BRST Quantization - The BRST Operator and Asymptotic Fock Space

The BRST Operator and Asymptotic Fock Space

Two important remarks about the BRST operator are due. First, instead of working with the gauge group one can use only the action of the gauge algebra on the fields (functions on the phase space).

Second, the variation of any "BRST exact form" with respect to a local gauge transformation is, which is itself an exact form.

More importantly for the Hamiltonian perturbative formalism (which is carried out not on the fiber bundle but on a local section), adding a BRST exact term to a gauge invariant Lagrangian density preserves the relation . As we shall see, this implies that there is a related operator on the state space for which —i. e., the BRST operator on Fock states is a conserved charge of the Hamiltonian system. This implies that the time evolution operator in a Dyson series calculation will not evolve a field configuration obeying into a later configuration with (or vice versa).

Another way of looking at the nilpotence of the BRST operator is to say that its image (the space of BRST exact forms) lies entirely within its kernel (the space of BRST closed forms). (The "true" Lagrangian, presumed to be invariant under local gauge transformations, is in the kernel of the BRST operator but not in its image.) The preceding argument says that we can limit our universe of initial and final conditions to asymptotic "states"—field configurations at timelike infinity, where the interaction Lagrangian is "turned off"—that lie in the kernel of and still obtain a unitary scattering matrix. (BRST closed and exact states are defined similarly to BRST closed and exact fields; closed states are annihilated by, while exact states are those obtainable by applying to some arbitrary field configuration.)

We can also suppress states that lie inside the image of when defining the asymptotic states of our theory—but the reasoning is a bit subtler. Since we have postulated that the "true" Lagrangian of our theory is gauge invariant, the true "states" of our Hamiltonian system are equivalence classes under local gauge transformation; in other words, two initial or final states in the Hamiltonian picture that differ only by a BRST exact state are physically equivalent. However, the use of a BRST exact gauge breaking prescription does not guarantee that the interaction Hamiltonian will preserve any particular subspace of closed field configurations that we can call "orthogonal" to the space of exact configurations. (This is a crucial point, often mishandled in QFT textbooks. There is no a priori inner product on field configurations built into the action principle; we construct such an inner product as part of our Hamiltonian perturbative apparatus.)

We therefore focus on the vector space of BRST closed configurations at a particular time with the intention of converting it into a Fock space of intermediate states suitable for Hamiltonian perturbation. To this end, we shall endow it with ladder operators for the energy-momentum eigenconfigurations (particles) of each field, complete with appropriate (anti-)commutation rules, as well as a positive semi-definite inner product. We require that the inner product be singular exclusively along directions that correspond to BRST exact eigenstates of the unperturbed Hamiltonian. This ensures that one can freely choose, from within the two equivalence classes of asymptotic field configurations corresponding to particular initial and final eigenstates of the (unbroken) free-field Hamiltonian, any pair of BRST closed Fock states that we like.

The desired quantization prescriptions will also provide a quotient Fock space isomorphic to the BRST cohomology, in which each BRST closed equivalence class of intermediate states (differing only by an exact state) is represented by exactly one state that contains no quanta of the BRST exact fields. This is the Fock space we want for asymptotic states of the theory; even though we will not generally succeed in choosing the particular final field configuration to which the gauge-fixed Lagrangian dynamics would have evolved that initial configuration, the singularity of the inner product along BRST exact degrees of freedom ensures that we will get the right entries for the physical scattering matrix.

(Actually, we should probably be constructing a Krein space for the BRST-closed intermediate Fock states, with the time reversal operator playing the role of the "fundamental symmetry" relating the Lorentz-invariant and positive semi-definite inner products. The asymptotic state space is presumably the Hilbert space obtained by quotienting BRST exact states out of this Krein space.)

In sum, no field introduced as part of a BRST gauge fixing procedure will appear in asymptotic states of the gauge-fixed theory. However, this does not imply that we can do without these "unphysical" fields in the intermediate states of a perturbative calculation! This is because perturbative calculations are done in the interaction picture. They implicitly involve initial and final states of the non-interaction Hamiltonian, gradually transformed into states of the full Hamiltonian in accordance with the adiabatic theorem by "turning on" the interaction Hamiltonian (the gauge coupling). The expansion of the Dyson series in terms of Feynman diagrams will include vertices that couple "physical" particles (those that can appear in asymptotic states of the free Hamiltonian) to "unphysical" particles (states of fields that live outside the kernel of or inside the image of ) and vertices that couple "unphysical" particles to one another.