Faddeev-Popov ghosts

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Ludwig Dmitrievich Faddeev (2009), Scholarpedia, 4(4):7389. doi:10.4249/scholarpedia.7389 revision #137028 [link to/cite this article]
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Curator: Ludwig Dmitrievich Faddeev

In the terminology of theoretical physics, the term "ghosts" is used to identify an object that has no real physical meaning. The name "Faddeev-Popov ghosts" is given to the fictitious fields that were originally introduced in the construction of a manifestly Lorentz covariant quantization of the Yang-Mills field. Later, these objects acquired more widespread application, including in string theory.

The necessity of ghosts is associated with gauge invariance. In gauge invariant theories, one usually has to deal with local fields, whose number exceeds that of physical degrees of freedom. For example in electrodynamics, in order to maintain manifest Lorentz invariance, one uses a four component vector potential \( A_{\mu}(x) \ ,\) whereas the photon has only two polarizations. Thus, one needs a suitable mechanism in order to get rid of the unphysical degrees of freedom. Introducing fictitious fields, the ghosts, is one way of achieving this goal.


Yang-Mills fields

Let us turn to the problem of quantization of the Yang-Mills theory. (Scholarpedia already has several articles related to this topic, see Gauge invariance, Gauge theories, Slavnov-Taylor identities, BRST symmetry and Zinn-Justin equation. However, I will repeat some basic ideas to make my point.)

The classical Yang-Mills field has a geometrical interpretation as a connection in the principal bundle \( E = M_{4} \times G \) with base \( M_{4} \ ,\) the Minkowskian space-time, and fiber \( G \ ,\) a compact Lie group of charges. In electrodynamics \( G=U(1), \) while in the description of isospin one uses \( G=SU(2). \) In the Standard Model \( G = U(1)\times SU(2) \times SU(3) \ ,\) where the three group factors correspond respectively to hypercharge, flavor and color. The explicit separation of space and charge variables in the fiber bundle structure is an important feature of the theory which stems from basic works of H. Weyl (1929) and V. Fock (1929). This feature contrasts with the Kaluza-Klein approach to the unification of the interactions (Kaluza 1921); (Klein 1926).

The Yang-Mills field is described by a multiplet of vector fields \( A_{\mu}(x)=A_{\mu}^{a}(x)t^a \) on \( M_{4} \ ,\) where \( \mu = 0,1,2,3 \) is the space-time index and \(t^a\) is a local orthonormal basis in the Lie algebra \( \mathcal{G} \) of \( G \ ,\) normalized by the Killing metric as \( <t_{a},t_{b}> = -\frac{1} {2}\delta_{ab}\ .\) (In this article we adopt the convention that repeated indices are summed.)

The matter fields are given as sections \( \psi(x) \) of the associated bundle with values in the space of a unitary representation \( \Gamma \) of the group \( G \ .\) They appear in the Lagrangian together with their covariant derivatives \[ \nabla_{\mu} \psi = \partial_{\mu} \psi - A_{\mu}^{a} \Gamma(t_{a}) \psi , \] where the matrices \( \Gamma(t_{a}) \) give a representation of the generators \( t_{a} \) of \( \mathcal{G} \ .\)

The gauge group \( \prod_{x}G \ ,\) consisting of elements \( \Omega(x) \) with values in \( G \ ,\) defines a local change of basis in the fiber \[ t_{a} \to \Omega^{-1}(x) t_{a}. \]

The action of the gauge group on the matter fields \( \psi \) and the connection \( A_{\mu}^{a} \) is given by \[\tag{1} \psi \to \psi^{\Omega} = \Gamma(\Omega) \psi , \]

\[\tag{2} A_{\mu} = A_{\mu}^{a}t_{a} \to A_{\mu}^{\Omega} = \Omega A_{\mu} \Omega^{-1} + \partial_{\mu} \Omega \Omega^{-1} .\]

From the transformation law of the connection, (2), follows the covariance of \( \nabla_{\mu}\psi ,\) \[ \nabla_{\mu} \psi \to \Gamma(\Omega) \nabla_{\mu} \psi . \]

The equivalence principle states that physics does not depend on the action of the gauge group. In particular, this implies that the basic Lagrangian should be invariant with respect to this action.

