In this chapter, after a brief review of the basics of differential and Riemannian geometry, we shall discuss some of the fundamental concepts of noncommutative geometry. After that, we shall illustrate with examples how quantum dynamical semigroups arise naturally in the context of classical and noncommutative geometry, and how they carry important information about the underlying classical or noncommutative geometric spaces. These semigroups are essentially the ‘heat semigroups’ with unbounded generator given by the Laplacian (or some variant of it) on the underlying space; and in the classical case, the dilation of such semigroups naturally involve a suitable Brownian motion on the manifold. While the classical theory of heat semigroup and Brownian motion on a manifold is well established and quite rich, there is not yet any general theory of their counterparts in noncommutative geometry. Neither the theory of quantum stochastic calculus nor noncommutative geometry are at a stage for developing a general theory connecting the two. Instead of a general theory, the present state of both subjects calls for an understanding of various examples available, and this is what we try to do in this chapter. We do so at two levels: first, at the semigroup level, and then at the level of quantum stochastic processes coming from dilation of the semigroups.

Basics of differential and Riemannian geometry

We presume that the reader is familiar with the basic concepts of differential geometry, including tangent, cotangent, differential forms etc. and at least the definition and elementary properties of Lie groups. Let us very briefly review the concepts of connection, curvature and also of Riemannian geometry.

In Chapter 6 we built a theory of quantum stochastic dilation ‘naturally’ associated with an arbitrary Q.D.S. on a von Neumann or C*-algebra with bounded generator. There the computations involved C* or von Neumann Hilbert modules, using the results of [24], map-valued quantum stochastic processes on modules and quantum stochastic integration with respect to them, developed in Chapter 5. It is now natural to consider the case of a Q.D.S. with unbounded generator and ask the same questions about the possibility of dilation. As one would expect, the problem is too intractable in this generality and we need to impose some further structures on it. In this chapter we shall consider a few classes of such Q.D.S. and try to construct H–P and E–H dilation for them. At first, we shall work under the framework of a Lie group action on the underlying algebra, and consider covariant Q.D.S. For H–P dilation, symmetry with respect to a trace is also assumed, whereas a general theory for E–H dilation has been built under the assumption of covariance under the action of a compact group, but without it being symmetric. Then, in the last section, we deal with a class of Q.D.S. on the U.H.F. algebra, described in Chapter 3. In this case, E–H dilation is constructed by a direct iteration using some natural estimates. However, what is common to the methods used in constructing dilation for the different kinds of Q.D.S. mentioned above is the use of a natural locally convex topology, in which the generator (unbounded in the norm topology) is continuous.

The motivations for writing the present monograph are three-fold: firstly from a physical point of view and secondly from two related, but different mathematical angles.

At the present time our mathematical understanding of a conservative quantum mechanical system is reasonably complete, both from the direction of a consistent abstract theory as well as from the one of mathematical theories of applications in many explicit physical systems like atoms, molecules etc. (see for example the books [12] and [108]). However, a nonconservative (open/dissipative) quantum mechanical system does not enjoy a similar status. Over the last seven decades there have been many attempts to make a theory of open quantum systems beginning with Pauli [104]. Some of the typical references are: Van Hove [126], Ford et al. [52], along with the mathematical monograph of Davies [35]. The physicists' Master equation (or Langevin equation) was believed to describe the evolution of a nonconservative open quantum (or classical) mechanical system, a mathematical description of which can be found in Feller's book [50].

Physically, one can conceive of an open system as the ‘smaller subsystem’ of a total ensemble in which the system is in interaction with its ‘larger’ environment (sometimes called the bath or reservoir). The total ensemble with a very large number of degrees of freedom undergoes (conservative) evolution, obeying the laws of standard quantum mechanics. However, for various reasons, practical or otherwise, it is of interest only to observe the system and not the reservoir, and this ‘reduced dynamics’ in a certain sense obeys the Master equation (for a more precise description of these, see [35]).

Let us now restrict ourselves to the case when the general locally convex space X is replaced by a C* or a von Neumann algebra A, and study the implications of the complete positivity of a semigroup Tt acting on it.

Definition 3.0.1 A quantum dynamical semigroup (Q.D.S) on a C*-algebra A is a contractive semigroup Tt of class C0 such that each Tt is a completely positive map from A to itself. Tt is said to be conservative if Tt (1) = 1 for all t ≥ 0.

Generators of uniformly continuous quantum dynamical semigroups: the theorems of Lindblad and Christensen–Evans

For a uniformly continuous semigroup on a von Neumann algebra A ⊆ B(h), we have the following result.

Lemma 3.1.1Let Tt = etL be a uniformly continuous contractive semigroup acting on A with L as the generator. Then Tt is normal for each t if and only if L is ultra-strongly (and hence ultra-weakly) continuous on any norm-bounded subset of A.

