Spin-statistics correletions in various noncommutative field theories

Abstract

Intuitive arguments involving standard quantum?mechanical uncertainty relations suggest that at length scales close to the Planck length, strong gravity effects will limit the spatial as well as temporal resolution to values not smaller than the fundamental length scale (?? ? Planck length). This leads to space–space as well as space–time uncertainties. Spacetime cannot be probed with a resolution beyond this scale; i.e., spacetime becomes “fuzzy” below this scale, resulting in noncommutative spacetime. Hence, it becomes important and interesting to study in detail the structure of such noncommutative spacetimes and the properties of quantum fields on them, because this not only improves our understanding of Planck?scale physics but also helps bridge standard particle physics with physics at the Planck scale. In this thesis, we study field theories defined on a particular model of noncommutative spacetime, the Groenewold–Moyal (GM) plane. We begin by briefly reviewing the novel features of field theories on the GM plane, e.g., the ??product, the restoration of Poincaré–Hopf symmetry, and twisted commutation relations. We then discuss our work on the renormalization of field theories on the GM plane. We show that any generic noncommutative theory consisting only of pure matter fields is renormalizable if the analogous commutative theory is renormalizable. We further show that all such noncommutative theories have the same fixed points and ??functions for their couplings as their commutative counterparts. A unique feature of these field theories is the twisted statistics obeyed by the particles. Motivated by this, we explore the possibility of twisted statistics arising from deforming internal symmetries instead of spacetime symmetries. We construct two different twisted theories that can be viewed as internal?symmetry analogues of the GM plane and of the dipole field theories which arise in the low?energy limit of certain string configurations. We further study their properties, including issues of causality and the scattering formalism. Having studied the mathematical properties of noncommutative and twisted internal symmetries, we move on to discuss their potential phenomenological signatures. We first analyze the noncommutative thermal correlation functions and show that, because of twisted statistics, all correlation functions except the two?point function are modified. Finally, we discuss modifications in the Hanbury–Brown–Twiss (HBT) correlation functions due to twisted statistics on the GM plane, and the potential for observing signatures of noncommutativity via an HBT correlation experiment using Ultra?High?Energy Cosmic Rays (UHECRs).

Plan of the Thesis Chapter 1: We review the basic concepts and well?known results of field theories defined on the GM plane. Chapter 2: We review the formalism of the noncommutative interaction picture and the noncommutative scattering theory. We then show the equivalence of the interaction picture and the Lehmann–Symanzik–Zimmermann (LSZ) approach for such theories. We further discuss renormalizability and show that any noncommutative field theory involving only matter fields is renormalizable, provided the corresponding commutative theory is renormalizable. All such theories are free from UV/IR mixing and have identical fixed points and ??functions as the commutative theory. Chapter 3: We discuss the construction of Poincaré?invariant field theories with twisted statistics, achieved by deforming the transformation properties of fields under a global SU(N) group. We construct two such twisted field theories and study causality and scattering. Chapter 4: We develop the Green’s?function formalism to compute correlation functions and adapt it to the noncommutative case. We show that, due to twisted commutation relations on the GM plane, all correlation functions except the two?point function are modified. Chapter 5: We explore potential signatures of noncommutativity in UHECRs. We analyze modifications in the noncommutative HBT correlation function due to twisted statistics and demonstrate that the commutative and noncommutative HBT correlation functions differ, with the difference increasing at higher energies. Thus, an HBT experiment using UHECRs may provide observable signatures of noncommutativity. We conclude the thesis with a summary of key results.