In the following, we concentrate on the pure Yang-Mills field and do not consider matter fields anymore. To introduce the basic Lagrangian, we need to introduce the curvature \[ F_{\mu\nu}(x) = \partial_{\mu}A_{\nu}(x) - \partial_{\nu}A_{\mu}(x) + [A_{\mu}(x),A_{\nu}(x)] , \] where, as above, \( A_{\mu} = A_{\mu}^{a}t_{a} \ .\) The above definition, written for the local components \( F_{\mu\nu}^{a} \ ,\) which are defined by \( F_{\mu\nu} = F_{\mu\nu}^{a} t_{a} \ ,\) looks as follows: \[ F_{\mu\nu}^{a} = \partial_{\mu}A_{\nu}^{a} - \partial_{\nu}A_{\mu}^{a} + f_{bc}^{a} A_{\mu}^{b} A_{\nu}^{c} , \] where the coefficients \( f_{bc}^{a} \) represent the structure constants of the Lie algebra \( \mathcal{G} \) (i.e., \([t_b,t_c]=f_{bc}^at_a\)). The curvature, or field strength, \( F_{\mu\nu} \) transforms homogeneously under the gauge action: \[ F_{\mu\nu} \to F_{\mu\nu}^{\Omega} = \Omega F_{\mu\nu} \Omega^{-1}, \] so that the combination \( \mathcal{L} = <F_{\mu\nu} F_{\mu\nu}> \) is gauge invariant. Note that \(L\) is quadratic in the derivatives of \( A_{\mu} \ .\) In the system of units in which \( \hbar=1, c=1 \) (the only free dimension being that of length \( [\text{L}] \)), we have evidently \[ [A_{\mu}] = [\text{L}]^{-1}, \quad [F_{\mu\nu}] = [\text{L}]^{-2}, \quad [\mathcal{L}] = [\text{L}]^{-4}, \] so that the integral \( \int \mathcal{L}(x) \, d^{4}x \) is dimensionless and can be taken as an action functional. The only freedom left is a positive dimensionless constant factor, which can be taken as proportional to the inverse squared coupling constant. Thus the action, proposed by C. N. Yang and R. Mills in 1954, is given by \[\tag{3} S = - \frac{1}{8g^{2}} \int \mathcal{L}(x) \, d^{4}x = \frac{1}{4g^{2}} \int (F_{\mu\nu}^{a})^{2} d^{4}x \]

Quantization of gauge theories

The action (3) is a natural generalization of the Maxwell action of electro-magnetism. However, in the case of a non-Abelian group \( G \) the action (3) contains not only the quadratic terms in the fields \(A_\mu^a\) but also "self-interaction" terms of the third and fourth degree in the fields. With the rescaling \[ A_{\mu} \to g A_{\mu}, \] the coupling constant disappears from the quadratic part in equation (3) and reappears as a factor \( g \) and \( g^{2} \ ,\) respectively, in front of the cubic and the quartic terms in the action. This rescaling is used to separate the self-interaction from the "free" part in perturbation theory (expansion in powers of \(g\)). The free, quadratic part is just a sum (over the index \(a\)) of Lagrangian terms, each one being analogous to those associated with the electromagnetic field: \[\tag{4} \frac{1}{4}\sum_a (\partial_{\mu}A_{\nu}^{a} - \partial_{\nu}A_{\mu}^{a})^2.\]

As in electrodynamics, the free part of the Lagrangian is a degenerate quadratic form in the derivatives of field components and, due to this degeneracy, the partial differential equation giving the classical vector potential in terms of its sources has multiple solutions. These solutions are related by gauge transformations. This means that the Green function of this differential equation is ill-defined and so is the propagator used in the construction of perturbation theory.

