Invariant Evolution Parameter Research Articles

Comparison of Two Competing Theories of 3-Flavor Neutrino Oscillations

Neutrino oscillation observations are used to compare two competing theories of 3-flavor neutrino oscillations. The two theories considered here are the standard model of neutrino oscillations, and parametrized Relativistic Quantum Theory (pRQT). pRQT is a manifestly covariant quantum theory with invariant evolution parameter. Recent data and a neutrino mass model from each theory are used to calculate neutrino masses. The models yield significantly different predictions of neutrino masses.

Parametrized Relativistic Quantum Theory in Curved Spacetime

The purpose of this paper is to present a parametrized relativistic quantum theory (pRQT) in curved spacetime. The formulation of pRQT in curved spacetime is developed and applied to free particle motion in flat and curved spacetime. It provides a theory for calculating probability amplitudes in curved spacetime. Unlike other formulations of parametrized relativistic dynamics (pRD), this work assumes that the metric tensor does not depend on the invariant evolution parameter.

Can particle appearance or disappearance be described by a quantum mechanical theory?

A common justification for replacing quantum mechanics with quantum field theory (QFT) is that the appearance or disappearance of particles cannot be described using quantum mechanics. We show that this justification for QFT is not generally true by presenting a counterexample: parametrized relativistic quantum mechanics (pRQM). We begin by outlining a pioneering formulation of QFT that includes an invariant evolution parameter. The introduction of an invariant evolution parameter helped guide the development of QFT and is a characteristic feature of pRQM. We then present a probabilistic formulation of pRQM that highlights features of the theory that make it suitable for modelling particle stability. Two examples of particle stability are then presented within the context of pRQM to show that a quantum mechanical theory can be applied to particle stability. The examples considered in this paper are exponential particle decay and neutrino oscillations.

Parametrized relativistic dynamical framework for neutrino oscillations

Mass state transitions are a key feature of parametrized relativistic dynamics (PRD). PRD is a manifestly covariant quantum theory with invariant evolution parameter. The theory has been applied to neutrino flavor oscillations between two mass states. It is generalized to transitions between three mass states and applied to the survival of electron neutrinos. The analysis shows that significant differences exist between theoretical results of the conventional model and the PRD model.

Quantization in relativistic classical mechanics: the Stückelberg equation, neutrino oscillation and large-scale structure of the Universe

Based on the Chetaev theorem on stable dynamical trajectories in the presence of dissipative forces, we obtain the generalized condition for stability of relativistic classical Hamiltonian systems (with an invariant evolution parameter) in the form of the Stückelberg equation. As is known, this equation is the basis of a competing paradigm known as parametrized relativistic quantum mechanics (pRQM).It is shown that the energy of dissipative forces, which generate the Chetaev generalized condition of stability, coincides exactly with Bohmian relativistic quantum potential. Within the framework of Bohmian RQM supplemented by the generalized Chetaev theorem and on the basis of the principle of least action for dissipative forces, we show that the squared amplitude of a wave function in the Stückelberg equation is equivalent to the probability density function for the number of particle trajectories, relative to which the velocity and the position of the particle are not hidden parameters.The conditions for reasonableness of trajectory interpretation of pRQM are discussed. Based on analysis of a general formalism for vacuum-flavor mixing of neutrino within the context of the standard and pRQM models we show that the corresponding expressions for the probability of transition from one neutrino flavour to another differ appreciably, but they are experimentally testable: the estimations of absolute value for neutrino mass based on modern experimental data for solar and atmospheric neutrinos show that the pRQM results have a preference. It is noted that the selection criterion of mass solutions relies on proximity between the average size of condensed neutrino clouds, which is described by the Muraki formula (29th ICRC, 2005) and depends on the neutrino mass, and the average size of typical observed void structure (dark matter + hydrogen gas), which plays the role of characteristic dimension of large-scale structure of the Universe.

