Tulsi Dass

Gauge fields, space-time geometry and gravity

A general framework for the gauge theory of the affine group and its various subgroups in terms of connections on the bundle of affine frames and its subbundles is given, with emphasis on the correct gauging of groups including space-time translations. For consistency of interpretation, the appropriate objects to be identified with gravitational vierbeins in such theories are not the translational gauge fields themselves, but their pull backs,via appropriate bundle homomorphisms, to the bundle of frames. This automatically solves the problems usually encountered in constructing a gauge theory of the conventional sort for groups containing translations. We give a consistent formulation of the Poincare gauge theory and also of the theory based on translational gauge invariance which, in the absence of matter fields with intrinsic spin, gives a local Lorentz invariant theory equivalent to Einstein gravity.

Augmented temporal logic formalism for histories-based generalized quantum mechanics

An axiomatic development of dynamics of systems in the framework of histories is given which contains the history versions of classical and traditional quantum mechanics as special cases. We consider theories which admit a quasitemporal structure (a generalization of the concept of time using partial semigroups) and whose “single time” propositions have the mathematical structure of a logic; isomorphism of logics at different “instants of time” is not assumed. The concept of directed partial semigroup is introduced to incorporate the concept of direction of flow of time in quasitemporal theories. Starting with a few simple axioms, the space of history propositions is explicitly constructed and shown to be an orthoalgebra (as envisaged in the scheme of Isham and Linden [J. Math. Phys. 35, 5452–5476 (1994)]); its subspace consisting of history filters (“homogeneous histories”) is a meet semilattice. The partial semigroups employed allow semi-infinite irreducible decompositions.

Symmetries and conservation laws in histories based theories

Abstract Symmetries are defined in histories-based theories, paying special attention to the class of history theories admitting quasi-temporal structure (a generalization of the concept of “temporal sequences” of “events” using partial semigroups) and logic structure for “single-time histories.” Symmetries are classified into orthochronous (those preserving the “temporal order” of events) and nonorthochronous. A straightforward criterion for the physical equivalence of histories is formulated in terms of orthochronous symmetries; this criterion covers various notions of physical equivalence of histories considered by Gell-Mann and Hartle (1990, in “Complexity, Entropy, and the Physics of Information” (W. Zurek, Ed.), SFI Studies in the Science of Complexity, Vol. 8, p. 425, Addison–Wesley, Reading, MA) as special cases. In familiar situations, a reciprocal relationship between traditional symmetries (Wigner symmetries in quantum mechanics and Borel-measurable transformations of phase space in classical mechanics) and symmetries defined in this work is established. In a restricted class of theories, definition of a conservation law is given in the history language which agrees with the standard ones in familiar situations; in a smaller subclass of theories, a Noether-type theorem (implying a connection between continuous symmetries of dynamics and conservation laws) is proved. The formalism evolved is applied to histories (of particles, fields, or more general objects) in general curved spacetimes. Sharpening the definition of symmetry so as to include a continuity requirement, it is shown that a symmetry in our formalism implies a conformal isometry of the spacetime metric.