# Matrix elements of stress-energy tensor $\langle q | T^{\mu\nu} |q\rangle$ in QFT?

In many QFTs we can define a stress tensor $$T^{\mu\nu}$$. What is the matrix element of $$T^{\mu\nu}$$ in momentum eigenstates? For instance, consider

$$\langle q | T^{\mu\nu} |q\rangle$$

in QCD, where $$|q\rangle$$ is a proton with momentum $$q$$. (According to Schwartz Eq. 32.106, this is $$= q^{\mu}q^{\nu}$$--- but why?).

One thing we may note is $$\int d^3x\, T^{0\nu} = P^{\nu}$$, so $$\int d^3x\, \langle q | T^{0\nu} |q\rangle = q^{\nu} \langle q | q\rangle$$; but in general I do not know how to find $$\langle q | T^{\mu\nu} |q\rangle$$.

• The stress-energy tensor is an operator given in terms of the field operator and its derivatives. Do you know how to find the expectation value of the field operator and/or its derivatives? – Void Jan 25 at 16:56
• @Void I suppose I could try to find $T^{\mu\nu}$ explicitly, plug in the creation/annihilation operator expansions and try to find the answer that way. But I was wondering if there is a general expression (such as what Schwartz posits, $\langle q | T^{\mu\nu} |q \rangle = q^{\mu}q^{\nu}$) in QFT which is independent of whatever crazy form $T^{\mu\nu}$ or the fields may take-- even if it's not completely general and requires a few assumptions. – Dwagg Jan 25 at 22:01