Relativity, Causality, Locality, Quantization and Duality in the $Sp(2M)$ Invariant Generalized Space-Time

TL;DR

This paper explores the properties of Sp(2M) invariant field equations in a generalized space-time, revealing how classical solutions define causality, enable quantization, and relate to standard Minkowski space, with implications for higher spin theories.

Contribution

It introduces a framework for analyzing Sp(2M) invariant field equations in generalized space-time, connecting them to known physics and extending duality and localization concepts.

Findings

Classical solutions define a causal structure in generalized space-time.

Quantization is possible with a positive definite Hilbert space.

Connections between generalized space-time and standard Minkowski space are established.

Abstract

We analyze properties of the Sp(2M) conformally invariant field equations in the recently proposed generalized $\half M(M+1)$-dimensional space-time $\M_M$ with matrix coordinates. It is shown that classical solutions of these field equations define a causal structure in $\M_M$ and admit a well-defined decomposition into positive and negative frequency solutions that allows consistent quantization in a positive definite Hilbert space. The effect of constraints on the localizability of fields in the generalized space-time is analyzed. Usual d-dimensional Minkowski space-time is identified with the subspace of the matrix space $\M_M$ that allows true localization of the dynamical fields. Minkowski coordinates are argued to be associated with some Clifford algebra in the matrix space $\M_M$. The dynamics of a conformal scalar and spinor in $\M_2$ and $\M_4$ is shown to be equivalent, respectively, to the usual conformal field dynamics of a scalar and spinor in the 3d Minkowski space-time and the dynamics of massless fields of all spins in the 4d Minkowski space-time. An extension of the electro-magnetic duality transformations to all spins is identified with a particular generalized Lorentz transformation in $\M_4$. The M=8 case is shown to correspond to a 6d chiral higher spin theory. The cases of M=16 (d=10) and M=32 (d=11) are discussed briefly.