In the previous chapter we have seen that departures from equilibrium generate a plethora of new phenomena in statistical physics. While our discussion so far has been limited to classical phenomena, in this chapter we want to ask how quantum mechanics interferes with the conditions of nonequilibrium. Experience shows that, in some cases, nonequilibrium quantum systems can be understood in terms of reasonably straightforward quantization of their classical limits. In others, an interplay of quantum coherence and nonequilibrium driving leads to entirely new phenomena – such as lasing, a paradigm for a steady state nonequilibrium quantum system. At any rate, we have every reason to anticipate that quantum nonequilibrium physics will be as diverse and colourful as its classical counterpart. Even minor departures from equilibrium call for a theoretical description entirely different from the ‘Matsubara formalism’ developed in earlier chapters of this book. The applicability of the latter is rigidly tied to the existence of a quantum grand canonical density operator, the hallmark of a many particle equilibrium system. But how then, can an functional theory of nonequilibrium quantum systems be developed? An obvious idea would be to once more go through the different elements of classical nonequilibrium theory developed in the previous chapter, and subject every one of them to an individual quantization scheme. Luckily, there are more efficient ways to achieve our goal: some decades ago, a many particle formalism suitable to describe nonequilibrium systems under the most general conditions was introduced by Keldysh.
In the previous chapter we have seen that departures from equilibrium generate a plethora of new phenomena in statistical physics. While our discussion so far has been limited to classical phenomena, in this chapter we want to ask how quantum mechanics interferes with the conditions of nonequilibrium. Experience shows that, in some cases, nonequilibrium quantum systems can be understood in terms of reasonably straightforward quantization of their classical limits. In others, an interplay of quantum coherence and nonequilibrium driving leads to entirely new phenomena – such as lasing, a paradigm for a steady state nonequilibrium quantum system. At any rate, we have every reason to anticipate that quantum nonequilibrium physics will be as diverse and colourful as its classical counterpart. Even minor departures from equilibrium call for a theoretical description entirely different from the ‘Matsubara formalism’ developed in earlier chapters of this book. The applicability of the latter is rigidly tied to the existence of a quantum grand canonical density operator, the hallmark of a many particle equilibrium system. But how then, can an functional theory of nonequilibrium quantum systems be developed? An obvious idea would be to once more go through the different elements of classical nonequilibrium theory developed in the previous chapter, and subject every one of them to an individual quantization scheme. Luckily, there are more efficient ways to achieve our goal: some decades ago, a many particle formalism suitable to describe nonequilibrium systems under the most general conditions was introduced by Keldysh.
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.
Perhaps due to its technical complexity, oftentimes one sees in research papers on nonequilibrium quantum field theory (NEqQFT) more emphasis placed on the field-theoretical formalisms than the ideas these sophisticated techniques attempt to capture, or the issues such problems embody. All the more so, we need some basic understanding of the important issues and concepts in nonequilibrium statistical mechanics (NEqSM), and how they are manifested in the context of quantum field theory. Many important advances in this field came from asking such questions and finding out how to answer them in the language of quantum field theory. Because of this somewhat skewed existing emphasis in NEqQFT, and since we do not assume the reader to have had a formal course on NEqSM before, we shall give a brief summary of the basic concepts of NEqSM relevant to the field-theoretical processes discussed in this book. Many fine monographs and reviews written on this subject take a more formal mathematical approach. approach. Since our purpose here is to familiarize readers with these issues and their subtleties, rather than training them to work in the rich field of NEqSM (which includes in addition to the traditional subject matter such as the projection operator formalism and open system concepts, also current topics at their foundation, such as dynamical systems and quantum chaos), we choose to approach these topics in a more intuitive and physical way, sacrificing by necessity rigor and completeness.
