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

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    $\begingroup$ 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? $\endgroup$ – Void Jan 25 at 16:56
  • $\begingroup$ @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. $\endgroup$ – Dwagg Jan 25 at 22:01

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