Search

We give a transparent algebraic formulation of our pictorial approach to the reflection positivity (RP), that we introduced in a previous paper. We apply this quantization to the 2 + 1 Levin–Wen model to obtain 1 + 1 anyonic/quantum spin chain theory on the boundary, possibly entangled in the bulk. The reflection positivity property has played a central role in both mathematics and physics, as well as providing a crucial link between the two subjects. In a previous paper we gave a new geometric approach to understanding reflection positivity in terms of pictures. Here we give a transparent algebraic formulation of our pictorial approach. We use insights from this translation to establish the reflection positivity property for the fashionable Levin–Wen models with respect both to vacuum and to bulk excitations. We believe these methods will be useful for understanding a variety of other problems.

We study a new set of duality relations between weighted, combinatoric invariants of a graph G. The dualities arise from a non-linear transform $$\mathcal{B}$$ , acting on the weight function p. We define $$\mathcal{B}$$ on a space of real-valued functions $$\mathcal{O}$$ and investigate its properties. We show that three invariants (the weighted independence number, the weighted Lovasz number, and the weighted fractional packing number) are fixed points of $$\mathcal{B}^2$$ , but the weighted Shannon capacity is not. We interpret these invariants in the study of quantum non-locality.

The classical de Finetti theorem in probability theory relates symmetry under the permutation group with the independence of random variables. This result has application in quantum information. Here we study states that are invariant with respect to a natural action of the braid group, and we emphasize the pictorial formulation and interpretation of our results. We prove a new type of de Finetti theorem for the four-string, double-braid group acting on the parafermion algebra to braid qudits, a natural symmetry in the quon language for quantum information. We prove that a braid-invariant state is extremal if and only if it is a product state. Furthermore, we provide an explicit characterization of braid-invariant states on the parafermion algebra, including finding a distinction that depends on whether the order of the parafermion algebra is square free. We characterize the extremal nature of product states (an inverse de Finetti theorem)., Comment: 25 pages

Significance Classical Fourier analysis, discovered over 200 years ago, remains a cornerstone in understanding almost every field of pure mathematics. Its applications in physics range from classical electromagnetism to the formulation of quantum theory. It gives insights into chemistry, engineering, and information science, and it underlies the theory of communication. Quantum Fourier analysis extends this perspective. It yields insights and inequalities associated with uncertainty principles for quantum symmetries. In this paper, we introduce this mathematical subject, we show how it can solve some theoretical problems, and we give some applications to quantum physics with bounds on entropy and the analysis of quantum entanglement. We believe that quantum Fourier analysis, now in its infancy, will have significant future impact., Quantum Fourier analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform F, as a map between suitably defined Lp spaces, leading to an uncertainty principle for relative entropy. We cite several applications of quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a topological inequality, and we outline several open problems.

Significance We reevaluate ways that one can use pictures, not only to gain mathematical insights, but also to prove mathematical theorems. As an example, we describe ways that the quon language, invented to study quantum information, sheds light on several other areas of mathematics. It results in proofs and algebraic identities of interest in several fields. Motivated by this success, we outline a picture-language program for further research., We give an overview of our philosophy of pictures in mathematics. We emphasize a bidirectional process between picture language and mathematical concepts: abstraction and simulation. This motivates a program to understand different subjects, using virtual and real mathematical concepts simulated by pictures.

We present a 3D, topological picture-language for quantum information. Our approach combines charged excitations carried by strings, with topological properties that arise from embedding the strings in the interior of a three-dimensional manifold with boundary. A quon is a composite that acts as a particle. Specifically a quon is a hemisphere containing a neutral pair of open strings with opposite charge. We interpret multi-quons and their transformations in a natural way. We obtain a new type of relation, a string-genus "joint relation," involving both a string and the 3D manifold. We use the joint relation to obtain a topological interpretation of the $C^{*}$ Hopf algebra relations, that are widely used in tensor networks. We obtain a 3D representation of the Controlled NOT or CNOT gate (that is considerably simpler than earlier work) and a 3D topological protocol for teleportation.

