One cannot do quantum mechanics without a thorough knowledge of the geometry of Hilbert space and the linear transformations on them. This book gives a good introduction to operators on Hilbert space, and could be read by a beginning graduate student of physics. The theory of operators on Hilbert spaces could be viewed as a generalization of the theory of matrix transformations on finite-dimensional vector spaces. This viewpoint is readily apparent in chapter 1, wherein the author introduces Hilbert spaces as infinite-dimensional vector spaces (over the complex numbers) with an inner product. The author shows how to handle infinite sums of vectors, which requires the notion of convergence, and how to guarantee an infinite sequence of vectors converges to a limit vector that is also in the Hilbert space: the famous Cauchy sequences of vectors. The notion of a linear functional is also introduced, the author proving the one-to-one correspondence between continuous linear functionals and vectors, and connects this with the Dirac bra-ket notation.
Observables in quantum mechanics are represented by operators on (separable) Hilbert spaces, and these are studied in chapter 2. It is straightforward to define a linear operator in finite dimensions, but in infinite dimensions one needs the notion of a continuous linear operator. The author proves that a linear operator is continuous if and only if it is bounded. Unitary operators, so crucial to the calculation of probabilities in quantum theory, are introduced in this chapter also. In addition, the author studies projection operators, which are very important in the measurement process in quantum mechanics. lastly, the author discusses unbounded operators, which are ubiquitous in quantum theory, especially in the theory of angular momenta.
Obtaining measurement results in quantum theory corresponds to obtaining an eigenvalue of a Hermitian linear operator. Thus one must develop a notion of diagonalization (or "spectral resolution") of these operators, and this takes place in chapter 3. In infinite dimensions a Hermitian or unitary operator need not have any eigenvalues or eigenvectors, but the author shows how to obtain a spectral resolution using spectral families of projection operators. He proves that a self-adjoint operator is bounded if and only if its spectrum is bounded, and also Stone's theorem, which gives a representation of a unitary operator as an exponential of a unique self-adjoint operator. Such a representation is expected from the standpoint of how time evolution is characterized in quantum mechanics.
Things become more abstract in chapter 4, wherein the author studies operator algebras. The goal of the chapter is to find conditions under which the functions of a set of noncommuting operators include all bounded operators. This problem motivates the definition of a von Neumann algebra or W*-algebra, this definition depending on the important notion of a weak topology on a set of bounded operators. It is this topology that is the most relevant for the connection of quantum theory with laboratory measurements.
In chapter 5, the author makes clearer the concept of a state in quantum mechanics, this being done using the concept of a density matrix. States specify expectation values of bounded operators, and the author shows how to represent the expectation value of a bounded operator using a unique density matrix. Probabilities in quantum-mechanical calculations are then viewed as expectation values for projection operators, and the author uses Gleason's theorem to justify that projection operators are sufficient to determine the representation of a state. Having set up all this formalism, the author then derives the uncertainty principle for a quantity represented by a Hermitian operator. He then shows that real quantities which are simultaneously measurable with unlimited precision are represented by commuting Hermitian operators. lastly, the author addresses the implicit assumption that every bounded Hermitian operator can represent a measurable quantity. He gives an example of a system that cannot, this occurring because of 'superselection rules'. An operator that commutes with every Hermitian operator which represents a measurable quantity, but is not a multiple of the identity operator is then called a 'superselection operator'. He also discusses, but does not prove in detail, the representation of the expectation value of an element of a von Neumann algebra in terms of a density matrix. When a superselection rule is in place, the density matrix is not always unique. The author then shows how these facts enable one to view a von Neumann algebra alternatively as a collection of bounded operators that commute with all the projection operators.
States of course evolve in time, and so do observables. In chapter 6 the author derives the equations of motion both for the states and the observables. For the states this is the 'Schrodinger picture', and for the observables the 'Heisenberg picture'. Wigner's theorem on unitary and antiunitary operators is used to show that the time evolution of states is linear. The Heisenberg picture is illustrated by an example of a single particle. A more complicated situation though is when the classical system is not integrable, and is still the topic of intense research. The author also includes, atypically for books at this level, a discussion of what happens to the Schrodinger picture when superselection rules are included.
The mathematical tools used in this chapter are used in chapter 7 to study Galilean and Newtonian space-time transformations of states.
