Spin, Anti-matter and all that jazz.

[QEDrpf33]

Hello,
Can someone explain to me- in a conceptual manner- why when Special Relativity and QM are combined (ie Dirac) anti-matter and spin naturally come out? I have seen and understand the mathematics, but I am looking for a more physical explanation.
Thanks

 

[Demystifier]

Contrary to frequent claims, neither spin nor antimatter necessarily arise from combination of relativity and QM. For example, Klein-Gordon equation describes particles without spin, and anti-photons do not exist.

 

[muppet]

anti-photons do not exist.

Sure they do... they're just ordinary photons :tongue:

 

[muppet]

Demystifier does have a point though, in that anti-matter is to an extent an interpretation grafted onto the mathematics of the problem. What does rear its ugly head is the existence of negative energy solutions. As the hamiltonian is the generator of time translations, these particles would go backwards in time. The fix of reinterpreting this as a different particle traveling forwards in time is known as the Feynman-Stuckelberg prescription. (For completeness, you also have to reverse the direction of the 3-momentum.)(Coming clean: similar solutions also exist at a classical level! There's a good reason that escapes me now as to why we can ignore them classically but not in QFT... I'll try and get back to you when I remember it, or where I read what it was. I seem to recall it having to do with the necessity of summing over states, or because it ends up being unavoidable that transitions to these states would occur, or somesuch quantum quirk, but can't remember properly or work it out right now.)

Spin is something I'm about do some work on understanding myself, so there's less guarantee of correctness in what follows than is usual even in my posts :tongue: The important point seems to be that it's a measure of how states transform under representations of the Lorentz group. Obviously, if you're messing around with spacetime you have to make allowances for the possibility that this will have some, albeit an indirect, effect on the state vector of some system that's supposed to live in that space. Furthermore, it makes sense that if you perform two successive transformations, equivalent to some third, single lorentz transformation, then the transformation you impose on your hilbert space shouldn't distinguish between physically equivalent states. My group theory isn't up to letting me progress much beyond that elementary level (which I'll hopefully be doing something about later today, or at least this week) but it at least makes sense to me that we need a bookkeeping device that tells us what representation our state transforms under. Spin-0 just means we have the trivial representation of the Lorentz group, i.e. the case where it actually makes no difference. That works for the scalar field, so if there was nothing in the universe apart from the Higgs, there'd be no need for the construction. Luckily for us, the universe turns out to be a more interesting place than that.

 

[shirosato]

I believe the reason why we can't ignore them in quantum is because we need our Hilbert space to be complete, which throwing away negative-energy solutions would mess with.

 

[PhilDSP]

Maybe the best way to understand this is to trace the development of the theory starting with the Pauli equation for the electron. It uses matrices that allow a mathematically viable and simple way of modeling potential spin or revolution of an object. He apparently attempted or desired to eliminate all degrees of freedom from the model so that all parameters of possible electron motion would be specified by the equation. But he wasn't successful in doing that where the results met all known experimental values.

It was the combination of ideas of Uhlenbeck, Goudsmit and Llewelyn Thomas which led to the development of first electron equation requiring spin in order to eliminate a crucial degree of freedom that Pauli had not resolved. There the kinematics of a revolving object operate in a way that when described by a Lorentz Transformation, the resulting momentum is twice the value one would naively expect. This is the famous "Thomas Precession":

http://www.rowdyboeyink.org/ehrenfest/images/Uhlenbeck.pdf
http://www.mathpages.com/rr/s2-11/2-11.htm

Dirac essentially found a new way to factor the equation into more primitive components where those components are effectively mirror images of each other with respect to time and spatial positions. Does that actually mean that certain particles experience time in reverse? That's hard to be certain about - maybe they only travel in such a way that it appears that they do. But certainly the polarity or phase propagation with respect to ...something... is reversed.

 

[haael]

Coming clean: similar solutions also exist at a classical level! There's a good reason that escapes me now as to why we can ignore them classically but not in QFT...

In the classical physics, you can say that negative energy states are simply unphysical.
But in quantum, a positive-energy particle can "fall" into a negative state (annihilation) or some other particle may excitate a particle from out of nothing, with a corresponding negative state (creation). That means: even if we start with all pure positive-energy states, the physical evolution will lead us into negative-energy states with nonzero probability.
That's why we can't simply ignore antimatter.

 

[dextercioby]

1. Antimatter is necessarily a result of QM and relativity working together.
2. The same can be said about spin.

For the second statement, relativity means Galilei or Minkowski, as for the first I'll investigate, as I don't have some articles at hand right now.

 

[PhilDSP]

The Lorentz Transformation is involved in eliminating (or at least specifying how to eliminate) a degree of freedom. But that preceded SR.