A path, in general, is defined by a determinate trajectory in time, from an initial to a final point. The classical trajectory is only one of the possible trajectories, and in quantum mechanics all the possible paths between the initial and final point come into play. Recall that the probability amplitude is a complex number that is assigned to each determinate path. Indeterminate paths are defined as a collection of determinate paths that are experimentally indistinguishable. In the Dirac–Feynman approach, the inherent indeterminacy of the quantum entity is realized by the degree of freedom – in undergoing time evolution – “taking” indeterminate paths [Baaquie (2013e)].

For a quantum degree of freedom evolving from an observed initial state to the observed final state – and with no other observations made – the Feynman path integral is a mathematical construction that computes the probability amplitudes by summing over all the allowed determinate paths of the degree of freedom – discussed in Feynman and Hibbs (1965), Zinn-Justin (1993), Zinn-Justin (2005) and Baaquie (2013e).

Probability amplitude and time evolution

Recall that the description of a quantum system, at a particular instant,isgiven by its state vector, namely |ψ>. To avoid confusion with the concept of a state vector, the term probability amplitude is used for describing a quantum entity undergoing transitions in time.

The simple harmonic oscillator is exactly soluble because it has a quadratic potential and yields a linear theory in the sense that the classical equation of motion is linear. Nonlinear path integrals have potentials that typically have a quartic or higher polynomial dependence on the degree of freedom, or the potential can be a transcendental function, such as an exponential. The techniques discussed for quadratic Gaussian path integrals needed to be further developed for nonlinear path integrals. In general, to solve these nonlinear systems, one usually uses either a perturbation expansion or numerical methods.

A perturbation expansion is useful if the theory has a behavior that is smooth about the dominant piece of the action or Hamiltonian; in practice a smooth expansion of the physical quantities yields an analytic series in some expansion parameter, say a coupling constant g, around g = 0.

There are, however, cases of physical interest for which nonperturbative effects change the qualitative behavior of the theory. Two examples where nonperturbative effects dominate are the following:

1. • Tunneling through a finite barrier. If one perturbs about the lowest lying eigen-states inside a well, one cannot produce the tunneling amplitude.

2. • The spontaneous breaking or restoration of a symmetry cannot be produced by perturbing about an incorrect ground state.

Tunneling and symmetry breaking are nonperturbative because these effects depend on g as exp{—1/g}, which is nonanalytic about g = 0, and hence cannot be obtained by perturbation theory.

The degrees of freedom studied so far have been either real or complex variables. These variables commute under multiplication, in the sense that two numbers a, b satisfy ab = ba; commuting variables are generically called bosonic variables, or bosonic degrees of freedom. Typical of the bosonic case are the degrees of freedom for a collection of quantum mechanical particles.

Interactions of fundamental particles are generally mediated by bosonic fields such as the Maxwell electromagnetic field, whereas mass is usually carried by particles that are fermions, the most familiar being the electron.

Two key features distinguish fermions from bosons:

• Fermions obey the Pauli exclusion principle, which states that no two fermions can occupy the same quantum state. This is the reason the concept of intensity does not apply to a fermion. A high intensity electric field is a reflection of the presence of a large number of photons, which are bosons, in the same quantum state; for photons, any number of photons can be in the same quantum state. In contrast, an electron is either in a quantum state or it is not; in particular, ignoring spin, an electron exists at a point or there is no electron there.

• The state function of a multi-bosonic system is totally symmetric in that the exchange of any two bosonic degrees of freedom yields the same state function. In contrast, a multi-fermion system is totally anti-symmetric: the exchange of any two fermion degrees of freedom gives the same state – but with a negative sign.

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.