Some Recent Advances in Loop Quantum Cosmology

Physical Review D, 1993

New non-perturbative methods for dealing with Hamiltonian systems are introduced. The derivation of these methods requires identities for rewriting exponentials of sums of operators which are different from the usual Campbell-Hausdorff formula. These identities allow one to derive approximations to e -6H which are correct to higher order in S and which contain fewer terms than the Campbell-Hausdorff formula.

1998

Schwinger's Closed-Time Path (CTP) formalism is an elegant way to insure causal-ity for initial value problems in Quantum Field Theory. Feynman's Path Integral on the other hand is much more amenable than Schwinger's differential approach (related to the Schwinger Dyson equations) for non-perturbative (in coupling constant) expansions such as the large-N expansion. By marrying the CTP formalism with a large-N expansion of Feynman's Path Integral approach, we are for the first time able to study the dynamics of phase transitions in quantum field theory settings. We review the Feynman Path Integral representation for the generating functional for the Green's functions described by an initial density matrix. We then show that the large-N expansion for the path integral forms a natural non-perturbative framework for discussing phase transitions in quantum field theory as well as for giving a space time description of a heavy ion collision. We review results for a tim...

Revista Brasileira de Ensino de Física, 2021

Lecture notes on path-integrals, suitable for an undergraduate course with prerequisites such as: Classical Mechanics, Electromagnetism and Quantum Mechanics. The aim is to provide the reader, who is familiar with the major concepts of Solid State Physics, to study these topics couched in the language of path integrals. We endeavor to keep the formalism to the bare minimum.

Journal of Physics A: Mathematical and Theoretical, 2009

The canonical operator quantisation formulation corresponding to the Klauder-Daubechies construction of the phase space path integral is considered. This formulation is explicitly applied and solved in the case of the harmonic oscillator, thereby illustrating in a manner complementary to Klauder and Daubechies' original work some of the promising features offered by their construction of a quantum dynamics. The Klauder-Daubechies functional integral involves a regularisation parameter eventually taken to vanish, which defines a new physical time scale. When extrapolated to the field theory context, besides providing a new regularisation of short distance divergences, keeping a finite value for that time scale offers some tantalising prospects when it comes to strong gravitational quantum systems.

Foundations of Physics Letters, 1995

ABSTRACT It is a well known result that the Feynman's path integral (FPI) approach to quantum mechanics is equivalent to Schrdinger's equation when we use as integration measure the Wiener-Lebesgue measure. This results in little practical applicability due to the great algebraic complexibity involved, and the fact is that almost all applications of (FPI) practical calculations — are done using a Riemann measure. In this paper we present an expansion to all orders in time of FPI in a quest for a representation of the latter solely in terms of differentiable trajetories and Riemann measure. We show that this expansion agrees with a similar expansion obtained from Schrdinger's equation only up to first order in a Riemann integral context, although by chance both expansions referred to above agree for the free particle and harmonic oscillator cases. Our results permit, from the mathematical point of view, to estimate the many errors done in practical calculations of the FPI appearing in the literature and, from the physical point of view, our results supports the stochastic approach to the problem.

1999

Physical Review A, 2005

We present a path-integral formulation of 't Hooft's derivation of quantum from classical physics. The crucial ingredient of this formulation is Gozzi et al. supersymmetric path integral of classical mechanics. We quantize explicitly two simple classical systems: the planar mathematical pendulum and the Rössler dynamical system.

Lecture Notes in Physics Monographs

These lectures are intended for graduate students who want to acquire a working knowledge of path integral methods in a wide variety of fields in physics. In general the presentation is elementary and path integrals are developed in the usual heuristic, non-mathematical way for application in many diverse problems in quantum physics. Three main parts deal with path integrals in non-relativistic quantum mechanics, manybody physics and field theory and contain standard examples (quadratic Lagrangians, tunneling, description of bosons and fermions etc.) as well as specialized topics (scattering, dissipative systems, spin & color in the path integral, lattice methods etc.). In each part simple Fortran programs which can be run on a PC, illustrate the numerical evaluation of (Euclidean) path integrals by Monte-Carlo or variational methods. Also included are the set of problems which accompanied the lectures and their solutions. Content 0. Contents First, an overview over the planned topics. The subsections marked by ⋆ are optional and may be left out if there is no time available whereas the chapters printed in blue deal with basic concepts. Problems from the optional chapters or referring to "Details" are marked by a ⋆ as well.

Journal of High Energy Physics, 2016

We define a (semi-classical) path integral for gravity with Neumann boundary conditions in D dimensions, and show how to relate this new partition function to the usual picture of Euclidean quantum gravity. We also write down the action in ADM Hamiltonian formulation and use it to reproduce the entropy of black holes and cosmological horizons. A comparison between the (background-subtracted) covariant and Hamiltonian ways of semi-classically evaluating this path integral in flat space reproduces the generalized Smarr formula and the first law. This "Neumann ensemble" perspective on gravitational thermodynamics is parallel to the canonical (Dirichlet) ensemble of Gibbons-Hawking and the microcanonical approach of Brown-York.