L. P. Horwitz

Quaternion quantum mechanics: Second quantization and gauge fields

Abstract Recent work on algebraic chromodynamics has indicated the importance of a systematic study of quaternion structures in quantum mechanics. A quaternionic Hilbert module, a closed linear vector space with many of the properties of a Hilbert space is studied. The propositional system formed by the subspaces of such a space satisfy the axioms of quantum theory. There is a hierarchy of scalar products and linear operators, defined in correspondence with the types of closed subspaces (with real, complex or quaternion linearity). Real, complex, and quaternion linear projection operators are constructed, and their application to the definition of quantum states is discussed. A quaternion linear momentum operator is defined as the generator of translations, and a complete description of the Euclidean symmetries is obtained. Tensor products of quaternion modules are constructed which preserve complex linearity. Annihilation-creation operators are constructed, corresponding to the second quantization of the quaternion quantum theory with Bose-Einstein or Fermi-Dirac statistics. The tensor product spaces provide representations for algebras with dimensionality increasing with particle number. The algebraic structure of the gauge fields associated with these algebras is precisely that of the semi-classical fields introduced by Adler.

Gibbs ensembles in relativistic classical and quantum mechanics

Abstract Classical and quantum Gibbs ensembles are constructed for equilibrium statistical mechanics in the framework of an extension to many-body theory of a relativistic mechanics proposed by Stueckelberg. In addition to the usual chemical potential in the grand canonical ensemble, there is a new potential corresponding to the mass degree of freedom of relativistic systems. It is shown that in the nonrelativistic limit the relativistic ensembles we have obtained reduce to the usual ones, and mass fluctuations for the free-particle gas approach the fluctuations in N . The ultrarelativistic limit of the canonical ensemble for the free-particle gas differs from the corresponding limit of the ensemble proposed by Juttner and Pauli. Due to the mass degree of freedom, the quantum counting of states is different from that of the nonrelativistic theory. If the mass distribution is sufficiently sharp, the thermodynamical effects of this multiplicity will not be large. There may, however, be detectable effects such as a shift in the Fermi level and the critical temperature for Bose-Einstein condensation, and some change in specific heats.

On the two aspects of time: The distinction and its implications

The contemporary view of the fundamental role of time in physics generally ignores its most obvious characteric, namely its flow. Studies in the foundations of relativistic mechanics during the past decade have shown that the dynamical evolution of a system can be treated in a manifestly covariant way, in terms of the solution of a system of canonical Hamilton type equations, by considering the space-time coordinates and momenta ofevents as its fundamental description. The evolution of the events, as functions of a universal invariant world, or historical, time, traces out the world lines that represent the phenomena (e.g., particles) which are observed in the laboratory. The positions in time of each of the events, i.e., the time of their potential detection, are, in this framework, controlled by this universal parameter τ, the time at which they are generated (and may proceed in the positive or negative sense). We find that the notion of thestate of a system requires generalization; at any given τ, it involves information about the system at timest(τ) ≠ τ. The correlation of what may be measured att(τ) with what is generated at τ is necessarily quite rigid, and is related covariantly to the spacelike correlations found in interference experiments. We find, furthermore, that interaction with Maxwell electromagnetism leads back to a static picture of the world, with no real evolution. As a consequence of this result, and the requirement of gauge invariance for the quantum mechanical evolution equation, we conclude that electromagnetism is described by a pre-Maxwell field, whose τ-integral (or asymptotic behavior as τ → ∞) may be identified with the Maxwell field. We therefore consider the world of events in space time, interacting through τ-dependent pre-Maxwell fields, as far as electrodynamics is concerned, as the objective dynamical reality. Our perception of the world, through laboratory detectors and our eyes, are based onintegration over τ over intervals sufficiently large to obtain an aposteriori description of the phenomena which coincides with the Maxwell theory. Fundamental notions, such as the conservation of charge, rest on this construction. The decomposition of the common notion of time into two essentially different aspects, one associated with an unvarying flow, and the second with direct observation subject to dynamical modification, has profound philosophical consequences, of which we are able to explore here only a few.

Off-shell electromagnetism in manifestly covariant relativistic quantum mechanics

Gauge invariance of a manifestly covariant relativistic quantum theory with evolution according to an invariant time τ implies the existence of five gauge compensation fields, which we shall call pre-Maxwell fields. A Lagrangian which generates the equations of motion for the matter field (coinciding with the Schrödinger type quantum evolution equation) as well as equations, on a five-dimensional manifold, for the gauge fields, is written. It is shown that τ integration of the equations for the pre-Maxwell fields results in the usual Maxwell equations with conserved current source. The analog of the O (3, 1) symmetry of the usual Maxwell theory is found to be O (3, 2) or O (4, 1), depending on the space-time Fourier spectrum of the field. We argue that the structure that is relevant to the description of radiation in interaction with matter evolving in a timelike sense is that of O (3, 2). The noncovariant form of the field equations is given; there are two fields of electric type and one (divergenceless) magnetic type field. The Noether currents are studied, and some remarks are made on second quantization.

