We are usually told that the world of antiparticles is like an exact copy of ours but with some peculiarities. The lack of evidence of large quantities of antimatter in the observable universe is an indication that the symmetry between particles and antiparticles is not perfect. However, from a mathematical perspective it is possible to argue about a significant asymmetry between particles and antiparticles.
Group theory tell us that the Dirac spinor is a double cover representation of the Special Lorentz Group denoted by SO(1,3). This group has a subgroup characterized for maintaining the direction of time. As a result, the operations in this group can be visualized as either spatial rotations or boosts of velocity. In other words, this subgroup has a simple and intuitive interpretation and can be associated with particles. Therefore, antiparticles are the complement of the total group with the subgroup of particles. This mathematical construction is called coset. Consequently the world of pure particles has the structure of a group while the world of pure antiparticles has the structure of a coset!
This argument can be illustrated in the following diagram where O_{+}(1,3) is the group that maintains the direction of time.
The book that contains a detailed description of the group structure of spinors is
* Pertti Lounesto, Clifford Algebras and Spinors, Cambridge University Press, May 3, 2001
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