Becchi-Rouet-Stora-Tyutin symmetry

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Author: Dr. Camillo Imbimbo, Genoa University, Italy
Author: Dr. Carlo Maria Becchi, Genoa University, Italy

BRST Symmetry

1 History
2 Maxwell's equations and Dirac's quantization
3 Landau's gauge quantization of Electrodynamics
4 Analysis of the Yang-Mills model
5 The Kugo-Ojima quartet mechanism
6 The quartet mechanism in quantum field theory
7 Yang-Mills theory in BRST formalism
8 References
9 Recommended reading
10 See also

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History

The BRST construction is the last episode of the long history of the quantization of fields, in particular of gauge fields. This history begins with Einstein's interpretation of the photoelectric effect based on the existence of the photon, a particle associated with the electromagnetic radiation which are described in terms of two polarization states. Indeed when Dirac and many others tried to quantize the electromagnetic field, using the canonical commutation relations they encountered an obvious difficulty. The canonical quantization needs a Lagrangian which is a function of the scalar and vector potential. Furthermore the quantization is based on a normal mode decomposition which associates to every field component a different elementary particle. Since the vector potential has four components one expects four different polarization states for the photon instead of two. As a matter of fact this does not happen since the Quantum Electrodynamics Lagrangian induces constraints. Dirac and Bergmann developed a very interesting theory of quantum constrained systems. In gauge theories, Quantum Electrodynamics and generalizations, the system of constraints generates a Lie algebra and is called a system of first class constraints. However in the interesting cases, that is in Quantum Electrodynamics, in its non-abelian Yang Mills theory extensions which include the Standard Model of Fundamental Interactions and General Relativity, this approach leads to a loss of explicit Lorentz covariance and locality (section 1) which is a problem for renormalization.

In Quantum Electrodynamics for a lucky and fortuitous reason the simplest choices of supplementary conditions on the potentials, called gauge fixing, which are introduced in order to reduce the number of degrees of freedom, are dynamically stable due to current conservation. Hence it is possible to quantize the theory by covariant and local canonical prescriptions albeit abandoning the Hilbert space character of the quantum state space. This, as shown by Gupta and Bleuler, becomes an indefinite norm space (section 2). The relation between stability of the gauge fixing and current conservation is made explicit by the well known Ward-Takahashi identities.

However this lucky condition does not persist in the extension to non-abelian Yang-Mills theory and to the other generalizations of Quantum Electrodynamics (section 3). Hence in these extensions one cannot use the same definition physical quantum state space as in Quantum Electrodynamics. After many efforts the problem was solved by Faddeev and Popov within the framework of Feynman's path integral construction of field theory by introducing a system of unphysical ghost fields which in the functional integral approach allow the insertion of a needed Jacobian into the functional integral. We shall see that in the usual state space framework these ghosts play the role of compensating degrees of freedom and the BRST symmetry is crucial to guarantee this compensation (section 4). It is on the basis of Faddeev and Popov's construction that 't Hooft and Veltman ('t Hooft et al. 1972) were able to get the first solid results on the renormalization of non-abelian gauge theories.

After Faddeev and Popov's paper Slavnov and Taylor studied how the Ward-Takahashi identities of Quantum Electrodynamics are deformed in non-abelian gauge theories. They introduced new identities which, as a matter of fact, led Becchi, Rouet and Stora (Becchi et al.1974 and 1976) and, independently, Tyutin to identify the property which at the quantum level guarantees the physical consistency of the theory and, in particular, unitarity. This is often called BRST symmetry; it is however not a physical symmetry since it acts trivially on observables.

BRS's work, devoted to renormalization, deals with the Green functions of the theory, the objects concerned by the renormalization program. The action of "BRST symmetry" on the space of states was carefully described by Kugo and Ojima (Kugo et al.1978), whose results are substantially used in the present text.

After the renormalization of non-abelian gauge theories the BRST construction was applied to the quantization of the gravitational field and, with more success, to String theory whose recent formulations heavily rely on this method.

A further important extension of the BRST construction introduced by Witten leads to topological field theory.

Finally a crucial remark is in order: BRST's construction is at the moment restricted to perturbation theory. It is not yet clear how it can be made consistent with non-perturbative effects and in particular with the appearence of Gribov copies.

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Maxwell's equations and Dirac's quantization

Since our discussion will mainly deal with relativistic quantum field theory we shall systematically use the fundamental unit system in which $\hbar=c=1 \$ . We shall also use the Minkowski metric with the sign choice $g^{00}=1$ and throughout leave the sum over repeated indices understood.

