Noether Currents and Maxwell-type Equations of Motion in Higher Derivative Gravity Theories

In general coordinate invariant gravity theories whose Lagrangians contain arbitrarily high order derivative fields, the Noether currents for the global translation and for the Nakanishi's IOSp(8|8) choral symmetry containing the BRS symmetry as its member, are constructed. We generally show that for each of those Noether currents a suitable linear combination of equations of motion can be brought into the form of Maxwell-type field equation possessing the Noether current as its source term.


Introduction
The equation of motion for the Yang-Mills field A a µ in the covariant gauge is given in the form where D ν F a µν is the covariant divergence of the field strength F a µν , j a µ is the color current from the matter field, and B a , c a andc a are Nakanishi-Lautrup (NL), Faddeev-Popov (FP) ghost and anti-ghost fields, respectively. This equation was first noted by Ojima [1] to be rewritten into the form of Maxwell-type equation of motion: Here, Q B is the BRS charge and J a µ in the RHS is the Noether current for the global gauge transformation (= color rotation) under which all the gauge field A a µ , NL and FP ghost fields, B a , c a ,c a , transform as adjoint representations, given by with (A × B) a ≡ f a bc A b B c . This form of YM field equation (1.2) is particular, firstly in the simple divergence form for the field strength, ∂ ν F a µν , and secondly in the BRS exact form for the NL and FP ghost contribution terms.
This form of YM field equation, which we call "Maxwell-type equation of motion", played very important rolls in discussing [2,3] 1. existence of elementary BRS quartet of asymptotic fields 2. spontaneous breaking of color symmetry and Higgs phenomenon 3. unbroken color symmetry and color confinement.
From the technical viewpoint also, it was useful to simplify the computations of equal time commutation (ETC) relations for some field variables as well as to derive Ward-Takahashi identities.
Also in gravity theory, there is a beautiful canonical formulation given by Nakanishi in a series of papers [4][5][6][7][8][9][10] based on the Einstein-Hilbert action with BRS gauge fixing in de Donder gauge. It is summarized in his textbook [12] co-authored with Ojima. He remarked there that the Einstein gravity field equation can also be rewritten in the form of Maxwell-type. [11] In this formulation, he also found a beautiful theorem [12] together with Ojima that the graviton can be identified with a Nambu-Goldstone (NG) massless tensor particle accompanying the spontaneous breaking of GL(4) symmetry down to SO(1, 3) Lorentz symmetry, thus proving the exact masslessness of graviton in Einstein gravity theory. 1 This is an gravitational extension of the Ferrari-Picasso's theorem [14] which proves that the photon is a NG vector boson accompanying the spontaneous breaking of a vector-charge Q µ symmetry, corresponding to the gauge symmetry with transformation parameter linear in x µ . Nakanishi also found in his GL(4)-invariant de Donder gauge that there exists an 8 + 8 dimensional Poincaré-like IOSp(8|8) supersymmetry which he called choral symmetry and contains (as its member) BRS and FP ghost scale symmetries as well as the GL(4) and rigid translation corresponding to the GC transformation with transformation parameter ε µ linear in x µ , ε µ = a µ ν x ν + b µ . However, this work is a formal theory based on the Einstein-Hilbert action. It is perturbative non-renormalizable and may not give a well-defined theory, although there is a possibility that it may satisfy the so-called asymptotic safety [15] and gives a UV complete theory. 2 On the other hand, however, there are many investigations of higher derivative gravity theories. In particular, quadratic gravity [17][18][19] attracted much attention in connection with the perturbative renormalizability [20], Weyl invariant theory [21][22][23], and asymptotic freedom [24][25][26].
These higher derivative theories suffer from the massive (negative metric) ghost problem in perturbative regime, although there have been many proposals for possible ways out (See e.g., Ref. [19] for the review). This ghost problem is, however, out of the scope of this paper.
Even if we are much less ambitious than to make gravity theory UV-complete, we still have several motivations to consider higher derivative gravity theories.
From the low energy effective field theory viewpoint, it is quite natural to consider the actions containing higher and higher order derivative fields, successively, from low to high energies. The Einstein-Hilbert action is the lowest derivative order, and the quadratic gravity actions are the next derivative order, and so on.
