Becchi-Rouet-Stora-Tyutin symmetry
Carlo Maria Becchi and Camillo Imbimbo (2008), Scholarpedia, 3(10):7135. | doi:10.4249/scholarpedia.7135 | revision #137143 [link to/cite this article] |
BRST Symmetry and the associated concept of BRST cohomology provide the most used covariant quantization method for constrained canonical systems such as gauge and string theories. These are systems quantized in indefinite-metric vector spaces. The positive norm physical states appear as cohomology classes of the BRST symmetry generator.
History
The BRST construction is the last episode of the long history of the quantization of fields and, 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 and characterized by two polarization states. When Dirac and many others tried to quantize the electromagnetic field using the canonical commutation relations they encountered a fundamental difficulty. Canonical quantization requires a Lagrangian which is a function of the four-vector potential. 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, its normal decomposition would seem to lead to four different polarization states rather than the actual two. The reason why this is not the case is that the Lagrangian of quantum electrodynamics induces constraints. Dirac and Bergmann developed a theory of quantum constrained systems (Sundermeyer K, 1982). In gauge theories, such as quantum electrodynamics and its generalizations, the system of constraints generates a Lie algebra and it is called a system of first class constraints. However in the physically interesting cases, that is in quantum electrodynamics, in its non-abelian Yang Mills theory extensions which include the Standard Model of Fundamental Interactions and in General Relativity, this approach leads to a loss of explicit Lorentz covariance and locality (section 1): this constitutes a major problem for carrying out renormalization.
To avoid loosing explicit Lorentz covariance and locality, one introduces covariant supplementary conditions on the potentials, called gauge fixing conditions which reduce the number of degrees of freedom. In quantum electrodynamics, for a fortuitous reason, the simplest choice of covariant gauge fixing conditions are dynamically stable due to current conservation. It is hence possible to quantize the theory by covariant and local canonical prescriptions albeit abandoning the Hilbert space character of the quantum state vector space. This space, as shown by Gupta and Bleuler, becomes an indefinite-metric space (section 2). The relation between the stability of the gauge fixing and current conservation is made explicit by the Ward-Takahashi identities.
This lucky condition does not persist in the extension of quantum electrodynamics to non-abelian Yang-Mills theory and to its other generalizations (section 3). Therefore one cannot use for these theories the same definition of physical quantum state vector space as in quantum electrodynamics. After many efforts the problem was solved by Faddeev and Popov (Faddeev LD and Popov VN,1967) within the framework of Feynman's path integral construction of field theory; they introduced a system of unphysical ghost fields which allow the insertion of a needed Jacobian matrix into the functional integral. In the state space framework these ghosts play the role of compensating degrees of freedom and the BRST symmetry is crucial to ensure this compensation (section 4). Building on Faddeev and Popov's construction, 't Hooft and Veltman ('t Hooft G and Veltman M, 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 led Becchi, Rouet and Stora (Becchi C, Rouet A and Stora R, 1974 and 1976) and, independently, Tyutin to identify the property which guarantees at the quantum level 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 T and Ojima I, 1978 a, b), 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.
Another important application of the BRST construction, introduced by Witten, leads to topological field theory.
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 appearance of Gribov copies (Gribov VN, 1978).
Maxwell's equations and Dirac's quantization
When dealing with relativistic quantum field theory it is customary to use a unit system in which \( \hbar=c=1\ .\) In the following the Minkowski metric with the sign choice \(\eta_{00}=1\) is adopted and the sum over repeated indices is understood.
The original historical problem is the quantization of electrodynamics. Maxwell's equations written in terms of the covariant field tensor are \[\tag{1} \partial_ \nu F^{ \mu \nu}=j^ \mu \quad \ , \quad \epsilon^{ \mu \nu \rho \sigma} \partial_ \nu F_{ \rho \sigma}=0 \]
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. The Lagrangian density of electrodynamics 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 of Maxwell's equations (1), while the equations associated to the tensor field lead to the others. 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 of 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 vector 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.
