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Fundamental theories of physics

Classical physics on a general phase space

Klaas Landsman

Published Jan 1, 2017• 0 citations
Publisher

Abstract

Passing from finite phase spaces X to infinite ones yields many fascinating new phenomena, some of which even seem genuinely “emergent” in not having any finite dimensional shadow, approximate or otherwise. Nonetheless, practically all results in the previous chapter remain valid, typically after the inclusion of some technical condition(s) that restrict the almost unlimited freedom allowed by infinite sets. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1007/978-3-319-51777-3_3

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Fundamental theories of physics•Jan 1, 2017•1 citation
Passing from finite phase spaces X to infinite ones yields many fascinating new phenomena, some of which even seem genuinely “emergent” in not having any finite dimensional shadow, approximate or otherwise. Nonetheless, practically all results in the previous chapter remain valid, typically after the inclusion of some technical condition(s) that restrict the almost unlimited freedom allowed by infinite sets. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This chapter gives an introduction to a chain of results attempting to exclude deeper layers underneath quantum mechanics that restore some form of classical physics: ‘[Such results] more or less illustrate the ways along which some opponents might hope to escape Bohr’s reasonings and von Neumann’s proof and the places where they are dangerously near breaking their necks.’ (Groenewold, 1946, p. 454). In so far as they are mathematically precise, such no-go results have their roots in von Neumann’s 1932 book, which gave rise to two traditions that were often in polemical opposition to each other. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Throughout this chapter, X is a finite set, playing the role of the configuration space of some physical system, or, equivalently (as we shall see), of its pure state space (in the continuous case, X will be the phase space rather than the configuration space). One should not frown upon finite sets: for example, the configuration space of N bits is given by X=2‾N‾X = \underline{2}^{\underline{N}}X=2​N​ , where for arbitrary sets Y and Z, the set YZY^{Z}YZ consists of all functions x:Z→Yx : Z \rightarrow Yx:Z→Y , and for any N∈NN \in \mathbb {N}N∈N we write N‾=1,2,…,N\underline{N} = {1,2,\ldots ,N}N​=1,2,…,N (although, following the computer scientists, 2‾\underline{2}2​ usually denotes {0,1}). More generally, if one has a lattice $$\varvec{\varLambda }\subset \mathbb {Z}^{d}$$ and each site is the home of some classical object (say a “spin”) that may assume N different configurations, then $$X = \underline{N}^{\varvec{\varLambda }}$$ , in that $$x : \varvec{\varLambda }\rightarrow \underline{N}$$ describes the configuration in which the “spin” at site $$\mathbf{n} \in \varvec{\varLambda }$$ takes the value x(n)∈N‾x(\mathbf{n}) \in \underline{N}x(n)∈N​ .
In this chapter we generalize the results of Chapter 2 to infinite-dimensional Hilbert spaces. So let H be a Hilbert space and let B ( H ) be the set of all bounded operators on H . Here a notable point is that linear operators on finite - dimensional Hilbert spaces are automatically bounded, whereas in general they are not. Thus we impose boundedness as an extra requirement, beyond linearity. This is very convenient, because as in the finite-dimensional case, B ( H ) is a C*-algebra, cf. §C.1. At the same time, assuming boundedness involves no loss of generality whatsoever, since we can alway replace closed unbounded operators by bounded ones through the bounded transform , as explained in §B.21. Nonetheless, even the relatively easy setting of bounded operators leads to some technical complications we have to deal with. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.