Pratul Bandyopadhyay

Stochastic quantization of a Fermi field: Fermions as solitons

It is shown that the stochastic quantization of a fermion introducing an anisotropy in the internal space so that this gives rise to two internal helicities corresponding to particle and antiparticle leads us to describe a fermion as a Skyrme soliton. The Skyrme term appears here as a consequence of this internal anisotropy and can be treated as a quantum effect. Some topological properties of this fermionization are then discussed.

Topological aspects of a fermion, chiral anomaly, and Berry phase

The relationship of the Berry phase with the topological aspects of a fermion and chiral anomaly is studied here. It is observed that this phase reflects the effect of quantum geometry when the quantization procedure of a Fermi field is considered from a classical system and may be considered as the manifestation of the multiply connected nature of quantum space‐time.

Topological aspects of a fermion and the chiral anomaly

It is here shown that the chiral anomaly is related to the topological properties of a fermion. The quantization procedure of a relativistic particle requires that the particle be an extended one, and to quantize a Fermi field, it is necessary to introduce an anisotropic feature in the internal space of the particle so that it gives rise to two internal helicities corresponding to a particle and an antiparticle. This specific quantum geometry of a Dirac particle gives rise to the solitonic feature as envisaged by Skyrme and the Skyrme term appears as an effect of quantization. When in the Lagrangian formulation the effect of this topological property is taken into account, it is found that the anomaly vanishes.

SL(2,C)-gauge theory,N=1 supergravity and torsion

It is shown that the SL(2,C)-gauge theory of gravitation may be considered to correspond toN = 1 supergravity and the conserved current gives rise to the Einstein-Cartan action. The torsion term here appears due to the spinorial variableθ, which is associated with the “internal helicity” giving rise to the isospin algebra from the conformal reflection group. In this sense, the internal symmetry of hadrons is found to take a dominant role in gravitational phenomena in the microlocal space-time region where the Einstein-Cartan action becomes significant.

HOLOMORPHIC QUANTUM MECHANICS, CONFORMAL REFLECTION AND THE INTERNAL SYMMETRY OF HADRONS

It is shown here that the holomorphic quantum mechanics in a complexified Minkowski space-time helps us to study the geometrical feature of the internal space of a particle and its relevance with conformal geometry. It is noted that the conformal reflection can be depicted in the formalism of an internal helicity which takes the value and for the particle and antiparticle state. This again can be described in the framework of holomorphic quantum mechanics in terms of the half-orbital angular momentum of a constituent in an anisotropic space in the sense of Minkowski space-time with a fixed lz value for the particle and antiparticle configuration when a composite system is considered. A massive or massless spinor moving with such characteristic in the configuration of a composite system can be depicted as a Cartan semispinor and behaves as a twistor. The doublet of such spinors with opposite helicities represent an eight-component conformal spinor. The internal symmetry group SU(3) for a composite system of hadrons can then be realized from the reflection group. This formalism reveals the microlocal region of a complexified Minkowski space-time as a twistor space.

Equivalence of stochastic and Klauder quantization and the concept of locality and nonlocality in quantum mechanics

Abstract The concept of locality and nonlocality in quantum mechanics is studied from the view point of the geometry involved in different quantization procedures. This analysis suggests that when we study quantum phenomena in classical space-time devoid of the specific features of quantum geometry, we come across nonlocality. On the other hand, when we take into account the geometry involved in the quantization procedure, we do not come across any difficulty with the concept of locality. This represents a departure from those mathematical formulations of locality that tend to identify locality with separability.

QUANTUM HALL EFFECT AND BERRY PHASE

It is shown here that a particle in an intense magnetic field may acquire the Berry phase and the topological features associated with this phase may be taken to be responsible for both the integrally and fractionally quantized Hall effect. The two different manifestations of quantum Hall effect have been realized in a unified scheme where the electrons associated with the fractional quantum Hall effect are found to be in an excited state having higher angular momentum.

On a unified theory of electromagnetism and weak interaction

A unified theory of weak and electromagnetic interaction has been developed on the basis of the assumption that the charge and mass of a lepton is of dynamical origin. According to this model, the spontaneous breakdown of symmetry generates the photon as a Goldstone boson.

Quantum gravity effect in torsion driven inflation and CP violation

A bstractWe have derived an effective potential for inflationary scenario from torsion and quantum gravity correction in terms of the scalar field hidden in torsion. A strict bound on the CP violating θ parameter, O10−10<θ<O10−9