Chapter 4 Spaces of Quantum Observables

Tilak Sinha

With the postulates already formulated in the language of an abstract inner product spaces, this chapter shows how an observable can be naturally expressed as a linear Hermitian operator acting on the inner product space where the states live. We start by introducing linear operators acting on vector spaces and their representations. Next, we discuss the systems of eigenvalues and eigenvectors associated with such operators. The important case of Hermitian operator and the properties of their eigenvalues and eigenvectors, relevant to quantum mechanics, are introduced. After developing the necessary background in linear algebra, the connection of Hermitian operators with quantum observables is established. We then proceed to explain what this connection actually accomplishes. In particular, we show how it translates the target questions of quantum mechanics into an eigenvalue problem. This makes it evident what the nature of the input to a quantum theory should be. This input enters the formalism through what are known as “quantization rules”. I discuss this in the last section of this chapter. An illustration using a disguised spin-half system winds up the chapter.

Vasily Evgenevich Tarasov

Kevin Costello, Owen Gwilliam

Sarah Durston, Ton Baggerman

Chapter 4 explored the idea that reality may be an interactive process leading to the stage of time and space rather than set on it, and – perhaps more speculatively – that there may be a role for us as observers in the process of causing reality to unfold. The last chapter investigated synchronicity and noted that there is an apparent overlap between phenomena, such as quantum entanglement and our experience of meaningful coincidences. Now Chapter 6 picks up the thread from Chapter 4 and asks why, if reality is a process, it does not appear random. What is the mechanism by which it appears stable to us, as a continuous stream of events?

Robert Alicki, M. Fannes

This chapter reminds, without entering into details, the main mathematical concepts and results relevant for finite system quantum mechanics. The basic postulates single out a Hilbert space of wave functions with self-adjoint linear operators corresponding to observables as was originally discovered by von Neumann. The chapter connects the contemporary terminology of linear Hilbert space operators with quantum physics. Important concepts like linear operators, measures, self-adjointness, spectral measures, density matrices, and tensor products are discussed and illustrated in the light of observables, probability for quantum systems and composite systems. A first example of a useful algebra of observables, the Weyl algebra, is described in detail and linked to the classical phase space of a point particle.