Landau's gauge quantization of electrodynamics
Landau's gauge quantization of quantum electrodynamics was suggested by Landau and refined by Nakanishi (Nakanishi N, 1974). It produces a quantum structure whose extension to non-abelian gauge theories gives a natural framework for the BRST construction.
In the following the quantization of quantum electrodynamics in Landau's gauge, a subject not particularly popular in textbooks, is sketched.
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 equations of motion of the charged matter fields.
The corresponding Euler-Lagrange equations are \[ \partial_ \mu A^ \mu(x)=0 \quad \ , \quad \partial_ \nu \partial^ \nu A_ \mu (x) \equiv \partial^2 A_ \mu (x)=j_ \mu (x)+ \partial_ \mu b(x) \quad \ , \quad \partial^2 b(x)=0 \ . \] The 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) and 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, which are obtained by taking \(j_ \mu(x)=0\ .\) In the free case this decomposition is based on the Fourier analysis of the solutions.
In relativistic quantum field theory free quantum fields play the physical role of asymptotic fields. Asymptotic fields are built with 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 known as the in and out fields. In Feynman's covariant perturbation theory one identifies the free quantized fields with the in-fields. Haag (Haag R, 1992) 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, respectively, in the far past and in the far future.
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 states with helicity \(h= \pm 1\ ;\) \( \alpha( \vec k)\) corresponds to the longitudinally polarized states and \( \beta( \vec k)\) is the Fourier transform of \(b\) Appendix: Landau’s_gauge_free_solutions.
In the quantized system the vector potential components 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 the general solution, translate into a set of commutation relations whose non-trivial part is \[[a( \vec k, h),a^+( \vec q,h')]=2|\vec k| \delta_{h,h'} \delta( \vec k- \vec q) \ \ , \ [ \alpha( \vec k ), \beta^+( \vec q)]=[ \beta( \vec k ), \alpha^+( \vec q)]=2|\vec k| \delta( \vec k- \vec q)\ . \] Note that the factor \(2 |\vec k| \) corresponds to a Lorentz invariant normalization of the operators. This normalization will be adopted in the following whenever dealing with field theory.
The second commutation prescription forbids the identification of \( \alpha^+( \vec k)\) and \( \beta^+( \vec k)\) with creation operators in a Fock-Hilbert space. This identification is natural both for \(a^+( \vec k, h)\) and for \(( \alpha^+( \vec k)+ \beta^+( \vec k))/ \sqrt{2}\) which satisfies the standard commutation rule. However, for \(( \alpha( \vec k)- \beta( \vec k))/ \sqrt{2}\) one obtains a commutator with the wrong sign. This means that if we insist, as we must do, on interpreting \(( \alpha^+( \vec k)- \beta^+( \vec k))/ \sqrt{2}\) as a creation operator, it creates negative norm states: hence the space underlying the quantization is not a Fock-Hilbert space but a vector space with indefinite inner product. Further details on such spaces can be found in Appendix: Indefinite Metric and BRST Cohomology where, in particular, the definition of the pseudo-adjoint is recalled. In the following the Fock-Hilbert adjoint is denoted by the dagger apex and the pseudo-adjoint by the \(+\) apex.
The appearance of negative norm states contrasts with the usual probabilistic interpretation of the inner product in Quantum Mechanics. The Gupta-Bleuler way out of this paradox consists in identifying the physical state vector space with the subset of the Fock space annihilated by \( \beta( \vec k)\ .\) This choice is justified by the fact that \(b(x)\) is a free field and hence the prescription selects a subspace of the Fock space which is physically invariant, i.e. invariant under the action of the observables. The physical invariant subspace is spanned by states generated by polynomials of \(a^+( \vec k,h)\) which are positive norm states and mixed polynomials of \(a^+( \vec k,h)\) and \( \beta^+( \vec k)\) which are 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 to generic gauge variations of the physical states.
