Quantum Measurement, Gravitation, and Locality *

This essay argues that when measurement processes involve energies of the order of the Planck scale, the fundamental assumption of locality may no longer be a good approximation. Idealized position measurements of two distinguishable spin-0 particles are considered. The measurements alter the space-time metric in a fundamental manner governed by the commutation relations [xi pj] = ih̄ δij and the classical field equations of gravitation. This in-principle unavoidable change in the space-time metric destroys the commutativity (and hence locality) of position measurement operators. This work was done under the auspices of the U. S. Department of Energy. E-mail address: [email protected] 1 The purpose of this brief essay is to make an in-principle remark on the fundamental assumption of locality in quantum field theories [1] and its interplay with the measurement process and gravitation. The essential philosophy of this essay is to enhance the quantum mechanical and gravitational effects and ignore the lowest-order classical effects (i.e., those effects that do not depend on h̄). We will see that the assumption of locality is deeply connected with gravitation and the measurement process. If all gravitational effects (irrespective of whether classical and quantum-mechanically-induced) are ignored, locality is recovered. The remarks that we present are seemingly trivial, but in view of their possible relevance, we take the liberty of presenting them in this brief essay. To give a precise definition to locality, let us note with Schwinger [2] that: “A localizable field is a dynamical system characterized by one or more operator functions of space-time coordinates, Φ(x) . Contained in this statement are the assumptions that the operators xμ , representing position measurements, are commutative, [ xμ , xν ] = 0 , (1) and furthermore, that they commute with the field operators, [ xμ , Φ α ] = 0 , (2) so that 〈x|Φ|x〉 = δ(x− x) Φ(x) . (3) The difficulties associated with current field theories may be attributable to the implicit hypothesis of localizability.” In reference to commutativity of the position measurements, expressed by Eq. (1), underlying the “hypothesis of localizability,” we consider two neutral spin-0 particles of masses m1 and m2 (> m1). For the purposes of the following discussion, it would be useful to keep the 2 following idealized picture of the world in view. The world consists of two particles. All measuring devices have no other effect except to introduce the quantum-mechanically-required perturbations consistent with the fundamental commutation relations: [xi pj ] = ih̄ δij . We now claim that we know, as a result of some appropriate measurement, M1, that particle-1 is confined to a sphere of radius R1 ≪ h̄/(m1c) centered at ~x1; while the space-time coordinates of particle-2 are completely unknown. Now a time ∆t ≪ h̄/(m1 c) later, we make a second measurement, M2, such that particle-2 is confined to a sphere of radius R2 ≪ h̄/(m2 c) centered at ~x2. The measurement M2, via the fundamental uncertainty relations [xi, pi] ∼ ih̄ δij , imparts certain momentum to particle-2 resulting in a local energy density ρ2(r2) > ∼ 3 θ(r2 − R2) 4πR 3 2 [ m 2 c + β h̄c R 2 2 1/2 , > ∼ 3 θ(r2 − R2) 4πR 3 2 √ βh̄c R2 , for R2 ≪ h̄/(m2c) , (4) where r2 equals the radial coordinate distance with ~x2 as origin, β is a geometrical factor of the order of unity, and θ(r) is the usual step function. We shall assume that the two particles have separations (of course, only after the measurements are made!) |~x1 − ~x2| > ∼ h̄/(m1c). The assumptions R1,2 ≪ h̄/(m1,2 c), etc., are made to keep possible quantum mechanical overlap of wave functions of particle-1 and -2 to a minimum and to enhance purely quantum mechanical effects arising solely from the measurement process. The assumption m2 6= m1 avoids complications that may arise from indistinguishability of the particles. The particles are assumed to have spin-0 to avoid (gravitational) Thirring-Lense [3] interaction. In order to keep our arguments as simple as possible, we refrain from incorporating uncertainties that arise from the specification of the time variable. The essential character of conclusion that follows is, however, expected to remain unaltered if all, or some of, these assumptions are relaxed. Define ρ1(r1) in a similar fashion to ρ2(r2) above. Consider the setup such that in the region r1 ≤ R1 and r2 ≤ R2 we have ρ2(r2) ≫ ρ1(r1). Then, as a result of inherently quantum mechanical perturbation in momentum of a particle by confining it to a finite region of space, 3 we are forced to induce a local modification of space-time structure. Explicitly, we see this via the classical field equations of Einstein and Eq. (4). In the spirit of the philosophy outlined in the beginning of this paper, if we neglect classical effects O[2Gm1,2/(cR1,2)], the space-time metric before the measurement M2, (in the notation of Ref. [4], and replacing r2 by r) can be written as dτ 2 = dt − dr − r dθ − r sin θ dφ , (5) After the measurement M2, this space-time metric is changed to dτ 2 = [