Special Relativity and the Particle-Antiparticle Problem

[Anamitra Palit]

In Quantum Mechanics we may have particles shooting across a space-
like interval (the amplitudes of the particles get canceled by those
of the antiparticles shooting across in the reverse direction). We
consider such an interval in the unprimed frame given by the space-
time coordinates (t1, x1) and (t2,x2) where,
(x2-x1)/(t2-t1)=k>c
We assume t2>t1
In the primed frame,
x1’=gamma (x1-vt1)
t1’=gamma(t1-(v/c^2)x1)
x2’=gamma (x2-vt2)
t2’=gamma(t2-(v/c^2)x2)
gamma=Sq-root[1/{1-v^2/c^2}]

t2’-t1’=gamma[(t2-t1)-(v/c^2)(x2-x1)]
=gamma(t2-t1)(1-k(v/c^2))
Temporal reversal takes place in the primed frame provided,
k>c^2/v
We consider an unprimed frame which belong to the category
C<k<c^2/v
Such frames pertain to space-like intervals in the unprimed frame
whose temporal coordinates do not get reversed in the primed frames.
Let us categorize such frames(in the primed systems) as ”Class A”
Query: How is the particle-antiparticle problem explained in relation
to the Class A frames? Do the particles and the antiparticles travel
in the same direction for these cases?Do these frames really cause any
confusion?
It is IMPORTANT to note that space-like separations in the unprimed
frame DO remain space-like separations in the primed frame in
relation to the “CLASS A “ category [despite maintenance of temporal
order] This may be easily verified by simple calculations.

Anamitra Palit