Formulation Of Relativistic Quantum Mechanics Research Articles

Fully Symmetric Relativistic Quantum Mechanics and Its Physical Implications

A new formulation of relativistic quantum mechanics is presented and applied to a free, massive, and spin-zero elementary particle in the Minkowski spacetime. The reformulation requires that time and space, as well as the timelike and spacelike intervals, are treated equally, which makes the new theory fully symmetric and consistent with the special theory of relativity. The theory correctly reproduces the classical action of a relativistic particle in the path integral formalism, and allows for the introduction of a new quantity called vector-mass, whose physical implications for nonlocality, the uncertainty principle, and quantum vacuum are described and discussed.

Euclidean formulation of relativistic quantum mechanics of N particles

A Euclidean formulation of relativistic quantum mechanics for systems of a finite number of degrees of freedom is discussed. Relativistic treatments of quantum theory are needed to study hadronic systems at sub-hadronic distance scales. While direct interaction approaches to relativistic quantum mechanics have proved to be useful, they have two disadvantages. One is that cluster properties are difficult to realize for systems of more than two particles. The second is that the relation to quantum field theories is indirect. Euclidean formulations of relativistic quantum mechanics provide an alternative representation that does not have these difficulties. More surprising, the theory can be formulated entirely in the Euclidean representation without the need for analytic continuation. In this work a Euclidean representation of a relativistic $N$-particle system is discussed. Kernels for systems of N free particles of any spin are given and shown to be reflection positive. Explicit formulas for generators of the Poincar\'e group for any spin are constructed and shown to be self-adjoint on the Euclidean representation of the Hilbert space. The structure of correlations that preserve both the Euclidean covariance and reflection positivity is discussed.

Time and energy operators in the canonical quantization of special relativity

Based on Lorentz invariance and Born reciprocity invariance, the canonical quantization of special relativity is shown to provide a unified origin for (i) the complex vector space formulation of quantum mechanics; (ii) the momentum and space commutation relations and the corresponding representations; (iii) the Dirac Hamiltonian in the formulation of relativistic quantum mechanics; (iv) the existence of a self adjoint time operator that circumvents Pauli’s objection.

Representations of relativistic particles of arbitrary spin in Poincaré, Lorentz, and Euclidean covariant formulations of relativistic quantum mechanics

Relativistic treatments of quantum mechanical systems are important for understanding hadronic structure and dynamics at sub-nucleon distance scales. Hadronic states in different inertial reference frames are needed to compute current matrix elements that probe hadronic structure and dynamics. Relativistic invariance is an important consideration as the resolution of the probe is increased. Many different treatments of relativistic dynamics are used in practice, including Poincar\'e covariant methods, Lorentz covariant methods, Euclidean covariant methods and methods based on quantum fields. Wave functions are typically matrix elements of interacting relativistic states in a basis of non-interacting relativistic states. The purpose of this work is to develop the relation between these different representations of relativistic states that are used in different applications from a unified point of view, starting with positive mass irreducible representations of the Poincar\'e group.

Entanglement and disentanglement in relativistic quantum mechanics

A satisfactory formulation of relativistic quantum mechanics requires that one be able to represent the entangled states of spacelike separated systems and describe how such states evolve. This paper presents two stories that one must be able to tell coherently in order to understand relativistic entangled systems. These stories help to illustrate why one׳s understanding of entanglement in relativistic quantum mechanics must ultimately depend on the details of one׳s strategy for addressing the quantum measurement problem.

Anisotropic magnetoresistance in an antiferromagnetic semiconductor.

Recent studies in devices comprising metal antiferromagnets have demonstrated the feasibility of a novel spintronic concept in which spin-dependent phenomena are governed by an antiferromagnet instead of a ferromagnet. Here we report experimental observation of the anisotropic magnetoresistance in an antiferromagnetic semiconductor Sr2IrO4. Based on ab initio calculations, we associate the origin of the phenomenon with large anisotropies in the relativistic electronic structure. The antiferromagnet film is exchange coupled to a ferromagnet, which allows us to reorient the antiferromagnet spin-axis in applied magnetic fields via the exchange spring effect. We demonstrate that the semiconducting nature of our AFM electrode allows us to perform anisotropic magnetoresistance measurements in the current-perpendicular-to-plane geometry without introducing a tunnel barrier into the stack. Temperature-dependent measurements of the resistance and anisotropic magnetoresistance highlight the large, entangled tunabilities of the ordinary charge and spin-dependent transport in a spintronic device utilizing the antiferromagnet semiconductor.