Analysis of the Yang-Mills model
The situation is completely different for non-abelian models such as Yang-Mills theories. In this case the field analogous to \(b\) is not a free field. 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 Lie group. The field equations are \[ \partial^2A^a_ \mu +gf^{abc} A^{b \nu} \left( \partial_ \nu A^c_ \mu- \partial_ \mu A^c_ \nu \right) -g^2f^{abc}f^{cde} A^d_ \mu A^{b \nu} A^e_ \nu= \partial_ \mu b^a \]
\[ \partial_ \mu A^{a \mu}=0 \ . \] These two equations do not imply anymore that \( \partial^2 b^a=0\) except when \(g=0\ .\) In this limit the above Lagrangian reduces to the sum of \(N\) pure quantum electrodynamics Lagrangians.
As recalled above, in perturbation theory one starts from the free theory describing the quantum mechanics of the asymptotic scattering states. Thus the space of asymptotic in-going states of Yang-Mills theory consists of the tensor product of \(N\) indefinite-metric Fock spaces analogous to that of quantum electrodynamics. The difficulty arises when one tries to identify a physical subspace with semi-definite norm. The natural extension of the Gupta-Bleuler choice valid for quantum electrodynamics does not identify a physically invariant subspace of the non-abelian theory since \(b^a\) is not a free field.
The solution to this difficulty was found by Faddeev and Popov, using the Feynman functional quantization method. The point they made is that the correct identification of the physical subspace requires an extension of the indefinite-metric space of states. This is obtained in a perfectly local approach by introducing ghost fields which compensate the unphysical degrees of freedom.
The Kugo-Ojima quartet mechanism
The idea of compensating fields, the Kugo-Ojima quartet mechanism, 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 described by introducing the complex canonical coordinates \(z=(x+iy)/ \sqrt{2}\) and \(P=(p_x-ip_y)/ \sqrt{2}\) together their complex conjugate coordinates \( \bar z\) and \( \bar P\ .\) The Hamiltonian is \[H_B= \bar P P+ \omega^2 \bar z z \ . \] This system is quantized through canonical commutation relations. The coordinates satisfy the harmonic oscillator equation. An explicit representation of the quantized system is usually given by means of the annihilation operators \(A\) and \( \bar A\) and their adjoint creation operators, as in the formalism of second quantization of bosons. The Hamiltonian of the bosonic oscillators writes as follows \[H_B= \omega (A^{ \dagger} A + \bar A^{ \dagger} \bar A +1) \ , \] 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 identified with the vacuum state of a Fock space.
One can quantize a system with the same Hamiltonian and the same equations of motion by replacing canonical commutators with anticommutators . Such a system will be named fermionic oscillator. To describe it one introduces pairs of coordinates \( \zeta \ , \ \bar \zeta \) and momenta \( \Pi \ , \ \bar \Pi\ ,\) the Hamiltonian \[H_F= \bar \Pi \Pi+ \omega^2 \bar \zeta \zeta \] and assumes canonical anticommutation relations. These imply that fermionic canonical coordinates are nilpotent operators.
In the formalism of second quantization of fermions 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 \(n,m=0,1\ .\)
Such an Hamiltonian and the canonical anticommutation relations induce the same equations of motion for the fermionic coordinate operators in the Heisenberg picture as those of the bosonic oscillator. The canonical coordinates, being nilpotent, cannot correspond to Hermitian operators. Indeed nilpotent operators have only null eigenvalues and therefore vanish if Hermitian.
The bosonic and the fermionic oscillators can be combined into a super-oscillator, the B-F oscillator, whose Hamiltonian is \[H_{B-F}= H_B+H_F= \omega(A^{ \dagger} A + \bar A^{ \dagger} \bar A + \bar a^{ \dagger} \bar a+a^{ \dagger} a)\ . \] 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 charge is nilpotent
\[ Q^2=0\, \]
and it is not Hermitian in the Fock-Hilbert space. The corresponding Hermitian conjugate charge
\[ Q^\dagger=i(A^\dagger a-\bar a^\dagger \bar A)\ \]
is also nilpotent and conserved. \( Q \) and \(Q^\dagger\) generate an extended supersymmetry algebra characterized by the anti-commutation relation
\[\tag{2} \{Q,Q^\dagger\}={H\over\omega} \ . \]
From this algebra it follows that the kernel of \( H\ ,\) \(ker \ H \) (i.e. the vector space annihilated by \( H \)), coincides with the intersection of the kernels of \( Q \) and \(Q^\dagger\ .\)
It is apparent that the Fock vacuum \(|0\rangle\) is the only vector of the Fock space of the B-F model invariant under the action of both \( Q \) and \(Q^\dagger\ .\) \( ker\ H\) for this model is therefore one-dimensional and it is generated by the Fock vacuum.
