Dirac Formalism Research Articles

Quantization of false-vacuum bubbles: A Hamiltonian treatment of gravitational tunneling.

We develop the quantization of spherically symmetric gravitational systems using the Dirac formalism and the WKB approximation. The particular application is the nucleation of false-vacuum bubbles, which may have been the origin of the inflationary universe and matter. Consistency between our results and those obtained by the Euclidean method requires that the latter include metrics with degeneracies. Our work also raises several other questions of principle, including the meaning of the wave function and the necessity of topology change. A tentative application of this work to quantum cosmology indicates that the "Hartle-Hawking wave function" is inconsistent.

Conserved constraints in three-plus-one Regge calculus

The problem of formulating a consistent set of constraints for discrete versions of general relativity is discussed. The application of the Dirac formalism to a slightly modified form of the 3+1 version of Regge calculus developed by Piran and Williams (1986) is described, and illustrated by consideration of a simple model universe.

Uncertainty principle and uncertainty relations: extension of a result of Uffink and Hilgevoord

A new four-space formulation of Dirac's equation motivates the extension of a result that expresses the uncertainty principle without using standard deviations.

Dirac optical-model analysis of polarized protons elastic scattering from12C,16,18O,20Ne and24,26Mg at 800 MeV

Proton elastic-scattering cross-sections and analysing powers at 800 MeV have been used to derive, within the Dirac formalism, optical-model parameters for six elements from12C to26Mg. The best set of Dirac optical parameters has been obtained by searching on all nine parameters which showed a large negative real scalar potential, a large positive real vector potential and a large negative imaginary vector potential. These calculations have been found to be consistent with an imaginary scalar potential set to zero (V s i =0). Considerable improvements were obtained by using the Dirac formalism compared to the traditional Schrodinger one with a similar number of parameters.

Four-space formulation of Dirac's equation

Dirac's equation is reviewed and found to be based on nonrelativistic ideas of probability. A 4-space formulation is proposed that is completely Lorentzinvariant, using probability distributions in space-time with the particle's proper time as a parameter for the evolution of the wave function. This leads to a new wave equation which implies that the proper mass of a particle is an observable, and is sharp only in stationary states. The model has a built-in arrow of time, which is associated with a restriction to positive-energy solutions. The usual solution for a Coulomb field is retained, though it now implies a slightly different charge distribution. The conventional nonstationary solutions become invalid. The new formulation appears to offer a resolution of difficulties that have been associated with Dirac's equation. It also predicts the occurrence of virtual pairs at a level that may be experimentally testable, and suggests a mechanism for self-cancellation of the vacuum energy.

Which Potentials have to be Surface Peaked to Reproduce Large Angle Proton Scattering at High Energy?

Corrections to the usual form factors of the optical potential are studied with a view to getting a better fit for proton elastic scattering at large angles on 40Ca at 497 and 800 MeV. When a real surface form factor is added to the central potential in the Schrbdinger formalism, the experimental data are as well reproduced as in the standard Dirac formalism. Coupling to the strong 3- collective state gives a better fit. The use of surface corrections to the imaginary Dirac potential also gives improved results. A slightly better fit is obtained by coupling to the 3- state with, at the same time, a weakening of these corrections. Further corrections to the potential do not give significant improvements.

Four-space formulation of Dirac's equation

Dirac's equation is reformulated in a way which, while preserving the usual energy levels, leads to predictions of vacuum polarization that may be experimentally testable.

Canonical normal coordinates

We examine the advantages and disadvantages of fixing the coordinate gauge to be that of the Riemann normal coordinates by applying the Dirac formalism for constrained canonical systems in general relativity.

Nonrigid quantization of the skyrmion

We present a nonrigid quantization scheme for rotating skyrmions, using the Dirac formalism for constrained systems. In this scheme, one can derive a systematic semiclassical expansion which yields a correct soft-pion limit and gives a proper treatment of retardation. We discuss the pion-baryon vertex and the baryon form factor.

A new derivation of the tensor form of Dirac's equation

In this paper we show that the classical Einstein-Yang-Mills-Dirac equations for the gravitational, W boson, and fermion fields can all be derived from a tetrad of vector fields on a Kaluza-Klein manifold.

A dynamical formalism of singular Lagrangian system with higher derivatives

The singular Lagrangian system with higher derivatives is analyzed with the aid of the Ostrogradski transformation and the Dirac formalism. The formulation of canonical theory is developed so that the equivalence between the Lagrange formalism and the Hamilton one is maintained. As a practical example, the acceleration-dependent potentials appearing in the Lagrangian of two-point particles interacting gravitationally are dealt with and the equivalence between the two Hamiltonians that follow from the two Lagrangians which are related by the coordinate transformations is shown. It is also shown, when the constraints are all first class, that a consistent generator of gauge transformation is constructed. Typical examples are given.

Two-dimensional model of the interaction of a nonrelativistic particle with scalar mesons in the strong-coupling limit

Bogolyubov transformations in strong-coupling theory lead to a singular Lagrangian containing constraints. Applying the Dirac formalism and a modified Feynman integral, we have succeeded in constructing an S matrix that correctly describes the symmetry properties of the system. The advantage of this method is that in such a formulation of the strong-coupling method it is not necessary, to construct the Hamiltonian, to fix a subsidiary condition from the very beginning, in contrast to what is done in [10] and other formulations.

Hamiltonian analysis of topologically massive Yang-Mills theory.

The Hamiltonian dynamics of three-dimensional topologically massive Yang-Mills theory is analyzed in both normal and null-plane coordinates using Dirac's constraint formalism. As expected, the physical degrees of freedom correspond to a single self-interacting massive mode. In null-plane coordinates, we obtain the standard result that the number of canonical degrees of freedom is half the number found using a spacelike initial-value surface.