In quantum electrodynamics, one uses the propagator \[\tag{5} G(k) = \frac{1}{k^{2}+i0} \bigl(\delta_{\mu\nu} - \frac{k_{\mu}k_{\nu}}{k^{2}}\bigr) + \beta(k) \frac{k_{\mu}k_{\nu}}{k^{2}} \]

with, as arbitrary "longitudinal" part, the second term in the right hand side of (5). The elements of the quantum scattering matrix among physical states do not depend on \( \beta(k) \ .\) This result has a long history whose details I will not discuss.

R. Feynman showed, by an explicit calculation, that the recipe of using the propagator (5) does not work for the Yang-Mills field. He performed his calculation as an exercise for a more elaborate work on the gravitational field, long before the Yang-Mills theory got its physical application. Feynman found that the naive one-loop diagram calculation with the propagator \( G(k) \) was not satisfactory. He then modified the calculation, reconstructing the one-loop answer from tree diagrams via unitarity and analyticity. He observed that this result differs from the diagrammatic one by the subtraction of a term, which can be interpreted as the contribution of some scalar particle. However, the minus sign in front of the contribution showed that this particle was a fermion. Feynman presented his result to the community of experts in gravitation. His lecture, the transcript of which is published in (Feynman 1963), was considered as a curiosity by most participants to the conference. However, B. DeWitt took it seriously and worked hard on the problem of full quantization. His final results were published in (DeWitt 1967). At the same time, my collaborator V. Popov and myself developed a more clear and intuitive approach, which was published in (Faddeev and Popov 1967). Our explanation of Feynman's fictitious particle was based on the proper treatment of Feynman's functional integral. It is a bit ironic that Feynman himself used his diagrammatic technique and not the functional integral, from which the diagrammatic technique originates. Our approach gained popularity when people understood the fundamental role of the Yang-Mills between the very end of the 60s and the early 70s. I do not know who first coined the term "Faddeev-Popov ghosts," which nowadays belongs to the common language of physics.

Feynman's Polish lecture (Feynman 1963) was the starting point for my work in quantum field theory in 1965. From a monograph of A. Lichnerowicz (1955), I learned about the interpretation of the Yang-Mills field as a connection. This was enough to realize that it is as geometric as Einstein's gravitational field, but easier to handle. Victor Popov was at that time a postgraduate student in our institute. His professor left the institute and Victor was put under my responsibility. Both of us knew about the functional integral approach to quantization and I began discussing with him the problem of the Yang-Mills field.

It was clear that the equivalence principle had to be taken into account. In the functional integral framework, the equivalence principle implies that one has to integrate over classes of gauge equivalent fields instead of integrating over all fields \(A_\mu^a\ .\)

The choice of the representatives in the classes of equivalent fields is realized by means of a gauge condition (gauge fixing), for instance, \[ \partial_{\mu} A_{\mu}^{a} = 0 . \] This condition defines a plane in the set of all fields, which is intersected by the gauge orbits defined above, (2).

In this context, the difference among Abelian and non-Abelian cases becomes clear. In the Abelian case, we take \( \Omega(x) = \exp{i\Lambda(x)} \) and a gauge orbit is defined by \[ A_{\mu} \to A_{\mu} + \partial_{\mu} \Lambda , \] which is just a linear shift. Thus all the Abelian orbits intersect the gauge surface at the same angle.

In the non-Abelian case, the gauge orbit equations are non-linear and the intersection angle depends on the field parameterizing the orbit. It is clear that this must be taken into account in the functional integral.

More explicitly, in the naive integral \[ \int \exp\{iS(A)\} \prod_{x} \delta(\partial_{\mu}A_{\mu}) \prod_{x,a} dA_{\mu}^{a} , \] where we integrate the Feynman weight \( \exp\{\frac{i}{\hbar} \, \text{action}\} \) over all fields, we should introduce a factor taking into account this angle. It is intuitively clear that this factor is a determinant of some operator. The only natural candidate operator for this determinant is the one obtained from an infinitesimal change of the gauge condition. An infinitesimal gauge transformation is defined via the function \( \epsilon(x) \) with values in the Lie algebra \( \mathcal{G} \ :\) \[\tag{6} \delta A_{\mu} = \partial_{\mu} \epsilon - [A_{\mu}, \epsilon] .\]

The corresponding change of the gauge condition is \[ \delta (\partial_{\mu}A_{\mu}) = M(A) \epsilon = \partial_{\mu}^{2} \epsilon -\partial_{\mu} [A_{\mu},\epsilon] . \] The dependence of \( M(A) \) on the field \( A \) is due to the second, non-linear, term in (6), which is absent in the Abelian case.

