The phenomenon of entanglement is a distinguishing feature of quantum theory and an increasingly important topic in contemporary physics, from the continuing debate of philosophical questions deriving from the papers of Einstein, Podolsky, and Rosen [1] to new technological applications in computation, communications, and radar [2,3,4]. The representation of entanglement as a superposition of states belonging to the direct product space $\mathcal{H}={\mathcal{H}}_1\otimes \cdots \otimes {\mathcal{H}}_n$ necessarily involves a precise and rigorous characterization of each Hilbert space ${\mathcal{H}}_k$. Moreover, a covariant theory of relativistic entanglement requires detailed attention to the basis eigenstates of the commuting operators formed from the generators of a unitary representation of the Poincaré group.

In particular, the treatment of temporal entanglement must be performed in a fully covariant relativistic framework that places space and time on an equal footing in a consistent manner. This can be seen in the comparative analysis of experiments showing spatial and temporal interference: just as a quantum wavefunction split in space by a pair of slits produces spatial interference fringes, a wavefunction split in time can produce temporal interference fringes.

The demonstration by Davisson and Germer [5] of the wavelike nature of particles in the double-slit experiment was a foundational contribution to quantum theory.

As shown schematically in Figure 1, an electron passes through a slit at either $y=y_1$ or $y=y_2$ leading to a spatial superposition producing interference fringes in space. In the temporal double-slit experiment shown schematically in Figure 2, Lindner [6] et. al. showed that an aperture opened and closed at temporal separation can produce interference in time. An ultra-short laser pulse ionizes an atom when $\mathit{E}\left(t\right)={\mathit{E}}_{max}$ causing an electron to be emitted at either $t=t_1$ or $t=t_2$ and accelerated toward the detector. The resulting wavefunctions were shown to interfere coherently, producing temporal fringes.

In analogy to (2), one might be tempted to describe this emission by a temporal superposition showing an energy state producing interference fringes in time. Although wavefunction interference is superficially similar for space and time separations, the nonrelativistic states defined at different times in (3) belong to different Hilbert spaces and cannot be said to form a coherent superposition. To see this, we simply recall that the time evolution of an unperturbed nonrelativistic state is determined by a Hamiltonian-generated unitary transformation

But expressing the proposed temporal superposition in the form there is no obvious way to define the unitary evolution of this state or formulate a nonrelativistic many-body theory. Another temporal interference experiment has been proposed by Palacios et. al. [7] involving a sequential double ionization of helium to produce a pair of entangled electrons in a singlet state. Horwitz and Arshansky [8,9] have discussed the conceptual difficulties associated with these experiments, and presented a rigorous relativistic analysis.

The challenges associated with formulating a consistent relativistic quantum mechanics for a single particle and its application to the description of many-body states are well-known. These difficulties are often reviewed in motivating the transition to quantum field theory (QFT). At minimum, constructing a consistent theory of superposition of states for the experiments of Lindner and Palacios requires a relativistic Hilbert space that treats space and time on the same footing. By definition, this space consists of the coherent eigenstates of a complete set of operators in a given representation defined with respect to a shared continuous parameterization. These operators are formed from the generators of the Poincaré group, with states characterized with respect to an induced representation of spin through the $SU\left(2\right)$ subgroup of the $SL(2,C)$ covering group for $O(3,1)$. As seen above, the time coordinate $x^0$ cannot be used to define the coherent states or parameterize their unitary evolution.

In this paper, we discuss the construction of relativistic Hilbert spaces in a framework developed by Stueckelberg, Horwitz, and Piron (SHP) to overcome the most serious difficulties associated with covariant quantum mechanics. This approach is described at length in a series of books [10,11,12,13] and only more recent references will be given explicitly in the following. Here we provide an updated review of the SHP framework, while introducing a novel representation of spin as a property of extra dimensions, organized as follows. Section 2 presents an short review of the SHP formalism for spinless particles, and briefly introduces a new approach to representation of spin as a characteristic of an extended phase space. The extension of the classical phase space to include extra dimensions is formalized in Section 3 and the electromagnetic interaction in the extended space is explored. Section 4 reviews the construction of unitary representations of the Poincaré group for quantum states, and presents the induced representation of spin in the extra dimensions. Although these developments follow the standard presentation given by Wigner, they are implemented in the extended space, along side usual 4D spacetime, and so lead to new interpretations. Quantum mechanics in the extended phase space is then developed in Section 5. This section generalizes the presentation by Horwitz and Arshansky, leading to an electromagnetic Hamiltonian in the new dimensions, along with solutions. In Section 6 we construct entangled bipartite states and specify the singlet state to a discussion of the Palacios experiment.

