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Modern Physics Letters A

PRINCIPLE OF GENERAL COVARIANCE AND QUANTUM THEORY

Τ. Padmanabhan

Published Mar 1, 1988• 1 citation
Publisher

Abstract

We emphasise the distinction between formal and operational notions of general covariance. Classically, formal covariance implies operational covariance. This is not true in quantum theory. Two observers may not agree on the results of measurement of a tensorial object like Tik (stress tensor) in quantum theory. In particular, one observer might conclude that the measured value is zero while another might attribute non-zero value to it.

DOI
10.1142/s0217732388000453

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We emphasise the distinction between formal and operational notions of general covariance. Classically, formal covariance implies operational covariance. This is not true in quantum theory. Two observers may not agree on the results of measurement of a tensorial object like Tik (stress tensor) in quantum theory. In particular, one observer might conclude that the measured value is zero while another might attribute non-zero value to it.
Measurements invariably determine the physical state of a given ``observed system'' with respect to an ``observer system.'' This implies an inescapable two-body relationality upon which physical theories must rely. There has been a growing effort to understand how quantum mechanics can accommodate the notion of reference frame in its substratum and whether such approach would respect general covariance. With regard to quantum resources, it has been shown that coherence and entanglement are not invariant upon changes of quantum reference frames. Here, we construct a theoretical framework within which a given combination of quantum resources, including not only coherence and entanglement, is shown to be a physical invariant. Under the premise of the universality of quantum phenomena, the proven covariance between quantum reference frames then implies that the total quantumness accessible to any observer in the universe is absolute.
Causality imposes strong restrictions on the type of operators that may be observables in relativistic quantum theories. In fact, causal violations arise when computing conditional probabilities for certain partial causally connected measurements using the standard non covariant procedure. Here we introduce another way of computing conditional probabilities, based on an intrinsic covariant relational order of the events, which differs from the standard one when these type of measurements are included. This alternative procedure is compatible with a wider and very natural class of operators without breaking causality. If some of these measurements could be implemented in practice, as predicted by our formalism, the non covariant, conventional approach should be abandoned.
In relativistic quantum field theory the notion of a local operation is regarded as basic: each open space-time region is associated with an algebra of observables representing possible measurements performed within this region. It is much more difficult to accommodate the notions of events taking place in such regions or of localized objects. But how can the notion of a local operation be basic in the theory if this same theory would not be able to represent localized measuring devices and localized events? After briefly reviewing these difficulties we discuss a strategy for eliminating the tension, namely by interpreting quantum theory in a realist way. To implement this strategy we use the ideas of the modal interpretation of quantum mechanics. We then consider the question of whether the resulting scheme can be made Lorentz invariant.
I propose now to examine Bohr's informal theory of measurement from a somewhat more formal point of view, and to consider how his theory relates to the orthodox theory of measurement in quantum mechanics. This is a difficult task, since Bohr nowhere discusses the theory of measurement in formal terms. My discussion, therefore, will be less closely tied to Bohr's own statements. I shall also consider some important objections to the theory. The objective-values theory of measurement Before we look into this theory, it will be useful to have a brief sketch of some of the relevant formal elements of quantum mechanics. In the Hilbert space formulation of quantum mechanics the measurable physical properties (‘observables’) of an object are represented by linear Hermitian operators A , B , … which have complete orthonormal sets of eigenvalues a n , b n , … corresponding to the subspaces (closed linear manifolds) of a Hilbert space H . These operators are in one-to-one correspondence with the projection measures P A , P B , … on the Hilbert space such that P A = Σ n a n P n , where P n are the projections of the eigenvectors ψ n belonging to the eigenvalues a n of A , and a n are discrete, non-degenerate values (for simplicity I shall deal only with operators which have such values, and shall denote observables and the corresponding operators by the same symbols).
Quantum mechanics (QM) is not just a theory of micro-objects: it is our current fundamental theory of motion. It expresses a deeper understanding of Nature than classical mechanics. Precisely as classical mechanics, the conventional formulation of QM describes evolution of states and observables in time. Precisely as classical mechanics, this is not sufficient to deal with general relativistic systems, because these systems do not describe evolution in time; they describe correlations between observables. Therefore, a formulation of QM slightly more general than the conventional one – or a quantum version of the relativistic classical mechanics discussed in the previous chapter – is needed. In this chapter I discuss the possibility of such a formulation. In the last section I discuss the general physical interpretation of QM. QM can be formulated in a number of more or less equivalent formalisms: canonical (Hilbert spaces and self-adjoint operators), covariant (Feynman's sum-over-histories), algebraic (states as linear functionals over an abstract algebra of observables) and others. Generally, but not always, we are able to translate these formalisms into one another, but often what is easy in one formulation is difficult in another. A generalrelativistic sum-over-histories formalism has been developed by Jim Hartle. Here I focus on the canonical formalism, because the canonical formalism has provided the mathematical completeness and precision needed to explicitly construct the mathematical apparatus of quantum gravity. Later I will consider alternative formalisms.