Or, alternatively, one may simply wants a gravity theory with a UV cut-off M valid only in the low energy region E < M . A simple momentum cut-off does not work here since it breaks the GC-invariance. Pauli-Villars regulators respecting the GC-invariance can be supplied by considering the covariant higher derivative terms. As noted by Stelle, the gravity field propagator behaves like ∼ 1/p 4 in the quadratic gravity, and sufficiently cut-off the UV contribution to make the theory renormalizable in 4D. As regulators, however, to work sufficiently enough to make all the quantities finite in 4D, the propagator must drop as fast as ∼ 1/p 6 . Such behavior would be supplied, for instance, by the quadratic term of covariant quantities which contain third order derivatives of gravity field.
In this paper we will consider a general gravity theory which is invariant under the general coordinate (GC) transformation and contains arbitrarily high order derivatives of gravity and matter fields, and we 1. derive a concrete form of the Noether current for the rigid translation, i.e., energy momentum tensor, and 2. derive the Maxwell-type gravity equation of motion in gauge unfixed, i.e., classical system, and 3. the Maxwell-type equation analogous to Eq. (1.2) in gauge fixed quantum system in de Donder-Nakanishi gauge.
Original motivation for the present author to consider this problem is to give a sound proof for the existence theorem [27] of massless graviton claiming that there should exist a spin 2 massless graviton in any GC invariant theory as far as it realizes a translational invariant vacuum with flat Minkowski metric. This is a generalization of the Ferrari-Picasso theorem for the massless photon and the Nakanishi-Ojima theorem for the massless graviton. Those theorems were proved explicitly assuming the renormalizable QED and Einstein gravity, respectively. To prove the existence theorem generally, however, it is necessary to have Maxwell-type gravity equation of motion in any GC invariant system assuming no particular form of action. This paper is organized as follows. In Section 2, we present totally general classical system containing arbitrarily high order derivative fields which is only assumed GC transformation invariant. To treat such a system, we introduce a series of generalized both-side derivatives and prove some formulas they satisfy. Based on them, we derive an expression for the energy-momentum tensor for such a general system as the Noether current for the translation invariance, and show that the gravity field equation of motion can be cast into the form of Maxwell-type equation. In Section 3, these results are generalized in the gaugefixed system by adopting the GL(4)-invariant de Donder gaugeà la Nakanishi. In Section 4, using the same technique we show that each of Noether currents of the IOSp(8|8) symmetry can be written in a form of the source current of a Maxwell-type equation. Section 5 is devoted to the conclusion. Some technical points on OSp transformations are discussed in Appendices A and B. In Appendix A, OSp(8|8) transformation of the gauge-fixing plus Faddeev-Popov term is computed for the x µ -dependent transformation parameter. In Appendix B, to get some familiarity with the OSp-symmetry, we briefly study the simplest model, OSp(2|2)-invariant scalar field system on flat Minkowski background; OSp(2|2) Noether current is derived and the IOSp(2|2) algebra are confirmed from the canonical (anti-)commutation relations.

Gravity Equation of Motion in a generic Higher Derivative System
We consider a generic system whose action contains higher order derivative fields up to N -th order ∂ µ1···µN : where φ stands for a collection of fields { φ j } (whose index i may be suppressed when unimportant) and we use abbreviations like The suffix 'weight 1' in the latter means that we keep always the weight to be one irrespectively of whether the n indices µ 1 , µ 2 , · · · , µ n take the same values or not; namely, for the case L = a µν ∂ µν φ, for instance, ∂L/∂(∂ 11 φ) = a 11 and ∂L/∂(∂ 12 φ) = a 12 + a 21 , but we define ∂L/∂(∂ µν φ) weight 1 = (a µν + a νµ )/2! always. The functional derivative of the action S with respect to φ j is given by where N is the highest order n of the derivative fields ∂ µ1···µn φ contained in L (so that L ;µ1···µn j = 0 for n ≥ N + 1), and, for the n = 0 case of empty set {µ 1 · · · µ n }, ∂ µ1···µn = 1 and L ;µ1···µn j = ∂L/∂φ j are understood. The Euler-Lagrange equations are given by δS/δφ j = 0.
