Spin and Entanglement in General Relativity

In a previous paper, we have shown that the classical and quantum relativistic dynamics of the Stueckelberg-Horwitz-Piron [SHP] theory can be embedded in general relativity (GR). We briefly review the SHP theory here, and, in particular, the formulation of the theory of spin in the framework of relativistic quantum theory. We show here how the quantum theory of relativistic spin can be embedded, using a theorem of Abraham, Marsden and Ratiu, and also explicit derivation, into the framework of GR by constructing a local induced representation. We then discuss entanglement for the spins of a two body system.


Introduction
The relativistic canonical Hamiltonian dynamics of Stueckelberg, Horwitz and Piron (SHP) [1] with scalar potential and gauge field interactions for single and many body theory can, by local coordinate transformation, be embedded into the framework of general relativity (GR) [2] [3] (to be called SHPGR). We first review the structure of this embedding and then discuss the introduction of spin, angular momentum and entanglement in this framework.
The theory was originally formulated for a single event (associated by its world line with a particle) by Stueckelberg in 1941[4][5] [6]. Stueckelberg envisaged the motion of an event along a world line in spacetime that can curve, due to interaction, and turn to flow backward in time, resulting in the phenomenon of pair annihilation in classical dynamics. Since this world line is not single valued in t, Stueckelberg parametrized it by an invariant monotonic parameter τ .
The theory was generalized by Horwitz and Piron in 1973 [7] to be applicable to many body systems by assuming that the parameter τ is universal (as for Newtonian time [8] [9]), enabling them to solve the two body central potential problem classically. A solution for the quantum case was found later by Arshanksy and Horwitz [10] [11] [12], both for bound states and scattering theory, for interaction represented by a central potential function.
Performing local coordinate transformations from the flat tangent space, for which we label coordinates and momenta ξ µ and π µ to coordinates on a general manifold, which we label x µ along with the corresponding transformation of the momenta (on the cotangent space of the original Minkowski manifold), which we label p µ , one obtains the SHP theory in the curved space of general coordinates and momenta with a canonical Hamilton-Lagrange (symplectic) [2] [3] structure. We shall refer to this generalization as SHPGR.
The invariance of the Poisson bracket* under local coordinate transformations provides a basis for the canonical quantization of the theory, for which the evolution under τ is determined by the Lorentz covariant form of the Stueckelberg-Schrödinger equation [1] [4].
This method was applied also to the many body case [13], for which the SHP Hamiltonian is a sum of terms quadratic in four momentum with a many body potential term. Each particle moves locally tangentially infinitesimally close to a flat Minkowski space, the tangent space of the general manifold of motions at that point; these local motions can then be mapped at each point x µ by coordinate transformation into the curvilinear coordinates in that neighborhood reflecting the curvature induced by the Einstein equations.
We assume a τ independent background gravitational field; the local coordinate transformations from the flat Minkowski space to the curved space are taken to be independent of τ , consistently with an energy momentum tensor that is τ independent. In a more dynamical setting, when the energy momentum tensor depends on τ , the spacetime evolves nontrivially; the transformations from the local Minkowski coordinates to the curved space coordinates then depend on τ ; this situtation was discussed in [2]and by Land [14], but will not be treated here.
The theory of intrinsic angular momentum of a particle in the framework of relativistic quantum theory for special relativity was worked out by Horwitz and Arshansky [12] (see also [1]) following the method of induced representations of Wigner [15] but with an inducing timelike vector n µ , transforming with the Lorentz group, independent of momentum. The necessity for this is that the induced representation of the angular momentum on the wave function depends on the inducing vector. When computing the expectation value of ξ µ , represented (in momentum space) by i ∂ ∂π µ in the relativistic quantum theory, as discussed below,this derivative would destroy the unitarity of a representation induced on π µ (as done by Wigner [15]). This expectation value would then not transform as a vector under the Lorentz group.
