- #1

Amir Livne

- 38

- 0

What I would like to ask, is what elementary quantities completely define the state of a single particle. In Newtonian gravitation these would be mass, position, and momentum. That's 7 real numbers, 6 of them unlimited and one restricted to positive values.

In QM things seem to be more difficult, and I understand much less. For a start, there is not isolated particle in vacuum, and even if there was, there would be no values for these properties. So instead - and my first question is whether this is an interesting property - I suggest counting operators whose eigenspaces are of dimension 1. These would include position and momentum, 6 operators with continuous spectrum. That's all there is for a free particle, meaning [tex]i\hbar\frac{\partial\Psi}{\partial t}=-\frac{\hbar^2}{2m}\nabla^2\Psi[/tex]

What do you get for QED? As I understand, you have mass and electric charge, which are operators with discrete spectrum but an infinite set of eigenvalues. If you consider a state with a single electron it's in an eigenspace with one specific value for mass and electric charge. How does spin relate to this? I read everywhere that spin can be measured along any direction and give 2 values. But how many linearly independent spin operators are there? What is the spectrum for a system with n electrons (my common sense tells me n+1, but I don't know really know the physics)? And is there any other observable independent of the above?

The same question goes for the theory of electro-weak interactions and for the Standard Model. Do these models simple tack on more and more degrees of freedom on each particle, or does something more basic change? What are those parameters? Can extensions to the SM be state along the same lines? Like - "instead of being 3 independent neutrino operators, we now believe in stealth neutrinos so there are 4"