= Hubbard–Stratonovich transformation
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{title2=$\int dM\,e^{-M^2/(2U)+Mj}\propto e^{Uj^2/2}$}
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= Hubbard–Stratonovich decoupling
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{synonym}
A Gaussian auxiliary-field identity replacing a quadratic expression in a composite field by a linear coupling to a new field. Completing the square in $M^2/(2U)-Mj$ yields an exponential $e^{Uj^2/2}$ after integrating $M$, with normalized Gaussian prefactors. It converts suitable quartic field interactions into bilinear couplings and permits fermionic fields to be integrated out.
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