On the Green-functions of the classical off-shell electrodynamics under the manifestly covariant relativistic dynamics of Stueckelberg

In previous papers derivations of the Green function have been given for 5D off-shell electrodynamics in the framework of the manifestly covariant relativistic dynamics of Stueckelberg (with invariant evolution parameter τ). In this paper, we reconcile these derivations resulting in different explicit forms, and relate our results to the conventional fundamental solutions of linear 5D wave equations published in the mathematical literature. We give physical arguments for the choice of the Green function retarded in the fifth variable τ.

Manifestly Covariant Quantum Theory with Invariant Evolution Parameter in Relativistic Dynamics

Manifestly covariant quantum theory with invariant evolution parameter is a parametrized relativistic dynamical theory. The study of parameterized relativistic dynamics (PRD) helps us understand the consequences of changing key assumptions of quantum field theory (QFT). QFT has been very successful at explaining physical observations and is the basis of the conventional paradigm, which includes the Standard Model of electroweak and strong interactions. Despite its record of success, some phenomena are anomalies that may require a modification of the Standard Model. The anomalies include neutrino oscillations, non-locality, and gravity.

Clifford Algebra Based Polydimensional Relativity and Relativistic Dynamics

Starting from the geometric calculus based on Clifford algebra, the idea that physical quantities are Clifford aggregates (“polyvectors”) is explored. A generalized point particle action (“polyvector action”) is proposed. It is shown that the polyvector action, because of the presence of a scalar (more precisely a pseudoscalar) variable, can be reduced to the well known, unconstrained, Stueckelberg action which involves an invariant evolution parameter. It is pointed out that, starting from a different direction, DeWitt and Rovelli postulated the existence of a clock variable attached to particles which serve as a reference system for identification of spacetime points. The action they postulated is equivalent to the polyvector action. Relativistic dynamics (with an invariant evolution parameter) is thus shown to be based on even stronger theoretical and conceptual foundations than usually believed.

The Mass Operator and Neutrino Oscillations

Recent work in parametrized relativistic quantum theory (PRQT) has shown that oscillations between mass states are predicted by an alternative formulation of relativistic quantum theory that uses an invariant evolution parameter. A PRQT model of flavor transitions is compared to the standard model. The resulting PRQT expression for the probability of survival of an incident neutrino differs significantly from the standard neutrino oscillation model. Neutrino oscillation measurements provide an experimental testing ground for two theories that are based on fundamentally different concepts of temporal evolution.

Fock-Schwinger proper time formalism for p-branes

The usual theory of constrained p-branes is embedded into a larger theory in which there are no constraints. In the latter theory the Fock-Schwinger proper time formalism is extended from point-particles to p-branes, considered as points in an infinite dimensional space M . The quantization appears to be straightforward and elegant. The conventional p-brane states are particular stationary solutions to the functional Schrödinger equation which describes the evolution of a membrane's state with respect to the invariant evolution parameter τ. It is also shown that states of a lower dimensional p-brane can be considered as particular states of a higher dimensional p-brane.

A chiral spin theory in the framework of an invariant evolution parameter formalism

We present a formulation for the construction of first-order equations which describe particles with spin, in the context of a manifestly covariant relativistic theory governed by an invariant evolution parameter; one obtains a consistent quantized formalism dealing with off-shell particles with spin. Our basic requirement is that the second-order equation in the theory is of the Schrödinger–Stueckelberg type, which exhibits features of both the Klein–Gordon and Schrödinger equations. This requirement restricts the structure of the first-order equation, in particular, to a chiral form. One thus obtains, in a natural way, a theory of chiral form for massive particles, which may contain both left and right chiralities, or just one of them. We observe that by iterating the first-order system, we are able to obtain second-order forms containing the transverse and longitudinal momentum relative to a timelike vector tμtμ=−1 used to maintain covariance of the theory. This timelike vector coincides with the one used by Horwitz, Piron, and Reuse to obtain an invariant positive definite space–time scalar product, which permits the construction of an induced representation for states of a particle with spin. We discuss the currents and continuity equations. The transverse and longitudinal aspects of the particle are complementary, and can be treated in a unified manner using a tensor product Hilbert space. Introducing the electromagnetic field we find an equation which gives rise to the correct gyromagnetic ratio, and is fully Hermitian under the proposed scalar product. Finally, we show that the original structure of Dirac’s equation and its solutions is obtained in the highly constrained limit in which pμ is proportional to tμ on mass shell. The chiral nature of the theory is apparent. We define the discrete symmetries of the theory, and find that they are represented by states which are pure left or right handed.