The generators of the Lorentz group acting both on {ξ µ } and {n µ }, in the relativistic quantum theory [1] [12], are where indices are raised and lowered by the Minkowski metric η µν = (−1, +1, +1, +1). Under the action of the group generated by this set of operators, M µν is a Lorentz tensor. By the equivalence principal, the Lorentz group acts in the locally flat freely falling frame (tangent space). It is therefore essential for the embedding of the special relativistic theory into GR that the set of local generators transform under the local embedding diffeomorphisms as covariant tensors. We shall see that this follows by an isomorphism theorem of Abraham, Marsden and Ratui [16] and by explicit calculation. In the following we briefly review the SHP theory and its imbedding into the curved space of GR [2][3](SHPGR) for a single spinless particle (the many body case in Minkowski * We provide an explicit proof of the invariance of the Poisson bracket in connection with our discussion of the spin in a later section. space was treated in ref. [13]) and then turn to discuss the representations of a particle with spin in GR. We then treat the manifestation of long range correlations in GR resulting in spin entanglement .

Classical Theory for a Single Particle in an External Potential
We write the SHP Hamiltonian [1] as where η µν is the flat Minkowski metric (− + ++) and π µ , ξ µ are the spacetime canonical momenta and coordinates in the local tangent space, following Einstein's use of the equivalence principle. The existence of a potential term (which may be a Lorentz scalar), representing nongravitational forces, implies that the "free fall" condition is replaced by a local dynamics carried along by the free falling system (an additional force acting on the particle within the "elevator" according to the coordinates in the tangent space).
The canonical equations arė where the dot here indicates d dτ , with τ the invariant universal "world time". Sincė or π ν = η νµ Mξ µ , the Hamiltonian can then be written as* We now transform the local coordinates (contravariantly) according to the diffeomorphism to relate small changes in ξ to corresponding small changes in the coordinates x on the curved space, so thatξ µ = ∂ξ µ ∂x λẋ λ .
(2.6) * Note that, as clear from (2.3), thatξ 0 = dt dτ has a sign opposite to π 0 which lies in the cotangent space of the manifold. The energy of the particle for a normal time-like particle should be positive (negative energy would correspond to an antiparticle [4] [17]). The physical momenta and energy therefore correspond to the mapping π µ = η µν π µ . back to the tangent space.
The Hamiltonian then becomes where V (x) is the potential at the point x corresponding to the point ξ in the tangent space that we have been considering, and Since V (x) has dimension of mass, one can think of this function as a scalar mass field, inducing forces acting in the local tangent space at each point. It may play the role of "dark energy" [18] [19]. The corresponding Lagrangian in the curved space is then In the locally flat coordinates in the neighborhood of x µ , the symplectic structure of Hamiltonian mechanics ( e.g. da Silva [20] ) implies that the momentum π µ *, lying in the cotangent space of the manifold {ξ µ }, transforms covariantly under the local transformation (1.5), i.e., as does ∂ ∂ξ µ , so that we may define This definition is consistent with the transformation properties of the momentum defined by the Lagrangian (2.9): (2.12) The second factor in the definition (2.8) of g µν in (2.12) acts onẋ ν ; with (2.6) we then have (as in (2.10)) (2.13) As we have remarked above for the locally flat space, the physical energy and momenta are given, according to the mapping, * We shall call the quantity π µ in the cotangent space a canonical momentum, although it must be understood that its map back to the tangent space π µ corresponds to the actual physically measureable energy momentum. back to the tangent space of the manifold, which also follows directly from the local coordinate transformation of (2.3).
It is therefore evident from (2.14) thaṫ We see thatṗ µ , which should be interpreted as the force acting on the particle, is proportional to the acceleration along the orbit of motion. From the coordinate transformation ofξ The procedure that we have carried out here provides a canonical dynamical structure for the motions in the curvilinear coordinates. The Poisson bracket remains valid for the coordinates {x, p} [2], as we show explicitly below, so that One also finds that so that p µ acts infinitesimally as the generator of translation along the coordinate curves (in a geodesically complete manifold these may be taken to be geodesic curves) and so that x µ is the generator of translations in p µ . The Poisson bracket structure of the mapping of the SHP theory into GR gives us a basis for quantization, following Dirac [17]. Properties of this quantum formulation were discussed in [2]. We now turn to the treatment of a particle with spin in this framework.