Our starting point is the historical one, that is, the quantization problem of electrodynamics. Let us begin by writing Maxwell's equations in terms of the covariant field tensor

$\partial_ \nu F^{ \mu \nu}=j^ \mu \quad \ , \quad \epsilon^{ \mu \nu \rho \sigma} \partial_ \nu F_{ \rho \sigma}=0$

In order to canonically quantize electrodynamics one has to identify Maxwell's equations with the Euler-Lagrange equations corresponding to some Lagrangian density and then deduce the appropriate canonical commutation relations from the chosen Lagrangian. In the present case the Lagrangian density is a function of the field tensor $F^{ \mu \nu}$ and of the four-vector potential $A_ \mu$

$L=- \partial_ \mu A_ \nu F^{ \mu \nu}+{1 \over 4} F_{ \mu \nu}F^{ \mu \nu}-A_ \mu j^ \mu \ .$ The four-vector potential Lagrange equation gives the first Maxwell's equation above, while the tensor field one gives the second equation.

The problem with quantization is that Maxwell's Lagrangian density describes a constrained mechanical system since $L$ is independent of $\partial_0 A^0$ . This induces Gauss's law as a constraint

$\vec \nabla \cdot \vec E= \rho$

The solution of Gauss's constraint requires a supplementary condition, e.g.

$\vec \nabla \cdot \vec A=0$

which induces the non-local canonical commutation relations

$[A^i( \vec r), E^j ( \vec r')]=-i \left( \delta^{ij} \delta( \vec r- \vec r')+ \nabla^i \nabla^j{1 \over4 \pi| \vec r - \vec r'|} \right) \ .$

If instead one insists on the covariant formulation beyond quantization and, even more importantly, on local commutation relations, one must abandon, as shown by Gupta and Bleuler, the usual framework of quantum mechanics and in particular the identification of the state space with a Hilbert space.

Gupta-Bleuler's results apply to Quantum Electrodynamics with Feynman's gauge fixing. An alternative and equivalent possibility, more suited for the presentation of the BRST formalism, is the quantization with Landau's gauge fixing.

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Landau's gauge quantization of Electrodynamics

Landau's gauge quantization of Quantum Electrodynamics was suggested by Landau and refined by Nakanishi. It produces a quantum structure whose extension to non-abelian gauge theories gives a natural framework for the BRST construction.

Here we sketch the quantization of Quantum Electrodynamics in Landau's gauge, a subject not particularly popular in text books.

The starting point is the Lagrangian density

$L=-{1 \over2} \partial^ \mu A^ \nu \left( \partial_ \mu A_ \nu- \partial_ \nu A_ \mu \right)+ b \ \partial_ \mu A^ \mu+A_ \mu j^ \mu \ .$ in which $j^ \mu(x)$ is the electromagnetic current density which is conserved due to the equation of charged matter fields.

The corresponding Euler-Lagrange equations are

$\partial_ \mu A^ \mu(x)=0 \quad \ , \quad \partial_ \nu \partial^ \nu A_ \mu (x) \equiv \partial^2 A_ \mu (x)=j_ \mu (x)+ \partial_ \mu b(x) \quad \ , \quad \partial^2 b(x)=0 \ .$ Here

Lorentz's gauge fixing condition is strongly implemented by the Euler-Lagrange equation of the $b$ field (first equation). This field gives an additive contribution to the current in the Euler-Lagrange equation of the vector potential (second equation), while $b$ satisfies the D'Alembert's wave equation (third equation) due to current conservation.

The standard field quantization procedure suited for Feynman's perturbation theory is based on the normal mode decomposition of the solutions of the free field equations, that is for $j_ \mu(x)=0$ . In the free case this decomposition is based on the Fourier analysis of the solutions.

It is important to remind here that in relativistic quantum field theory free quantum fields play the physical role of asymptotic fields. These asymptotic fields are built with the creation and annihilation operators of the Fock space of scattering states. Since there are two kinds of scattering states, the in-going and the out-going ones, there also are two different kinds of asymptotic free fields which are called ‘’in’’ and ‘’out’’ fields. In Feynman's covariant perturbation theory one identifies the free quantized fields one starts from with the in-fields. Haag has shown in a general framework based on a natural set of axiomatic conditions that the quantum interacting fields evolve towards the in and out fields at asymptotic times in the past and, respectively, in the future.

In the present case the Fourier analysis of the general solution $A_\mu(x)$ involves four complex coefficients. Two of them, $a( \vec k,h)$ , are associated with the physical, helicity $h= \pm 1$ , states, while $\alpha( \vec k)$ corresponds to the longitudinally polarized states and $\beta( \vec k)$ corresponds to the Fourier transform of $b$ ./Landau’s gauge free solutions

In the quantized system the complex coefficients correspond to operators which satisfy the equal-time canonical commutation relations : $[A_i( \vec r), \partial_0A_j( \vec r')- \partial_jA_0( \vec r')]=i \delta_{i,j} \delta( \vec r- \vec r') \quad \ , \quad [A_0( \vec r), b( \vec r') ]=i \delta( \vec r- \vec r')$ where the $i,j$ indices label the space components of four-vectors.

