Search Results (6)

In his foundational work on classical and quantum electrodynamics, Stueckelberg introduced an external evolution parameter, $\tau$, in order to overcome difficulties associated with the problem of time in relativity. Stueckelberg particle trajectories are described by the evolution of spacetime events under the monotonic advance of $\tau$, the basis for the Feynman–Stueckelberg interpretation of particle–antiparticle interactions. An event is a solution to $\tau$-parameterized equations of motion, which, under simple conditions, including the elimination of pair processes, can be reparameterized by the proper time of motion. The $4+1$ formalism in general relativity (GR) extends this framework to provide field equations for a $\tau$-dependent local metric ${\gamma }_{\mu \nu }(x,\tau )$ induced by these Stueckelberg trajectories, leading to $\tau$-parameterized geodesic equations in an evolving spacetime. As in standard GR, the linearized theory for weak fields leads to a wave equation for the local metric induced by a given matter source. While previous attempts to solve the wave equation have produced a metric with the expected features, the resulting geodesic equations for a test particle lead to unreasonable trajectories. In this paper, we discuss the difficulties associated with the wave equation and set up the more general ADM-like $4+1$ evolution equations, providing an initial value problem for the metric induced by a given source. As in the familiar $3+1$ formalism, the metric can be found as a perturbation to an exact solution for the metric induced by a known source. Here, we propose a metric, ansatz, with certain expected properties; obtain the source that induces this metric; and use them as the initial conditions in an initial value problem for a general metric posed as a perturbation to the ansatz. We show that the ansatz metric, its associated source, and the geodesic equations for a test particle behave as required for such a model, recovering Newtonian gravitation in the nonrelativistic limit. We then pose the initial value problem to obtain more general solutions as perturbations of the ansatz. Full article

A consistent theory of quantum entanglement requires that constituent single-particle states belong to the same Hilbert space, the coherent eigenstates of a complete set of operators in a given representation, defined with respect to a shared continuous parameterization. Formulating such eigenstates for a single relativistic particle with spin, and applying them to the description of many-body states, presents well-known challenges. In this paper, we review the covariant theory of relativistic spin and entanglement in a framework first proposed by Stueckelberg and developed by Horwitz, Piron, et al. This approach modifies Wigner’s method by introducing an arbitrary timelike unit vector $n^{\mu }$ and then inducing a representation of $SL(2,C)$, based on $p^{\mu }$ rather than on the spacetime momentum. Generalizing this approach, we construct relativistic spin states on an extended phase space $\{(x^{\mu },p^{\mu }),({\zeta }^{\mu },{\pi }^{\mu })\}$, inducing a representation on the momentum ${\pi }^{\mu }$, thus providing a novel dynamical interpretation of the timelike unit vector $n^{\mu }={\pi }^{\mu }/M$. Studying the unitary representations of the Poincaré group on the extended phase space allows us to define basis quantities for quantum states and develop the gauge invariant electromagnetic Hamiltonian in classical and quantum mechanics. We write plane wave solutions for free particles and construct stable singlet states, and relate these to experiments involving temporal interference, analogous to the spatial interference known from double slit experiments. Full article

In a series of recent papers we developed a formulation of general relativity in which spacetime and the dynamics of matter evolve with a Poincaré invariant parameter $\tau$. In this paper, we apply the formalism to derive the metric induced by a ‘static’ event evolving uniformly along its t-axis at the spatial origin $\mathbf{x}=0$. The metric is shown to vary with t and $\tau$, as well as spatial distance r, taking its maximum value for a test particle at the retarded time $\tau =t-r/c$. In the resulting picture, an event localized in space and time produces a metric field similarly localized, where both evolve in $\tau$. We first derive this metric as a solution to the wave equation in linearized field theory, and discuss its limitations by studying the geodesic motion it produces for an evolving event. By then examining this solution in the 4+1 formalism, which poses an initial value problem for the metric under $\tau$-evolution, we clarify these limitations and indicate how they may be overcome in a solution to the full nonlinear field equations. Full article

