# Special Relativity and the Particle-Antiparticle Problem

1. Aug 15, 2010

### Anamitra

In Quantum Mechanics we have the possibility of particles shooting across a space-like interval (the probability amplitudes of the particles get canceled by those of the antiparticles shooting across in the reverse direction as seen by an observer in the boosted frame). I have found this fact in
1) Michael Peskin's "An Introduction to Quantum Field Theory" Chapter 2[The Klein Gordon Field],Subsection 2.1,The Necessity of the Field View Point.
2)Steven Weinberg's "Gravitation and Cosmology", Chapter 2[Special Relativity],Section 13,Temporal Order and Antiparticles

My Problem:
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)

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”

My queries: How do they explain the particle-antiparticle problem in relation to the Class A frames? Do the particles and the antiparticles travel in the same direction for these cases and still their amplitudes cancel?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.

Last edited: Aug 15, 2010
2. Aug 20, 2010

### Anamitra

I am not getting any reply even after a long time.I am feeling more and more confused!

3. Aug 20, 2010

### bcrowell

Staff Emeritus
It would help if you would mark up your math in LaTeX. Personally, I'm disinclined to spend a lot of time deciphering a long, mathematical post that isn't rendered in my browser as equations.