Zinn-Justin equation

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Jean Zinn-Justin (2009), Scholarpedia, 4(1):7120. doi:10.4249/scholarpedia.7120 revision #147047 [link to/cite this article]
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Curator: Jean Zinn-Justin

Quantum field theories in a naive formulation lead to physical results plagued with infinities due to short distance singularities and require a regularization, operation by which their short-distance structure below a cut-off scale is modified in an unphysical way. For a class of quantum theories called, therefore, renormalizable, it is possible to construct a theory finite when the cut-off is removed by rendering the parameters of the initial Lagrangian cut-off dependent, a mathematical procedure called renormalization and whose deep meaning can only be understood in the framework of the renormalization group.

Non-Abelian gauge theories are quantum theories at the basis of the Standard Model of particle physics (that describes fundamental interactions at the microscopic scale); their mathematical consistency requires their renormalizability and the preservation of some form of gauge invariance by the renormalization process. At the beginning of the 1970s, much effort was devoted to the proof of the perturbative renormalizability of non-Abelian gauge theories. Initial arguments based on Feynman diagrams ('t Hooft and Veltman 1972), and Lee-Zinn-Justin's proof (1972), based on Slavnov-Taylor identities (Slavnov 1972), (Taylor 1971), were simplified and generalized with the use of the BRST symmetry, discovered by Becchi, Rouet, Stora (1974,1975,1976) and by Tyutin (1975), that generalizes gauge invariance in this context. A general proof of renormalizability of non-Abelian gauge theories is based on the master equation, also called Zinn-Justin equation (Zinn-Justin, 1974). The Zinn-Justin equation (ZJ equation) is a quadratic equation satisfied both by the so-called one particle irreducible generating functional of Green's functions (or correlation functions) and by the quantized action. The ZJ equation can be shown to be perturbatively stable under renormalization. The general solution of this equation, taking into account locality, power counting and ghost number conservation, gives the general form of the renormalized action. In particular, the ZJ equation implies independence of physical observables from the gauge fixing procedure required to construct the quantum theory.


Non-Abelian gauge theories: Classical field theory

A classical non-Abelian gauge theory is a generalization of Maxwell's Electrodynamics, in which the gauge invariance is based on a non-Abelian gauge group \(G \) in place of the Abelian \(U(1)\) group underlying Maxwell's theory. We consider here only local field theories, that is, theories in which the action is the space-time integral of a function of fields and their derivatives (the Lagrangian density or simply Lagrangian).

A classical gauge theory is a classical field theory whose action is invariant under gauge transformations. A gauge transformation is a space-time dependent representation of a matrix complex Lie group \(G\ ,\) acting on the fields of the theory. If the group \(G\) is non-Abelian, one speaks of non-Abelian gauge transformations and non-Abelian gauge theories. In what follows, in view of physics applications, we restrict the discussion to unitary groups (but orthogonal groups require trivial modifications) and 3+1 space-time dimensions.

Gauge transformations and gauge fields

We assume that matter fields \(\phi(x)\) form complex vectors that, in a gauge transformation, transform like \[\tag{1} \phi(x)\mapsto\phi^{\mathbf{g}}(x)= \mathbf{g}(x)\phi(x),\quad \mathbf{g}(x)\in G \quad\forall x\,,\]

where \(\mathbf{g}(x)\) smoothly maps the space time to the group \(G\ .\)

To construct a gauge theory, it is necessary to introduce a gauge field (or connection) \(\mathbf{A}_\mu(x)\ ,\) also known as Yang-Mills field, which, for each space-time index \(\mu=0,1,2,3 \) takes values in the Lie algebra of \(G\ .\) Gauge transformations act on the gauge field as \[\tag{2} \mathbf{A}_{\mu}(x) \mapsto \mathbf{A}_{\mu}^{\mathbf g}(x) := \mathbf{g}(x) \mathbf{A}_{\mu}(x) \mathbf{g}^{-1}(x) + \mathbf{g}(x) \partial_{\mu} \mathbf{g}^{-1}(x). \]

When \( \mathbf{g}(x)\) is constant for all \( x\ ,\) the gauge field \(\mathbf{A}_\mu(x)\) transforms under the adjoint representation of the group \(G\ .\) For generic \( \mathbf{g}(x)\) the transformed field \(\mathbf{A}_{\mu}^{\mathbf g}(x)\) is no longer linear in \(\mathbf{A}_\mu(x)\) but affine. The form (2) ensures that \(\mathbf{A}_{\mu}^{\mathbf g}(x)\) is still valued in the Lie algebra of \(G\ .\)

Covariant derivatives and curvature

In a local, gauge invariant, field theory ordinary derivatives must be replaced by covariant derivatives that, because they transform linearly under gauge transformations, ensure the gauge invariance of the action. Covariant derivatives are constructed using the gauge connection \(\mathbf{A}_{\mu} \ .\) Their explicit form depends on the representation under which fields are transforming.

For example, for the matter fields that transform like in (1), the covariant derivative takes the form \( \mathbf{D}_{\mu}= \mathbf{1}\,\partial_{\mu} + \mathbf{A}_{\mu}\) and transforms like \[\tag{3} \mathbf{D}_{\mu} \mapsto \mathbf{D}_{\mu}^{\mathbf{g}} =\mathbf{1}\,\partial_{\mu} + \mathbf{A}_{\mu}^{\mathbf{g}} = \mathbf{g}(x) \mathbf{D}_{\mu} \mathbf{g}^{-1}(x) ,\]

where the product has to be understood as a product of differential and multiplicative operators. Covariant derivatives of fields then transforms as \( \mathbf{D}_{\mu}^{\mathbf{g}}\, \mathbf{g}(x)\phi(x) =\mathbf{g}(x) \mathbf{D}_{\mu}\phi(x)\ .\)

As a consequence of the property (3), the commutator \[ \mathbf{F}_{\mu\nu}(x) = \left[ \mathbf{D}_{\mu},\mathbf{D}_{\nu}\right] = \partial_{\mu} \mathbf{A}_{\nu}(x) - \partial_{\nu} \mathbf{A}_{\mu}(x) + \left[ \mathbf{A}_{\mu}(x),\mathbf{A}_{\nu}(x)\right] ,\] which is no longer a differential operator, is also a tensor (the curvature of the connection) for gauge transformations: \[\tag{4} \mathbf{F}_{\mu\nu}(x) \mapsto \mathbf{F}_{\mu\nu}^{\mathbf{g}}(x)= \mathbf{g}(x) \mathbf{F}_{\mu\nu}(x) \mathbf{g}^{-1}(x). \]

It then follows from (4) that the local action for the gauge field \[\tag{5} \mathcal{S}_{\mathrm{class.}}(\mathbf{ A})= {1\over 4 e^{2}} \int \mathrm{d}^4 x\, \mathrm{tr}\sum_{\mu,\nu} \mathbf{F} _{\mu \nu} (x ) \mathbf{F}^{\mu \nu} (x )\,,\]

is gauge-invariant. (\(e\) in (5) will characterize the strength of the interaction in the presence of matter fields.) More generally, it is important to realize that physical observables are related to gauge-invariant polynomials in the fields.