The Lagrangian generally changes under an infinitesimal transformation φ → φ + δφ, as where summation over the repeated j is also implied. We consider the system which is invariant under the gauge transformation taking the form For the GC transformation x ρ → x ′ρ = x ρ − ε ρ (x), this field transformation reads more explicitly 3 For a general tensor field φ j = T ν1···νp σi···σq , the symbol φ j µ ρ is defined by [12] T ν1···νp (2.7) The GC invariance of the system implies that the Lagrangian is a scalar density so that the change of L is given by a total divergence δL = ∂ µ (Lε µ ) . (2.8) For the GC transformation δφ in Eq. (2.5), we can equate this expression (2.8) for δL with Eq. (2.4) and obtain an identity This equation, if expanded in a power of derivatives ∂ µ on the gauge transfor- 3 Here, we are taking ε ρ with opposite sign to Nakanishi's so that the definitions of G j ρ , T jµ ρ and φ j µ ρ all have opposite sign to Nakanishi's.
where K k 's (k = 0, 1, 2, · · · ) are defined by with n C k denoting the binomial coefficient n k = n!/k!(n − k)!. Note that the first summation term Since the functions ε ρ , ∂ µ ε ρ , · · · , ∂ µ1···µ k ε ρ are mutually independent, the coefficients should vanish separately, implying the following N + 2 identities which we shall refer to as δL-identities below: (2.14) Here Sym {ν1···ν k µ} implies the totally symmetric part with respect to the k + 1 indices ν 1 , · · · , ν k and µ. Note that only the totally symmetric part of K k ν1···ν k µ ρ should vanish since it vanishes when multiplied by the totally symmetric function ∂ ν1···ν k µ ε ρ (x). Note also that K k ν1···ν k µ ρ is manifestly symmetric with respect to the first k indices ν 1 , · · · , ν k as is clear from the defining Eq. (2.15) First, a linear combination −(δS/δφ j )G j ρ of the equations of motion is rewritten into the following form by adding the first δL-identity (2.12): (2.16) To rewrite this more concisely, we introduce a generalized 'both-side' derivative defined for n ≥ 0 by [28] for arbitrary two functions F and G, with understanding ←− − − −→ ∂ µ1µ2···µn = 1 when n = 0. This derivative is no longer symmetric under permutation of the indices but satisfies a useful formula [28] 18) for any totally symmetric function F µα1···αn with respect to the n + 1 indices { µ, α 1 , · · · , α n }. Applying this formula we can rewrite the identity (2.16) as This J µ ρ is the Noether current for the global GC transformation with xindependent ε ρ (= translation), i.e., energy-momentum tensor, for the higher derivative system. This identity shows that it is indeed conserved when the equations of motion δS/δφ j = 0 are satisfied. Now in order to derive various identities from the rest of the δL-identities, (2.13) and (2.14), we need to introduce generalized both-side derivatives and some formulas for them.
We define k-th both-side derivative k ←→ by induction both in the number k and the differential order n: It is easy to see that the k=0 ←→ is just the same as the original 'both-side' derivative ←→ introduced above in Eq. (2.17); indeed, it satisfies the above recursive defining relation (2.20) for k = 0 as follows: Then, as a generalization of the k = 0 formula (2.18), we have the following formula which holds for all k ≥ 0, n ≥ 0 and for any totally symmetric function F µα1···αn with respect to the n + 1 indices { µ, α 1 , · · · , α n }: The proof easily goes by induction in the number N ≡ k + n in the region k ≥ 0 and n ≥ 0. First note that this formula holds at k = 0 boundary as shown above for ∀n ≥ 0, and clearly hold also at n = 0 boundary with ∀k ≥ 0 since So it is sufficient to prove the formula only for k ≥ 1 and n ≥ 1. Now assume that the formula (2.23) holds for all k ≥ 0 and n ≥ 0 values in the region k + n ≤ ∃N , and let us evaluate the LHS of the formula for any k ≥ 1 and n ≥ 1 with k + n = N + 1. If we use the defining Eq. (2.20) the two terms on the RHS have lower values k + n = N by one than the LHS, to which we can apply the formula by the induction assumption, so that If we note an identity (of Pascal's triangle) n+k C k + n+k C k−1 = n+k+1 C k and apply again the defining Eq. (2.20) with k → k − 1, then we see that the last expression is just reproducing the RHS of the formula (2.23), finishing the proof. Now we are ready to derive the Maxwell-type form of gravity equation of motion. For that purpose let us introduce the following quantity J k for k = 0, 1, 2, · · · : The first of this quantity J 0 µ ρ with k = 0 is a combination of equation of motion, (δS/δφ j )T jµ ρ , Lagrangian L and the energy-momentum tensor J µ ρ : This can be seen from Eqs. (2.19) and (2.15) which are rewritten by using the definition of the k-th both-side derivative with k = 0 and −1, respectively, into Owing to the general formula (2.23), the two quantities, J k introduced here (2.27) and K k defined previously in Eq. (2.11), satisfy the following recurrence relation: When applying the formula (2.23) to derive this equality, we should note that the summation over the set of n + 1 dummy indices {τ, α 1 · · · α n } contained in the RHS quantity ∂ τ J k+1 τ ν1···ν k µ ρ is identified with the summation over the set This implies that the n = 0 terms existing in the summations n=0 in K k and J k on the LHS do not appear on the RHS. However, the n = 0 terms in K k and J k are seen to be the same, so canceling themselves on the LHS.
