Alternative interpretation of relativistic time-reversal and the time arrow

It is well known that the 4-rotation in four-dimensional space-time is equivalent to the $CPT$ transformation ($C$ is the charge conjugation, $P$ is the space inversion, and $T$ is the time reversal). The standard definition of the $T$ reversal includes the change of the sign of the time variable and replacement of the initial state of the particle (system of particles) by the final state and vice versa. Since the time-reversal operation changes the state of a particle, the particle's wave function cannot be the eigenfunction of the corresponding operator with a certain eigenvalue, as in the case of space parity. Unlike the $CPT$ transformation, the separate $P$, $T$, or $C$ transformations cannot be reduced to any 4-rotation. The extended Lorentz group incorporates all the separate $C$, $P$, or $T$ transformations which do not bring the time axis out of the corresponding light cone. The latter restriction is included in the standard definition of the time reversal. In the present Letter, we ignore this restriction. This allows us to introduce the “time arrow” operator and characterize every particle by the new quantum number—the “time arrow” value. The wave functions of all particles are eigenfunctions of this operator with eigenvalues equal to time arrow values. The particles with the time arrow values opposite to the time arrow value in our universe form another universe (antiuniverse). The existence of an antiuniverse can be confirmed, in principle, by laboratory (atomic) experiments. The antiuniverse may be also considered as a candidate for the role of dark matter.

Figure 1

The two-dimensional images of the four-dimensional light cones. (a) The time axis transformations allowed in the standard extended Lorentz group. (b) Additional transformations of the time axis in the additionally extended Lorentz group.

Figure 2

The Feynman graph corresponding to the process of photon emission (absorption) in the theory with an alternative description of the time reversal. The solid lines as in the standard QED denote the electrons (positrons). The wavy line denotes the the photon. The indices $A$ correspond to the time arrow quantum numbers. The indices for all other quantum numbers are omitted. The indices $x,\tau$ at the vertex refer to the space-time variable $x=(t,\mathbf{r})$ and to the “time arrow variable” $\tau$. The integration over $x$ and summation over $\tau$ is assumed.