Sitter vacuum state. The latter is given by Araki’s perturbation theory of modular automorphisms: e−πHΩ◦Zπ ke−πHΩ◦k,H := L◦+0r cos ψ dψ :P(ϕ(0, ψ)): , withP a real valued polynomial, bounded from below. The modular group for the pairA◦(W1), Ωprovides a one-parameter group which leaves the algebraA◦(W1) invariant. Since Ω lies in the natural positive coneP♯(A◦(W1), Ω◦), we have J◦∆1/2AΩ = A∗Ω ,A∈ A◦(W1) , with J◦the modular conjugation for the pair (A◦(W1), Ω◦). We can thus interpret t7→ ∆itW1as a new group of Lorentz boosts and, in fact, together with the (free) rotations U◦(R0(α)), α∈ [0, 2π), they generate a new representation U(Λ) of SO0(1, 2). Form a more technical perspective, several mathematical challenges have to be overcome in order to guarantee that the operator sum in the second equation in (2) is well-defined. Local symmetric semi-groups techniques [6,10] are used to establish the operator sums Z (3)L := L◦+ V ,V =r cos ψ dψ :P(ϕ(0, ψ)): . S1 The theory of virtual representations [7] is used to prove that the newly defined one-parameter unitary groups actually give rise to a representation of SO(1, 2). The resulting unitary representation Λ7→ U(Λ) induces a group of automorphisms . αΛ(V (h))= U (Λ)V (h)U (Λ)−1,h∈ H , Λ ∈ O(1, 2) , representing the interacting dynamics. The sum given in (3) provides the crucial link between the free and the interacting quantum field theory, as L is the generator of the modular group which leaves the von Neumann algebraA◦(W1) for the free field associated to the wedge W1invariant. We can now proceed just as before: i.) for the wedge W1, setA(W1)=.A◦(W1) ; ii.) for an arbitrary wedge W = ΛW1, setA(W )= α.ΛA W1T; iii.) for a causally complete, convex regionO ⊂ dS, set A(O) =O⊂WA W. The mapO 7→ A(O) is the net of local von Neumann algebras for the P(ϕ)2model. The de Sitter vacuum state, induced by the vector Ω, is uniquely characterised by the geodesic KMS condition. Due to thermalisation effects introduced by the curvature of space-time, it is unique even for large coupling constants, despite the fact that different phases occur in the limit of curvature to zero (i.e., the Minkowski limit). 3304Oberwolfach Report 55/2017 References |
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| [130] | A. Klein and L. Landau, {\it Construction of a unique selfadjoint generator for a symmetric} {\it local semigroup}, J. Funct. Anal. 44 (1981) 121-137. Reflection positivity for parafermions Bas Janssens (joint work with Arthur Jaffe) We describe reflection positivity for parafermions or anyons. Whereas bosons and fermions receive a plus or minus sign upon exchanging particles, parafermions receive a factor q = exp(2πi/p), where p∈ N is the order. Note that parafermions of order 1 are bosons, and parafermions of order 2 are fermions. Parafermions of order 3 or more behave in a qualitatively different fashion, as in this case q is not the same as q−1. In a discrete setting, parafermions are described by the *-algebraA(Z, p) with generators cilabelled by i∈ Z +12, satisfying the parafermion relations (1)cicj=qcjcifori < j (2)cpi=1 (3)c∗i=c−1i. If A = cni11· · · cnikk, then the number of parafermions n1+ . . . + nkis well-defined only modulo p. It is called the degree of A, and denoted by|A| ∈ Zp. 1. The setting Abstracting from the discrete situation, we move to the more general setting where parafermions are described by a Zp-graded, unital algebraA. In order for the exponential series to make sense, we will assume thatA is a locally convex topological algebra, for which multiplication is separately continuous. Further, we will take Reflection Positivity3305 the Hamiltonian H to be a neutral (degree zero) element ofA. We require that the exponential series X∞1 exp(A) =An n! n=0 converges, and that it defines a continuous map exp :A → A. Note that we do not require thatA is a *-algebra, and we do not require H to be hermitean. A reflection is an antilinear homomorphism θ :A → A that squares to the identity, and reverses the grading. We assume that our algebraA is the q-double of a Zp-graded subalgebraA+. This means thatA = A−A+withA−:= θ(A+), and that the two halves of the system paracommute, in the sense that (4)A−A+= q|A−||A+|A+A− for all A+∈ A+and A−∈ A−. For example, in the algebraA(Z, p) introduced above, we can take θ(ci) = c−1−i, withA+(Z, p) the algebra generated by the parafermion operators ciwith i > 0. 2. Reflection positivity Let τ0be a neutral continuous linear functional onA, meaning that τ0(A) = 0 if A is of pure degree|A| 6= 0. We consider τ0as a ‘background state’, and we are interested in the Bolzmann functionals τβH(A) := τ0(e−βHA), where β≥ 0. In the context of parafermions, we define τβHto be reflection positive if (5)ζ|A+|2τβH(θ(A+)A+)≥ 0 for all A+∈ A+, where ζ∈ C is a square root of q with ζp2= 1. Note that if τβH is reflection positive, then the Hermitian form hA+, B+i := ζ|A+|2τβH(θ(A+)B+) is positive definite onA+. Since the above expression is zero for|A+| 6= |B+|, the closureH+ofA+is a Zp-graded Hilbert space. Note that alternatively, reflection positivity can be formulated in terms of the convex coneK spanned by elements of the form ζ|A+|2θ(A+)A+with A+∈ A+. By definition, τβHis reflection positive if and only if τβH(K) ⊆ R≥0. Since τβHis continuous, this is equivalent to τβH(K) ⊆ R≥0, whereK is the closure of K. The following theorem [10,11] gives sufficient conditions on H in order for τβH to be reflection positive, extending well-known results in the bosonic and fermionic case [1,2,3,4,5,6,7,8,9]. Under mild additional assumptions, one can prove that these conditions are not just sufficient, but also necessary [11]. Theorem 1. Suppose that H = H−+H0+H+, where H±∈ A±with θ(H+) = H−, and where−H0is in the closureK of K. Then reflection positivity of τ0implies reflection positivity of τβHfor all β > 0. The cornerstone of the proof – and the reason for the appearance of ζ|A+|2in equation (5) – is the fact thatK is closed under multiplication. Indeed, the product of ζ|A+|2θ(A+)A+and ζ|B+|2θ(B+)B+is equal to ζ|A+B+|2θ(A+B+)A+B+, and 3306Oberwolfach Report 55/2017 hence an element ofK again. To see this, use the paracommutation relation (4) in the expression θ(A+)A+θ(B+)B+to exchange A+∈ A+and θ(B+)∈ A−. Since |θ(B+)| = −|B+|, this yields a factor q|A+||B+|, which combines with ζ|A+|2+|B+|2 to form ζ|A+B+|2. Since this argument extends to convex combinations, one finds that the cone K is multiplicatively closed. The same conclusion for its closure K can be derived from separate continuity of the multiplication. Indeed, since left multiplication by K∈ K is continuous and K · K ⊆ K, we find that K · K ⊆ K. Similarly, since right multiplication by K′∈ K is continuous, we find that K · K′⊆ K for all K′∈ K, and hence thatK · K ⊆ K. To prove Theorem1, consider first the special case H = H0, where HP−and H+ vanish. SinceK is multiplicatively closed, −βH0∈K implies thatnk=0k!1(−βH0)k is inK for every n, and hence that e−βH0∈ K. As K is multiplicatively closed, we have e−βH0K ⊆ K, and hence τβH(K) = τ0(e−βH0K) ⊆ R≥0. It follows that τβH is reflection positive for every β > 0. Next, consider the case where H+and H−are nonzero. Let Hε:= H− ε2H−H+−ε121 . Since H0∈ −K, and since Hεcan be written as Hε= H0−θ(1ε1−εH+)(1ε1−εH+), we have Hε∈ −K. By the above reasoning, we thus find that the functional τβHε is reflection positive for all β > 0. Since Hε′:= H− ε2H−H+differs from Hεby an additive constant, reflection positivity of τβHεimplies reflection positivity of τβHε′. It follows that τβH(K) = limε↓0τβHε′(K)≥ 0 for all K ∈ K, so that τβHis reflection positive. References |
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| [141] | A. Jaffe and B. Janssens, {\it Reflection positive doubles}, Journal of Functional Analysis 272 (2017), 3506-3557. Reflection positivity, Lax-Phillips theory, and spectral theory Palle Jorgensen (joint work with Karl-Hermann Neeb, Gestur Olafsson, and Feng Tian) We review reflection-positivity (Osterwalder-Schrader positivity, O.S.-p.) as it is used in the study of renormalization questions in physics. In concrete cases, this refers to specific Hilbert spaces that arise before and after the reflection. Our focus is a comparative study of the associated spectral theory, now referring to the canonical operators in these two Hilbert spaces. We analyze in detail a number of geometric and spectral theoretic properties connected with axiomatic reflection positivity, as well as their probabilistic counterparts; especially the role of the Markov property. In rough outline: It is possible to express OS-positivity purely in terms of a triple of projections in a fixed Hilbert space, and a reflection operator. For such three projections, there is a related property, often referred to as the Markov property; and it is well known that the latter implies the former; i.e., when the reflection is given, then the Markov property implies O.S.