Time Reversal Symmetry Not Exist Actually In the Theories of Particle Interactions

It is pointed out that the judgment condition of time reversal symmetry in the processes of particle interaction is that the Hamiltioians of systems satisfy the relation Ĥ(t) = Ĥ∗(−t). But this condition can not be satisfied actually in the current interaction theories, so the symmetry of time reversal does not exist actually in the current theories of particle interactions. In the current proof of time reversal symmetry, the relation Ĥ(t) = Ĥ∗(−t) is supposed to exist in advance, then the translation relations between the time reversal operator and the field quantities are deduced. So the current proof about the symmetry of time reversal is not a real one. PACS number: 1130 In quantum mechanics, the transformation of time reversal is carried out in the light of following procedure. Let t → −t in the Schordinger’s equation. i ∂ ∂t ψ(x, t) = Ĥ(t)ψ(x, t) (1) and suppose that the Hamiltonian Ĥ is unchanged when t → −t with Ĥ(t) = Ĥ(−t). Thus, we get. − i ∂ ∂t ψ(x,−t) = Ĥ(t)ψ(x,−t) (2) Then, take the complex conjugation of Eq.(2) and suppose Ĥ(t) = Ĥ(t) again, we have i ∂ ∂t ψ(x,−t) = Ĥ(t)ψ(x,−t) (3) Comparing Eq.3 with Eq.1, it can been seen that ψ∗(x,−t) and ψ(x, t) satisfy the same equation. So we define time reversal operator T as follows when it is acted on the wave function Tψ(x, t) = ψ(x,−t) (4) In this way, ψ∗(x,−t) represents the wave function of time reversal process. Eq.(4) shows that the operator of time reversal is an antiunitary operator, which can be defined as generally T (λψ1 + Xψ2) = λTψ1 + X Tψ2 (5) Because wave function ψ(x, t) and ψ∗(x,−t) satisfy the same equation, we have ψ∗(x,−t) = bψ(x, t). Here b is a constant. We can take b = 1 for simplification. But ψ(x,−t) 6= ψ(x, t) in general. It seems that the positive process is different from the reversal process. However, because the wave function can not be directly measured, what can be done is the possibility density ρ. Suppose ρ(x, t) = ψ(x, t)ψ(x, t) is the possibility density of the positive process, the possibility density of the time reversal process is ρ(x,−t) = ψ∗(x,−t)ψ(x,−t). If the motion equation of quantum mechanics is unchanged under the time reversal, we have ψ∗(x,−t) = ψ(x,−t) so that the possibility densities of the positive and opposite processes become the same with ρ(x,−t) = ψ∗(x,−t)ψ(x,−t) = ψ(x, t)ψ(x, t) = ρ(x, t). Therefore, the process is considered as the reversible or symmetrical for