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Дата индексирования: Mon Oct 1 22:51:24 2012
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Gauge fixing terms.

In order to quantize the gauge theory one must add to (1) gauge fixing term and the corresponding Faddeev-Popov term. The first term breaks the gauge symmetry and in this way removes the divergence of the functional integral. The second term improves the integration measure to provide correct predictions for gauge invariant observables.

The general form of the gauge fixing term is

$$L_{FG}(x)= -\, \frac{1}{2}\sum_{\alpha} (\Phi^{\alpha}(x))^2\;.$$ (2)

The corresponding Faddeev-Popov term is

$$L_{FP}= - \bar{c}_{\alpha}(x) (D(c)[\Phi^{\alpha}])(x) \;,$$ (3)

where $c^{\alpha}(x)$ and $\bar{c}_{\alpha}(x)$ are the auxiliary anti-commutative fields. They are called the Faddeev-Popov ghosts. Note that we may multiply (3) by an arbitrary factor which can be hidden in the definition of ghost fields. As a rule it is chosen in such a way to provide a convenient form of the ghost propagator.

The well-known choice of the gauge fixing terms, the Feynman-like gauge, is

$$L_{GF}= -\,\frac{1}{2g^2}\sum_{\alpha} (\partial^{\mu} A^{\alpha}_{\mu} +...)^2\;.$$

In this case the quadratic part of gauge field Lagrangian takes the simplest form

$$-\,\frac{1}{2g^2} \partial_{\mu}A_{\nu}\partial^{\mu}A^{\nu}\;.$$ (4)

The corresponding Faddeev-Popov Lagrangian is

$$L_{FP}= - \bar{c}_{\alpha}(x) (\Box c^{\alpha}(x) + \partial... ...ha}_{\beta \gamma} A^{\beta}_{\mu}(x) c^{\gamma}(x)) + ...)\;.$$

The normalization of $L_{FP}$ is chosen to have the Faddeev-Popov ghost propagator equal to the propagator of scalar particle:

$$T(\bar{c}_{\alpha}(p),c^{\beta}(q))= \delta_{\alpha}^{\beta} \delta(p+q) \frac{1}{i(2\pi)^4}\;\frac{1}{-p^2}\;.$$