As far as i know the BRST symmetry is an infinitesimal (and expanded) version of gauge symmetry. Recently I read the following: "when QFT was reformulated in fiber bundle language for application to problems in the topology of low-dimensional manifolds, did it become apparent that the BRST 'transformation' is fundamentally geometric" I am aware of how ghosts are Maurer-Cartan form on the (infinite dimensional) group of gauge transfprmations of one's principle bundle... Now the above quote continues, "The relationship between gauge invariance and "BRST invariance" forces the choice of a Hamiltonian system whose states are composed of "particles" according to the rules familiar from the canonical quantization formalism. This esoteric consistency condition therefore comes quite close to explaining how quanta and fermions arise in physics to begin with."

Does anyone know what this second half of the quote is talking about? E.g. what "relationship", what "esoteric consistency condition, anr which special "form of Hamiltonian" is forced on which (presumably upon quantization) gives rise to particles ...? If the whole thing makes sense, does anyone know any references to this matter? (preferably, original sources...)


I) First of all, note that although gauge theory and BRST formulation originally only referred to Yang-Mills theory (and hence QED), they nowadays apply to general theories with so-called local gauge symmetry, cf. e.g. this Phys.SE post.

The Lagrangian and Hamiltonian BRST formalism are known as Batalin-Vilkovisky (BV) formalism and Batalin-Fradkin-Vilkovisky (BFV) formalism, respectively.

II) The full story is explained in e.g. the book Quantization of Gauge Systems by M. Henneaux and C. Teitelboim. But in a nutshell, given an infinitesimal gauge transformation of the form

$$ \delta_{\varepsilon}\varphi^i(x) ~=~ \int d^d y \ R^i{}_a (x,y)\varepsilon^a(y),$$

where $\varphi^i$(x) are the original fields; where $R^i{}_a (x,y)$ are the gauge generator; and where $\varepsilon^a(y)$ are infinitesimal gauge parameters, then the corresponding Grassmann-odd BRST transformation is

$$ {\bf s} \varphi^i(x) ~=~ \int d^d y \ R^i{}_a (x,y)c^a(y), $$

where $c^a(y)$ are Faddeev-Popov ghost fields. The $c^a(y)$ carry opposite Grassmann-parity as compared to the infinitesimal gauge parameters $\varepsilon^a(y)$.

In that sense a BRST transformation is nothing but a systematic reformulation of a gauge symmetry. BRST formalism is not needed for simple gauge theories such as QED, but for more complicated gauge theories, say with reducible and/or open gauge algebra, the BRST formalism quickly becomes an indispensable tool.

III) As for the quote:

  1. The quote "esoteric consistency condition" undoubtedly refers to the fact that the BRST transformation is nilpotent (=squares to zero), which encodes a (possibly infinite) tower of consistency relations.

  2. The quote "comes quite close to explaining how quanta and fermions arise in physics to begin with" refers to the fact that physical states are counted/classified by BRST cohomology, and that BRST formulation relies vitally on the use of Grassmann-odd fields, respectively.

  • $\begingroup$ Thank you. Any idea what this "forcing of a hamiltonian system" where the "states are particles" etc...? $\endgroup$ – Jan Vandervoort Apr 3 '16 at 10:53

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.