From this relation (2.30), we find, suppressing the tensor indices, Since L ;ν1···ν k = 0 for k > N , J k as well as K k vanish for k ≥ N + 1. So we find the following expression for J 0 µ ρ reviving the tensor indices: We now insert Eq. (2.28) into J 0 on the LHS, then, noting that the term δ µ ρ L there cancels the K k=0 term on the RHS due to the second δL-identity (2.13), K 0 µ ρ = δ µ ρ L, we find the gravity field equation in the form This is still not the final form. The last summation term can be written as a divergence form of a 'field-strength' tensor F νµ ρ , but it is not yet νµ antisymmetric: However, thanks to the remaining δL-identities (2.14), we can modify it into an νµ antisymmetric field strength F νµ ρ satisfying As noted before, the tensor K k ν1···ν k µ ρ defined in Eq. (2.11) is manifestly totally symmetric with respect to the first k-indices {ν 1 , · · · , ν k }. The δLidentities (2.14) say that it vanishes if further symmetrized including the last index µ; namely, taking the cyclic permutation of the k+1 indices {ν 1 , · · · , ν k , µ} If we act k-ple divergence ∂ ν1···ν k on this, the ν j indices become dummy, and, since the manifest total symmetry among the first k indices of K k , the k terms from the second to the last yield the same quantity and we get Or, taking k → k + 1 and renaming ν k+1 → ν in the first term and ν 1 → ν in the second term, we have This means that K ν1···ν k νµ k+1 ρ can be replaced by a νµ anti-symmetric tensor which we can define as Indeed the difference between K k+1 and K k+1 is given by ρ , (2.40) whose (k + 1)-ple divergence ∂ ν ∂ ν1···ν k is guaranteed to vanish by Eq. (2.38). Thus we find that the 'field-strength' F νµ ρ in Eq. (2.34) can be replaced by the νµ anti-symmetric one: With this antisymmetric field strength, the gravity equation of motion is finally written in the desired form of the Maxwell-type equation: This is an equation for the gauge-unfixed classical system. Here, we note a more explicit expression for K k+1 in terms of the Lagrangian. Substituting the expression (2.11) for K k into the definition (2.39), we note that the G j ρ -proportional part contained in K k+1 ν1···ν k νµ is νµ symmetric so that only the T jµ ρ -proportional part contributes to K k+1 , and obtain for k = 0, 1, 2, · · · .
with h ≡ √ −g andg µν ≡ hg µν . Here the usual BRS transformation δ B (obtained by replacing −ε µ (x) → κc µ (x) for the usual gravity/matter fields) is given by a sum of Nakanishi's BRS δ N and the translation −κc λ ∂ λ : We call this gauge specified by the gauge-fixing and FP term L GF+FP in (3.1) "de Donder-Nakanishi gauge". It corresponds to the de Donder-Landau gauge possessing no αη µν B µ B ν term violating GL(4) invariance by the use of η µν . Since the present L GF+FP for the de Donder-Nakanishi gauge is given in a usual BRS exact form for the de Donder-Landau gauge up to a total derivative term as shown in the last expression in Eq. (3.1), it is also invariant under the usual BRS transformation δ B . The use of Nakanishi's BRS δ N , which represents the tensorial transformation part of the usual BRS transformation δ B , and the use of the b µ = i −1 δ Ncµ field, in particular, make manifest the existence of much larger IOSp(8|8) symmetry, called choral symmetry by Nakanishi, which contains symmetries of energy-momentum, GL(4), BRS, FP-ghost scale transformation etc as will be discussed explicitly in the next section. We still consider the GC transformation x ρ → x ′ρ = x ρ − ε ρ (x) in this quantum theory to derive identities. The gravity/matter fields φ j are transformed in the same way as before: 3) We call the newly added fields b µ ,c µ and c µ ghost fields collectively, and treat them all as scalar fields under GC transformation; namely denoting ghost fields by φ M = (b µ ,c µ , c µ ) collectively, Of course, L GF+FP is not invariant under the GC transformation, but we can easily calculate the change by noting the structure of the L GF+FP , which is written formally as a scalar density: If the ghost part tensor E µν truly behaved as a µν covariant tensor, L GF+FP were a scalar density transforming only into the total