Thus, my proposal was to modify the functional integral to \[ \int \exp\{iS(A)\} \prod_{x} \delta(\partial_{\mu}A_{\mu}) \det M(A) \prod_{x,a} dA_{\mu}^{a} . \] In the Abelian case, the determinant \( \det M \) does not depend on \( A \) and gives a constant factor, which does not influence physical quantities. In the non-Abelian case, this determinant adds to the perturbative expansion some new terms (new Feynman diagrams), which exactly correspond to Feynman's subtraction in the one-loop case.

A couple of days after my proposal, Victor Popov came up with an intuitive derivation, which now is called "insertion of \(1\)", obtained by averaging the gauge condition over the gauge group. Let \( \Delta(A) \) be given by \[ \Delta(A)^{-1} = \int \prod \delta (\partial_{\mu}A_{\mu}^{\Omega}) \prod_{x} d \Omega . \] This integral is apparently gauge invariant and, on the gauge surface \( \partial_{\mu} A_{\mu} = 0 \ ,\) it is equal to \[ \int \prod \delta(M(A)\epsilon) \prod d\epsilon = \frac{1}{\det M(A)} . \] Thus we have \(\Delta(A)= \det M(A)\) and the identity \[ \det M(A) \int \delta(\partial_{\mu}A_{\mu}^{\Omega}) \prod_{x}d\Omega =1\,, \] which we can substitute into the expression for the naive functional integral. Changing variables \( A \to A^{\Omega^{-1}} \) and taking into account the gauge invariance of the action \( S(A) \ ,\) that of the factor \( \Delta(A) \) and of the measure \( \prod_{x,a} dA_{\mu}^{a} \ ,\) we get the conjectured expression multiplied by \( \int \prod_{x}d\Omega \ ,\) a constant factor which is just the volume of the gauge orbit and should be canceled.

We wrote a short article (Faddeev and Popov, 1967) for Physics Letters and then we published a long preprint in the journal of the new Institute of Theoretical Physics in Kiev (Popov and Faddeev 1967). There was no hope to publish it as a journal article in the USSR since at that time quantum field theory was considered to be dead. At the beginning of the 70s this text was translated into English by B. Lee and published as a Fermi Lab preprint. However, it appears clearly from Lee's introduction that it was known in the West since 1968.

In (Popov and Faddeev 1967) the functional integral was rewritten with the introduction of the fictitious fields (ghosts). We used the integral representation of a determinant as a Gaussian integral over anticommuting (Grassmann) variables, introduced by F. Berezin (1966). In the case of a finite dimensional matrix \( A_{m,n}, m,n=1,\dots N \ ,\) one finds the representation \[ \det A = \int \exp \{\bar{c}_{m}A_{mn}c_{n}\} \prod_{n} d\bar{c}_{n} dc_{n}, \] where \( (\bar{c}_{n}, c_{n}) \) constitute the set of generators of a Grassmann algebra and satisfy the anticommutation relations \[ c_{n} c_{m} = -c_{m} c_{n} , \quad \bar{c}_{n} \bar{c}_{m} = - \bar{c}_{m} \bar{c}_{n}, \quad \bar{c}_{n} c_{m} = -c_{m}\bar{c}_{n}. \] Berezin's integral is a functional on this algebra defined as a multiple integral by the elementary relations \[ \int c_{n} dc_{n} =1 , \quad \int dc_{n} = 0 \] and the similar relations for generators \( \bar{c}_{n} \ .\) It is clear that this functional integral is equal to the coefficient of the term of the highest degree, \( \bar{c}_{1}\ldots \bar{c}_{N}c_{1}\ldots c_{N} \ ,\) in the integrand. This, for the integrand \( \exp(\bar{c}Ac) \ ,\) is equal to the determinant \( \det A \ .\)

