On Klauder's Path: A Field Trip

The following sections are included:

• PREFACE

• CONTENTS

• THE PUBLICATIONS OF JOHN R. KLAUDER

We prove the essential self–adjointness on a natural domain for the Dirichlet operators H associated with Gibbs measures describing statistical mechanical lattice models with continuous-valued variables having polynomial self-interaction and short range quadratic coupling. This is used to prove that H has a gap at the lower end of its spectrum and the logarithmic Sobolev inequality holds. As a consequence we have the ergodic property of the process solving the corresponding stochastic differential equation (stochastic quantization equation).

We consider a class of perturbed XY–models which are related to quasifree Fermion system via Jordan-Wigner Transsformation, recently treated by Carey and Hume. In contrast to their examples, in which failure of return to equilibrium under perturbed dynamics and multiple ground states of the unperturbed dynamics occur in the same range of parameters, we present a similar and general class of examples in which return to equilibrium under perturbed dynamics holds and multiple ground states occur under unperturbed as well as perturbed dynamics.

We construct the characteristic functional E(f) of ½ : j² : (f) , where j^μ is a free conserved current in 1 + 1 dimensions, and discuss its relevance for conformal quantum field theories in two dimensions.

Let be a C*-dynamical system with G a topological group. Let be the subset of elements of on which acts in a strongly continuous manner. It is shown that is the pre-dual of a von Neumann-algebra ⊂ . The algebra consists, except for some trivial reduction, of those elements of which map, by right and left multiplication, the submodule of into itself. The algebra is covariant under the automorphisms .

Wavelets have many roots in older theories. One of these roots is given by the affine coherent states of Aslaksen and Klauder. This paper presents a short historic review of wavelets and their links to physics, mathematics and engineering.

A brief review of coherent states and of wavelets is given and then several relations among the two structures are indicated. It is pointed out that many aspects of wavelets in physics are variants of old ideas in the coherent states.

It is demonstrated with a simple model of non-interacting Bose system in one-dimensional space that, contrary to prevailing expectations, the transformation mediating the wave- and the particle-pictures does not exist in the Fock space in the continuum limit.

It happened before [1] that I have concluded my publication with a special dedication to John R. Klauder. Then, the reason was John's PhD thesis [2] and the questions (perhaps outdated in the eyes of the band-wagon jumpers, albeit still retaining their full vitality [3]): (i) What are the uses of the classical (c-number, non-Grassmann) spinor fields, especially nonlinear ones, what are they for at all ? (ii) What are, if any, the classical partners for Fermi models and fields in particular ? The present dedication, even if not as conspicuously motivated as the previous one by John's research, nevertheless pertains to investigations pursued by John through the years and devoted to the analysis of random noise. Sometimes, re-reading old papers and re-analysing old, frequently forgotten ideas might prove more rewarding than racing the fashions. Following this attitude, let us take as the departure point Schrödinger's original suggestion [4] of the existence of a special class of random processes, which have their origin in the Einstein-Smoluchowski theory of the Brownian motion and its Wiener's codification. The original analysis due to Schrodinger of the probabilistic significance of the heat equation and of its time adjoint in parallel, remained unnoticed by the physics community, and since then forgotten. It reappeared however in the mathematical literature as an inspiration to generalise the concept of Markovian diffusions to the case of Bernstein stochastic processes. But, it stayed without consequences for a deeper understanding of the possible physical phenomena which might underly the corresponding abstract formalism. Schrödinger's objective was to initiate investigations of possible links between quantum theory and the theory of Brownian motion, an attempt which culminated later in the so-called Nelson's stochastic mechanics [8] and its encompassing formalism [7] in which the issue of the Brownian implementation of quantum dynamics is placed in the framework of Markov-Bernstein diffusions…

Some problems and peculiarities of complex Langevin are discussed. Special attention is given to the equivalence of the 2n-variable Fokker-Planck equation and the n-variable pseudo Fokker-Planck equation with a complex drift term.

The following sections are included:

• Introduction

• Quantum Networks

• Network Matrix and Single Valuedness

• Network Reduction

• Flux Quantization

• Current Quantization

• Network Energy and Meissner Effect

• Summary and Discussion

• References

After a general introduction we discuss the role of solitons in one-dimensional polymers. Methods which allow to solve certain nonlinear evolution equations can be applied to the large class of ultralocal models. The solvability of vertex models can be traced back to solutions of the Yang-Baxter relation. Recently the notion of a symmetry has been enlarged. Models showing instead of a Lie-algebra symmetry a quantum group symmetry are discussed in the last section of the paper.

