Mark S. Swanson

Transition elements for a non-Hermitian quadratic Hamiltonian

The non-Hermitian quadratic Hamiltonian H=ωa†a+αa2+βa†2 is analyzed, where a† and a are harmonic oscillator creation and annihilation operators and ω, α, and β are real constants. For the case that ω2−4αβ⩾0, it is shown using operator techniques that the Hamiltonian possesses real and positive eigenvalues. A generalized Bogoliubov transformation allows the energy eigenstates to be constructed from the algebra and states of the harmonic oscillator. The eigenstates are shown to possess an imaginary norm for a large range of the parameter space. Finding the orthonormal dual space allows the inner product to be redefined using the complexification procedure of Bender et al. for non-Hermitian Hamiltonians. Transition probabilities governed by H are shown to be manifestly unitary when the complexification procedure is followed. A specific transition element between harmonic oscillator states is evaluated for both the Hermitian and non-Hermitian cases to identify the differences in time evolution.

Glucocorticoid binding to rat liver microsomal fractions in vitro.

Abstract Specific binding of free glucocorticoids, dexamethasone or 9α-fluoroprednisolone (9FP), to rat liver microsomal subfractions was studied in cell free systems. Binding was of lower affinity but higher capacity than that reported for cytosol corticoid receptor and was stable to freezing for long periods. Binding at 4 C varied among the subtractions in the early time course of binding, in apparent affinities, and in effects of some competitors. [ 3 H]-9FP binding to all fractions was strongly inhibited by unlabelled 9FP, dexamethasone or cortisol, less strongly by corticosterone, and not by 19β-estradiol, diethylstilbesterol or Vitamin B 1 . Progesterone blocked 9FP binding to RER, less so to polysomes, and not to SER fractions. Testosterone blocked 9FP binding only in the RER fractions. Specific binding to purified subfractions was abolished at 22°C, but was increased in whole microsome fractions. There was no specific binding of [ 3 H]-9FP to microsomes of rat thymus, spleen, kidney or lung in the cell free systems used.

CANONICAL TRANSFORMATIONS AND PATH-INTEGRAL MEASURES

This paper is a generalization of previous work on the use of classical canonical transformations to evaluate Hamiltonian path integrals for quantum-mechanical systems. Relevant aspects of the Hamiltonian path integral and its measure are discussed, and used to show that the quantum-mechanical version of the classical transformation does not leave the measure of the path-integral invariant, instead inducing an anomaly. The relation to operator techniques and ordering problems is discussed, and special attention is paid to incorporation of the initial and final states of the transition element into the boundary conditions of the problem. Classical canonical transformations are developed to render an arbitrary power potential cyclic. The resulting Hamiltonian is analyzed as a quantum system to show its relation to known quantum-mechanical results. A perturbative argument is used to suppress ordering-related terms in the transformed Hamiltonian in the event that the classical canonical transformation leads to a nonquadratic cyclic Hamiltonian. The associated anomalies are analyzed to yield general methods to evaluate the path integrals prefactor for such systems. The methods are applied to several systems, including linear and quadratic potentials, the velocity-dependent potential, and the time-dependent harmonic oscillator.

Quantum Mechanical Path Integrals

This chapter discusses the reformulation of the standard version of nonrelativistic quantum mechanics in terms of a path integral. The path integral is derived as a limiting expression from the fundamental structure of nonrelativistic quantum mechanics and is a representation of quantum mechanical amplitudes equivalent to the usual wave mechanical or the matrix formulations of these amplitudes. The chapter reviews the relevant aspects of basic quantum mechanics, which are further used to find the path integral form for quantum mechanical amplitude. The chapter then presents the idea of the path integral as a “sum over histories” as a conceptual generalization of the results obtained from its derivation.

