# Relativistic quantum field theory:antiparticles!

Avodyne
Getting back to the original question, here is a heuristic explanation of why quantum mechanics plus special relativity requires antiparticles.

Suppose a particle goes from point x1 at time t1 to point x2 at time t2. If the points have a "timelike separation", (x2-x1)^2 < c^2 (t2-t1)^2, this is classically allowed; if the points have a "spacelike separation", (x2-x1)^2 > c^2 (t2-t1)^2, it is classically disallowed; the particle would have to move faster than light.

But in quantum mechanics, all paths make a contribution to the probability amplitude, and so these paths contribute.

For points that are spacelike separated, their temporal order (that is, whether t1 > t2 or t2 > t1) is frame dependent, and can be different for different inertial observers. So, a process that looks to one observer like a particle going from x1 to x2, looks to another like a particle going from x2 to x1. (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.

What happens in a QFT which obeys SR to cause more particles to emerge? Do we have a modern interpretaton? Why does SR invariance, a geometric constraint, cause antiparticles to emerge in QFT?
Naty

The equivalence of energy and rest mass ultimately arises from an invariance of the form of classical laws with respect to inertial frame and the invariance of the speed of light with respect to inertial frame. The antiparticle ultimately arise from the invariance of quantum mechanical laws with respect to inertial frame and the invariance of the speed of light with respect to inertial frame.

atyy
well, Charles Seife in his book says it DOES...but maybe you are right in which case that would address my question....That's the same reaction I had when when I read Seife's explanation!!!! It still sounds "crazy" but so does time dilation.

Ah, yes it does. I didn't read your question properly. Particle creation and destruction can occur non-relativistically, but antiparticles only occur relativistically, pretty much as others explained that you go from E=p2 to E2=p2+m2.

strangerep
$$\Box = \eta ^{\mu\nu}\partial _{\nu} \partial _{\mu} = \frac{1}{2}\gamma ^{\mu}\gamma^{\nu} \partial _{\nu} \partial _{\mu} +\frac{1}{2}\gamma ^{\nu}\gamma^{\mu} \partial _{\nu} \partial _{\mu}$$
Huh? Was that a rhetorical question?
$$\frac{1}{2}\gamma ^{\mu}\gamma^{\nu} \partial _{\nu} \partial _{\mu} ~=~ \frac{1}{2} \gamma ^{\mu} \partial _{\mu} \gamma^{\nu} \partial _{\nu} ~=~ \frac{1}{2} (\gamma \partial)^2$$
and similarly for the second term. (Uses commutativity of partial derivatives
and constancy of gamma matrices).

malawi_glenn
Homework Helper
Ok, I think I get it "why" one in principle can write $$\Box ^{1/2}= \gamma _{\mu}\partial ^{\mu}$$

But, here are my thouhgts on why one can not think of the Dirac equation as simply the "square root of the KG eq".

i) KG is for SCALAR fields $$\phi$$ :

$$\partial _\mu \partial ^\mu \phi = -m^2\phi \qquad (\partial^2 _t + \partial ^2_x +\partial ^2_y + \partial ^2_x ) \phi = -m^2\phi$$

This represents $$E^2 = p^2 + m^2$$, 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:
$$\partial _\mu \partial ^\mu \phi = -m^2\phi$$

The Dirac Equation is for spin 1/2 particles, with 4 compotnent spinor $$\Psi$$, i.e. a 4 component vector.

What Dirac did was to find an equation which is linear in derivatives and reduces to Pauli equation in non rel limit.

$$\left(\beta mc^2 + \sum_{k = 1}^3 \alpha_k p_k \, c\right) \psi (\mathbf{x},t) = i \hbar \frac{\partial\psi(\mathbf{x},t) }{\partial t}$$
With constraints on the p_k's and alpha_k's

And indeed, he also started from the $$E = (p^2 + m^2)^{1/2}$$ and found the gamma matrices..

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

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.)

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?

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(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.

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.