These rules when expressed in terms of the Fourier coefficients of general solution trnslate into a set of commutation relations whose non-trivial part is : $[a( \vec k, h),a^+( \vec q,h')]=2k_0 \delta_{h,h'} \delta( \vec k- \vec q) \ \ , \ [ \alpha( \vec k ), \beta^+( \vec q)]=[ \beta( \vec k ), \alpha^+( \vec q)]=2k_0 \delta( \vec k- \vec q)\ .$ The second commutation prescription contrasts with the possibility of identifying $\alpha^+( \vec k)$ and $\beta^+( \vec k)$ with invariantly normalized creation operators in a Fock-Hilbert space, while this identification is natural for $a^+( \vec k, h)$ . Indeed, one has for $( \alpha( \vec k)+ \beta( \vec k))/ \sqrt{2}$ a commutation rule analogous to that of $a( \vec k, h)$ , for $( \alpha( \vec k)- \beta( \vec k))/ \sqrt{2}$ one gets a commutator with the wrong sign. This means that if we insist, as we must do, interpreting $( \alpha^+( \vec k)- \beta^+( \vec k))/ \sqrt{2}$ as a creation operator, it creates negative norm states and hence the space underlying the quantization is not a Fock-Hilbert space but a Fock space with indefinite norm. Further details on Fock spaces with indefinite norm will be given in BRST Symmetry/Indefinite Metric and BRST Cohomology where we shall present in particular the definition of the pseudo-adjoint. In the following we shall keep the dagger apex for the Fock-Hilbert adjoint using the $+$ apex for the pseudo-adjoint operator.

The appearance of negative norm states contrasts with the usual probabilistic interpretation of the inner product[1] in Quantum Mechanics. The Gupta-Bleuler way out of this paradox consists in identifying the physical state space with the subset of the Fock space annihilated by $\beta( \vec k)$ . This is a choice justified by the fact that $b(x)$ is a free field and hence the prescription selects a [[Superselection Sector]] of the Fock space. Furthermore with this definition the physical space is spanned by states generated by the action of polynomials of $a^+( \vec k,h)$ which are positive norm states and mixed polynomials of $a^+( \vec k,h)$ and $\beta^+( \vec k)$ which turn out to be orthogonal to the rest of the physical space and have zero norm. These states can be freely added to the positive norm physical states without changing their inner products and correspond a generic gauge variation of the physical states.

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Analysis of the Yang-Mills model

The situation changes completely in the for the non-abelian theory, as for example the Yang-Mills model. In this case the field analogous to $b$ is not a free field anymore. Disregarding for simplicity the possible couplings to matter, one starts from the Lagrangian density

$L =-{1 \over4} \left( \partial_ \mu A^a_ \nu- \partial_ \nu A^a_ \mu+ g f^{abc} A^b_ \mu A^c_ \nu \right) \left( \partial^ \mu A^{a \nu}- \partial^ \nu A^{a \mu}+ g f^{abc} A^{b \mu} A^{c \nu} \right)+ b^a \partial_ \mu A^{a \mu} \ ,$ where the coefficients

$f^{abc}$ are the structure constants of a $N$ -parameter semi-simple group. The field equations are

$\partial^2A^a_ \mu +gf^{abc} A^{b \nu} \left( \partial_ \nu A^c_ \mu- \partial_ \mu A^c_ \nu \right) -g^2f^{abc}f^{cde} A^d_ \mu A^{b \nu} A^e_ \nu= \partial_ \mu b^a$

$\partial_ \mu A^{a \mu}=0 \ .$

It is apparent that the combination of these two equations does not imply anymore that $\partial^2 b^a=0$ except in the $g=0$ limit where the above Lagrangian reduces to the sum of $N$ pure Quantum Electrodynamics Lagrangians.

Still, as we have already said, in perturbation theory one starts from the free theory describing the quantum mechanics of the asymptotic scattering states. Thus in the present case the space of asymptotic in-going states consists in the tensor product of $N$ indefinite norm Fock spaces perfectly analogous to that of Quantum Electrodynamics. The difficulty arises when one tries to identify a physical subspace with semi-definite norm. Indeed the natural extension of Quantum Electrodynamics, that is, the selection of states annihilated by all the annihilation operators $\beta^a( \vec k)$ , does not identify, as it should do, a super-selection sector since $b^a$ is not anymore a free field.