A further consequence of the supersymmetry algebra (2) concerns the structure of \(ker \ Q\ .\) Since \(Q\) is nilpotent, \(ker \ Q\) contains \(im \ Q\ ,\) the image of \(Q\) (i.e. the subspace of vectors of the form \(Q|s\rangle\)). Moreover, the orthogonal complement of \( im\ Q \) in the total Fock-Hilbert space (i.e. the subspace of vectors \(|y\rangle\) such that \(\langle y | Q x\rangle=\langle Q^\dagger y |x\rangle=0 \) for any \(|x\rangle\)) is evidently annihilated by \( Q^\dagger \ .\) Hence, the orthogonal complement of \( im \ Q \) inside \( ker\ Q \) is contained in the intersection of \( ker\ Q \) and \( ker\ Q^\dagger \ .\) It is therefore also contained in \( ker\ H \ .\) Considering that \( ker\ H \) is orthogonal to \( im \ Q \ ,\) we conclude that the kernel \( ker\ Q \) admits the following orthogonal decomposition
\[ ker\ Q = ker\, H \oplus im\ Q \]
This relation implies that the full Hilbert space of states admits the orthogonal decomposition \( ker\, H \oplus im\ Q \oplus im\ Q^\dagger \ ,\) which is the Hodge decomposition associated with the nilpotent operator \( Q\ .\)
An obvious way of disregarding the unwanted excited states of the B-F oscillator is to select supersymmetric invariant vectors, that is, vector states annihilated by both \( Q \) and \( Q^\dagger \ .\) However, in the field theory context, this choice --- which turns out to be equivalent to the Dirac-Bergmann construction --- is an exceedingly rigid constraint which contrasts with explicit Lorentz covariance.
If, on the contrary, we weaken our condition, by selecting the set of the vector states annihilated by only one of the two charges, say \(Q\ ,\) too many unphysical vector states --- the elements of \(im\ Q\) --- survive the selection.
The way to overcome this difficulty is to void \( im\ Q \) of physical content. This is accomplished by introducing an indefinite inner product in the Fock space (see Appendix: Indefinite Metric and BRST Cohomology), relative to which \( Q \) is required to be pseudo-Hermitian. Under this condition, \( im\ Q \) is pseudo-orthogonal to \( ker \ Q \ .\) Indeed for any state \(|i\rangle=Q|t\rangle\) and any state \( |s\rangle \) in the kernel of \( Q\) one has \[ \langle s | i \rangle = \langle s |Q|t \rangle = \langle t|Q|s \rangle ^*=0 \] Hence \(ker\ Q\) is not a Hilbert space. However the quotient space
\(H_{phys}= ker \ Q/im \ Q ,\)
known as the BRST cohomology space, is the space which is naturally identified with the space of physical states. States in \( ker \ Q\) which differ by an element of \( im \ Q\) belong to the same equivalence class. They correspond to the same physical state in \(H_{phys} \ .\) The Hodge decomposition establishes an isomorphism between \(H_{phys} \) and \( ker\ H \ .\)
We have seen above that, in the case of the B-F model, \( ker \ Q \) coincides with the equivalence class of the vacuum state and \( H_{phys} \) is indeed a one-dimensional Hilbert space. This means that a complete compensation of degrees of freedom occurs in this particular model.
A less trivial system can be obtained by coupling the B-F oscillator to some further mechanical system, e.g. to a bosonic one-dimensional, physical, harmonic oscillator with Hamiltonian \[h={1 \over2}[p^2+ \Omega^2 x^2]= \Omega (A_P^{ \dagger}A_P +{1 \over2}) \ . \] The space of states of this model is the tensor product of the B-F Fock space with that of the physical oscillator. This is an indefinite-metric vector space. The physical 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.
