Schulman Replies: Casati et al. [1] have failed to understand the goals of my article [2]. They and I find ourselves in a situation that is common in discussions of the "arrow of time," namely no disagreement on technical issues and no agreement on basic assumptions. Instead of criticizing their goals (presumably "to clear out the too-long standing confusion around (ir)reversibility of statistical laws" [1]), which occupy many workers in this area [3], I will try to make clear what problem I do address.About 40 years ago Gold [4] argued that the thermodynamic arrow of time followed the cosmological one. This is a thesis of beauty and scope, but I found [5] that for logical consistency one needed to introduce a new element into the argument: the use of boundary conditions at two (usually remote) times. This is because the use of macroscopic initial conditions already fixes the thermodynamic arrow. A further benefit of this conceptual point is that if we are in a big-bang/big-crunch universe and if, relieved of the initial conditions perspective, one would give roughly symmetric boundary conditions, one would then have a framework within which to look for physical effects arising from the future big crunch. See [6], where these arguments are spelled out in greater detail and the relevant literature, particularly due to J. A. Wheeler, is presented.A consequence of this framework is that one can imagine subsystems that, because of a particular history of isolation, survive the era of maximal expansion with arrow intact. Such a scenario has been the basis for attacks on Gold's thesis [7]. It is this isolation and survival phenomenon that is captured by the boundary conditions given in my article [2]. The two times given there are some intermediate times, not particularly near the end points, such that at those times one system is dominated by the conditioning at one end, the other at the other end. If one desired a more complete picture, one could (in principle) arrange boundary conditions near the big bang and big crunch such that relatively isolated chunks of matter would pass the point of maximal expansion with arrow intact. (Obviously this requires greater spatial and dynamical richness than the cat map.) If the time interval under study is [t 1 , t 2 ], then one system would have low entropy at t 1 , one at t 2 (and both times would be significantly distant from the end points). I modeled this by restricting each subsystem to a single coarse grain in its phase space at its respective "initial" time. But as part of a larger dynamical scheme (i.e., as a slice out of big bang to big crunch boundary value problem), each system would carry a great deal more information encoded in its microscopic coordinates. This further information ensures that each system possesses its arrow "before" (by its own clock) its encounter with the other. In terms of the remarks of Casati et al. this microscopic information prevents the bi-directional entropy increase envisioned in their criticism.The goal of many researchers in this area...
Altland-Simons have a good discussion in §3.3 Applications of the Feynman Path Integral to the quantization of spin, which is defined by the commutation relations Ŝ j ,Ŝ k = −i jk Ŝ , 123 = − 12 3 = +1 ,Ŝ ×Ŝ = iŜ .
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