As known, in QFT, the conserved currents, such as the energy-momentum tensor, can be derived from the Noether's theorem and expressed as the product of the field operators. These conserved currents can be later on employed in the linear response theory for calculating the transport coefficients, such as viscosity, where the field operators certainly do not obey the classical equations of motion. However, in the derivation of the Noether's theorem, the validation of the classical euler-lagrange equation is assumed. How to understand the contradiction here?
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It is indeed possible that a classical symmetry does not survive quantization. This is called an "anomaly". Well known example: Chiral anomaly, responsible for the decay $\pi^0 \to \gamma \gamma$. – Hyperon Jan 18 '23 at 14:24
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Noether's theorem is a classical statement, so we assume OP is talking about classical theory. (However for quantum theory, see e.g. this Phys.SE post.)
Indeed, the conservation law in Noether's first theorem typically only holds on-shell, i.e. if the EL equations are satisfied. Take e.g. the example of energy conservation because of no explicit time dependence. Energy is typically only conserved on-shell.
Perhaps OP's discomfort is related to the fact that the expression for the Noether current (such as e.g. the canonical SEM tensor) often leads to quantities that is defined outside the conserved realm where they were derived in the first place.
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