Quantum Theory
Our discussion so far has been primarily classical. The Poisson brackets, as mentioned above, provide a basis [2] for the corresponding qauntum theory on the curved space as discussed in SHPGR, for which the canonical commutation relations are and therefore and The scalar product for wave functions in the Hilbert space The operator −ih∂/∂x µ is not Hermitian in this scalar product. However, is self-adjoint (somewhat in analogy to the Newton-Wigner position operator [21] in momentum space in Klein-Gordon theory), and satisfies the canonical commutation relations (3.1). In coordinate space, the operator p µ in (3.6) is used everywhere in our analysis except where specified, as we shall discuss, in what we shall call the Foldy-Wouthuysen [22] representation, where it takes on the form −ih∂/∂x µ . The states {ψ τ (x)} satisfy the Schrödinger-Stueckelberg (see also Schwinger [23] and DeWitt [24]) equation The spin of a particle is an essentially quantum mechanical property. In the nonrelativistic quantum theory, the lowest non-trivial representation of the rotation group corresponds to the spin degrees of freedom of the particle. However, for a particle decribed in the framework of special relativity, the Lorentz group O(3, 1) or its covering SL(2, C) acts on the wave function. Wigner [15] showed that representations of SU (2) in the relativistic case, in particular, for spin 1/2, can be constructed by starting with a particle at rest so that its four-momentum has just one non-zero component, π 0 = m, where m is the mass of the particle (assumed nonzero; the zero mass case, such as for the photon,must be treated separately). In the four dimensional Minkowsi space this vector lies along the time axis. The elements of the Lorentz group that leave this vector invariant lie in the subgroup SO(3) or its covering SU (2), and therefore provide a representation of spin in that frame. Under a Lorentz boost the vector (π 0 , 0, 0, 0) may move to a general timelike four vector π µ , but the action of the group remains the same about this new vector, i.e., it remains in SU (2). This so-called induced representation is then identified by Wigner with the intrinsic spin of the particle.
In the relativistic dynamics of SHP, however, which provides a quantum mechanical Hilbert space for the description of quantum states, this construction is not adequate [12] since the wave functions would transform under a unitary transformation that depends explicitly on the momentum of the state (in momentum representation). The expectation value of the operator ξ µ = ih ∂ ∂π µ would then not be covariant. This problem was solved in SHP by Arshansky and Horwitz [12] (see also [1]), who constucted an induced representation on a time-like vector n µ instead of on the fourmomentum. For this vector to transform under the Lorentz group, the generators must have the form We start, as for the method of Wigner [15], in a frame for which n µ = (1, 0, 0, 0). The subgroup of the Lorentz group O(3, 1) (SL(2, C)) which leaves this n µ invariant is SO(3) (SU (2)). Under a general Lorentz transformation Λ µ ν , n µ takes on general timelike values in the upper light cone; as shown in [1] [12], the wave function then transforms at any point n on the orbit, as where σ, σ ′ = ±1 are the spin indices and D σ ′ ,σ (Λ, n) is the Wigner D-function [15] D((Λ, n) = L −1 (n)ΛL(Λ −1 n), (3.11) with L(n) the transformation bringing (1, 0, 0, 0) to n µ . Since the transformatiion on the wave function (in ξ or π representation) is indepedent of π, the expectation value of x µ is covariant.In discussing the two body case later, we remark that this formulation may be applied to any spin (constructed with Clebsch-Gordan products in the spin space [25]). We now wish to imbed this structure into the manifold of GR.

Spin of a Particle in SHPGR
The rather straighforward method we have descrihed above for achieving representations of spin in the framwework of the SHP theory is not adequate for general relativity. Although the orbital part of M µν can be assumed to transform under local diffeomorphims for ξ µ in a small local region, we must explicitly assume that the vector n µ , whose properties are at our disposal, also has this local covariance property.
