Relating field theories via stochastic quantization

This note aims to subsume several apparently unrelated models under a common framework. Several examples of well-known quantum field theories are listed which are connected via stochastic quantization. We highlight the fact that the quantization method used to obtain the quantum crystal is a discrete analog of stochastic quantization. This model is of interest for string theory, since the (classical) melting crystal corner is related to the topological A-model. We outline several ideas for interpreting the quantum crystal on the string theory side of the correspondence, exploring interpretations in the Wheeler–De Witt framework and in terms of a non-Lorentz invariant limit of topological M-theory.     

Transition Effect Matrices and Quantum Markov Chains

A transition effect matrix (TEM) is a quantum generalization of a classical stochastic matrix. By employing a TEM we obtain a quantum generalization of a classical Markov chain. We first discuss state and operator dynamics for a quantum Markov chain. We then consider various types of TEMs and vector states. In particular, we study invariant, equilibrium and singular vector states and investigate projective, bistochastic, invertible and unitary TEMs.     

Review of invariant time formulations of relativistic quantum theories

The purpose of this paper is to review relativistic quantum theories with an invariant evolution parameter. Parametrized relativistic quantum theories (PRQT) have appeared under such names as constraint Hamiltonian dynamics, four-space formalism, indefinite mass, micrononcausal quantum theory, parametrized path integral formalism, relativistic dynamics, Schwinger proper time method, stochastic interpretation of quantum mechanics and stochastic quantization. The review focuses on the fundamental concepts underlying the theories. Similarities as well as differences are highlighted, and an extensive bibliography is provided.     

Heat transport in quantum spin chains

We discuss the problem of heat conduction in quantum spin chain models. To investigate this problem it is necessary to consider the finite open system connected to heat baths. We describe two different procedures to couple the system with the reservoirs: a model of stochastic heat baths and the quantum trajectories solution of the quantum master equation. The stochastic heat bath procedure operates on the pure wave function of the isolated system, so that it is locally and periodically collapsed to a quantum state consistent with a boundary nonequilibrium state. In contrast, the quantum trajectories procedure evaluates ensemble averages in terms of the reduced density matrix operator of the system. We apply these procedures to different models of quantum spin chains and numerically show their applicability to study the heat flow.     

Zeno dynamics for open quantum systems

In this paper, we formulate limit Zeno dynamics of general open systems as the adiabatic elimination of fast components. We are able to exploit previous work on adiabatic elimination of quantum stochastic models to give explicitly the conditions under which open Zeno dynamics will exist. The open systems formulation is further developed as a framework for Zeno master equations, and Zeno filtering (that is, quantum trajectories based on a limit Zeno dynamical model). We discuss several models from the point of view of quantum control. For the case of linear quantum stochastic systems, we present a condition for stability of the asymptotic Zeno dynamics.     

Translational Symmetry Breaking and Soliton Sectors for Massive Quantum Spin Models in 1 + 1 Dimensions

We consider the classification of pure infinite volume ground states and that of soliton sectors for 1+1 dimensional massive quantum spin models. We obtain a proof that non-translationally invariant ground state cannot exist for a class of translationally invariant Hamiltonians including the spin 1 AKLT (Affleck Kennedy Lieb Tasaki) antiferromagnetic spin model. We also obtain a complete classification of soliton sectors (up to unitary equivalence) for certain massive models (e.g. ferromagnetic XXZ models). Received: 13 January 1997 / Accepted: 11 March 1997     

The chiral potts model and its associated link invariant

A new link invariant is derived using the exactly solvable chiral Potts model and a generalized Gaussian summation identity. Starting from a general formulation of link invariants using edge-interaction spin models, we establish the uniqueness of the invariant for self-dual models. We next apply the formulation to the self-dual chiral Potts model, and obtain a link invariant in the form of a lattice sum defined by a matrix associated with the link diagram. A generalized Gaussian summation identity is then used to carry out this lattice sum, enabling us to cast the invariant into a tractable form. The resulting expression for the link invariant is characterized by roots of unity and does not appear to belong to the usual quantum group family of invariants. A table of invariants for links with up to eight crossings is given.     

Stochastic quantization and detailed balance in Fokker-Planck dynamics

The path integral and operator formulations of the Fokker-Planck equation are considered as stochastic quantizations of underlying Euler-Lagrange equations. The operator formalism is derived from the path integral formalism. It is proved that the Euler-Lagrange equations are invariant under time reversal if detailed balance holds and it is shown that the irreversible behavior is introduced through the stochastic quantization. To obtain these results for the nonconstant diffusion Fokker-Planck equation, a transformation is introduced to reduce it to a constant diffusion Fokker-Planck equation. Critical comments are made on the stochastic formulation of quantum mechanics.     

