Canonical Effective Methods for Quantum Systems

Jani Lukkarinen

We develop a regularization of the quantum microcanonical ensemble, called a Gaussian ensemble, which can be used for derivation of the canonical ensemble from microcanonical principles. The derivation differs from the usual methods by giving an explanation for the, at the first sight unreasonable, effectiveness of the canonical ensemble when applied to certain small, isolated, systems. This method also allows a direct identification between the parameters of the microcanonical and the canonical ensemble and it yields simple indicators and rigorous bounds for the effectiveness of the approximation. Finally, we derive an asymptotic expansion of the microcanonical corrections to the canonical ensemble for those systems, which are near, but not quite, at the thermodynamical limit and show how and why the canonical ensemble can be applied also for systems with an exponentially increasing density of states. The aim throughout the paper is to keep mathematical rigour intact while attempting to produce results which are both physically and practically interesting.

Gerard ’t Hooft

Philipp O. J. Scherer

Sean Crowe

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.

Bekir Baytaş, Martin Bojowald A, Sean Crowe

Canonical variables for the Poisson algebra of quantum moments are introduced here, expressing semiclassical quantum mechanics as a canonical dynamical system that extends the classical phase space. New realizations for up to fourth order in moments for a single classical degree of freedom and to second order for a pair of classical degrees of freedom are derived and applied to several model systems. It is shown that these new canonical variables facilitate the derivation of quantum-statistical quantities and effective potentials. Moreover, by formulating quantum dynamics in classical language, these methods result in new heuristic pictures, for instance, of tunneling, that can guide further investigations.

Huey-Wen Lin, James E. Gubernatis, Harvey Gould +1 more