Causal non-local character of quantum statistics

Developing Tersoff and Bayers ideas one shows that quantum statistics result from distinguishable particles correlated by causal non-local actions-at-a-distance. The difference between classical and quantum statistics thus results from the uncontrollable non-local character of stochastic interactions connecting particles imbedded in Dirac's random aether.     

Solitons and supertranslations

We construct the quantum theory for a two-dimensional supersymmetric theory around a supertranslated soliton solution. This is done by using Dirac's method for constrained systems. The fermionic parameter of supertranslation is treated as a collective co-ordinate. The path-integral expression is also obtained.     

Colored magnetic monopoles

We extend Dirac's theory of magnetic monopoles to the case of non-Abelian color gauge groups. Exact classical solution is obtained by making use of the gauge-independent method of Yang-Mills field. The case of broken gauge symmetry is also briefly discussed.     

Classical and quantum mechanics of free relativistic membranes

Motivated by the theory of relativistic strings, the theory of a two-dimensional relativistic membrane whose action is proportional to the three-dimensional area it traces out in space-time is investigated both in Lagrangian and Hamiltonian formalisms. The quantum theory is developed using Dirac's method for constrained systems and the question of gauge choices is considered in some detail.     

A one-potential formulation of the quantum field theory of magnetic poles

We formulate a new approach to the quantum field theory of electric and magnetic charges. Its advantages relative to existing formulations are that (a) it is a one-potential lagrangian one, (b) it has a canonical structure and (c) it representys a natural generalization of Dirac's quantum mechanical theory to field theory. Our formulation is non-local in coordinate space, but yields Feynman rules that are local in momentum space.     

Eigenvalue Spectrum of a Dirac Particle in Static and Spherical Complex Potential

It has been observed that a quantum theory need not be Hermitian to have a real spectrum. We study the non-Hermitian relativistic quantum theories for many complex potentials, and obtain the real relativistic energy eigenvalues and corresponding eigenfunctions of a Dirac-charged particle in complex statically and spherically symmetric potentials. Complex Dirac–Eckart, complex Dirac–Rosen–Morse II, complex Dirac–Scarf and complex Dirac–Poschl–Teller potential are investigated.     

The supersymmetry extended Weyl algebra and Casalbuoni's G4 model

We briefly review the classical version of Casalbuoni's G₄ supersummetric model (i.e. the single particle version of the scalar chiral supermultiplet) with particular emphasis on the role played by the chirality. We show that the off-mass-shell commutators of the quantum model can be derived from the Lie algebra of the Weyl (i.e. Poincaré plus dilatations) group extended by supersymmetry. The proper-time wavefunctions of the off-mass-shell states satisfy equations which clarify the role of the “auxiliary fields” of quantum field theory.     

Quantum behavior of deterministic systems with information loss: Path integral approach

’t Hooft’s derivation of quantum from classical physics is analyzed by means of the classical path integral of Gozzi et al. It is shown how the key element of this procedure—the loss of information constraint—can be implemented by means of Faddeev–Jackiw’s treatment of constrained systems. It is argued that the emergent quantum systems are identical with systems obtained in Blasone et al. [Phys. Rev. A 71 (2005) 052507] through Dirac–Bergmann’s analysis. We illustrate our approach with two simple examples—free particle and linear harmonic oscillator. Potential Liouville anomalies are shown to be absent.     

The quantization of a massless relativistic string in a time-like gauge

The massless relativistic string is quantized in a time-like orthonormal gauge. The results are completely equivalent to those obtained in the light-like case. In particular, a strictly canonical quantization procedure succeeds only when the dimension of space-time is 26. The discussion is based on Dirac's formalism for contrained systems and the introduction of classical variables analogous to those defined by Del Giudice, Di Vecchia and Fubini in the dual model.     

The equivalence of Bell's inequality and the Nash inequality in a quantum game-theoretic setting

The interaction of competing agents is described by classical game theory. It is now well known that this can be extended to the quantum domain, where agents obey the rules of quantum mechanics. This is of emerging interest for exploring quantum foundations, quantum protocols, quantum auctions, quantum cryptography, and the dynamics of quantum cryptocurrency, for example. In this paper, we investigate two-player games in which a strategy pair can exist as a Nash equilibrium when the games obey the rules of quantum mechanics. Using a generalized Einstein–Podolsky–Rosen (EPR) setting for two-player quantum games, and considering a particular strategy pair, we identify sets of games for which the pair can exist as a Nash equilibrium only when Bell's inequality is violated. We thus determine specific games for which the Nash inequality becomes equivalent to Bell's inequality for the considered strategy pair.     