-p., but not conversely. In this paper we shall prove two theorems which flesh out a much more precise relationship between the two. We show that for every OS-positive system (E+, θ), the operator E+θE+has a canonical and universal factorization. Our second focus is a structure theory for all admissible reflections. Our theorems here are motivated by Phillips’ theory of dissipative extensions of unbounded operators (see e.g., [21]). The word “Markov” traditionally makes reference to a random walk process where the Markov property in turn refers to past and future: Expectation of the future, conditioned by the past. By contrast, our present initial definitions only make reference to three prescribed projection operators, and associated reflections. Initially, there is not even mention of an underlying probability space. This in fact only comes later. The notion “reflection-positivity” came up first in a renormalization question in physics: “How to realize observables in relativistic quantum field theory (RQFT)?” This is part of the bigger picture of quantum field theory (QFT); and it is based on a certain analytic continuation (or reflection) of the Wightman distributions (from the Wightman axioms). In this analytic continuation, Osterwalder-Schrader (OS) axioms induce Euclidean random fields; and Euclidean covariance. (See, e.g., [25,26,5,6,13,8,9].) For the unitary representations of the respective symmetry groups, we therefore change these groups as well: OS-reflection applied to the Poincar´e group of relativistic fields yields the Euclidean group as its reflection. 3308Oberwolfach Report 55/2017 The starting point of the OS-approach to QFT is a certain positivity condition called “reflection positivity.” Now, when it is carried out in concrete cases, the initial function spaces change; but, more importantly, the inner product which produces the respective Hilbert spaces of quantum states changes as well. Before reflection we may have a Hilbert space of functions, but after the time-reflection is turned on, then, in the new inner product, the corresponding completion, magically becomes a Hilbert space of distributions. The motivating example here is derived from a certain version of the SegalBargmann transform. For more detail on the background and the applications, we refer to two previous joint papers [15] and [16], as well as [17,18,19,11,12,22, 14,10,3]. Our present purpose is to analyze in detail a number of geometric properties connected with the axioms of reflection positivity, as well as their probabilistic counterparts; especially the role of the Markov property. In rough outline: It is possible to express Osterwalder-Schrader positivity (O.S.p.) purely in terms of a triple of projections in a fixed Hilbert space, and a reflection operator. For such three projections, there is a related property, often referred to as the Markov property. It is well known that the latter implies the former; i.e., when the reflection is given, then the Markov property implies O.S.-p., but not conversely. For the readers benefit we have included the following citations [23,24,20] on Markov random fields. We begin by recalling the fundamentals in the subject. 1. A characterization of the Markov property: Markov vs O.-S. positivity In the classical case of Gaussian processes, the question of reflection symmetry and reflection positivity is of great interest; see, e.g., [7,14,8], and also [17,18,19]. Let H be a given (fixed) Hilbert space; e.g., H = L2(Ω, F , P), square integrable random variables, where Ω is a set (sample space) with a σ-algebra of subsets F (information), and P a given probability measure on (Ω, F ). But the question may in fact be formulated for an arbitrary Hilbert space H , and possible inseparable generally. Recall that θ : H→ H is a reflection if it satisfies θ∗= θ, and θ2= IH. Definition 1.1. Given a Hilbert space H , let Ref (H ) be the set of all reflections in H , i.e., Ref (H ) =θ : H→ H ; θ∗= θ, θ2= IH. Question. (1) Given ε ={E±}, what is R (ε)? (2) Given θ, what is E (θ)? Definition 1.2. Suppose ε = (E0, E±) is given, and θ∈ R (ε). (1) We say that reflection positivity holds iff (Def.) (1)E+θE+≥ 0, also called Osterwalder-Schrader positivity (O.S.-p). Reflection Positivity3309 (2) Given ε, we say that it satisfies the Markov property iff (Def.) (2)E+E0E−= E+E−. (3) We set EOS(θ) ={(E0, E±) ; E+θE+≥ 0}. Lemma 1.3. Suppose (2) holds (the Markov property), and θ∈ R (ε), then (3)E+θE+≥ 0, i.e., the O.S.-positivity condition (1) follows. Recall the definition of R (ε) and R (ε, U ). Lemma1.3can be reformulated as: Lemma 1.4. For all θ∈ R (ε), we have (4)E(M arkov)∩ E (θ) ⊆ EOS(θ) . (See Definitions1.1and eq. (3).) Question. Let ε = (E0, E±) be given, and suppose E+θE+≥ 0, for all θ ∈ R (ε), then does it follow that E+E0E−= E+E−holds? Theorem 1.5. Given an infinite-dimensional complex Hilbert space H , let the setting be as above, i.e., reflections, Markov property, and O.S.-positivity defined as stated. Then (5)EOS(θ) = E (M arkov) . θ∈R(ε) Remark 1.6. If (E±, E0, U ) is Markov, then (5) also holds with θ, U . The idea in (5) is that when a system ε of projections is fixed as specified on the RHS in the formula, then on the LHS, we intersect only over the subset of reflections θ subordinated to this ε-system. And similarly when both ε and U are specified, we intersect over the smaller set of jointly ε, U subordinated reflections θ. Example 1.7 (Markov property). Let H = L2(Ω, F , P), where • Ω: sample space; • F : total information; • F−: information from the past (or inside); • F+: information from the future (predictions), or from the outside; • F0: information at the present. Let E (· | F0), E (· | F±) be the corresponding conditional expectations, and the Markov property (2) then takes the form E0H= H0, E±H= H±. The Markov process is a probability system: (6)E(E (ψ+| F−)| F0) = E (ψ+| F−) , for∀ψ+(random variables conditioned by F+= the future); or, if F0⊆ F−, it simplifies to: (7)E(ψ+| F−) = E (ψ+| F0) ,∀ψ+∈ H+. For more details on this point, see Section2below. 3310Oberwolfach Report 55/2017 Question. Do we have analogies of O.S.-positivity (see (1)) in the free probability setting? That is, in the setting of free probability and non-commuting random variables. 2. Markov processes and Markov reflection positivity In the above, we considered systems H , E0, E±, θ, and U , where H is a fixed Hilbert space; E0, E±are then three given projections in H , θ is a reflection, and U is a unitary representation of a Lie group G. The axioms for the system are as follows: (1) θE0= E+; (2) E+θE−= θE−; (3) E−θE+= θE+; (4) the O.S.-positivity holds, i.e., (8)E+θE+≥ 0; (5) θU θ = U∗, or θU (g) θ = U (g−1). It is further assumed that, for some sub-semigroup S⊂ G, we have U (s) H+⊂ H+,∀s ∈ S; or equivalently, (9)E+U (s) E+= U (s) E+, s∈ S. 2.1. Probability Spaces. By a probability space we mean a triple (Ω, F , P), where Ω is a set (the sample space), F is a σ-algebra of subsets (information), and P is a probability measure defined on F . Measurable functions ψ on (Ω, F ) are called random variables. If ψ is a random variable in L2(Ω, F , P), we say that it has finite second moment. An indexed family of random variables is called a stochastic process, or a random field. Let (Ω, F , P) be a fixed probability space. The expectation will be denoted Z (10)E(ψ) =ψ dP, Ω if ψ is a given random variable on (Ω, F , P). We shall be primarily interested in the L2(Ω, F , P) setting. If ψ is a random variable (or a random field) then (11)ψ−1(B)⊆ F , where B is the Borel σ-algebra of subsets of R. For every sub-σ-algebra G⊂ F , there is a unique conditional expectation (12)E(· | G ) : L2(Ω, F , P)−→ L2(Ω, F , P) . In fact G defines a closed subspace in L2(Ω, F , P), the closed span of the indicator functions{χS; S∈ G }, and E (· | G ) in (12) will then be the projection onto this subspace. If G⊂ F is as in (11) then, for random variables ψ1∈ L2(G , P), and ψ2∈ L2(F , P), we have E (ψ1ψ2) = E (ψ1E(ψ2| G )). If ψ1is also in L∞(G , P), then E(ψ1ψ2| G ) = ψ1E(ψ2| G ). Reflection Positivity3311 The following property is immediate from this: If Gi, i = 1, 2, are two sub-σalgebras with G1⊆ G2, then for all ψ∈ L2(Ω, F , P) we have E (E (ψ| G2)| G1) = E(ψ| G1). Let{ψt}t∈Rbe a random process in the given probability space (Ω, F , P). For t∈ R, set Ft:= the σ-algebra (⊆ F ) generated by the random variables {ψs; s≤ t}. When t is fixed, we set Bt:= the σ-algebra generated by the random variable ψt. We say that{ψt}t∈Ris a Markov-process iff (Def.), for every t > s, and every measurable function f , we have (13)E(f◦ ψt| Fs) = E (f◦ ψt| Bs) where E (· | Fs), and E (· | Bs), refer to the corresponding conditional expectations. It is well known that the Markov property is equivalent to the following semigroup property: Set (14)(Stf ) (x) := E (f◦ ψt| ψ0= x) , then, for all t, s≥ 0, we have (15)St+s= StSs. So the semigroup law (15) holds if and only if the Markov property (13) holds. 2.2. The covariance operator. Now let V be a real vector space; and assume that it is also a LCTVS, locally convex topological vector space. Let G be a Lie group, U a unitary representation of G; and let{ψv,g}(v,g)∈V ×Gbe a real valued stochastic process s.t. ψv,g∈ H = L2(Ω, F , P), and E (ψv,g) = 0, (v, g)∈ V × G. We further assume that a reflection θ is given, and that θ (ψv,g) = ψv,g−1, (v, g)∈ V× G. Let (vi, gi), i = 1, 2, be given, and set (16)E(ψv1,g1ψv2,g2) =hv1, r (g1, g2) v2i whereh·, ·i is a fixed positive definite Hermitian inner product on V . Hence (16) determines a function r on G× G; it is operator valued, taking values in operators in V . This function is called the covariance operator. To sketch the setting for the Markov property, we shall make two specializations (these may be removed!): (i) G = R, S = R+∪ {0} = [0, ∞), and (ii) the process is stationary; i.e., referring to (16) we assume that the covariance operator r is as follows: (17)E(ψv1,t1ψv2,t2) =hv1, r (t1− t2) v2i , ∀t1, t2∈ R, ∀v1, v2∈ V. In this case, the O.S.