Proof:

Let us first note that L is norm-bounded. If L is ultra-strongly continuous on bounded sets, then clearly etL is ultra-strongly continuous on bounded sets for each t, and hence normal. For the converse, first note that for any t ≥ 0 and x ∈ A, we have

Hence it is not difficult to see that

Now suppose that xα is a net of elements in A such that xα strongly converges to x ∈ A and there exists positive constant M such that ∥xα∥ ≤ M for all α. Fix u ∈ h and ∈ > 0. Choose t0 small enough so that ∥L∥2M∥u∥t0 ≤ ∈.

In this chapter we introduce the idea of Hilbert modules and briefly discuss some useful results on them. For a more detailed account on this subject, the reader is referred to [81], [90], [98] (and [122] for von Neumann modules).

Hilbert C*-modules

A Hilbert space is a complex vector space equipped with a complex-valued inner product. A natural generalization of this is the concept of Hilbert module, which has become quite an important tool of analysis and mathematical physics in recent times.

Definition 4.1.1 Given a *-subalgebra A ⊆ B(h) (where h is a Hilbert space), a semi-Hilbert A-module E is a right A-module equipped with a sesquilinear map 〈., .〉 : E × E → A satisfying 〈x, y〉* = 〈y, x〉, 〈x, ya〉 = 〈x, y〉a and 〈x, x〉 ≥ 0 for x, y ∈ E and a ∈ A. A semi-Hilbert module E is called a pre-Hilbert module if 〈x, x〉 = 0 if and only if x = 0; and it is called a Hilber C*-module if furthermore A is a C*-algebra and E is complete in the norm x → ∥〈x, x〉∥1/2 where ‖.‖ the C*-norm of A.

It is clear that any semi-Hilbert A-module can be made into a Hilbert module in a canonical way: first quotienting it by the ideal {x : 〈x, x〉 = 0} and then completing the quotient.

Let us assume that A is a C*-algebra. The A-valued inner product 〈., .〉 of a Hilbert module shares some of the important properties of usual complexvalued inner product of a Hilbert space, such as the Cauchy–Schwartz inequality, which we prove now.

On the one hand, in almost all the scientific areas, from physical to social sciences, biology to economics, from meteorology to pattern recognition in remote sensing, the theory of classical probability plays a major role and on the other much of our knowledge about the physical world at least is based on the quantum theory [12]. In a way, quantum theory itself is a new kind of theory of probability (in the language of von Neumann and Birkhoff) (see for example [106]) which contains the classical model, and therefore it is natural to extend the other areas of classical probability theory, in particular the theory of Markov processes and stochastic calculus to this quantum model.

There are more than one possible ways (see for example [127]) to construct the above-mentioned extension and in this book we have chosen the one closest to the classical model in spirit, namely that which contains the classical theory as a submodel. This requirement has ruled out any discussion of areas such as free and monotone-probability models. Once we accept this quantum probabilistic model, the ‘grand design’ that engages us is the ‘canonical construction of a *-homomorphic flow (satisfying a suitable differential equation) on a given algebra of observables such that the expectation semigroup is precisely the given contractive semigroup of completely positive maps on the said algebra’.

This problem of ‘dilation’ is here solved completely for the case when the semigroup has a bounded generator, and also for the more general case (of an unbounded generator) with certain additional conditions such as symmetry and/or covariance with respect to a Lie group action.

In this chapter we shall introduce all the basic materials and preliminary notions needed later on in this book.

C* and von Neumann algebras

For the details on the material of this section, the reader may be referred to [125], [40] and [76].

C*-algebras

An abstract normed *-algebra A is said to be a pre C*-algebra if it satisfies the C*-property : ‖x*x‖ = ‖x‖2. If A is furthermore complete under the norm topology, one says that A is a C*-algebra. The famous structure theorem due to Gelfand, Naimark and Segal (GNS) asserts that every abstract C*-algebra can be embedded as a norm-closed *-subalgebra of B(H) (the set of all bounded linear operators on some Hilbert space H). In view of this, we shall fix a complex Hilbert space H and consider a concrete C*-algebra A inside B(H). The algebra A is said to be unital or nonunital depending on whether it has an identity or not. However, even any nonunital C*-algebra always has a net (sequence in case the algebra is separable in the norm topology) of approximate identity, that is, an nondecreasing net eμ of positive elements such that eμa → a for all a ∈ A. Note that the set of compact operators on an infinite dimensional Hilbert space H, to be denoted by K(H), is an example of nonunital C*-algebra.

We now briefly discuss some of the important aspects of C*-algebra theory. First of all, let us mention the following remarkable characterization of commutative C*-algebras.