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arXiv:1010.3969

Parameter Independence and Outcome Independence in Dynamical Collapse Theories

Travis Norsen

Published Oct 19, 2010• 0 citations
arXivHTMLPDF

Abstract

Ghirardi, Grassi, Butterfield, and Fleming have previously argued that two distinct formulations of dynamical reduction (GRW) theories have distinct non-local properties: one (the "non-linear" formulation) violates Parameter Independence while the other (the "linear + cooking" formulation) violates Outcome Independence. The claim is then that, since theories violating Outcome Independence are compatible with relativity in a way that theories violating Parameter Independence are not, the "linear + cooking" dynamical reduction theories hold the key to reconciling quantum non-locality with relativity. I review and assess this claim, arguing that, while ultimately misleading, there is an interesting and important lesson to be extracted.

DOI
10.48550/arxiv.1010.3969

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Ghirardi, Grassi, Butterfield, and Fleming have previously argued that two distinct formulations of dynamical reduction (GRW) theories have distinct non-local properties: one (the "non-linear" formulation) violates Parameter Independence while the other (the "linear + cooking" formulation) violates Outcome Independence. The claim is then that, since theories violating Outcome Independence are compatible with relativity in a way that theories violating Parameter Independence are not, the "linear + cooking" dynamical reduction theories hold the key to reconciling quantum non-locality with relativity. I review and assess this claim, arguing that, while ultimately misleading, there is an interesting and important lesson to be extracted.
Ever since we have been in the possession of quantum theories without observers, such as Bohmian mechanics or the Ghirardi‐Rimini‐Weber (GRW) theory of spontaneous wave function collapse, a major challenge in the foundations of quantum mechanics is to devise a relativistic quantum theory without observers. One of the difficulties is to reconcile nonlocality with relativity. I report about recent progress in this direction based on the GRW model: A relativistic version of the model has been devised for the case of N noninteracting (but entangled) particles. A key ingredient was to focus not on the evolution of the wave function but rather on the evolution of the matter in space‐time as determined by the wave function.
Abstract I examine G.B. Baga’s arguments for the Ghirardi-Rimini-Weber (GRW) interpretation of non-relativistic quantum mechanics as ideally suited for Whitehead’s philosophy. Much of Bagci’s claims are in response to Michael Epperson, who argues in the same vein in favor of decoherence accounts (Omnès; Zureck). Pace Epperson, I do not think that decoherence is the final arbiter here, and instead I contrast GRW with several other accounts addressing foundational problems of quantum theory (Finkehtein; Green; Peres and Temo; etc.), which aho account for relativistic covariance, while GRW does not. I argue that such latter research programs align themselves in a more convincing manner with Whitehead’s scheme, in epistemic as well as metaphysical senses, than GRW.
Einstein was never satisfied with quantum mechanics. He argued that quantum mechanics was incomplete for two main reasons; it violated the locality principle and the separability principle. The violation of separability is an unavoidable consequence of quantum interactions. Non-separability can be seen in quantum entanglement. Non-locality, however, is more controversial. Einstein and his associates published the EPR paper in order to argue for the incompleteness of quantum mechanics. Years later, John Bell formulated what became known as the Bell Inequalities in response to the EPR paper. The Bell Inequalities are seen as a major obstacle for quantum locality. I will argue that non-locality is not a necessary implication of the Bell Inequalities. The Bell Inequalities were developed using Bell’s locality requirement as a major premise. Bell’s locality requirement can be described in terms of two conditions, parameter independence and outcome independence. A violation of either condition will lead to a violation of the Bell inequalities. Parameter independence is not violated by the results of experimental quantum physics. So, it can be argued that violations of the Bell inequalities are caused by the violation of outcome independence. Such a violation of outcome independence does not imply non-locality if we accept some form of holism or non-separability. Thus, by including some form of holism or non-separability into our picture of the quantum realm we can develop a theory that does not conflict with locality. This paper will discuss different types of holism and non-separability and how they can be used to help understand quantum phenomena.
Bell proved that no theory of pre-existing local values can reproduce the predictions of quantum mechanics, but his proof leaves the culprit ambiguous: one may reject either of two independence conditions, Outcome Independence or Parameter Independence, and most commentators have found Outcome Independence the safer sacrifice. The first part of this paper presents, in a form adapted to spin-1/2 particles in the singlet state, an argument (Forster 2014) that removes the ambiguity: using a chained family of experiments in which quantum mechanics predicts an extreme pattern of correlations -- the quantum Sorites phenomenon -- a contradiction is derived without ever assuming Outcome Independence. Under the resulting theorem, anyone who holds that hidden variables could improve on the quantum probabilities must give up Parameter Independence itself. That looks like a heavy price, because Parameter Independence appears to be protected twice over: rejecting it seems to put superluminal influences into spacetime, and its statistical shadow -- the No-Signaling condition -- is experimentally beyond reproach The second part of the paper shows that the price is payable. In the random-matrix collapse dynamics proposed by Kryukov, measurement is a random walk of the quantum state, and the hidden variable is not a stock of values fixed at the source but the random stream that drives the walk -- like the stored random numbers of a computer simulation, with the measurement settings playing the role of seeds. In that framework Parameter Independence is false while Outcome Independence and No-Signaling are both true, and one can say exactly how the Sorites argument is blocked, why the violation involves no process propagating in spacetime, and why the influence of one wing's setting on the other wing's outcome -- demonstrated here in a simulation -- can never be used to send a message.
We provide general guidelines for generalizing dynamical reduction models to curved spacetimes and propose a class of generally covariant relativistic versions of the Ghirardi-Rimini-Weber model. We anticipate that the collapse operators of our class of models may play a role in a yet-to-be-formulated theory of semiclassical gravity with collapses. We show explicitly that the collapse operators map a dense domain of states that are initially Hadamard to final Hadamard states---a property that we expect will be needed for the construction of such a semiclassical theory. Finally, we provide a simple example in which we explicitly compute the violations in energy-momentum due to the state reduction process and conclude that this violation is of the order of a parameter of the model---supposed to be small.
A class of theories alternative to standard quantum mechanics, including that of Ghirardi et al ('GRWP'), postulates that when a quantum superposition becomes amplified to the point that the superposed states reach some level of 'macroscopic distinctness', then some non-quantum-mechanical principle comes into play and realizes one or other of the two macroscopic outcomes. Without specializing to any particular theory of this class, I ask how far such 'macrorealistic' theories are generically constrained, if one insists that the physical reduction process should respect Einstein locality, by the results of existing EPR-Bell experiments. I conclude that provided one does not demand that the prescription for reduction respects Lorentz invariance, at least some theories of this type, while in principle inevitably making some predictions that conflict with those of standard quantum mechanics, are not refuted by any existing experiment.