-condition (8) is considered for the following three sub-σalgebras A0, A±in F : 0= the σ-algebra generated by{ψv,0}v∈V, A+= the σ-algebra generated by{ψv,t}, and v∈V,t∈[0,∞) −= the σ-algebra generated by{ψv,t}v∈V,t∈(−∞,0]. 3312Oberwolfach Report 55/2017 The corresponding conditional expectations will be denoted as follows: (18)E0(ψ) = E (ψ| A0) , and E±(ψ) = E (ψ| A±) . The corresponding closed subspaces in H = L2(Ω, F , P) will be denoted H0, H±, respectively, and we consider the positivity conditions (8) O.S.-p, and Markov, in this context. 3. Extension from stationary (Gaussian) to stationary-increment processes A novelty is a new extension of known results from stationary Gaussian processes Xtindexed by t∈ R to a much more realistic class of Gaussian processes, the stationary-increment processes Xt, i.e., satisfying E (Xt) = 0, t∈ R, and r (|t|) + r (|s|) − r (|s − t|) 2; in particular, E|Xt− Xs|2= r (|t − s|), where r is a function on [0, ∞). This part is based on a joint work with D. Alpay et al. (see [2,4,1]). In particular, we recall that the stationary-increment processes are indexed by tempered measures ν on (R, B) via r = r(ν), where Z r (t) =1− eitx+itxdν (x), t R1 + x2x2∈ R. In this case X(ν)is first constructed, and indexed by the Schwartz-spaceS, with E X(ν)(ϕ)2=Z| ˆϕ (x)|2dν (x) , ϕ∈ S, R where ˆϕ denotes the usual Fourier transform. References |
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| [167] | Konrad Osterwalder and Robert Schrader. Axioms for Euclidean Green’s functions. II. {\it Comm. Math. Phys.}, 42:281-305, 1975. With an appendix by Stephen Summers. Conformal field theory and operator algebras Yasuyuki Kawahigashi We give a review of recent progress of operator algebraic studies of chiral conformal field theory. See [1] and [2] for more details and references. Chiral conformal field theory arises from a decomposition of a 2-dimensional conformal field theory. A local conformal net is an operator algebraic object describing a chiral conformal field theory. A local conformal net is an assignment of a von Neumann algebra A(I) to each interval I contained in the circle S1which plays the role of “spacetime”. The “spacetime symmetry” group is Diff(S1), the group of orientation preserving diffeomorphisms of S1. Then we impose physically 3314Oberwolfach Report 55/2017 natural axioms such as isotony, locality, conformal covariance, positivity of the conformal Hamiltonian and existence of the vacuum on the family{A(I)}. Our important tool to study local conformal nets is a representation theory in the style of Doplicher-Haag-Roberts. The notion of a dimension of a representation is given by the Jones index of a subfactor. We are often interested in the situation where we have only finitely many irreducible representations. Such a case is called rational. An operator algebraic characterization of rationality was given by Kawahigashi-Longo-M¨uger in terms of finiteness of certain Jones index. In the rational case, we have a modular tensor category for representations. We have a machinery of α-induction introduced by Longo-Rehren. Works of Xu and Ocneanu were unified by B¨ockenhauer-Evans-Kawahigashi and general properties of modular invariance were proved. Using these methods, KawahigashiLongo gave a complete classification of local conformal nets with central charge less than 1. (A central charge is a numerical invariant arising from a representation of the Virasoro algebra.) The classification list contains four exceptionals and one of them does not seem to arise from any other known construction. A vertex operator algebra is an algebraic axiomatization of Fourier expansions of operator-valued distributions on S1. It arose from studies of the celebrated Moonshine conjecture of Borcherds and Frenkel-Lepowsky-Meurman. Since a local conformal net and a vertex operator algebra both give mathematical axiomatizations of chiral conformal field theory, it is natural to expect a direct relation between the two. We now present such a relation due to Carpi-Kawahigashi-Longo-Weiner. First, we need so-called unitarity of a vertex operator algebra. Then we impose a physically natural condition called strong locality. We do not know any example of a unitary vertex operator algebra which does not satisfy strong locality and we have a simple sufficient condition for strong locality. Note that if there should exist a unitary vertex operator algebra without strong locality, it would not correspond to physical chiral conformal field theory. If we have strong locality, we can construct a corresponding local conformal net through a construction of smeared vertex operators. We can also come back to the original vertex operator algebra from the local conformal net we construct. This is due to an idea of Fredenhagen-J¨orss and the Tomita-Takesaki theory. References |
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| [169] | Y. Kawahigashi, {\it Conformal field theory, vertex operator algebras and operator algebras}, to appear in Proceedings of ICM 2018, arXiv:1711.11349. Reflection Positivity3315 Modular Localization and Constructive Algebraic QFT Gandalf Lechner This talk was the second in a series of three (by R. Longo, myself, and Y. Tanimoto, respectively) on applications of Tomita-Takesaki modular theory in algebraic quantum field theory. The focus of my talk was the construction of examples of algebraic quantum field theories. A central notion in this context is that of a Borchers triple. In the simplest two-dimensional situation, this consists of a von Neumann algebraM on a Hilbert spaceH, a unitary strongly continuous positive energy representation U(x+, x−) = eix+P+eix−P−of R2such that adU (x+, x−)(M) ⊂ M for x+≥ 0, x−≤ 0, and a unit vector Ω∈ H that is invariant under U and cyclic and separating for M (see [2], and [5] for a review). Given a Borchers triple (M, U, Ω), the representation U extends from R2to the proper Poincar´e groupP by a theorem of Borchers [1]. The algebraM can be interpreted as being localized in the wedge region W ={x ∈ R2: x+> 0, x−< 0}, and by a canonical procedure, the triple defines a map from open subsetsO ⊂ R2to von Neumann algebrasA(O) ⊂ B(H) that is inclusion-preserving, local, covariant under the representation U , and fixed byA(W ) = M. If this map has also the property that Ω is cyclic forA(O) for every nonemptyO, these data describe a quantum field theory in its vacuum representation. One is therefore interested in finding examples of Borchers triples. Free field theory examples of Borchers triples are well-known. In this talk, I reviewed two procedures for constructing examples of Borchers triples. The first [2] is related to Rieffel deformations of C∗-algebras [7] and is based on the notion of a warped convolution, a deformation procedure for operators in B(H). Let (M, U, Ω) be a Borchers triple (for example, one given by a free field theory), and let A∈ M be an operator that is smooth w.r.t. U, i.e. such that x7→ ad(U(x))(A) is smooth in norm. As a deformation parameter, consider a 2×2 matrix Q antisymmetric w.r.t. the Minkowski inner product. Then the warped convolution of A is defined as ZZ AQ= (2π)−2dpdx e−ip·xad(U (Qp))(A) U (x). R2R2 This integral exists in an oscillatory sense on the dense domain of smooth vectors inH and extends to a bounded operator on all of H. Denote the von Neumann algebra generated by all AQ, A∈ M smooth, by MQ. Then the main theorem in this context, a consequence of various properties of the map A7→ AQ, is that the triple (MQ, U, Ω) is again a Borchers triple if Q satisfies a positivity condition related to the spectrum of U . One thus obtains a family of Borchers triples, indexed by Q, from a given one. Triples with different parameters are inequivalent, but they all have the special property that the modular data ofMQare independent of Q, i.e. coincide with the modular data of the original von Neumann algebraM0=M. It is an open problem whether the 3316Oberwolfach Report 55/2017 “local” von Neumann algebrasAQ(O) generated from (MQ, U, Ω) are non-trivial or not. The second procedure for constructing Borchers triples is based on the notion of a crossing-symmetric R-matrix. One starts from the single particle Hilbert space H1= L2(R, dθ)⊗K, where K is a separable (often times finite-dimensional) Hilbert space for internal degrees of freedom, and L2(R, dθ) carries the usual realization of the unitary massive irreducible positive energy representation U1ofP. In this context, a crossing-symmetric R-matrix is a function R from R to unitaries on K ⊗ K satisfying a number of properties. In particular, R(θ) is required to satisfy the Yang-Baxter equation with spectral parameter θ and a symmetry condition that ensures