When in the transformation (2), \( \mathbf{g}(x)\) is close to the identity, that is, when \(\mathbf{g}(x)=\mathbf{1}+\omega(x) +O(\|\omega\|)^2\ ,\) \(\omega(x)\) being a `small' smooth map valued in the Lie algebra of \(G\ ,\) the gauge transformation takes the form \[\tag{6} \mathbf{A}_{\mu}^{\mathbf g}(x)-\mathbf{A}_{\mu}(x)=-\mathbf{D}_\mu \omega(x)+O(\|\omega\|)^2\quad \mathrm{with}\quad \mathbf{D}_\mu \omega(x)\equiv \partial_\mu \omega(x)+[\mathbf{A}_{\mu}(x),\omega(x)].\]

Equation (6) gives the form of covariant derivative \(\mathbf{D}_\mu\) when it is applied to fields valued in the Lie algebra of \(G\ ,\) like \(\omega(x)\ .\)

Non-Abelian gauge theories: The quantized action

Due to gauge invariance, in non-Abelian gauge theories like in Quantum Electrodynamics (QED), not all components of the gauge field are dynamical and a simple canonical quantization is impossible. However, in non-Abelian gauge theories the methods required for the construction of a quantized theory are more involved than in QED. The construction of local, relativistic-covariant quantum non-Abelian gauge theories, involves the so-called Faddeev-Popov determinant and the introduction of ghost fields and relies on manipulations of field integrals (Faddeev and Popov 1967). BRST symmetry emerges from this formalism. In what follows, as a slight simplification, we discuss only gauge theories without matter, the modifications due to the addition of matter fields being simple (even though, in the case of fermions and chiral gauge invariance it may lead to obstruction to quantization in the form of gauge anomalies).

The quantized gauge action without matter

Following Feynman (1948), one would naively expect the quantum evolution operator to be given by an integral over classical gauge fields of the form \[\mathcal{U}=\int[\mathrm{d}\mathbf{A}_\mu]\exp\left({i\over \hbar}\mathcal{S}_{\mathrm{class.}}(\mathbf{A})\right),\] where \(\mathcal{S}_{\mathrm{class.}}(\mathbf{A})\) is the classical action (5). However, as a consequence of gauge invariance, the integrand is constant along gauge orbits (the trajectories obtained by starting from one gauge field and acting on it with all gauge transformations) and thus the integral is not defined. The idea is then to introduce a surface (section) that cuts all gauge orbits once and to restrict the integral to this section with a measure that ensures that all choices of sections are equivalent. This surface is defined by a (Lie algebra valued) constraint of the form \(\mathbf{G}(\mathbf{A},x)=0\ ,\) known as gauge fixing condition or simply gauge fixing. The appropriate integration measure has been identified by Faddeev and Popov and involves the determinant of a linear operator \(\mathbf{M}(\mathbf{A})\) defined by \[[\mathbf{M}(\mathbf{A})\omega](x) =\Delta_\omega \mathbf{G}(\mathbf{A},x),\] where \(\Delta_\omega \mathbf{G} \) is the variation of \(\mathbf{G}\) at first order in \(\omega\) corresponding to the variation of \(\mathbf{A}_\mu\) in (6): \[\Delta_\omega \mathbf{G}(\mathbf{A},x)=\mathbf{G}(\mathbf{A}-\mathbf{D}\omega,x)-\mathbf{G}(\mathbf{A},x)+O(\|\omega\|)^2.\] In relativistic-covariant gauges \(\mathbf{M}(\mathbf{A})\) is typically a differential operator. For example, in Landau's gauge, \[\tag{7} \mathbf{G}(\mathbf{A},x)\equiv\sum_\mu \partial_\mu \mathbf{A}^\mu(x)=0 \ \Rightarrow \ [\mathbf{M}(\mathbf{A})\bar{\mathbf{C}}](x)=-\sum_\mu\partial_\mu {\mathbf D}^\mu \bar{\mathbf{C}}(x)\,.\]

It follows from the rules of integration in Grassmann algebras that determinants can be represented by integrals over generators of Grassmann algebras, which are anti-commuting variables (see the section #The origin of BRST symmetry and equation (30) in particular). In gauge theories, this corresponds to introducing two spinless fermion fields \(\mathbf{C}(x)\) and \(\bar{\mathbf{C}}(x)\ ,\) often called Faddeev-Popov ghosts, which are matrices belonging to the Lie algebra of \(G\ .\) Such spinless fermions are unphysical because they violate the spin-statistics theorem. In addition, one introduces a scalar field \(\boldsymbol{\lambda}(x)\ ,\) again belonging to the Lie algebra, which is used to enforce the gauge condition \(\mathbf{G}(\mathbf{A},x)=0\) in the field integral (i.e., it acts as a Lagrange multiplier).

