arXiv subjects

Pushpa Kalauni

Publications and source records attributed to Pushpa Kalauni.

12 recordsLinked to original sources

arXiv2023

A symmetry perspective of the Riemann zeros

We study the relationship between the zeros of the Riemann zeta function and physical systems exhibiting supersymmetry, $PT$ symmetry and $SU(2)$ group symmetry. Our findings demonstrate that unbroken supersymmetry is associated with the presence of non-trivial zeros of the zeta function. However, in other cases, supersymmetry is spontaneously broken and the ground state energy of the system is not zero. Moreover, we have established the manifestation of PT symmetry invariance within our supersymmetric system. In addition, our findings provide insights into a $SU(2)$ symmetry that arises within these systems, with the Hilbert space having a two-level structure.

quant-ph↗

arXiv2023

Supersymmetric quantum mechanics and the Riemann hypothesis

We construct a supersymmetric quantum mechanical model in which the energy eigenvalues of the Hamiltonians are the products of Riemann zeta functions. We show that the trivial and nontrivial zeros of the Riemann zeta function naturally correspond to the vanishing ground state energies in this model. The model provides a natural form of supersymmetry.

hep-th↗

arXiv2019

Supersymmetry and the Riemann zeros on the critical line

We propose a new way of studying the Riemann zeros on the critical line using ideas from supersymmetry. Namely, we construct a supersymmetric quantum mechanical model whose energy eigenvalues correspond to the Riemann zeta function in the strip $0< {\rm Re}\, s < 1$ (in the complex parameter space) and show that the zeros on the critical line arise naturally from the vanishing ground state energy condition in this model.

math.GM↗

arXiv2019

Remarks on the Casimir Self-Entropy of a Spherical Electromagnetic $δ$-Function Shell

Recently the Casimir self-entropy of an electromagnetic $δ$-function shell was considered by two different groups, with apparently discordant conclusions, although both had concluded that a region of negative entropy existed for sufficiently weak coupling. We had found that the entropy contained an infrared divergence, which we argued should be discarded on physical grounds. On the contrary, Bordag and Kirsten recently found a completely finite self-entropy, although they, in fact, have to remove an infrared divergence. Apart from this, the high- and low-temperature results for finite coupling agree precisely for the transverse electric mode, but there are significant discrepancies in the transverse magnetic mode. We resolve those discrepancies here. In particular, it is shown that coupling-independent terms do not occur in a consistent regulated calculation, they likely being an artefact of the omission of pole terms. The results of our previous analysis, especially, the existence of a negative entropy region for sufficiently weak coupling, are therefore confirmed. Finally, we offer some analogous remarks concerning the Casimir entropy of a thin electromagnetic sheet, where the total entropy is always positive. In that case, the origin of the analogous discrepancy can be explicitly isolated.

hep-th↗

arXiv2018

Bogoliubov transformation and the thermal operator representation in the real time formalism

It has been shown earlier \cite{brandt,brandt1} that, in the mixed space, there is an unexpected simple relation between any finite temperature graph and its zero temperature counterpart through a multiplicative scalar operator (termed thermal operator) which carries the entire temperature dependence. This was shown to hold only in the imaginary time formalism and the closed time path ($σ=0$) of the real time formalism (as well as for its conjugate $σ=1$). We study the origin of this operator from the more fundamental Bogoliubov transformation which acts, in the momentum space, on the doubled space of fields in the real time formalisms \cite{takahashi,umezawa,pushpa}. We show how the ($2\times 2$) Bogoliubov transformation matrix naturally leads to the scalar thermal operator for $σ=0,1$ while it fails for any other value $0<σ<1$. This analysis also suggests that a generalized scalar thermal operator description, in the mixed space, is possible even for $0<σ<1$. We also show the existence of a scalar thermal operator relation in the momentum space.

hep-th↗

arXiv2017

Casimir Self-Entropy of a Spherical Electromagnetic $δ$-Function Shell

In this paper we continue our program of computing Casimir self-entropies of idealized electrical bodies. Here we consider an electromagnetic $δ$-function sphere ("semitransparent sphere") whose electric susceptibility has a transverse polarization with arbitrary strength. Dispersion is incorporated by a plasma-like model. In the strong coupling limit, a perfectly conducting spherical shell is realized. We compute the entropy for both low and high temperatures. The TE self-entropy is negative as expected, but the TM self-entropy requires ultraviolet and infrared subtractions, and, surprisingly, is only positive for sufficiently strong coupling. Results are robust under different regularization schemes.

