The problem of giving a theoretical description of sequential decays is reconsidered. It is shown that the usual quantum description of such processes cannot be reconciled with the classical one. This fact suggests a more critical analysis of the actual physical situation occurring in such processes. By taking into consideration the remarkable successes which have been obtained in the description of nonsequential decays by considering the unavoidable interactions of the decaying system with the measuring apparatuses, an analogous formalism is developed for the problem under consideration. The description of the process is given in the language of the density operator. The evolution equation for such an operator has the standard form of the dynamical equations in the theory of quantum dynamical semigroups. Also in the present case, as for the nonsequential decay, the more realistic description which takes into account the actual set-up of a decay experiment turns out to yield results which agree completely with the classical ones. Some interesting problems arising within the present formalism are discussed.
The problem of giving a theoretical description of sequential decays is reconsidered. It is shown that the usual quantum description of such processes cannot be reconciled with the classical one. This fact suggests a more critical analysis of the actual physical situation occurring in such processes. By taking into consideration the remarkable successes which have been obtained in the description of nonsequential decays by considering the unavoidable interactions of the decaying system with the measuring apparatuses, an analogous formalism is developed for the problem under consideration. The description of the process is given in the language of the density operator. The evolution equation for such an operator has the standard form of the dynamical equations in the theory of quantum dynamical semigroups. Also in the present case, as for the nonsequential decay, the more realistic description which takes into account the actual set-up of a decay experiment turns out to yield results which agree completely with the classical ones. Some interesting problems arising within the present formalism are discussed.
The present review article is aimed at a clear formulation of the basic problematics of the decay and at a description of the various recent attempts to solve this delicate problem, illustrating both their successes and limitations. The organisation of the article is as follows. After a short summary of the classical description of the decay, a detailed study of the quantum non-decay probability is given. Then the peculiar dynamical situation leading to the formation of an unstable system is identified and the formal and physical aspects of the process are discussed. A detailed analysis is then made of the preparation procedure of an unstable system which is shown to amount to a localisation of the decay fragments within a distance of the order of the range of the forces acting between them. After this the discussion of the general properties of the non-decay probability at all times is completed.
Basic notions and theWigner–Weisskopf theory Although the notions of bound states, scattering and quantum transitions are well defined in the quantum theory, the description of an unstable system, involved in the process of decay, has remained an outstanding issue for many years. The problem is fundamental, since it concerns the nature of irreversible processes, one of the most important issues in statistical mechanics and the theme that is central to this book. The theory of decay is intimately connected with scattering theory and necessarily contains mathematical ideas and methods. We shall try to explain these points carefully as we get to them. We treat elsewhere in the book the ideas of Boltzmann, Van Hove and Prigogine on irreversible phenomena. The tools that are developed there are basically approximate, although very useful. One can argue that the basic rigorous characteristic of an irreversible process is that, as represented in terms of the evolution of a state in the Hilbert space of the quantum theory, it must be a semigroup. This type of evolution, resulting in an operation Z ( t ) on a state ψ , should satisfy the property Z ( t 2 ) Z ( t 1 ) = Z ( t 1 + t 2 ). (17.1) The argument is as follows. If the system evolves in time t 1 and is stopped, then evolves further at time t 2 , since the process has no memory, the total evolution should be as if the system evolved from the initial state to a state at t 1 + t 2 independently of the fact that it was done in two stages (Piron, 1976). Since the process is irreversible, the operator Z ( t ) may have no inverse. Such an evolution is called a semigroup . As we shall see, it is not possible to obtain such an evolution law in the framework of the standard quantum theory (Horwitz et al. , 1971), but recently much work has been done, and methods have been developed, based on ideas of Sz.-Nagy and Foias (1976), such as the theory of Lax and Phillips (1967) and its extension to the quantum theory (Strauss et al. , 2000) in which semigroup evolution can be achieved.