The evolution of the quantized gauge theory is then described in terms of an integral over four types of fields, \(\mathbf{A}_\mu(x),\boldsymbol{\lambda}(x),\mathbf{C}(x),\bar{\mathbf{C}}(x),\) of the form \[\tag{8} \mathcal{U}=\int[\mathrm{d}\mathbf{A}_\mu][\mathrm{d}\boldsymbol{\lambda}][[\mathrm{d}\mathbf{C}\mathrm{d}\bar{\mathbf{C}}]\exp\left( {i\over \hbar}\mathcal{S}_{\mathrm{quant.}}(\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})\right),\]

where \(\mathcal{S}_{\mathrm{quant.}}\) is a local action that reads \[\tag{9} \mathcal{S}_{\mathrm{quant.}}(\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})=\mathcal{S}_{\mathrm{class.}}(\mathbf{A})+\hbar\mathcal{S}_{\mathrm{gauge}} (\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})\]

with \[\tag{10} \mathcal{S}_{\mathrm{gauge}}(\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})=\int\mathrm{d}^4 x\,\mathrm{tr} \left[\boldsymbol{\lambda}(x)\mathbf{G}(\mathbf{A},x)-\mathbf{C}(x)\mathbf{M}(\mathbf{A})\bar{\mathbf{C}}(x)\right].\]

Finally, instead of using a constraint of the form \(\mathbf{G}(\mathbf{A},x)=0\ ,\) it is often convenient to impose \(\mathbf{G}(\mathbf{A},x)=\mathbf{s}(x)\) and to integrate over the field \(\mathbf{s}(x)\) with a Gaussian distribution of width \(\xi\ .\) This procedure corresponds to extending the field integral to a whole neighbourhood of the gauge section and, after the integration over \(\mathbf{s}(x)\ ,\) it amounts to adding a term quadratic in \(\boldsymbol{\lambda}\) to the gauge action, which becomes \[\tag{11} \mathcal{S}_{\mathrm{gauge}}(\mathbf{A},\bar{\mathbf{C}},\mathbf{C},\boldsymbol{\lambda})=\int\mathrm{d}^4 x\,\mathrm{tr} \left[\boldsymbol{\lambda}(x)\mathbf{G}(\mathbf{A},x)-\mathbf{C}(x)\mathbf{M}(\mathbf{A})\bar{\mathbf{C}}(x) +{\xi\over 2} \boldsymbol{\lambda}^2(x)\right].\]

In the limit \(\xi\rightarrow 0\ ,\) Landau's gauge fixing and the correspondent gauge action (10) are recovered.

BRST symmetry of the quantized action

BRST symmetry

Remarkably enough, the quantized action (9), (11), which is no longer gauge-invariant, is invariant, independently of the choice of the gauge condition \(\mathbf{G}(\mathbf{A},x)\ ,\) under the following transformations: \[\tag{12} \mathbf{A}_\mu(x)\mapsto \mathbf{A}_\mu(x)+\varepsilon \Delta \mathbf{A}_\mu(x),\quad \bar{\mathbf{C}}(x)\mapsto \bar{\mathbf{C}}(x)+ \varepsilon\Delta \bar{\mathbf{C}}(x),\quad \mathbf{C}(x)\mapsto \mathbf{C}(x)+ \varepsilon\Delta \mathbf{C}(x),\quad \boldsymbol{\lambda} (x)\mapsto \boldsymbol{\lambda} (x)+\varepsilon \Delta\boldsymbol{\lambda} (x)\]

where \(\varepsilon\) is a Grassmann constant (it thus anticommutes with \(\bar{\mathbf{C}}(x)\) and \(\bar{\mathbf{C}}(x)\ ,\) and \(\varepsilon^2=0\)) and \[\Delta \mathbf{A}_\mu(x) =-\mathbf{D}_\mu \bar{\mathbf{C}}(x)\,,\quad \Delta \bar{\mathbf{C}}(x)=\bar{\mathbf{C}}^2(x),\quad \Delta\mathbf{C}(x) =\boldsymbol{\lambda} (x)\,,\quad \Delta\boldsymbol{\lambda} (x)=0\,.\] These transformations, in which bosons are transformed into fermions and conversely, are called BRST transformations and reflect the BRST symmetry of the action. Another way of expressing the symmetry is based on the introduction of a functional differential operator. Expanding all fields on a basis of (anti-hermitian) matrices \(\mathbf{t}^{a}\) generating the Lie algebra of \(G\ ,\) \[\tag{13} \mathbf{A}_\mu(x)=\sum_{a} A^{a}_\mu(x)\mathbf{t}^{a},\quad \bar{\mathbf{C}}(x)=\sum_{a} \bar C^{a}(x)\mathbf{t}^{a}, \quad \mathbf{C}(x)=\sum_{a} C^{a}(x)\mathbf{t}^{a}, \quad \boldsymbol{\lambda} (x)=\sum_{a} \lambda^{a}(x)\mathbf{t}^{a}, \]

as well as \[\Delta \mathbf{A}_\mu(x)=\sum_{a} \Delta A^{a}_\mu(x)\mathbf{t}^{a}, \quad \Delta \bar{\mathbf{C}}(x)= \sum_{a} \Delta\bar C^{a}(x)\mathbf{t}^{a},\] one can write it in the form (\(\delta\) denotes functional differentiation) \[\tag{14} \mathcal{D}=\int\mathrm{d}^4x\sum_{a}\left[\sum_\mu \Delta A^{a}_\mu(x) {\delta\over \delta A^{a}_\mu(x)}+ \Delta\bar C^{a}(x){ \delta\over\delta \bar C^{a}(x)}+\lambda^{a}(x) {\delta\over\delta C^{a}(x)}\right]\,. \]

Then, the quantized action satisfies \[\tag{15} \mathcal{D}\mathcal{S}_{\mathrm{quant.}}=0\,.\]

In the field integral implementation of BRST symmetry, one must also verify the invariance of the integration measure in (8). This leads to the conditions \[\tag{16} \sum_{ {a},\mu} {\delta \Delta A^{a}_\mu(x)\over \delta A^{a}_\mu(x)} =0\,,\quad \sum_{a} {\delta\Delta\bar C^{a}(x)\over\delta \bar C^{a}(x)} =0\,,\]

that is, that the traces of the generators of the Lie algebra of the group \(G\) in the adjoint representation (corresponding to the Lie algebra structure constants) must vanish, a property satisfied by compact Lie groups. Note, however, that in presence of matter fields, this condition may apply to generators in other representations and is then only satisfied for semi-simple Lie algebras.

