Show that the path integral is invariant under BRST transformations, i.e. show that the Jaco- bian determinant of the transformation is unity.

QFT II Problem Set 9.

FS 2019 Prof. M. Grazzini https://www.physik.uzh.ch/en/teaching/PHY552/FS2019.html Due by: 6/5/2019

Exercise 1. BRST Jacobian

Show that the path integral is invariant under BRST transformations, i.e. show that the Jaco- bian determinant of the transformation is unity.

(a) Write down the Jacobian matrix for a BRST transformation.

(b) Write the Jacobian matrix as

J = A D C B

,

and find the determinant of J, the Jacobian determinant, using:

detJ = detB det(A − DB ⁻¹ C),

where A and B are commuting matrices of the form 1 + θM , where θ is a Grassman variable, and C and D are anticommuting matrices.

Hint. T aylor expand the determinants in powers of θ, remembering that θ ² = 0 and hence that all higher powers also vanish. Finally use det(M ) = exp[Tr(ln(M))].

Exercise 2. The origin of the BRST symmetry

We want to discuss the origin of the BRST symmetry in a more general context, and show that, by quoting Zinn-Justin, the ”Slavnov Taylor identities in gauge theories owe less to gauge symmetry than to gauge fixing”. Let ϕ α be a set of dynamical variables that fulfil a set of constraints F _α (ϕ) = 0. For simplicity we use a discrete index α. Furthermore we assume that the functions F _α (ϕ) are smooth and that there exists a one-to-one mapping E _α = E _α (ϕ) in the neighbourhood of E α = 0, so that we can find an inverse ϕ α = ϕ α (E). In particular there is a unique solution ϕ ^s _α ≡ ϕ _α (0). We start from the trivial identity

σ(ϕ ^s ) = Z (

Y

α

dF α δ(F α ) )

σ(ϕ(F)) , (1)

valid for any function σ(ϕ). Note that this expression does not require us to explicitly solve the constraint.

(a) Perform a change of variables from F _α to ϕ _α . What does the translational invariance (F α → F α + ν α , ν α const.) of the measure Q

α dF α imply for the transformation properties of ϕ α ?

(b) Replace the δ-function by its Fourier representation, Y

α

δ[F _α (ϕ)] = Z

Y

α

dλ α

2iπ e ^−λ

^F

^(ϕ) , (2)

where the integration is carried out over the imaginary axis, and the Jacobi determinant by its Grassmann integral representation in terms of the variables c _α and ¯ c _α . Identify the action S in the obtained path integral.

1

(c) Show that the symmetry transformations

δϕ _α = εc _α , (3)

δ¯ c _α = ελ _α , (4)

δc α = 0 , (5)

δλ α = 0 , (6)

can be expressed using the Grassmann differential operator s = c _α ∂

∂ϕ _α + λ _α ∂

∂¯ c _α (7)

acting on the fields, i.e. δΦ = εsΦ for Φ = ϕ _α , c _α , ¯ c _α , λ _α . Show that s ² Φ = 0 (nilpotency).

(d) Show that S can be written as S = s S ˜ and that this implies invariance of S under above transformations.

2