Re: I think I can understand....

Posted by DickT on December 16, 2002 at 14:02:50:

In Reply to: I think I can understand.... posted by TonyC on December 16, 2002 at 12:36:50:

Tony,

It's difficult to find one that doesn't assume too much. Maybe this one? http://arxiv.org/abs/hep-ph/9705477.

The idea is this. You are in a relativistic space time and you have fixed one of the components, say x as the direction of motion (this is all local). So the intersection of the light cone with the x-t plane is the pair of lines x=ct and x=-ct. If you are in units where c=1, as physicists nearly always are, these are x = t and x = -t. Looking at a spacetime diagram these are lines through the origin x=0, t=0 at 45^o to the x-axis and the t-axis.

Now, for the moment only, you replace x and t with the two forward pointing unit vectors along these lines, calling them x₊ and x_-. Restating your dynamical equations in terms of these, they become more tractible. But you must always bear in mind that you have broken the Lorentz symmetry by treating x differently from y and z; your result will be frame dependent. When your calculation is done, you must do work to translate it into a Lorentz invariant result.

Then suppose you are in a gauge invariant theory. This says any variation of the gauge group on your equations is as good as any other, and this is death on calculation. It's like the personality test motto, "There isn't any right answer. Put down whatever appeals to you".

The way out of this mirror maze is to fix the gauge. You establish some relationship you hold as true, so that it picks out just one example of the physics for you to compute with. Maybe you've seen the Lorentz gauge in electrodynamics: Div Phi = 0, where Phi is the vector potential.

Insisting on the non relativistic light cone coordinates, and then setting x₊ = 0 and x_- = 0 is a way to fix the gauge in quantum field systems, and it's especially used in string theory. This is called the light cone gauge, but it isn't itself a gauge; it's a gauge fixing prescription. It freeze-frames some of the action. But the peculiarity of the action thus frozen is that it has two less dimensions than the full theory.

Regards,
Dick

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Follow Ups: (Reload page to see most recent)

• Re: I think I can understand.... -- TonyC 12/18/02 (1)
  • Re: I think I can understand.... -- DickT 12/18/02 (0)

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