Physics Reports Chern–Simons modified general relativity

S. Alexander, N. Yunes / Physics Reports 480 (2009) 1–55 11approach in 1 + 1 dimensions [49,50], generalized to d + 1 dimensions. Since amplitudes and currents can be generatedfrom the path integral, Fujikawa realized that anomalies arise from the non-invariance of the fermionic measure in thepath integral under an arbitrary fermionic field redefinition. For concreteness let us consider a massless fermion coupled toelectromagnetism in 3 + 1 dimensions. The action and partition function for this theory are the following:∫ (S = d 4 x − 1)4e F abF ab + i ¯ψγ a D 2 a ψ(46)∫Z = DADψD ¯ψe iS[A a,ψ, ¯ψ] , (47)where ψ is a Dirac fermion, A a is a gauge field, F ab is the electromagnetic field strength tensor, e is the coupling constant orcharge of the Dirac fermions and γ a are Dirac matrices, where the overhead bar stands for complex conjugation. Such a toytheory is invariant under a chiral transformation of the formψ → e iαγ 5ψ = ψ + iαγ 5 ψ + · · · , (48)where α is a real number and γ 5 is the chiral Dirac matrix. Such an invariance leads to the U(1) global Noether currentj Axiala= ¯ψγ a γ 5 ψ. In the path integration approach, current conservation is exhibited by studying the Ward identities, whichcan be derived by requiring that the path integral be invariant under an arbitrary phase redefinition of the Dirac fermions.The non invariance of the fermionic measure is precisely the main ingredient that encodes the chiral anomaly. In orderto study this effect, we must first define the measure precisely. For this purpose, it is helpful to expand ψ in terms oforthonormal eigenstates of iγ a D a :andγ a D a φ m = λ m φ m (49)ψ(x) = ∑ ma m φ m (x), (50)where a m are Grassmann variable multiplying the c-number eigenfunctions φ m (x) and D a is the gauge covariant derivative.The measure is then defined asDψD ¯ψ = ∏ nda n dā n . (51)Let us now demand invariance of the partition function:∫∫DADψD ¯ψe iS[a,ψ, ¯ψ] = DA ′ Dψ ′ D ¯ψ ′ e iS′ [a,ψ, ¯ψ] , (52)where the fermions transform as followsψ(x) → ψ ′ (x) + ɛ(x), ¯ψ(x) → ¯ψ ′ (x) + ¯ɛ(x) (53)for a chiral transformation ɛ(x) = iα(x)γ 5 ψ(x). ¯ We see then that the Lagrangian transforms as∫∫d 4 x( ¯ψ ′ iγ a D a ψ ′ ) = d 4 x[ ¯ψiγ a D a ψ − (∂ a α) ¯ψiγ a γ 5 ψ]. (54)Assuming that the measure is invariant, integrating by parts and varying the action with respect to α, we recover the Wardidentity∂ a 〈 ¯ψγ a γ 5 ψ〉 = 0, (55)which is nothing but the statement of axial current conservation.Naively, we might conclude that the classical global current carries over to the quantum one, but Fujikawa [49,50] realizedthat such a reasoning assumes that the path integral measure is invariant. A more careful analysis then reveals that a changeof variables in the measure affects the coefficient of the Dirac fermion eigenstate expansion viaa ′ = ∑ mnwhere∫B mn = i(δ mn + B mn )a n (56)d 4 x φ Ď m αγ 5φ n . (57)Using the Grassmanian properties of a m , this transformation returns the Jacobian in the measureDψ ′ D ¯ψ ′ = [det(1 + B)] −2 DψD ¯ψ, (58)