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A. K. Kapoor

Publications and source records attributed to A. K. Kapoor.

At least 19 recordsLinked to original sources
arXiv2024

Parisi-Wu Quantization, CP violation and Beyond the Standard Model

The Parisi-Wu scheme of quantization opens up the possibility of using anomalous fermionic gauge theories. An analysis of ultra-violet divergences reveals that the structure of counter terms is different from what is expected in conventional quantization schemes. In this letter it is argued that there exists a possible mechanism of CP violation that requires at least three generations of quarks, a result well known from a phenomenological analysis of mass mixing of quarks. A few observations on possible ways of going beyond the standard model are included.

hep-ph↗
arXiv2021

Stochastic Quantization of Axial Vector Gauge Theories

The stochastic quantization scheme proposed by Parisi and Wu in 1981 is known to have differences from conventional quantum field theory in higher orders. It has been suggested that some of these new features might give rise to a mechanism to explain tiny fermion masses as arising due to radiative corrections. In view of importance for need of going beyond the standard model, in this article some features of U(1) axial vector gauge theory in Parisi Wu stochastic quantization scheme are reported. Renormalizability of a massive axial vector gague theory coupled to a massless fermion appears as one of the important conclusions.

hep-th↗
arXiv2017

Non-Commutative space-time and Hausdorff dimension

We study the Hausdorff dimension of the path of a quantum particle in non-commutative space-time. We show that the Hausdorff dimension depends on the deformation parameter $a$ and the resolution $Δx$ for both non-relativistic and relativistic quantum particle. For the non-relativistic case, it is seen that Hausdorff dimension is always less than two in the non-commutative space-time. For relativistic quantum particle, we find the Hausdorff dimension increases with the non-commutative parameter, in contrast to the commutative space-time. We show that non-commutative correction to Dirac equation brings in the spinorial nature of the relativistic wave function into play, unlike in the commutative space-time. By imposing self-similarity condition on the path of non-relativistic and relativistic quantum particle in non-commutative space-time, we derive the corresponding generalised uncertainty relation.

hep-th↗
arXiv2016

Quantum Mechanics in Pseudotime

Based on some results on reparmetrisation of time in Hamiltonian path integral formalism, a pseudo time formulation of operator formalism of quantum mechanics is presented. Relation of reparametrisation of time in quantum with super symmetric quantum mechanics is established. We show how some important concepts such as shape invariance and tools like isospectral deformation appear in pseudo time quantum mechanics.

hep-th↗
arXiv2014

Shape Invariant Potentials in Higher Dimensions

In this paper we investigate the shape invariance property of a potential in one dimension. We show that a simple ansatz allows us to reconstruct all the known shape invariant potentials in one dimension. This ansatz can be easily extended to arrive at a large class of new shape invariant potentials in arbitrary dimensions. A reformulation of the shape invariance property and possible generalizations are proposed. These may lead to an important extension of the shape invariance property to Hamiltonians that are related to standard potential problems via space time transformations, which are found useful in path integral formulation of quantum mechanics.

quant-ph↗
arXiv2012

Uniformly accelerating observer in $κ$-deformed space-time

In this paper, we study the effect of $κ$-deformation of the space-time on the response function of a uniformly accelerating detector coupled to a scalar field. Starting with $κ$-deformed Klein-Gordon theory, which is invariant under a $κ$-Poincaré algebra and written in commutative space-time, we derive $κ$-deformed Wightman functions, valid up to second order in the deformation parameter $a$. Using this, we show that the first non-vanishing correction to the Unruh thermal distribution is only in the second order in $a$. We also discuss various other possible sources of $a$-dependent corrections to this thermal distribution.

hep-th↗
arXiv2012

A proof of the nodal structure of the wave functions of supersymmetric partner potentials

Quantum Hamilton-Jacobi formalism is used to give a proof for Gozzi's criterion that for eigenstates of the supersymmetric partners, corresponding to same energy, the difference in the number of nodes is equal to one when supersymmetry (SUSY) is unbroken and is zero when SUSY is broken. We show that this proof is also applicable to the case, where isospectral deformation is involved.

math-ph↗
arXiv2011

Exceptional orthogonal polynomials, QHJ formalism and SWKB quantization condition

We study the quantum Hamilton-Jacobi (QHJ) equation of the recently obtained exactly solvable models, related to the newly discovered exceptional polynomials and show that the QHJ formalism reproduces the exact eigenvalues and the eigenfunctions. The fact that the eigenfunctions have zeros and poles in complex locations leads to an unconventional singularity structure of the quantum momentum function $p(x)$, the logarithmic derivative of the wave function, which forms the crux of the QHJ approach to quantization. A comparison of the singularity structure for these systems with the known exactly solvable and quasi-exactly solvable models reveals interesting differences. We find that the singularities of the momentum function for these new potentials lie between the above two distinct models, sharing similarities with both of them. This prompted us to examine the exactness of the supersymmetric WKB (SWKB) quantization condition. The interesting singularity structure of $p(x)$ and of the superpotential for these models has important consequences for the SWKB rule and in our proof of its exactness for these quantal systems.

