Faddeev-Popov procedure

From Physics wiki

1 Gauge fixing
2 Path integral
3 Determinant
4 Gauge-fixing terms
5 See also

Gauge fixing

Suppose we have a number of (bosonic) fields $\phi_a\,$ and that our action is invariant under some gauge transformation $\phi_a \to f_a( \phi, \xi^\alpha)\,$ , which we denote by $\phi_a \to \phi_a^{\xi}\,$ 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 $\phi_a^{\xi^\alpha}\,$ obtained by varying the parameters $\xi^\alpha\,$ is known as a gauge orbit. The path integral sums over many physically equivalent configurations (parameterized by $\xi^\alpha\,$ ), 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:

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

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

Now consider inserting

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

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

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

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 $\int\!d\xi\,\delta[ G_a(\phi^\xi) ]\,$ , which is perhaps easier to see.

Path integral

The path integrall is now

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

since $\phi\,$ is merely a dummy variable. Therefore the potentially divergent term $\textstyle\left( \int\mathcal{D}\xi\right)\,$ 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

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

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

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

Then

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

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

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

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 $\xi^\alpha\,$ 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 $G_a(\phi) = 0\,$ . For example, the Lorenz gauge is given by $G_\mu(A) = \partial^\mu A_\mu\,$ . We can then write

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

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 $\det \left|\frac{\delta G_a}{\delta \xi} \right| = \det \left|\frac{\delta \tilde{G_a}}{\delta \xi} \right|\,$ ,

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

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,

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

where $\alpha\,$ is arbitrary.

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

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

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

Faddeev-Popov procedure

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Contents

Gauge fixing

Path integral

Determinant

Gauge-fixing terms

See also

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