One verifies the very important property that the differential operator (14) satisfies \(\mathcal{D}^2=0\) (nilpotency) so that \( \mathcal{D}\) can be identified with a cohomology operator. In cohomological terminology, equation (15) states that the quantized action \( \mathcal{S}_{\mathrm{quant.}}\) is BRST closed. Quantities of the form \( \mathcal{D}\Phi\) are said to be BRST exact. The nilpotency of \( \mathcal{D}\) implies that BRST exact quantities are BRST closed. One verifies that \[\mathcal{S}_{\mathrm{gauge}}=\mathcal{D}\int\mathrm{d}^4 x\,\mathrm{tr}\,\mathbf{C}(x)\left(\mathbf{G}(\mathbf{A},x)+\textstyle{ {\xi\over2} }\boldsymbol{\lambda} (x)\right)\,,\] that is, that the gauge dependent part of the quantum action is BRST exact. This property is very important: it allows proving that gauge-invariant observables are insensitive to a modification of the gauge fixing function \(\mathbf{G}(\mathbf{A},x)\ ,\) by using a simple integration by parts in the field integral. However, integration by parts implies acting with \(\mathcal{D}\) on the left and, thus, commuting derivatives with coefficients in \(\mathcal{D}\ .\) This commutation yields again the conditions (16), which we have discussed above. Gauge independence, in particular, is essential to prove that physical observables satisfy the unitarity requirement, a property that is not obvious for a quantized gauge theory.

BRST invariant solutions

Without entering into too many details, let us point out that one observation facilitates the construction of general BRST invariant polynomials in the fields. Setting, \[\mathcal{D}=\mathcal{D}_+ + \mathcal{D}_-\quad\mathrm{with}\quad \mathcal{D}_+ =\int\mathrm{d}^4x\sum_{a}\left[ \Delta A^{a}_\mu(x) {\delta\over \delta A^{a}_\mu(x)}+ \Delta\bar C^{a}(x){ \delta\over\delta \bar C^{a}(x)}\right],\ \mathcal{D}_-= \int\mathrm{d}^4x\sum_{a}\,\lambda^{a}(x) {\delta\over\delta C^{a}(x)},\] one verifies that \[\mathcal{D}_+ ^2=\mathcal{D}_-^2=0\,,\quad \mathcal{D}_+ \mathcal{D}_- + \mathcal{D}_- \mathcal{D}_+=0\,.\] One then uses the property that Grassmann algebras are graded algebras, that \(\mathcal{D}_+\) increases the degree in \(\bar{\mathbf{C}}\) while \(\mathcal{D}_-\) leaves it unchanged, to expand an equation of the form \(\mathcal{D}\Phi=0\) in powers of \(\bar{\mathbf{C}}\ .\)

Master (Zinn-Justin) equation


The initial field integral (8) of the quantized gauge theory is not defined, even in the sense of a perturbative expansion: the perturbative expansion is generated by keeping in the exponential the part of the classical action that is quadratic in the fields (free action) and expanding in a formal power series the exponential of the remaining part. This expansion is usefully described in terms of Feynman diagrams, each one representing a Feynman integral contribution to the perturbative series. In perturbation theory, the field integral (8) is not defined because, at space-time dimension \(d=4\ ,\) divergent Feynman integrals (one also speaks of UltraViolet, UV divergences) arise at all orders, corresponding to short distance or large momentum singularities. A first necessary step is called regularization, in which one modifies in some unphysical way the classical action to render the expansion finite. The construction of quantum gauge theories and practical calculations are much simplified if the regularization preserves the BRST symmetry, even if, strictly speaking, this requirement is not mandatory. Different methods are available: one can modify the theory at short distance, in the continuum at a scale specified by a cut-off or by introducing a space-time lattice (lattice gauge theories). In theories without chiral fermion fields a BRST invariant regularization can also be achieved by using the so called dimensional regularization, which is based on the formal analytic continuation of Feynman integrals to arbitrary complex values of the space-time dimension \(d\ .\) When \(d\rightarrow 4\) poles appear in the dimensionally regularized Feynman integrals: these singularities are related to the initial short distance divergences. Renormalization consists into adding to the initial action so-called counter-terms, that is, space-time integrated monomials in the fields and their derivatives, multiplied by constant coefficients diverging when \(d\to4\ .\) The coefficients are then fixed order by order in perturbative expansion in such a way that all singularities are cancelled. When the number of different field monomials, required to render finite the perturbative expansion, is bounded at all orders by some fixed number, one calls the quantum field theory (perturbatively) renormalizable by power counting. Gauge theories with properly chosen gauge sections satisfy this criterion, like, for example, Landau's gauge in (7). However, in gauge theories one still has to prove that renormalization can be achieved without spoiling the geometric structure that ensures that physical results do not depend on the choice of the gauge section. Initial proofs of gauge independence of the renormalized gauge theory were based on the Slavnov-Taylor identities, suitably extended to the situation of spontaneous symmetry breaking by Lee-Zinn-Justin. A more general and more transparent proof was then given by Zinn-Justin using BRST symmetry and the Zinn-Justin (ZJ) equation. Note that all these proofs apply to compact Lie groups with semi-simple Lie algebras; the simpler Abelian case has to be dealt with by different methods.

Let us point out that if renormalization would preserve the BRST symmetry in the explicit form (12), gauge independence could be proved easily. However, this is not the case because counter-terms have the effect of rescaling fields, for example, \(\mathbf{A}_\mu \mapsto Z^{1/2}_A \mathbf{A}_\mu\ ,\) where \(Z_A\) is a divergent constant. Because the gauge transformation of \(\mathbf{A}_\mu\) is affine, the form of the gauge transformation is modified for the rescaled fields. Other fields are similarly renormalized. Moreover, if the gauge fixing function \(\mathbf{G}(\mathbf{A},x)\) is not linear in the fields, the gauge fixing equation is also modified and counter-terms quartic in the ghost fields are generated.