that R generates a unitary representation of the symmetric group SnonH⊗n1. Denoting byHn⊂ H⊗n1the subspace on which this representation acts trivially, the Hilbert space of the Borchers triple to be constructed is thenL H :=nHn. The representation U is defined by second quantization of U1, and Ω is defined as the Fock vacuum ofH. Similar to a usual Bose Fock space, alsoH carries canonical creation / annihilation operators, and their sum defines a quantum field ϕ [6]. These field operators generate a von Neumann algebraM such that (M, U, Ω) is a Borchers triple if R extends to a bounded analytic function on the complex strip 0 < Imθ < π and the crossing symmetry hk1⊗ k2, R(iπ− θ) k3⊗ k4i = hk2⊗ Γk4, R(θ) Γk1⊗ k3i,θ∈ R, reminiscent of the KMS condition. Here k1, . . . , k4are arbitrary vectors inK, and Γ is an antiunitary involution onK related to the modular conjugation of M (see [3]). The local algebras are non-trivial if R satisfies further regularity conditions. To make connections to other talks given at this meeting, let us make the following two observations. (1) The crossing symmetry of R can be depicted graphically as which suggests a relation to the string Fourier transform presented in the talk by A. Jaffe. (2) The domain of the one-particle component of the modular operator ∆1/2 becomes a complex Hilbert space when closed in its graph norm. For Reflection Positivity3317 the setting of the second example, this Hilbert space coincides with the classical Hardy space on the strip 0 < Imθ < π [4], and thus connects to the reproducing kernel Hilbert space setting discussed by P. Jorgensen. References |
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| [176] | M.A. Rieffel, Deformation quantization for actions of Rd, Memoirs A.M.S., 506, 1-96 (1993). Reflection positivity and operator theoretic correlation inequalities Tadahiro Miyao This talk consists of the following three sections: Section 1: Operator theoretic correlation inequalities Section 2: Spin-reflection positivity Section 3: Universality in the Hubbard model 1. Operator theoretic correlation inequalities In Section 1, operator theoretic correlation inequalities [7,8] are introduced as follows. Let H be a complex Hilbert space. By a convex cone, we understand a closed convex set P⊂ H such that tP ⊆ P for all t ≥ 0 and P ∩ (−P) = {0}. Definition 1.1.• The dual cone of P is defined by P†={η ∈ H | hη|ξi ≥ 0∀ξ ∈ P}. We say that P is self-dual if P = P†. • A vector ξ is said to be positive w.r.t. P if ξ ∈ P. We write this as ξ ≥ 0 w.r.t. P. • A vector η ∈ P is called strictly positive w.r.t. P whenever hξ|ηi > 0 for all ξ∈ P{0 |
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| [185] | K. Osterwalder, R. Schrader, Axioms for Euclidean Green’s functions. Comm. Math. Phys. 31 (1973), 83-112. Axioms for Euclidean Green’s functions. II. With an appendix by Stephen Summers. Comm. Math. Phys. 42 (1975), 281-305. The role of positivity in generalized coherent state transforms Jos´e Mour˜ao (joint work with T. Baier, W. Kirwin, J. P. Nunes and T. Thiemann) From the rather messy quantization ambiguity of a symplectic manifold (M, ω) on a real polarization one gets a very nice, infinite dimensional, geodesically convex space of K¨ahler quantizations. This is done, in geometric quantization, by allowing the preferred local observables defining a polarization to be complex-valued while restricting them to a class satisfying adequate positivity conditions. For systems with one degree of freedom this is illustrated by changing from the (infinite-dimensional) family of real quantizations with quantum Hilbert-spaces of L2-functions of the variables, ztf= q + t f (p) ,t∈ R to the family of complex observables obtained by letting t to enter the upper half plane zτ f= q + τ f (p) ,τ∈ C, ℑ(τ) > 0 . It turns out that if f′(p) > 0 then several simplifying facts occur: 1. Complex Structure: There is a unique complex structure Jτ fon R2for which zτ fis a global holomorphic coordinate. Reflection Positivity3321 2. K¨ahler Metric: The (standard) symplectic form together with the complex structure Jifdefine on R2the K¨ahler metric 1 f′(p)dq2+ f′(p) dp2, with scalar curvature, ′′ S(γf) =−1. f′(p) 3. Quantum Hilbert space: much better defined than in the case of quantizations based on real observables: no HQf=Ψ(q, p) = ψ(zif) e−kf(p)/2,||Ψ|| < ∞ R where ψ is a Jif-holomorphic function and kf(p) = pf (p)−f (p)dp is the K¨ahler potential for (1). Things improve even further if one takes into account the half-form correction. 4. Reality conditions: The inner product is fixed by geometric quantization and resolves the problem of “reality conditions”. With the half-form correction, in our example, takes the form, Z < Ψ1, Ψ2>=ψ1(zif)ψ2(zif) e−kf(p)(f′(p))1/2dqdp . R2 In this formalism the space of quantizations becomes the space of K¨ahler structures on a given symplectic manifold. In the complex picture, in which one fixes a complex structure and lets the symplectic form to change, on a compact manifold, the space of K¨ahler forms with fixed cohomology class is H/R =ωϕ= ω + i∂ ¯∂ ϕ , ϕ∈ C∞(M ) : γϕ= ωϕ(·, J·) > 0, and can be interpreted as (part of the) space of K¨ahler quantizations of (M, ω). Despite being an open subset of C∞(M ), the space of K¨ahler potentialsH with Mabuchi metric,Z ωnϕ hf1, f2iϕ=f1f2, Mn! has the very rich geometry of an infinite dimensional symmetric space. In particular its geodesics, described by the complex homogeneous Monge-Amp“ere (CHMA) equation are given by one-parameter “groups” of imaginary time canonical transformations [4,3,14], (2)eisXH. In [14] a method, based on the Gr¨obner theory of Lie series, to construct these imaginary time flows has been proposed, which effectively reduces the analytic Cauchy problem for the CHMA equation to analytically continuing to complex time the corresponding Hamiltonian flow. The coherent state transforms (CST) correspond to appropriate liftings of eisXHto the quantum bundle, (3)VisH:HQ0−→ HQis. 3322Oberwolfach Report 55/2017 Examples are given by Hall generalizations [7] of the Coherent State Segal– Bargmann transforms to complexifications GCof compact Lie groups G (see also [8,10,6,9]), U : L2(G, dx)−→ HL2(GC, dν(g)) ∆ (4)U=C ◦ e2, where GCis the unique complexification of G,HL2means holomorphic L2functions, ν is the averaged heat kernel measure on GC. Following the introduction of the concept of “complexifiers” in [15,16] the CST (4) has been shown to be equivalent to geometric quantization transforms of the form (3) in [11,12,5,13] with the complexifier H being the norm square of the moment map, ||µ||2. (5)H = 2 The CST (3) with Hamiltonians H given by (5) (or other convex function of µ) play an important in tropical geometry [1] and in representation theory and algebraic geometry [2]. References |
| [186] | T. Baier, C. Florentino, J. M. Mour˜ao, and J. P. Nunes, {\it Toric K¨}{\it ahler metrics seen from} {\it infinity, quantization and compact tropical amoebas}, J. Diff. Geom. 89 (2011), 411-454. · Zbl 1261.53081 |
| [187] | T. Baier, J. M. Mour˜ao, and J. P. Nunes, {\it work in progress}. |
| [188] | D. Burns, E. Lupercio and A. Uribe, {\it The exponential map of the complexification of }H{\it am} {\it in the real-analytic case}, arXiv:1307.0493. · Zbl 1489.53095 |
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| [190] | J. N. Esteves, J. M. Mourao and J. P. Nunes, {\it Quantization in singular real polarizations:} {\it Kaehler regularization, Maslov correction and pairings, }J. Phys. A 48 (2015), 22FT01. · Zbl 1316.81039 |
| [191] | C. Florentino, P. Matias, J. Mour˜ao and J. P. Nunes, {\it On the BKS pairing for K¨}{\it ahler} {\it quantizations of the cotangent bundle of a Lie group}, J. Funct. Anal. 234 (2006) 180-198. · Zbl 1102.22007 |
| [192] | B. Hall, {\it The Segal-Bargmann “coherent-state” transform for Lie groups}, J. Funct. Anal. 122 (1994), 103-151. · Zbl 0838.22004 |
| [193] | B. Hall, {\it Geometric quantization and the generalized Segal-Bargmann transform for Lie} {\it groups of compact type}, Comm. Math. Phys. 226 (2002) 233-268. · Zbl 1007.53070 |
| [194] | B. Hall and W. D. Kirwin, {\it Adapted complex structures and the geodesic flow}, Math. Ann. 350 (2011) 455-474. · Zbl 1228.53097 |
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| [196] | W. Kirwin, J. Mour˜ao and J.P. Nunes, {\it Complex symplectomorphisms and pseudo-Kahler} {\it islands in the quantization of toric manifolds}, Math Annalen (2015), 1-28. |
| [197] | W. Kirwin, J. Mour˜ao and J.P. Nunes, {\it Coherent state transforms and the Mackey-Stone-} {\it Von Neumann theorem}, Journ. Math. Phys. 55 (2014) 102101. · Zbl 1314.81119 |
| [198] | W. Kirwin, J. Mour˜ao, J.P. Nunes and T. Thiemann, Hyperbolic analogs of the SegalBargmann transform, work in progress. Reflection Positivity3323 |
| [199] | J. Mour˜ao and J.P. Nunes, {\it On complexified analytic Hamiltonian flows and geodesics on} {\it the space of K¨}{\it ahler metrics}, Int. Math. Res. Not. 2015, 10624-10656. · Zbl 1333.58003 |