There is a theorem stated in Abraham, Marsden and Ratiu [16] asserting that: Under the C r map ϕ, for X, Y elements of an algebra on an r-manifold, X → X ′ and Y → Y ′ , f a function on the manifold, which establishes an algebraic isomorphism. We give an explicit proof for our construction in the following. In our case ϕ : where N is the mapping, defined below, of n into the manifold. We define the angular momentum in a small neighborhood so that the variables ξ can be considered to be very small. Under the local diffeomorphism The Lorentz algebra therefore remains under these local diffeomorphisms, and we can follow the construction of the induced representation for spin, defined in the algera of SU (2), just as in the flat Minkowski space.
In the following, we show explicitly how the theorem of Abraham, Marsden and Ratiu works in our case for the algebra of commutation relations obeyed by where Mapping (4.4) into the manifold by local diffeomorphims, remains of the same form, where we assume that the vectors n µ and m µ transform under local diffeomorphism in the same way (the properties of n µ are at our disposal) as the coordinates and momenta, i.e., and dπ µ = ∂x λ ∂ξ µ dp λ . We also define dn µ = ∂ξ µ ∂x λ dN λ (4.9) and dm µ = ∂x λ ∂ξ µ dM λ . (4.10) To show show the consistency of these definitions, we define a Poisson bracket on the variables {ξ, n, π, m}. Generally the motivation for defining the Poisson bracket is in the construction of the τ derivative of a function F (ξ, π) with the use of Hamilton's equations to show that this derivative is given by the Poisson bracket of F with the Hamiltonian K.In our case, K generally does depend on n, but not, in our discussions so far, on m, and thereforeṅ would be zero. In any case, we may define a Poisson bracket For brevity, let us define {ξ µ , n µ } = ζ ν and {π µ , m µ } = η µ , and the images With the transformation laws we see that for (summing over both sets of variables) (4.16) The Poisson brackets of x, p therefore remain of the form of ξ, π and of N, M , the same form as n, m. The N, M commutator. as for n, m imply unbounded spectrum for N and M . We shall be concerned with the constraint, in N representation, that (on the spectrum) N µ N µ = −1, invariant under the Lorentz algebra generated by (4.6), enabling us to proceed with the program described in (3.9) − (3.11), with n µ replaced by N µ (also commuting with all dynamical observables).
We emphasize that the generators we have constructed cannot be simply integrated to form a group on the curved spacetime. However, the spin representations we construct are entirely within the local infinitesimal algebra, sufficient to define spin as a local intrinsic structure of the particle (with generators satisfyig the Pauli spin algebra). Constructing higher representations from direct product with Clebsch-Gordan coefficients (coordinate independent) can also be done in the same small neighborhood, as can the composition of spins of different particles [25], as we shall do in our discussion of entanglement.
Independently of the coordinate system, N µ transforms as a vector under the Lorentz algebra. We may then, as for the flat space, construct a representation of SU (2) as the stability subalgebra of N µ in SL(2, C) . For the definition of the Hilbert space, we remark that there are two fundamental representations of SL(2, C) which are inequivalent [26]. Multiplication by, i.e., the operator σ · p of a two dimensional spinor representing one of these results in an object transforming like the second representation. Such an operator could be expected to occur in a dynamical theory, and therefore the state of lowest dimension in spinor indices of a physical system should contain both representations (for the rotation subgroup, both of the fundamental representations yield the same SU (2) matrices up to a unitary transformation). the defining relation for the fundamental SL(2, C) matrices is (on he spectrum of the operator N µ ) where σ µ = (σ 0 , σ); σ 0 is the unit 2 × 2 matrix, and σ are the Pauli matrices. Since the determinant of σ µ N µ is the Lorentz invariant N 0 2 − N 2 , and the determinant of Λ is unity