ZJ equation

In non-Abelian gauge theories, two BRST variations, \(\Delta\mathbf{A}_\mu\) and \(\Delta\bar{\mathbf{C}}\ ,\) are not linear in the dynamical fields. Local polynomials in the fields of degree larger than one are called composite operators. They generate new divergences and require new types of counter-terms. The renormalization of composite operators can be best discussed by introducing source fields that generate, by functional differentiation, their multiple insertions in correlation (or Green's) functions. We denote by \(\mathbf{K}^\mu\) and \( \mathbf{L}\) the sources for the \(\Delta\mathbf{A}_\mu\) and \(\Delta\bar{\mathbf{C}}\ ,\) respectively. Then, we consider \[ \mathcal{S}=\mathcal{S}_{\mathrm{quant.}}-\hbar\int\mathrm{d}^4 x \,\mathrm{tr}\left(\sum_\mu \mathbf{K}^\mu(x) \Delta\mathbf{A}_\mu(x) +\mathbf{L}(x)\Delta\bar{\mathbf{C}}(x)\right)\,,\] such that \[\tag{17} {\delta {\mathcal S}\over \delta \mathbf{K}^\mu{} (x) }=-\hbar\Delta\mathbf{A}_\mu(x),\qquad{\delta {\mathcal S} \over \delta \mathbf{L}(x)}=-\hbar\Delta\bar{\mathbf{C}}(x)\,.\]

The source \(\mathbf{K}^\mu(x)\) is a Grassmann (anticommuting) field while \(\mathbf{L}(x)\) is a complex field, and both are matrices that belong to the Lie algebra.

Using \(\Delta\mathbf{A}_\mu=\mathcal{D}\mathbf{A}\ ,\) \(\Delta\bar{\mathbf{C}}=\mathcal{D}\bar{\mathbf{C}}\ ,\) \(\mathcal{D}^2=0\) and the invariance of the sources under BRST transformation, it follows that \[\tag{18} \mathcal{D}\mathcal{S}=0\ .\]

We can expand \(\mathbf{K}^\mu(x)\) and \(\mathbf{L}(x)\) on the basis of generators of the Lie algebra as in (13). The ZJ equation then follows directly form BRST symmetry of \({\mathcal S}\) and from relations (17), and can be written in the form \[\tag{19} \int \mathrm{d}^4 x \sum_{a} \left( \sum_\mu {\delta {\mathcal S}\over \delta K^\mu{}^{a} (x) } {\delta {\mathcal S}\over \delta A_\mu^{a} (x) } +{\delta {\mathcal S} \over \delta L^{a}(x)}{\delta {\mathcal S}\over \delta \bar C^{a}(x)} -\lambda^{a}(x) {\delta {\mathcal S}\over \delta C^{a}(x)} \right)=0\,.\]

To discuss in simpler terms equation (19), it is sometimes convenient to add to \(\mathcal{S}\) the source for the \(\boldsymbol{\lambda}\) field\[\mathcal{S}\mapsto \mathcal{S}-\hbar\int\mathrm{d}^4 x \,\mathrm{tr}\boldsymbol{\lambda} (x)\boldsymbol{\mu}(x)\ .\] Equation (19) then takes the purely quadratic form \[\tag{20} \int \mathrm{d}^4 x \sum_{a} \left( \sum_\mu {\delta {\mathcal S}\over \delta K^\mu{}^{a} (x) } {\delta {\mathcal S}\over \delta A_\mu^{a} (x)} +{\delta {\mathcal S} \over \delta L^{a}(x)} {\delta {\mathcal S}\over \delta \bar C^{a}(x)} +{\delta {\mathcal S}\over\delta \mu^{a}(x)} {\delta {\mathcal S}\over \delta C^{a}(x)} \right)=0\,.\]

In contrast with equation (15) where BRST transformations (12) are explicit, equations (19), (20) can be proved to be stable under renormalization, that is, the renormalized action \( {\mathcal S}_\mathrm{ren.}={\mathcal S}+{\mathcal S}_\mathrm{CT}\ ,\) sum of the initial quantized action and properly chosen counter-terms, still satisfies equation (19). Its explicit solution, using locality (the action is a space-time integral over functions of fields and derivatives), power counting, which is a form of dimensional analysis, and ghost number conservation (if one assigns a ghost charge \(+1\) to \(C\) and \(-1\) to \(\bar C\ ,\) the action has total charge 0), yields the general form of the renormalized action. In particular, the renormalized action has still a BRST symmetry but with renormalized fields and parameters. In the example of gauge-fixing functions \(\mathbf{G}\) linear in the fields, the \(\boldsymbol{\lambda}\)-quantum equation of motion yields additional relations between counterterms. In the simple example of the gauge (7) and in the absence of matter fields, the renormalized action \( {\mathcal S}_\mathrm{ren.}\) is then obtained from \( {\mathcal S}\) by the simple substitutions \[e\mapsto Z_e^{1/2} e\,,\quad\mathbf{A}_\mu \mapsto Z^{1/2}_A \mathbf{A}_\mu\,,\quad \mathbf{C}\bar{\mathbf{C}} \mapsto Z_C \mathbf{C}\bar{\mathbf{C}},\quad \xi\mapsto Z_A \xi\,.\] (Since only the product \(\mathbf{C}\bar{\mathbf{C}}\) always appears, only the renormalization of \(\mathbf{C}\bar{\mathbf{C}}\) is defined.)

By contrast, when the gauge-fixing function \(\mathbf{G}\) is not linear in the fields, renormalization generates terms quartic in the ghost fields and, thus, the integration over \(\mathbf{C}\) and \(\bar{\mathbf{C}}\) no longer yields a simple determinant. Nevertheless, the ZJ equation still implies gauge independence (i.e., independence of the gauge section) of physical quantities and, thus, the field theory has the same physical properties. The ZJ equation can also be used to discuss the renormalization properties of gauge-invariant operators, which are related to physical observables.

Finally, note that a (simpler) quadratic equation somewhat analogous to (19), (20) appears in the renormalization of the non-linear sigma model, a quantum field theory renormalizable in 1+1 space-time dimensions with an \(O(N)\) orthogonal symmetry, in which the field belongs to a sphere \(S_{N-1}\ .\)

A few properties

What distinguishes most the ZJ equation (20) (or equation (19)) from equation (18) is its quadratic structure, because the transformations of fields depend on the action itself. Thus, several properties of the equation can be understood by studying the more general equation \[\tag{21} \int\mathrm{d}^4 x\,\mathrm{tr}\left({\delta \mathcal {S}\over\delta \mathbf{Q}(x)} {\delta \mathcal{S}\over\delta \mathbf{\Pi}(x)}\right)=0\,,\]

where, with respect to the preceding section, the field \(\mathbf{Q}(x)\) plays the role of the set of commuting fields \(\{\mathbf{A}_\mu,\mathbf{L},\boldsymbol{\mu}\}\) and the field \(\mathbf{\Pi}(x)\) the role of the set of anticommuting fields \(\{\mathbf{K}_\mu,\bar{\mathbf{C}},\mathbf{C}\}\ .\)

For notational simplicity, we replace equation (21) by a formally identical but simpler equation; the generalization to field theory is then straightforward. We assume that \(S\) is a smooth function of \(N\) real variables \(q_i\) and \(N\) generators \(\pi_i\) of a Grassmann algebra\[\pi_i\pi_j+\pi_j\pi_i=0\ ,\] which belongs to the commuting subalgebra and satisfies the equation \[\tag{22} \sum_i{\partial S\over \partial q_i}{\partial S\over \partial \pi_i} =0\,.\]

The index \(i\) plays the role of space-time coordinate together with Lorentz and group indices in equation (21). Summation over \(i\) replaces integration and summation.

