Space-Time Symmetry and Conservation Laws as Organizing Principles of Matter and Fields

The self-assembly of the universe was constrained by the symmetries of space-time and the resulting conservation laws. Symmetries are defined as length preserving transformations in vector spaces. The Lagrangian and the Hamiltonian are introduced, along with the Euler–Lagrange equations of motion and Noether’s Theorem. Momentum conservation results from the homogeneity of space. Temporality gives rise to energy conservation. The isotropy of space-time leads to the conservation of angular momentum. Physicality generates special relativity. The mathematics of symmetry is that of Group Theory, which is explored for both classical symmetries of space and time, as well as for relativistic symmetries of 4D space-time. Lie groups and Lie algebras are introduced for the rotation group SO(3) and the special unitary group SU(2). It is found that the only objects allowed in 4D space-time are irreducible representations of the double cover of the Poincaré group. The eigenfunctions of the Casimir operators for the Poincaré group describe Fermions and Bosons; the particles and fields that make up all of the Universe.