[Paper Review] New quantum numbers for the Dirac equation in curved spacetime

Marco Cariglia|arXiv (Cornell University)|May 17, 2003

Noncommutative and Quantum Gravity Theories4 citations

TL;DR

This paper introduces new quantum numbers for the Dirac equation in curved spacetime by constructing operators that (anti)commute with the Dirac operator using Yano tensors, even in the absence of isometries. The key result is that such operators exist for Yano tensors of arbitrary rank, ensuring anomaly freedom and genuine quantum mechanical supersymmetries, with a complete structure derived when Killing spinors are present.

ABSTRACT

We show that, on spacetimes which admit Yano tensors, it is possible to construct operators that (anti)commute with the Dirac operator, thus providing extra quantum numbers even when isometries are not present. This is the main result obtained and is valid for Yano tensors of arbitrary rank. It implies that the theory of the spinning particle in such spacetimes has no anomalies and admits genuine quantum mechanical extra supersymmetries. If a Killing spinor is present, that is, the spacetime has almost special holonomy, then it is possible to construct a tower of Yano tensors of different rank from it. We give a full description of this structure and its relation to Hodge duality and the conformal Yano equation. As a concrete application of our result, we construct Yano operators on maximally symmetric spacetimes, where the underlying group structure greatly simplifies the calculation. The high symmetry tightly constrains the form of Yano tensors in these spacetimes: they are spanned by tensor products of Killing vectors.

Motivation & Objective

• To identify new conserved quantum numbers for the Dirac equation in curved spacetimes lacking isometries.
• To establish the existence of operators that (anti)commute with the Dirac operator using Yano tensors of arbitrary rank.
• To explore the implications of these operators for quantum anomalies and supersymmetry in spinning particle theories.
• To describe the algebraic structure of Yano tensors derived from Killing spinors and their relation to Hodge duality and the conformal Yano equation.
• To provide explicit constructions of Yano operators in maximally symmetric spacetimes, where group structure simplifies the tensor forms.

Proposed method

• Utilizes Yano tensors on spacetimes that admit them to define operators commuting (anti)commuting with the Dirac operator.
• Applies the conformal Yano equation to characterize the tensorial structure of these operators.
• Derives the relationship between Yano tensors and Hodge duality, particularly in the context of almost special holonomy spacetimes.
• Constructs a tower of Yano tensors of increasing rank from a single Killing spinor, leveraging the underlying geometric structure.
• Employs the group-theoretic properties of maximally symmetric spacetimes to constrain and simplify the form of Yano tensors to tensor products of Killing vectors.
• Analyzes the algebraic closure and consistency of the resulting quantum numbers under the Dirac equation dynamics.

Experimental results

Research questions

• RQ1 Can new quantum numbers be defined for the Dirac equation in curved spacetimes without isometries, using geometric structures beyond Killing vectors?
• RQ2 How do Yano tensors of arbitrary rank contribute to the (anti)commutation relations with the Dirac operator?
• RQ3 What is the algebraic and geometric structure of Yano tensors when derived from a Killing spinor, and how does it relate to Hodge duality?
• RQ4 How do the conformal Yano equation and the underlying spacetime symmetry constrain the form of Yano tensors in maximally symmetric spacetimes?
• RQ5 What are the implications of these new quantum numbers for quantum anomalies and supersymmetry in spinning particle theories?

Key findings

• Operators that (anti)commute with the Dirac operator can be constructed using Yano tensors, even in spacetimes without isometries, providing new quantum numbers.
• The existence of such operators ensures the absence of quantum anomalies in the theory of spinning particles on these spacetimes.
• When a Killing spinor exists, a complete tower of Yano tensors of different ranks can be systematically generated from it.
• The Yano tensors in maximally symmetric spacetimes are constrained to be tensor products of Killing vectors due to the high underlying symmetry.
• The structure of Yano tensors is deeply connected to Hodge duality and satisfies the conformal Yano equation, revealing a hidden algebraic consistency.
• The construction leads to genuine quantum mechanical supersymmetries, not just classical or approximate ones, in the specified spacetime geometries.