Comment by jostylr | Hacker News Reader

Comment by jostylr | Hacker News Reader

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In physics, particles not being labelled by anything other than their trajectories is a very natural starting point. When one uses the natural configuration space, one without labels in which a configuration is a set of n points in physical space rather than an ordered n-tuple, then the complex-valued wave functions on that space are exactly those of boson type. To get the fermions, one replaces the value space with a 1 dimensional complex bundle over the configuration space, one which twists in the right way. A paper I coauthored explores this in a general context: [1]

The "you" is then a rough set of particles whose trajectories roughly coincide with your macroscopic trajectory. Their identity is just given by where they are.

As for representations, I feel like I can easily understand how to get momentum or temperature from particles with their time evolution (trajectories), but I do not see how, say, to get positions of particle just from knowing what their momentums were and their time evolution.

1: https://arxiv.org/abs/quant-ph/0601076

But we don't have even a set of definite trajectories. If we see e.g. an electron coming towards a hydrogen atom and then an electron moving away from it, not only do we not know whether the incoming electron "bounced off" or whether it settled into the orbital and "kicked" the electron that was already there out, but in a fundamental physics sense what occurred is some weighted average of both (in the same way that we don't merely "not know" which of the two slits an electron went through but in an important physical sense it partially went through both).

It depends on the theory. The Bohmian theory, which is what I have been using, is one in which electrons have actual positions that change over time along trajectories. We may not have access to that data, but that is fine. Certainly in simulations one would be able to see which scenario happened. For some, it might be the same electron moving away, for others it would be kicking one out. One could definitively say which one is happening in the simulation. In experiments, we cannot say that because our access to the knowledge is limited by quantum equilibrium. The quantum formalism is very much like thermodynamics in that regard; the individual details are missing, but the larger picture can be computed. Nevertheless, in a Bohmian world, the electrons have their distinct identities as distinguished by, and only by, their trajectories.

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