Roads to the past: how to go backwards in time in quantum mechanics

CRISTIAN LÓPEZ

Exeter

Congreso; EPSA 2017; 2017

European Philosophy of Science Assoiation

Intuitively we believe time goes by. We have learned whilst future is something which ?everyone reaches at the rate of sixty minutes an hour, whatever he does, whoever he is? (as C.S. Lewis has written), past remains untouchable, only reachable by memory. Time seems to be asymmetric and directed: it always ?flows? from past to future, and never the other way around. Philosophical enquiry starts out when we raise the question about the very bedrock of such daily belief: what does ?time passes by? mean? What is our human experience about a passing time grounded in? Loads of ink have been spilled in the field of philosophy of physics in addressing the asymmetric and directed nature of time from our fundamental physical theories? perspective. According to this view, the human experience of time would mirror an objective feature of the world: if time is to any extent physical, world?s temporal features should be picked out by our best physical theories. In this way, our experience of time would take roots in the reality itself. In the literature, this topic is commonly referred to as ?the problem of the arrow of time?.The problem of the arrow of time in physics is usually posed in terms of invariance under time-reversal, that is, whether our fundamental physical theories offer symmetric descriptions of the world when time is inverted. The usual way to mathematically represent a backward-in-time evolution is via a time-reversal operator (T), which is just a mathematical gadget used to see whether the theory meets the symmetry under scrutiny. The time-reversal operator T performs at least the transformation on dynamical equations (laws). If a dynamical equation is invariant under time-reversal, then if is a solution, is also a solution, each being the temporal mirror image of the other. Thus, if a dynamical equation is not time-reversal invariant, then either or is not a solution, and this introduces a clear difference between the two directions of time, a difference time-asymmetry can be grounded in.If the problem of the arrow of time is the question at issue, time-reversal invariance is expected to be a symmetry that theories may or may not meet ?we have to outwardly see the formal structure of a given theory and to check out whether time-reversal invariance holds. Nevertheless, there is a broad consensus among philosophers and physicists that our fundamental physical theories seem to be invariant under time-reversal: statistical classical mechanics, electromagnetism, relativistic and non-relativistic quantum mechanics are seemingly blind to the distinction between the past-to-future and the future-to-past directions (e.g. Gibson and Polland 1978, Price 1996, Wallace 2012). However, I think, conclusions are drawn too quickly, since philosophers and physicists usually disagree on which the relevant properties for a time-reversal operator should be. This situation leads to a coexistence of a myriad of different criteria and, thus, of several operators standing for, allegedly, the same symmetry (Peterson 2015). The crucial point is that theories turn out to be time-reversal or non-time-reversal invariant depending on what operator we come to conceive as appropriate.Despite the discussion having been particularly heated in classical electromagnetism (see Horwich 1987, Albert 2000, Arntzenius and Greaves 2009) there is an akin situation in non-relativistic quantum mechanics, though scarcely explored and discussed in the literature, comparatively. According to most textbooks, the Schrödinger equation is time-reversal invariant. Nevertheless, for instance, Craig Callender (2000) cautiously casts doubts on that statement and claims that there is one sense in which it is non-timereversal invariant. The point of disagreement is they are appealing to different criteria to figure out what the time-reversal operator must look like: whereas textbooks and most philosophers and physicists usually turn to the anti-unitary Wigner operator in order to test time-reversal invariance in non-relativistic quantum mechanics (which not only transforms t into ?t but also introduces complex conjugation), Callender relies on an unitary time-reversal operator that flips the sign of the Hamiltonian (below the theoretical bound). So, which one must we work with? Which one represents time-reversal correctly?In this presentation I would like to shed some light on two questions that are critical for tackling the problem of the arrow of time in non-relativistic quantum mechanics. Firstly, I shall characterize this messy situation and try to settle the problem rightly: there seems to be at least two mathematical possibilities for a time-reversal operator, leading to two quite different scenarios. So there is no obvious univocal and objective way to define a time-reversal operator. On the one hand, literature (mainly physics books) is prone to embrace an account of time-reversal that builds up a time-reversal operator preserving the theory time symmetric: T is supposed to be anti-unitary in order to have the same properties and behavior as in classical mechanics (Messiah 1966, Gibson and Polland 1976, Ballentine 1998, Roberts 2016). On the other hand, an off the beaten track account for time-reversal defines time-reversal in a purer way, with no reference to any physical theory in particular (Jauch and Rohrlich 1959, Costa de Beauregard 1980, Callender 2000). In this case, T is supposed to be unitary and defined from theory-independent, all-embracing intuitions about what time-reversal means. After that, I shall argue these two criteria face different types of problems: whereas the former runs the risk of circularity inasmuch as the time-reversal operator is built to preserve the very symmetry that it intends to prove, the latter is grounded on intuitions difficut to explain and that somebody might not share. Finally, I shall put forward a desideratum regarding what a time-reversal operator should look like in order specifically to make sense and to properly deal with the problem of the arrow of time in physics in general.