Special Relativity and Its Newtonian Limit from a Group Theoretical Perspective

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.

Quantum Origin of (Newtonian) Mass and Galilean Relativity Symmetry

“…Of course at the representation level, all the operator sum/product combinations are well defined. The Casimir element 1 2 T ij T ij can be seen as sitting in for the 1 2 J ij J ij of SO(3) which gives our familiar angular momentum as a Casimir invariant. Introducing physics is not commonly addressed.…”

“…The 'spin' of the full Ŝij , i.e. 1 2 Ŝij Ŝij , is effectively a Casimir invariant. This 'spin' s for the composite has a list of admissible values coming from the standard angular momentum addition picture, as the sum of the three parts.…”

“…In the presence of interaction, the physical Hamiltonian or the energy observables need to have an additional dynamical potential term which depends typically only on Ri Ri , in relation to having 1 2 Ŝij Ŝij as a Casimir invariant. In any case, intuitively, one can see that the interaction potential has to be independent of Xi and Pi .…”

“…The H R (3), as well as its classical counterpart [3], does not give a direct notion of t as it does for the x i . However, the natural symplectic structure from the Lie group on the phase space, quantum or classical, at least gives all possible dynamics under admissible generic Hamiltonians as its symmetries [1,3]. One can argue that a picture of Newtonian space-time in itself without dynamics is quite meaningless.…”

“…The Galilean symmetry for 'nonrelativistic' physics has been firstly taken as the symmetry for the Newtonian space-time which is a representation/coset space of the Galilei group G(3). The phase space for a Newtonian particle can also be identified with a G(3) coset space [1]. The Hilbert space for the states for a quantum particle is known as the representation space of the Heisenberg-Weyl symmetry H(3), as usually appreciated through the commutator algebra of the position and momentum operators.…”