arXiv subjects

Lawrence P. Horwitz

Publications and source records attributed to Lawrence P. Horwitz.

9 recordsLinked to original sources

arXiv2020

Spin and Entanglement in General Relativity

In a previous paper, we have shown that the classical and quantum relativistic dynamics of the Stueckelberg-Horwitz-Piron [SHP] theory can be embedded in general relativity (GR). We briefly review the SHP theory here, and, in particular, the formulation of the theory of spin in the framework of relativistic quantum theory. We show here how the quantum theory of relativistic spin can be embedded, using a theorem of Abraham, Marsden and Ratiu, and also explicit derivation, into the framework of GR by constructing a local induced representation. The relation to the work of Fock and Ivanenko is also discussed. We show that in a gravitational field there is s highly complex structure for the spin distribution in the support of the wave function. We then discuss entanglement for the spins of a two body system.

math-ph↗

arXiv2015

Induced Representations of Tensors and Spinors of any Rank in the SHP Theory

We show that a modification of Wigner's induced representation for the description of a relativistic particle with spin can be used to construct spinors and tensors of arbitrary rank, with invariant decomposition over angular momentum. In particular, scalar and vector fields, as well as the representations of their transformations, are constructed. The method that is developed here admits the construction of wave packets and states of a many body relativistic system with definite total angular momentum. Furthermore, a Pauli-Lubanski operator is constructed on the orbit of the induced representation which provides a Casimir operator for the Poincaré group and which contains the physical intrinsic angular momentum of the particle covariantly.

quant-ph↗

arXiv2014

Quantum Field Theory of Classically Unstable Hamiltonian Dynamics

We study a class of dynamical systems for which the motions can be described in terms of geodesics on a manifold (ordinary potential models can be cast into this form by means of a conformal map). It is rigorously proven that the geodesic deviation equation of Jacobi, constructed with a second covariant derivative, is unitarily equivalent to that of a parametric harmonic oscillator, and we study the second quatization of this oscillator. The excitations of the Fock space modes correspond to the emission and absorption of quanta into the dynamical medium, thus associating unstable behavior of the dynamical system with calculable fluctuations in an ensamble with possible thermodynamic consequences.

math-ph↗

arXiv2005

On the significance of a recent experiment demonstrating quantum interference in time

I comment on the interpretation of a recent experiment showing quantum interference in time. It is pointed out that the standard nonrelativistic quantum theory, used by the authors in their analysis, cannot account for the results found, and therefore that this experiment has fundamental importance beyond the technical advances it represents. Some theoretical structures which consider the time as an observable, and thus could, in principle, have the required coherence in time, are discussed briefly, and the application of Floquet theory and the manifestly covariant quantum theory of Stueckelberg are treated in some detail. In particular, the latter is shown to account for the results in a simple and consistent way.

quant-ph↗

arXiv2004

Classical Gravity as an Eikonal Approximation to a Manifestly Lorentz Covariant Quantum Theory with Brownian Interpretation

We discuss in this Chapter a series of theoretical developments which motivate the introduction of a quantum evolution equation for which the eikonal approximation results in the geodesics of a four dimensional manifold. This geodesic motion can be put into correspondence with general relativity. One obtains in this way a quantum theory on a flat spacetime, obeying the rules of the standard quantum theory in Lorentz covariant form, with a spacetime dependent Lorentz tensor $g_{μν}$, somewhat analogous to a gauge field, coupling to the kinetic terms. Since the geodesics predicted by the eikonal approximation, with appropriate choice of $g_{μν}$, can be those of general relativity, this theory provides a quantum theory which could be underlying to classical gravitation, and coincides with it in this classical ray approximation. In order to understand the possible origin of the structure of this equation, we appeal to the approach of Nelson in constructing a Schroedinger equation from the properties of Brownian motion. Extending the notion of Browninan motion to spacetime in a covariant way, we show that such an equation follows from correlations between spacetime dimensions in the stochastic process.

gr-qc↗

arXiv2003

Global unitary fixing and matrix-valued correlations in matrix models

We consider the partition function for a matrix model with a global unitary invariant energy function. We show that the averages over the partition function of global unitary invariant trace polynomials of the matrix variables are the same when calculated with any choice of a global unitary fixing, while averages of such polynomials without a trace define matrix-valued correlation functions, that depend on the choice of unitary fixing. The unitary fixing is formulated within the standard Faddeev-Popov framework, in which the squared Vandermonde determinant emerges as a factor of the complete Faddeev-Popov determinant. We give the ghost representation for the FP determinant, and the corresponding BRST invariance of the unitary-fixed partition function. The formalism is relevant for deriving Ward identities obeyed by matrix-valued correlation functions.

hep-th↗

arXiv1999

Structure and Properties of Hughston's Stochastic Extension of the Schrödinger Equation

Hughston has recently proposed a stochastic extension of the Schrödinger equation, expressed as a stochastic differential equation on projective Hilbert space. We derive new projective Hilbert space identities, which we use to give a general proof that Hughston's equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We discuss the relation of Hughston's equation to earlier work on norm-preserving stochastic equations, and show that Hughston's equation can be written as a manifestly unitary stochastic evolution equation for the pure state density matrix. We discuss the behavior of systems constructed as direct products of independent subsystems, and briefly address the question of whether an energy-based approach, such as Hughston's, suffices to give an objective interpretation of the measurement process in quantum mechanics.

quant-ph↗

arXiv1999

State Vector Collapse Probabilities and Separability of Independent Systems in Hughston's Stochastic Extension of the Schrödinger Equation

We give a general proof that Hughston's stochastic extension of the Schrödinger equation leads to state vector collapse to energy eigenstates, with collapse probabilities given by the quantum mechanical probabilities computed from the initial state. We also show that for a system composed of independent subsystems, Hughston's equation separates into similar independent equations for the each of the subsystems, correlated only through the common Wiener process that drives the state reduction.

quant-ph↗

arXiv1996

Time and the Evolution of States in Relativistic Classical and Quantum Mechanics

A consistent classical and quantum relativistic mechanics can be constructed if Einstein's covariant time is considered as a dynamical variable. The evolution of a system is then parametrized by a universal invariant identified with Newton's time. This theory, originating in the work of Stueckelberg in 1941, contains many questions of interpretation, reaching deeply into the notions of time, localizability, and causality. Some of the basic ideas are discussed here, and as an example, the solution of the two body problem with invariant action-at-a-distance potential is given. A proper generalization of the Maxwell theory of electromagnetic interaction, implied by the Stueckelberg-Schrödinger dynamical evolution equation and its physical implications are also discussed. It is also shown that a similar construction occurs, applying similar ideas, in the theory of gravitation.

hep-ph↗