Relativistic quantum field theory:antiparticles

[strangerep]

So the situation to go from KG to Dirac is more
subtle than "just taking the square root of KG" ;-)

Indeed. It's better to do the Wignerian thing and think in terms of unirreps
of the Poincare group. $\Box^2$ is just a representation for wave functions
of the Poincare casimir $P^2$ (4-momentum squared). But one should
also think about $J^2$ (total angular momentum squared), which often
is not introduced in basic RQM textbooks until much later.

(BTW, $(-\Delta +m^2)^{1/2}$ is only a Foldy-Wouthuysen
transformation away from the usual Dirac operator anyway, so the
usual objections about nonlocality are perhaps less convincing than
they appear.)

 

[cesiumfrog]

ephemereal nature of life

But in quantum mechanics, all paths make a contribution to the probability amplitude, [including space-like paths ... and for consistency we must interpret backward-in-time paths as antiparticles]

This explanation (for relativity implying creation and annihilation) seems much more fundamental (than the happen-stance that negative energies are not disallowed by Dirac, Klein & Gordon's attempts at writing relativistic wave equations). Is it really true that space-like paths must contribute (and is there a heuristic explanation why space-like paths shouldn't just be ignored from the outset)?

Also, is there an analogous explanation regarding phonons and a speed of sound?

 

[RedX]

(Again, this only happens for non-classical, faster-than-light paths). But, if the particle carries a charge (say +1), then to the first observer, the charge decreases at x1 when the particle leaves, but to the second observer, it looks like the charge increases at x1 when the particle arrives. This is inconsistent, so it must be that the second observer sees a particle arriving with charge -1. Obviously this can only happen if such a particle exists, and so there must be antiparticles.

If an observer observes a particle in one frame, then a boosted frame should also observe a particle, and not an antiparticle, so this shouldn't be taken literarly.

It is true that mathematically, an antiparticle behaves as a negative energy particle moving backwards in time, but that's really confusing to think about physically.

 

[hagi]

If one wants to write $$E = (p^2 + m^2)^{1/2}$$ in operator form:
$$(-\Delta +m^2)^{1/2}\phi = \partial_t \phi$$
This is NONLOCAL! That's why one sticks with:
$$\partial _\mu \partial ^\mu \phi = -m^2\phi$$

I was tought that if you expand the square root in $$E = (p^2 + m^2)^{1/2}$$ and rewrite the operators in position space, then we have one time differentiation on the left but a polynomial of space differentiations on the right hence this equation cannot be Lorentz covariant.