Scattering and reflection positivity in relativistic Euclidean quantum mechanics

In this paper I discuss a formulation of relativistic few-particle scattering theory where the dynamical input is a collection of reflection-positive Euclidean covariant Green functions. This formulation of relativistic quantum mechanics has the advantage that all calculations can be performed using Euclidean variables. The construction of the Hilbert space inner product requires that the Green functions are reflection positive, which is a non-trivial constraint on the dynamics. In this paper I discuss the structure of reflection positive Green functions. I show that scattering wave operators exist when the dynamics is given by a Green function with the structure discussed in this work. Techniques for computing scattering observables in this Euclidean framework are discussed.

Relativistic Few-Body Physics

I discuss different formulations of the relativistic few-body problem with an emphasis on how they are related. I first discuss the implications of some of the differences with non-relativistic quantum mechanics. Then I point out that the principle of special relativity in quantum mechanics implies that the quantum theory has a Poincaré symmetry, which is realized by a unitary representation of the Poincaré group. This representation can always be decomposed into direct integrals of irreducible representations and the different formulations differ only in how these irreducible representations are realized. I discuss how these representations appear in different formulations of relativistic quantum mechanics and discuss some applications in each of these frameworks.

Euclidean formulation of relativistic quantum mechanics

In this paper, we discuss a formulation of relativistic quantum mechanics that uses model Euclidean Green functions or their generating functional as input. This formalism has a close relation to quantum field theory, but as a theory of linear operators on a Hilbert space, it has the advantages of quantum mechanics. One interesting feature of this approach is that matrix elements of operators in normalizable states on the physical Hilbert space can be calculated directly using the Euclidean Green functions without performing an analytic continuation. The formalism is summarized in this paper. We discuss the motivation, advantages, and difficulties in using this formalism. We discuss how to compute bound states, scattering cross sections, and finite Poincar\'e transformations without using analytic continuation. A toy model is used to demonstrate how matrix elements of ${e}^{\ensuremath{-}\ensuremath{\beta}H}$ in normalizable states can be used to construct sharp-momentum transition-matrix elements.

Alternative formulation of relativistic quantum mechanics

Alternative formulation of relativistic quantum mechanics

Relativistic quantum mechanics and relativistic entanglement in the rest-frame instant form of dynamics

A new formulation of relativistic quantum mechanics is proposed in the framework of the rest-frame instant form of dynamics, where the world-lines of the particles are parametrized in terms of the Fokker-Pryce center of inertia and of Wigner-covariant relative 3-coordinates inside the instantaneous Wigner 3-spaces, and where there is a decoupled (non-covariant and non-local) canonical relativistic center of mass. This approach: (a) allows us to make a consistent quantization in every inertial frame; (b) leads to a description of both bound and scattering states; (c) offers new insights on the relativistic localization problem; (d) leads to a non-relativistic limit with a Hamilton-Jacobi treatment of the Newton center of mass; (e) clarifies non-local aspects (spatial non-separability) of relativistic entanglement connected with Lorentz signature and not present in its non-relativistic treatment.

Canonical quantization, twistors and relativistic wave equations

A formulation of relativistic quantum mechanics in twistor space is studied. The phase-space description of the classical theory leads (via canonical quantization) to all known wave equations describing massless particles. A natural link between the classical and the quantum description is provided by the Wigner function defined on the space of twistors.

Canonical Quantization, Twistors and Relativistic Wave Equations

A formulation of relativistic quantum mechanics in twistor space is studied. The phase-space description of the classical theory leads (via canonical quantization) to all known wave equations describing massless particles. A natural link between the classical and the quantum description is provided by the Wigner function de®ned on the space of twistors.