The Dirac-Nambu-Gotop-branes as particular solutions to a generalized, unconstrained theory

The theory of the usual, constrained p-branes is embedded into a larger theory in which there is no constraints. In the latter theory the Fock-Schwinger proper time formalism is extended from point-particles to membranes of arbitrary dimension. For this purpose the tensor calculus in the infinite dimensional membrane space M is developed and an action which is covariant under reparametrizations in M is proposed. The canonical and Hamiltonian formalism is elaborated in detail. The quantization appears to be straightforward and elegant. No problem with unitarity arises. The conventional p-brane states are particular stationary solutions to the functional Schroedinger equation which describes the evolution of a membrane's state with respect to the invariant evolution parameter tau. A tau-dependent solution which corresponds to the wave packet of a null p-brane is found. It is also shown that states of a lower dimensional membrane can be considered as particular states of a higher dimensional membrane.

On Feynman’s approach to the foundations of gauge theory

In 1948, Feynman showed Dyson how the Lorentz force law and homogeneous Maxwell equations could be derived from commutation relations among Euclidean coordinates and velocities, without reference to an action or variational principle. When Dyson published the work in 1990, several authors noted that the derived equations have only Galilean symmetry and so are not actually the Maxwell theory. In particular, Hojman and Shepley proved that the existence of commutation relations is a strong assumption, sufficient to determine the corresponding action, which for Feynman’s derivation is of Newtonian form. In a recent paper, Tanimura generalized Feynman’s derivation to a Lorentz covariant form with scalar evolution parameters, and obtained an expression for the Lorentz force which appears to be consistent with relativistic kinematics and relates the force to the Maxwell field in the usual manner. However, Tanimura’s derivation does not lead to the usual Maxwell theory either, because the force equation depends on a fifth (scalar) electromagnetic potential, and the invariant evolution parameter cannot be consistently identified with the proper time of the particle motion. Moreover, the derivation cannot be made reparameterization invariant; the scalar potential causes violations of the mass-shell constraint which this invariance should guarantee. Tanimura’s derivation is examined in the framework of the proper time method in relativistic mechanics, and the technique of Hojman and Shepley is used to study the unconstrained commutation relations. It is shown that Tanimura’s result then corresponds to the five-dimensional electromagnetic theory previously derived from a Stueckelberg-type quantum theory in which one gauges the invariant parameter in the proper time method. This theory provides the final step in Feynman’s program of deriving the Maxwell theory from commutation relations; the Maxwell theory emerges as the ‘‘correlation limit’’ of a more general gauge theory, in which it is properly contained.

The Zeeman effect for the relativistic bound state

In the context of a relativistic quantum mechanics with invariant evolution parameter, solutions for the relativistic bound state problem have been found, which yield a spectrum for the total mass coinciding with the nonrelativistic Schr\"odinger energy spectrum. These spectra were obtained by choosing an arbitrary spacelike unit vector $n_\mu$ and restricting the support of the eigenfunctions in spacetime to the subspace of the Minkowski measure space, for which $(x_\perp )^2 = [x-(x \cdot n) n ]^2 \geq 0$. In this paper, we examine the Zeeman effect for these bound states, which requires $n_\mu$ to be a dynamical quantity. We recover the usual Zeeman splitting in a manifestly covariant form.