One can couple the physical oscillator to the B-F oscillator, e.g. through the interaction \[V=i \lambda \{Q,x \bar z \zeta \} \equiv \lambda \ x(z \bar z+ \bar \zeta \zeta) \] which is pseudo-Hermitian and, being an anticommutator \( \{Q,X \}\ ,\) commutes with \(Q\ .\) The dynamics induced by the complete Hamiltonian \[H_C=H_{B-F}+h+V \] on the full unphysical Fock space is affected by \(V\ .\) However the dynamics in \(H_{phys}\) does not depend on 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 one concludes that bosonic and fermionic degrees of freedom of the F-B oscillator compensate each other.
In QED Kugo-Ojima \( Q\) invariance condition reduces to the Gupta-Bleuler condition in the case of electrodynamics and BRST equivalence is the analogue of the gauge equivalence of the Gupta-Bleuler formalism.
The quartet mechanism in quantum field theory
This compensation mechanism is naturally extended to quantum field theory. As an example, consider a neutral scalar field \( \Phi\) interacting with a couple of unphysical complex, scalar fields \( \phi\) and \( \psi\) quantized with opposite statistics.
Let the Lagrangian density be \[L=L ( \Phi)+ \partial_ \mu \bar \phi \partial^ \mu \phi-m^2 \bar \phi \phi + \partial_ \mu \bar \psi \partial^ \mu \psi-m^2 \bar \psi \psi+g \Phi \left( \bar \phi \phi + \bar \psi \psi \right) \] In view of a perturbation expansion in powers of \(g\ ,\) one begins quantizing the free (\(g=0\)) theory. For each value \( \vec k\) of the spatial momentum, one finds four oscillators corresponding to the fields \( \phi \ , \bar \phi \ , \psi \ \) and \( \bar \psi\) plus one further oscillator associated with the field \( \Phi\ .\) For any \( \vec k\) the system composed of the first four oscillators is equivalent to the B-F oscillator described above, while the oscillator associated with \( \Phi\) plays the role of the physical oscillator.
One defines a nilpotent BRST operator \[ Q=i \int_{k_0= \sqrt{| \vec k|^2 +m^2}} {d \vec k \over 2 k_0}( \bar A^{ \dagger}( \vec k) \bar a( \vec k)-A( \vec k) a^{ \dagger}( \vec k)) \ , \] where \( A\ ,\bar A\) are the annihilation operators of \( \phi\ ,\bar\phi\) while \( a\ ,\bar a\) are those of \( \psi\ ,\bar\psi\ .\)
The BRST formalism requires the conservation of \(Q\ ,\) which can indeed be verified by evaluating the commutator \([Q,H]\ ,\) where \(H\) is the Hamiltonian operator corresponding to the Lagrangian density given above. To short-cut this analysis one observes that \(Q\) is the generator of field transformations which leave the Hamiltonian invariant.
Symmetries in relativistic quantum field theory are conveniently studied by considering the action \[S= \int d^4x \ L(x) \] instead of the Hamiltonian. The action is a functional of the fields.
In quantum field theory the exponential \( \exp(iS)\ ,\) when evaluated on suitable classical solutions of field equations, gives the eikonal approximation to the relativistic scattering amplitudes. Renormalization theory deals with the quantum corrections to the scattering amplitudes. Thus in a relativistic framework it is natural to introduce conserved charges as operators on field functionals which leave invariant the classical action. The task of renormalization theory is extending this property to all orders of perturbation theory. The renormalization of BRST symmetry is, in this framework, a completely well-posed problem, but its analysis goes beyond the scope of this article.
In the present case the classical fields generate a Grassmann 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 Grassmann algebra. The operator \(Q\) can be interpreted as the generator of following field transformations \[ \delta \ \bar \phi(x)= \epsilon Q \bar \phi(x)= \epsilon \bar \psi(x) \quad \ , \quad \delta \ \psi(x)= \epsilon Q \psi(x)= \epsilon \phi(x) \] the other fields remaining unchanged. The parameter \( \epsilon\) should be considered anticommuting.