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| [201] | T. Thiemann, {\it Modern canonical quantum general relativity}, Cambridge University Press, Cambridge, 2007. Generalized transfer matrices and automorphic functions Anke Pohl As known for a long time, transfer matrix techniques prove to be powerful in the study of lattice spin systems, for example for deriving exact solutions of one- and two-dimensional systems such as the Onsager solution. Reflection positivity in lattice spin systems is intimately related to the existence of self-adjoint positive definite transfer matrices. Also known for a long time, the correspondence principle of quantum mechanics suggests close relations between geometric and spectral entities of Riemannian manifolds (and, more generally, of Riemannian orbifolds). In particular, one expects strong interdependencies between geodesics (classical mechanical objects) on the one side and L2-eigenfunctions and L2-eigenvalues of the Laplacian, and more generally, resonances and resonant states (quantum mechanical objects) on the other side. Over the last century much effort was spend on establishing instances of such interdependencies in a mathematically rigorous way. An ever increasing number of results were found and were seen to be of great importance for various areas of mathematics, including dynamical systems, spectral theory, harmonic analysis, representation theory, number theory, and mathematical physics. However, the full scope and depth of the relation between geometric and spectral objects of Riemannian orbifolds is still mysterious. In the talk we restricted to the case of non-elementary hyperbolic surfaces X = Γ\H with at most finitely many ends (of finite and infinite area). Here, Hdenotes the hyperbolic plane, and Γ is a discrete non-cyclic geometrically finite subgroup of the M¨obius group PSL(2, R). For these spaces, a relation between periodic geodesics and resonances is shown by the Selberg zeta function which is the dynamical zeta function given by YY��� (1)ZX(β) :=1− e−(β+k)ℓ(β∈ C, Re(β) ≫ 1) ℓ∈L(X)k=0 where L(X) denotes the primitive geodesic length spectrum of X, counted with multiplicities. The infinite product in (1) converges if Re β is sufficiently large, and it has a meromorphic continuation to all of C. The zeros of the Selberg zeta function consist of the resonances of X and some well-understood ‘trivial’ zeros (of rather topological nature). Thus, the Selberg zeta function ZXestablishes a relation between the geodesic length spectrum and the Laplace spectrum of X, or, 3324Oberwolfach Report 55/2017 in other words, a relation between geodesics and resonant states on the spectral level. We discussed a construction of generalized transfer matrices and showed that these allow us to establish a relation between periodic geodesics and L2-eigenfunctions beyond the spectral level, thereby improving on the connection provided by means of the Selberg zeta function. The construction of the generalized transfer matrices rely on a good choice of a discretization for the geodesic flow on X. We took advantage of the discretizations provided in [16], which are particularly wellsuited for our purposes. Each such discretization provides a discrete dynamical system F : D→ D on a union of certain intervals in R that is semi-conjugate to the geodesic flow on X and that branches into finitely many ‘submaps’ given by the M¨obius action of some element in Γ. The associated generalized transfer matrix with parameter β∈ C (transfer operator in the sense of Ruelle and Mayer) is X Lβf (x) :=e−β ln |F′(y)|f (y), y∈F−1(x) acting on functions f : D→ C. Some of the major results regarding the role of these transfer operators in the study of the interdependencies of geodesics and eigenfunctions and resonant states of X are roughly as follows: • If X has finite area and at least one cusp, that is, an end of finite area, and if Re β∈ (0, 1) then the space of rapidly decaying L2-eigenfunctions on X (Maass cusp forms) is isomorphic to the space of sufficiently regular eigenfunctions with eigenvalue 1 ofLβ[12,15,14,18,13]. The isomorphism is given by an explicit integral transform. Up to date, transfer operator techniques are the only tool known to provide such a deep relation between geometric and spectral entities of hyperbolic surfaces. • If X has finite or infinite area and at least one cusp then an induction procedure of the discretization of the geodesic flow used for the construction ofLβprovides a uniformly expanding, infinitely branched discrete dynamical system. The associated transfer operator eLβacts on a certain Banach space of holomorphic functions. As such it is nuclear of order zero and hence admissible for the thermodynamic formalism. Its Fredholm determinant equals the Selberg zeta function ZX(β) = det 1− eLβ. The possibility to represent ZXas a Fredholm determinant of a transfer operator family suggests that many results obtained with the help of the Selberg zeta function and the Selberg trace formula should follow as a ‘shadow’ from results obtained via transfer operators. Moreover, transfer operator techniques provide an alternative proof of meromorphic extendability of the Selberg zeta function. See [12,18,17,19] for all of these results. Reflection Positivity3325 • Eigenfunctions with eigenvalue 1 of Lβand eLβare isomorphic, see [1] for Hecke triangle groups and forthcoming manuscripts for general Γ. This result together with the previously mentioned allows us to recover already a part of the spectral interpretations of the zeros of the Selberg zeta function without relying on the Selberg trace formula. • Twists by finite-dimensional unitary representations can easily be accommodated by the transfer operators as additional weights. The results on the connection between Selberg zeta functions and eLβas well as on the relation betweenLβand eLβextend to the twisted objects [19,1]. • Also twists by finite-dimensional representations χ with non-expanding cusp monodromies (representations which are not necessarily unitary but have controlled behavior in cusps) can be accommodated by transfer operators. Transfer operator techniques are currently the only known method to prove meromorphic extendability of the χ-twisted Selberg zeta functions [8]. • These results recover, illuminate and refine the seminal transfer operator techniques for the modular surface PSL(2, Z)\H by Mayer [10,11], Chang– Mayer [3], Efrat [7], Lewis-Zagier [9], Bruggeman [2], and its extension to certain finite-index subgroups of PSL(2, Z) [4,5,6]. It is expected that the mentioned results, in particular the isomorphism between eigenfunctions of transfer operators and Maass cusp forms, can be generalized to eigenfunctions of other regularity, to (Γ, χ)-twisted and vector-valued eigenfunctions, and to general resonant states. Moreover, generalizations to more general locally symmetric spaces are expected. The relation between these generalized transfer matrices and reflection positivity remains to be understood. References |
| [202] | A. Adam and A. Pohl, {\it A transfer-operator-based relation between Laplace eigenfunctions} {\it and zeros of Selberg zeta functions}, arXiv:1606.09109. · Zbl 1475.11153 |
| [203] | R. Bruggeman, {\it Automorphic forms, hyperfunction cohomology, and period functions}, J. reine angew. Math. 492 (1997), 1-39. · Zbl 0914.11023 |
| [204] | C.-H. Chang and D. Mayer, {\it The transfer operator approach to Selberg’s zeta function and} {\it modular and Maass wave forms for }PSL(2, Z), Emerging applications of number theory (Minneapolis, MN, 1996), IMA Vol. Math. Appl., vol. 109, Springer, New York, 1999, pp. 73– 141. · Zbl 0982.11049 |
| [205] | , {\it Eigenfunctions of the transfer operators and the period functions for modular} {\it groups}, Dynamical, spectral, and arithmetic zeta functions (San Antonio, TX, 1999), Contemp. Math., vol. 290, Amer. Math. Soc., Providence, RI, 2001, pp. 1-40. · Zbl 1037.11032 |
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| [209] | K. Fedosova and A. Pohl, {\it Meromorphic continuation of Selberg zeta functions with twists} {\it having non-expanding cusp monodromy}, arXiv:1709.00760. 3326Oberwolfach Report 55/2017 · Zbl 1453.11118 |
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| [214] | A. Pohl, {\it Symbolic dynamics and Maass cusp forms for cuspidal cofinite Fuchsian groups}, in preparation. |
| [215] | , {\it A dynamical approach to Maass cusp forms}, J. Mod. Dyn. 6 (2012), no. 4, 563-596. · Zbl 1273.37016 |
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| [217] | , {\it Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orb-} {\it ifolds}, Discrete Contin. Dyn. Syst., Ser. A 34 (2014), no. 5, 2173-2241. · Zbl 1312.37033 |
| [218] | , {\it A thermodynamic formalism approach to the Selberg zeta function for Hecke tri-} {\it angle surfaces of infinite area}, Commun. Math. Phys. 337 (2015), no. 1, 103-126. · Zbl 1348.37042 |
