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

TL;DR

This work advances a geometric, group-theoretic framework in which Sp$(2M)$-invariant field equations live on a generalized matrix space-time ${\cal M}_M$ with coordinates $X^{\alpha\beta}$. It demonstrates causal propagation, a positive-frequency decomposition, and a consistent quantization within a positive-definite Hilbert space, while revealing Minkowski space as a localized subspace tied to Clifford-algebra structure. For $M=2$ the theory reduces to 3d conformal dynamics; for $M=4$ it encodes the full tower of massless 4d fields of all spins and links electromagnetic duality to generalized Lorentz transformations. The paper also develops extended supersymmetry, the geometric origin of ${\cal M}_M$ via Sp$(2M)$, and outlines higher-$M$ extensions (e.g., $M=8$ 6d, $M=16$ 10d, $M=32$ 11d) with potential connections to M-theory and holography. Overall, it provides a unified, geometrical picture where locality, causality, duality, and higher-spin spectra emerge from a single generalized space-time and its Clifford-algebraic localization properties.

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.