Faddeev-Popov procedure

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Contents

Gauge fixing

Suppose we have a number of (bosonic) fields \phi_a\, and that our action is invariant under some gauge transformation \phi_a \to f_a( \phi, \Omega^\alpha)\,, which we denote by \phi_a \to \phi_a^{\Omega}\, for brevity. Furthermore, let us demand that the path integral measure \mathcal{D}\phi\, also remain invariant, so that this is a true symmetry of the theory.

The set of configurations \phi_a^{\Omega^\alpha}\, obtained by varying the parameters \Omega^\alpha\, is known as a gauge orbit. The path integral sums over many physically equivalent configurations (parameterized by \Omega^\alpha\,), and this overcounting potentially leaves the sum divergent in a way that is difficult to keep track of. Furthermore, the equations of motion are not deterministic, often giving rise to further divergences.

We can instead pick a representative of each gauge orbit. This is done by choosing a gauge condition G_a(\phi) = 0\, and restricting the sum to configurations that satisfy this condition. Of course, we should require that the surface G_a(\phi) = 0\, intersect each orbit only once. Naïvely, we might consider inserting \delta[ G_a(\phi) ]\, into the path integral. Actually, this does not leave the path integral invariant under changes in G\,.

For ordinary functions, a property of the delta function gives

\delta(x-x_0) = \left|\frac{df(x)}{dx}\right|_{x=x_0}\delta(f(x))\,

assuming f(x)\, has only one zero at x=x_0\, and is differentiable there. Integrating both sides gives

1 = \left|\frac{df(x)}{dx}\right|_{x=x_0}\int\!dx\,\delta(f(x))\,.

Extending over n variables, suppose f(x^i) = 0\, for some x^i_0\,. Then, replacing \delta(x-x_0)\, with \prod_i^n \delta^i(x^i-x^i_0)\,

1 = \left(\prod_i \left|\frac{\partial f(x^i)}{\partial x^i}\right|\right) \int\!\left(\prod_i dx^i\right)\,\delta(f(x^i))\,.

Recognizing the first factor as the determinant of the diagonal matrix \frac{\partial f(x^i)}{\partial x^i}\delta^{ij}\, (no summation implied), we can generalize to the functional version of the identity:

1 = \det\left|\frac{\delta G}{\delta \Omega}\right|_{G=0} \int\!\mathcal{D}\Omega\,\delta[G_a(\phi^\Omega)]\,,

where \Delta_F[\phi] \equiv \det\left|\frac{\delta F}{\delta g}\right|_{F=0}\, is the Faddeev-Popov determinant.

Now consider inserting

1 = \Delta_{FP}(\phi) \,\int\!\mathcal{D}\Omega\,\delta[ G_a(\phi^\Omega) ]\,,

into the path integral, where \Delta_{FP}(\phi)\, has been chosen to make the combination equal to unity. If the transformations \phi\to \phi^\Omega\, act as a group, i.e., \left( \phi^\Omega \right)^{\Omega'} = \phi^{\Omega' \Omega}\,, where \Omega' \Omega\, is itself an allowable transformation, then consider

\Delta_{FP}^{-1}(\phi^{\Omega'}) = \int\!\mathcal{D}\Omega\,\delta[ G_a(\phi^{\Omega'\Omega}) ] = \int\!\mathcal{D}{(\Omega'\Omega)}\,\delta[ G_a(\phi^{\Omega'\Omega}) ] = \int\!\mathcal{D}\Omega''\,\delta[ G_a(\phi^{\Omega''}) ] = \Delta_{FP}^{-1}(\phi)\,,

where we have used the fact that group multiplication is equivalent to a translation in the group. Thus \Delta_{FP}\, is gauge invariant, and consequently so is \int\!d\Omega\,\delta[ G_a(\phi^\Omega) ]\,, which is perhaps easier to see.

Path integral

The path integrall is now

Z\, = \int\!\mathcal{D}\phi\,e^{i S[\phi]}\,,
=\int\mathcal{D}\Omega\, \int\!\mathcal{D}\phi\, \Delta_{FP}(\phi) \delta[ G_a(\phi^{\Omega}) ] e^{i S[\phi]}\,,
= \int\mathcal{D}\Omega\,\int\!\mathcal{D}\phi^\Omega \Delta_{FP}(\phi^\Omega) \, \delta[ G_a(\phi^{\Omega}) ] e^{i S[\phi^\Omega]}\,,
= \int\mathcal{D}\Omega\, \int\!\mathcal{D}\phi^\Omega \Delta_{FP}(\phi^\Omega) \, \delta[ G_a(\phi^{\Omega}) ] e^{i S[\phi^\Omega]}\,,
= \left( \int\mathcal{D}\Omega\right) \int\!\mathcal{D}\phi \,\Delta_{FP}(\phi) \delta[ G_a(\phi) ] e^{i S[\phi]}\,,

since \phi\, is merely a dummy variable. Therefore the potentially divergent term \textstyle\left( \int\mathcal{D}\Omega\right)\, has been neatly isolated and accounted for.

Determinant

Let us formally write

\delta[ G_a(\phi) ] = \int\!\mathcal{D}\varphi\, e^{i \varphi_a G_a(\phi)}\,,

which is just the Fourier transform of the delta function (up to an irrelevant normalization). Then

\Delta_{FP}^{-1}(\phi) = \int\!\mathcal{D}\Omega\, \mathcal{D}\varphi\, e^{i \varphi_a G_a(\phi^\Omega)}\,.