We define a planar para algebra, which arises naturally from combining planar algebras with the idea of $\mathbb{Z}_{N}$ para symmetry in physics. A subfactor planar para algebra is a Hilbert space representation of planar tangles with parafermionic defects, that are invariant under para isotopy. For each $\mathbb{Z}_{N}$, we construct a family of subfactor planar para algebras which play the role of Temperley-Lieb-Jones planar algebras. The first example in this family is the parafermion planar para algebra (PAPPA). Based on this example, we introduce parafermion Pauli matrices, quaternion relations, and braided relations for parafermion algebras which one can use in the study of quantum information. An important ingredient in planar para algebra theory is the string Fourier transform (SFT), that we use on the matrix algebra generated by the Pauli matrices. Two different reflections play an important role in the theory of planar para algebras. One is the adjoint operator; the other is the modular conjugation in Tomita-Takesaki theory. We use the latter one to define the double algebra and to introduce reflection positivity. We give a new and geometric proof of reflection positivity, by relating the two reflections through the string Fourier transform., Comment: 41 pages

The article Holographic software for quantum networks, written by Arthur Jaffe, Zhengwei Liu & Alex Wozniakowski, was originally published electronically on the publisher’s internet portal (currently SpringerLink) on 14/02/2018 without open access. With the author(s)’ decision to opt for Open Choice the copyright of the article changed in March 2018 to © The Author(s) 2018 and the article is forthwith distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, duplication, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The original article has been corrected.

We investigate stochastic quantization as a mathematical tool for quantum field theory. We test the method for the free scalar field. We find that the usual method of stochastic quantization is incompatible with establishing a Hilbert-space interpretation for transition probabilities in quantum theory. In particular, we prove that for any finite stochastic time, the standard probability measure violates reflection positivity. As a consequence, if one desires to use stochastic quantization in constructive quantum field theory, one needs to find a more robust procedure than the standard one.

We study complex, classical, scalar fields within a new framework introduced in a previous work. We replace the usual functional integral by a complex functional arising from a boosted Hamiltonian. We generalize the Feynman–Kac relation to this setting, and use it to establish the spectral condition on a cylinder. We consider also positive-temperature states.

We establish reflection positivity for Gibbs trace states defined by certain Hamiltonians that describe the interaction of Majoranas on a lattice. These Hamiltonians may include many-body interactions, as long as the signs of the associated coupling constants satisfy certain restrictions. We show that reflection positivity holds on an even sub-algebra of Majoranas., Annales Henri Poincaré, 16 (1), ISSN:1424-0661, ISSN:1424-0637

We introduce a pictorial approach to quantum information, called holographic software. Our software captures both algebraic and topological aspects of quantum networks. It yields a bi-directional dictionary to translate between a topological approach and an algebraic approach. Using our software, we give a topological simulation for quantum networks. The string Fourier transform (SFT) is our basic tool to transform product states into states with maximal entanglement entropy. We obtain a pictorial interpretation of Fourier transformation, of measurements, and of local transformations, including the $n$-qudit Pauli matrices and their representation by Jordan-Wigner transformations. We use our software to discover interesting new protocols for multipartite communication. In summary, we build a bridge linking the theory of planar para algebras with quantum information., 48 pages. Accepted for publication in SCIENCE CHINA Mathematics

A description of Jim's contributions to mathematics would take us far beyond our expertise. Jim was never content to spend too long in any one area of mathematics. Operator algebras, hyperbolic systems of conservation laws, uid dynamics, quantum field theory and statistical mechanics and the computational aspects of applied mathematics are just a few of the areas to which Jim has contributed. Here we relate a few impressionistic recollections of personal interactions, together with some ideas about his work in quantum field theory and statistical mechanics. In spite of the non-technical character of these observations, we hope nevertheless that they will provide pleasure for our dear friend and for other readers as well.