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| [220] | , {\it Symbolic dynamics, automorphic functions, and Selberg zeta functions with unitary} {\it representations}, Contemp. Math. 669 (2016), 205-236. Local nets of von Neumann algebras in the sine-Gordon model Kasia Rejzner In my talk I presented recent results on construction of local nets of von Neumann algebras using perturbative algebraic quantum field theory (pAQFT). The talk is based on [BR17,BFR17]. 1. Perturbative AQFT Algebraic quantum field theory (AQFT) is a convenient framework to investigate conceptual problems in QFT. It started as the axiomatic framework of HaagKastler [HK64]: a model is defined by associating to each regionO of Minkowski spacetime an operator algebra A(O) of observables that can be measured in O. The physical notion of subsystems is realized by the condition of Isotony, i.e.: O1⊂ O2⇒ A(O1)⊂ A(O2). Other axioms include: Einstein causality and the Time-slice axiom. Perturbative algebraic quantum field theory (pAQFT) is a mathematically rigorous framework that allows to build interacting AQFT models, where A(O) is now a formal power series in topological∗-algebras. It combines Haag’s idea of local quantum physics with methods of perturbation theory. Main ingredients: • Free theory obtained by the formal deformation quantization of Poisson (Peierls) bracket: ⋆-product [DF01,BF00,BDF09]. • Interaction introduced in the causal approach to renormalization due to Epstein and Glaser [EG73]. • Generalization to curved spacetime [BFV03,HW01,FV12]. Reflection Positivity3327 2. sine-Gordon model in pAQFT In pAQFT we need the following physical input: • A globally hyperbolic spacetime M. For the sine-Gordon model in two dimensions, M = M2, the 2-dimensional Minkowski spacetime. • Configuration space E(M): choice of objects we want to study in our theory (scalars, vectors, tensors,. . . ). For the scalar field:E(M) ≡ C∞(M, R). • Dynamics: we use a modification of the Lagrangian formalism. Observables are functionals onE(M). For the free massless scalar field the equation of motion is P ϕ = 0, where P =−✷ is (minus) the wave operator. For M globally hyperbolic, P admits retarded and advanced Green’s functions ∆R, ∆A. They satisfy: P◦ ∆R/A= idD(M), ∆R/A◦ (P) = id D(M)D(M)and supp(∆R)⊂ {(x, y) ∈ M2|y ∈ J−(x)} , supp(∆A)⊂ {(x, y) ∈ M2|y ∈ J+(x)} . For the massless scalar field in 2D: ∆R(x) =−1θ(t− |x|)∆A(x) =−1θ(−t − |x|) , x = (t, x) ∈ M2. 22 . Their difference is the Pauli-Jordan “function”: ∆= ∆R− ∆A. The Poisson bracket of the free theory is DE {F, G}=F(1), ∆G(1). We define the ⋆-product (deformation of the pointwise product): X∞ nDE . (F ⋆ G)(ϕ)=F(n)(ϕ), W⊗nG(n)(ϕ), n! n=0 where W is the 2-point function of a Hadamard state and it differs from2i∆ by a symmetric bidistribution, denoted by H. For the massless scalar field in 2D it is convenient to use the Hadamard parametrix W (x) =(∆R(x)− ∆A(x)) + H(x) =−ln−x · x + iεt 24πµ2 where µ > 0 is the scale parameter that we need to fix. The Hadamard Parametrix W differs from the 2-point function of a Hadamard state by a smooth symmetric function v, Wv=2i(∆R−∆A)+Hv= W +v. Define ⋆vas the star product induced by Wv, while ⋆ denotes the star product induced by the parametrix W . These products are intertwined by a “gauge transformation” αDEH= e. 2DH, whereDH=. H,δϕδ22=RH(x, y)δϕ(x)δϕ(y)δ2dxdy. Hence ⋆, and ⋆vare equivalent products. The free QFT is defined as A0(M )= (F(M)[[ ]], ⋆, ∗), where F∗(ϕ)= F (ϕ). andF(M) is an appropriate functional space (some wavefront set conditions on F(n)(ϕ)s induced by W ). 3328Oberwolfach Report 55/2017 For construction of interacting fields we need the time-ordered product. We define time-ordering operatorT as: X∞1DE T F (ϕ)=n!F(2n)(ϕ), ( 2∆F)⊗n, n=0 where ∆F=2i(∆A+ ∆R) + H and H = W−i2∆. FormallyT corresponds to the operator of convolution with the oscillating Gaussian measure “with covariance i ∆F”, Z T F (ϕ)formal=F (ϕ− ϕ) dµi ∆F(ϕ) . We define the time-ordered product·Tby: . F·TG=T (T−1F· T−1G) In the sine-Gordon model, prominent role is played by the vertex operators. These are defined as Va(g)=.Rexp(iaΦx)g(x)dx, where Φx(ϕ)= ϕ(x) is the evaluation. functional at x. Note that we are constructing the abstract algebra first, with no reference to Fock space. Using the commutative product·Twe define the S-matrix : . S(V )= eiV / T=T (eT−1(iV / )) . Interacting fields are defined by the formula of Bogoliubov: RV(F )= (eiV / T)⋆−1⋆ (eiV / T·TF ) =−i dS(V )−1S(V + µF ) dµµ=0 Passing from ⋆ to ⋆vmeans changing the Wick ordering. Denote α−1HvF= :F :.v. The expectation value of the product of two normally-ordered observables F, G in the quasi-free Hadamard state with 2-point function Wvis: . ωv(:F :v⋆ :G:v)= αv(:F :v⋆ :G:v)(0) = (F ⋆vG)(0) . . Similar for the S-matrix: ωv(S(λ :V :v))= αveiλ :V :Tv/ (0) = eiλV / T(0). Here v ·Tvis the time-ordered product corresponding to ⋆v. Theorem 2.1. The formal S-matrix αv◦ S(λ :V :v) = eiλV / Tin the-sine Gordon v model with V =12(Va(f )+V−a(f )) and 0 < β = a2/4π < 1, f∈ D(M), converges as a functional on the configuration space in the appropriate topology (related to H¨ormander topology on distribution spaces). In [BFR17] it was shown that by choosing appropriate Hadamard states (based on the construction of [DM06]) one can show that the interacting fields form a net of von Neumann algebras. These Hadamard states are, moreover, locally normal to vacuum states for scalar field theories with non-zero mass. Reflection Positivity3329 References [BDF09] R. Brunetti, M. D¨utsch, and K. Fredenhagen, {\it Perturbative algebraic quantum field} {\it theory and the renormalization groups}, Adv. Theor. Math. Phys. 13 (2009), no. 5, 1541-1599. [BF00]R. Brunetti and K. Fredenhagen, {\it Microlocal analysis and interacting quantum field} {\it theories}, Commun. Math. Phys. 208 (2000), no. 3, 623-661. [BFR17] D. Bahns, K. Fredenhagen, and K. Rejzner, {\it Local nets of von Neumann algebras in the} {\it sine-Gordon model}, [arXiv:math-ph/1712.02844]. [BFV03] R. Brunetti, K. Fredenhagen, and R. Verch, {\it The generally covariant locality principle—} {\it A new paradigm for local quantum field theory}, Commun. Math. Phys. 237 (2003), 31-68. [BR17]D. Bahns and K. Rejzner, {\it The quantum Sine Gordon model in perturbative AQFT}, Commun. Math. Phys. (2017), https://doi.org/10.1007/s00220-017-2944-4. [DF01]M. D¨utsch and K. Fredenhagen, {\it Perturbative algebraic field theory, and deformation} {\it quantization}, Mathematical Physics in Mathematics and Physics: Quantum and Operator Algebraic Aspects 30 (2001), 1-10. [DM06] J. Derezi´nski and K. A. Meissner, {\it Quantum massless field in 1+ 1 dimensions}, pp. 107– 127, Springer, 2006, In {\it Mathematical Physics of Quantum Mechanics}, J. Asch, A. Joye Eds. [EG73]H. Epstein and V. Glaser, {\it The role of locality in perturbation theory}, AHP 19 (1973), no. 3, 211-295. [FV12]C. J. Fewster and R. Verch, {\it Dynamical locality and covariance: What makes a physical} {\it theory the same in all spacetimes?}, Annales Henri Poincar´e 13 (2012), no. 7, 1613-1674. [HK64]R. Haag and D. Kastler, {\it An algebraic approach to quantum field theory}, Journal of Mathematical Physics 5 (1964), no. 7, 848-861. [HW01] S. Hollands and R. M. Wald, {\it Local Wick polynomials and time ordered products of} {\it quantum fields in curved spacetime}, Commun. Math. Phys. 223 (2001), no. 2, 289-326. Half-sided modular inclusions (and free products in AQFT) Yoh Tanimoto (joint work with Roberto Longo, Yoshimichi Ueda) 1. Conformal nets on the circle In two-dimensional conformal field theory on the Minkowski space, important observables such as currents and stress-energy tensor decompose into the so-called chiral components, which are quantum fields living on one of the lightrays. These chiral components extends further to the circle (the one-point compactification of the lightray) (the L¨uscher-Mack theorem, see [FST89, Section 3.2]). It turns out that operator algebras are useful in constructing such chiral components. In the operator-algebraic approach, such a chiral component is realized as a M¨obius covariant net on S1: a family{A(I)}I⊂S1of von Neumann algebras parametrized by open, proper, connected and non empty intervals in S1, a unitary representation U of the M¨obius group PSL(2, R) and the “vacuum” vector Ω invariant for U , which satisfy the Haag-Kastler axioms, namely, isotony, locality, M¨obius covariance, positivity of energy and cyclicity of the vacuum [GF93, Definition 2.5]. 