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<strong><nowiki>Slavnov-Taylor identities</nowiki></strong> are relations between Green functions, which
provide the gauge invariance of quantum theory, including non-Abelian gauge fields.

==Introduction==

The principle of relativity in the internal space (the principle of gauge invariance), 
according to which two field configurations are related by a [[gauge transformation]] 
describe the same physical situation, determines uniquely the form of a minimal 
classical Lagrangian in gauge theories and constraints the structure of composite 
operators corresponding to physical observables, e.g. the energy-momentum tensor.

To quantize a gauge invariant theory one has to choose a
representative in a class of gauge equivalent configurations, for example by imposing
on the fields '''the Coulomb gauge condition''' <math>\partial_iA_i=0</math> (i=1,2,3).
Although a gauge condition breaks explicit gauge invariance of the classical theory,
in accordance with the relativity principle in internal space all observables, for 
example scattering aplitudes, are independent on the particular choice of a gauge.

Quantum theory cannot be constructed only in terms of observable quantities. To formulate 
some basic principles, like causality, it is necessary to introduce objects which are 
not directly observable and therefore may depend on a gauge. The most important objects 
of this kind are so called [[Green functions]] that is the vacuum expectation values of 
chronologically ordered field products <math><T\bar{\psi}(x_1) \ldots
\psi(x_m)A_{\mu}(x_{m+1}) \ldots A_{\nu}(x_n)></math> where the symbol <math>T</math>
means chronological ordering of the field arguments. Although these functions cannot
be measured directly in any experiment, they are related to scattering amplitudes 
and to provide the gauge independence of observables they must satisfy certain relations. 
These relations are Slavnov-Taylor identities.

In Abelian theories like quantum electrodynamics the relations between Green functions 
providing the gauge independence of observables were obtained by J.C. Ward and later in 
a more general form by E.S. Fradkin and Y. Takahashi. In quantum electrodynamics imposing a
gauge condition does not break the conservation of the electromagnetic current, which
is a source of the gauge field, and the [[Ward identities]] follow directly from
the conservation of this current. In non-Abelian models, like the Yang-Mills
theory, fixing a gauge breaks the conservation of the current which is the source of
the gauge field. Moreover, fixing a gauge in non-Abelian theories must be accompanied
by a further modification of the classical Lagrangian. This modification may be
presented as a new interaction of the gauge fields with fictitious anticommuting
scalar particles, [[Faddeev-Popov ghosts]]. This additional interaction also
breaks gauge invariance. As a result in non-Abelian models the simple Ward identities do
not hold.

==Necessary and sufficient conditions for gauge independence of observables in non-Abelian gauge theories==

The relations between the Green functions in the quantum
Yang-Mills theory, which are necessary and sufficient conditions of the gauge
independence of observables were obtained by A.A. Slavnov and J.C. Taylor. They may be
written in the form of the [[path integral]] as follows
:<math>\label{1}
\int \exp\{i\int[L_{ef}(x)+J_{\mu}^a(x)A_{\mu}^a(x)]dx\}\{-
\frac{1}{\alpha}\partial_{\mu}A_{\mu}^a(y)
+\int \bar{c}^a(y)J_{\mu}^b(z)[D_{\mu}c(z)]^b dz\}\prod_x dA_{\mu}(x)d
\bar{c}(x)dc(x)=0

</math>

where the effective Lagrangian is given by the equation
:<math>\label{2}
L_{ef}= - \frac{1}{4}F_{\mu\nu}^aF_{\mu\nu}^a+
\frac{1}{2\alpha}(\partial_{\mu}A_{\mu})^2-\partial_{\mu} \bar{c}^a(D_{\mu}c)^a

</math>

Here <math>A_{\mu}^a</math> is the Yang-Mills field, <math>\bar{c}^a, c^b</math> are Faddeev-Popov ghosts, 
<math>F_{\mu\nu}^a=\partial_{\mu}A_{\nu}^a-\partial_{\nu}A_{\mu}^a+gt^{abc}A_{\mu}^bA_{\nu}^c</math>
is the Yang-Mills [[stress tensor]],
<math>(D_{\mu}c)^a=\partial_{\mu}c^a+gt^{abd}A_{\mu}^bc^d</math> is the [[covariant  derivative]] of the ghost field, and <math>t^{abc}</math> are the structure constants of the gauge group. <math>J_{\mu}</math> is the classical external source.