The radial path integral is a Besselian path integral, which is sufficiently ugly even for the harmonic oscillator but has been evaluated by iterating Weber's integral formula. Since there are renewed interests in calculating the radial path integral, a new way of calculating the Besselian path integral for the n-dimensional harmonic oscillator is reported.

Thermodynamics interrelates the discontinuities associated with first order phase transitions. I discuss the relation between the jump in the quark condensate and the latent heat.

We review the concept of a q-deformation in connection with quantum groups and a q-deformed Heisenberg-Weyl algebra. With these concepts we construct q-analogs of the canonical coherent states and the associated q-harmonic oscillator states. We briefly formulate q-analog vector coherent states and with them construct representations of the quantum group SU_q(2).

We calculate the dynamical entropy of quasifree automorphisms by showing that their abelian counterpart already gives an optimal abelian model.

This article reports on a program to obtain and understand coherent states for general systems. Most recently this has included supersymmetric systems. A byproduct of this work has been studies of squeezed and supersqueezed states. To obtain α physical understanding of these systems has always been a primary goal. In particular, in the work on supersymmetry an attempt to understand the role of Grassmann numbers in quantum mechanics has been initiated.

Using renormalization group techniques, we examine several interesting relations among masses and mixing angles of quarks and leptons in the Standard Model of Elementary Particle Interactions as a function of scale. We extend the analysis to the minimal Supersymmetric Extension to determine its effect on these mass relations. For a heavy top quark, and minimal supersymmetry, most of these relations can be made to agree at one unification scale.

Hamiltonians with perturbations of vanishing support are discussed. It is shown that there are generalizations to Friedman's theorem. Applications to physics, however, may be limited.

A class of representations of the canonical commutation relations is presented which naturally subsumes coherent and symplectic (i.e. quasifree) representations. Necessary and sufficient conditions are presented for the unitary inequivalence of these representations with the Fock representation. Hence a new class of representations for bosonic systems with infinitely many degrees of freedom is opened up.

The inner Bogoliubov automorphisms of the von Neumann Clifford algebra were characterized by Blattner; we offer a new proof of his result, elaborating upon an idea of Carey.

Although photons are usually associated with light, their arrival times at a given target can be simulated on a computer for demonstration purposes. The statistics depends of course on the “preparation” of the light beam. For incoherent photons with a geometric (Bose–Einstein) number distribution, the random time intervals t₁ between adjacent photons can be generated from a uniform random variable u₁ in the interval (0,1] by setting . The arrival times of the nth photon are generated by setting , where the u_k are independent uniform random variables.

The following sections are included:

• The Model

• Sketch of ideas underlying the proofs

• A further remark and extension

• References

The components of the position operator, at a fixed time, for a massless and spinning particle with given helicity λ described in terms of bosonic degrees of freedom have an anomalous commutator proportional to λ. The position operator generates translations in momentum space. We show that a ray-representation for these translations emerges due to the non-commuting components of the position operators and relate this to the Berry-phase for photons. The Tomita-Chiao experiment then gives support for this relativistic and quantum mechanical description of photons in terms of non-commuting position operators.

We construct a C*-algebra, associated with the free Dirac equation, which has a local structure obeying the Haag-Kastler axioms. We show that the algebra possesses no representations having positive energy. This provides a simple proof that the positive-energy axiom is independent of the other axioms, a result first proved by Doplicher, Regge and Singer in 1968.

We propose a new method for stochastically quantizing systems with bottomless action functionals, based on a field-dependent kerneled Langevin equation. We prove that our kerneled Langevin equation describes a substitute stochastic process with a thermal equilibrium state. The proof makes clear the relation between our method and the conventional path-integral quantization. We also demonstrate numerically that dynamical evolution leads to the desired equilibrium state. We check that our method gives the same perturbative results as in the conventional path-integral quantization. We also show that our method is applicable to numerical calculation of nonperturbative expectation values that are consistent with the perturbative results in the case of a small coupling constant.

It is shown that for a very simple quantum chain a dynamical symmetry algebra may be realized by generators of corner transfer matrices recently studied in the literature of exactly soluble models in statistical mechanics. As by—products we obtain some new relationships for orthogonal polynomials and some extensions of the Yang-Baxter equations.

A mechanism is discussed for the creation of quantum correlations in a pure state of a system consisting of two independent parts sharing vacuum. This mechanism of vacuum correlation is studied for the cases of a two mode squeezed state, a black hole and thermal physics. The physical role of the second freedom in thermo field dynamics (TFD) is analyzed. The phase operator and time operator in TFD are reviewed.