Quantum Potential in Relativistic Dynamics

The experimental confirmation of nonlocality has renewed interest in Bohm's quantum potential. The construction of quantum potentials for relativistic systems has encountered difficulties which do not arise in a parametrized formulation of relativistic quantum mechanics known as Relativistic Dynamics. The purpose of this paper is to show how to construct a quantum potential in the relativistic domain by deriving a relativistically invariant quantum potential using Relativistic Dynamics. The formalism is applied to three relativistic scalar particle models: a single particle interacting with a scalar potential; N particles interacting with a scalar potential; and a single particle interacting with an electromagnetic 4-vector potential.

Scaling for deuteron structure functions in a relativistic light-front model.

Scaling limits of the structure functions [B.D. Keister, Phys. Rev. C 37, 1765 (1988)], ${W}_{1}$ and ${W}_{2}$, are studied in a relativistic model of the two-nucleon system. The relativistic model is defined by a unitary representation, $U(\ensuremath{\Lambda}, a)$, of the Poincar\'e group which acts on the Hilbert space of two spinless nucleons. The representation is in Dirac's [P.A.M. Dirac, Rev. Mod. Phys. 21, 392 (1949)] light-front formulation of relativistic quantum mechanics and is designed to give the experimental deuteron mass and $n\ensuremath{-}p$ scattering length. A model hadronic current operator that is conserved and covariant with respect to this representation is used to define the structure tensor. This work is the first step in a relativistic extension of the results of Hueber, Gl\"ockle, and Boemelburg [D. Hueber et al. Phys. Rev. C 42, 2342 (1990)]. The nonrelativistic limit of the model is shown to be consistent with the nonrelativistic model of Hueber, Gl\"ockle, and Boemelburg. The relativistic and nonrelativistic scaling limits, for both Bjorken and $y$ scaling are compared. The interpretation of $y$ scaling in the relativistic model is studied critically. The standard interpretation of $y$ scaling requires a soft wave function which is not realized in this model. The scaling limits in both the relativistic and nonrelativistic case are related to probability distributions associated with the target deuteron.

Interpretation of the evolution parameter of the Feynman parametrization of the Dirac equation

The Feynman parametrization of the Dirac equation is considered in order to obtain an indefinite mass formulation of relativistic quantum mechanics. It is shown that the parameter that labels the evolution is related to the proper time. The Stückelberg interpretation of antiparticles naturally arises from the formalism.

The problem of time in parametrized theories

A common feature of reparametrization invariant theories is the difficulty involved in identifying an appropriate evolution parameter and in constructing a Hilbert space on states. Two well known examples of such theories are the relativistic point particle and the canonical formulation of quantum gravity. The strong analogy between them (specially for minisuperspace models) is considered in order to stress the correspondence between the ``localization problem'' and the ``problem of time,'' respectively. A possible solution for the first problem was given by the proper time formulation of relativistic quantum mechanics. Thus, we extrapolate the main outlines of such a formalism to the quantum gravity framework. As a consequence, a proposal to solve the problem of time arises.

A quaternion expression for the quantum mechanical probability and current densities

A simple quaternion expression is proposed for the quantum mechanical four-current of a single particle. It is shown that the equation of continuity is obeyed and the behaviour under a Lorentz transformation is correct. The expression is almost equivalent to that emanating from the Dirac theory. It differs, however, from those of other authors who have considered the quaternion formulation of relativistic quantum mechanics.

Relativistic many-body systems: Evolution-parameter formalism

The complexity of the field theoretic methods used for analyzing relativistic bound state problems has forced researchers to look for simpler computational methods. Simpler methods such as the relativistic harmonic oscillator method employed in the description of extended hadrons have been investigated. They are considered phenomenological, however, because they lack a theoretical basis. A probabilistic basis for these methods is presented here in terms of the four-space formulation of relativistic quantum mechanics (FSF). The single-particle FSF is reviewed and its physical meaning is examined. The many-body single-parameter formalism is then developed. Applications are presented to illustrate use of the many-body formalism and demonstrate the ease with which relativistic bound state problems can be handled. A multiple-parameter formalism is constructed in the Appendix.

The transformation properties of world lines in relativistic quantum mechanical Hamiltonian models

The supposition of the manifest covariance of average trajectory world lines is violated in Hamiltonian formulations of relativistic quantum mechanics. This is due to the nonlinear appearance of particle dynamical variable operators in the Heisenberg picture boosted position, velocity, and momentum operators. The magnitude of this deviation from world line manifest covariance is found to be exceedingly small for a number of common time of flight experiments.