To understand the meaning of this equation it is instructive to consider the scattering matrix in the Yang-Mills theory. In terms of the path integral the scattering matrix may be written as follows
:<math>\label{3}
S= \int \exp \{i \int L_{eff}(x)dx \} \Pi_xdA_{\mu}(x)d \bar{c}(x)dc(x)

</math>

Here the effective Lagrangian is given by the eq.(2) and depends explicitly on the gauge parameter 
<math> \alpha\ .</math> Nevertheless the value of the integral (3) which determines the scattering amplitudes for the Yang-Mills field is gauge independent. This is a consequence of the relativity principle in the internal space according to which the fields related by a gauge transformation describe the same physical situation, and therefore in the path integral for the scattering matrix one should integrate over the classes of the gauge equivalent fields. In practice that means that one has to select a single representative in the gauge equivalent class by imposing some gauge condition and normalising correctly the integration measure by adding the integration over Faddeev-Popov ghost fields. The result does not depend on the particular choice of the gauge because the transformation from one gauge to another may be accomplished by a simple change of variables in the path integral.

Contrary to the scattering matrix the Green functions are not directly observable and do depend on the gauge. Any Green function may be presented as a variational derivative of the Green function generating functional given by the following path integral
:<math>\label{4}
Z(\eta)= \int \exp \{i \int[L_{eff}(x)+J_{\mu}^a(x)A_{\mu}^a(x)]dx \}dA_{\mu}d \bar{c}dc

</math>


:<math>\label{5}
<TA_{\mu}^a(x_1) \ldots A_{\nu}^c(x_n)>= \frac{\delta^nZ}{\delta J_{\mu}^a(x_1) \ldots J_{\nu}^c(x_n)}_{J=0}

</math>

One sees that the integrand in the formula for the Green functions generating functional differs of the corresponding integrand for the scattering matrix by the presence of the gauge non invariant term <math> \int J_{\mu}^a(x)A_{\mu}^a(x)dx</math> in the exponent. When one passes from one gauge to another by the change of variables this functional is changed. Its variation is given by the left hand side of the eq.(1). As a change of variables does not influence the value of integral this variation is equal to zero. Hence the Slavnov-Taylor identities (1) describe the response of the Green function generating functional to the change of gauge conditions.

Differentiating the identity \eqref{1} with respect to external source <math>J_{\mu}</math> and
putting <math>J_{\mu}=0</math> one obtains the relations between Green functions with different
number of external lines, which guarantee independence of the observables on the
choice of the gauge, and hence physical equivalence of different gauge conditions. The simplest identity is
:<math>\label{6}
\alpha^{-1}<T \partial_{\mu}A_{\mu}^a(x)A_{\nu}^b(y)>=-i<T \bar{c}^a(x)(D_{\nu}c)^b(y)>

</math>

Differentiating this equation with respect to <math> \nu</math> and taking into account the equation of motion for the ghost field we conclude that this equation implies the absence of radiative corrections to the longitudinal part of the two point Green function
:<math>\label{7}
\partial_{\mu} \partial_{\nu}G_{\mu \nu}^{ab}(x,y)= \alpha \delta^{ab} \delta (x-y)

\ .</math>


==Slavnov-Taylor identities and renormalization==

A quantum theory includes a new element: [[renormalization]]. Calculation of
radiative corrections in any four dimensional theory leads to appearance of divergent
integrals. A consistent treatment of these integrals is provided by the
renormalization procedure which may be considered as modification of the Lagrangian by
introducing the new terms ([[counterterms]]) and is equivalent to redefinition of
the parameters (masses, charges) of the classical Lagrangian and multiplicative renormalization of the fields
:<math>\label{8}
A_{\mu}^a \rightarrow \sqrt{z_2}\,A_{\mu}^a, \quad \bar{c}^a \rightarrow \sqrt{\tilde{z}_2} \,\bar{c}^a, \quad c^a \rightarrow \sqrt{\tilde{z}_2}\,c^a

</math>

 
To guarantee  the gauge independence of observables this procedure must respect the principle of relativity in
the internal space, and hence the Slavnov-Taylor identities must impose certain
conditions on possible counterterms. 