With this recipe we can rewrite the basic functional integral in the form \[ \int \exp\{i\tilde{S}(A,\bar{c},c)\} \prod_{x} \delta(\partial_{\mu}A_{\mu}) \prod_{x} dA_{\mu} d\bar{c} dc , \] where the new action has the form \[ \tilde{S}(A,\bar{c},c) = \frac{1}{4g^{2}} \int (F_{\mu\nu}^{a})^{2} dx + \int \bar{c}^{a}(x)\partial_{\mu}(\partial_{\mu}c - [A_{\mu},c])^{a}dx \ .\] It depends on the Yang-Mills field and on the set of scalar fermionic fields \( \bar{c}(x), c(x) \) with values in the Lie algebra \( \mathcal{G} \ .\) These fictitious fields, which satisfy Fermi statistics and hence realize Feynman's idea, now are called in the literature Faddeev-Popov ghosts. It is apparent that these scalar ghosts satisfying Fermi statistics violate the standard spin-statistics relation and hence have no physical meaning.

One more improvement was introduced by 't Hooft (t'Hooft, 1971, see appendix A). We can modify the gauge condition to \[ \partial_{\mu} A_{\mu} + f(x) = 0, \] where \( f(x) \) is an arbitrary function. The operator \( M(A)\) and the ghost fields do not change, but now we have a modified the \( \delta \)-function in the functional integral. Let us use the simple relation \[ \int e^{i\frac{\alpha}{2g^{2}}\int f^{2}(x)dx} \prod_{x} \delta(\partial_{\mu}A_{\mu}+f) \prod_{x} df = e^{i\frac{\alpha}{2g^{2}}\int(\partial_{\mu}A_{\mu}^{a})^{2}dx} \] and substitute it into the functional integral. We get a new expression of the proper Feynman type \[ \int \exp\{iS(A,\bar{c},c)\} \prod dA d\bar{c}dc , \] where the final action has the form \[ S=\int\bigl(\frac{1}{4g^{2}}(F_{\mu\nu}^{a})^{2} +\bar{c}^{a} \partial_{\mu}(\partial_{\mu}c^{a}-f^{abc}A_{\mu}^{b}c^{c}) + \frac{\alpha}{2g^{2}}(\partial_{\mu}A_{\mu})^{2})dx . \] The last term here makes the quadratic part of the Yang-Mills field non-degenerate, as in the Gupta-Bleuler approach to quantum electrodynamics, and one can use the propagator written above (\(G(k)\)) with \( \beta= \frac{1}{\alpha k^{2}}\ .\)

This action is the starting point of the analysis of renormalization of gauge theories, which is investigated in the articles BRST symmetry and Zinn-Justin equation of Scholarpedia; moreover, it is presented in modern textbooks (where it is proven essentially with the same derivation as given above).

However, I was personally not satisfied with the above construction. My understanding of the functional integral is based on its Hamiltonian derivation, where one uses an action in the form \[ S = \int dt\bigl(\sum p_{i}\dot{q}_{i} - H(p,q)\bigr) \] and integrates over the trajectories in phase space. The more conventional functional integral with an action depending only on \( q \) and \( \dot{q} \) follows from the Hamiltonian integral only when the Hamiltonian is quadratic in the momenta \(p_i\ .\)

