2.2. Hilbert Space

Hilbert spaces are mentioned in most textbooks on quantum mechanics and functional analysis [3] . Therefore we will only mention some features, which are not found almost everywhere. We will also not have to go into the subtleties of topologies, continuous spectra, or unbounded operators, because throughout this course, we can assume that all Hilbert spaces are finite dimensional. Modifications in the infinite dimensional case will be mentioned in the notes. Our standard notation is <\phi ,\psi > for the scalar product of the vectors \phi ,\psi \in H, ||\phi ||=<\phi ,\phi >^1/2 for the norm, and B(H) for the algebra of bounded linear operators on H. Of course, all linear operators on a finite dimensional space are bounded anyway, and the B is used mostly for conformity with the infinite dimensional case.

For many purposes it is sufficient to take the Hilbert space associated with some type of systems as an abstract Hilbert space. That is to say, one needs to refer only to vectors, and the basic operations of sum, multiplication by scalars, scalar product <.,.>, and limits, but not to any particular mathematical realization of these vectors (e.g., as square integrable functions, or in terms of their components in a basis). This abstract point of view allows us to give concise formulations of general facts and arguments. However, for detailed calculations it is often necessary to choose a basis, and express everything in terms of components in this basis, or to take H=L²(X,µ), the space of square integrable functions on a "configuration space" X with respect to a measure µ (modulo almost everywhere equality). The space L²(X,µ) is called a concrete Hilbert space. Examples are, of course, the standard spaces used for the description of non-relativistic particles, with X=R³, interpreted as either position or momentum space, and µ the Lebesgue measure. Since we will look at finite dimensional spaces only, the configuration space X also has to be finite, and without loss we can take the measure µ to be counting measure. Thus in a standard notation L²(X,µ)=\ell ²(X), is the set of functions \psi :X->C with scalar product <\psi ,\phi >=\sum _x\overline {\psi (x)}\phi (x).

Abstract and concrete Hilbert spaces are completely analogous to "abstract vectors", and "vectors written out in components" in ordinary analytical geometry: the passage between the two amounts to a choice of basis, and often a computation can be simplified substantially by choosing a basis adapted to the problem. In Hilbert space theory the word "basis" is reserved for orthonormal basis, i.e., a collection of vectors \phi _x (for x\in X) spanning the space with the property that <\phi _x,\phi _y>=\delta _xy. The expansion of a vector \Psi in such a basis reads \Psi =\sum _x<\phi _x,\Psi >\phi _x. In a more suggestive form, this is written as \Psi =\sum _x|\phi _x><\phi _x,\Psi > or, introducing the "ketbra" operators |\phi ><\chi | [4] , 1=\sum _x|\phi _x><\phi _x|. The components <\phi _x,\Psi > of \Psi with respect to the basis can then be considered as the values of a function in \ell ²(X). More formally, we get a unitary operator U:H->\ell ²(X) with (U\Psi )(x)=<\phi _x,\Psi >. Unitary operators are the Hilbert space isomorphisms: they can be considered as "renaming" operations, setting up a translation table between the vectors of two spaces preserving all of the Hilbert space structure. The above operator U makes this passage between the abstract Hilbert space H and one of its concrete versions \ell ²(X). The dimension of a Hilbert space is the cardinal number of elements in any basis or, equivalently the cardinality of X, when H\isomorphic\ell ²(X). Hilbert spaces of the same dimension are thus isomorphic.

We will frequently have to construct Hilbert spaces from certain other data. A typical pattern of such constructions is summarized the following Lemma, known as Kolmogorov's Dilation Theorem for positive definite kernels. The idea is that we specify certain vectors and their scalar products, and construct a Hilbert space around these vectors. Clearly, with the given vectors, our new Hilbert space has to contain all their linear combinations. By linearity of the scalar product the scalar products (and hence the norms) of these linear combinations are likewise determined. Finally, a Hilbert space has to be complete, so it also has to contain the limits of all Cauchy sequences. Again the scalar product extends to these limits in a canonical way. These are all the vectors, which have to be there given the original vectors, their scalar products, plus the information that we have a Hilbert space. Moreover, the structure of this "minimal" Hilbert space is completely determined by the data. Only one thing can go wrong in this construction: it may turn out that one of the expressions <\phi ,\phi > comes out negative, which is obviously impossible for a Hilbert space. This becomes a constraint on the scalar products of the basic vectors. To summarize, we have the following Theorem:

Lemma: ("Kolmogorov's Dilation Theorem") Let X be a set, and q:X×X->C a function. Then the following statements are equivalent

• q is a "positive definite kernel", i.e., for each choice of finitely many x₁,...,x_n \in X the n×n-matrix M_ij=q(x_i,x_j) is positive definite.
• There is a Hilbert space H, with vectors v(x)\in H, for each x\in X, such that <v(x),v(y)>=q(x,y) for all x,y\in X.