Relativisticp-branes without constraints and their relation to the wiggly extended objects

The invariant evolution parameter τ is often used in the formulation of a so-called unconstrained relativistic quantum theoryof a point particle. Such a theory is very elegant, and contains the usual Klein-Gordon or the Dirac particle as a special case. In the present paper we extend the unconstrained theory to describe a continuous set of point particles forming a string or, in general, a membrane of arbitrary dimension p.The action of this system is not invariant with respect to reparametrizations of the evolution parameter τ and therefore there is no constraints among the dynamical variables, and no need for ghosts in the quantized theory. Then we show that such an unconstrained membrane is equivalent to a wiggly membranewhich has variable tension on its (p+1)-dimensional worldsheet. The equations of motion admit the usual well-known Dirac-Nambu-Goto membranes as particular solutions.

Selection rules for dipole radiation from a relativistic bound state

Recently, in the framework of a relativistic quantum theory with invariant evolution parameter, solutions have been found for the two-body bound state, whose mass spectrum agrees with the nonrelativistic Schrodinger energy spectrum. In this paper, we study the radiative transitions of these states in the dipole approximation and find that the selection rules are identical with those of the usual nonrelativistic theory, expressed in a manifestly covariant form. In addition to the transverse and longitudinal polarizations of the nonrelativistic theory, we find a “scalar” transition, induced by the relative time coordinate, which is of the same type as the longitudinal transition, expressing the Lorentz covariance of the theory.

Evaluating the validity of parametrized relativistic wave equations

We wish to determine the correct partial differential equation(s) for describing a relativistic particle. A physical foundation is presented for using a parametrized wave equation with the general form $$i\frac{{\partial \psi }}{{\partial s}} = K\psi$$ where s is the invariant evolution parameter. Several forms have been proposed for the generatorKof evolution parameter translations. Of the proposed generators, only the generatorK2which is proportional to the inner productPμPμof fourmomentum operators can be derived from first principles, notably a probabilistic basis. Although experimental tests must be made to establish the validity ofK2,we conclude thatK2is the leading theoretical candidate for the form of a generator of evolution parameter translations.

Review of invariant time formulations of relativistic quantum theories

The purpose of this paper is to review relativistic quantum theories with an invariant evolution parameter. Parametrized relativistic quantum theories (PRQT) have appeared under such names as constraint Hamiltonian dynamics, four-space formalism, indefinite mass, micrononcausal quantum theory, parametrized path integral formalism, relativistic dynamics, Schwinger proper time method, stochastic interpretation of quantum mechanics and stochastic quantization. The review focuses on the fundamental concepts underlying the theories. Similarities as well as differences are highlighted, and an extensive bibliography is provided.

Relativistic quantum mechanics and quantum field theory with invariant evolution parameter

Relativistic quantum mechanics with the invariant evolution parameter or historical time τ is reconsidered. The theory is obtained by quantizing the classical theory in which mass is not fixed but enters later as a constant of motion. There are no constraints among the components of the canonical momentumPμ, but the theory is still relativistic and reparametrization invariant. The Feynman propagator of a free particle is obtained in a very elegant way. The theory is then secondly quantized. The Hamiltonian for the τ-evolution is constructed and it is shown that the Heisenberg equations are identical to the field equations. The formalism is manifestly Poincare covariant at each step and refers in general to the states with indefinite mass. On the mass shell it contains the known results of the relativistic field theory of a free charged particle.

Group manifold analysis of the structure of relativistic quantum dynamics

We present a group law, derived as a contraction of the conformal group, from which we obtain by using a canonical procedure a relativistic quantum system with an invariant evolution parameter (the proper time) and where the position operator belongs to the Lie algebra of the group. The restriction of the theory to the mass shell breaks part of the symmetry; of the previous 15 generators, only 10 remain which generate an action of the Poincaré group defining an orbit in the former group manifold. Some comments on the relativistic position operator are also made.