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Reparametrization-Invariant Effective Action in Field-Antifield Formalism

We introduce classical and quantum antifields in the reparametrization-invariant effective action, and derive a deformed classical master equation.


Introduction
It is well-known that the basic/standard notion of effective action Γ(Φ) in quantum field theory is not reparametrization-invariant, cf. Sections 2-3. A remedy was proposed by Vilkovisky [1,2,3] by using a connection Γ γ αβ on the field configuration manifold M, cf. Sections 4-5 and Appendix B. In this paper, we amend the reparametrization-invariant construction with antifields, and develop the corresponding field-antifield formalism [4,5,6,7]. We derive a Ward identity (6.8) and a deformed classical master equation (6.10), cf. Section 6 and Appendix A. The resulting approach works in principle for an arbitrary gauge theory. A manifest superfield approach is considered in Appendix C.
In the following we will use DeWitt condensed notation.

Legendre Transformation
In quantum field theory, one often performs a Legendre transformation to change variables J α ↔ Φ α from sources J α to classical fields Φ α . Here W c ≡ i ln Z is the generating action for connected diagrams and Γ(Φ) is the effective action. One usually takes n implicit relations to be of the form We stress that although the relations (2.3) are the most natural choice of implicit relations (2.2), they are not the only possibility, as we shall see in Section 5.

Standard Partition Function and Effective Action
The standard (non-reparametrization-invariant) partition function Z(J) depends on sources J α e i W c (J) ≡ Z(J) := dµ e i (W (ϕ)+J α ϕ α ) . (3.1) The quantum average is dµ e i (W (ϕ)+J α ϕ α ) F , F = F (ϕ) . Here W = W (ϕ) is a (gauge-fixed) quantum action, and ϕ α is the quantum field/integration variable of Grassmann parity ε α . The level-zero * measure in the path integral (3.1) is The effective action is defined as , (3.4) where the implicit relations (2.2) are given by the standard Legendre relations (2.3). Standard reasoning yields that and Moreover, Here explicit dependence "expl" means dependence that is not via the implicit relations (2.2). Note that in the standard Legendre transformation (2.3), total and explicit differentiations are the same.
Let Φ be a fixed base point. Let V ⊆ T Φ M be an sufficiently small open neighborhood of the zero (velocity) vector 0 ∈ T Φ M. The exponential map Exp Φ : V ⊆ T Φ M → M takes a (velocity) vector v Φ ∈ V and maps it to the unique point ϕ ∈ M on the manifold that is reached along a geodesic γ : [t 0 , t 1 ] → M, i.e., These formulas are invariant under affine reparametrizations t → at+b of the geodesic γ. The geodesic differential equation reads For a point ϕ sufficiently close to the fixed point Φ ∈ M (technically speaking, for points in a so-called normal neighborhood ϕ ∈ U (Φ) ⊆ M), there exists a unique geodesic γ : [t 0 , t 1 ] → M that goes from Φ to ϕ. One defines the logarithmic map Ln Φ : U (Φ) → T Φ M as the inverse of the exponential map, i.e., it has the corresponding initial velocity vector as output, Often in the literature, the logarithmic map (i.e., the initial velocity vector) is denoted as The coordinate functions −σ α (Φ, ϕ) are also known as the Riemann normal coordinates based at Φ. Note that the bi-local coordinate function σ α (Φ, ϕ) behaves geometrically as a vector with respect to the point Φ and as a scalar with respect to the point ϕ. A short-distance expansion of the logarithmic map reads If the Christoffel symbols Γ α βγ = 0 vanish identically in the neighborhood U (Φ), then the logarithmic map is simply given by The logarithmic map satisfies the differential equation

Reparametrization-Invariant Effective Action
The standard effective action (3.4) is not invariant under reparametrizations of the quantum field ϕ α and the classical field Φ α .
The source J α behaves by definition as a co-vector (scalar) under reparametrizations of the point Φ (the point ϕ), respectively. In particular the term J α Φ α is not a scalar under reparametrizations Φ α → Φ ′β = f β (Φ). Since we want to maintain eq. (2.1), it therefore becomes impossible to make both W c and Γ reparametrization-invariant quantities simultaneously. We will focus on the latter, i.e., the effective action Γ.
A reparametrization-invariant effective action can be achieved by using the logarithmic map [1,2] The quantum average is .

