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Throughout this chapter, 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, will be the phase space rather than the configuration space). One should not frown upon finite sets: for example, the configuration space of bits is given by , where for arbitrary sets and , the set consists of all functions , and for any we write (although, following the computer scientists, usually denotes {0,1}). More generally, if one has a lattice and each site is the home of some classical object (say a “spin”) that may assume different configurations, then , in that describes the configuration in which the “spin” at site takes the value .

The quantum analogue of a finite set (in its role as a configuration space in classical mechanics) is the finite-dimensional Hilbert space , by which we mean the vector space of functions , equipped with the inner product

Passing from finite phase spaces 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.

In this chapter we generalize the results of Chapter 2 to infinite-dimensional Hilbert spaces. So let be a Hilbert space and let () be the set of all operators on . Here a notable point is that linear operators on - 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, () 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 , as explained in §B.21. Nonetheless, even the relatively easy setting of bounded operators leads to some technical complications we have to deal with.

Roughly speaking, a of some mathematical object is an invertible transformation that leaves all relevant structure as it is. Thus a symmetry of a set is just a bijection (as sets have no further structure, whence invertibility is the only demand on a symmetry), a symmetry of a topological space is a homeomorphism, a symmetry of a Banach space is a linear isometric isomorphism, and, crucially important for this chapter, a symmetry of a Hilbert space is a , i.e., a linear map satisfying one and hence all of the following equivalent conditions:

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.

Limits are essential to the asymptotic Bohrification program. It was recognized at an early stage in the development of quantum mechanics that the limit of Planck’s constant going to zero should play a role in the derivation of classical physics from quantum theory, and later on also the thermodynamic limit (which often means “”, where N is the number of particles in the system) became a subject of interest in quantum statistical mechanics.

Beside the limit , we consider the limit , where could be the principal quantum number labeling orbits in atomic physics (as in Bohr’s Correspondence Principle), or the number of particles or lattice sites, or the number of identical experiments in a long run measuring the relative frequencies of possible outcomes.

In §3.9 we defined symmetries of classical physics as symmetries of either Poisson manifolds or Poisson algebras; these notions are equivalent.

As we shall see, the undeniable natural phenomenon of spontaneous symmetry breaking (SSB) seems to indicate a serious mismatch between theory and reality. This mismatch is well expressed by what is sometimes called Earman’s Principle: ‘While idealizations are useful and, perhaps, even essential to progress in physics, a sound principle of interpretation would seem to be that no effect can be counted as a genuine physical effect if it disappears when the idealizations are removed.’ (Earman, 2004, p. 191)