The cure to this sickness of the standard covariant approach was found, after many failed attempts, by Faddeev and Popov, using the Feynman functional quantization method. The point is to identify a new selection of the physical subspace which however requires an extension of the indefinite norm Fock space. This is obtained in a perfectly local approach introducing ghost fields compensating the unphysical degrees of freedom.

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The Kugo-Ojima quartet mechanism

The idea of compensating fields, the Kugo-Ojima quartet mechanism (Kugo et al. 1978), is based on the extension to field theory of the following example with a finite number of degrees of freedom. Consider a two-dimensional harmonic oscillator with rotational invariance in the x-y plane. It is easily described introducing the complex canonical coordinates $z=(x+iy)/ \sqrt{2}$ and $P=(p_x-ip_y)/ \sqrt{2}$ together their complex conjugate ( adjoint ) coordinates $\bar z$ and $\bar P$ . The Hamiltonian is

$H_B= \bar P P+ \omega^2 \bar z z \ .$

This system is quantized through the canonical commutation relations.The coordinates satisfy the harmonic oscillator equation. An explicit representation of the quantized system is usually given in the framework of the second quantization of bosons formalism through the annihilation operators $A$ and $\bar A$ and their adjoint creation operators. The Hamiltonian is given by : $H_B= \omega (A^{ \dagger} A + \bar A^{ \dagger} \bar A +1) \ ,$ its spectrum is given by $E_{(n,m)} = \omega(n+m+1)$ with $n,m=0,1,2, \dots, \infty$ . The ground state $|0 \rangle$ is usually called and will be called in the following the [[vacuum state]] understanding that it belongs to a Fock space.

One might wonder at quantizing a system with the same Hamiltonian and the same equations of motion replacing canonical commutators by anti-commutators[2]. We call this new system fermionic oscillator while the former will be called bosonic oscillator. Building the fermionic oscillator we introduce pairs of coordinates $\zeta \ , \ \bar \zeta$ and momenta $\Pi \ , \ \bar \Pi$ , and the Hamiltonian

$H_F= \bar \Pi \Pi+ \omega^2 \bar \zeta \zeta$

assuming canonical anti-commutation relations. These imply that canonical coordinates correspond to nilpotent operators.

In the framework of the second quantization of fermions formalism the Hamiltonian is written in terms of the annihilation operators $a$ and $\bar a$ and their adjoint creation operators

$H_F= \omega[ \bar a^{ \dagger} \bar a+a^{ \dagger} a -1]$

its spectrum is given by $E_{(n,m)} = \omega(n+m-1)$ with $n,m=0,1$ .

It is clear that these relations and Hamiltonian induce for the coordinate operators in the Heisenberg picture the same equations of motion as above. It is also fairly obvious that the canonical coordinates, being nilpotent, cannot correspond to Hermitian operators in the usual sense. Indeed nilpotent operators have only null eigenvalues and therefore they vanish if they are Hermitian.

The bosonic and the fermionic oscillators can be combined into a super-oscillator, that we call B-F oscillator, with Hamiltonian

$H_{B-F}= H_B+H_F= \omega(A^{ \dagger} A + \bar A^{ \dagger} \bar A + \bar a^{ \dagger} \bar a+a^{ \dagger} a)\ .$

The resulting theory is invariant under transformations interchanging bosonic and fermionic degrees of freedom generated by the BRST operator (also called BRST charge)

$Q=i( \bar A^{ \dagger} \bar a- a^{ \dagger} A) \ .$

This operator is nilpotent $Q^2=0 \ ,$ and is not Hermitian in the Fock-Hilbert space. This model has a supersymmetry generated by $Q$ and its adjoint $Q^\dagger$ . However in this in the Fock-Hilbert space there is compensation of degrees of freedom.

There is alternative possibility. One asks for $Q$ to be ‘’Pseudo-Hermitian’’ and interprets the vector space of states as an indefinite norm space. With this choice the full state space is an indefinite norm space and hence it is not compatible with conventional probabilistic interpretations of the scalar (pseudo-inner) product. /Indefinite Metric and BRST Cohomology

However one can recover a physically consistent interpretation by limiting oneself to a physical subspace of the indefinite norm Fock space as long as:

This subspace has a semi-positive definite norm.
It coincides with a super-selection sector.

If $Q$ commutes with all the observables, its kernel , $ker \ Q$ , is a super-selection sector. However, since $Q$ is nilpotent, $ker \ Q$ contains $im \ Q$ , the image of $Q$ , which is pseudo-orthogonal to $ker \ Q$ as $Q$ is pseudo-Hermitian.

