A Theory of Spontaneous $T$ Violation

A theory of spontaneous $T$ violation is presented. The total Lagrangian is assumed to be invariant under the time reversal $T$ and a gauge transformation (e.g., the hypercharge gauge), but the physical solutions are not. In addition to the spin-1 gauge field and the known matter fields, in its simplest form the theory consists of two complex spin-0 fields. Through the spontaneous symmetry-breaking mechanism of Goldstone and Higgs, the vacuum expectation values of these two spin-0 fields can be characterized by the shape of a triangle and their quantum fluctuations by its vibrational modes, just like a triangular molecule. $T$ violations can be produced among the known particles through virtual excitations of the vibrational modes of the triangle which has a built-in $T$-violating phase angle. Examples of both Abelian and non-Abelian gauge groups are discussed. For renormalizable theories, all spontaneously $T$-violating effects are finite. It is found that at low energy, below the threshold of producing these vibrational quanta, $T$ violation is always quite small.