Nonrelativistic counterparts of twistors and the realizations of Galilean conformal algebra

TL;DR

This paper introduces Galilean twistors as nonrelativistic counterparts of relativistic twistors, explores their quantum representations, and finds limitations on positive-definite Hamiltonian realizations within Galilean conformal algebra.

Contribution

It develops a new nonrelativistic twistor framework for Galilean conformal algebra and constructs explicit quantum-mechanical operator representations.

Findings

Positive-definite Hamiltonian representations do not exist for GCA.

Hermitian Galilean N-twistor realizations are constructed for N≥2.

Explicit N=2 Galilean twistor realization is provided.

Abstract

Using the notion of Galilean conformal algebra (GCA) in arbitrary space dimension d, we introduce for d=3 quantized nonrelativistic counterpart of twistors as the spinorial representation of O(2,1){\oplus}SO(3) which is the maximal semisimple subalgebra of three-dimensional GCA. The GC-covariant quantization of such nonrelativistic spinors, which shall be called also Galilean twistors, is presented. We consider for d=3 the general spinorial matrix realizations of GCA, which are further promoted to quantum-mechanical operator representations, expressed as bilinears in quantized Galilean twistors components. For arbitrary Hermitian quantum-mechanical Galilean twistor realizations we obtain the result that the representations of GCA with positive-definite Hamiltonian do not exist. For non-positive H we construct for N{\geq}2 the Hermitian Galilean N-twistor realizations of GCA; for N=2 such realization is provided explicitly.