Abstract
In this pedagogical article, we explore a powerful language for describing the notion of spacetime and particle dynamics intrinsic to a given fundamental physical theory, focusing on special relativity and its Newtonian limit. The starting point of the formulation is the representations of the relativity symmetries. Moreover, that seriously furnishes—via the notion of symmetry contractions—a natural way in which one can understand how the Newtonian theory arises as an approximation to Einstein’s theory. We begin with the Poincaré symmetry underlying special relativity and the nature of Minkowski spacetime as a coset representation space of the algebra and the group. Then, we proceed to the parallel for the phase space of a spin zero particle, in relation to which we present the full scheme for its dynamics under the Hamiltonian formulation, illustrating that as essentially the symmetry feature of the phase space geometry. Lastly, the reduction of all that to the Newtonian theory as an approximation with its space-time, phase space, and dynamics under the appropriate relativity symmetry contraction is presented. While all notions involved are well established, the systematic presentation of that story as one coherent picture fills a gap in the literature on the subject matter.
Highlights
Over the past century, the notion of symmetry became an indispensable feature of theoretical physics
Textbooks typically use x μ, what we show below is that we should start with tμ as coordinates for Minkowski spacetime, as we have done above, if we want to directly and naturally recover t and xi as coordinates of the representation space of Newtonian physics in the Newtonian limit, i.e., under the symmetry contraction described
With an understanding of how the principle of relativity informs our notion of physical spacetime and the theory of particle dynamics behind us, we can move on to the important connection this language provides us between different theories from the relativity symmetry perspective
Summary
The notion of symmetry became an indispensable feature of theoretical physics. The term relativity symmetry, though much like introduced into physics by Einstein, is a valid notion for Newtonian mechanics too. It just has a different relativity symmetry, the Galilean symmetry. This is augmented by a continuation of the exploration of special relativity, and in particular, the way in which the Newtonian limit is to be understood within this context, before giving some concluding remarks in the last section. All that illustrate well the value of looking at the well-known theories from a somewhat different point of view seriously, as done here
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