Dirac equation and structure of spacetime

[sandy stone]

I've been reading about the Dirac equation, and most authors eventually make some statement to the effect that the fact of spin and antiparticles falling out of the equation reflects a deep connection to the structure of spacetime. Is the implication that the math requires four particle states (or types, spin up/down, matter/antimatter) because there are four dimensions of spacetime? And that this tells us something important about the way the universe is constructed and is not just an artifact of the mathematics?

 

[DEvens]

Kind of.

Spin is angular momentum. That gets you into rotation. That gets you into group theory and representations of rotations.

It goes a little something like this. I am leaving out just HUGE amounts of mathematics here. If you are keen on this, you should get yourself a textbook or two and read about group theory and quantum mechanics.

There is a scalar spin-zero representation of rotations. In that case, rotations are "trivial" in that scalars are invariant under rotations. Next is a "spin half" representation. That is related to the Pauli matrices. But in relativistic quantum field theory, you get relationships between particles and anti-particles. So instead of just the 2x2 Pauli matrices, you get 4x4 Dirac matrices. Next is a "spin 1" representation. That is related to vector fields such as photons. The Dirac equation with a photon interaction shows you how the spin-half electron interacts with a spin-1 photon. The electron (along with its anti-particle) sit in the 4 component $\psi$ spin-half rep of rotations. The photons sit in the vector $A_\mu$ spin-1 rep.

The reason these representations are important is because a particle has to rotate onto itself, so to speak. If you have a thing in one coordinate reference frame, then you rotate to another frame, you have to get back the same thing in some sense. So if you rotate the coordinates with which you view an object, and that object winds up splitting up into different objects in the new system, then the original view of the object does not seem to be fundamental.

To motivate this, imagine you had a 2-dimensional object. If you had this rotated in such a way that it projected a non-zero distance on all three coordinate axis, you might be misled to think it had extent in three dimensions. But if you then rotated it to the appropriate angles, it would have zero projection in one direction. Angular momentum representations are, very vaguely and hazily, a little like that. If you have a spin-half object and rotate it, it behaves like a spin-half object. If you have a spin-one object it behaves like a spin-one object. With the correct mathematics, you can produce a projection operator that projects out the spin-half part and the spin-one part.

To motivate this a tiny bit farther, consider odd and even functions. An odd function is such that f(-x) = - f(x). And an even function is such that g(-x) = g(x). So consider the following projection operations on some generic function H(x).

f(x) = 1/2 ( H(x) - H(-x))
g(x) = 1/2 (H(x) + H(-x))

It is clear that f(x) here is odd, and g(x) is even. So we can find the odd and even parts of a generic function. And if we do the operation x -> -x, the odd parts transform onto odd parts, and the even onto even parts. (Maybe with a sign change.) We have, in effect, separated out the odd sub-space from the even sub-space of functions.

So too we can find the parts that correspond to different spins in quantum mechanics. And that is what is shown in the Dirac equation. The spin-half parts are the electrons, and the spin one parts are the photons. And this is a fundamental property of space-time, since it relates to rotational symmetry and to time reflection symmetry through particle-anti-particle relationships.

As I said, there is just a HUGE amount of math peaking out here.

 

[sandy stone]

Thanks for your reply. I have been slogging through the texts by Klauber and Srednicki, supplemented by whatever I might find on the web. I had come away with the impression that the gamma matrices had to be 4X4 to satisfy the requirements of Dirac's formulation, and that the solutions to the equation took the form of 4-component vectors because of the dimensions of the matrices as well as the fact that there were 4 terms, corresponding to the 4 dimensions of spacetime, in the equation itself. I see there is plenty more to learn.

 

[ZealScience]

I think the connection between spin and space-time is torsion. In elementary general relativity, we usually assume torsionless situations in which algebra could be simplified. The components are as you said because of the matrices. 4-dimensional matrices are the smallest dimensional representation of gammas. This is probably more on mathematics of spinors.