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 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.
In this chapter we assume that H is a Gaussian Hilbert space. We will use H to define and study several operators on L 2 (Ω, F(H) , P). These operators are important in quantum physics, but we will not go into any such applications here; cf. for example Segal (1956), Glimm and Jaffe (1981, Chapter 6) and Meyer (1993). There are also other applications of these operators, and we will use some of the results below in Chapter 15. Some of the operators will be studied again in Chapters 15 and 16, where we also consider actions on L p for p ≠ 2. The reader may note that the operators treated here can be defined for any abstract Fock space based on an arbitrary Hilbert space, and that many of the results make sense in this generality, cf. for example Baez, Segal and Zhou (1992), Meyer (1993) and Parthasaraty (1992). Nevertheless, we will exclusively consider the Gaussian case here, where the extra structure is both helpful and, we hope, illuminating. (Of course, any result that can be stated for an abstract Fock space is valid in general as long as it is valid for the concrete realization treated here for Gaussian spaces.) We warn the reader that several of the operators defined below will be unbounded and defined only on a dense subset of L 2 .
The Hilbert-Space Description By the static description of a physical system we mean the rules that assign specified mathematical objects to the states and to the physical quantities of the system, and the prescriptions for calculating the probability distribution of the possible values of every physical quantity when the state of the system is given. In the usual Hilbert-space formulation of quantum mechanics, to each physical system is attached a separable Hilbert space (generally infinite-dimensional) over the complex field. To every physical quantity is associated a linear, self-adjoint, not necessarily bounded operator on. If one deals with a strictly quantum system, then the converse is also generally assumed: every self-adjoint operator on represents some physical quan- tity. The restriction “strictly quantum” is necessary: if the system retains some nonquantum (i.e. classical) feature, so that it requires the algorithm of so-called “superselection rules” (see Chapter 5), then there are self-adjoint operators on that do not represent physical quantities. It should also be stressed that even in the strictly quantum case, most of the self-adjoint operators actually do not represent “interesting” physical quantities: only a few of them represent physical quantities that are useful and meaningful for the description of the physical system (e.g., energy, momentum, position, angular momentum). Therefore, by asserting that every self-adjoint operator on corresponds to some physical quantity we mean there is no a priori impossibility of devising such a correspondence, but we do not claim that this correspondence is present in the real work of experimental physics.
Quantum theory, in its conventional formulation, is built on the theory of Hilbert spaces and operators. In this chapter we go through this basic material, which is central for the rest of the book. Our treatment is mainly intended as a refresher and a summary of useful results. It is assumed that the reader is already familiar with some of these concepts and elementary results, at least in the case of finite-dimensional inner product spaces. We present proofs for propositions and theorems only if the proof itself is considered to be instructive and illustrative. This gives us the freedom to present the material in a topical order rather than in the strict order of mathematical implication. Good references for this chapter are the functional analysis textbooks by Conway [45], Pedersen [113] and Reed and Simon [121]. These books also contain the proofs that we skip here. Hilbert spaces As an introduction, before a formal definition is given, one may think of a Hilbert space as the closest possible generalization of the inner product spaces c d to infinite dimensions. Actually, there are no finite-dimensional Hilbert spaces (up to isomorphisms) other than c d spaces. The crucial requirement of completeness becomes relevant only in infinite-dimensional spaces. This defining property of Hilbert spaces guarantees that they are well-behaved mathematical objects, and many calculations can be done almost as easily as in c d .
We present in this paper a set of postulates for a physical system and deduce from these the main general features of the quantum theory of stationary states. Our theory is strictly operational in the sense that only the observables of the physical system are involved in the postulates. The collection of all bounded self-adj oint operators on a Hilbert space, which has previously been used as a mathematical model for the observables in quantum mechanics, satisfy the postulates, as do a variety of considerably more general mathematical structures. We show that for any two observables there exists a pure state of the system (this term being defined essentially as done by von Neumann and Weyl) in which they have different (expectation) values. Each observable has a spectral resolution, and a state of the system induces, in a natural way, a probability distribution on the range of spectral values of each observable. A number of observables are simultaneously observable if and only if they commute (although the product of two observables is not defined in general, a reasonable notion of commutativity can be introduced). Inasmuch as Hilbert space plays no role in our theory, our proofs are necessarily of a different character from the proofs of these results for the case of the system of all bounded selfadjoint operators. Actually, Hilbert space appears to be somewhat inadequate as a state space even for the latter system, in that there exist pure states of the system which cannot be represented in the usual way by rays of the Hilbert space. The postulates are partly algebraic and partly metric. The algebraic postulates require essentially that an observable can be multiplied by real numbers and raised to integral powers, and that any two observables can be added. It is assumed that the usual algebraic laws are satisfied so that (1) the observables can be treated like the elements of a linear space, and (2) the usual rules for dealing with polynomials in one variable with real coefficients remain valid when the variable is replaced by an observable. It is not assumed that two observables have a product. The metric postulates require that for each observable there be defined a kind of maximum numerical value, which plays the part of a norm, and has various properties in accord with its physical significance. While this norm is quite essential to the development of the theory, an interesting consequence of the theory is that the norm can be (uniquely) defined in a purely algebraic fashion. This shows that the objective features of a physical system,-the spectral values and probability distributions of the observables, and the pure states,-are completely determined by the algebra of observables, i.e., by the rules for addition, scalar multiplication, and powers, of observables.