Abstract
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 \(X = \underline{2}^{\underline{N}}\), where for arbitrary sets Y and Z, the set \(Y^{Z}\) consists of all functions \(x : Z \rightarrow Y\), and for any \(N \in \mathbb {N}\) we write \(\underline{N} = {1,2,\ldots ,N}\) (although, following the computer scientists, \(\underline{2}\) 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 \(x(\mathbf{n}) \in \underline{N}\).
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Landsman, K. (2017). Classical physics on a finite phase space. In: Foundations of Quantum Theory. Fundamental Theories of Physics, vol 188. Springer, Cham. https://doi.org/10.1007/978-3-319-51777-3_1
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DOI: https://doi.org/10.1007/978-3-319-51777-3_1
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