The dominance of the Copenhagen interpretation is the result of a historical accident (Cushing, 1994). At the Solvay conference in 1927, Louis de Broglie put forward the deterministic pilot wave theory, described later. Never slow to criticize, Pauli jumped up and demolished de Broglie’s proposal, and de Broglie was so discouraged by this that he did not develop his interpretation any further. The Copenhagen interpretation was then developed by Bohr, Heisenberg, Pauli and many others, as described in the previous chapter. It was strongly supported by von Neumann’s proof of the impossibility of hidden variables. Years later, in 1952, Bohm published a deterministic hidden variable theory, showing that there must be something wrong with von Neumann’s proof. The error was found by John Bell in 1966, as described in the last chapter, and he also showed how Pauli’s objections to de Broglie’s original idea can be answered. If this had been done at the conference in 1927, quantum mechanics could have been interpreted deterministically from the beginning. The essential mistake in the Copenhagen interpretation is to treat it as a complete account of the behaviour of each individual system. It is frequently claimed that it enables us to calculate, at least in principle, everything that can be measured. Thus Hooft (1997, p. 11) said that ‘The laws of quantum mechanics have been formulated very accurately. We know exactly how to compute anything we would like to know.’ Also Peierls (1997, p. 25) has said that quantum mechanics ‘had become a complete and consistent scheme capable of giving a unique answer to any questions relating to actual or possible observations’. Even if this were true, it would not imply that it is the final complete theory, for it is always possible that some new phenomenon might be found that cannot be calculated quantum-mechanically. However, it is not true: there are many phenomena such as the time of decay of a radioactive nucleus, or the direction a particle is scattered by a nucleus, that cannot be calculated, even in principle. Quantum mechanics is therefore an incomplete theory. We can only calculate the statistical properties of these systems, such as the half-life of a radioactive decay or the differential scattering cross-section that gives the probabilities that a particle is scattered through various angles. All measurements of quantum systems are of this statistical character. It might be thought that the possibility of making measurements on a single electron provides an exception to this. However, even in such cases the electron is continually bathed in a fluctuating background radiation from nearby atoms. Since this is variable, that electron is a member of an ensemble of electrons subject to different fluctuations. All these examples clearly demonstrate the statistical character of quantum mechanics. ‘I am rather firmly convinced’, Einstein (in Schilpp, 1949, pp. 666, 671) remarked, ‘that the essentially statistical character of contemporary quantum theory is solely to be ascribed to fact that this theory operates with an incomplete description of physical systems’. As a result, ‘the c-function is to be understood as the description not of a single system but of an ensemble of systems’. So, ‘if the statistical quantum theory does not pretend to describe the individual system completely, it appears unavoidable to look elsewhere for a complete description of the individual systems’. Thus ‘the difficulties of theoretical interpretation disappear, if one views the quantum-mechanical description as a description of an ensemble of systems’. If this is achieved, ‘the statistical theory would, within the framework of future physics, take an approximately analogous position to statistical mechanics within the framework of classical mechanics’. ‘Thus, essentially, nothing has changed since Galileo or Newton or Faraday concerning the status of the ‘‘observer’’ or our ‘‘consciousness’’ or of our ‘‘information’’ in physics’ (Popper, 1982, p. 46). Once this is accepted, the quantum paradoxes that plagued the Copenhagen interpretation are easily resolved, as described below. In any textbook of quantum mechanics problems such as the calculation of the energy levels of the hydrogen atom are solved as if that hydrogen atom is alone in the universe. As Feynman (1972) remarked, ‘when we solve a quantum-mechanical problem, what we really do is to divide the universe into two parts – the system in which we are interested and the rest of the universe. We then usually act as if the system in which we are interested comprised the whole universe.’ In fact, however, every hydrogen atom on which measurements are made is surrounded by other atoms and exposed to their radiations. Quantum mechanics somehow takes this into account and gives the average behaviour of an ensemble of hydrogen atoms. A more detailed theory takes these influences into account. One attempt to do this is stochastic electrodynamics, described in a later section.
The theory of sequential decays of an unstable system is studied. Examples include the sequential emission of two or more photons by an excited atom which reaches its ground state via one or more intermediate levels, and the decay of an unstable particle into other unstable particles. To describe these phenomena, a factorization of the Green's function is introduced. This leads to a simple, and intuitively obvious, description of sequential decays. It also makes possible an assessment of the accuracy of this description.
In the models of quantum computing usually considered, some quantum data is manipulated by means of operations which are controlled in an essentially classical way. Controlling these operations in a quantum way is actually possible, but has been much less studied. In particular, quantum control misses a formalism in which one could represent it in a simple way in order to efficiently reason on processes involving it. The first contribution of this thesis is to lay the foundations of a formal framework dedicated to quantum control, in the form of a graphical language. Our main result about this language is the introduction of a complete equational theory, that is, a set of equations that makes it possible, by successive local rewriting, to transform a given diagram into any other diagram representing the same program or physical process. A second contribution is to apply this formalism, on the one hand, to the problem of resource optimisation of processes involving quantum control, and on the other hand, to the characterisation of the observational equivalence of quantum communication channels. A third contribution of this thesis is to introduce a language for linear optical circuits. We equip this language with a complete equational theory, together with a simple normal form, reachable via a strongly normalising and confluent rewriting system. The last contribution of this thesis, maybe the most significant one, is to introduce a complete equational theory for the language of quantum circuits. We obtain this result by exploiting a correspondence between quantum circuits and optical circuits, which allows us to transfer the equational theory already obtained for optical circuits.
We consider the extended Hubbard model and introduce a corresponding Heisenberg-like problem written in terms of spin operators. The derived formalism is reminiscent of Anderson's idea of the effective exchange interaction and takes into account nonlocal correlation effects. The results for the exchange interaction and spin susceptibility in the magnetic phase are expressed in terms of single-particle quantities. This fact not only can be used for realistic calculations of multiband systems but also allows us to reconsider a general description of many-body effects in the most interesting physical regimes, where the physical properties of the system are dominated by collective (bosonic) fluctuations. In the strongly spin-polarized limit, when the local magnetic moment is well defined, the exchange interaction reduces to a standard expression of the density functional theory that has been successfully used in practical calculations of magnetic properties of real materials.