The 4+1 formalism in general relativity (GR) prescribes field equations for the spacetime metric ${\gamma }_{\mu \nu }\left(x,\tau \right)$ which is local in the spacetime coordinates x and evolves according to an external “worldtime” $\tau$. This formalism extends to GR the Stueckelberg Horwitz Piron (SHP) framework, developed to address the various issues known as the problem of time as they appear in electrodynamics. SHP field theories exhibit a formal 5D symmetry on $(x,\tau )$ that is strategically broken to 4+1 representations of the Lorentz group, resulting in a manifestly covariant canonical formalism describing the $\tau$-evolution of spacetime structures as an initial value problem. Einstein equations for ${\gamma }_{\mu \nu }\left(x,\tau \right)$ are found by constructing a 5D pseudo-manifold (combining 4D geometry and $\tau$-dynamics) and performing the natural foliation under broken 5D symmetry. This paper discusses weak gravitation in the 4+1 formalism, demonstrating the natural decomposition of the field equations into first-order evolution equations for the unconstrained 4D metric, and the propagation of constraints associated with the Bianchi identity. Full article

We propose a field theory for the local metric in Stueckelberg–Horwitz–Piron (SHP) general relativity, a framework in which the evolution of classical four-dimensional (4D) worldlines $x^{\mu }\left(\tau \right)$ ($\mu =0,1,2,3$) is parameterized by an external time $\tau$. Combining insights from SHP electrodynamics and the ADM formalism in general relativity, we generalize the notion of a 4D spacetime $\mathcal{M}$ to a formal manifold ${\mathcal{M}}_5=\mathcal{M}\times R$, representing an admixture of geometry (the diffeomorphism invariance of $\mathcal{M}$) and dynamics (the system evolution of $\mathcal{M}\left(\tau \right)$ with the monotonic advance of $\tau \in R$). Strategically breaking the formal 5D symmetry of a metric $g_{\alpha \beta }(x,\tau )$ ($\alpha ,\beta =0,1,2,3,5$) posed on ${\mathcal{M}}_5$, we obtain ten unconstrained Einstein equations for the $\tau$-evolution of the 4D metric ${\gamma }_{\mu \nu }(x,\tau )$ and five constraints that are to be satisfied by the initial conditions. The resulting theory differs from five-dimensional (5D) gravitation, much as SHP U(1) gauge theory differs from 5D electrodynamics. Full article

In classical Maxwell electrodynamics, charged particles following deterministic trajectories are described by currents that induce fields, mediating interactions with other particles. Statistical methods are used when needed to treat complex particle and/or field configurations. In Stueckelberg–Horwitz–Piron (SHP) electrodynamics, the classical trajectories are traced out dynamically, through the evolution of a 4D spacetime event $x^{\mu }\left(\tau \right)$ as $\tau$ grows monotonically. Stueckelberg proposed to formalize the distinction between coordinate time $x^0=ct$ (measured by laboratory clocks) and chronology $\tau$ (the temporal ordering of event occurrence) in order to describe antiparticles and resolve problems of irreversibility such as grandfather paradoxes. Consequently, in SHP theory, the elementary object is not a particle (a 4D curve in spacetime) but rather an event (a single point along the dynamically evolving curve). Following standard deterministic methods in classical relativistic field theory, one is led to Maxwell-like field equations that are $\tau$ -dependent and sourced by a current that represents a statistical ensemble of instantaneous events distributed along the trajectory. The width $\lambda$ of this distribution defines a correlation time for the interactions and a mass spectrum for the photons emitted by particles. As $\lambda$ becomes very large, the photon mass goes to zero and the field equations become $\tau$ -independent Maxwell’s equations. Maxwell theory thus emerges as an equilibrium limit of SHP, in which $\lambda$ is larger than any other relevant time scale. Thus, statistical mechanics is a fundamental ingredient in SHP electrodynamics, and its insights are required to give meaning to the concept of a particle. Full article