What is the physical meaning of states pseudo-orthogonal to the rest of $ker \ Q$ such as those in $im \ Q$ .
Whether the states in $ker \ Q$ have non-negative norm.

One proves that the quotient space $H_{phys}= ker \ Q/im \ Q$ is a Hilbert space which is known as the BRST cohomology space and it is naturally identified with the space of physical states.

The choice of different states in $ker \ Q$ belonging to the same equivalence class should be considered analogous to a " gauge transformation equivalence " in the Gupta-Bleuler quantization of electrodynamics and hence physically irrelevant.

Notice that $H_{phys}$ for the B-F oscillator is [[Dimension (vector space)|one-dimensional]. This means that we have a ‘’complete compensation of degrees of freedom’’ and this model is physically trivial.

A less trivial system can be obtained if one couples our model to some further mechanical system whose simplest choice is a bosonic one-dimensional, physical, harmonic oscillator with Hamiltonian

$h={1 \over2}[p^2+ \Omega^2 x^2]= \Omega (A_P^{ \dagger}A_P +{1 \over2}) \ .$

Now the space of states is enlarged to the tensor product of the B-F Fock space with that of the physical oscillator. This is an indefinite norm space with the metric induced by $J$ . In the enlarged Fock space $ker \ Q/im \ Q$ , coincides with the equivalence classes of the vector space spanned by $(A_P^{ \dagger})^n|0 \rangle$ for any $n \geq 0$ . Thus $H_{phys}$ is equivalent to the Hilbert-Fock space of the physical oscillator.

Let us now couple the physical oscillator with the former B-F oscillator through the e.g. interaction

$V=i \lambda \{Q,x \bar z \zeta \} \equiv \lambda \ x(z \bar z+ \bar \zeta \zeta)$

which is pseudo-Hermitian and, as any anticommutator $\{Q,X \}$ , commutes with $Q$ .

Considering the dynamics induced by the complete Hamiltonian

$H_C=H_{B-F}+h+V$

in the enlarged Fock space, the bosonic and fermionic degrees of freedom of the two dimensional B-F oscillator couple to the physical oscillator and hence their unphysical dynamics is affected by $V$ . On the contrary the dynamics in $H_{phys}$ is not affected by the coupling $V$ since all the matrix elements $\langle p |V|p' \rangle$ with $p \ , \ p' \in H_{phys}$ vanish. Therefore the dynamics of the coupled physical oscillator in $H_{phys}$ is perfectly equivalent to that of the uncoupled physical oscillator and we can conclude that bosonic and fermionic degrees of freedom of the F-B oscillator compensate each other.

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The quartet mechanism in quantum field theory

This compensation mechanism is naturally extended to quantum field theory.

In order to give a simple example, let us consider a neutral scalar field $\Phi$ interacting with a couple of complex scalar un-physical fields $\phi$ and $\psi$ quantized with opposite statistics.

Let the Lagrangian density be

$L=L ( \Phi)+ \partial_ \mu \bar \phi \partial^ \mu \phi-m^2 \bar \phi \phi + \partial_ \mu \bar \psi \partial^ \mu \psi-m^2 \bar \psi \psi+g \Phi \left( \bar \phi \phi + \bar \psi \psi \right)$

Once again, in view of a perturbation expansion in powers of $g$ , one begins quantizing the free ( $g=0$ ) theory decomposing the fields into normal modes in analogy with the example of Quantum Electrodynamics discussed above. In the present case, in correspondence with any momentum $\vec k$ , one finds a system of four oscillators associated with the fields $\phi \ , \bar \phi \ , \psi \$ and $\bar \psi$ and a further oscillator associated with the field $\Phi$ . For any $\vec k$ the system of normal modes of the first four oscillators turns out to be perfectly equivalent to the B-F oscillator described above, while that associated with $\Phi$ corresponds to the physical oscillator above. Let the invariantly normalized asymptotic annihilation operators of the un-physical fields $\phi \ , \bar \phi \ , \psi \$ and $\bar \psi$ be $A( \vec k), \bar A( \vec k), a( \vec k)$ and $\bar a( \vec k)$ , one can define a nilpotent BRST operator

$Q=i \int_{k_0= \sqrt{| \vec k|^2 +m^2}} {d \vec k \over 2 k_0}( \bar A^{ \dagger}( \vec k) \bar a( \vec k)-A( \vec k) a^{ \dagger}( \vec k)) \ .$

The crucial point to be proven in order to apply the BRST formalism to this example is the conservation of $Q$ . This requires a lengthy analysis of the commutator $[Q,H]$ , where $H$ is the Hamiltonian operator corresponding to the Lagrangian density we have started from. There is however a kind of short-cut through this analysis which is based on the fact that, if one interprets $Q$ as the generator of field transformations, these are symmetry transformations and leave invariant the Hamiltonian.

