# Beta functions and the energy-momentum tensor

#### muppet

Hi all,

In Polchinski's string theory text he asserts (volume 1, section 3.4) that the trace of the energy-momentum tensor of a classically scale -invariant theory becomes proportional in the quantum theory to the beta function of the coupling, as a general point of QFT. This makes a kind of intuitive sense, but does anyone know of a reference that demonstrates the proportionality more precisely?

Can I also ask if anyone is aware of a similar connection in general between the trace of the energy-momentum tensor and the beta function of an arbitrary (i.e. non-conformal) theory?

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#### ManyNames

I presume from what i've just read of the OP - ( and i'm no string theorist, so correct me if i am wrong) - to be related to the Yang Mills coupling - if my memory serves me correctly.

Such work leads to mathematical expressions of quantum contributions to the beta function. The proportionality for instance, where $T_{\mu}^{\mu}$ is the momentum energy tensor and with the Kaluza-Klein models of particles, after much math i cannot be bothered to write, can determine coupling constants.

As i said, i'm not a string theorist, so anyone who see's a mistake in that, please correct me.

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