Stochastic processes in conformal Riemann-Cartan-Weyl gravitation

We construct the Riemann-Cartan geometries with torsion generated by the action of the conformal Weyl group. We study the wave operators associated to these structures, which, in addition to the usual Laplace-Beltrami operator, have a term which is a gradient vector field conjugate to the one-form given by the torsion potential derived from the Weyl group, and which we associate with a relativistic extension of the drift vector field in Nelson's construction of stochastic mechanics. In fact, our construction is valid for configuration spaces of any dimension. We sketch the construction of the stochastic processes on space-time generated by these operators, where the invariant measure is found to be defined by the conformal structure. We discuss briefly the relation with the theory of Dirichlet forms and D. Bohm's quantum potential in the theory of hidden variables, which in this setting acquire a gauge-geometric status previously unknown.     

Gauged Laplacians on Quantum Hopf Bundles

We study gauged Laplacian operators on line bundles on a quantum 2-dimensional sphere. Symmetry under the (co)-action of a quantum group allows for their complete diagonalization. These operators describe ‘excitations moving on the quantum sphere’ in the field of a magnetic monopole. The energies are not invariant under the exchange monopole/antimonopole, that is under inverting the direction of the magnetic field. There are potential applications to models of quantum Hall effect.     

Stochastic methods in quantum field theory and hydrodynamics

We give a review of recent work in quantum field theory and hydrodynamics, in which methods of stochastic analysis, in particular of stochastic equations, are used. The review includes the discussion of the canonical formalism for models of quantum fields as well as the discussion of statistical solutions of the Euler equations for the motion of an inviscid fluid.     

Lévy models and scale invariance properties applied to Geophysics

In this work we study scale invariant functions and stochastic Lévy models and we apply them to geophysical data. We show that a pattern arises from the scale invariance property and Lévy flight models that may be used to estimate parameters related to some major event–major earthquakes.     

From quantum mechanics to classical statistical physics: Generalized Rokhsar-Kivelson Hamiltonians and the “Stochastic Matrix Form” decomposition

Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar-Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with stochastic classical systems described by a Master equation of the matrix type, hence their name. It then follows that the equilibrium partition function of the stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models.     

On Isospectral Deformations of an Inhomogeneous String

For an anyon model in two spatial dimensions described by a modular tensor category, the topological S-matrix encodes the mutual braiding statistics, the quantum dimensions, and the fusion rules of anyons. It is nontrivial whether one can compute the S-matrix from a single ground state wave function. Here, we define a class of Hamiltonians consisting of local commuting projectors and an associated matrix that is invariant under local unitary transformations. We argue that the invariant is equivalent to the topological S-matrix. The definition does not require degeneracy of the ground state. We prove that the invariant depends on the state only, in the sense that it can be computed by any Hamiltonian in the class of which the state is a ground state. As a corollary, we prove that any local quantum circuit that connects two ground states of quantum double models (discrete gauge theories) with non-isomorphic abelian groups must have depth that is at least linear in the system’s diameter. As a tool for the proof, a manifestly Hamiltonian-independent notion of locally invisible operators is introduced. This gives a sufficient condition for a many-body state not to be generated from a product state by any small depth quantum circuit; this is a many-body entanglement witness.     

Remarks on some new models of interacting quantum fields with indefinite metric

We study quantum field models in indefinite metric. We introduce the modified Wightman axioms of Morchio and Strocchi as a general framework of indefinite metric quantum field theory (QFT) and present concrete interacting relativistic models obtained by analytical continuation from some stochastic processes with Euclidean invariance. As a first step towards scattering theory in indefinite metric QFT, we give a proof of the spectral condition on the translation group for the relativistic models.     

Coherent Population Trapping and Partial Decoherence in the Stochastic Limit

We use the stochastic limit technique to predict a new phenomenon concerning a two-level atom with degenerate ground state interacting with a quantum field. We show, that the field drives the state of the atom to a stationary state, which is non-unique, but depends on the initial state of the system through some conserved quantities. This non uniqueness follows from the degeneracy of the ground state of the atom, and when the ground subspace is two-dimensional, the family of stationary states will depend on a one-dimensional parameter. Only one of the stationary states in this family is a pure state and it coincides with the known trapped state. This means that by controlling the initial state (input) we can control the final state (output). The quantum Markov semigroup obtained in the limit admits an invariant pure state, but it is not true that all the extremal invariant states are pure. This is an interesting phenomenon also from mathematical point of view and its meaning will be discussed in a future paper. PACS numbers: 31.15.-p, 31.15.Gy, 32.80.Pj, 32.80.Qk     

Stochastic Schrödinger equation from an interaction with the environment

We consider a class of models describing a quantum oscillator in interaction with an environment. We show that models of continuous spontaneous localization based on a stochastic Schrödinger equation can be derived as an approximation to purely deterministic Hamiltonian systems.     

Pion correlations and resonance effects in \bar{p}p annihilation to 4\pi^0 at rest

We study correlations in the exclusive reaction at rest with complete reconstruction of the kinematics for each event. The inclusive distribution is fairly flat at small invariant mass of the pion pair while a small enhancement in the double differential distribution is observed for small invariant masses of both pion pairs. Dynamical models with resonances in the final state are shown to be consistent with the data while the stochastic HBT mechanism is not supported by the present findings. Received: 26 February 2002 / Revised version: 22 July 2002 / Published online: 30 August 2002