Dirac equation and gamma factor

[exponent137]

I am reading about Dirac's equation for relativistic electron in Feynman's book "Quantum Electrodynamics". Factor \gamma =(1-v^2)^{-1/2} (units c=1) is almost always presented in non quantum calculations of Special relativity. But in his book I also find it on page 44 in lecture "Relativistic invariance", when he shows Lorentz transformations of his matrix \gamma_{x,y,z,t}
1. But, I still ever do not understand, what is quantum analog of equation p=\gamma mv.
2. Is this analog \alpha p, or only p. \alpha is matrix.
3. I also wish examples which uses the above \gamma =(1-v^2)^{-1/2} in calculations?

 

[Bill_K]

exponent137, There are at least three answers, depending on how complicated you want to get. If you only want the gamma factor, it's easy, the equation to use is the relativistic expression for energy, E = γmc², implying γ = E/mc².

If you want to know what happens to p = γmv in the framework of the Dirac equation, it's a little more complicated. The difficulty is in how to define v, and requires a discussion of "zitterbewegung" and the "Newton-Wigner position operator" (which is usually denoted by X to distinguish it from the normal position operator x.)

The velocity is dX/dt, and I quote the result from a big book on relativistic quantum field theory: "Within the manifold of positive energy solutions the time derivative of X is the operator pc²/E, which can be identified with the velocity of the particle."

Sure enough, using the special relativity formulas, pc²/E = (γmvc²)/(γmc²) = v.

If you really want the full expression, here it is! dX/dt = (pc²/E²) (βmc² + α·p c), where α, β are the Dirac matrices.