Bloch's Paradox and the Nonlocality of Chance

Abstract

I show how an almost exclusive focus on the simplest case—the case of a single particle—along with the commonplace conception of the single‐particle wave function as a scalar field on spacetime contributed to the perception, first brought to light by I. Bloch, that there existed a contradiction between quantum theory with instantaneous state collapses and special relativity. The incompatibility is merely apparent since treating wave‐function values as hypersurface dependent avoids the contradiction. After clarifying confusions which fueled the perception of a paradox, I elaborate on an analysis of the wave function due to Wayne Myrvold to show that nothing special, or ad hoc, is required in treating wave‐function values, even in the single‐particle case, as hypersurface–dependent; rather, the hypersurface dependence of these values is the natural development of nonlocal entanglement in the context of the relativity of simultaneity. Properly understood, what Bloch’s paradox reveals is that the combination of nonlocal entanglement together with a hypersurface‐dependent process of state collapse conflicts with the thesis of spatiotemporal separability and, in particular, with the idea that chances are local matters of fact.

Acknowledgements

I would like to thank Robert Batterman, Wayne Myrvold, and two anonymous referees for their comments on this paper. I am also very grateful to Jeffrey Barrett, Gordon Fleming, and David Malament for discussions on this and related topics. Research for this paper was supported by the Rotman Family Foundation.

Notes

[1] For detailed treatments of the quantum measurement problem, see Albert (1992) and Barrett (1999).

[2] On the standard formulation, the collapse is a transition to an eigenstate of the dynamical variable of interest (assuming that the dynamical variable has eigenstates). One might allow for a more relaxed notion of collapse (as is done in the GRW theory) just so long as it can account for our experience of a determinate measurement result.

[3] The expression ‘postulate of instantaneous reduction’ comes from Aharonov and Albert (1981, 364).

[4] For the purpose of giving a precise and uncomplicated description, two nonessential features of the collapse process are assumed here: (i) the operator representing the dynamical variable A has eigenvectors (this is not essential to the idea of collapse, so one might consider a more sophisticated version of this postulate which encompasses a broader range of dynamical variables); and (ii) the new quantum state appears at time t0. If the new quantum state ought to appear only after t ₀, a revised formulation would end with the following: ‘so that for some ϵ > 0 and all δ ϵ (0, ϵ], ψ _S (t ₀ + δ)= ϕ A’.

[5] The only person I am aware of who has explicitly called this ‘Bloch’s paradox’ is Gordon Fleming (1989, 114). However, Aharonov and Albert called the failure of Lorentz covariance for the standard ‘instantaneous’ reduction postulate a paradox (1981). And Breuer and Petruccione, reviewing the work of Aharonov and Albert, described it likewise, as ‘The paradox of the instantaneity of the state vector reduction’ (1999, 3).

[6] Aharonov and Albert’s example is repeated with discussion by Malin (1982), Breuer and Petruccione (1999), and Barrett (2003).

[7] These are the typical assumptions made for this discussion—see, for example, Bloch (1967, 1381), Aharonov and Albert (1984, fn. 1), and Myrvold (2002, 437–438, 2003, 495).

[8] It is not clear what to say about the state of the particle at the time t = τ_R of the collapse. Is it the pre‐collapse state? The post–collapse state? Or, is it the case that the system has no definite state at all then, perhaps as a result of the measurement interaction? In any case, it does not matter for our purposes.

[9] For a nice discussion on this point, see Maudlin (1996, 290–293).

[10] A thorough discussion of this option and the reasons for rejecting it would take us beyond the scope of Bloch’s paradox. One of Aharonov and Albert’s reasons for rejecting the light cone collapse proposal was that they believed it would result in a failure of charge conservation in every frame (1980, 3322–3324). For a criticism of a different sort, see Maudlin (1994, 195–201).

[11] For discussions on separability, see Healey (1991), Howard (1989; 1997), and Maudlin (1994).

[12] The EPR spin singlet state for a pair of spin‐1/2 particles labelled a and b is typically written as , where the kets correspond to the spin measurement outcomes ‘up’ (+) and ‘down’ (−) in the z direction.

[13] Similarly, contrary to Barrett’s (2003, 1214) criticism of hyperplane‐ and hypersurface‐dependent approaches toward state collapse, no ‘exotic structures for the physical world’ need to be employed, if by ‘exotic’ he means simply ‘nonstandard’.

[14] One might question this result by calling into question the few meager assumptions also involved in Bell‐type theorems—for example, the assumption that the choices of which measurements to make at spacelike separation are independent. Giving up this assumption amounts to embracing deep conspiracies in the structure of the universe. Without any further positive reasons for believing that such is the case, the assumption seems an extremely plausible one.

[15] Or, in other words, with purely unitary evolution, the reduced density operator representing the state of a local region is the same no matter which spacelike hypersurface through the region the global state is defined on. See Myrvold (2003, 480–486).

[16] Myrvold (2003) makes a similar point in his development of a more general condition he calls the local evolution condition. Moreover, this point seems to be the main point at issue when one considers Barrett’s claim that the standard formulation of quantum mechanics is inconsistent with the principle of relativity because the former ‘requires mutually incompatible states for different inertial frames’ (2003, 1214). Barrett’s argument carries weight only if the principle of relativity requires states on intersecting, non‐parallel hyperplanes to be active transformations of one another by means of unitary representations of appropriate members of the Lorentz group of isometries. But, as a requirement, this would be to ignore what has actually occurred to the state of the originally entangled system in the region between the hyperplanes.

Additional information

Notes on contributors

Brian A. Woodcock

Brian A. Woodcock is at the Department of Philosophy, University of Western Ontario.