We also consider the differential operator \[\tag{23} \mathcal{D} =\sum_i \left({\partial \mathcal{S} \over \partial \pi_i}{\partial \over \partial q_i}+{\partial {\mathcal S} \over \partial q_i}{\partial \over \partial \pi_i}\right)\,.\]

One verifies that equation (22) is the necessary and sufficient condition for \( \mathcal{D}\) to be a BRST cohomology operator, that is, for \( \mathcal{D}\) to satisfy \(\mathcal{D}^2=0\ .\)

Special solutions

It is possible to characterize all solutions of equation (22) of the special form \[\mathcal{S}=\Sigma^{(0)}(q)+\sum_{i,j}\Sigma^{( 2)}_{ij}(q)\pi_i\pi_j \,,\] where \(\Sigma^{(0)}\) and \(\Sigma^{( 2)} _{ij}=-\Sigma^{( 2)}_{ji}\) are analytic functions of the \(q_i\)s. Equation (22) is equivalent to a system of two equations, corresponding to the vanishing of the terms of degree one and three in the generators \(\pi_i\) in (22). The first equation, coming from the linear term, can be more easily expressed by introducing the differential operator \[\mathrm{d}_i:=\sum_k \Sigma^{( 2)}_{ki}(q){\partial\over\partial q_k}\,.\] It then takes the form \[\tag{24} \mathrm{d}_i \Sigma^{(0)}(q)=0\quad \forall\, i\,.\]

The second equation takes the form \[\tag{25} \sum_l \left\{ {\partial \Sigma^{( 2)}_{jk}\over\partial q_l}\Sigma^{( 2)}_{li}\right\}_{ijk}=0 \quad \forall \,i,j,k\,,\]

where the notation means antisymmetrized over \(ijk\ .\) From the latter equation, one derives the commutation relation \[\tag{26} [\mathrm{d}_i,\mathrm{d}_j]=\sum_k {\partial\Sigma^{( 2)}_{ij} \over\partial q_k}\mathrm{d}_k\,.\]

Equation (26) is the compatibility condition for the linear differential system (24). It also implies that the operators \(\mathrm{d}_i\) are the generators of a Lie algebra in some non-linear representation. Finally, if \(\Sigma^{( 2)}_{ij}\) is a first degree polynomial, \({\partial\Sigma^{( 2)}_{ij} \over\partial q_k}\) are the structure constants of the Lie algebra and equation (25) contains the corresponding Jacobi identity.

Perturbative solutions

We assume that we have found a solution \(\mathcal{S}^{(0)}\) of equation (22), to which is associated a BRST operator \(\mathcal{D}_0\) like in (23), and we look for solutions that can be expanded in terms of a real parameter \(\kappa\) in the form \[\mathcal{S}(\kappa)=\sum_{n \ge0}\kappa^n \mathcal{S}^{(n)} .\] Expanding equation (22) at order \(\kappa\ ,\) one obtains the condition \(\mathcal{D}_0 \mathcal{S}_1=0\ .\) Thus, one has to find \(\mathcal{D}_0\) closed solutions. More generally, at order \(\kappa^n\) the equation can be written as \[\mathcal{D}_0\mathcal{S}^{(n)}=-\sum_{1\le m\le n-1}\sum_i\left({\partial\mathcal{S}^{(m)}\over\partial q_i}{\partial\mathcal{S}^{(n-m)}\over\partial \pi_i}\right).\] This reduces the problem of the recursive determination of the coefficients \(\mathcal{S}^{(n)}\)to an investigation of the properties of the \(\mathcal{D}_0\) operator.

Canonical invariance

Equation (22) has properties reminiscent of those of the symplectic form \(\mathrm{d} p\wedge \mathrm{d q}\) of classical mechanics; in particular it is invariant under some generalized canonical transformations. Indeed, after the change of variables \((\pi ,q ) \mapsto (\pi ' ,q' )\ ,\) \[\tag{27} q_i = {\partial \varphi \over \partial \pi_i} (\pi,q' ), \quad \pi'_i = {\partial \varphi \over \partial q '_i} (\pi, q' ),\]

in which \(\varphi (\pi, q' )\) is a function belonging to the anticommuting part of the Grassmann algebra, one recovers equation (22) in the new variables: \[\sum_i { \partial \mathcal{ S} \over \partial \pi '_i}{\partial \mathcal{ S} \over \partial q '_{i}}=0\,.\] The proof goes in two steps, which both involve the anticommutation of \(\pi_i\) and \(\pi_j\) or of the corresponding derivatives. One first goes from \(q_i\) to \(q'_i\) at \(\pi_i\) fixed. One finds \[ \left.{\partial S\over\partial q'_i}\right|_\pi=\sum_j {\partial q_j \over \partial q'_i}\left.{\partial S\over \partial q_j}\right|_\pi=\sum_j {\partial^2 \varphi \over\partial q'_i \partial \pi_j} \left.{\partial S\over\partial q_j}\right|_\pi\,,\quad \left.{\partial S\over \partial \pi_i}\right|_{q'}=\left.{\partial S\over \partial \pi_i}\right|_{q}\,.\] Then one changes from \( \pi_i\) to \(\pi'_i\ :\) \[\left.{\partial S\over\partial \pi_i}\right|_{q'}=\sum_j {\partial^2\varphi \over \partial \pi_i\partial q'_j}\left.{\partial S \over\partial\pi'_j}\right|_{q'}\,,\quad \left.{\partial S\over \partial q'_i}\right|_\pi =\left.{\partial S\over \partial q'_i}\right|_{\pi'}+\sum_j {\partial^2 \varphi \over \partial \pi_i\partial q'_j}\left.{\partial S \over \partial \pi'_j}\right|_{q'}\,. \] Collecting all terms, one verifies the property.

