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.

This post imported from StackExchange Physics at 2014-09-17 10:15 (UCT), posted by SE-user PhysiXxx