in SL(2, C), the transformation represented on the left hand side of (4.17) must induce a Lorentz transformation on N µ The inequivalent second fundamental representation may be constructed by using this defining relation with σ µ replaced by σ µ ≡ (σ 0 , −σ). For every Lorentz transformation Λ acting on N µ , this defines an SL(2, C) matrix Λ (we use the same symbol for the Lorentz transformation on a four-vector as for the corresponding SL(2, C) matrix acting on the 2-spinors). Since, then, both fundamental representations of SL(2, C) should occur in the general quantum wave function representing the state of the system, the norm in each N -sector of the Hilbert space must be defined as 18) whereψ N transforms with the first SL(2, C) andφ N with the second. From the construction of the little group it follows that L(n)ψ N transforms with Λ, and L(n)φ n transforms with Λ. Making this replacement in (4.18), and using the fact, obtained from the defining relation (4.17), that where γ · N ≡ γ µ N µ (for which (γ · N ) 2 = −1), and the matrices γ µ are the Dirac matrices as defined in the book of Bjorken and Drell [27]. Here, the four-spinor ψ N (x) is defined by 29) and the sign ∓ corresponds to N µ in the positive or negative light cone. The wave function then transforms as and S(Λ) is a (nonunitary) transformation generated infinitesimally, as in the standard Dirac theory (see, for example, Bjorken and Drell [27]), by Σ µν ≡ i 4 [γ µ , γ ν ]. However, in our formulation, in the dynamics of SHP, we do not obtain the Dirac equation as a factorization of the Klein Gordon equation, but rather a second order equation with Hermition interaction between spin and electromagnetism.
The Dirac operator γ · p is not Hermitian ( with p the Hermitian operator defined in (4.27)) in the (invariant) scalar product associated with the norm (4.20). To construct a Hamiltonian for the evolution of the wave function consider the Hermitian and anti-Hermitian parts of γ · p: where K µ = Σ µν n ν , and we have introduced the factor γ 5 = iγ 0 γ 1 γ 2 γ 3 , which anticommutes with each γ µ and has square −1 so that K T is Hermitian and commutes with the Hermitian K L . Since and we may consider to pose an eigenvalue problem analogous to the second order mass eigenvalue condition for the free Dirac equation (the Klein Gordon condition). For the Stueckelberg equation of evolution corresponding to the free particle, we may therefore take In the presence of electromagnetic interaction, gauge invariance under a spacetime dependent gauge transformation, the expressions for K T and K L given in (4.31), in gauge covariant form, then imply, in place of (4.35), (4.37) and the γ µ N are defined below in (4.41). The expression (4.36) is quite similar to that of the second order Dirac operator; it is, however, Hermitian and has no direct electric coupling to the electromagnetic field in the special frame for which N µ = (1, 0, 0, 0) in the minimal coupling model we have given here (note that in his calculation of the anomalous magnetic moment Schwinger [23] puts the electric field to zero; a non-zero electric field would lead to a non-Hermitian term in the standard Dirac propagator, the inverse of the Klein-Gordon square of the interacting Dirac equation). The matrices Σ µν N are, in fact, a relativistically covariant form of the Pauli matrices.
To see this, we note that the quantities K µ and Σ µν N satisfy the commutation relations (4.38) Since K µ N µ = N µ Σ µν N = 0, there are only three independent K µ and three Σ µν N . The matrices Σ µν N are a covariant form of the Pauli matrices, and the last of (4.38) is the Lie algebra of SU (2) in the spacelike surface orthogonal to N µ . The three independent K µ correspond to the non-compact part of the algebra which, along with the Σ µν N provide a representation of the Lie algebra of the full Lorentz group.
In our construction of the Dirac matrices by studying the spin on the manifold of general relativity in an induced representation, we may see a relation with the work of Fock and Ivanenko [28], discussing the geometrical meaning of the Dirac matrices, and their reference to the "vierbeins" of Ricci [29].