Quantum physics consists of a formal core that is universally agreed upon (basically being a piece of mathematics with suggestive names for concepts matching related concepts of experimental culture) and an interpretational halo (trying to make this relation more precise) that remains highly disputed even after more than 90 years of modern quantum physics. The latter is the subject of the interpretation of quantum mechanics (see Part II), where many interpretations coexist and compete for the attention of those interested in what quantum physics really means. In this chapter, an axiomatic introduction to the undisputed formal core of quantum physics is given. The axioms described present nonrelativistic quantum statistical mechanics in the Schrödinger picture. The relativistic case is outside the scope of these axioms, as it must be treated by quantum field theory; see Chapter 7. As in any axiomatic setting (necessary for a formal discipline), there are a number of different but equivalent sets of axioms or postulates that can be used to define formal quantum physics. Since they are equivalent, their choice is a matter of convenience. The choice presented here is a formulation featuring three aspects: It emphasises the similarity of quantum mechanics and classical mechanics, gives most direct access to statistical mechanics, and is free from allusions to measurement. The reason for the first is that this similarity serves as a guide to meaning. From the axiomatic setting for quantum mechanics presented below, classical mechanics (in the Koopman setting discussed in Section 7.10) is obtained by simply restricting the operators to be diagonal, so that all operations happen pointwise on the diagonal elements. Thus, multiplication is commutative, and one can identify operators and functions. In particular, the density operator degenerates into a probability density. Therefore, the quantum case appears as a generalization of the classical case allowing for noncommutativity, so that both q-observables and the density operator are (usually infinite-dimensional) operators. The reason for the second is that statistical mechanics is the main tool for applications of quantum physics to the macroscopic systems we are familiar with. The reason for the third is that realistic measurements constitute a complex process involving macroscopic detectors, hence should be explained by quantum statistical mechanics rather than be part of the axiomatic foundations themselves. (This is in marked contrast to other foundations, and distinguishes the present system of axioms.)
In the two preceding chapters of this book we have analyzed many interesting physical phenomena in magnetic and superconducting systems which could adequately be described by phenomenological dynamical equations in terms of collective classical variables. One unavoidable consequence of this approach is that, as we are always dealing with variables that describe only part of the whole system, the interaction with the remaining degrees of freedom shows up through the presence of non-conservative terms which describe the relaxation of those variables to equilibrium. Those phenomenological equations are able to describe a very rich diversity of physical phenomena, in particular, those which can be studied in the context of quantum mechanics. Since these are genuine dynamical equations, there is no reason why they should be restricted to classical physics. However, as we do not yet know how to treat dissipative effects in quantum mechanics, we have deliberately neglected those terms when trying to describe quantum mechanical effects of our collective variables. In this chapter we will describe the general approach to dealing with dissipation in quantum mechanics. However, before we embark on this enterprise we should spend some time learning a little bit about the classical behavior of dissipative systems. In this way we can develop some intuition on how systems evolve during a dissipative process and, hopefully, this will be useful later on when we deal with quantum mechanical systems. The immediate problem we have to face concerns the choice of dissipative system to be studied.
Scope of this book Quantum mechanics is a difficult subject, and this book is intended to help the reader overcome the main difficulties in the way to understanding it. The first part of the book, Chs. 2–16, contains a systematic presentation of the basic principles of quantum theory, along with a number of examples which illustrate how these principles apply to particular quantum systems. The applications are, for the most part, limited to toy models whose simple structure allows one to see what is going on without using complicated mathematics or lengthy formulas. The principles themselves, however, are formulated in such a way that they can be applied to (almost) any nonrelativistic quantum system. In the second part of the book, Chs. 17–25, these principles are applied to quantum measurements and various quantum paradoxes, subjects which give rise to serious conceptual problems when they are not treated in a fully consistent manner. The final chapters are of a somewhat different character. Chapter 26 on decoherence and the classical limit of quantum theory is a very sketchy introduction to these important topics along with some indication as to how the basic principles presented in the first part of the book can be used for understanding them. Chapter 27 on quantum theory and reality belongs to the interface between physics and philosophy and indicates why quantum theory is compatible with a real world whose existence is not dependent on what scientists think and believe, or the experiments they choose to carry out. The Bibliography contains references for those interested in further reading or in tracing the origin of some of the ideas presented in earlier chapters.
Calculations of nonequilibrium processes become increasingly feasable in quantum field theory from first principles. There has been important progress in our analytical understanding based on 2PI generating functionals. In addition, for the first time direct lattice simulations based on stochastic quantization techniques have been achieved. The quantitative descriptions of characteristic far-from-equilibrium time scales and thermal equilibration in quantum field theory point out new phenomena such as prethermalization. They determine the range of validity of standard transport or semi-classical approaches, on which most of our ideas about nonequilibrium dynamics were based so far. These are crucial ingredients to understand important phenomena in high-energy physics related to collision experiments of heavy nuclei, early universe cosmology and complex many-body systems.