The method of faithful realizations is defined and explored within the context of several physical problems. We constructed a general procedure for deriving faithful realizations from arbitrary semi-classical truncations, as well as applying this procedure to several explicit examples. This method is useful in systems which have a strong time dependence or which lack a ground state, because in these situations one cannot use the standard tools, such as the effective action.
Using the faithful canonical realizations we developed, as well as an all orders closure, we studied the problem of tunneling times, which has been recently debated in the literature. Our definition of tunneling times, based on the canonical realizations, always gives a time delay. The usage of a canonical coordinate system also allowed us to use the moments as a tool to study the purity of a quantum state. That is, given a set of moments, a faithful canonical realization will have some parameters that correspond to the purity of the state. Moreover, canonical realizations facilitate the usages of methods from statistical mechanics in the realm of canonical effective methods, allowing one to consider ensemble averages of semi-classical quantities. They also facilitate the usage of the powerful techniques of canonical transformations in semi-classical physics. Using the framework of realization equivalence, we were also able to explore a link between the model of loop quantum cosmology as well as a model of group field cosmology. This link has led to several implications on the group field side of the equivalence, with regards to the formations of singularities and of quantization ambiguities.
The method of faithful realizations is defined and explored within the context of several physical problems. We constructed a general procedure for deriving faithful realizations from arbitrary semi-classical truncations, as well as applying this procedure to several explicit examples. This method is useful in systems which have a strong time dependence or which lack a ground state, because in these situations one cannot use the standard tools, such as the effective action.
Using the faithful canonical realizations we developed, as well as an all orders closure, we studied the problem of tunneling times, which has been recently debated in the literature. Our definition of tunneling times, based on the canonical realizations, always gives a time delay. The usage of a canonical coordinate system also allowed us to use the moments as a tool to study the purity of a quantum state. That is, given a set of moments, a faithful canonical realization will have some parameters that correspond to the purity of the state. Moreover, canonical realizations facilitate the usages of methods from statistical mechanics in the realm of canonical effective methods, allowing one to consider ensemble averages of semi-classical quantities. They also facilitate the usage of the powerful techniques of canonical transformations in semi-classical physics. Using the framework of realization equivalence, we were also able to explore a link between the model of loop quantum cosmology as well as a model of group field cosmology. This link has led to several implications on the group field side of the equivalence, with regards to the formations of singularities and of quantization ambiguities.
A careful study of the classical/quantum connection with the aid of coherent states offers new insights into various technical problems. This analysis includes both canonical as well as closely related affine quantization procedures. The new tools are applied to several examples including: (1) A quantum formulation that is invariant under arbitrary classical canonical transformations of coordinates; (2) A toy model that for all positive energy solutions has singularities which are removed at the classical level when the correct quantum corrections are applied; (3) A fairly simple model field theory with non-trivial classical behavior that, when conventionally quantized, becomes trivial, but nevertheless finds a proper solution using the enhanced procedures; (4) A model of scalar field theories with non-trivial classical behavior that, when conventionally quantized, becomes trivial, but nevertheless finds a proper solution using the enhanced procedures; (5) A viable formulation of the kinematics of quantum gravity that respects the strict positivity of the spatial metric in both its classical and quantum versions; and (6) A proposal for a non-trivial quantization of [Formula: see text] that is ripe for study by Monte Carlo computational methods. All of these examples use fairly general arguments that can be understood by a broad audience.
This dissertation addresses the problem of constructing true degrees of freedom in quantum mechanical systems with constraints. The method developed relies on assuming the behavior of the system of interest is ``nearly classical', or semiclassical, in a precise quantitative sense. The approximation is formulated as a perturbative hierarchy of the expectation values of quantum observables and their non-linear combinations, such as spreads and correlations. We formulate the constraint conditions and additional quantum gauge transformations that arise directly on the aforementioned quantities. We specialize this framework to several situations of particular interest. The first situation considered is the case of a constraint that commutes with all quantum observables, which is appropriate for quantization of a Lie algebra with a single Casimir polynomial. Through explicit order-by order counting argue that the true degrees of freedom are captured correctly at each order of the approximation. The rest of the models considered are motivated by the canonical approach to quantizing general relativity. The homogeneous sector of general relativity splits into classes of models, according to topology, which are described by a finite number of degrees of freedom and can therefore be quantized as quantum mechanical systems, making our construction directly applicable. In the Hamiltonian formulation the dynamics of these systems is governed by the Hamiltonian constraint, which, in the quantum theory, gives rise to several conceptual and technical issues collectively known as the Problem of Time. One of the aspects of this problem that does not possess a general solution is the dynamical interpretation of the theory. We use our construction, truncated at the leading order in quantum corrections, together with the intuition gained from dealing with constrained systems in classical mechanics, to define local notion of dynamics relative to a chosen configuration clock variable. We consider a class of models, where the chosen clock in not globally valid, eventually leading to singular dynamics. Within these models we construct an explicit transformation between dynamical evolution relative to two distinct clocks, thus providing a consistent local dynamical interpretation of the true degrees of freedom of the quantum theory.