Studying symmetries in relativistic quantum field theory it is convenient, due to Lorentz invariance, to consider the action $S= \int d^4x \ L(x)$ instead of the Hamiltonian. The action is a functional of the fields understood as vector valued functions on the four-dimensional space-time. Its role in quantum field theory is due to the fact that its exponential $\exp(iS) \ ,$ evaluated on suitable classical solutions of field equations gives the eikonal approximation to the relativistic scattering amplitudes. Renormalization theory gives the quantum corrections to the scattering amplitudes.

Thus in a relativistic framework it is natural to introduce a conserved charge as an operator on field functionals leaving invariant the classical action. Then it is the task of renormalization theory to extend this property to all orders of the perturbation. The BRST symmetry renormalization is now a completely settled problem, but its analysis is far from the aims of the present note. For obvious reasons we must remain at the semiclassical level, the starting point of renormalization theory, in which field equations coincide with their classical approximation.

In the present case the classical fields generate an exterior algebra, due to their partial anticommutativity. The fields $\Phi$ , $\phi(x)$ and $\bar \phi(x)$ are commuting variables, while $\psi(x)$ and $\bar \psi(x)$ are anticommuting variables, that is odd elements of the exterior algebra. The operator $Q$ can be interpreted as the generator of field transformations which can be written in the form

$\delta \ \bar \phi(x)= \epsilon Q \bar \phi(x)= \epsilon \bar \psi(x) \quad \ , \quad \delta \ \psi(x)= \epsilon Q \psi(x)= \epsilon \phi(x)$

the other fields remaining unchanged. The parameter $\epsilon$ should be considered anticommuting, that is an odd element of the exterior field algebra. Introducing the field functional derivative one has an alternative definition of $Q$ given by

$Q=-i \int d^4x \left[ \bar \psi(x) { \delta \over \delta \bar \phi(x)} + \phi(x) { \delta \over \delta \psi (x)} \right] \ .$

It is apparent that $Q$ is nilpotent due to the odd character of the fields $\psi$ and $\bar \psi \ , \$ .

Now our purpose is to verify that $Q$ annihilates the classical action : $Q \int dx L(x)=0 \ .$ This follows from the equation

$L= L( \Phi)+ i \ Q [ \partial_ \mu \bar \phi \partial^ \mu \psi-m^2 \bar \phi \psi+g \Phi \bar \phi \psi]$

and from the nilpotent character of $Q$ .

Then $Q$ , defined as an operator on the asymptotic Fock space, is Pseudo-Hermitian, conserved and nilpotent and one can straightforwardly repeat the analysis of the physical content of the coupled oscillators. One selects in the whole indefinite norm Fock space the kernel of $Q$ which is a super-selection sector since $Q$ is conserved. The physical state-space is then identified with the linear set of equivalence classes $ker \ Q/im \ Q$ in whole indefinite norm Fock space. This is a Hilbert space since it coincides with the set of equivalence classes of the states the asymptotic Fock space of the field $\Phi$ where the metric operator is acts as the identity.

Notice that in the perturbation theory based on Feynmam diagrams, the $n-{ \Phi}$ couplings induced by the unphysical fields are given by the combined amplitudes corresponding to one loop diagrams built with either $\phi$ or $\psi$ internal lines, that is by the vacuum expectation values

$g^n \langle 0|T \left( \left( \bar \phi(x_1) \phi(x_1) + \bar \psi(x_1) \psi(x_1) \right) \dots \left( \bar \phi(x_n) \phi(x_n) + \bar \psi(x_n) \psi(x_n) \right) \right)|0 \rangle,$

they however vanish since the contributions from the fermionic, ghost, fields $\psi$ exactly cancel those from the bosonic ones $\phi$ .

We can thus say that $\phi$ and $\psi$ compensate each other.

Considering this point of view with more care one sees that the Feynman diagram expression for $n-{ \Phi}$ couplings with $n \leq 2$ is ill-defined since the corresponding one-loop diagrams are "divergent". In a more rigorous approach to renormalization they are defined up to additive terms depending on three free coefficients. Therefore the exact cancellation of bosonic and fermionic loop contributions requires the renormalization prescription that the above mentioned free coefficients compensate each other in much the same way as the finite parts of the loop amplitudes.

This is a clear, however simple, example of the role of renormalization in BRST formalism.