(5.2)
Since we assume the Legendre relation (2.1), the partition function becomes Let us now elaborate on the status of the implicit dependence (2.2).
If one uses the standard Legendre relations (2.3), one gets However, we shall here not use the standard Legendre relations (5.4) and (5.5).
Instead we shall impose n reparametrization-invariant implicit conditions as advocated by Vilkovisky [1,2,3]. The main point is that condition (5.6) is covariant (invariant) under reparametrizations of the classical field Φ α (quantum field ϕ α ), respectively. The condition (5.6) implies that the total and explicit differentiations of the effective action Γ with respect to the classical field Φ α are the same Note that the condition (5.6) means that the classical field Φ α and the quantum average ϕ α may differ (even at the classical level). In particular, the classical decomposition formula (3.8) may no longer hold.
Finally, let us mention that one could in principle perform a change of integration variables to bring the the path integral (5.1) back to the form (3.4) (and similarly bring the average (5.6) back to eq. (3.5)), with the caveat that the new action W ′ (Φ, ϕ) = W (Φ, σ(Φ, ϕ)) and measure ρ ′ (Φ, ϕ) would depend on the classical fields Φ.
(6.1) (The vertical line notation on the right-hand side of eq. (6.1) means that the two formulas to the right of the vertical line should be substituted into the path integral. By definition the substitution J = J(Φ, Φ * ) counts as implicit dependence, while the substitution ϕ * = ∂ Ψ ∂ϕ counts as explicit dependence.) The quantum average is The n implicit relations J α = J α (Φ, Φ * ) come by definition from the n conditions cf. condition (5.6). The extended gauge-fixing Fermion Ψ is assumed to be affine in the classical antifields The Fermion Ψ is a scalar under reparametrizations of both ϕ and Φ. We stress that the antifieldfree part Ψ = Ψ(Φ, ϕ) of the gauge-fixing Fermion Ψ is allowed to depend on the classical fields Φ, in contrast to the construction in Sections 3 and 5. The condition (6.3) implies that the total and explicit differentiations of the effective action Γ with respect to the classical field Φ α and antifield Φ * α are the same The quantum master equation [4,5,6] reads with the odd Laplacian † The Ward identities read and where we have defined a deformed antibracket of classical variables as The deformed classical master equation (6.10) is our main result. The antibracket (6.11) may in general violate the Jacobi identity (even at the classical level), cf. eq. (A. 16).
Finally, the change of integration variables (5.9) is now part of a type-2 anticanonical transformation with a type-2 Fermionic generator

A Extended Formalism
In this Appendix A, we promote (for technical rather than fundamental/profound reasons) the quantum fields ϕ α , the classical fields Φ α and antifields Φ * α to superfields where θ is a Fermionic parameter. Note that the superpartners of the classical antifields Φ * α are (minus) the sources J α . Our primary aim in this Appendix A is not to create a superfield formalism, but merely a convenient platform to derive the pertinent Ward identities (6.8) and (6.9). (A treatment from a manifest superfield perspective is developed in the next Appendix C.) Our sign convention for the Berezin integral is dθ θ = 1 . and action is the usual quantum master action, and is given by Later in eq. (A.21) we will introduce n implicit relations J α = J α (Φ, Φ * , Λ). In anticipation of this, we will already now begin to distinguish between total and explicit derivative. Note e.g., that Extended Ward identity for Y : The extended Ward identity (A.11) can be seen by shifting integration variable θ → θ + θ 0 in the formula (A.9) for Y , and collecting terms proportional to the Fermionic constant θ 0 .
Extended Ward identity for Γ: Proof of eq. (A.12): Expansion of the extended Ward identity (A.13) around Λ = 0 to second order in Λ: Extended quantum average Expansion of the extended Ward identity (A.12) around Λ = 0 to second order in Λ: The n implicit constraints reads The effective action Γ = Γ(Φ, Φ * ) from eq.

B Metric and Synge's World Function
In the main text we assumed that field configuration manifold M is equipped with a torsionfree connection ∇. In this Appendix B we will additionally assume that field configuration manifold M is equipped with a (pseudo) Riemannian metric g αβ , and that ∇ is the corresponding Levi-Civita connection. We will follow the sign conventions of Ref. [18].

B.2 Levi-Civita Connection
The torsion tensor is just an antisymmetrization of the Christoffel symbol Γ β αγ with respect to the lower indices, In particular, the Christoffel symbol is symmetric with respect to the lower indices when the connection is torsionfree. A connection ∇ is called metric, if it preserves the metric Here we have lowered the Christoffel symbol with the metric Γ αβγ := g αδ Γ δ βγ (−1) ε γ . (B.11) ‡ Vilkovisky [1,2] assumes that the field configuration manifold M is Bosonic. Our superconventions are related to those of DeWitt [3] via g (here) αβ , and Γ γ (here) The metric condition (B.10) reads in terms of the contravariant inverse metric Here we have introduced a reordered Christoffel symbol The Levi-Civita connection is the unique connection ∇ that is both torsionfree T = 0 and metric (B.10). The Levi-Civita formula for the lowered Christoffel symbol in terms of derivatives of the metric reads (B.14) with Lagrangian L(t) given by a normalized squared distance

B.3 Synge's World Functional
The Synge world functional Σ[γ] is invariant under affine reparametrizations t → at + b of the curve γ. The corresponding (on-shell) Euler-Lagrange equation is precisely the geodesic eq. (4.2). The momentum p α (t) reads Since there is no explicit t-dependence, the corresponding energy function does not depend on time t on-shell, cf. Noether's theorem. In particular, the Lagrangian L(t) can be pulled outside the action integral Σ[γ] ≈ (t 1 − t 0 )L(t) on-shell. (Here the ≈ symbol means equality modulo the geodesic eq. (4.2). We do not use the ≈ symbol in the main text.)

B.4 Synge's World Function
Let there be given two points Φ, ϕ ∈ M that are linked by a unique geodesic γ : [t 0 , t 1 ] → M, so that The Synge world function between the two points Φ and ϕ is defined [22] as the value of the Synge world functional Σ[γ] along the geodesic γ. In other words, the Synge world function σ is the associated on-shell action function for the off-shell action functional Σ. It follows that the Synge world function σ(Φ, ϕ) = σ(ϕ, Φ) is numerically precisely half the square of the geodesic distance from Φ to ϕ, Here in eq. (B.21) it is implicitely understood that the curve γ is the geodesic between Φ to ϕ.
If one adds the action term (C.5) to the quantum master action as if one introduces an odd vector field Finally let us mention, that if one introduces an odd vector field