Moreover, there is only one Hilbert space H with these properties (up to unitary equivalence) such that the linear span of the vectors v(x) is dense in H.

Sketch of proof: Suppose that the second condition holds, i.e., v:X->H exists. Then, for any collection x₁,...,x_n \in X, and complex c₁,...,c_n \in X, we find \sum _ij\overline {c_i}c_jM_ij=<\sum _ic_iv(x_i),\sum _jc_jv(x_j)>=||\sum _ic_iv(x_i)||²>=0. Hence M is positive.

For the converse, we start by building a vector space around X. The construction is known as the "free vector space" "lin(X)" over X, and can be described in various ways. The simplest is to say that lin(X) consists of all "formal linear combinations" of elements of X, i.e., expressions of the form \sum _ic_ix_i with x_i\in X. (Another possibility is to work directly with the coefficients c, i.e., to consider lin(X) as the space of functions c:X->C vanishing on all but finitely many points). We can immediately put a kind of scalar product on lin(X), by setting <\sum _ic_ix_i,\sum _jd_jx_j>=\sum _ij\overline {c_j}d_j q(x_i,x_j). (If the two vectors in a scalar product are the sum of different sets of x_i, we can just join the two sequences, and set the additional coefficients zero). The positive definiteness of the kernel q is then equivalent to the inequality <\Phi ,\Phi >>=0, for arbitrary \Phi \in lin(X).

Thus lin(X) with its scalar product is almost the space we are looking for, except that it may happen that <\Phi ,\Phi >=0 even for non-zero \Phi , and that the space lin(X) need not be complete, in the sense that it may contain non-convergent Cauchy sequences. There is a standard way to remedy both defects at the same time: the completion construction.

In this construction (which is described in detail in almost every book on functional analysis or topology [\R@BourTop ]) we consider the space H₀ of Cauchy sequences in lin(X), i.e., the set of sequences \Phi _n\in lin(X) such that ||\Phi _n-\Phi _m|| becomes arbitrarily small for n,m both sufficiently large. Vector space operations in H₀ are defined as the vector space operations of lin(X), applied n-wise. A special class of convergent sequences consists of the "null sequences", for which ||\Phi _n||->0. Call two Cauchy sequences equivalent, if they differ by a null sequence. Then one defines H, the completion of lin(X), as the space of equivalence classes. When \phi ,\psi denote the equivalence classes of the sequences \Phi ,\Psi , we define \phi +\psi as the class of \Phi +\Psi , and <\phi ,\psi >=lim_n->\infty <\Phi _n,\Psi _n>. The space H is now complete, and ||\phi ||=<\phi ,\phi >^1/2 is a proper norm, in the sense that ||\phi ||=0 implies \phi =0. There is a canonical "embedding" map from lin(X) into H, taking each \phi \in lin(X) to the equivalence class of the constant sequence \Phi _n=\phi . In particular, the elements of X are mapped to vectors v(x)\in H, and these obviously have the prescribed scalar products. One should note, of course, that v need not be injective, i.e., it is possible that v(x)=v(y) but not x=y.

In order to show the uniqueness, suppose that v(x)\in H and v'(x)\in H' satisfy the conditions of the Lemma, and generated the respective spaces. Then define the operator U:H->H' by Uv(x)=v'(x), and extend it to linear combinations. This is well-defined, because \sum _ic_iv(x_i)=0 implies ||\sum _ic_iv'(x_i)||²=||\sum _ic_iv(x_i)||²=0, and the same computation shows that U preserves the norm, so U^*U=1. Reversing the roles of H and H', we find that U is invertible, so U is even unitary. QED

Please send comments and corrections to the author, R.F. Werner.
Document created: Nov. 8, 1996; last updated: Nov. 10, 1998