Abstract
A widespread view in physics holds that the implementation of time reversal in standard quantum mechanics must be given by an anti-unitary operator. In foundations and philosophy of physics, however, there has been some discussion about the conceptual grounds of this orthodoxy, largely relying on either its obviousness or its mathematical-physical virtues. My aim in this paper is to substantively change the traditional structure of the debate by highlighting the philosophical commitments underlying the orthodoxy. I argue that the persuasive force of the orthodoxy can benefit from a relational metaphysics of time and a by-stipulation view on symmetries. Within such philosophical background, I submit, the orthodoxy of time reversal in standard quantum mechanics could find a fertile terrain to lay the groundwork for a more thorough conceptual justification.
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Notes
I will be mainly following Sakurai and Napolitano (2011, pp. 289–293), Ballentine (1998, pp. 380–381), Gasiorowicz (1966, pp. 25–30) Gibson and Pollard (1978, pp. 179–180). I will also introduce some insights from Bigi and Sanda (2016, pp. 27–30), Jauch and Rohrlich (1959, pp. 88–91), Sachs (1987, pp. 32–36), and Messiah (1966, pp. 664–674).
This argument is troublesome because it somehow assumes that the Schrödinger equation defines the Hamiltonian, when in general it “is defined independently as an operator that acts on the x dependence of a state function” [see Laue (1996) for discussion].
Beauregard refers to “Racah’s operator” as opposed to Wigner’s (Costa de Beuregard 1980, p. 524 and further references therein).
It can be pointed out that there is a third key argument, to wit, that momentum changes its sign under time reversal. Certainly, this is one of the most salient features of the time reversal implementation in classical mechanics. For the most part though, the reasons why the sign of momentum should change under time reversal in QM follow the lines of the other two arguments. In the particular case of momentum, reasons swing back and forth from preserving certain smooth continuity between the classical mechanics and QM to achieving the representation of motion and reversal and appealing to its obviousness. Some authors just claim that the transformation follows by definition (Messiah, 1966, p. 667; Sachs, 1987; Ballentine, 1998, pp. 377–378). A more philosophically refined discussion can be found in Callender (2000) and Roberts (2018). In addition, it can be argued that the transformation of momentum plays a paramount role in the semi-classical limit, mainly in relation to Ehrenfest’s theorem. However, this argument, and various versions thereof, does not add anything substantive to the point I want to make in this paper, so I will set it aside.
One could argue that a general time-reversal transformation will never change the sign of the position operator, because it is not the right sort of transformation that time reversal is expected to carry out. I do not find any equally stronger argument for transition probabilities, though Uhlhorn’s theorem could, after some assumptions, do the work (see below).
From a philosophical viewpoint, I think it is not trivial. Even though most symmetries physicists are interested in are required at minimum to transform states into states, we could want to leave some room for metaphysically possible scenarios in which some transformation fails to transform a state into a physical state. Primitivists with respect to time could argue that time is fundamental and defines not only the dynamics of a physical theory, but also is constitutive of its kinematics. Consider, for instance, Maudlin’s argument about doppelgängers and mental states (Maudlin 2002: 271): if we suppose that time reversal acts in such a way that the time-reversed mental states are still mental states, we are unjustifiably assuming that the direction of time does not play any role in making a mental state what it is. How do we know that when reversing time, we will still end up with something like mental states, and not something completely different? An analogous argument, I think, can be run here: Why should the substantivalist assume that time does not play a role in defining what is a physical state? Does the requirement of preserving the notion of state when time is reversed not discard, from the outset, the notion of time as fundamental? I am not defending this viewpoint here, but I just want to draw the attention towards the non-obviousness of the assumption from a metaphysical viewpoint.
It is worth clarifying that the predicates “positive” or “negative” for the energy spectrum, or “unbounded from below/from above” for Hamiltonians are conventional. So, the argument could not hinge upon which predicate we adopt to describe the system properly. The problem is not exactly whether the Hamiltonian is unbounded from below. The problem is that if we start with a Hamiltonian unbounded from above (but bounded from below) and end up with a Hamiltonian unbounded from below (but bounded from above) after a transformation. A specific Hamiltonian must be bounded (either from above or from below), and the problem will come up if one adopts a transformation that turns a Hamiltonian unbounded from above (bounded from below) into a Hamiltonian unbounded from below (bounded from above).
I thank an anonymous reviewer for this observation.
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Acknowledgements
I am deeply grateful to Olimpia Lombardi, Michael Esfeld, Karim Thébault, Carl Hoefer, Bryan Roberts, Christian Sachse and María José Ferreira for so enriching and substantive discussions, criticisms, and suggestions on this manuscript and earlier versions. I also thank to Dustin Lazarovici, Frida Trotter, Andrea Oldofredi, Federico Benitez, Manuel Gadella, Federico Holik and Sebastián Fortín for suggestions and comments on some ideas expressed here. Finally, I would like to thank the anonymous reviewers: their reports significantly improved the original version. This work was supported by a FRS-FNRS (Fonds de la Recherche Scientifique) Postdoctoral Fellowship and made possible through the support of the Grant #61785 from the John Templeton Foundation. The opinions expressed in this publication are those of the author and do not necessarily reflect the views of the John Templeton Foundation.
Funding
This research was partially funded by a FNRS Postdoctoral fellowship, by the Université de Lausanne, and by the Grant ID# 61785 from the John Templeton Foundation.
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López, C. The physics and the philosophy of time reversal in standard quantum mechanics. Synthese 199, 14267–14292 (2021). https://doi.org/10.1007/s11229-021-03420-0
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DOI: https://doi.org/10.1007/s11229-021-03420-0