A last comment is in order here concerning the B-F oscillator and its field theory extension. Let us introduce in the B-F case the ghost number operator : $N_g \equiv (a^{ \dagger} a- \bar a^{ \dagger} \bar a)/2$ it commutes with $H_c$ and satisfies $[Q,N_g]=Q$ . Thus $N_g$ is conserved and leaves the subspace $ker \ Q$ invariant. This implies that $N_g$ introduces a grading into $ker \ Q$ which appears as the direct sum of eigen-spaces of $N_g$ corresponding to integer eigen-values of $N_g$ ranging from $0$ to $\infty$ . In the present case only the ghost number zero sub-space is of physical interest, the rest of $ker \ Q$ belonging to $im \ Q$ .

The same happens in the field theory extension where : $N_g \equiv \int d^4x \left[ \psi(x) { \delta \over \delta \psi (x)} - \bar \psi(x) { \delta \over \delta \bar \psi(x)} \right]$

The presence of a conserved ghost number defining a grading into the Fock space of states and into that of functionals is a further basic ingredient of BRST formalism.

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Yang-Mills theory in BRST formalism

We now apply what we have learned from the former toy models to the Yang-Mills theory in the Landau gauge . We start from the asymptotic free theory ( $g=0$ ) which corresponds to the sum of the free Landau Lagrangian densities depending on the fields $A^a_ \mu$ and $b^a$ for $a=1, \dots, N$ . Following the compensating field strategy we introduce two more sets of odd scalar ghost fields, that we label by $c^a(x)$ and $\bar c^b(x)$ . Proceeding in strict analogy with the B-F oscillator we add to the Lagrangian density the compensating term $- \bar c^a \partial^2 c^a$ .

The normal mode decomposition of the ghost field gives

$c^a(x)={1 \over(2 \pi)^{3/2}} \int_{k_0= | \vec k|} {d \vec k \over 2 k_0}( \gamma^a( \vec k)e^{-ik \cdot x}+ \gamma^a( \vec k)^+( \vec k)e^{ik \cdot x})$

and for the anti-ghost

$\bar c^a(x)={1 \over(2 \pi)^{3/2}} \int_{k_0= | \vec k|} {d \vec k \over 2 k_0}( \bar \gamma^a( \vec k)e^{-ik \cdot x}- \bar \gamma^a( \vec k)^+( \vec k)e^{ik \cdot x}) \ .$

We introduce the anti-commutation rules for the coefficients

$\ \{ \gamma^a( \vec k ),( \bar \gamma^b)^+( \vec q) \}= \{ \bar \gamma^a( \vec k ),( \gamma^b)^+( \vec q) \}=2k_0 \delta^{ab} \delta( \vec k- \vec q)$

which, together with the Landau commutation rules

$\ [ \alpha^a( \vec k ),( \beta^b)^+( \vec q)]=[ \beta^a( \vec k ),( \alpha^b)^+( \vec q)]=2k_0 \delta( \vec k- \vec q) \ ,$

and those for $a^a( \vec k,h)$ , complete the canonical quantization prescriptions of the asymptotic theory. Then we introduce the BRST operator $Q$

$Q=-i \int_{k_0= | \vec k|} {d \vec k \over 2 k_0}(( \beta^a)^+( \vec k) \gamma^a( \vec k)-( \gamma^a)^+( \vec k) \beta^a( \vec k)) \ .$

In the functional formalism one finds

$Q=-i \int d^4x( \partial_ \mu c^a(x){ \delta \over \delta A^{a, \mu}(x)}-b^a(x){ \delta \over \delta \bar c^a(x)}) \ .$

It is apparent that $Q$ annihilates the action functional corresponding to the Lagrangian density

$L_0 =-{1 \over2} \partial^ \mu A^{a \nu} \left( \partial_ \mu A^a_ \nu- \partial_ \nu A^a_ \mu \right)-iQ( \bar c^a \ \partial_ \mu A^{a \mu}) \ .$ Indeed the first term in $iQ$ is just the generator of an infinitesimal abelian gauge transformation $A^a_ \mu \to A^a_ \mu+ \partial_ \mu c^a$ which leaves invariant the first term in $L_0$ , while the second term in $L_0$ is annihilated by $Q$ which is nilpotent.

Therefore, for what concerns the asymptotic theory, we can say that $ker \ Q/im \ Q$ is a Hilbert physical state corresponding to the Fock space of gluons with helicity $\pm1$ .

However we remain with the problem of extending this result to the fully interacting theory. Indeed the space time integral of the Lagrangian density

$L = -{1 \over4}G^{a \mu \nu}G^{a}_{ \mu \nu}-iQ( \bar c^a \ \partial_ \mu A^{a \mu}) \ .$

with $G^{a}_{ \mu \nu}= \partial_ \mu A^a_ \nu- \partial_ \nu A^a_ \mu+ g f^{abc} A^b_ \mu A^c_ \nu \ ,$ is not annihilated by $Q$ since $-{1 \over4}G^{a \mu \nu}G^{a}_{ \mu \nu}$ is not left invariant by the above abelian gauge transformation.