3330Oberwolfach Report 55/2017 It follows from these axioms that Ω is cyclic and separating for each algebra A(I) (Reeh-Schlieder property), hence we can define the modular objects: SI:A(I)Ω ∋ xΩ 7−→ x∗Ω, 1 whose closure is still denoted by SIand SI= JI∆I2is the polar decomposition. Let us recall that the lightray R is identified as a subset of S1by the stereographic projection. With this identification, the positive half-line R+is an interval in S1. For a M¨obius covariant net, the Bisognano-Wichmann property holds [GF93, Theorem 2.19]: ∆itR= U (DR(2πt)), where D ++R+is the one-parameter subgroup of dilations of PSL(2, R), which clearly preserves R+. Now, observe that, by covariance, it holds that Ad ∆it(A(R++ 1))⊂ A(R++ 1) for t≥ 0. This inclusion of algebras, called a half-sided modular inclusion, turns out to contain much information of the given net. 2. Half-sided modular inclusions LetN ⊂ M be von Neumann algebras and Ω be a vector cyclic and separating for bothN and M. We can then define the modular group ∆itMofM with respect to Ω. This triple (N ⊂ M, Ω) is said to be a half-sided modular inclusion (HSMI) if Ad ∆itM(N ) ⊂ N for t ≥ 0. From this simple object, many interesting properties follow, among which is that ∆itMand ∆isNgenerate a positive-energy representation of the translation-dilation group [Wie93] [AZ05, Theorem 2.1]. Moreover, if Ω is cyclic forN′∩ M (N′is the set of all bounded operators commuting withN ), this inclusion is said to be standard. Let (A, U, Ω) be a M¨obius covariant net. It is said to be strongly additive if A(I1)∨A(I2) =A(I) holds, where I1and I2are two intervals made from an interval I by removing one point, andA(I1)∨ A(I2) denotes the von Neumann algebra generated byA(I1) andA(I2). There is a one-to-one correspondence between • Standard half-sided modular inclusions (N ⊂ M, Ω) • Strongly additive M¨obius covariant nets (A, U, Ω) and the correspondence is given byN = A(R++ 1),M = A(R+) [GLW98, Corollary 1.9]. This correspondence allows one to construct new M¨obius covariant nets from standard HSMIs. For example, consider the Virasoro nets Vircwith c > 1 (the nets generated by the stress-energy tensor only). They are known not to be strongly additive [BSM90, Section 4], therefore, from the standard HSMI (N = Virc(R++ 1),M = Virc(R+), Ω) one obtains a M¨obius covariant netA which is different from Virc. These nets have not been identified with any known M¨obius covariant net. Another construction goes as follows: for a given M¨obius covariant netA, one considers its restrictionA|Rto the real line R. If one takes a KMS state ϕ onA|R and makes the GNS representation πϕwith the GNS vector ΩSϕS, with respect to the translation group, then the inclusion (πϕ(I⋐R+A(I)) ⊂ πϕ(I⋐RA(I))′′, Ωϕ) is a standard HSMI, and hence one can construct a M¨obius covariant net [Lon01, Proposition 3.2]. We have constructed continuously many differerent KMS states Reflection Positivity3331 on Virc, c≥ 1 [CLTW12, Section 5], therefore, from these standard HSMIs one can construct (possibly new) M¨obius covariant nets. Problem 2.1. Determine whether these M¨obius covariant nets are isomorphic to any known net. 3. Non standard HSMI from free products IfA is a Haag-Kastler net in 1 + 1 or more dimensions with the BisognanoWichmann property (meaning that the modular group is the Lorentz boost), then (A(W + a) ⊂ A(W ), Ω) is a half-sided modular inclusion, where W is a wedge region and a is a past lightlike vector in W . It is in general a difficult problem to determine whether such a given HSMI is standard or not, ifA is not the free field net. Some successful cases are those coming from two-dimensional interacting nets [Tan14], and they are some fixed point subnets of the tensor product of the so-called U(1)-current net [BT15, Section 5.3]. Many other cases are open. Problem 3.1. Determine whether the HSMIs coming from the interacting nets of [Lec08] are standard. Now it is a natural question whether there is a HSMI (N ⊂ M, Ω) where N′∩M is trivial, i.e. containing only the multiples of the identity operator. We answer this positively by constructing an example from free products [LTU17]. A free product of a family of von Neumann algebras{Mk}k∈Kwith respect to cyclic and separating vectors{Ωk}k∈Kis a large von Neumann algebra containing isomorphic images of allMk’s which are highly non commutative, and equipped with a cyclic separating vector Ω [Voi85]. One can determine the modular objects of (M, Ω) in terms of those of (Mk, Ωk) [Bar95, Lemma 1]. Let (N0⊂ M0, Ω0) be a standard HSMI and{(Nk⊂ Mk, Ωk)}k∈Kisomorphic copies of (N0⊂ M0, Ω0). We can then construct an inclusion of the free product von Neumann algebras (N ⊂ M, Ω). As the modular objects are known, it is immediate to see that this is a HSMI. Theorem 1. If|K| = ∞, then for the free product HSMI (N ⊂ M, Ω), N′∩ M is trivial. Problem 3.2. DetermineN′∩ M when |K| < ∞, and if it is nontrivial, study the corresponding M¨obius covariant net. 3332Oberwolfach Report 55/2017 References [AZ05]Huzihiro Araki and L´aszl´o Zsid´o. Extension of the structure theorem of Borchers and its application to half-sided modular inclusions. {\it Rev. Math. Phys.}, 17(5):491-543, 2005.https://arxiv.org/abs/math/0412061. [Bar95]Lance Barnett. Free product von Neumann algebras of type III. {\it Proc. Amer. Math.} {\it Soc.},123(2):543-553,1995.http://www.ams.org/journals/proc/1995-123-02/ S0002-9939-1995-1224611-7/ S0002-9939-1995-1224611-7. pdf. [BT15]Marcel Bischoff and Yoh Tanimoto. Integrable QFT and Longo-Witten endomorphisms. {\it Ann. Henri Poincar´}{\it e}, 16(2):569-608, 2015.https://arxiv.org/abs/1305. 2171. [BSM90]Detlev Buchholz and Hanns Schulz-Mirbach. Haag duality in conformal quantum field theory. {\it Rev. Math. Phys.}, 2(1):105-125, 1990.https://www.researchgate.net/ publication/246352668. [CLTW12] Paolo Camassa, Roberto Longo, Yoh Tanimoto, and Mih´aly Weiner. Thermal States in Conformal QFT. II. {\it Comm. Math. Phys.}, 315(3):771-802, 2012.https://arxiv. org/abs/1109.2064. [FST89]P. Furlan, G. M. Sotkov, and I. T. Todorov. Two-dimensional conformal quantum field theory. {\it Riv. Nuovo Cimento (3)}, 12(6):1-202, 1989.link.springer.com/ content/pdf/10.1007/BF02742979.pdf. [GF93]Fabrizio Gabbiani and J¨urg Fr¨ohlich. Operator algebras and conformal field theory. {\it Comm. Math. Phys.}, 155(3):569-640, 1993.http://projecteuclid.org/euclid. cmp/1104253398. [GLW98]D. Guido, R. Longo, and H.-W. Wiesbrock. Extensions of conformal nets and superselection structures. {\it Comm. Math. Phys.}, 192(1):217-244, 1998.https://arxiv.org/ abs/hep-th/9703129. [Lec08]Gandalf Lechner. Construction of quantum field theories with factorizing Smatrices. {\it Comm. Math. Phys.}, 277(3):821-860, 2008.http://arxiv.org/abs/ math-ph/0601022. [Lon01]Roberto Longo. Notes for a quantum index theorem. {\it Comm. Math. Phys.}, 222(1):45– 96, 2001.https://arxiv.org/abs/math/0003082. [LTU17]Roberto Longo, Yoh Tanimoto, and Yoshimichi Ueda. Free products in AQFT. 2017. https://arxiv.org/abs/1706.06070. [Tan14]Yoh Tanimoto. Construction of two-dimensional quantum field models through Longo-Witten endomorphisms. {\it Forum Math. Sigma}, 2:e7, 31, 2014.https://arxiv. org/abs/1301.6090. [Voi85]Dan Voiculescu. Symmetries of some reduced free product C∗-algebras. In {\it Operator} {\it algebras and their connections with topology and ergodic theory (Bu¸steni, 1983)}, volume 1132 of {\it Lecture Notes in Math.}, pages 556-588. Springer, Berlin, 1985.http:// dx.doi.org/10.1007/BFb0074909. [Wie93]Hans-Werner Wiesbrock. Half-sided modular inclusions of von-Neumann-algebras. {\it Comm. Math. Phys.}, 157(1):83-92, 1993.https://projecteuclid.org/euclid.cmp/ 1104253848. Reflection Positivity3333 Wick rotation for quantum field theories on degenerate Moyal space/-time Rainer Verch (joint work with Harald Grosse, Gandalf Lechner, Thomas Ludwig) This talk is based on joint a article of Harald Grosse, Gandalf Lechner, Thomas Ludwig and Rainer Verch [1]. The main result is an extension of the OsterwalderSchrader theorem [2,3] which establishes a (bijective) correspondence between Euclidean quantum field theories on d-dimensional Euclidean space Rdand Wightman quantum field theories on d-dimensional Minkowski spacetime R1,d−1, to the case where the underlying space/-time is non-commutative as the result of a Moyal deformation. For the methods used to be applicable, it is important that the Moyal deformation does not act on the time coordinate, or the Euclidean coordinate with respect to which the condition of reflection positivity is imposed for the Euclidean quantum field theory. In establishing this result, we rely on a generalization, or reformulation, of the Osterwalder-Schrader theorem to the operatoralgebraic framework of quantum field theory due to Schlingemann [4] which in turn relies substantially on the theory of virtual group representations based on a certain type of reflection positivity as another branch of generalization of the Osterwalder-Schrader theorem, developed in works by Fr¨ohlich, Osterwalder and Seiler [5], Klein and Landau [6,7], and Jorgensen and Olafsson [8]. Schlingemann’s result states that, ifM denotes a Minkowski spacetime quantum field theory in the operator algebraic setting (with properties such as Poincar´e covariance, locality, spectrum condition and with a vacuum state), and ifE denotes a Euclidean quantum field theory in the operator algebraic setting (characterized by Euclidean covariance, commutativity, and existence of a reflection-positive functional), and if the theories satisfy a property referred to as time-zero condition (reminiscent of the condition that solutions to a hyperbolic field equation are determined by their Cauchy data), then a Euclidean quantum field theoryE uniquely determines a Minkowski spacetime quantum field theoryM = M(E); this is an abstract form of what in physics is called “Wick rotation”. For any operator algebraA that carries a group action τx, x∈ Rd, of Rdby automorphisms, one can obtain a new operator algebraAθby endowingA with a new operator product×θ, the Moyal-Rieffel product [9], given by Z 1 (2π)ddp dx ei(p,x)τθp(A)τx(B)(A, B∈ A) . where θ = (θµν)dµ,ν=1is a real, anti-symmetric matrix and (p, q) is the Euclidean scalar product (or the Minkowski metric product if the underlying Rdis interpreted as Minkowski spacetime). For concreteness, focus on the case d = 4 (generalizations to other dimensions d 3334Oberwolfach