hep-th↗

arXiv2016

Casimir Self-Entropy of an Electromagnetic Thin Sheet

Casimir entropies due to quantum fluctuations in the interaction between electrical bodies can often be negative, either caused by dissipation or by geometry. Although generally such entropies vanish at zero temperature, consistent with the third law of thermodynamics (the Nernst heat theorem), there is a region in the space of temperature and separation between the bodies where negative entropy occurs, while positive interaction entropies arise for large distances or temperatures. Systematic studies on this phenomenon in the Casimir-Polder interaction between a polarizable nanoparticle or atom and a conducting plate in the dipole approximation have been given recently. Since the total entropy should be positive according to the second law of thermodynamics, we expect that the self-entropy of the bodies would be sufficiently positive as to overwhelm the negative interaction entropy. This expectation, however, has not been explicitly verified. Here we compute the self-entropy of an electromagnetic $δ$-function plate, which corresponds to a perfectly conducting sheet in the strong coupling limit. The transverse electric contribution to the self-entropy is negative, while the transverse magnetic contribution is larger and positive, so the total self-entropy is positive. However, this self-entropy vanishes in the strong-coupling limit. In that case, it is the self-entropy of the nanoparticle that is just sufficient to result in a nonnegative total entropy.

hep-th↗

arXiv2016

Operator description for thermal quantum field theories on an arbitrary path in the real time formalism

We develop an operator description, much like thermofield dynamics, for quantum field theories on a real time path with an arbitrary parameter $σ\,(0\leqσ\leqβ)$. We point out new features which arise when $σ\neq \fracβ{2}$ in that the Hilbert space develops a natural, modified inner product different from the standard Dirac inner product. We construct the Bogoliubov transformation which connects the doubled vacuum state at zero temperature to the thermal vacuum in this case. We obtain the thermal Green's function (propagator) for the real massive Klein-Gordon theory as an expectation value in this thermal vacuum (with a modified inner product). The factorization of the thermal Green's function follows from this analysis. We also discuss, in the main text as well as in two appendices, various other interesting features which arise in such a description.

hep-th↗

arXiv2016

Negative Entropies in Casimir and Casimir-Polder Interactions

It has been increasingly becoming clear that Casimir and Casimir-Polder entropies may be negative in certain regions of temperature and separation. In fact, the occurrence of negative entropy seems to be a nearly ubiquitous phenomenon. This is most highlighted in the quantum vacuum interaction of a nanoparticle with a conducting plate or between two nanoparticles. It has been argued that this phenomenon does not violate physical intuition, since the total entropy, including the self-entropies of the plate and the nanoparticle, should be positive. New calculations, in fact, seem to bear this out at least in certain cases.

quant-ph↗

arXiv2016

Stress tensor for a scalar field in a spatially varying background potential: Divergences, "renormalization," anomalies, and Casimir forces

Motivated by a desire to understand quantum fluctuation energy densities and stress within a spatially varying dielectric medium, we examine the vacuum expectation value for the stress tensor of a scalar field with arbitrary conformal parameter, in the background of a given potential that depends on only one spatial coordinate. We regulate the expressions by incorporating a temporal-spatial cutoff in the (imaginary) time and transverse-spatial directions. The divergences are captured by the zeroth- and second-order WKB approximations. Then the stress tensor is "renormalized" by omitting the terms that depend on the cutoff. The ambiguities that inevitably arise in this procedure are both duly noted and restricted by imposing certain physical conditions; one result is that the renormalized stress tensor exhibits the expected trace anomaly. The renormalized stress tensor exhibits no pressure anomaly, in that the principle of virtual work is satisfied for motions in a transverse direction. We then consider a potential that defines a wall, a one-dimensional potential that vanishes for $z<0$ and rises like $z^α$, $α>0$, for $z>0$. The full finite stress tensor is computed numerically for the two cases where explicit solutions to the differential equation are available, $α=1$ and 2. The energy density exhibits an inverse linear divergence as the boundary is approached from the inside for a linear potential, and a logarithmic divergence for a quadratic potential. Finally, the interaction between two such walls is computed, and it is shown that the attractive Casimir pressure between the two walls also satisfies the principle of virtual work (i.e., the pressure equals the negative derivative of the energy with respect to the distance between the walls).

hep-th↗

arXiv2015

Proper time method in de Sitter space

We use the proper time formalism to study a (non-self-interacting) massive Klein-Gordon theory in the two dimensional de Sitter space. We determine the exact Green's function of the theory by solving the DeWitt-Schwinger equation as well as by calculating the operator matrix element. We point out how the one parameter family of arbitrariness in the Green's function arises in this method.

hep-th↗

arXiv2014

Supersymmetry, shape invariance and the hypergeometric equation

It has been shown earlier that the solubility of the Legendre and the associated Legendre equations can be understood as a consequence of an underlying supersymmetry and shape invariance. We have extended this result to the hypergeometric equation. Since the hypergeometric equation as well as the hypergeometric function reduce to various orthogonal polynomials, this study shows that the solubility of all such systems can also be understood as a consequence of an underlying supersymmetry and shape invariance. Our analysis leads naturally to closed form expressions (Rodrigues' formula) for the orthogonal polynomials.

hep-th↗