The Hamiltonian construction for the Yang-Mills field is based on the first order formalism where \( A_{\mu}^{a} \) and \( F_{\mu\nu}^{a} \) are considered as independent field variables \[ \mathcal{L} = <\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu} + [A_{\mu},A_{\nu}],F_{\mu\nu}> - \frac{1}{2} <F_{\mu\nu},F_{\mu\nu}> . \] After separation of the space and time indices, we get an expression in terms of the variables \( E_{k}, A_{k}, A_{0} \) \[ \mathcal{L}=<E_{k},\partial_{0}A_{k}>-\frac{1}{2}(E_{k}^{2}+H_{k}^{2}) - A_{0}(\partial_{k}E_{k}+[A_{k},E_{k}]) , \] where \[ E_{k} = F_{0k}, \quad H_{i} = \frac{1}{2}\epsilon_{ikj} F_{kj}=\epsilon_{ikj} \partial_k A_j \] and we consider the magnetic field \( H_{i} \) as a function of \( A_{k} \ .\) We see that \( E_{k},A_{k} \) constitute a canonical pair whose Poisson bracket is defined by \[ \{E_{k}^{a}(x), A_{l}^{b}(y)\} = \delta_{kl} \delta_{ab} \delta^{3}(x-y). \] Moreover \( A_{0} \) plays the role of Lagrangian multiplier associated with the constraint \[ C(\vec{x}) = \partial_{k}E_{k} + [A_{k},E_{k}] . \] This constraint satisfies the relation \[ \{C^{a}(\vec{x}), C^{b}(\vec{y})\} = f^{abc} C^{c}(\vec{x}) \delta^{(3)}(\vec{x}-\vec{y}) \ ,\] which encodes the Lie algebra of the time independent gauge transformations. According to Dirac definition (Dirac, 1950), \(\{C^{a}\}\) is a set of first class constraints since their brackets vanish on the constraint surface \( C^{a}(\vec{x}) = 0 \ .\) In (Faddeev 1969) I showed that, given an action with first class constraints \( \phi_{\alpha}(p,q)\ ,\) where \[ \{\phi_{\alpha},\phi_{\beta}\} = C_{\alpha\beta}^{\gamma}(\phi) \phi_{\gamma} \] and a constrained action \[ l = \sum p\dot{q} - H - \sum\lambda^{\alpha}\phi_{\alpha}(p,q) , \] the corresponding unitary functional integral is given by \[ \int e^{i\int l\,dt} \delta(\xi_{\alpha})\det(\{\xi_{\alpha},\phi_{\beta}\}) dp \,dq \, d\lambda , \] where \( \xi_{\alpha}(p,q) \) is a set of "subsidiary conditions", that is, arbitrary functions such that the matrix \( \{\xi_{\alpha}, \phi_{\beta}\} \) is non-degenerate.

The geometric meaning of this procedure in terms of what is now called "symplectic reduction" is given in the appendix of (Faddeev 1969).

Now it is easy to derive the main functional integral from the above considerations, and I leave it as an exercise for the reader. In a monograph with A. Slavnov (Faddeev and Slavnov, 1991), we give a detailed discussion of the derivation. One can take the set of functions \( \xi^{a}(x) = \partial_{k}A_{k}(x) \) as subsidiary conditions, take into account that \[ \{C^{a}(\vec{x}),\xi^{b}(\vec{y})\} = (M^{(3)})^{ab} \delta^{(3)}(\vec{x}-\vec{y}) , \] where \( M^{(3)} \) is a three dimensional analogue of operator \( M \) \[ M^{(3)} f(x) = \partial_{k}\partial_{k} f(x) - [A_{k}, \partial_{k} f(x)] \ ,\] and then show that the expressions \( \det M \delta(\partial_{\mu}A_{\mu})\prod dA \) and \( \det M^{(3)} \delta(\partial_{k}A_{k}) \prod dA \) give a realization of the same measure on the manifold of the gauge orbits \( \mathcal{A}/\Omega \ ,\) where \( \mathcal{A} \) is the set of all fields \( A_{\mu}^{a} \) and \( \Omega \) is a gauge group.

In conclusion of this rather personal account of the origin of Faddeev-Popov ghosts, I present an alternative interpretation of the \( \det M^{4} \ ,\) which was developed by O. Babelon and C.M. Viallet (1979) and A.S. Schwarz (1989). A similar approach was used by A.M. Polyakov in his treatment of the bosonic string (Polyakov, 1981). In this interpretation, the \( \det M^{4} \) appears via construction of the Riemannian metric on the gauge orbit, induced from the Euclidean metric \[ d s^{2} = \int dx \sum_{\mu,\alpha} d A_{\mu}^{a}(x) d A_{\mu}^{a}(x) \] on the linear space of vector fields \( A_{\mu}^{a}(x) \ .\)

