arXiv subjects
Publications and source records attributed to Pankaj Sharan.
Variational formalism in the extended phase space for fields is applied to gravity. It is shown that the requirement of invariance under arbitrary local inertial frames implies a coupling of torsion to a 3-form of matter fields on the one hand and to a 3-form (related to the Einstein tensor) on the other. Gravitational dynamics is restricted to torsion zero surface in the extended phase space for Einstein-Hilbert action.
It is shown that the nature of physical time requires the extended phase-space in mechanics to have a bundle structure with time as the 1-dimensional base manifold and the phase space as the fiber. This bundle picture of the extended phase space is then applied to fields in a covariant, `directly' Hamiltonian formalism. Variational principle in the covariant phase-space is discussed, Noether currents calculated for symmetry fields and a new bracket analogous to the Peierls bracket is defined.
Necessary and sufficient condition for the existence of a minimum uncertainty state for an arbitrary pair of observables is given.
The `directly Hamiltonian' field theory in the extended phase space is applied to gauge fields in curved spacetime background. These fields being differential 1-forms, have canonical momenta which are 2-forms. The Poincare-Cartan 4-forms for matter and gauge fields have to be modified with the exterior derivatives replaced by the covariant derivative for maintaining gauge invariance.
Poincare-Cartan form for scalar field is constructed as a differential 4-form in a `directly Hamiltonian' formalism which does not use a Lagrangian. The canonical momentum $p$ of a scalar field $ϕ$ is a 1-form and the Poincare-Cartan 4-form $Θ$ is $(*p)\ww dϕ-H$ where the Hamiltonian $H$ is a suitable 4-form made from $ϕ$ and $p$ using the Hodge star operator defined by the Riemannian metric of the background spacetime. An allowed field configuration is a 4-dimensional surface in the 9-dimensional extended phase space such that its tangent vectors annihilate $Ω=-dΘ$. Relation of this to variational principle, symmetry fields and conserved quantities is worked out. Observables are defined as differential 4-forms constructed from field and momenta smeared with appropriate test functions. A bracket defined by Peierls long ago is found to be the suitable candidate for quantization.
Relation between the Peierls and the Poisson bracket is derived in classical mechanics of time-dependent systems. Equal-time Peierls brackets are seen to be the same as the Poisson brackets in simple cases but a proof for a general Hamiltonian is lacking.
We recast Dirac's Lagrangian in quantum mechanics in the language of vector bundles and show that the action is an operator-valued connection one-form. Phases associated with change of frames of reference are seen to be total differentials in the transformation of the action. The relativistic case is discussed and we show that it gives the correct phase in the non-relativistic limit for uniform acceleration.
Dynamical evolution is described as a parallel section on an infinite dimensional Hilbert bundle over the base manifold of all frames of reference. The parallel section is defined by an operator-valued connection whose components are the generators of the relativity group acting on the base manifold. In the case of Galilean transformations we show that the property that the curvature for the fundamental connection must be zero is just the Heisenberg equations of motion and the canonical commutation relation in geometric language. We then consider linear and circular accelerating frames and show that pseudo-forces must appear naturally in the Hamiltonian.
We examine two singular Lagrangian systems with constraints which apparently reduce the phase space to a 2-dimensional sphere and a 2-dimensional hyperboloid. Rigorous constraint analysis by Dirac's method, however, gives 2-dimensional open disc and an infinite plane with a hole in the centre respectively as the reduced phase spaces. Upon canonical quantisation the classical constraints show up as restrictions on the Hilbert space.
Complete constraint analysis and choice of gauge conditions consistent with equations of motion is done for non-abelian Chern-Simons field interacting with N-component complex scalar field. Dirac-Schwinger condition is satisfied by the reduced phase-space Hamiltonian density with respect to the Dirac bracket.
Complete constraint analysis and choice of gauge conditions consistent with equations of motion is done for Abelian Chern Simons field interacting minimally with a complex scalar field. The Dirac-Schwinger consistency condition is satisfied by the reduced phase space Hamiltonian density with respect to the the Dirac bracket. It is shown that relativistic invariance under boosts can be obtained only if gauge conditions were chosen consistent with the equations of motion. Moreover all gauge invariant quantities are shown to be free of transformation anomaly.
Using a scheme proposed earlier we set up Hamiltonian path integral quantization for a particle in two dimensions in plane polar coordinates.This scheme uses the classical Hamiltonian, without any $O(\hbar^2)$ terms, in the polar varivables. We show that the propagator satisfies the correct Schrödinger equation.
A method for deriving the Schrodinger equation for Lagrangian path integral with scaling of local time is given.
We briefly review a hamiltonian path integral formalism developed earlier by one of us. An important feature of this formalism is that the path integral quantization in arbitrary co-ordinates is set up making use of only classical hamiltonian without addition of adhoc $\hbar^2$ terms. In this paper we use this hamiltonian formalism and show how exact path integration may be done for several potentials.
Inspired by the usefulness of local scaling of time in the path integral formalism, we introduce a new kind of hamiltonian path integral in this paper. A special case of this new type of path integral has been earlier found useful in formulating a scheme of hamiltonian path integral quantization in arbitrary coordinates. This scheme has the unique feature that quantization in arbitrary co-ordinates requires hamiltonian path integral to be set up in terms of the classical hamiltonian only, without addition of any adhoc $ O(\hbar ^2) $terms. In this paper we further study the properties of hamiltonian path integrals in arbitrary co-ordinates with and without local scaling of time and obtain the Schrodinger equation implied by the hamiltonian path integrals. As a simple illustrative example of quantization in arbitrary coordinates and of exact path integration we apply the results obtained to the case of Coulomb problem in two dimensions.