This leads to the alternative definition of \(Q\) written in terms of the fields functional derivatives \[ Q=-i \int d^4x \left[ \bar \psi(x) { \delta \over \delta \bar \phi(x)} + \phi(x) { \delta \over \delta \psi (x)} \right] \ . \] \(Q\) is nilpotent due to the anticommuting character of the fields \( \psi\) and \( \bar \psi \ \ .\)
The invariance of the classical action \[ Q \int dx L(x)=0 \] 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 \(Q\ .\)
This shows that \(Q\ ,\) defined as an operator on the asymptotic Fock space, is Pseudo-Hermitian, conserved and nilpotent. Therefore the physical content of this theory can be analysed in much the same way as done above for the physical oscillator coupled to the B-F oscillator. The kernel of \(Q\) is physically invariant since \(Q\) is conserved. The physical state space is identified with the linear set of equivalence classes \(ker \ Q/im \ Q \ .\) It is a Hilbert space since it coincides with the set of equivalence classes of the states of the asymptotic Fock space associated to the field \(\Phi\ .\)
Notice that in the perturbation theory based on Feynman diagrams, the \({\Phi}\) self-interactions induced by the unphysical fields correspond to one loop diagrams built with either \( \phi\) or \( \psi\) internal lines. In particular the \({\Phi}^n\) coupling is given by the sum of two \(n\)-vertex polygonal diagrams with sides corresponding either to \( \phi\ ,\) or to \( \psi\ ,\) lines. This sum vanishes for any \( n\) since the contributions from the fermionic fields \( \psi\) exactly cancel those from the bosonic ones \( \phi\ .\) In this sense \( \phi\) and \( \psi\) compensate each other.
Considering this point of view with more care one sees that the Feynman diagram expression for \({\Phi}^n\) couplings with \(n \leq 2\) are ill-defined since the corresponding one-loop diagrams are divergent. In renormalization theory they are defined up to additive terms depending on three free coefficients. Therefore the exact cancellation of bosonic and fermionic loop contributions requires a renormalization prescription for which the above mentioned free coefficients compensate each other.
This is a clear, however simple, example of the role of renormalization in the BRST formalism.
One last basic ingredient of the BRST construction is the ghost number operator. For the B-F oscillator this is defined as follows \[N_g \equiv (a^{ \dagger} a- \bar a^{ \dagger} \bar a)/2 \] \(N_g\) commutes with \(H_C\) and satisfies \([Q,N_g]=Q\ .\) Thus it is conserved and it leaves the subspace \(ker \ Q\) invariant. This implies that \(N_g\) induces a grading on \(ker \ Q\) which decomposes into the direct sum of eigenspaces of \(N_g\) corresponding to integer eigenvalues of \(N_g\) ranging from \(0\) to \( \infty\ .\) For this model only the ghost number zero subspace is of physical interest, the rest of \(ker \ Q\) belonging to \(im \ Q\ .\)
The ghost number operator for the field theory extension is an operator acting on field functionals: \[ N_g \equiv \int d^4x \left[ \psi(x) { \delta \over \delta \psi (x)} - \bar \psi(x) { \delta \over \delta \bar \psi(x)} \right] \ . \]
Yang-Mills theory in BRST formalism
One starts 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 one introduces two more sets of anticommuting scalar fields: the ghost field \(c^a(x)\) and the anti-ghost field \( \bar c^b(x)\ .\) Proceeding in strict analogy with the B-F oscillator one adds to the Lagrangian density the compensating term \(- \bar c^a \partial^2 c^a\ .\)