Infinitesimal canonical transformations

We now consider infinitesimal canonical transformations of type (27). The function \(\varphi(\pi ,q' ) =\sum_i\pi_iq'_i\) corresponds to the identity. We then write the function \(\varphi \) in terms of a real parameter \(\kappa\) as \[\varphi(\pi ,q' ) =\sum_i\pi_{i}q'_{i}+\kappa \psi (\pi ,q' ). \] The variation of \(S\) at first order in \(\kappa\) is \[\mathcal {S} (\pi ' ,q' )-\mathcal {S} (\pi ,q )=\kappa\sum_i \left({ \partial \psi \over \partial q _i}{\partial \mathcal {S} \over \partial \pi_i}- { \partial \psi\over \partial \pi_i}{\partial \mathcal {S} \over \partial q_i}\right)+O\left(\kappa^2\right)=-\kappa\, \mathcal {D}\psi +O\left(\kappa^2\right)\, .\] One thus finds that an infinitesimal canonical transformation generates a BRST exact contribution and, conversely, any infinitesimal addition to \( \mathcal{S}\) of a BRST exact term can be generated by a canonical transformation acting on \( \mathcal{S}\ .\) One then verifies that, indeed, the additional contributions to the action due to the gauge-fixing procedure can also be generated by such a canonical transformation with \[\kappa\, \psi (\pi ,q' )\mapsto \int\mathrm{d}^4x\,\mathrm{tr}\,\mathbf{C}(x)\left(\mathbf{G}(\mathbf{A},x)+\textstyle{ {1\over2} }\lambda(x)\right),\] acting on \[\mathcal{S}_{\mathrm{class.}}(\mathbf{A}_\mu)-\hbar\int\mathrm{d}^4 x \,\mathrm{tr}\left(\sum_\mu \mathbf{K}^\mu(x) \Delta\mathbf{A}_\mu(x) +\mathbf{L}(x)\Delta\bar{\mathbf{C}}(x)+\boldsymbol{\mu}(x)\boldsymbol{\lambda} (x)\right).\]

The origin of BRST symmetry

One might be surprised that quantized gauge theories have this peculiar BRST symmetry. In fact, BRST symmetry is an automatic property of constraint systems handled in a specific way as we explain now. In particular, in gauge theories it is induced by the constraint of the gauge section (7) but its form is complicated by the choice of coordinates because the equation of the section applies to group elements \(\mathbf{g}(x)\ ,\) the gauge transformations.

Let \(\varphi^{a}\) be a set of real quantities satisfying a system of real equations, \[\tag{28} E_{a} (\varphi)=0\,, \]

where the functions \(E_{a} (\varphi)\) are smooth, and \(E_{a}=E_{a} (\varphi)\) is a one-to-one map in some neighbourhood of \(E_{a}=0\ ,\) which can be inverted in \(\varphi^{a}=\varphi^{a}(E).\) In particular, this implies that equation (28) has a unique solution \(\varphi_{\rm s}^{a}\equiv \varphi^{a}(0)\ .\) In the neighbourhood of \(\varphi_{\rm s}\ ,\) the determinant \(\det\mathbf{E}\) of the matrix \(\mathbf{ E}\) with elements \[E_{ {a}{b}}\equiv {\partial E_{a} \over\partial \varphi^{b} } \,,\] does not vanish and thus we choose \(E_{a}(\varphi)\) such that it is positive.

For any continuous function \(F(\varphi)\) we now derive a formal expression for \(F (\varphi_{ {\mathrm s} } )\) that does not involve solving equation (28) explicitly. We start from the simple identity \[F(\varphi_{\mathrm{s}} )=\int\left\{ \prod_{ {a} } \mathrm{d} E_{a} \, \delta ( E_{a} ) \right\} F\bigl(\varphi(E)\bigr) , \] where \(\delta(E)\) is Dirac's \(\delta\hbox{-function}\ .\) We then change variables \(E \mapsto \varphi\ .\) This generates the Jacobian \(\mathcal{ J}(\varphi)=\det\mathbf{ E}>0\ .\) Thus, \[\tag{29} F(\varphi_{\mathrm{s}} )=\int\left\{ \prod_{ {a} } \mathrm{d} \varphi^{a} \, \delta \left[ E_{a}(\varphi) \right] \right\} \mathcal{ J}(\varphi)\, F(\varphi).\]

We replace the \(\delta\)-function by its Fourier representation: \[\prod_{ {a} }\delta \left[E_{a} (\varphi)\right] = \int \prod_{ {a} }{\mathrm{d}\bar\varphi^{a} \over 2i\pi} \exp\left(-\sum_{a}\bar\varphi^{ {a} } E_{a}(\varphi)\right), \] where the \(\bar\varphi\) integration runs along the imaginary axis. Moreover, a determinant can be written as an integral over Grassmann variables (i.e., generators of a Grassmann or exterior algebra) \(\bar c^{a}\) and \(c^{ {a} }\) in the form \[\tag{30} \det\mathbf{ E}= \int \prod_{ {a} } \left(\mathrm{d} \bar c^{a} \mathrm{d} c^{ {a} } \right)\exp\left(\sum_{ {a},{b} }c^{ {a} }E_{ {a}{b} } \bar c^{ {b} }\right).\]

Expression (29) then becomes \[\tag{31} F(\varphi_{\rm s}) =\mathcal{N} \int \prod_{ {a} } \left(\mathrm{d} \varphi^{a}\mathrm{d} \bar\varphi^{ {a} }\mathrm{d} \bar c^{a} \mathrm{d} c^{ {a} } \right)F(\varphi) \exp\left[-S (\varphi,\bar\varphi, c,\bar c)\right], \]

in which \(\mathcal{N}\) is a constant normalization factor and \[\tag{32} S (\varphi ,\bar\varphi ,c ,\bar c)=\sum_{ {a} }\bar\varphi^{ {a} }E_{a} (\varphi)-\sum_{ {a},{b} } c^{ {a} }E_{ {a}{b} }(\varphi) \bar c^{b} \,, \]

is a commuting element of the Grassmann algebra.

Somewhat surprisingly, the function \(S\) has a new type of symmetry, the BRST symmetry that we now describe.