math-ph↗
arXiv2010

Newton's Equation on the kappa space-time and the Kepler problem

We study the modification of Newton's second law, upto first order in the deformation parameter $a$, in the $κ$-space-time. We derive the deformed Hamiltonian, expressed in terms of the commutative phase space variables, describing the particle moving in a central potential in the $κ$-space-time. Using this, we find the modified equations of motion and show that there is an additional force along the radial direction. Using Pioneer anomaly data, we set a bond as well as fix the sign of $a$. We also analyse the violation of equivalence principle predicted by the modified Newton's equation, valid up to first order in $a$ and use this also to set an upper bound on $a$.

hep-th↗
arXiv2009

QHJ, WKB and exact quantisation

We present a simple derivation of the WKB quantisation condition using the quantum Hamilton-Jacobi formalism and propose an exact quantisation condition within this formalism for integrable models in higher dimensions.

quant-ph↗
arXiv2008

An explicit realization of fractional statistics in one dimension

An explicit realization of anyons is provided, using the three-body Calogero model. The fact that in the coupling domain, $-1/4<g<0$, the angular spectrum can have a band structure, leads to the manifestation of the desired phase in the wave function, under the exchange of the paticles. Concurrently, the momentum corresponding to the angular variable is quantized, exactly akin to the relative angular momentum quantization in two dimensional anyonic system

quant-ph↗
arXiv2008

Construction of localized atomic wave packets

It is shown that highly localized solitons can be created in lower dimensional Bose-Einstein condensates (BEC), trapped in a regular harmonic trap, by temporally varying the trap frequency. A BEC trapped in such a trap can be effectively used to construct a pulsed atomic laser emitting coherent atomic wave packets. It is also shown that one has complete control over the spatio-temporal dynamics of the solitons. The dynamics of these solitons are compared with those constructed in a BEC where the trap frequency is constant.

cond-mat.other↗
arXiv2008

Soliton response to transient trap variations

The response of bright and dark solitons to rapid variations in an expulsive longitudinal trap is investigated. We concentrate on the effect of transient changes in the trap frequency in the form of temporal delta kicks and the hyperbolic cotangent functions. Exact expressions are obtained for the soliton profiles. This is accomplished using the fact that a suitable linear Schrodinger stationary state solution in time can be effectively combined with the solutions of non-linear Schrodinger equation, for obtaining solutions of the Gross-Pitaevskii equation with time dependent scattering length in a harmonic trap. Interestingly, there is rapid pulse amplification in certain scenarios.

cond-mat.other↗
arXiv2006

Designing bound states in a band as a model for a quantum network

We provide a model of a one dimensional quantum network, in the framework of a lattice using Von Neumann and Wigner's idea of bound states in a continuum. The localized states acting as qubits are created by a controlled deformation of a periodic potential. These wave functions lie at the band edges and are defects in a lattice. We propose that these defect states, with atoms trapped in them, can be realized in an optical lattice and can act as a model for a quantum network.

quant-ph↗
arXiv2005

Bound States and Band Structure - a Unified Treatment through the Quantum Hamilton - Jacobi Approach

We analyze the Scarf potential, which exhibits both discrete energy bound states and energy bands, through the quantum Hamilton-Jacobi approach. The singularity structure and the boundary conditions in the above approach, naturally isolate the bound and periodic states, once the problem is mapped to the zero energy sector of another quasi-exactly solvable quantum problem. The energy eigenvalues are obtained without having to solve for the corresponding eigenfunctions explicitly. We also demonstrate how to find the eigenfunctions through this method.

quant-ph↗
arXiv2004

Bound State Wave Functions through the Quantum Hamilton - Jacobi Formalism

The bound state wave functions for a wide class of exactly solvable potentials are found utilizing the quantum Hamilton-Jacobi formalism. It is shown that, exploiting the singularity structure of the quantum momentum function, until now used only for obtaining the bound state energies, one can straightforwardly find both the eigenvalues and the corresponding eigenfunctions. After demonstrating the working of this approach through a number of solvable examples, we consider Hamiltonians, which exhibit broken and unbroken phases of supersymmetry. The natural emergence of the eigenspectra and the wave functions, in both the unbroken and the algebraically non-trivial broken phase, demonstrates the utility of this formalism.

quant-ph↗