Symmetry and Conservation in Spacetime

The concept of groups originally came from several branches of mathematics, including geometry, number theory, and the theory of algebraic equations, and gradually led into the development of formal group theory beginning in the late 18th Century (O’Connor and Robertson, 1996). Since its formulation, group theory has been applied to almost every branch of physics. It is probably most well known for its usefulness in quantum mechanics, but group theory can also be used to find and explore symmetries of classical and relativistic physical systems. For example, the group of translations in space is a symmetry group for classical space because of the homogeneity of space. The conservation of linear momentum can be derived from this symmetry. The group of translations in time is also a symmetry group for classical time, because of the homogeneity of time. This symmetry can be used to derive the conservation of energy. The group of rotations of space, which is a symmetry group of space because of the isotropy of space, can be used to derive the conservation of angular momentum. Additionally, the group of Lorentz transformations is a symmetry group for spacetime which is used in special relativity. 1.1 Basic Group Theory A group is a set G with an operation ∗ which associates any given ordered pair of elements a, b ∈ G with a product a ∗ b which is also an element of G, such that the following conditions are satisfied