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Why in most QFT books when author discusses of non-invariance of measure of path integral (massless fermions interact with gauge fields) $$ \int D\bar{\Psi} D\Psi \to |\Psi \to U\Psi , \quad \bar{\Psi} \to \bar{\Psi} \bar{U},\quad U = e^{i \alpha (x) \gamma_{5}t}, \quad t^{\dagger} =t| \to \int D\bar{\Psi} D\Psi \det \left( U \right)^{-2} $$ and looks only on infinitesimal transformations $U \approx 1 + i\alpha(x)\gamma_{5}t$ he doesn't introduce something like fictive particles (fictive, but formally independent) called ghosts in non-abelian gauge theories? Instead of this he introduces something like monster (like in Weinberg's "QFT" Vol. 2) $$ \int D\bar{\Psi} D\Psi \to \int D\bar{\Psi} D\Psi e^{-2i\int d^{4}xTr \left[\gamma_{5}tf \left( \frac{(\gamma_{\mu}D^{\mu}_{x})^{2}}{M^{2}}\right)\delta (x - y)\right]_{y \to x}}, $$ which is equal to summation over eigenstates of $\gamma_{\mu}D^{\mu}$ operator without introducing the new fields.

I don't understand this.

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    $\begingroup$ Could you be more specific what you don't understand? Why do you think that the ghosts should be discussed here? It is the fermion measure that turns out to be non-invariant, not the ghost measure. Also, why should we need them for the eigenstates of the Dirac operator? $\endgroup$ – ACuriousMind Sep 16 '14 at 20:53

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