Shailesh Dhasmana

A bstract: Conformal higher-spin gravity is the log-divergent part of the effective action of the scalar field coupled to background fields via higher-spin currents, as was defined by Segal and Tseytlin, which can be worked out over the flat space background. We revisit the problem of the scalar field in a higher-spin background and propose a manifestly covariant version thereof. The construction utilizes the Fedosov quantization of the cotangent bundle and the action is written with the help of the trace on a curved phase space that is provided by the Feigin-Felder-Shoikhet cocycle. The same construction allows one to formulate quantum mechanics on a curved space, the phase space being the cotangent bundle.

A bstract: We propose a realisation of partially-massless higher spin algebras in four dimensions in terms of bosonic and fermionic oscillators, using Howe duality between sp (4, ℝ) ≅ so (2, 3) and osp (1|2( ℓ − 1), ℝ). More precisely, we show that the centraliser of osp (1|2( ℓ − 1), ℝ) in the Weyl-Clifford algebra generated by 4 bosonic and 8( ℓ − 1) fermionic symbols, modulo osp (1|2( ℓ − 1), ℝ) generators, is isomorphic to the higher spin algebra of the type-A _ℓ theory whose spectrum contains partially-massless fields of all spins and depths t = 1, 3,…, 2 ℓ − 1. We also discuss the possible existence of a deformation of this algebra, which would encode interaction for the type-A _ℓ theory.

The Koopman-von Neumann (KvN) mechanics is an approach that was formulated long ago to answer the question regarding the existence of a Hilbert space representation of classical mechanics. KvN mechanics is a non-relativistic theory, and it is not clear how spin can be included in it, since spin is widely regarded as a relativistic property. In Eur. Phys. J. Spec. Top. {\bf 227}, 2195 (2019) it was argued that the Spohn equation [Ann. Phys. {\bf 282}, 420 (2000)] is the correct classical framework for the Koopman-von Neumann theory corresponding to the Dirac equation. However, after Lévy-Leblond's seminal work on this topic, it became clear that spin naturally arises also from the Galilean invariant wave equations, without any need of relativistic considerations. Inspired by this, we propose another possibility of including spin in the KvN formalism: the Lévy-Leblond equation in the Koopman-von Neumann formalism can be obtained as a null reduction of the massless Dirac equation in the Eisenhart-Duval lift metric. To illustrate the idea, we implement it for a one-dimensional classical system without magnetic interactions.

The Koopman–von Neumann (KvN) mechanics is an approach that was formulated long ago to answer the question regarding the existence of a Hilbert space representation of classical mechanics. KvN mechanics is a non-relativistic theory, and it is not clear how spin can be included in it, since spin is widely regarded as a relativistic property. Cabrera et al., in Eur. Phys. J. Spec. Top. 227, 2195 (2019) argued that the Spohn equation [Spohn, Ann. Phys. 282, 420 (2000)] is the correct classical framework for the Koopman–von Neumann theory corresponding to the Dirac equation. However, after Lévy-Leblond’s seminal work on this topic, it became clear that spin naturally arises also from the Galilean invariant wave equations, without any need of relativistic considerations. Inspired by this, we propose another possibility of including spin in the KvN formalism: the Lévy-Leblond equation in the Koopman–von Neumann formalism can be obtained as a null reduction of the massless Dirac equation in the Eisenhart–Duval lift metric. To illustrate the idea, we implement it for a one-dimensional classical system without magnetic interactions.

We propose a new (chiral) description of partially-massless fields in 4d, including the partially-massless graviton, that is similar to the pure connection formulation for gravity and massless higher spin fields, the latter having a clear twistor origin. The new approach allows us to construct complete examples of higher spin gravities with (partially-)massless fields that feature Yang-Mills and current interactions.

The Eisenhart lift establishes a fascinating connection between non-relativistic and relativistic physics, providing a space-time geometric understanding of non-relativistic Newtonian mechanics. What is still little known, however, is the fact that there is a Hilbert space representation of classical mechanics (also called Koopman-von Neumann mechanics) that attempts to give classical mechanics the same mathematical structure that quantum mechanics has. In this article, we geometrize the Koopman-von Newmann (KvN) mechanics using the Eisenhart toolkit. We then use a geometric view of KvN mechanics to find transformations that relate the harmonic oscillator, linear potential, and free particle in the context of KvN mechanics.

It is widely known in quantum mechanics that solutions of the Schr\"{o}inger equation (SE) for a linear potential are in one-to-one correspondence with the solutions of the free SE. The physical reason for this correspondence is Einstein's principle of equivalence. What is usually not so widely known is that solutions of the Schr\"{o}dinger equation with harmonic potential can also be mapped to the solutions of the free Schr\"{o}dinger equation. The physical understanding of this equivalence is not known as precisely as in the case of the equivalence principle. We present a geometric picture that will link both of the above equivalences with one constraint on the Eisenhart metric.

ISIM(2) symmetry group of Cohen and Glashow's very special relativity is unstable with respect to small deformations of its underlying algebraic structure and according to Segal's principle cannot be a true symmetry of nature. However, like special relativity, which is a very good description of nature thanks to the smallness of the cosmological constant, which characterizes the deformation of the Poincare group, the very special relativity can also be a very good approximation thanks to the smallness of the dimensionless parameter characterizing the deformation of ISIM(2).