The normal mode decompositions of the ghost and anti-ghost fields are \[ c^a(x)={1 \over(2 \pi)^{3/2}} \int_{k_0= | \vec k|} {d \vec k \over 2 k_0}( \gamma^a( \vec k)e^{-ik \cdot x}+ (\gamma^a)^+( \vec k)e^{ik \cdot x}) \] and \[ \bar c^a(x)={1 \over(2 \pi)^{3/2}} \int_{k_0= | \vec k|} {d \vec k \over 2 k_0}( \bar \gamma^a( \vec k)e^{-ik \cdot x}- (\bar\gamma^a)^+( \vec k)e^{ik \cdot x}) \ . \] The anticommutation rules for the coefficients \[ \ \{ \gamma^a( \vec k ),( \bar \gamma^b)^+( \vec q) \}= \{ \bar \gamma^a( \vec k ),( \gamma^b)^+( \vec q) \}=2|\vec k| \delta^{ab} \delta( \vec k- \vec q) \] together with the Landau commutation rules \[ \ [ \alpha^a( \vec k ),( \beta^b)^+( \vec q)]=[ \beta^a( \vec k ),( \alpha^b)^+( \vec q)]=2|\vec k| \delta( \vec k- \vec q) \ , \] and those for \(a^a(\vec k,h)\) complete the canonical quantization prescriptions of the asymptotic theory. The BRST operator \(Q\) is: \[ Q=-i \int_{k_0= | \vec k|} {d \vec k \over 2 k_0}(( \beta^a)^+( \vec k) \gamma^a( \vec k)-( \gamma^a)^+( \vec k) \beta^a( \vec k)) \ . \] or, in the functional formalism, \[Q=-i \int d^4x( \partial_ \mu c^a(x){ \delta \over \delta A^{a, \mu}(x)}-b^a(x){ \delta \over \delta \bar c^a(x)}) \ . \] It is apparent that \(Q\) annihilates the action functional corresponding to the Lagrangian density \[\tag{3} L_0 =-{1 \over2} \partial^ \mu A^{a \nu} \left( \partial_ \mu A^a_ \nu- \partial_ \nu A^a_ \mu \right)-iQ( \bar c^a \ \partial_ \mu A^{a \mu}) \ . \]
Indeed the first term in \(iQ\) is 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 the nilpotent \(Q\ .\)
Therefore, for what concerns the asymptotic theory, \( ker \ Q/im \ Q\) is a Hilbert physical state corresponding to the Fock space of gluons with helicity \( \pm1\ .\)
One is left with the problem of extending this result to the fully interacting theory. The space time integral of the Lagrangian density \[\tag{4} L = -{1 \over4}G^{a \mu \nu}G^{a}_{ \mu \nu}-iQ( \bar c^a \ \partial_ \mu A^{a \mu}) \ . \]
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 invariant under the abelian gauge transformations.
The natural solution to this problem consists in replacing, at the interacting level, the abelian generator with the non-abelian one which reads \[X(c) \equiv \int d^4x \left( \partial_ \mu c^c(x)+ g f^{abc}c^a(x) A^b_{ \mu}(x) \right){ \delta \over \delta A^c_ \mu (x)}= \int d^4x D_ \mu c^c(x) { \delta \over \delta A^c_ \mu (x)} \equiv \int d^4x c^a(x) X^a(x) \] The BRST operator is deformed accordingly \[Q \to-i( X(c)- \int d^4x \ b^a(x) \ { \delta \over \delta \bar c^a(x)}) \] \(-{1 \over4}G^{a \mu \nu}G^{a}_{ \mu \nu}\) is gauge invariant and hence is annihilated by \(X(c)\ .\) The second term in \(Q\) is nilpotent and \(X(c)\) anticommutes with it. However one is not finished yet, since \(X(c)\ ,\) is not nilpotent. Indeed, due to the non-abelian character of the gauge transformations, the generators \( X^a(x)\) satisfy non-trivial commutation relations \[ \left[ X^a(x) , X^b(y) \right]= \delta(x-y)gf^{abc}X^c(y) \ . \] Therefore \[X^2(c)=X(c \wedge c/2)\] 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 \[ D^2(c)=0\ . \] \( D(c)\) annihilates \(-{1 \over4}G^{a \mu \nu}G^{a}_{ \mu \nu}\) since \(X(c)\) does it. In mathematics \( D(c)\) is identified with the coboundary operator of Chevalley gauge Lie algebra cohomology.