The Stueckelberg, Horwitz, Piron (SHP) formalism [10,11,12,13] is a framework for special and general relativity in the classical and quantum realms developed to overcome the difficulties associated with the problem of time. In describing an antiparticle as a particle moving backward in time, Stueckelberg introduced an external parameter $\tau$, similar to the Newtonian time in nonrelativistic physics, so that the spacetime event $x^{\mu }\left(\tau \right)$ can change direction in coordinate time $x^0$ while $\tau$ advances monotonically. In order to permit the event velocity ${\dot{x}}^{\mu }\left(\tau \right)=d{x}^{\mu }/d\tau$ to evolve continuously through the spacelike region between the forward and reverse lightlike regions, the 8D phase space $(x^{\mu },{\dot{x}}^{\mu })$ must be unconstrained, so that the usual constraint ${\dot{x}}^2=1$ is reduced to the status of a conservation law that applies under appropriate interactions. Unlike the convention in SHP theory, we take the flat metric to be ${\eta }_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$ for consistency with familiar descriptions of spin. Horwitz and Piron extended the Stueckelberg framework into a fully covariant canonical theory with many-body interactions, whose applications include solutions to relativistic bound state and scattering problems under central forces, a microdynamical theory of electrodynamics, and more recently, motion in curved spacetime and evolution of the gravitational field [14,15,16].

A free classical event in SHP theory is described by the Lagrangian and equivalent Hamiltonian where $p_{\mu }=\partial L/\partial {\dot{x}}^{\mu }$, with Euler-Lagrange and Hamilton equations to which interactions may be added in much the same manner as in nonrelativistic theory. The Stueckelberg-Schrodinger equation describes a single quantum mechanical event, and may also describe the reduced motion of a pair of interacting events after separation of the free center of mass motion. Solutions to this equation undergo unitary evolution in $\tau$ as

By generalizing the central force problem for particles of spacelike separation $x^2<0$ as

Horwitz and Arshansky found relativistic generalizations of the classical bound state solutions. States with the correct spectrum and multiplicity are obtained by choosing an arbitrary spacelike vector $n^{\mu }$ and solving (8) in a coordinate parameterization that guarantees $x^2\le 0$ and $x_{\perp }^2\le 0$, where $x_{\perp }^{\mu }{n}_{\mu }=0$. These solutions lower the spectrum by breaking the $O(3,1)$ symmetry of the original system to $O(2,1)\subset O(3,1)$. As a result, wavefunctions that transform under the full Lorentz group must be constructed as an induced representation with respect to $n^{\mu }$, leading to generators of $O(3,1)$ in a form that includes $n^{\mu }$ and $\partial /\partial n^{\mu }$. The Lie algebra for these transformations was found and new wavefunctions were then obtained as eigenstates of the commuting operators formed from these generators, providing an induced representation of $O(3,1)$. Although the value of $n^{\mu }$ was chosen arbitrarily, Land and Horwitz showed that it shifts under radiative transitions. By treating $n^{\mu }\left(\tau \right)$ as a dynamical quantity and extending the classical phase space to include $(n^{\mu },{\dot{n}}^{\mu })$, the gauge invariant Hamiltonian acquires an interaction term in which the electromagnetic field is coupled in a natural way to the generators of the induced $O(3,1)$. Thus, the Zeeman and Stark effects were found by Land and Horwitz in manifestly covariant form as a first order perturbation, reproducing the expected lifting of energy degeneracy.

Equations (6) to (8) describe a spinless particle. In standard relativistic quantum theory [17] massive particles may be characterized in their rest frame, where $p=(p^0,\mathbf{0})$. Spin is introduced, following Wigner, by studying the unitary representations of the Poincaré group, involving an induced representation of $O(3,1)$ with respect to the momentum $p^{\mu }$. But in the SHP formalism $p^2$ is unconstrained and may vanish or be negative for real particles, precluding the identification of a rest frame. Therefore, Horwitz, Piron, and Reuse modified Wigner’s method by specifying a position state and inducing a representation of $SL(2,C)$ with respect to an arbitrarily chosen timelike unit vector designated $n^{\mu }$. The resulting spin operators act in the spacelike hypersurface normal to $n^{\mu }$, characterizing the chosen $SU\left(2\right)$ subgroup of $SL(2,C)$. Because a singlet state belongs to an irreducible representation of the direct product of identical spin-1/2 states in the same representation, the entanglement requires that the individual wavefunctions be defined at the same value of $\tau$ and with the same value of $n^{\mu }$.

