The BRST global supersymmetry introduced in the mid-1970s was quickly understood to rationalize the introduction of these Faddeev–Popov ghosts and their exclusion from "physical" asymptotic states when performing QFT calculations.
Crucially, this symmetry of the path integral is preserved in loop order, and thus prevents introduction of counterterms which might spoil renormalizability of gauge theories.
– Discuss] a few years later related the BRST operator to the existence of a rigorous alternative to path integrals when quantizing a gauge theory.
Its significance for rigorous canonical quantization of a Yang–Mills theory and its correct application to the Fock space of instantaneous field configurations were elucidated by Taichiro Kugo and Izumi Ojima.
There are other kinds of "sanity checks" that can be performed on a quantum field theory to determine whether it fits qualitative phenomena such as quark confinement and asymptotic freedom.
In the early days of QFT, one would have had to say that the quantization and renormalization prescriptions were as much part of the model as the Lagrangian density, especially when they relied on the powerful but mathematically ill-defined path integral formalism.
The BRST method provided the calculation techniques and renormalizability proofs needed to extract accurate results from both "unbroken" Yang–Mills theories and those in which the Higgs mechanism leads to spontaneous symmetry breaking.
It has proven rather more difficult to prove the existence of non-Abelian quantum field theory in a rigorous sense than to obtain accurate predictions using semi-heuristic calculation schemes.
The Lorenz gauge is a great simplification relative to Maxwell's field-strength approach to classical electrodynamics, and illustrates why it is useful to deal with excess degrees of freedom in the representation of the objects in a theory at the Lagrangian stage, before passing over to Hamiltonian mechanics via the Legendre transformation.
Because the definition of the Hamiltonian involves a unit time vector field on the base space, a horizontal lift to the bundle space, and a spacelike surface "normal" (in the Minkowski metric) to the unit time vector field at each point on the base manifold, it is dependent both on the connection and the choice of Lorentz frame, and is far from being globally defined.
, in the form of a power series in the coupling constant g; it is the principal tool for making quantitative predictions from a quantum field theory.
It suffers from Gribov ambiguities and from the difficulty of defining a gauge fixing constraint that is in some sense "orthogonal" to physically significant changes in the field configuration.
By the stationary phase approximation on which the Feynman path integral is based, the dominant contribution to perturbative calculations will come from field configurations in the neighborhood of the constraint surface.
If one ignores the problem and attempts to use the Feynman rules obtained from "naive" functional quantization, one finds that one's calculations contain unremovable anomalies.
In the second view, a gauge transformation is a change of coordinates along the entire fiber (arising from multiplication by a group element g) which induces a vertical diffeomorphism of the principal bundle.
For concreteness and relevance to conventional QFT, much of this article sticks to the case of a principal gauge bundle with compact fiber over 4-dimensional Minkowski space.
Identifying local gauge transformations with a particular subspace of vector fields on the manifold P provides a better framework for dealing with infinite-dimensional infinitesimals: differential geometry and the exterior calculus.
The proper generalization of Clairaut's theorem to the non-trivial manifold structure of P is given by the Lie bracket of vector fields and the nilpotence of the exterior derivative.
This provides an essential tool for computation: the generalized Stokes theorem, which allows integration by parts and then elimination of the surface term, as long as the integrand drops off rapidly enough in directions where there is an open boundary.
The Ward and BRST operators are related (up to a phase convention introduced by Kugo and Ojima, whose notation we will follow in the treatment of state vectors below) by
This implies that the universe of initial and final conditions can be limited to asymptotic "states" or field configurations at timelike infinity, where the interaction Lagrangian is "turned off".
As is conventional for second quantization, the Fock space is provided 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.
This ensures that any pair of BRST-closed Fock states can be freely chosen out of the two equivalence classes of asymptotic field configurations corresponding to particular initial and final eigenstates of the (unbroken) free-field Hamiltonian.
This requires that 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.
It is enlightening to inspect their variant of the BRST transformation, which emphasizes the hermitian properties of the newly introduced fields, before proceeding from an entirely geometrical angle.
are unspecified, as is left the form of the Ward operator on it; these are unimportant so long as the representation of the gauge algebra on the matter fields is consistent with their coupling to
and the Lagrangian has a gauge breaking term that is equal, up to a divergence, to Likewise, there are two kinds of quanta that will lie entirely in the image of the BRST operator: those of the Faddeev–Popov ghost
i.e. its quantization prescription, which ignores the spin–statistics theorem by giving Fermi–Dirac statistics to a spin-0 particle—will be given by the requirement that the inner product on our Fock space of asymptotic states be singular along directions corresponding to the raising and lowering operators of some combination of non-BRST-closed and BRST-exact fields.
While the BRST symmetry, with its corresponding charge Q, elegantly captures the essence of gauge invariance, it presents a challenge for path integral quantization.
The ghost fields, originally introduced to compensate for the unphysical degrees of freedom, now play a crucial role in maintaining the unitarity of the theory in the quantized version.