In conclusion the nilpotent BRST operator \(Q\) of the fully interacting theory is \[Q = -i (D(c) - \int d^4y \ b^a(y){ \delta \over \delta \bar c ^a(y)}) \ , \] which, at the semiclassical level, is both conserved and nilpotent. The Faddeev-Popov Lagrangian density is given by equation (4) in which \(Q\) is the interacting one. The first term is the original gauge invariant Yang-Mills term; the second \(Q\)-trivial term is the gauge-fixing term implementing the compensation mechanism which relies on the nilpotency of \( Q\ .\)
Replacing the \(Q\)-trivial term in the Lagrangian density with 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 unphysical part of the asymptotic Fock space: it does not change the physical results. For example, the parameter \( \xi\) does not affect the physics.
This construction extends to the full quantum theory through renormalization. This is a non-trivial result which will not be reviewed here.
The BRST formalism applies directly to arbitrary gauge theories and to Lagrangian systems whose constraints are first class and form a Lie algebra. 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.
To summarize, the relation between the BRST construction and gauge-invariance is the following. Covariant quantization of gauge fields requires the compensation of unphysical degrees of freedom. This is ensured by the Kugo-Ojima quartet compensation mechanism, which is not related to any gauge symmetry, but is based on the existence of a physically invariant subspace identified with the kernel of the BRST operator \(Q\ .\) In turn, the existence of such a BRST operator in the interacting theory relies on the gauge invariance of the underlying classical theory.
References
- Sundermeyer, K (1982). Constrained Dynamics. Lecture Notes in Physics Vol. 169. Springer-Verlag, Berlin. ISBN 978-3-540-11947-0
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- Kugo(1978 b). Manifestly Covariant Canonical Formulation of the Yang-Mills Field Theories. I. Progr. Theor. Phys. 60: 1869. doi:10.1143/ptp.60.1869.
- Gribov, V N (1978). Quantization of Nonabelian Gauge Theories. Nucl. Phys. B139: 1. doi:10.1016/0550-3213(78)90175-x.
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- Haag, R (1992). Local Quantum Physics: Fields, Particles, Algebras. Springer-Verlag, Berlin. ISBN 3540610499
Internal references
- Carlo Rovelli (2008) Quantum gravity. Scholarpedia, 3(5):7117.
- Andrei A. Slavnov (2008) Slavnov-Taylor identities. Scholarpedia, 3(10):7119.
Further reading
- A general reference on quantum field theory is: Zinn Justin J (1989) Quantum Field Theory and Critical Phenomena Claredon Press (Oxford)
- Concerning in particular gauge field quantization in the functional formalism: Faddeev LD, Slavnov AA (1980) Gauge Fields. Introduction to Quantum Theory, The Benjamin/Cummings Publishing Company Inc. (Reading MA)
- Gauge field quantization in the operator formalism can be found in: Nakanishi N, Ojima I (1990) Covariant Operator Formalism of Gauge Theories and Quantum Gravity World Scientific Lecture Notes in Physics Vol. 27 World Scientific Publishing Company (Singapore)
- For the application of eikonal approximations to gauge theories see: Becchi C, Ridolfi G (2005) An introduction to relativistic processes and the Standard Model of electro-weak interactions Springer-Verlag (Berlin)
- A general reference on string theory is: Polchinski J (1998)String Theory Cambridge University Press (Cambridge)
- A list of further related subject on BRST symmetry is presented in: Abe M, Nakanishi N, Ojima I editors (1996) BRS symmetry Universal Academic Press Inc. (Tokyo)
- Introduction to Gauge Theories, C. Becchi, Lectures given at the Triangle Graduate School 96, Charles University, Prague September 2-11, 1996, arXiv:hep-ph/9705211.
- Introduction to BRS symmetry, C. Becchi, Lectures given at the ETH, Zurich, May 22-24, 1996 (revised version: December 2008) arXiv:hep-th/9607181v2.
See Also
Faddeev-Popov action, Gauge invariance, Gauge theories, Slavnov-Taylor identities, Zinn-Justin equation