We give an intuitive method — using local, cyclic replica symmetry — to isolate exponential tree decay in truncated (connected) correlations. We give an expansion and use the symmetry to show that all terms vanish, except those displaying replica condensation. The condensation property ensures exponential tree decay. We illustrate our method in a low-temperature Ising system, but expect that one can use a similar method in other random field and quantum field problems. While considering the illustration, we prove an elementary upper bound on the entropy of random lattice surfaces.

We study derivatives on an interval of length ℓ (or the associated circle of the same length), and certain pseudo-differential operators that arise as their fractional powers. We compare different translations across the interval (around the circle) that are characterized by a twisting angle. These results have application in

A twist field on a cylindrical space–time has the defining property that translation about a spatial circle results in multiplying the field by a phase. In this paper we investigate how such multivalued twist fields fit into the framework of constructive quantum field theory. Twisted theories have an interest in their own right; the twists also serve as infrared regulators that partially preserve the underlying symmetries of the Hamiltonian. The main focus of this paper is to investigate the extent that boson–fermion twist-field systems are compatible with the Lie symmetry and with the N=2 supersymmetry that one expects in the same examples without twists. We consider free systems and nonlinear boson–fermion interactions that arise from a holomorphic, quasihomogeneous, polynomial superpotential. We choose the twisting angles to lie on a chosen line in twist parameter space (leaving one free twist parameter). Doing this, we can obtain Lie symmetry and half the number of supersymmetry generators that one expects in our examples without the twists. We also show that the Hamiltonians for scalar twist fields yield twisted, positive-temperature expectations with the “twist-positivity” property. This is important because it justifies the existence of a functional integral representation for twisted, positive-temperature trace functionals. We regularize these systems in a way that preserves symmetry to the maximal extent. We pursue elsewhere other aspects and applications of this method, including bounding the extent of supersymmetry breaking.

In this paper we give a new expansion, based on cyclicity of the trace, to study regularity properties of twisted expectations 〈X(s)〉=Tr H (γU(θ) X(s)). Here X(s)=X0e−s0Q2X1e−s1Q2…Xke−skQ2 is a product of operators Xj, regularized by heat kernels e−sjQ2 with sj>0. The twist groups γ∈ Z 2 and U(θ)∈U(1) are commuting symmetries of Q2. The name “holonomy expansion” arises from picturing 〈X(s)〉 as a circular graph, with vertices in the graph representing the operators Xj, in the order that they appear in the product, and the linesegment following Xj representing the heat kernel e−sjQ2. The trace functional is cyclic, so the graph is circular. We generate our expansion by “transporting” a vertex Xk around the circle, ending in its original position. We choose an Xk that transforms under a one-dimensional representation of Z 2×U(1). For θ in the complement of the discrete set ϒsing (where the group Z 2×U(1) acts trivially on Xk) we obtain an identity between the original expectation and some new expectations. We study an example from supersymmetric quantum mechanics, with a Dirac operator Q(λ) depending on a parameter λ and with a U(1) group of symmetries U(θ). We apply our expansion to invariants Z (λ; θ)= Z (Q(λ); θ) suggested by non-commutative geometry. These invariants are sums of expectations of the form above. We investigate this example as a first step toward developing an expansion to evaluate related invariants arising in supersymmetric quantum field theory. We establish differentiability of Z (λ; , θ) in λ for λ∈(0, 1] and show Z (λ; θ) is independent of λ. We wish to evaluate Z (λ; θ) at the endpoint λ=0, but Z (0; θ) is ill-defined. We regularize the endpoint, while preserving the U(θ)-symmetry, by replacing Q(λ)2 with H(e, λ)=Q(λ)2+e2 |z|2. The regularized function Z (e, λ; θ) depends on all three variables e, λ, θ; for fixed θ, it is differentiable in the unit (e, λ) square, except at the origin. Using the holonomy expansion, we prove for fixed θ∉ϒsing that Z (e, λ; θ) is also jointly continuous in (e, λ), at the origin. As a consequence, if θ∉ϒsing, then we can interchange limits and Z (λ; θ)=lime→0 Z (e, 0; θ). We observe that the joint continuity of Z (e, λ; θ) in (e, λ) is not uniform in θ, and Z (e, λ; θ) is not jointly continuous for θ∈ϒsing. But the limiting function Z (λ; θ) is continuous in θ; so the e-limit also determines Z (λ; θ) for all θ, including for θ∈ϒsing. We use these facts to calculate Z (λ; θ). Our regularization destroys supersymmetry, but the holonomy expansion gives quantitative bounds on the error terms.

