The Superfield Quantisation of a Superparticle Action With an Extended Line Element

Abstract

A massive superparticle action based on the generalised line element in N=1 global superspace is quantised canonically. A previous method of quantising this action, based on a Fock space analysis, showed that states existed in three supersymmetric multiplets, each of a different mass. The quantisation procedure presented uses the single first class constraint as an operator condition on a general N=1 superwavefunction. The constraint produces coupled equations of motion for the component wavefunctions. Transformations of the component wavefunctions are derived that decouple the equations of motion and partition the resulting wavefunctions into three separate supermultiplets. Unlike previous quantisations of superparticle actions in N=1 global superspace, the spinor wavefunctions satisfy the Dirac equation and the vector wavefunctions satisfy the Proca equation.The off-shell closure of the commutators of the supersymmetry transformations, that include mass parameters, are derived by the introduction of auxiliary wavefunctions.To avoid the ghosts arising in a previous Fock space quantisation an alternative conjugation is used in the definition of the current, based on a Krein space approach.

On possible quantization of the fundamental tensor in the relativistic regime

“…Alternatively, noncommutation relations, Section 2.2, between $$$ {dx}^{\mu } $$$ and the fundamental tensor on both Finsler and Riemann manifolds, should be invented. Despite the conclusion drawn in Martinetti (2009), that neither the fundamental tensor nor $$$ {dx}^{\mu } $$$ seems to have an obvious noncommutation translation, with noncommutative differential calculus Dubois‐Violette (2001), Madore (2000), noncommutative metric tensor Ulhoa et al (2015) would represent an alternative procedure toward a noncommutative line element FitzGerald (2005). All these research directions could be followed elsewhere.…”

Discretized Finsler Structure: An Approach to Quantizing the First Fundamental Form

“…In this regard, we emphasize that neither the metric tensor nor the 1-form, dx µ , has noncommutative translations [22]. Alternatively, we might recall noncommutative metric tensor [23] and noncommutative differential calculus [24,25] to define a noncommutative line element [26]. This could be performed elsewhere.…”

A Quantum Geometric Approach for the Generalization of Spacetime Curvature

“…On the other hand, defining a noncommutative differential calculus [57,58] and a noncommutative metric tensor [59], could be conducted. This is conjectured to allow for integrating both procedures in defining a noncommutative measure of the line element [60]. Furthermore, the modified relativistic kinematics, which is geometrically described by Finsler geometry, describes conceivable vacuum state of quantum gravity, at low energies [61].…”