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$$

\hat{a}^\dagger = \frac{1}{\sqrt{2}}\left(- \frac{\mathrm{d}}{\mathrm{d}q} + q \right),\;\;\;\hat{a} = \frac{1}{\sqrt{2}}\left(\frac{\mathrm{d}}{\mathrm{d}q} + q \right)

$$

however I cannot find any explicit terms for these in QFT. My question is, is it possible to formulate an expression for them in terms of a differential operator or do we just assume that they exist in QFT? I am particularly interested in the Massive Thirring Model (Dirac field in 1+1D with self interactions).