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Edizioni della Normale eBooks

Mathematics of quantum theory

Alexander Semenovich Holevo

Published Jan 1, 2011• 2 citations
Publisher

Abstract

In the previous chapter the quantum statistical model was introduced in its simplest finite-dimensional form. However to describe many interesting and important properties of quantum-mechanical objects the infinitedimensional generalization of this model is needed, where matrices are replaced by operators in a Hilbert space. 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
10.1007/978-88-7642-378-9_2

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In the previous chapter the quantum statistical model was introduced in its simplest finite-dimensional form. However to describe many interesting and important properties of quantum-mechanical objects the infinitedimensional generalization of this model is needed, where matrices are replaced by operators in a Hilbert space. 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.
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.
In this chapter we introduce the Hilbert space which plays the central role in quantum mechanics. For a more detailed discussion of this subject we refer to the books of Balakrishnan [5], Richtmyer [137], Sewell [146], Stakgold [152], Steeb [163], Weidmann [181], Yosida [185]. Moreover the proofs of the theorems given in this chapter can be found in these books. We assume that the reader is familiar with the notation of a linear space. First we introduce the pre-Hilbert space. 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.
When the movement of a particle with noise was introduced in Chapter I, we were interested in the source of the noise, but postponed further discussion about it, accepting that the noise is presumably caused by the structure of the vacuum itself, and also by our mathematical method of treating the problem. We now return to the subject. On the other hand, we have seen in Chapter IV that the single particle interpretation of quantum mechanics is problematic, since in exited states of an electron, say in a hydrogen atom, the ergodic decomposition occurs at the zero sets of distribution densities (see Theorems 5.2.2 and 5.4.2). In other words, since the tunnelling is not allowed at the zero sets, the long-time average of a sample path cannot realize the predicted distribution of an electron in an excited state. This indicates that quantum mechanics must be a statistical theory. To clarify this point we need an additional mathematical structure. A statistical model was discussed in chapters VII and VIII of Nagasawa (1993) to solve this problem. We will explain it briefly, and apply it to relativistic quantum particles. 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.
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.
As was repeatedly seen in the last chapter, for computing most of the usual matrix elements only a few of the many coordinates of an N -particle wavefunction are used, where due to the symmetry properties it is irrelevant which ones. Density matrices are a tool to extract the relevant information out of such monstrous constructs as are N -particle wavefunctions. (For reviews, see [Erdahl and Smith, 1987, Coleman, 1963, McWeeny, 1960]; in the text below we try to keep the notation as canonical as possible.) 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.