A manifestly covariant relativistic Boltzmann equation for the evolution of a system of events

A quantum mechanical derivation of a manifestly covariant relativistic Boltzmann equation is given in a framework in which the fundamental dynamical constituents of the system are a family of N events with motion in space-time parametrized by an invariant “historical time” τ. The relativistic analog of the BBGKY hierarchy is generated. Approximating the effect of correlations of second and higher order by two event collision terms, one obtains a manifestly covariant Boltzmann equation. The Boltzmann equation is used to prove the H-theorem for evolution in τ. For ensembles containing only positive energy (or only negative energy) states, a precise H-theorem is also proved for increasing t. In the nonrelativistic limit, the usual H-theorem is recovered. It is shown that a covariant form of the Maxwell-Boltzmann distribution is obtained in the equilibrium limit; an examination of the energy momentum tensor for the free gas yields the ideal gas law in this limit. It is shown that in local equilibrium the internal energy is defined by a Lorentz transformation to a local rest frame. We study the conserved quantities in this theory. There is a new nontrivially conserved quantity corresponding to the particle mass. We obtain continuity equations for these quantities.

Particles vs. events: The concatenated structure of world lines in relativistic quantum mechanics

The dynamical equations of relativistic quantum mechanics prescribe the motion of wave packets for sets of events which trace out the world lines of the interacting particles. Electromagnetic theory suggests thatparticle world line densities be constructed from concatenation of event wave packets. These sequences are realized in terms of conserved probability currents. We show that these conserved currents provide a consistent particle and antiparticle interpretation for the asymptotic states in scattering processes. The relation between current conservation and unitarity is used to establish relations between pair production and annihilation amplitudes and scattering. The discrete symmetriesC, T, P are studied and it is shown that no Dirac sea (for fermions where such a construction is possible, or bosons where it is not) is required for consistency of the theory. These currents, furthermore, represent the discrete symmetries in a way consistent with their interpretation as particle currents.

The quantum relativistic two‐body bound state. I. The spectrum

In the framework of a manifestly covariant quantum theory on space‐time, it is shown that the ground state mass of a relativistic two‐body system with O(3,1) symmetric potential is lower when represented by a wave function with support in an O(2,1) invariant subspace of the spacelike region. The wave functions for the relativistic bound states are obtained explicitly. Coulomb type binding, the harmonic oscillator, and the relativistic square well are treated as examples. The mass spectrum is determined by a differential equation in the invariant spacelike interval ρ, which can be put into correspondence with the radial part of a nonrelativistic Schrodinger equation with potential of the same form, where r is replaced by ρ. In the case that the binding is small compared to the particle masses, the mass spectrum (bounded below) is well‐approximated by the results of the nonrelativistic theory. The eigenfunctions transform under the full Lorentz group as elements of an induced representation with O(2,1) litt...

On Feynman’s approach to the foundations of gauge theory

In 1948, Feynman showed Dyson how the Lorentz force law and homogeneous Maxwell equations could be derived from commutation relations among Euclidean coordinates and velocities, without reference to an action or variational principle. When Dyson published the work in 1990, several authors noted that the derived equations have only Galilean symmetry and so are not actually the Maxwell theory. In particular, Hojman and Shepley proved that the existence of commutation relations is a strong assumption, sufficient to determine the corresponding action, which for Feynman’s derivation is of Newtonian form. In a recent paper, Tanimura generalized Feynman’s derivation to a Lorentz covariant form with scalar evolution parameters, and obtained an expression for the Lorentz force which appears to be consistent with relativistic kinematics and relates the force to the Maxwell field in the usual manner. However, Tanimura’s derivation does not lead to the usual Maxwell theory either, because the force equation depends o...

Geometry of Hamiltonian chaos.

The characterization of chaotic Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor in the structure of the Hamiltonian is extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential model when a transition is made to an associated manifold. We find, in this way, a direct geometrical description of the time development of a Hamiltonian potential model. The second covariant derivative of the geodesic deviation in this associated manifold results in (energy dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We discuss some examples of unstable Hamiltonian systems in two dimensions.

On the definition and evolution of states in relativistic classical and quantum mechanics

Some of the problems associated with the construction of a manifestly covariant relativistic quantum theory are discussed. A resolution of this problem is given in terms of the off mass shell classical and quantum mechanics of Stueckelberg, Horwitz and Piron. This theory contains many questions of interpretation, reaching deeply into the notions of time, localizability and causality. A proper generalization of the Maxwell theory of electromagnetic interaction, required for the well-posed formulation of dynamical problems of systems with electromagnetic interaction is discussed, and some of the significance of recently found (classical) relativistic chaotic behavior is pointed out. Many results of a technical nature have been achieved in this framework; in this paper, some of these are reviewed, but I shall concentrate on a discussion of the basic ideas and foundations of the theory.