If you haven't found something strange: A quick proof of the C-theorem

Give me a quantum field theory in 2 dimensions that is rotation invariant, unitary and has stress energy tensor conservation. In return I will give you a function that is decreasing along renormalization group flow and on the fixed points, it will give the value for the central charge.

$\langle T(z,\bar{z}) T(0,0) \rangle = \frac{F}{z^4}$

$\langle T(z,\bar{z}) \Theta(0,0) \rangle = \frac{G}{z^3\bar{z}}$

$\langle \Theta(z,\bar{z}) \Theta(0,0) \rangle = \frac{H}{z^2\bar{z}^2}$

T is the component of spin 2, $\Theta$ is the component of 0 spin and $\tilde{T}$ is the one with eigenvalue -2.

Conservation of stress energy tensor translates to the following equations:

$\dot{F}+\frac{1}{4}(\dot{G}-3G)=0$

$\dot{G}-G+\frac{1}{4}(\dot{H}-2H)=0$

Now define

$C \equiv 2F - G - \frac{3}{8}H$

We get

$\dot{C}=-\frac{3}{4}H$

but H is positive, see the definition of it. Therefore this function decreases.

At the conformal points, $\Theta$ vanishes and sense so do G and H making C=2F. But by OPEs F=c/2. Therefore we get a precise match of our defined function C and the central charge at the fixed points of the renormalization flow.

If you haven't found something strange

Sunday, January 8, 2012

A quick proof of the C-theorem

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