Field theories from physical requirements: Noether's first theorem, energy-momentum tensors and the question of uniqueness
Abstract
An axiomatic approach to physics is proposed for obtaining classical gauge theories from a common set of physical requirements based on standard features of special relativistic field theories such as gauge invariance, conformal invariance and being in four dimensions. This approach involves the use of Noether's first theorem to directly obtain a unique, complete set of equations from the symmetries of the action. However, implementation of this procedure is obstructed by issues of ambiguity and non-uniqueness associated with the conserved tensors in the majority of special relativistic field theories. In the introductory chapter, we outline the three major problems which are considered in this thesis. Each of these three problems are addressed separately in the three central chapters of the thesis, which consist of eight integrated articles. These three problems are (i) the failure of the canonical Noether energy-momentum tensor to obtain known physical conservation laws, and the ad-hoc ``improvement'' of the energy-momentum tensors occurring in the literature, (ii) the ambiguities and non-uniqueness associated with multiple different methods for derivation of the energy-momentum tensor, and (iii) the procedure required for converting a set of axioms to a set of Lagrangian densities. The concluding chapter summarizes our major results, such as proper variational ``Noetherian'' symmetries for several completely gauge invariant models using the Bessel-Hagen method, a formal disproof of the equivalence of the Noether and Hilbert energy-momentum tensors in Minkowski spacetime, a proof that there are infinitely many solutions for energy-momentum tensors in linearized gravity obtained from the ``improvement'' method, and a derivation of the curvature tensors of higher spin gauge theories without referring to the symmetry properties of the Riemann curvature tensor. Future research that could follow from our results is discussed.