Re: Question about Paul Dirac

Posted by DickT on December 07, 2002 at 18:15:29:

In Reply to: Re: Question about Paul Dirac posted by TonyC on December 07, 2002 at 12:34:27:

Tony,

From Polchinski Volume 1.

Working on the free spectrum of the open string (pp 22 f) he arrives at a formula for the mass squared of the associated particle

m^2 = (1/a')(N + (2-D)/24)

Where a' is the tension of the string, N is the number of particles represented, and D is the dimension of spacetime. The numerator D-2 arises because he is working in light cone gauge so the time coordinate and 1 space coordinate have been gauged away and only D-2 dimensions are left to vibrate in.

Now he considers exciting a one particle state, N = 1.

Then m^2 = (26 - D)/24a'

Now he uses an argument from Lorentz invariance. In light cone gauge you destroy Lorentz invariance, and it has later to be put back in by hand. Counting L.I. states he shows there are D-1 of them if the particle is massive, but only D-2 if it is massless. But there are in this case only D-2 states. He concludes that L.I. requires the free N=1 states to be massless, hence D = 26.

This is the first proof he gives of D = 26, and it is the easiest to follow. (Yes I know, if this is the easy one, what can the others be like? superstrings is not for the mathematically faint of heart).

Regards,
Dick

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