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Open Quantum Systems and Feynman Integrals (Fundamental Theories of Physics): P. Exner

Every part of physics offers examples of non-stability phenomena, but probably nowhere are they so plentiful and worthy of study as in the realm of quantum theory. The present volume is devoted to this problem: we shall be concerned with open quantum systems, i.e. those that cannot be regarded as isolated from the rest of the physical universe. It is a natural framework in which non-stationary processes can be investigated.

There are two main approaches to the treatment of open systems in quantum theory. In both the system under consideration is viewed as part of a larger system, assumed to be isolated in a reasonable approximation. They are differentiated mainly by the way in which the state Hilbert space of the open system is related to that of the isolated system — either by orthogonal sum or by tensor product. Though often applicable simultaneously to the same physical situation, these approaches are complementary in a sense and are adapted to different purposes. Here we shall be concerned with the first approach, which is suitable primarily for a description of decay processes, absorption, etc. The second approach is used mostly for the treatment of various relaxation phenomena. It is comparably better examined at present; in particular, the reader may consult a monograph by E. B. Davies.

In the existing literature, unstable systems are seldom studied from first principles. From the practitioner's point of view, this is not necessary as we have a functioning method, developed in the early years of quantum theory, by which we can satisfactorily describe the observed decays and often predict their characteristics. However, in a theory pretending to be a complete and coherent description of the micro-world, such a method should admit derivation from the postulates. This task is fraught with peculiar difficulties, mainly because the mathematics here is somewhat more involved than in the 'stationary' quantum mechanics, and formal considerations sometimes yield misleading or seemingly paradoxical results. One of the aims of this volume is to prevent the permanent rediscovering of these 'paradoxes'. At the same time, a careful analysis of the problem can provide a deeper insight into some of the fundamental problems of quantum theory. By the methods used, the book is addressed primarily to those engaged in mathematical physics, but it is hoped that the work will be accessible and useful to a wider audience among mathematicians, physicists and quantum chemists. The choice of scope reflects the author's opinion that mathematical physics is not a part of mathematics or physics but rather a bridge between the two — a bridge whose existence must be defended at any expense.

The book can be divided into two parts. The first three chapters contain what one might call a sketch of the general theory of unstable quantum systems. Here we examine the Hilbert-space kinematics of decays, the dynamical mechanism that governs most of the practically interesting decay processes, their symmetries, etc. Particular attention is paid to a possible influence of interaction with the environment to the decay laws. These considerations illustrate how the commonly used semigroup ansatz for time evolution of the unstable systems arises. We know that it is necessarily approximative, but this is nothing to be afraid of if we understand the nature of this approximation and if we are able to estimate its degree of accuracy.

In the second part, we treat the semigroup time evolution from the viewpoint of its infinitesimal generator, thus giving a more rigorous meaning to the phenomenological non-self adjoint Hamiltonians widely used in some branches of quantum physics. For these operators, which we call pseudo-Hamiltonians, a sort of generalized quantum mechanics can be constructed. Various relevant results can be found in the mathematical literature; usually one has only to add the dissipativity condition which makes the natural physical interpretation possible. A particular interest concerns the class which appears most frequently in the applications; namely, the Schrodinger-type operators with complex-valued potentials. Notice that aside from its primary physical aim, the theory of pseudo-Hamiltonians also has some technical merit. For instance, it may sometimes be useful to regularize a mathematical construction for a given Schrodinger operator by introducing a suitable absorptive term to the potential instead of imposing some truncations or constraints. Rather than a detailed exposition of all the known results on non-selfadjoint second-order differential operators, we present a new method of treating the Schrodinger pseudo-Hamiltonians. This method is based on extending to complex potentials the (rigorous version of) Feynman's path-integral solution to the Schrodinger equation. As a preliminary, we include an extensive and more or less self-contained chapter where the Feynman path integrals are examined. This problem has attracted attention for over thirty years, during which time many people have tried to set this appealing concept on a firm mathematical basis. In fact, a physicist need not master any such theory. It is sufficient for him to know that his considerations can lean on a mathematically sound formalism which might be consulted in the case of ambiguities. A comparison with the theory of distributions arises naturally, but the present problem is further complicated by the fact that it admits various mathematical approaches of which none, until now, could win a dominating role. Hence, it is likely that we shall still not have a satisfactory and commonly accepted theory of Feynman path integrals. Nevertheless, considerable progress has been achieved in the last decade, and some of these new results will be reported here.

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