Math 2.0

Quantum theory Definition

BRST quantization

Builds on Lie algebraExterior algebraChevalley–Eilenberg complexLie algebra cohomologyFaddeev–Popov ghost

Definition

Let 𝔤\mathfrak{g} be a finite-dimensional real Lie algebra with basis T1,…,TrT_1, \dots, T_r and structure constants [Ta,Tb]=∑cfabcTc[T_a, T_b] = \sum_c f^c_{ab} T_c. Let ρ\rho be a representation of 𝔤\mathfrak{g} on a complex vector space VV of states. Let Λ𝔤∗\Lambda \mathfrak{g}^* be the exterior algebra of the dual space, graded by degree, with dual basis c1,…,crc^1, \dots, c^r of 𝔤∗\mathfrak{g}^*, the ghosts. Let bab_a be the contraction of Λ𝔤∗\Lambda\mathfrak{g}^* with TaT_a, the antighosts, so that bacb+cbba=δabb_a c^b + c^b b_a = \delta^b_a. The BRST operator on the BRST complex C=Λ𝔤∗⊗ℝVC = \Lambda \mathfrak{g}^* \otimes_\R V is Q=∑aca ρ(Ta)−12∑a,b,cfabc cacb bc,Q = \sum_{a} c^a\, \rho(T_a) - \frac12 \sum_{a, b, c} f^c_{ab}\, c^a c^b\, b_c, where cac^a acts by left multiplication on the first factor. Then Q2=0Q^2 = 0, QQ raises the ghost degree by 11, and the space of BRST-physical states is the cohomology H0(Q)=ker⁡(Q∣C0)H^0(Q) = \ker(Q|_{C^0}), and more generally Hk(Q)=ker⁡(Q∣Ck)/Q(Ck−1)H^k(Q) = \ker(Q|_{C^k}) / Q(C^{k-1}).

Remarks

Here QQ is the differential of the Chevalley–Eilenberg complex of 𝔤\mathfrak{g} with coefficients in VV, so H∙(Q)H^\bullet(Q) is the Lie algebra cohomology H∙(𝔤,V)H^\bullet(\mathfrak{g}, V). On v∈V=C0v \in V = C^0 one has (Qv)(Ta)=ρ(Ta)v(Qv)(T_a) = \rho(T_a)v, so H0(Q)H^0(Q) is the space V𝔤V^{\mathfrak{g}} of invariant vectors, the gauge-invariant states. On a 11-form θ∈𝔤∗\theta \in \mathfrak{g}^* with VV trivial, the second term gives (Qθ)(x,y)=−θ([x,y])(Q\theta)(x, y) = -\theta([x, y]).

Example: let 𝔤=𝔰𝔬(3)\mathfrak{g} = \mathfrak{so}(3) act on V=ℂ3V = \C^3 by the standard representation. Then H0(Q)=0H^0(Q) = 0, since no nonzero vector of ℂ3\C^3 is fixed by all rotations.

Non-example: the operator ∑acaρ(Ta)\sum_a c^a \rho(T_a) alone does not square to zero when 𝔤\mathfrak{g} is non-abelian; its square is 12∑fabc cacbρ(Tc)\frac12 \sum f^c_{ab}\, c^a c^b \rho(T_c), and the ghost term is what cancels it.

In gauge field theory the same construction is applied to the Lie algebra of infinitesimal gauge transformations. It is combined with extra antighost and auxiliary fields from gauge fixing, and the ghosts are then the Faddeev–Popov ghosts.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Chevalley–Eilenberg complex, concept, no evidence levelChevalley–Eilenberg c…conceptExterior algebra, concept, no evidence levelExterior algebraconceptFaddeev–Popov ghost, concept, no evidence levelFaddeev–Popov ghostconceptLie algebra, concept, no evidence levelLie algebraconceptLie algebra cohomology, concept, no evidence levelLie algebra cohomologyconceptBRST quantization, definition, no evidence levelBRST quantizationdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Chevalley–Eilenberg complex concept
  2. Exterior algebra concept
  3. Faddeev–Popov ghost concept
  4. Lie algebra concept
  5. Lie algebra cohomology concept
  6. BRST quantization definition

Further reading

  1. Web https://en.wikipedia.org/wiki/BRST_quantization
About this page
TagS84R3CH6YD
URIm2://S84R3CH6YD
Kinddefinition
WritingW0 stub
MSC81T70, 81T13, 17B56
DefinesBRST quantization
UsesLie algebra, Exterior algebra, Chevalley–Eilenberg complex, Lie algebra cohomology, Faddeev–Popov ghost
Changed3 November 2026

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