Contrary to the Ward identities in quantum electrodynamics the Slavnov-Taylor
identities include composite operators, such as <math>(D_{\mu}c)^a\ .</math> This leads to considerable differences in
renormalization of quantum electrodynamics and non-Abelian gauge theories. In
particular in the Yang-Mills theory the Slavnov-Taylor identities impose the following
conditions on the possible counterterms
:<math>\label{9}
\delta m=0; \quad  z_1z_2^{-1}= \tilde{z}_1 \tilde{z}_2^{-1}: \quad z_4=z_1^2z_2^{-1}

</math>

where <math>\delta m</math> is the mass renormalization of the Yang-Mills field, <math>z_1</math> and <math>z_4</math> are the renormalization constants of the
Yang-Mills Green functions with three and four external lines, and <math>\tilde{z}_1</math> is
the renormalization constant of the vertex function describing the interaction of the
Yang-Mills field with the ghost fields. These relations provide the gauge invariance
of the renormalized Lagrangian, however the parameters of the gauge transformation
depend on the renormalization constants.

==Alternatives and generalizations==

It was noted by C. Becchi, A. Rouet, R. Stora and by I. Tyutin that the Slavnov-Taylor
identities may be interpreted as consequence of a charge conservation. However this
charge is not related to the source of the gauge field, as it happens in quantum
electrodynamics, but includes also the contribution from Faddeev-Popov ghosts. Its
conservation follows from the invariance of the effective action with respect to some
supersymmetry transformations (Becchi-Rouet-Stora-Tyutin symmetry). Similar identities may be
also written for the strongly connected Green functions (B.W. Lee; J. Zinn-Justin). (See also [[Zinn-Justin equation]]).
Slavnov-Taylor identities play the key role in the proof of the gauge invariant
renormalizability of non-Abelian gauge theories.

As was shown by G t'Hooft the quantization of the gauge invariant models with
spontaneously broken symmetry may be performed along the same lines as the
quantization of the Yang-Mills theory leading to renormalizability of such models, in
particular Weinberg-Salam model. The Slavnov-Taylor identities also may be generalized
in a straightforward way to spontaneously broken models (B.W. Lee, J. Zinn-Justin), and
their fulfillment guarantees the gauge invariance and hence unitarity of renormalized
gauge models with spontaneously broken symmetries.

==References==

* {{bibitem article 3| |Phys.Lett.|B52|1974|344|Becchi|C M|Rouet|O|Stora|R}}
* {{bibitem article 3| |Comm.Math.Phys.|42|1975|127|Becchi|C M|Rouet|O|Stora|R}}
* {{bibitem article 2| |Phys.Lett.|B25|1967|30|Faddeev|L D|Popov|V N}}
* {{bibitem article 1| |JETP|29|1955|288|Fradkin|E S}}
* {{bibitem article 1| |Nucl.Phys.|B35|1971|167|'t Hooft|G}}
* {{bibitem article 1| |Phys.Rev.|D9|1974|933|Lee|B W}}
* {{bibitem article 2| |Phys.Rev.|D5|1972|3137|Lee|B W|Zinn-Justin|J}}
* {{bibitem book 1 |Elementary Particle Theory.|editors= Svartholm, N editor.|Almquist Ferlag AB|Stockholm|1968|Salam|A}}
* {{bibitem article 1| |Theor.Math.Phys.|10|1972|99|Slavnov|A A}}
* {{bibitem article 1| |Nuovo Cimento.|6|1957|370|Takahashi|Y}}
* {{bibitem article 1| |Nucl.Phys.|B33|1971|436|Taylor|J C}}
*  Tyutin, I; 1975. Preprint of Lebedev Physical Institute, 39, 1975.
* {{bibitem book 1 |Lecture Notes in Physics, 37.|Springer-Verlag|Berlin|1974|Zinn-Justin|J}}
* {{bibitem article 1| |Phys.Rev.|77|1950|2931|Ward|J C}}
* {{bibitem article 1| |Phys.Rev.Lett.|19|1967|1264|Weinberg|S}}

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<b>Internal references</b>

* Carlo Maria Becchi and Camillo Imbimbo (2008) [[Becchi-Rouet-Stora-Tyutin symmetry]]. Scholarpedia, 3(10):7135.


==Further reading==

*{{bibitem book 2|Gauge Fields. Introduction to quantum theory. Second edition.|Addison-Wesley Publishing Company|T|1991|Faddeev|L D|Slavnov|A A|isbn=ISBN  0201524724.}}
*{{bibitem book 1|Quantum Field Theory and Critical Phenomena. 4th edition.|Oxford University Press|USA|2002|Zinn-Justin|J|isbn=ISBN 0198509235.}}

==See also==
[[Gauge invariance]], [[Gauge theories]], [[Becchi-Rouet-Stora-Tyutin symmetry]], [[Zinn-Justin equation]]

[[Category:Physics]]
[[Category:Quantum and Statistical Field Theory]]
[[Category:Quantum Field Theory (Foundations)]]
[[Category:Eponymous]]