BRST symmetry

The function \(S\) defined by equation (32) is invariant under the BRST transformations \[\tag{33} \varphi^{a} \mapsto \varphi^{a}+\delta_{\rm BRST}\varphi^{a} \ \mathrm{with}\ \delta_{\rm BRST}\varphi^{a}=\varepsilon \bar c^{a}, \quad \bar c^{a} \mapsto \bar c^{a} \,,\]

and \[ c^{a}\mapsto c^{a} +\delta_{\rm BRST}c^a\ \mathrm{with}\ \delta_{\rm BRST}c^a= \varepsilon\bar\varphi^{a}\, ,\quad \bar\varphi^{a} \mapsto \bar\varphi^{a} ,\] where \( \varepsilon\) is an anticommuting constant, an additional generator of the Grassmann algebra such that \( \varepsilon^{2}=0\, ,\quad \varepsilon \bar c^{a} + \bar c^{a} \varepsilon =0 \, ,\quad \varepsilon c^{ {a} }+ \varepsilon c^{ {a} }=0 \ .\) Moreover, the integration measure in (31) is also invariant.

The BRST transformation is clearly nilpotent (of vanishing square) since the variation of the variation always vanishes. Note that the transformations of \(c^{a}\) and \(\bar\varphi^{a}\) are identical to the transformations of the fields \(\mathbf{C}(x)\) and \(\boldsymbol{\lambda}(x)\) in (12).

The BRST transformation can also be represented by a Grassmann differential operator \(\mathcal{D}\) acting on functions of \(\{\varphi,\bar\varphi,c,\bar c\}\ :\) \[\mathcal{D} = \mathcal{D}_+ +\mathcal{D}_-\,,\quad \mathcal{ D}_+=\sum_{a} \bar c^{a} {\partial \over \partial \varphi^{a} }\,,\quad \mathcal{ D}_-=\sum_{a} \bar\varphi^{ {a} }{\partial \over \partial c^{ {a} } }\, .\] Then, \[\mathcal{D}S=0\,.\] One verifies immediately that \[\mathcal{D}_+ ^2=\mathcal{D}_-^2= \mathcal{D}_+ \mathcal{D}_- + \mathcal{D}_- \mathcal{D}_+ =0\,.\] The nilpotency of the BRST transformation follows: \[\tag{34} \mathcal{ D}^2=0\,. \]

The differential operator \(\mathcal{ D}\) is a cohomology operator, generalization of the exterior differentiation of differential forms. In particular, the first term \(\mathcal{D}_+\) in the BRST operator is identical to the exterior derivative of differential forms in a formalism in which the Grassmann variables \(\bar c^{a} \) generate the corresponding exterior algebra.

Equation (34) implies that all quantities of the form \(\mathcal{ D} Q(\varphi ,\bar\varphi ,c,\bar c)\ ,\) (BRST exact), are BRST closed since \(\mathcal{ D} ( \mathcal{ D} Q(\varphi ,\bar\varphi ,c,\bar c))=0\ .\) One verifies that the function \(S\) (defined in equation (32)) is not only BRST closed but also BRST exact: \[S=\mathcal{ D} \left(\sum_{a} c^{ {a} }E_{a} (\varphi) \right).\] The reciprocal property, the meaning and implications of the BRST symmetry follow from some simple arguments based on BRST cohomology.

BRST symmetry and group elements

We now assume that the variables \(\varphi^{a}\) parametrize locally elements \( \mathbf{g}(\varphi)\) of a group \(G\) in some matrix representation. It is then natural to parametrize the BRST variation of \( \mathbf{g}\) in terms of an element \(\bar{\mathbf{C}}\) of the Lie algebra (being also a generator of a Grassmann algebra) in the form \[\tag{35} \delta_{\rm BRST}\mathbf{g}=\varepsilon\,\bar{\mathbf{C}}\,\mathbf{g}\,.\]

Calculating directly the variation of \( \mathbf{g}\) from the variation (33) of \(\varphi^a\ ,\) one obtains the relation \[\bar{\mathbf{C}}\,\mathbf{g}=\sum_a {\partial\mathbf{g}\over\partial\varphi^a}\bar c^a\ \Rightarrow\ \bar{\mathbf{C}}=\sum_a {\partial\mathbf{g}\over\partial\varphi^a}\mathbf{g}^{-1}\bar c^a \,.\] The BRST variation of the matrix \(\bar{\mathbf{C}}\) is \[\delta_{\rm BRST}\bar{\mathbf{C}}=\varepsilon\sum_{a,b}\left( {\partial^2\mathbf{g}\over\partial\varphi^a\partial\varphi^b} \mathbf{g}^{-1} -{\partial\mathbf{g}\over\partial\varphi^a}\mathbf{g}^{-1}{\partial\mathbf{g}\over\partial\varphi^b}\mathbf{g}^{-1}\right)\bar{c}^b \bar{c}^a=\varepsilon\,\bar{\mathbf{C}}^2 ,\] where the anticommutation of \(\bar c^a\) and \(\bar c^b\) has been used. One recognizes the transformation of the ghost fields in non-Abelian gauge theories (second equation in (12)). Applying then the transformation (35) to the \(\mathbf{g}(x)\) in (2), one can derive the first equation in (12).


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Internal references

  • Jean Zinn-Justin and Riccardo Guida (2008) Gauge invariance. Scholarpedia, 3(12):8287.

Further reading

  • Faddeev, L D and Slavnov, A A (1991). Gauge Fields. Introduction to quantum theory. (2nd edition). Addison-Wesley Publishing Company, T. ISBN 0201524724.
  • Itzykson, C and Zuber, J B (2006). Quantum Field Theory. Dover Publications, New York. ISBN 0486445682
  • Lai, C H ed. (1981). Gauge Theory of Weak and Electromagnetic Interactions. World Scientific Publishing, Singapore. ISBN 978-9971830236
  • Weinberg, S (1996). The quantum theory of fields. Vol. 2: Modern Applications. Cambridge University Press, Cambridge. ISBN 0521550025
  • Zinn-Justin, J (2002). Quantum Field Theory and Critical Phenomena (4th edition). Oxford University Press, Oxford. ISBN 0198509235

See also

Gauge invariance, Gauge theories, Slavnov-Taylor identities, Becchi-Rouet-Stora-Tyutin symmetry

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