Report 55/2017 are obvious), and take 0000 θ=0010 0−100 0000 Then the operator algebrasM and E, both carrying automorphic actions of the translation group, can be Moyal-Rieffel deformed intoMθandEθ. It turns out thatEθstill fulfills the conditions of Schlingemann’s result (although with a smaller covariance group) and thus one can assign a Minkowski spacetime theoryM(Eθ) to Eθ. On the other hand, one can also Moyal-Rieffel deform the Minkowski spacetime quantum field theoryM = M(E) into M(E)θ. One of the important results of [1] is thatM(Eθ) andM(E)θare isomorphic. (For a commutative diagram depicting this, see [1].) We mention that there are various motivations to consider quantum field theories on non-commutative spaces and spacetimes. In approaches to quantum gravity, there is some motivation for uncertainty relations between spacetime coordinates akin to the uncertainty relations between position and momentum in quantum mechanics [10,11]. On the other hand, the Moyal-Rieffel deformation (and other ways of making spacetime non-commutative) has a delocalizing effect and may be helpful in “softening” the short-distance/ultraviolet problems encountered in the construction of interacting quantum field theories. In fact there are some promising results in that direction [12,13,14]. Actually, this was the original motivation for introducing quantum fields on deformed Minkowski spacetime by Snyder in 1947 [15]. It is at present unclear if, or how, the result can be extended to the case that θ is invertible and hence there are noncommutativity relations between space and time coordinates. It is known that Wick rotation cannot be done naively in this situation [16]. References |
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| [236] | D. Bahns, {\it Schwinger functions in noncommutative quantum field theory}, Annals Henri Poincar´e 11 (2010) 1273-1283 Reflection positivity in large-deformation limits of noncommutative field theories Raimar Wulkenhaar (joint work with Harald Grosse, Akifumi Sako) 1. Quantum field theory of matrix models We investigate the possibility to construct quantum field theories as limits of models defined on some Euclidean noncommutative space. Such models are essentially matrix models with action S(Φ) = tr(EΦ2+ pol(Φ)), understood as limit of finite matrices. Here E is a positive selfadjoint operator which defines a dimensionP D = inf{p > 0 : tr((1+E)−p/2) <∞}, and pol(Φ) =rk=1λkΦk. The task could be to give a meaning to the limit measureZ1e−V S(Φ)dΦ, where V > 0 represents the volume. We do not suppose that the limit can be constructed. Instead we derive (for N × N -matrices) equations between moments of the measure, simplify them by further Ward-Takahashi identities [1] resulting from the U (N )-group action, take√ the limit of the equations (which requires renormalisation Φ7→ZΦ and suitable dependence Z(N ), λk(N ) on N , D) and look for exact solutions of these SchwingerDyson equations. This strategy was developed and investigated first for pol(Φ) =λ4Φ4in D = 4 and in a special limitN , V → ∞, withVN2/D= Λ2fixed, followed by Λ→ ∞ [2]. In examples, this limit corresponds to a large-deformation limit of noncommutative geometries. We proved that this approach collapsed the tower of Schwinger-Dyson equations into a closed non-linear integral equation for the matricial 2-point function and a hierarchy of affine integral equations for all higher correlation functions. In fact higher functions were algebraically expressable in terms of fundamental building blocks, which in particular proved that the β-function in this matricial 3336Oberwolfach Report 55/2017 λΦ4-model is identically zero (perturbatively proved in [1]). The equation for the 2-point function was reduced to a fixed point problem (for which we proved existence of a solution) for the boundary 2-point function R+∋ x 7→ G(x, 0). Recent highlight is the λΦ3model in D = 2 [3] and D∈ {4, 6} [4] where renormalisation requires (for D = 6) to consider (1)S(Φ) = tr ZEΦ2+ (κ + νE + ζE2)Φ +12µ2bareΦ2+13λbareΦ3. BPHZ normalisation conditions were directly implemented in the Schwinger-Dyson equation, leading to exact formulae for Z, κ, ν, ζ, µ2bare, λbareas function ofN , V and the given spectrum of E. It turns out that λbareis Z32times a running coupling constant which corresponds to positive β-function for real λ, λbare. Nevertheless there is no Landau ghost; the model can be solved up to any scale Λ. After renormalisation we obtain a closed non-linear equation for the 1-point function. This equation is exactly solvable similar to the Makeenko-Semenoff solution [5] of f2(x) + λ2Rabdt ρ(t)f (x)x−f(t)−t= x by f (x) =√x + c +λ22Rab(√x+c+dt ρ(t)√t+c)√t+c (together with a consistency condition on c). We prove: Let the spectrum of E converge in the considered limit to a positive function e(x), and let X(x) := (2e(x) + 1)2. Then for D = 6 the 1-point function reads in the considered limit p√ (X+c)(1+c)−c−X (2)G(x) = Z2λ√√ λ∞dT (e−1(T2−1))2(X+c−√1+c)2 41√T e′(e−1(√T−1))(√X+c+√T +c)(√1+c+√T +c)2√T +c, 2 R∞√ with c(λ) the implicit solution of−c = λ21√√dT (e−1(T −12))2 T e′(e−1(T −12))(√1+c+√T +c)3√T +c. We checked that Taylor expansion toO(λ3) agrees with renormalised Feynman graph computation. See [4]. Higher N -point functions can be viewed as representations of the permutation group. Every permutation is a product of cycles. Collecting permutations of the same cycle lengths (N1, . . . , NB) leads to a decomposition of (total) N -point functions into N1+ . . . +NB-point functions G(x11, . . . , x1N1| . . . |xB1, . . . , xBNB). It was straightforward [3] to reduce them to 1+ . . . +1-point functions: (3)G(x11, . . . , x1N| . . . |xB, . . . , xB) 11NB XN1XNBYBYNβ λB· · ·G(x|1k1| . . . |x{zBkB})(2e(xβ)+1)24λ, k1=1kB=1β=1lβ =1kβ− (2e(xβlβ)+1)2 (∗)lβ 6=kβ where for B = 1 one should read e(x1k1) +12+ λG(x1k1) instead of (∗). Solving the equations for the 1+ . . . +1-functions is a difficult combinatorial problem. Making Reflection Positivity3337 essential use of Bell polynomials we proved (with X(xi) = (2e(xi) + 1)2): G(x1|x2) =ppp4λ2p X(x1) + cX(x2) + c(X(x1) + c +X(x2) + c)2 G(x1| . . . |xB) =dB−3 (−2λ)3B−411! dtB−3(R(t))B−2pX(x1)+c−2t3· · ·pX(xB)+c−2t3 t=0 for B = 2 and B≥ 3, respectively, where R(t) is an explicit integral [3, eq. (4.9)], which depends on D, λ, e(.). 2. Schwinger functions and reflection positivity It was speculated that space-time might be a noncommutative manifold. In its Euclidean formulation, a scalar field would be an element of a noncommutative algebraA which in many cases is approximated by matrices. A convenient example is the D-dimensional Moyal space, the Rieffel deformation of Schwartz functionsR by translation, (f ⋆ g)(ξ) =R2Dd(k,y)(2π)Df (ξ +12Θk)g(ξ + y)eihk,y, where Θ is a skewsymmetric real D×D-matrix. We describe the transition to matrices in D = 2 with Θ =−θ 00 θ. The functions fmn(z) = 2(−1)mqm!n!q2θzn−mLnm−m2|z|θ2e−|z|θ, with z∈ C ≡ R2, satisfy (fmn⋆ fkl)(z) = δnkfml(z) andRCdz fmnP(z) = 2πθδmn. Therefore, expanding an action functional for a scalar field ϕ =m,nΦmnfmn QD/2 on Moyal space, where m = (m1, . . . , mD/2) and fmn(z) =i=1fmini(zi), in this matrix basis leads back to the starting point (1) of a matricial QFT model. Connected Schwinger N -point functions on Moyal space can thus be obtained via (4) X(−i)N∂Nlog ˆZ(J) Sc(z1, . . . , zN) := limfm1n1(z1)· · · fmNnN(zN), mi,ni∂Jm1n1. . . ∂JmNnNJ=0 RP where formally ˆZ(J) =DΦ exp(−S(Φ) + iVabΦabJab). We proved that only the diagonals G(x1, . . . , x1| . . . |xB, . . . , xB) of the rigorously constructed matricial correlation functions (3) contribute to the limit. Inserting the explicit formulae we were able to check reflection positivity [6]. It should be known that reflection positivity implies the following for the momentum space Schwinger functions ˆS: the temporal Fourier transform from independent energies p0jto time differences τj> 0 is, for all spatial momenta pj, a positive definite function on Rm+. By the Hausdorff-Bernstein-Widder theorem, (i) positive definiteness is equivalent to (ii) being Laplace transform of a positive measure and to (iii) being a completely monotonic function. The latter property is what we check. We find that the 2-point function of λΦ3Don Moyal space is reflection positive iff D = 4, 6 (not D = 2!) and λ∈ R (where the partition function does not define a measure). The K¨all´en-Lehmann measure was explicitly computed; it√ consists of a ‘broadened mass shell’ of width 2µ2−c centred at p2= µ2(with 3338Oberwolfach Report 55/2017 c given after (2),|λ| ≤ λcexpressed in terms of the Lambert W -function) and a ‘scattering part’ supported on p2≥ 2µ2. See [4, Thm 6.1+6.2]. In unpublished work we prove that the projection to diagonal matricial correlation functions violates reflection positivity in higher Schwinger N -point functions. Hence, the above limit procedure needs modification. A natural suggestion would be to replace in (4) the pointwise product fm1n1(z1)· · · fmNnN(zN) by a state ωz1,...,zN(fm1n1⊗ · · · ⊗ fmNnN) onA⊗N. It would be interesting to study whether the choice of state permits enough flexibility to rescue reflection positivity. References |
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