Let us write \( A_{\mu}(x) = A_{\mu}^{a}(x) t^{a} \) via gauge transformation from the point \( B_{\mu}(x) \) on the orbit \[ A_{\mu}(x) = g^{-1} B_{\mu} g - g^{-1} \partial_{\mu} g . \] We have \[ d A_{\mu} = g^{-1} (\nabla_{\mu} \omega +C_{\mu}) g , \] where \( \omega \) and \( C_{\mu} \) are one forms \[ C_{\mu} = d B_{\mu} , \quad \omega = d g g^{-1} \] and \[ \nabla_{\mu} \omega = \partial_{\mu} \omega - [B_{\mu},\omega] . \] Thus the metric gets the expression \[ (d A)^{2} = \int \mbox{tr} \bigl(C_{\mu}^{2} + 2 C_{\mu}\nabla_{\mu}\omega +(\nabla_{\mu} \omega)^{2}\bigr) d^{4} x \] and the corresponding volume element gets the form \[ \prod_{x,\mu,a} d A_{\mu}^{a}(x) = f(B) \prod_{x,a,\mu} C_{\mu}^{a}(x) \prod_{x,a} \omega^{a}(x) \ .\] The last factor is the invariant measure over the gauge group, which we separated by a different method above. We omit it and concentrate on the calculation of the factor \( f(B) \ .\) For that we concretize our gauge fixing and we take Landau gauge \[ \partial_{\mu} B_{\mu}^{a} = 0 \] as before. So we have also \[ \partial_{\mu} C_{\mu} = 0 \] and the term \( \mbox{tr}(C_{\mu}\partial_{\mu}\omega) \) disappears. Thus \( (dA)^{2} \) is given by quadratic form \[ N = \begin{pmatrix} P & PB \\ B^{*}P & (\partial^{*} -B^{*})(\partial -B) \end{pmatrix} , \] where we use notations \[ \begin{matrix} \left (B\omega\right)_{\mu} = \left[B_{\mu}, \omega\right ], & \quad B^{*}C_{\mu} = \left [B_{\mu},C_{\mu}\right ] \\ \left (\partial\omega\right )_{\mu} = \partial_{\mu} \omega, & \partial^{*}C_{\mu} = \partial_{\mu} C_{\mu} \end{matrix} \] and \( P \) is projector on the transverse fields \[ P = I - \partial \frac{1}{\Box} \partial^{*} , \quad \Box = \partial^{*} \partial . \]

To accomplish our goal we calculate \( \det N \) and take the square root of it. For that we use Gauss decomposition \[ N = \begin{pmatrix} P & 0 \\ B^{*}P & X \end{pmatrix} \begin{pmatrix} P & BP \\ 0 & Y \end{pmatrix}, \] where operators \( X \) and \( Y \) should be obtained from \[ (\partial^{*}-B^{*})(\partial -B) = B^{*}PB + XY . \] Denote \[ \partial^{*} B = B^{*}\partial = T . \] The Faddeev-Popov operator \( M^{4} \) is given by \[ M^{4} = \Box - T . \] The solution of condition for \( X \) and \( Y \) is given by \[ X = \Box^{1/2} - T \Box^{-1/2} , \quad Y = \Box^{1/2} - \Box^{-1/2} T , \] so that up to constant factor \[ \det N \sim (\det M)^{2} , \] which realizes the interpretation we looked for.


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Internal references

  • Jean Zinn-Justin and Riccardo Guida (2008) Gauge invariance. Scholarpedia, 3(12):8287.
  • Jean Zinn-Justin (2009) Path integral. Scholarpedia, 4(2):8674.

Further reading

  • Faddeev, L D (1976). Les Houches School 1975 in Methods in Field Theory, Balian R and Zinn-Justin J eds. Elsevier Science Publishing, Amsterdam. ISBN 978-0720404333
  • Faddeev, L D and Slavnov, A A (1991). Gauge Fields. Introduction to quantum theory. (2nd edition). Addison-Wesley Publishing Company, T. ISBN 0201524724.

See also

BRST symmetry, gauge invariance, gauge theories, path integral, Slavnov-Taylor identities, Zinn-Justin equation

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