Gauge Independence in a Higher-Order Lagrangian Formalism via Change of Variables in the Path Integral

In this paper we work out the explicit form of the change of variables that reproduces an arbitrary change of gauge in a higher-order Lagrangian formalism.


Introduction
It is a standard lore in the path integral formalism, that any result (such as, e.g., the Schwinger-Dyson equations, the Ward identities, etc.), that can be (formally) proven via change of integration variables, can equivalently be (formally) obtained via an integration by parts argument. And vice-versa. The latter method is typically the simplest. In 1996 it was shown in Ref. [1], by using integration by parts, how to formulate a higher-order field-antifield formalism that is independent of gauge choice. In this paper we work out the explicit form of the change of variables that reproduces a given change of gauge in an higher-order formalism. Perhaps not surprisingly, the construction relies on identifying appropriate homotopy operators.

The ∆ Operator
From a modern perspective [2] the primary object in the Lagrangian field-antifield formalism [3,4,5] is the ∆ operator, which is a nilpotent Grassmann-odd differential operator and which depends on antisymplectic variables z A and their corresponding partial derivatives ∂ B .
Their commutator * reads 3 Sp(2)-Symmetric Formulation We mention for completeness that there also exists an Sp(2)-symmetric Lagrangian field-antifield formulation [6]. This formulation is endowed with two Grassmann-odd nilpotent, anticommuting ∆ a operators ∆ {a ∆ b} = 0 , ε(∆ a ) = 1 , a, b ∈ {1, 2} . (3.1) Often (but not always!) the resulting Sp(2)-symmetric formulas look like the standard formulas with Sp(2)-indices added and symmetrized in a straighforward manner. In this paper, we will focus on the standard formulation and usually only mention the corresponding Sp(2)-symmetric formulation when it deviates in a non-trivial manner.
the operator, respectively. We mention for later convenience the superadditivity of Planck number grading where the uppercase letters F and G denote operators.

Higher-Order ∆ Operator
In the standard field-antifield formalism [3,4,5], the ∆ operator is a second-order operator. (See also Section 18.) In the higher-order generalization [1], which is the main topic of this paper, the ∆ operator is assumed to have Planck number grading [7] Pl(∆) ≥ −2 .
Evidently, the Planck number inequality (5.1) means that the normal-ordered ∆ operator is of the following triangular form ‡ The higher-order terms in the ∆ operator can e.g., be physically motivated as quantum corrections, which arise in the correspondence between the path integral and the operator formalism.

2.
A gauge-generating quantum master action W , which satisfies the quantum master equation (QME) The path integral (6.1) will in general depend on W , since W contains all the physical information about the theory, such as, e.g., the original action, the gauge generators, etc. [15,16]. The ‡ In contrast to the original proposal [1], we also allow the three terms ∆ −2,0 , ∆ −1,0 and ∆ −1,1 with negative n in eq. (5.2). The two last terms arise naturally in the Sp(2)-symmetric formulation [6,12]. The two first terms affect the classical master eq. See also Sections 18-19 for the second-order case.
§ The parenthesis in eq. (6.2) is here meant to emphasize that the QME is an identity of functions (as opposed to differential operators), i.e., the derivatives in ∆ do not act outside the parenthesis. Note however that similar parenthesis will not always be written explicitly in order not to clog formulas. In other words, it must in general be inferred from the context whether an equality means an identity of functions or an identity of differential operators. triangular form (5.2) of the ∆ operator implies that the QME (6.2) is perturbative in Planck's constant , i.e., Besides the triangular form (5.1), which is imposed to ensure perturbativity, there are additional "boundary" and rank conditions to guarantee the pertinent classical ¶ master equation and proper classical master action S [15,16].
3. A gauge-fixing quantum master action X, which satisfies the transposed quantum master equation The path integral (6.1) will in general not depend on X, cf. Section 13 and Section 16.
The transposed operator F T has the property that Here the lowercase letters f, g, . . . denote functions, while the upper case letters F, G, . . . denote operators. One can construct any transposed operator by successively apply the following rules The transposed derivative ∂ T A satisfies a modified Leibniz rule: Let us mention for completeness that the ∆ operator (which takes functions to functions) and the W -X-formalism can be recast in terms of Khudaverdian's operator ∆ E (which takes semidensities to semidensities) [17,18,19,20,21,22,23,24,25].