Following previous works by E. Prugovečki [Physica A 91A, 202 (1978) and Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)] on common features of classical and quantum mechanics, we develop a unified mathematical framework for classical and quantum mechanics (based on L2-spaces over classical phase space), in order to investigate to what extent quantum mechanics can be obtained as a simple modification of classical mechanics (on both logical and analytical levels). To obtain this unified framework, we split quantum theory in two parts: (i) general quantum axiomatics (a system is described by a state in a Hilbert space, observables are self-adjoints operators, and so on) and (ii) quantum mechanics proper that specifies the Hilbert space as L2(Rn); the Heisenberg rule [pi,qj]=−iℏδij with p=−iℏ∇, the free Hamiltonian H=−ℏ2Δ/2m and so on. We show that general quantum axiomatics (up to a supplementary “axiom of classicity”) can be used as a nonstandard mathematical ground to formulate physical ideas and equations of ordinary classical statistical mechanics. So, the question of a “true quantization” with “ℏ” must be seen as an independent physical problem not directly related with quantum formalism. At this stage, we show that this nonstandard formulation of classical mechanics exhibits a new kind of operation that has no classical counterpart: this operation is related to the “quantization process,” and we show why quantization physically depends on group theory (the Galilei group). This analytical procedure of quantization replaces the “correspondence principle” (or canonical quantization) and allows us to map classical mechanics into quantum mechanics, giving all operators of quantum dynamics and the Schrödinger equation. The great advantage of this point of view is that quantization is based on concrete physical arguments and not derived from some “pure algebraic rule” (we exhibit also some limit of the correspondence principle). Moreover spins for particles are naturally generated, including an approximation of their interaction with magnetic fields. We also recover by this approach the semi-classical formalism developed by E. Prugovečki [Stochastic Quantum Mechanics and Quantum Space–time (Reidel, Dordrecht, 1986)].
Although classical mechanics and quantum mechanics are separate disciplines, we live in a world where Planck’s constant ℏ > 0, meaning that the classical and quantum world views must actually coexist . Traditionally, canonical quantization procedures postulate a direct linking of various c -number and q -number quantities that lie in disjoint realms, along with the quite distinct interpretations given to each realm. In this paper we propose a different association of classical and quantum quantities that renders classical theory a natural subset of quantum theory letting them coexist as required. This proposal also shines light on alternative linking assignments of classical and quantum quantities that offer different perspectives on the very meaning of quantization. In this paper we focus on elaborating the general principles, while elsewhere we have published several examples of what this alternative viewpoint can achieve; these examples include removal of singularities in classical solutions to certain models, and an alternative quantization of several field theory models that are trivial when quantized by traditional methods but become well defined and nontrivial when viewed from the new viewpoint.
The study of electrical currents in a regime where a description as a continuous flow fails and the particle nature becomes important, but where notions such as particle density or currents continue to be meaningful, will be at the heart of the present dissertation. Given such a mesoscopic system through which electrons flow, either actively through quantum pumping or passively due to a potential difference, we investigate quantities such as transferred charge, noise, current fluctuations, and relations between them. More generally we shall examine the statistics of particle transport. All discussions will concentrate on compact devices transferring charge between two one-dimensional reservoirs (the leads) of independent electrons. Firstly, we consider the generating function of the moments of transferred charge and we prove that it is given by the determinant of a single-particle operator. It is formally equivalent to a formula of Lesovik and Levitov, but is shown to be well-defined in the situations under consideration, whereas the former is not, in general. The mechanism leading to regularization is typical for mesoscopic systems: we consider a limiting case that reveals the fundamental structure of the problem. In fact, as suited for an open system, the leads contain an infinite number of charges. It follows mathematically that the second quantization ought to be carried out upon the Fermi sea and not upon the vacuum, a procedure known as the GNS construction. We shall rely heavily on operator algebraic techniques. The set of observables, the C*-algebra of canonical anticommutation relations, admits different representations for systems in the thermodynamic limit, which are described by their reduced one-particle density matrix. An important question will be the existence of operators such as the charge or the time evolution in a given representation, which induce so-called Bogoliubov transformations. These prerequisites at hand, we shall pose the problem of counting statistics clearly, prove that the regularized determinant emerges naturally, and show that it describes the transport of integer charges with non-negative probabilities. Secondly, we introduce a model of instantaneous scattering in which various examples can be computed explicitely. In particular, we show that the above theorem applies for scattering through a smoothly operating pump over a finite time interval. We also exhibit situations where the propagator does exist but the generating function is ambiguous. In a more computational approach, we then turn to the relation of individual moments of transferred charge to current correlators. It turns out that the former can indeed be expressed as integrals of the latter over time, provided the current operators are ordered according to Matthews’ T∗ ordering. It is given by first arranging the charge operators according to their time indices, and only then taking the time derivatives, which induce contact terms whose singular support lies along time coincidences. We shall give two derivations of this result, because the first one is transparent, and the second one is related to contradictory computations in the literature where the system is considered to be measured by an external spin. In the model of constant instantaneous scattering, we then argue from the Lesovik-Levitov determinant that the transport can be interpreted in terms of a binomial process in the long time limit. In a parallel computation, we concentrate on the third cumulant as defined by integration of current correlators and show how the contact terms of the T∗ ordering play an essential role in providing the expected binomial result, too. To close this thesis, we suggest an algebraic framework that lays the foundations for a rigorous discussion of the Casimir effect. The problem of infinites arising in the comparison of two physical situations for a bosonic field is interpreted as the non-equivalence of states on the algebra of canonical commutation relations. We exhibit sufficient conditions on the one-particle Hamiltonians which ensure that the Casimir energy is finite and give an expression for it. We discuss its formal equivalence with the result obtained from the standard mode summing method.