The integrand may be equivalently be evaluated where G_a(\phi^\Omega) = 0\,, and for a particular configuration \phi_a\,, we may translate \Omega^\alpha\, such that \left.G_a(\phi^\Omega)\right|_{\Omega^\alpha = 0} = 0\,. Thus

G_a(\phi^\Omega) = \left.G_a(\phi^\Omega)\right|_{\Omega^\alpha=0} + \frac{\partial}{\partial \Omega^\beta} \left.G_a(\phi^\Omega)\right|_{\Omega^\alpha = 0} \Omega^\beta + ...\,.

Then

\Delta_{FP}^{-1}(\phi) = \int\!\mathcal{D}\Omega\, \mathcal{D}\varphi\, \exp\left(i \varphi_a\frac{\partial}{\partial \Omega^\beta} \left.G_a(\phi^\Omega)\right|_{\Omega^\alpha = 0} \Omega^\beta \right)\,.

Denote \mathcal{G}_{a\beta} = \frac{\partial}{\partial \Omega^\beta} \left.G_a(\phi^\Omega)\right|_{\Omega^\alpha = 0}\,. Then

\Delta_{FP}^{-1}(\phi) = \int\!\mathcal{D}\Omega\, \mathcal{D}\varphi\, e^{i \varphi_a \mathcal{G}_{a\beta} \Omega^\beta }\,.

From the theory of path integrals, we recognize this as

\operatorname{det}^{-1} \mathcal{G}\,,

which was of course by construction. We can find the inverse of this result by replacing \varphi_a\, and \Omega^\alpha\, with Grassmann numbers b_a\, and c^\alpha\,. Then

\Delta_{FP}(\phi) = \int\!\mathcal{D}b\,\mathcal{D}c\, e^{i b_a \mathcal{G}_{a\beta} c^\beta }\,.

The anticommuting fields b\, and c\, are known as Faddeev-Popov ghosts.

Gauge-fixing terms

Our aim is to select a representative from each class of field configurations using the gauge condition G_a(\phi) = 0\,, perhaps from a family of gauge conditions G_a(\phi; \xi) = 0\, parameterized by some parameter \xi\,. For example, the Lorenz gauge is given by G_\mu(A) = \partial^\mu A_\mu\,. We can then write

Z = \int\!\mathcal{D}\Omega\int\!\mathcal{D}\phi \det \left|\frac{\delta G_a}{\delta \Omega} \right| \delta[G_a(\phi)]e^{i S[\phi]}\,.

Since Z\, is gauge invariant, the choice of G_a(\phi)\, is arbitrary. Starting with a particular choice for G_a(\phi)\,, we could then replace G_a(\phi)\, with \tilde{G_a}(\phi) = G_a(\phi) - C(x)\, where C(x)\, is an arbitrary function independent of \phi\,. Then, since \det \left|\frac{\delta G_a}{\delta \Omega} \right| = \det \left|\frac{\delta \tilde{G_a}}{\delta \Omega} \right|\,,

Z = \int\!\mathcal{D}\Omega\int\!\mathcal{D}\phi \det \left|\frac{\delta G_a}{\delta \Omega} \right| \delta[G_a(\phi)-C(x)]e^{i S[\phi]}\,,

where we have dropped the tilde. It is important to note that Z\, is independent of the choice of C(x)\,. Furthermore, we can multiply Z\, by any constant (before normalizing) without affecting the dynamics of the system,

Z\, \to \int\!\mathcal{D}\Omega\int\!\mathcal{D}\phi \det \left|\frac{\delta G_a}{\delta \Omega} \right| \delta[G_a(\phi;\xi)]e^{i S[\phi]} \left( \int\!\mathcal{D}C e^{-\frac{i}{2\xi} \int\!dx C^2(x)}\right)\,,
= \int\!\mathcal{D}\Omega\int\!\mathcal{D}C\,\mathcal{D}\phi \det \left|\frac{\delta G_a}{\delta \Omega} \right| \delta[G_a(\phi;\xi)] e^{i S[\phi]} e^{-\frac{i}{2\xi} \int\!dx C^2(\phi)}\,,
= \int\!\mathcal{D}\Omega\int\!\mathcal{D}C\,\mathcal{D}\phi \det \left|\frac{\delta G_a}{\delta \Omega} \right| \delta[G_a(\phi;\xi)-C(x)] e^{i S[\phi]} e^{-\frac{i}{2\xi} \int\!dx C^2(\phi)}\,,
= \int\!\mathcal{D}\Omega\int\!\mathcal{D}\phi \det \left|\frac{\delta G_a}{\delta \Omega} \right| e^{i S[\phi]} e^{-\frac{i}{2\xi} \int\!dx G_a^2(\phi;\xi)}\,,

where \xi\, is arbitrary.

Therefore we obtain the same dynamics if we replace \mathcal{L}\, by the gauge fixed Lagrangian density \mathcal{L} + \mathcal{L}_{gf} = \mathcal{L} -\frac{1}{2\xi} G_a^2(\phi;\xi)\,. For instance, the Lorenz gauge follows from the substitution:

\mathcal{L} \to \mathcal{L} -\frac{1}{2\xi}(\partial^\mu A_\mu)^2\,.

See also

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