Chapter 3 Physical observables and representations

Susumo Okubo

N. J. B. Green

This chapter discusses some of the fundamental principles in quantum mechanics. Towards the end of the nineteenth century, it had been thought that all the fundamental laws of nature were well-understood. However, the next 20 years saw the development of both quantum mechanics and relativity. The chapter then looks at the representation of a particle by a wavefunction. It also considers operators, observables, eigenfunctions, and eigenvalues, as well as commutation and uncertainty. Since it is not possible for a wavefunction to have characteristic values of two observables with non-commuting operators, it is not possible to specify exact simultaneous values for these observables. It should be noted that the uncertainty principle is rather deeper than the usual physical explanation might imply, it is actually inherent in the description of matter in terms of wavefunctions. Finally, the chapter explains expansion in eigenfunctions; the variational principle; and transition probabilities.

Emmanuel E. Haven, Andrei Yu. Khrennikov

Introduction In previous chapters, we discussed the conjecture of quantum-like (QL) processing of information in the brain. See, for instance, Chapter 1 (Section 1.16) and Chapter 3 (Section 3.1). In general, such processing need not be based on the physical quantum brain (see, for instance, Hameroff [1] [2] and Penrose [3] [4]), which is the quantum physical carrier of information. As we remarked in Khrennikov [5], the brain created the QL representation (QLR) of information in Hilbert space. Such representation uses quantum information rules in decision making. The existence of such QLR was (at least preliminary) confirmed by experimental data from cognitive psychology (please see Chapter 8). The violation of the law of total probability in these experiments is an important sign of the non-classicality of data. In the constructive wave function approach (see Section 8.16 of Chapter 8), such data can be represented by complex amplitudes. We also presented the QL model of decision making in Chapter 9. In this chapter we reproduce (with only slight modifications) the paper by Khrennikov [6]. The next natural step is to try to find possible physical realizations of the QLR in the brain. One of the possibilities is to appeal to the quantum physics of the microprocesses in the brain - the quantum brain. However, quite surprisingly, it is possible to proceed within the classical field framework, i.e. to create a classical wave model producing the QLR.

A.G. Sitenko

Laura Ruetsche

This chapter explicates the Hamiltonian scheme for quantizing classical mechanical theories by finding a Hilbert space representation of the appropriate canonical commutation relations. The chapter also reviews the canonical anticommutation relations, which encapsulate the quantum mechanics of spin systems. Having catalogued reasons to regard unitary equivalence as a robust criterion of physical equivalence for quantum theories obtained by finding representations of the CCRs/CARs, the chapter presents a pair of theorems—the Stone-von Neumann and Jordan-Wigner theorems—that establish that, provided only finitely many degrees of freedom are involved, all representations of the CCRs/CARs for a given quantum theory are unitarily (and so presumptively physically) equivalent.