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Physical Review A

Manifestly Lorentz covariant formulation of the Einstein-Podolsky-Rosen problem using the Tomonaga-Schwinger formalism

Partha Ghose, Dipankar Home

Published Jun 1, 1991• 7 citations
Publisher

Abstract

We show that the Tomonaga-Schwinger formalism in quantum field theory provides a manifestly Lorentz covariant description of an Einstein-Podolsky-Rosen (EPR) correlated state defined on a curved-space-like surface. This avoids the notion of a universal time and clearly demarcates between the completion of the measuring process on a member of an EPR pair and its nonlocal effect on the state of its partner. Being reciprocal and restricted to a spacelike surface, the latter is deterministic but not causal and is consistent with Lorentz invariance.

DOI
10.1103/physreva.43.6382

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We show that the Tomonaga-Schwinger formalism in quantum field theory provides a manifestly Lorentz covariant description of an Einstein-Podolsky-Rosen (EPR) correlated state defined on a curved-space-like surface. This avoids the notion of a universal time and clearly demarcates between the completion of the measuring process on a member of an EPR pair and its nonlocal effect on the state of its partner. Being reciprocal and restricted to a spacelike surface, the latter is deterministic but not causal and is consistent with Lorentz invariance.
The Einstein–Podolsky–Rosen (EPR) argument was originally posed as a challenge to the completeness of quantum mechanics, rather than its empirical adequacy. While subsequent developments—most notably Bell’s theorem and experimental tests of nonlocal correlations— have constrained classes of hidden-variable theories, they have not resolved the underlying on- tological question raised by EPR: whether the quantum state exhausts physical reality. In this paper, we revisit the EPR criterion of completeness from the perspective of a temporally grounded physical framework, the Temporal Inertia Framework (TIF). We argue that the orig- inal EPR dilemma rests on an implicit assumption that physical reality must be specified on spacelike hypersurfaces, an assumption shared by both EPR and Bohr. By relaxing this assump- tion and allowing physical correlations to arise from shared temporal structure rather than local spatial variables or superluminal influence, one may satisfy the EPR realism criterion without violating relativistic causality or Bell-type constraints. Quantum mechanics is thus interpreted as operationally complete but ontologically incomplete, with quantum indeterminacy emerging from dynamical interactions with a proposed underlying temporal structure.
Abstract This chapter develops the theory of measurements in relativistic quantum mechanics. The presentation is based on the idea to consider the state of a relativistic quantum system as a functional on the set of space-like hypersurfaces in Minkowski space. This idea leads to a manifest covariant time-evolution equation, known as Schwinger–Tomonaga equation, and allows a relativistic formulation of the projection postulate of quantum measurement theory. This postulate is used to study a number of applications to local and non-local measurements, to the erasing of local information, and to the restrictions on the measurability of non-local quantities imposed by the causality principle. The relativistic projection postulate further enables the discussion of many important problems from a unified perspective, such as EPR-type experiments, measurements of Bell state operators, exchange measurements, and quantum teleportation.
Einstein, Podolsky, and Rosen pointed out that the quantum-mechanical description of “physical reality” implies an unphysical, instantaneous action between distant measurements. To avoid such an action at a distance, these three concluded that quantum mechanics had to be incomplete. However, its extensions involving additional “hidden variables,” allowing for the recovery of determinism and locality, have been disproved experimentally (Bell’s theorem). In this paper, an opposite solution of the paradox is presented, based on the greater indeterminism of the modern quantum field theory (QFT) description of particle physics, which prevents the preparation of any state having a definite number of particles. The resulting uncertainty in photon radiation has interesting consequences in quantum information theory (e.g., cryptography and teleportation). Moreover, since it allows for fewer elements of Einstein-Podolsky-Rosen (EPR) physical reality than the old non-relativistic quantum mechanics, QFT satisfies the EPR condition of completeness without the need for hidden variables. The residual physical reality never violates locality; thus, the unique objective proof of “quantum nonlocality” is removed in an interpretation-independent way. At the same time, the supposed nonlocality of the EPR correlations turns out to be a problem in interpretation of the measurement process. If we do not rely on hidden variables or new physics beyond QFT, the viable interpretation is a minimal statistical one, which preserves locality and Lorentz symmetry.
Quantum mechanics is formulated on a Hilbert space that is assumed to be separable. However, there seems to be no clear reason justifying this assumption. Does it have physical implications? We answer in the positive by proposing a test that witnesses the non-separability of the Hilbert space, at the expense of requiring measurements with uncountably many outcomes. In the search for a less elusive manifestation of non-separability, we consider the original Einstein-Podolsky-Rosen (EPR) state as a candidate for possessing nonlocal correlations stronger than any state in a separable Hilbert space. Nevertheless, we show that, under mild assumptions, this state is not a vector in any bipartite space, even non-separable, and therefore cannot be described within the standard Hilbert space formalism.