Tensorial q-number commutator function for abelian gauge ﬁelds in quantum Einstein gravity

A tensorial version of the gravitational Pauli–Jordan D function is introduced in the manifestly covariant canonical formalism of quantum gravity. This function is deﬁned as a bilocal operator that relates to the 4D commutation relation between an abelian gauge ﬁeld and itself. Some properties of this function and the gravitational Pauli–Jordan D function are investigated.


Introduction
In 1982, Nakanishi [1] extended the Pauli-Jordan D function to a q-number and introduced it in the manifestly covariant canonical formalism of quantum gravity. This is called the quantum-gravity D function [2], and is defined by the following q-number version of a Cauchy problem [1][2][3]: Here, g μν (x) is the gravitational field, h ≡ √ − det g μν , δ 3 denotes the spatial delta function 3 k=1 δ(x k − y k ), a symbol | 0 means to set x 0 = y 0 , and D(x, y) the quantum-gravity D function; we use Greek small letters for G L(4) indexes, and raise or lower them by g μν or g μν . Arguments of space-time functions are often omitted for simplicity.
The quantum-gravity D function D(x, y) is a bilocal operator and is not a function of x − y alone, since it is not translationally invariant [2,4]. We must strictly distinguish ∂ y from −∂ x . Hence Kanno and Nakanishi [3] proved Using D(x, y), we have an integral representation for massless fields that satisfies the quantumgravitational d'Alembert equation [1,3]. For example [2,4], the field equation for the electromagnetic PTEP 2013, 123B03 R. Yoshida B-field, ∂ μ (hg μν ∂ ν B) = 0, (1.6) gives us in the quantum gravi-electrodynamics [5]. This is a quantum-gravitational version of an integral representation in QED [2], is the Pauli-Jordan invariant D function. Both the right-hand sides of Eqs. (1.7) and (1.8) are independent of z 0 . The 4D commutation relation between the B-field B(x) in (1.7) and the electromagnetic field A ν (y) is given by Here, C ν (x, y; B) is a linear functional [1] of the B-field and contains commutation relations between A ν , and D or ∂D. Equation (1.9) is a quantum-gravitational version of the 4D commutation relation in QED [2], on the correspondence between (1.7) and (1.8).
Comparing (1.9) with (1.10), we find that the 4D commutation relation between A μ (x) and A ν (y) is given by a form of (1.11) The first term on the right-hand side should be reduced to the 4D commutation relation in the free QED [2], where η μν is the Minkowski metric, α a gauge parameter, and E(x − y) a dipole-ghost invariant D-function. The second term, C μν (x, y; A λ , B), is a linear functional of A λ and of B that contains commutation relations between A μ , and D μν , ∂D μν , D, or ∂D. The purpose of the present paper is to investigate D μν (x, y) in (1.11). For this purpose, we analogously use the properties of the quantum-gravity Pauli-Jordan D function.
This paper is organized as follows. In the next section, we review the manifestly covariant canonical quantum theory of the interacting system of an abelian gauge field and the gravitational one. In Sect. 3, we introduce D μν and investigate its time derivatives. In Sect. 4, we show the transformation properties of D μν and D. Some remarks are made in the last section.

Covariant operator formalism for abelian gauge fields
The Lagrangian density [5] for an abelian gauge field A μ (x) interacting with the gravitational one g μν (x) is given by is an auxiliary scalar field, C and C are the Faddeev-Popov ghost scalar fields, and α denotes a gauge parameter. We treat A μ as the electromagnetic field without any currents in this paper. In order to take A μ , C, and C as the canonical variables, we replace L A byL A [5] as follows: Here the discarded term is a total divergence that is given by 3) The field equations related to the electromagnetic field are The field equation (1.6) for B(x) follows from (2.4). The Lagrangian density (2.2) is invariant under the electromagnetic BRST transformations [5], except for the total divergence in (2.3). The electromagnetic BRST charge [5] is defined by The action integral ofL A in (2.2) is invariant under the gravitational BRST transformations [2], where κ is Einstein's gravitational constant, and c λ is the gravitational Faddeev-Popov ghost field. The gravitational BRST charge [2] is defined by The canonical conjugates of A λ , C, and C are defined by respectively. Here, the functional derivative with respect to C or C is made from the left of each operand. The equal-time canonical commutation and anti-commutation relations are set as follows: In the above and subsequently, a prime attached to a space-time function means that its argument is not x λ but y λ where it is understood that x 0 = y 0 . Using the field equations, the canonical conjugates, and the equal-time canonical (anti-) commutation relations, we obtain various commutation relations [5], e.g., Note that we use the de Donder condition [2], The (anti-)commutation relations [5] between the electromagnetic BRST charge Q B , and the fields A μ , B, C, and C are given by The charge Q B commutes with g μν and b ρ , and anti-commutes with the gravitational Faddeev-Popov ghost fields. The (anti-)commutation relations [2] between the gravitational BRST charge Q G , and the matter fields A μ , B, C, and C are given by Here note the distinction between the BRST transformation in Eqs. (2.13)-(2.16) and that in Eqs. (2.33)-(2.36). The former is the intrinsic BRST transformation [2], which is not commutative with ∂ μ . The latter is the total BRST transformation [2], which is realizable as an algebraic transformation at the operator level. The above BRST charges give us the subsidiary conditions, to define the physical subspace of the indefinite-metric Hilbert space.

Introducing D μν (x, y)
In order to define D μν (x, y) in the right-hand side of (1.11), we form a q-number Cauchy problem for it, which is analogous to Eqs.
respectively. Comparing (1.9) with (1.11), we see a correspondence  Next, we set the following conditions for D μν (x, y) at x 0 = y 0 : Equation (3.6) relates to the commutability between A μ (x) and A ν (y) in (1.11) at x 0 = y 0 . Equation (3.7) relates to the fact that the differentiation of (1.11) with respect to x 0 should coincide with (2.25) at x 0 = y 0 .
We consequently obtain a q-number Cauchy problem (3.4)-(3.7) for D μν (x, y). The bilocal operator D μν (x, y) is not a function of x − y alone, because of the same reason for the quantum-gravity D function D(x, y). In addition, the index μ of D μν (x, y) relates to the vector property at the point x while the index ν does to that at the point y. These facts come from the form of the left-hand side of (1.11).

Transformation properties
In the quantum gravi-electrodynamics [5], there are the affine, the electromagnetic BRST, and the gravitational BRST symmetry. We investigate the transformation properties of D μν (x, y) and D(x, y) with respect to these three symmetries.

Affine transformation
LetP λ andM κ λ be the translation generator and the G L(4) one [2], respectively. The transformation properties [1] of the quantum-gravity D function are given by The proof is done by showing that Eqs. In a similar way, we show that the transformation properties of D μν (x, y) are given by We treat only (4.4), sinceP λ can formally be regarded asM 5 λ with x 5 = y 5 = 1, and δ 5 μ = δ 5 ν = 0. We define the difference between both sides of (4.4) as follows: y).
we obtain