Cosmic numbers and quantum theory

The cosmic numbers are considered, with emphasis on the relationN ². (HereN is the number of nucleons in the universe, and, its radius in atomic units.) This relation is interpreted in terms of a quantum mechanical model.     

Complex energies in relativistic quantum theory

A new four-component spin-1/2 wave equation for ordinary mass is discussed. It is shown that this equation has a conserved current not easily identified with a transition probability, only pure imaginary energy states, and is covariant. A tachyon-like Klein-Gordon equation is satisfied by this equation, but rest states are explicitly constructed.     

Quasiparticle excitations in relativistic quantum field theory

We analyze the particle-like excitations arising in relativistic field theories in states different than the vacuum. The basic properties characterizing the quasiparticle propagation are studied using two different complementary methods. First we introduce a frequency-based approach, wherein the quasiparticle properties are deduced from the spectral analysis of the two-point propagators. Second, we put forward a real-time approach, wherein the quantum state corresponding to the quasiparticle excitation is explicitly constructed, and the time-evolution is followed. Both methods lead to the same result: the energy and decay rate of the quasiparticles are determined by the real and imaginary parts of the retarded self-energy, respectively. Both approaches are compared, on the one hand, with the standard field-theoretic analysis of particles in the vacuum and, on the other hand, with the mean-field-based techniques in general backgrounds.     

Mass splitting in relativistic quantum field theory

Null plane integrals of certain classes of tensor densities, conserved or non-conserved, may define symmetric operators on dense subspaces of the in and out states. These operators annihilate the vacuum and may satisfy a Lie algebra. In particular, the possibility that a finite number of null plane charges, which includes the Poincaré generators, close on an algebra whose irreducible representations contain particles with different masses is considered. The situation in which the Lie algebra is defined on a dense domain which is not from the in and out states is discussed. Some algebraic hypotheses other than that of a Lie algebra in the usual sense are briefly considered; in these cases there can be no mass splitting.Supported in part by the U.S.-Israel Binational Science Foundation (B.S.F.), Jerusalem, Israel.     

Fractional quantum numbers via complex orbifolds

This paper studies both the conductance and charge transport on 2D orbifolds in a strong magnetic field. We consider a family of Landau Hamiltonians on a complex, compact 2D orbifold Y that are parametrised by the Jacobian torus J(Y) of Y. We calculate the degree of the associated stable holomorphic spectral orbibundles when the magnetic field B is large and obtain fractional quantum numbers as the conductance and a refined analysis also gives the charge transport. A key tool studied here is a nontrivial generalisation of the Nahm transform to 2D orbifolds.

     

Linking numbers in local quantum field theory

Linking numbers appear in local quantum field theory in the presence of tensor fields, which are closed two-forms on Minkowski space. Given any pair of such fields, it is shown that the commutator of the corresponding intrinsic (gauge-invariant) vector potentials, integrated about spacelike separated, spatial loops, are elements of the center of the algebra of all local fields. Moreover, these commutators are proportional to the linking numbers of the underlying loops. If the commutators are different from zero, the underlying two-forms are not exact (i.e. there do not exist local vector potentials for them). The theory then necessarily contains massless particles. A prominent example of this kind, due to J.E. Roberts, is given by the free electromagnetic field and its Hodge dual. Further examples with more complex mass spectrum are presented in this article.

     

On the relativistic invariance of a quantum theory based on beables

We discuss the question of the relativistic invariance of a quantum theory based on beables, and we suggest the general outlines of one possible form of such a theory.     

On the geometric structure of a parity-conserving relativistic quantum field theory

Wheeler's conjecture that there might exist a ‘principle’ which rules out parity-non-conserving spaces is analysed. The following result has been obtained: A local relativistic quantum field theory is parity-conserving if the following conditions hold:

1. The fields are derived from geometry, i.e. they are represented by quantised currents (in the sense of de Rham); and
2. The theory may be defined on a connected and, under certain restrictions, on a disconnected orientable space-time continuumM ⁴.