Ambiguity on the imaginary potentials in the Dirac formalism for the elastic and the inelastic scattering of nucleons

The Dirac formalism for the elastic scattering of nucleons has been extended to a coupled channel formalism taking into account collective excited states. Calculations for the scattering of 500 MeV protons on 40Ca shows large second-order effects on the elastic scattering. The imaginary scalar potential and the imaginary vector potential can be varied widely without giving so large effects on the elastic scattering. This large ambiguity is not limited to this case.

First order formalism for quantum gravity

We develop a first order formalism for the quantization of gravity. We take as canonical variables both the induced metric and the extrinsic curvature of the (d − 1)-dimensional hyper- surfaces obtained by the foliation of the d-dimensional spacetime. After solving the constraint algebra we use the Dirac formalism to quantize the theory and obtain a new representation for the Wheeler-DeWitt equation, defined in the functional space of the extrinsic curvature. We also show how to obtain several different representations of the Wheeler-DeWitt equation by considering actions differing by a total divergence. In particular, the intrinsic and extrinsic time approaches appear in a natural way, as do equivalent representations obtained by functional Fourier transforms of appropriate variables. We conclude with some remarks about the construction of the Hilbert space within the first order formalism.

Supersymmetry in quantum mechanics

We give some illustrations and interpretations of supersymmetry in quantum mechanics in simple models. We show that the value of 2 for the g factor of the electron expresses the presence of supersymmetry in the Hamiltonian for an electron in a uniform magnetic field. The problem is considered both in the Schrödinger and Dirac formulations. We also show that the radial Coulomb problem with orbital angular momentum l, nuclear charge Z, and principal quantum number n, is supersymmetrically linked to the similar problem with charge Z(1−1/n) and quantum number n−1. Thereby the dependence of Coulomb energies only on the combination Z/n is seen as a manifestation of the supersymmetry in the radial Coulomb problem. Other examples of supersymmetry we consider are the Morse potential, the three-dimensional isotropic oscillator, the states of the helium atom and those of the hydrogen atom in an extremely strong magnetic field.

Three-dimensional gauge theory in Dirac formalism

The Hagen model [C. R. Hagen, Ann. Phys. (NY) 157, 342 (1984); Phys. Rev. D 31, 331 (1985)] is studied using the method of constrained Hamiltonian formalism developed by Dirac [P. A. M. Dirac, Can. J. Math. 2, 129 (1950); Lectures on Quantum Mechanics (Yeshiva U. P., New York, 1964)]. The results recently obtained by Burnel and Van Der Rest-Jaspers [A. Burnel and M. Van Der Rest-Jaspers, J. Math. Phys. 26, 3155 (1985)] are reexamined and modified. There appear two second-class constraints and their choice is not crucial. The equivalence of different gauges is proved without referring to the current conservation law.

A mathematical interpretation of Dirac's formalism: Part c: free field operators

In the present paper we give a mathematical regorization and interpretation of the canonical (anti-) commutation relations which occur in the theory of the quantized free field. In our set-up, with respect to the physics, we felt inspired by the paper [Ro] of B. Robertson. His paper covers all essential elements of the subject under consideration. The main part of [Ro] will be mathematically justified and even generalized. In this connection we mention also the monograph [Sh] of A.S. Shvarz. With respect to the involved mathematics the papers [Co] of J. M. Cook and [KMP] of P. Kristensen et al. have been a source of inspiration. Our set-up is closely related to two of our earlier papers [EG 1–2] in which we presented a mathematical interpretation of Dírac's formalism. This interpretation is based on two new mathematical concepts: the concept of Dirac basis ans the concept of bracket. The bracket is no longer regarded as a “number” but as an analytic function on the open right half of the complex plane. As in [EG 1–2], we use the theory of generalized functions which has been constructed by one of us (cf. [G]). Quantum theory and distribution theory seem to be dissolubly connected. One might wonder whether the choice of the distribution theory plays an essential role in a mathematical interpretation of Dirac's formalism and related physical concepts. The answer is threefold: There are strong indications that similar interpretations can be derived in a very wide class of distribution theories (which include Schwarz's theory of tempered distributions). This class has been described in our papers [EK] and [EGK]. On the other hand, the distribution theory employed here, introduces generalized functions in a way very close to the physical intuitive view on improper functions. Moreover, it seems natural to adapt the used distribution theory to the concrete quantum mechanical system under consideration.

Covariant poisson brackets for classical fields

Poisson brackets that are spacetime covariant are presented for a variety of relativistic field theories. These theories include electromagnetism, general relativity, and general relativistic fluids and plasmas in Eulerian representation. The examples presented suggest the development of a general theory; the beginnings of such a theory are presented. Our covariant bracket formalism provides a general setting for, amongst other things, clarifying the transition from the covariant formalism to the dynamical 3 + 1 Hamiltonian formalism of Dirac and Arnowitt, Deser, and Misner. We illustrate the relevant procedures with electromagnetism.

On the space‐time structure and its relation to quantum mechanics

AbstractThe Weyl covariant formulation of Dirac's equation by means of a rest mass the Weyl type of which is — 1, implies the introduction of a Riemann structure in addition to the Weyl structure. The Ehlers‐Pirani‐Schild attempt to specify the space‐time structure and the lowest order of the WKB approximation are compatible with each other. Only the requirement that the connection to which the Dirac field couples minimally is identical to the connection the autoparallels of which are the world lines of structure‐less test particles, leads to a reduction of the Weyl structure to the Riemannian one.