The covariance of this representation follows from In the special frame for which N µ = (1, 0, 0, 0)), Σ i,j N become the Pauli matrices 1 2 σ k with (i, j, k) cyclic, and Σ 0j N = 0. In this frame there is no direct electric interaction with the spin in the minimal coupling model (4.37). We remark that there is, however, a natural spin coupling which becomes pure electric in the special frame, given by It is a simple exercise to show that the value of this commutator reduces to ∓eσ · E in the special frame for which N µ (1.0, 0, 0) this operator is Hermitian and would correspond to an electric dipole interaction with the spin. The matrices γ µ N = γ λ π λµ , (4.41) where the projection π λµ = g λµ + N λ N µ , (4.42) appearing in (4.38), plays an important role in the description of the dynamics in the induced representation. In (4.36), the existence of projections on each index in the spin coupling term implies that F µν can be replaced by F N µν in this term, a tensor projected into the foliation subspace.

Entanglement
For two particles, for example, with spin 1/2, one can construct a singlet state with the properties of the Einstein-Podolsky-Rosen (EPR) [30] construction. After the initial state is formed, say, by ionization of He, the two particle state may evolve coherently along geodesic curves,* with motion generated coherently by the free Hamiltonian. We can see this by multiplying the two body wave function by the unitary map for ǫ 1 , ǫ 2 a set of infinitesimal shifts along selected geodesic curves. Then, where x 1 ′ , x 2 ′ are points along a classical geodesic curve, x 1 , x 2 translated separately by the sequence of maps in (5.1) corresponding to the free motion of the wave packets in a two body system with given initial conditions.
In order that the the vector N for the induced representation be an intrinsic property of the wave function, we may consider that this vector is parallel transported throughout on its support (we have assumed the manifold geodesically complete), and the spin representations remain as in the initial state, preserving their correlation. The two body wave function, defined at a given N µ therefore maintains correlations with respect to this vector.
If we measure spin in a singlet state in the direction n, for example (any direction may be chosen), and find a particle with spin oriented along this direction at some spacetime point A, then at some point B along a geodesic curve conncted to A, we are sure to find the spin in the −n direction providing us with an EPR [30] situation.
In this way, two particles initially in a spin zero state in the spin space, with wave packets moving coherently along geodesic curves, should maintain the EPR correlations in spacetime, where correlations are maintained for small relative time differences, as discussed in [32].
It is interesting to consider the possible effect of a gravitational field on this two body spin correlation. Since the identity of N µ on the support of the wave function (generally unbounded) can be defined through parallel transport, transport from A to B could, if there is a closed geodesic curve in the support of the wave function, close around a curve containing a gravitational field (such as in a wave function with support around a black hole as discussed in [33]). The point B may be the same spacetime point as A, but the vector N µ would not coincide in spacetime with initial vector at A before transport, with a difference proportional to the integral over the field enclosed [34]. Nevertheless, the EPR correlation must remain. In spacetime, the "spin up" and "spin down" outcomes will not * Deng et al [31], for example, have observed quantum correlations over a distance of about 150 × 10 6 kilometers. equal and opposite, but will differ by some angle (as in the usual EPR when we measure at two different angles), leading to possibly observable effects particularly when we look for EPR correlations in the presence of strong gravitational fields.

Conclusions
We have shown that the method of induced representations developed for relativistic quantum mechanics on the Minkowski manifold can be applied as well to the construction of an induced representation for the spin of a particle on the manifold of general relativity. We have cited the theorem of Abraham, Marsden and Ratiu which assures the existence of an isomporphic mapping, and shown explicitly, using the invariance of Poisson brackets, that the algebraic structure of the induced representation for spin can be mapped into the quantum theory on the manifold of general relativity. As in the relativistic quantum theory on the Minkowski manifold, the wave function is labelled by a timelike vector, which we have called N µ , which is the stability vector for the algebra of the little group that constitutes the spin.
The association of the vector N µ with the wave function on all of its support (which is generally unbounded) can be constructively defined by parallel transport.
In the presence of a gravitational field of non-vanishing curvature, however, parallel transport of a vector around a closed geodesic curve in spacetime, if such a configuration occurs in the support of the wave function, would bring us to a different vector after the circuit, and therefore the correlated spin of the second particle would be aligned along a direction different from that of the orientation of the measurement of the first particle. The possibility of observable effects of this phenomenon is under study.