Higher-Order Quantum BRST Operators
The quantum BRST operators σ W and σ X take operators into functions (i.e., left multiplication operators). They are defined as respectively, where F is an operator. They are nilpotent, Grassmann-odd, and perturbative in the sense that In the Sp(2)-symmetric formulation the quantum BRST operators σ a W and σ a X carry an Sp(2)-index since the ∆ a operator does. ¶ The word classical means here independent of Planck's constant .

Higher quantum antibrackets
The 1-quantum antibracket is The n-quantum antibracket consists of nested commutators of n operators with the ∆-operator [26,27,28,29,30,31,32]. We will not need the full definition here, but it can in principle be deduced uniquely via polarization of the following recursive formula [32] Φ n ∆ (B, . . . , B) n arguments (8.2) Philosophically speaking, the n-quantum antibrackets (8.2) are secondary/derived objects, which can be obtained from the underlying concept of a fundamental ∆-operator. The pertinent Lie bracket is the 2-quantum antibracket / derived bracket [26,27,28] The 1-quantum antibracket D generates the 2-quantum antibracket [28] [ The 3-quantum antibracket is defined as The Jacobi identity for the 2-quantum antibracket is satisfied up to D-exact terms 9) or equivalently, in the polarized language [32] Proof of eq. (8.10): 9 Grassmann-even Sp(2) quantum brackets In the Sp(2)-symmetric case, besides the Sp(2)-symmetric higher quantum antibrackets (which we will not discuss here), there is a tower of Grassmann-even quantum brackets. The pertinent 1-quantum bracket is The 2-quantum bracket is defined as (Hopefully it does not lead to confusion that we use the same notation for the Grassmann-even quantum brackets D and [[·, ·]] in this Section 9 as we use for the Grassmann-odd quantum antibrackets D and [[·, ·]] in the previous Section 8.) Up to D-exact terms, the 2-quantum bracket is 3) The 2-quantum bracket is Grassmann-even The Jacobi identity for the 2-quantum antibracket is satisfied up to D-closed terms  , respectively). In detail, the latter reads Here we have used the Jacobi identity (8.9).

Maximal Deformation
One may formally argue [7] that any two solutions to the QME (6. The same story holds for X instead of W if we replace the operator ∆ with the transposed operator ∆ T , e.g., When discussing X (as opposed to W ) we will implicitly assume that the pertinent quantum (anti)brackets are generated by the transposed operator ∆ T .

Gauge-Independence via Integration by Parts
The gauge-independence of the path integral can be formally proved via integration by parts [1] δZ ≡ Z X+δX − Z X 14 Homotopy Operator The pertinent homotopy operator → h A (∆) is best explained for operators ∆ on antinormal-ordered form The definition (14.2) is extended to an arbitrary operator ∆ by linearity. The homotopy operator (14.2) satisfies the following homotopy property for antinormal-ordered operators (14.1). Two homotopy operators (14.2) commute: Given a function f and an operator ∆, the bilinear homotopy operator B A (f, ∆) is defined via . One may prove that the bilinear homotopy operator B A (f, ∆) has the following important homotopy property

Gauge-Independence via Change of Variables
The infinitesimal change δz A of (passive) coordinates z A can be viewed as an infinitesimal vector field One may show that the Planck number Pl(δz A ) ≥ −1 of the vector field is greater than or equal to −1, as it should be. The Boltzmann density (= integrand) of the path integral (6.1) is ρwx. The divergence of the vector field (16.1) with respect to the Boltzmann density is On one hand, an infinitesimal change of integration variables in path integral cannot change the value of path integral. On the other hand, it induces an infinitesimal Jacobian factor. Hence

1)
We are here and below guilty of infusing some active picture language into a passive picture, i.e., properly speaking, the active vector field has the opposite sign.
In particular, the 2-antibracket (f, g) of two function f and g is defined as

Second-Order ∆ operator
It is natural to ponder how to build a nilpotent ∆-operator, that takes scalar functions in scalar functions, from the following given geometric data: 1. An anti-Poisson structure A density ρ with ε(ρ) = 0 and Pl(ln ρ) ≥ −1.

A Grassmann-odd vector field
with ε(V ) = 1 and Pl(V ) ≥ −2, that is compatible with the anti-Poisson structure: Often we assume that the antibracket (18.1) is non-degenerate/invertible. Then the vector field is locally a Hamiltonian vector field V = (H, ·). This Hamiltonian H can be absorbed into the density by redefining the density ρ = ρe 2H .