Doctor of Philosophy
D.G.C. (Gerry) McKeon
S.V. (Sergei) Kuzmin
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.
Summary for Lay Audience
Physics is a science which focuses on quantifying observed natural phenomena. To do this for a classical system, physicists use equations that can be solved, subject to initial conditions corresponding to the dynamics of a particular observed phenomena; these equations are known as equations of motion. For the fundamental interactions of electromagnetism and gravity, the accepted equations of motion describing the dynamics of these theories are Maxwell's equations and Einstein's field equations, respectively. These equations have not been replaced or changed in over 100 years (although some modifications have been proposed). Other equations may be needed to complete a theory, such as in electrodynamics where conservation of energy and the force law are described by Poynting's theorem and the Lorentz force law. Using what is known as the action, the equations of motion of the theory can be straightforwardly obtained using the Euler-Lagrange equation; this equation ensures that this action has a minimum value. For conservation laws however, this is not as straightforward --- multiple methods which contradict each other exist for obtaining them. In addition, some of these methods fail to obtain all known physical laws in a straightforward manner. These issues are the focus of the first two chapters of this thesis. The basis for our approach is Noether's first theorem, a fundamental result that shows that symmetries present in a physical system results in there being ``conserved'' quantities (quantities whose value remains constant as the system evolves). In Chapter 1 we clarify a straightforward methodology for obtaining physical conservation laws using the Bessel-Hagen approach to Noether's first theorem, and then in Chapter 2 we clarify the status of the other methods that contradict this approach. This use of Noether's first theorem renders the complete set of physical equations, for example in electrodynamics, as implicit information contained in the Lagrangian. For a set of physical theories, only the set of Lagrangians are required. In Chapter 3 we ask if all of these Lagrangians can be obtained from a common set of axioms, so that even the set of Lagrangians are implicit information to the imposed physical requirements. We show that such an approach can be used to obtain electrodynamics, linearized Gauss-Bonnet gravity, and other ``higher-spin" gauge theories found in the physics literature. Future research that could follow from our results is discussed.
Baker, Mark Robert, "Field theories from physical requirements: Noether's first theorem, energy-momentum tensors and the question of uniqueness" (2021). Electronic Thesis and Dissertation Repository. 7802.
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