     

A relativistic quantum field theory for rotating nuclear matter

The relativistic quantum field theory of Walecka is extended to rotating nuclear systems using a mean-field Thomas-Fermi approximation. Self-bound systems exhibit centrifugal stretching and a maximum angular frequency. Systems constrained to a cylindrical box develop central holes for large angular frequencies.     

Additive conservation laws in local relativistic quantum field theory

We consider generatorsQ of symmetry transformations acting additively on asymptotic particle states according to (1.1). [This equation can be derived forQ defined as integral over a conserved local current!]. For simplicity, we consider only the case that all asymptotic fields are scalar. Assuming that elastic scattering occurs at least in an open subset of the scattering manifold we show thatQ is at most alinear combination of generators of the Poincaré group and internal symmetries.     

Landau levels as mass excitations in relativistic quantum theory

We show that, in a manifestly covariant quantum theory, the mass spectrum of a single charged particle moving in a constant magnetic field contains a discrete part, corresponding to mass excitations associated with the field, and a continuous part associated with the free motion parallel to the field.     

Generalized quaternion formulation of relativistic quantum theory in curved space

A survey is presented of the essential principles for formulating relativistic wave equations in curved spacetime. The approach is relatively simple and avoids much of the philosophical debate about covariance principles, which is also indicated. Hypercomplex numbers provide a natural language for covariance symmetry and the two important kinds of covariant derivative.     

Quantum mechanics: From complex to complexified quaternions

This paper is an attempt to simplify and clarify the mathematical language used to express quaternionic quantum mechanics (QQM). In our quaternionic approach the choice of “complex” geometries allows an appropriate definition of momentum operator and gives the possibility to obtain consistent formulations of standard theories. Barred operators represent the key to realizing a set of translation rules between quaternionic and complex quantum mechanics (QM). These translations enable us to obtain a rapid quaternionic counterpart of standard quantum mechanical results.     

Properties of finite nuclei in a relativistic quantum field theory

A renormalizable relativistic quantum field theory is used to study finite nuclei in the mean-field Thomas-Fermi approximation. Mass formula parameters are calculated and proton densities illustrated for ⁴⁰Ca and ²⁰⁸Pb. The Dirac equation is solved to determine single-particle spectra.     

Five- and eight-vector extensions of relativistic quantum theory: The preferred reference frame

The complex quaternion approach to relativistic quantum theory is extended to include time, in the preferred cosmic reference frame, as a fifth dimension. Particle mass appears as an eigenvalue of this invariant operator. Closing the extended algebra also suggests three additional space-time components, possibly related to microscopic rotational degrees of freedom and the scale setting fundamental length. The Dirac, Klein-Gordon, Maxwell, and Einstein equations are generalized to accommodate 5- and 8-vector space-time, including gravitational curvature. The usual conserved probability density is shown to harbor fundamental difficulties.     

Renormalized Hartree-Fock equations in a nuclear relativistic quantum field theory

Renormalized Hartree-Fock equations are derived for an infinite system of mesons and baryons in the framework of a relativistic quantum field theory. Direct and exchange diagrams in the baryon propagator are summed self-consistently to all orders, and the effects of occupied negative-energy states in the Dirac sea are included. The required counterterm subtractions are defined by conventional renormalization conditions, but they need not be evaluated explicitly. The result is a set of finite nonlinear integral equations for the baryon self-energy that includes vacuum fluctuation effects from virtual N$\text{N}$ pairs in the many-body wavefunction at finite density.     

On stochastic Einstein locality in algebraic relativistic quantum field theory

This paper assesses Miklós Rédei's [1991] proof of the proposition that algebraic relativistic quantum field theory is stochastic Einstein local. The conclusion is that either Rédei's proof is spurious, in that it does not really prove what it intends to establish, or that the proof is fallacious. The paper is self-contained in the sense that the few ingredients of algebraic quantum theory that go into Rédei's proof are first summed up. Then Hellman's definition of stochastic Einstein locality is discussed, a detailed exposition is offered of Rédei's proof, and finally the author's refutation is explicated.