Since its very beginning, quantum mechanics has been developed to deal with systems on the atomic or sub-atomic scale. For many decades, there has been no reason to think about its application to macroscopic systems. Actually, macroscopic objects have even been used to show how bizarre quantum effects would appear if quantum mechanics were applied beyond its original realm. This is, for example, the essence of the so-called Schrödinger cat paradox (Schrödinger, 1935). How-ever, due to recent advances in the design of systems on the meso- and nanoscopic scales, as well as in cryogenic techniques, this situation has changed drastically. It is now quite natural to ask whether a specific quantum effect, collectively involving a macroscopic number of particles, could occur in these systems.

In this book it is our intention to address the quantum mechanical effects that take place in properly chosen or built “macroscopic” systems. Starting from a very naïve point of view, we could always ask what happens to systems whose classical dynamics can be described by equations of motion equivalent to those of particles (or fields) in a given potential (or potential energy density). These can be represented by a generalized “coordinate” φ(r, t) which could either describe a field variable or a “point particle” if it is not position dependent, φ(r, t) = φ(t).

Having achieved this point we hope to have accomplished, at least partly, our main aim which was to convince the reader that once appropriate systems have been found (or built) they can present a very peculiar combination of microscopic parameters in such a way that quantum mechanics should be applied to general macroscopic variables to describe the collective effects therein. Moreover, the very nature of these macroscopic variables does not allow them to be treated in an isolated fashion. They must rather be considered coupled to uncontrollable microscopic degrees of freedom which is the ultimate origin of dissipative phenomena. The latter, at least in the great majority of cases, play a very deleterious role in the dynamics of the macroscopic variables and we hope to have introduced minimal phenomenological techniques in order to quantify this.

We have concentrated our discussions on questions originating from a few examples of superconducting or magnetic systems where quantum mechanics and dissipative effects coexist. In particular, superconducting devices which present the possibility of displaying several different quantum effects (quantum interference, decay by quantum tunneling, or coherent tunneling) are of special importance, as we will see below. Prior to development of the modern cryogenic techniques and/or the ability to build nanometric devices, it was unthinkable to imagine the existence of subtle quantum mechanical effects such as the entanglement of macroscopically distinct quantum states.

Superconductivity was discovered by Onnes (1911), who observed that the electrical resistance of various metals dropped to zero when the temperature of the sample was lowered below a certain critical value Tc, the transition temperature, which depends on the specific material being dealt with.

Another equally important feature of this new phase was its perfect diamagnetism which was discovered by Meissner and Ochsenfeld (1933), the so-called Meissner effect. A metal in its superconducting phase completely expels the magnetic field from its interior (see Fig. 3.1). The very fact that many molecules and atoms are repelled by the presence of an external magnetic field is quite well known, as we have already seen in the preceding chapter. The difference here lies in the perfect diamagnetism, which means that it is the whole superconducting sample that behaves as a giant atom!

This effect persists (for certain kinds of metal) until we reach a critical value of the external magnetic field, H = Hc(T), above which the superconducting sample returns to its normal metallic state. Moreover, at fixed temperature, this effect is completely reversible, suggesting that the superconducting phase is an equilibrium state of the electronic system. The temperature dependence of the critical field is such that Hc(Tc) = O and Hc(0) attains its maximum value as shown in Fig. 3.2.

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.

On deciding to write this book, I had two main worries: firstly, what audience it would reach and secondly, to avoid as far as possible overlaps with other excellent texts already existing in the literature.

Regarding the first issue I have noticed, when discussing with colleagues, super-vising students, or teaching courses on the subject, that there is a gap between the standard knowledge on the conventional areas of physics and the way macroscopic quantum phenomena and quantum dissipation are presented to the reader. Usually, they are introduced through phenomenological equations of motion for the appropriate dynamical variables involved in the problem which, if we neglect dissipative effects, are quantized by canonical methods. The resulting physics is then interpreted by borrowing concepts of the basic areas involved in the problem – which are not necessarily familiar to a general readership – and adapted to the particular situation being dealt with. The so-called macroscopic quantum effects arise when the dynamical variable of interest, which is to be treated as a genuine quantum variable, refers to the collective behavior of an enormous number of microscopic (atomic or molecular) constituents. Therefore, if we want it to be appreciated even by more experienced researchers, some general background on the basic physics involved in the problem must be provided.

In order to ill this gap, I decided to start the presentation of the book by introducing some very general background on subjects which are emblematic of macroscopic quantum phenomena: magnetism and superconductivity.

In this central and essential chapter, we develop the dynamic renormalization group approach to time-dependent critical phenomena. Again, we base our exposition on the simple O(n)-symmetric relaxational models A and B; the generalization to other dynamical systems is straightforward. We begin with an analysis of the infrared and ultraviolet singularities appearing in the dynamic perturbation expansion. Although we are ultimately interested in the infrared critical region, we first take care of the ultraviolet divergences. Below and at the critical dimension dc = 4, only a finite set of Feynman diagrams carries ultraviolet singularities, which we evaluate by means of the dimensional regularization prescription, and then eliminate via multiplicative as well as additive renormalization (the latter takes into account the fluctuation-induced shift of the critical temperature). The renormalization group equation then permits us to explore the ensuing scaling behavior of the correlation and vertex functions of the renormalized theory upon varying the arbitrary renormalization scale. Of fundamental importance is the identification of an infrared-stable renormalization group fixed point, which describes scale invariance and hence allows the derivation of the critical power laws in the infrared limit from the renormalization constants determined in the ultraviolet regime. This program is explicitly carried through for the relaxational models A and B. For model B, we derive a scaling relation connecting the dynamic exponent to the Fisher exponent η. The critical exponents ν, η, and z are computed to first non-trivial order in an ∊ expansion around the upper critical dimension dc = 4.