Baryon-antibaryon asymmetry in a model with domain structure and soft CP violation

Abstract We consider a model with an abelian gauge symmetry, a Higgs potential involving two scalar fields, and two spinor fields coupled to the scalars through Yukawa couplings. The model accomodates soft violation of charge conjugation, and a domain structure of the universe with two different types of domains, which have identical energy but are governed by different effective lagrangians. The effective lagrangian has complex c -number coefficients that become parts of effective coupling constants, and these are different in the two kinds of domains. In spite of that fact the model neither predicts any domain-dependent effects, nor any particle-antiparticle asymmetries within domains.

Charge–monopole trajectories and the WKB approximation

The classical nonrelativistic motion of an electric charge in the presence of a magnetic monopole is reviewed. Using general properties of these trajectories it is shown that the path integral form of the WKB approximation for the charged particle propagator develops an anomalous form for the transition probability density unless the Dirac quantization condition is imposed. This result is shown to be independent of the specific form of the vector potential used to represent the magnetic monopole. The general result is then verified for the specific case that the standard Dirac string solution is used in the path integral. This result is interpreted as an indirect demonstration of the Dirac condition. Other properties of the solutions and the associated action are briefly discussed.

The 1+1 SU(2) Yang?Mills path integral

The path integral for SU(2) invariant two-dimensional Yang–Mills theory is recast in terms of the chromoelectric field strength by integrating the gauge fields from the theory. Implementing Gausss law as a constraint in this process induces a topological term in the action that is no longer invariant under large gauge transformations. For the case that the partition function is considered over a circular spatial degree of freedom, it is shown that the effective action of the path integral is quantum mechanically WKB exact and localizes onto a set of chromoelectric zero modes satisfying antiperiodic boundary conditions. Summing over the zero modes yields a partition function that can be reexpressed using the Poisson resummation technique, allowing an easy determination of the energy spectrum, which is found to be identical to that given by other approaches.

Of ghosts, gauge volumes, and Gauss's law

The relationship between the canonical operator and the path integral formulation of quantum electrodynamics is analyzed with a particular focus on the implementation of gauge constraints in the two approaches. The removal of gauge volumes in the path integral is shown to match with the presence of zero-norm ghost states associated with gauge transformations in the canonical operator approach. The path integrals for QED in both the Feynman and the temporal gauges are examined and several ways of implementing the gauge constraint integrations are demonstrated. The upshot is to show that both the Feynman and the temporal gauge path integrals are equivalent to the Coulomb gauge path integral, matching the results developed by Kurt Haller using the canonical formalism. In addition, the Faddeev–Popov form for the Feynman gauge and temporal gauge Lagrangian path integrals are derived from the Hamiltonian form of the path integral.

Evaluating the Path Integral

This chapter explains the evaluation of the various forms of the path integral. The basic types of approach for evaluating the path integral form of the transition amplitude can be broken into two groups: discrete methods and continuum methods. The chapter describes how the free-particle propagator is calculated from the corresponding form and shown to be the exact result founded by the operator approach. The chapter also discusses how the free particle is given a time-dependent driving force and referred to as a “source.” The resulting form can be combined with functional derivatives to analyze time-ordered products of the Heisenberg picture position operator. The chapter further discusses how the harmonic oscillator is analyzed using quasi-continuous methods and how an analysis of the Jacobian associated with the change of variables is made. Using Poisson resummation techniques, the path integral for a particle on the ring is analyzed to reveal its underlying topological structure.

Gauge Field Theory

This chapter explains the development of the gauge field theory at the classical and quantum levels for both the abelian and nonabelian cases. It discusses the dual aspects of gauge invariance and ghosts and explains the development of the path integral formulation of quantum electrodynamics as an example of a constrained path integral. This technique is the generalization to the field theory of the quantum mechanical method for implementing the gauge condition developed in the chapter. The chapter also discusses the properties of Lie algebras and the nonabelian extension of the Yang–Mills field. In the chapter, the technique of minimal coupling is developed for nonabelian gauge-matter interactions. The chapter further explains the application of the technique of constrained path integrals to the path integral representation of the Yang–Mills field. The chapter also introduces the concept of the gauge field as a one-form and discusses its topological properties.