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Arthur Jaffe

Publications and source records attributed to Arthur Jaffe.

At least 55 records · Page 3Linked to original sources
arXiv2015

Characterization of Reflection Positivity: Majoranas and Spins

We study linear functionals on a Clifford algebra (algebra of Ma- joranas) equipped with a reflection automorphism. For Hamiltonians that are functions of Majoranas or of spins, we find necessary and sufficient conditions on the coupling constants for reflection positivity to hold. One can easily check these conditions in concrete models. We illustrate this by discussing a number of spin systems with nearest-neighbor and long-range interactions.

math-ph↗
arXiv2015

Quantum Fields, Stochastic PDE, and Reflection Positivity

We investigate stochastic quantization as a method to go from a classical PDE (with stochastic time $λ$) to a corresponding quantum theory in the limit $λ\to\infty$. We test the method for a linear PDE satisfied by the free scalar field. We begin by giving some background about the importance of establishing the property of {\em reflection positivity} for the limit $λ\to\infty$. We then prove that the measure determined through stochastic quantization of the free scalar field violates reflection positivity (with respect to reflection of the physical time) for every $λ<\infty$. If a non-linear perturbation of the linear equation is continuous in the perturbation parameter, the same result holds for small perturbations. For this reason, one needs to find a modified procedure for stochastic quantization, in order to use that method to obtain a quantum theory.

math.PR↗
arXiv2014

Reflection Positivity for Parafermions

We establish reflection positivity for Gibbs trace states for a class of gauge-invariant, reflection-invariant Hamiltonians describing parafermion interactions on a lattice. We relate these results to recent work in the condensed-matter physics literature.

quant-ph↗
arXiv2013

Topological Order and Reflection Positivity

The interplay between the two fundamental concepts of topological order and reflection positivity allows one to characterize the ground states of certain many-body Hamiltonians. We define topological order in an appropriate fashion and show that certain operators have positive expectation value in all ground states. We apply our method to vortex loops in a model relevant to topological quantum memories.

quant-ph↗
arXiv2013

Vortex Loops and Majoranas

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.

quant-ph↗
arXiv2013

Reflection Positivity for Majoranas

We establish reflection positivity for Gibbs trace states defined by a certain class of 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 subalgebra of Majoranas.

math-ph↗
arXiv2013

Complex Classical Fields: A Framework for Reflection Positivity

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.

math-ph↗
arXiv2013

Quantization Domains

We study the quantization of certain classical field theories using reflection positivity. We give elementary conditions that ensure the resulting vacuum state is cyclic for products of quantum field operators, localized in a bounded Euclidean space-time region O at positive time. We call such a domain a quantization domain for the classical field. The fact that bounded regions are quantization domains in classical field theory is similar to the "Reeh-Schlieder" property in axiomatic quantum field theory.

math-ph↗
arXiv2013

Complex Classical Fields and Partial Wick Rotations

We study some examples of complex, classical, scalar fields within the new framework that we introduced in a previous work. In these particular examples, we replace the usual functional integral by a complex functional arising from partial Wick rotation of a quantum field. We generalize the Feynman-Kac relation to this setting, and use it to establish the spectral condition on a cylinder. We also consider positive-temperature states.

math-ph↗
arXiv2007

Replica Condensation and Tree Decay

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 {\em 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.

math-ph↗
arXiv2007

Reflection Positivity and Monotonicity

We prove general reflection positivity results for both scalar fields and Dirac fields on a Riemannian manifold, and comment on applications to quantum field theory. As another application, we prove the inequality $C_D \leq C_N$ between Dirichlet and Neumann covariance operators on a manifold with a reflection.

math-ph↗
arXiv2007

Quantum Field Theory on Curved Backgrounds. II. Spacetime Symmetries

We study space-time symmetries in scalar quantum field theory (including interacting theories) on static space-times. We first consider Euclidean quantum field theory on a static Riemannian manifold, and show that the isometry group is generated by one-parameter subgroups which have either self-adjoint or unitary quantizations. We analytically continue the self-adjoint semigroups to one-parameter unitary groups, and thus construct a unitary representation of the isometry group of the associated Lorentzian manifold. The method is illustrated for the example of hyperbolic space, whose Lorentzian continuation is Anti-de Sitter space.

hep-th↗
arXiv2006

Quantum Field Theory on Curved Backgrounds, I. The Euclidean Functional Integral

We give a mathematical construction of Euclidean quantum field theory on certain curved backgrounds. We focus on generalizing Osterwalder-Schrader quantization, as these methods have proved useful to establish estimates for interacting fields on flat space-times. In this picture, a static Killing vector generates translations in Euclidean time, and physical positivity is played by positivity under reflection of Euclidean time. We discuss the quantization of flows which correspond to classical space-time symmetries, and give a general set of conditions which imply that broad classes of operators in the classical picture give rise to well-defined operators on the quantum-field Hilbert space. In particular, Killing fields on spatial sections give rise to unitary groups on the quantum-field Hilbert space, and corresponding densely-defined self-adjoint generators. We construct the Schrödinger representation using a method which involves localizing certain integrals over the full manifold to integrals over a codimension-one submanifold. This method is called sharp-time localization, and implies reflection positivity.

hep-th↗
arXiv2000

Twist Positivity for Lagrangian Symmetries

We prove twist positivity and positivity of the pair correlation function for combined spatial and internal symmetries of free bosonic Lagrangians. We work in a general setting, extending the results obtained in Twist Positivity [1].

math-ph↗
arXiv1998

Twist Positivity

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.

math-ph↗
arXiv1998

Quantum Invariants

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.

hep-th↗
arXiv1997

Quantum Harmonic Analysis and Geometric Invariants

We develop from scratch a theory of invariants within the framework of non-commutative geometry. Given an operator Q (a supercharge in physics language) and an operator a (whose square equals the identity I), we derive a general formula for an invariant Z(Q,a) depending on Q and a. In case a=I, our formula reduces to the McKean-Singer representation of the index of Q. The function Z is invariant in the following sense: if Q=Q(s) depends on a parameter s, and if Z(Q(s),a) is differentiable in s, then in fact Z(Q(s),a) is independent of s. We give detailed conditions on Q(s) for which Z(Q(s),a) is differentiable in s. At the end of this paper, we consider a 2-dimensional generalization of our theory motivated by space-time supersymmetry. In the case that expectations are given by functional integrals, Z(Q,a) has a simple integral representation. We also explain in detail how our construction relates to Connes' entire cyclic cohomology, as well as to other frameworks.

math-ph↗