A mathematical interpretation of Dirac's formalism part a: Dirac bases in trajectory spaces

In this paper the notion of Dirac basis will be introduced. It is the continuous pendant of the discrete basis for Hilbert spaces. The introduction of this new notion is closely related to the theory of generalized functions. Here De Graaf's theory will be employed. It is based on the triplet $\text{S}{}_{\mathrm{<mtext>X,</mtext><mtext>A</mtext>}}\text{?X?}\text{T}{}_{\mathrm{<mtext>X,</mtext><mtext>A</mtext>}}$ where X is a Hilbert space. In a well specified way any member of $\text{T}{}_{\mathrm{<mtext>X,</mtext><mtext>A</mtext>}}$ can be expanded with respect to a Dirac basis. Both the introduction of Dirac bases and a new interpretation of Dirac's bracket notion will lead to a mathematical rigorization of various aspects of Dirac's formalism for quantum mechanics. This rigorization goes much beyond earlier proposals.     

Canonical quantization of the electromagnetic field with Dirac's monopoles

From a lagrangian theory of charge-monopole electrodynamics which uses strings but is free from Dirac's veto, a hamiltonian theory is derived which describes the interaction between electric and magnetic point particles and photons.     

Quantum theory of collective motions and an application to theory of extended hadrons

It is shown that the theory of collective motions can be reduced to a gauge problem in Dirac's generalized Hamiltonian formalism. As an example, we study the φ⁴ theory in two-dimensional space-time by using our formalism.     

Relativistic quark potential for the vector mesons

We extend the heavy quark potential recently proposed by Richardson to the light quarks by using Dirac's theory of constraints. Our relativistic version reproduces the ground state spectrum of the light vector mesons while slightly improving Richardson's own results for the ψ and ? systems.     

Exact classical and quantum mechanics of a particle coupled to a membrane

The exact dynamical solution is given for a harmonically bound particle of finite size (the oscillator) coupled rigidly to a finitely extended two-dimensional mechanical continuum transmission line (the membrane). A quantum mechanical ultraviolet divergence in the particle's momentum fluctuations is discussed. In the limit of a very large membrane and a very small particle the oscillator's classical friction coefficient is obtained explicitly.     

Extended systems and generalized London equations

We propose a classical action for interacting extended objects in n spacetime dimensions. The theory of membranes closely resembles the classical Schwinger model. We explain this analogy and suggest a natural trapping mechanism for quarks according to Nambu's “electric” confinement scheme.     

Quarkonium and hydrogen spectra with spin-dependent relativistic wave equation

The non-linear non-perturbative relativistic atomic theory introduces spin in the dynamics of particle motion. The resulting energy levels of hydrogen atom are exactly the same as that of Dirac theory. The theory accounts for the energy due to spin-orbit interaction and for the additional potential energy due to spin and spin-orbit coupling. Spin angular momentum operator is integrated into the equation of motion. This requires modification to classical Laplacian operator. Consequently, the Dirac matrices and the k operator of Dirac’s theory are dispensed with. The theory points out that the curvature of the orbit draws on certain amount of kinetic and potential energies affecting the momentum of electron and the spin-orbit interaction energy constitutes a part of this energy. The theory is developed for spin-1/2 bound state single electron in Coulomb potential and then extended further to quarkonium physics by introducing the linear confining potential. The unique feature of this quarkonium model is that the radial distance can be exactly determined and does not have a statistical interpretation. The established radial distance is then used to determine the wave function. The observed energy levels are used as the input parameters and the radial distance and the string tension are predicted. This ensures 100% conformance to all observed energy levels for the heavy quarkonium.     

Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations

Newton-Leibniz integration rule only applies to commuting functions of continuum variables, while operators made of Dirac’s symbols (ket versus bra, e.g., |q〉〈q| of continuous parameter q) in quantum mechanics are usually not commutative. Therefore, integrations over the operators of type |〉〈| cannot be directly performed by Newton-Leibniz rule. We invented an innovative technique of integration within an ordered product (IWOP) of operators that made the integration of non-commutative operators possible. The IWOP technique thus bridges this mathematical gap between classical mechanics and quantum mechanics, and further reveals the beauty and elegance of Dirac’s symbolic method and transformation theory. Various applications of the IWOP technique, including constructing the entangled state representations and their applications, are presented.     

Some non-perturbative semi-classical methods in quantum field theory (a pedagogical review)

A pedagogical introduction is given to non-perturbative semi-classical methods for finding solutions to quantum field theories. Both the weak coupling method based on a time-independent classical solution, and the WKB method based on all periodic orbits are developed in detail, proceeding ffrom elementary quantum mechanics to field theory in stages. Both methods are then illustrated in model field theories. The [λø⁴]₂ theory to which the weak coupling method is applied yields a new family of “kink” states whose properties are discussed.The WKB method is illustrated by quantizing “soliton” and “doublet” solutions of the two-dimensional sine-Gordon theory. The results are compared to consequences of Coleman's equivalence proof relating this system to the massive Thirring model. The speculation that solitons may be fermions is discussed, along with indications that the WKB method may ne yielding exact mass ratios for this theory.A final section is devoted to solutions of more realistic four-dimensional models containing fermions, internal symmetry etc. Some quark-confinement models and vortex type solutions come under this category. General remarks are made on this family of solutions, and illustrated using 't Hooft's monopole solution.