The natural solution to this problem consists in replacing at the fully interacting level the abelian generator with the non-abelian one, that, in the present situation, is

$X(c) \equiv \int d^4x \left( \partial_ \mu c^c(x)+ g f^{abc}c^a(x) A^b_{ \mu}(x) \right){ \delta \over \delta A^c_ \mu (x)}= \int d^4x D_ \mu c^c(x) { \delta \over \delta A^c_ \mu (x)}$

thus deforming the BRST operator according to

$Q \to-i( X(c)- \int d^4x \ b^a(x) \ { \delta \over \delta \bar c^a(x)})$

As a matter of fact $-{1 \over4}G^{a \mu \nu}G^{a}_{ \mu \nu}$ is gauge invariant and hence it is annihilated by $X(c)$ . However our problem is not yet solved since $X(c)$ , however anticommuting with the second term in $Q$ which is nilpotent, is not nilpotent. Indeed, due to the non abelian character of the gauge transformations the generators $X^a(x)$ satisfy non-trivial commutation relations

$\left[ X^a(x) , X^b(y) \right]= \delta(x-y)gf^{abc}X^c(y) \ ,$

and hence

$X^2(c)=X(c \wedge c/2)$ where : $(c \wedge c)^a (x)=g f^{abc}c^b(x) c^c(x)$

It follows that the operator

$D(c) \equiv X(c)-{g \over2} \int d^4xf^{abc}c^a(x) c^b(x) { \delta \over \delta c^c(x)} \ ,$

is nilpotent : $D^2(c)=0 \ ,$ and, of course, annihilates $-{1 \over4}G^{a \mu \nu}G^{a}_{ \mu \nu}$ since $X(c)$ does. As a matter of fact the translation of the functional formalism into a more rigorous Mathematical framework leads to the identification of $D(c)$ with the coboundary operator of Chevalley gauge Lie algebra cohomology[3].

We have thus identified the nilpotent BRST operator $Q$ of the fully interacting theory: $Q = D(c) + \int d^4y \ b^a(y){ \delta \over \delta \bar c ^a(y)} \ ,$ which, at the semiclassical level is conserved and nilpotent. The Faddeev-Popov Lagrangian density corresponds to the above expression built with the full $Q$ . One immediately identifies the original gauge invariant Yang-Mills term and the gauge-fixing term which implements the compensation mechanism thanks to the nilpotent action of $Q$ .

It is an important remark at this point that changing the $Q$ -trivial term in the Lagrangian density, that is $-iQ( \bar c^a \ \partial_ \mu A^{a \mu})$ , into a different term with the same structure and ghost number, e.g. $-iQ( \bar c^a \ ( \partial_ \mu A^{a \mu}+ \xi b^a)$ , corresponds to a change in the un-physical part of the asymptotic Fock space, but does not change the physical results. Indeed e.g. a change of $\xi$ acts on the asymptotic states within their equivalence class.

Our construction is directly transfered to the full quantum level through renormalization theory. This is however not a trivial result.

It should appear clearly that our construction has a direct generalization to the quantization of Lagrangian systems with a Lie algebra of first class constraints and, in particular, to any gauge field theory. Further generalizations are possible, e.g. to cases in which the algebra of constraints is not a Lie algebra, since its structure constants are field dependent, however there is no room left here for them.

Let us conclude with a short comment on the relation between the BRST construction and gauge invariance. As we have seen the covariant quantization of gauge fields presents the problem of the compensation of un-physical degrees of freedom. This problem is solved by the Kugo-Ojima quartet compensation mechanism, which not related to any gauge symmetry, but is based on a super-selection rule induced by the BRST operator $Q$ . Finally the existence of this BRST operator in the fully interacting theory relies on the gauge invariance of the theory.

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Invited by:	Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the free peer reviewed encyclopedia
Invited by:	Dr. Riccardo Guida, Institut de Physique Théorique, CEA, IPhT, F-91191 Gif-sur-Yvette, France; CNRS, URA 2306, F-91191 Gif-sur-Yvette, France

Becchi-Rouet-Stora-Tyutin symmetry

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Maxwell's equations and Dirac's quantization

Landau's gauge quantization of Electrodynamics

Analysis of the Yang-Mills model

The Kugo-Ojima quartet mechanism

The quartet mechanism in quantum field theory

Yang-Mills theory in BRST formalism

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