In this paper, we will treat $n^{\mu }\left(\tau \right)$ as a dynamical quantity, as was done for the induced representation of $O(3,1)$ in the analysis of the Zeeman and Stark effects for relativistic bound states. More specifically, we introduce an auxiliary momentum ${\pi }^{\mu }=M{n}^{\mu }$ conjugate to a position ${\zeta }^{\mu }$ and study the classical and quantum dynamics on the extended phase space $\{\left(x^{\mu },p^{\nu }\right),\left({\zeta }^{\mu },{\pi }^{\nu }\right)\}$. Applying Wigner’s method, we construct a unitary representation of the Poincaré group for quantum states, leading to a representation of spin induced on ${\pi }^{\mu }$ rather than $p^{\mu }$. Since a superposition can only be formed from coherent states in the same Hilbert space, characterized by a representation of spin, a singlet state formed from pair of free particles must have identical values of ${\pi }_i^{\mu }$, even as this extended momentum evolves dynamically. In the following, we explore the implications of this requirement.

We write the classical phase space and introduce the fields $A^{\mu }\left(x,\zeta \right)$ and ${\chi }^{\mu }\left(x,\zeta \right)$ to write the classical Lagrangian for which the action is invariant under the gauge transformations

Variation with respect to $x^{\mu }$ and ${\zeta }^{\mu }$ leads to the Lorentz force where are the field strengths.

We may define the canonical momenta and perform a Legendre transformation on the classical Lagrangian (12) to obtain the classical gauge invariant scalar Hamiltonian

In Section 5 we will see how the quantum Hamiltonian acquires a spin interaction term. As discussed in Section 2, a pair of quantum particles must have the same value of ${\pi }^{\mu }$ in order to enter a singlet state. To get a sense of how this quantity behaves in an external field, we consider two classical particles with initial conditions and write the Hamilton equations

In the case that $A^{\mu }=A^{\mu }\left(x\right)$ and ${\chi }^{\mu }={\chi }^{\mu }\left(\zeta \right)$ we have and note that by (16), $H^{\mu \nu }=0$ in this case. Since the forces and initial conditions are the same for each particle, they will evolve without divergence in their values of ${\pi }^{\mu }$. We note that transforming the interaction terms in the Lagrangian (12) as and introducing kinetic terms for the fields, we may write a field Lagrangian as

Variation of this Lagrangian with respect to the potentials $A^{\mu }$ and ${\chi }^{\mu }$ leads to Maxwell-like field equations for which the sources are $J_{\mu }^x\left(x\right)$ and $J_{\mu }^{\zeta }\left(\zeta \right)$, suggesting that the field solutions will be of the type $A^{\mu }=A^{\mu }\left(x\right)$ and ${\chi }^{\mu }={\chi }^{\mu }\left(\zeta \right)$

Nevertheless, if either potential depends on both x and $\zeta$ then we have so that the forces on ${\pi }_1$ and ${\pi }_2$ may differ at spacetime separated locations $x_1$ and $x_2$, leading to a relative change in their values. In the quantum context, divergent evolution in ${\pi }^{\mu }$ may indicate a mechanism of decoherence.

Basis quantities for quantum states are found as eigenstates of the commuting operators formed from the generators of a unitary representation of the Poincaré group for $\Lambda \in O(3,1)$ and translation a. The spacetime Lorentz transformations $x^{\prime \mu }={\Lambda }_{\nu }^{\mu }{x}^{\nu }$ and translations $x^{\mu }⟶x^{\prime \mu }=x^{\mu }+a^{\mu }$ have generators from which one may write unitary transformations where $P_{\mu }$ and $L_{\mu \nu }$ satisfy the well-known Lie algebra leading to spinor representations.

We treat the extra dimensions as an inequivalent representation of the Poincaré group, and so in the extended phase space, the transformations of ${\zeta }^{\mu }$ and ${\pi }^{\mu }$ are generated by defined in analogy to (28) but independent of $X^{\mu }$ and $P_{\mu }$. With these, we may consider extended unitary transformations of the quantum states of the type whose generators satisfy the Poincaré algebra. Although the spacetime 4-momentum is unconstrained and $p^{\mu }{p}_{\mu }$ is not necessarily non-negative definite, we now stipulate that the momentum ${\pi }^{\mu }$ must satisfy the mass-shell constraint ${\pi }^2=M^2$.