In super-symmetric quantum theory, or in string theory, (including generalizations of these theories to underlying quantum spaces) we study a certain partition function Z(Q,A,g). Here Q denotes a supercharge, A denotes an observable with the property A^2 = I, and g denotes an element of a symmetry group of Q. The supercharge may depend on a parameter lambda, namely Q = Q(lambda). We give an elementary argument to show that Z, as defined, does not actually depend on lambda., 13 pages, Latex

We identify a positivity property for partition functions in quantum systems with a unitary symmetry group, and we call this "twist positivity." The existence of Feynman-Kac measures and the existence of zero-mass limits are both related to this property. Twist positivity arises from the occurrence of complex conjugate representations on an energy eigenspace, and ultimately reflects a particle interpretation of the quantum system., Comment: 51 pages

Definissant les mathematiques comme une science proche de la physique dans leur volonte de comprendre la nature, l'A. montre que la verification scientifique et la preuve mathematique different dans leur degre de certitude. Soulevant la question d'un changement des standards generaux et des buts de la physique theorique, l'A. montre que l'idee de preuve mathematique a evolue ces dernieres annees, sous la double impulsion d'une interraction entre les mathematiques et la physique, d'une part, et de l'apparition de nouveaux modes de communication avec internet, d'autre part

We investigate the role that vortex loops play in characterizing eigenstates of interacting Majoranas. We first give some general results, and then we focus on ladder Hamiltonian examples to test further ideas. Two methods yield exact results: i.) We utilize the mapping of spin Hamiltonians to quartic interactions of Majoranas and show under certain conditions the spectra of these two examples coincide. ii) In cases with reflection-symmetric Hamiltonians, we use reflection positivity for Majoranas to characterize vortices. Aside from these exact results, two additional methods suggest wider applicability of these results: iii.) Numerical evidence suggests similar behavior for certain systems without reflection symmetry. iv.) A perturbative analysis also suggests similar behavior without the assumption of reflection symmetry., 28 pages

We explore a framework for complex classical fields, appropriate for describing quantum field theories. Our fields are linear transformations on a Hilbert space, so they are more general than random variables for a probability measure. Our method generalizes Osterwalder and Schrader's construction of Euclidean fields. We allow complex-valued classical fields in the case of quantum field theories that describe neutral particles. From an analytic point-of-view, the key to using our method is reflection positivity. We investigate conditions on the Fourier representation of the fields to ensure that reflection positivity holds. We also show how reflection positivity is preserved by various space-time compactifications of Euclidean space., 30 pages

Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation., 13 pages

We interpret the cocycle condition in quantum field theory as a set of integrated Ward identities for non-commutative geometry.

We prove that the Chern character of quantum algebras is invariant under a class of deformations of the Dirac operator. We also extend the definition of the Chern character to include certain unbounded operators.

From the point of view of physics, there are essentially two problems in the study of quantum field models. The first problem is to find the phenomena of interest to physics (e.g. particles, bound states, resonances, broken symmetries, and asymptotic power series expansions) in the two and three dimensional quantum field models.1 This problem includes obtaining the mathematical structure underlying these phenomena, and determining qualitatively which interactions and/or parameter values give rise to these phenomena and which do not. We believe this problem is feasible at the present time; indeed the initial progress is the subject of the present lectures. The second problem is the construction of nontrivial fields in four space time dimensions. Here it seems that some new ideas are needed to supplement those which have already been used in the construction of quantum fields.

We establish a useful identity for intertwining a creation or annihilation operator with the heat kernel of a self-interacting bosonic field theory., 7 pages, Latex