divergence ∂ µ (L GF+FP ε µ ). This is actually true for the FP ghost part iκ · ∂ µcρ · ∂ ν c ρ in E µν since c ρ and c ρ are regarded as scalars so that their simple derivatives ∂ µcρ and ∂ ν c ρ behave as µ and ν vectors, giving the desired µν tensor as a product. But the NL field part ∂ µ b ν transforms just as a µ vector since b ν is regarded as a scalar, so that the ν leg rotation part of the transformation ofg µν , i.e., δg µν ⊃ −g µρ ∂ ρ ε ν , is not canceled. We thus see So, the total Lagrangian in our quantum gravity theory changes under the GC transformation as Namely, this differs from Eq. (2.8) in the classical system case only in the point that the κ −1gµν ∂ ν b ρ term is added in the first order derivative term ∝ ∂ µ ε ρ . Therefore, the δL-identities in the previous section almost all remain the same and only the first order ∝ ∂ µ ε ρ identity (2.13) is slightly changed into Note that we should now understand that that L is the total Lagrangian containing the ghost part L GF+FP also and the fields φ j cover not only the gravity/matter fields φ j but also the ghost fields φ M = (b µ ,c µ , c µ ).
So Eq. (2.28) for J 0 holds unchanged. The identity (2.32) also holds as it stands. In going from Eq. (2.32) to the gravity equation (2.33), however, the Lδ µ ρ term from J 0 now does not totally cancel the first term K k=0 µ ρ = Lδ µ ρ + κ −1gµν ∂ ν b ρ but leaves the κ −1gµν ∂ ν b ρ term. Thus the Eq. (2.33) is now replaced by Note here that the implicit summation over φ j also contains the ghost fields φ M which contribute only to the n = 0 terms since the ghost fields appear only in the first order derivatives in the de Donder-Nakanishi gauge Lagrangian (3.1). The final form of Maxwell-type gravity field equation is therefore given by in place of previous classical one (2.42). The expressions Eq. (2.41) for the field-strength F µν ρ and Eq. (2.43) for the quantities K ν1···ν k νµ k+1 ρ remain the same as before. Here L is understood to be the total Lagrangian but actually only the classical Lagrangian part contributes there since all the ghost fields φ M = {b µ ,c µ , c µ } have vanishing contributions since T M µ ρ = 0 for them. That is, the field-strength is in fact the same as that in the classical theory with Lagrangian L cl .
One may wonder that the final Maxwell-type gravity equation of motion (3.12) is slightly different from the Yang-Mills case since the present ghost field term κ −1gµν ∂ ν b ρ is not written in a BRS exact form like {Q B , D µc } in the latter. It is actually possible to rewrite Eq. (3.12) into such a form. Indeed, the term κ −1gµν ∂ ν b ρ is in fact BRS exact up to a divergence of an antisymmetric tensor: (3.13) The gravity field equation (3.12), therefore, can be rewritten into quite a similar form as the Maxwell-type YM equation: where we have written δ B (· · · ) = {iQ B , · · · } in terms of the BRS charge Q B and defined a modified field strength F µν ρ : This form of Maxwell-type equation (3.14) with the BRS exact term was also derived for the Einstein theory case by Nakanishi [11].

Noether Current for the Choral symmetries in a generic Higher Derivative System
BRS symmetry, or more generally, choral symmetries IOSp(8|8) exist for any GC transformation invariant systems if one adopts the gauge-fixing Lagrangian (3.1) of de Donder-Nakanishi gauge [12]. This is because the currents of the choral symmetries are conserved as far as the equations of motion hold for the 16 component 'fields' (=4d coordinate x µ and three fields) [9,10] Indeed, this equation of motion for the coordinate X M ∝ x ρ actually implies the de Donder condition on the gravity field: And the FP ghost equations of motion directly follow from the gauge-fixing Lagrangian (3.1), implying the equations for X M = c ρ andc ρ . The equation for X M = b ρ may be a bit non-trivial, but we now already know the Maxwell-type gravity equation of motion (3.12), the divergence ∂ µ of which immediately leads to where η MN is the OSp(8|8) metric given by Note the symmetry property of this (c-number) metric where the statistics index |M | is 0 or 1 when X M is bosonic or fermionic, respectively. This property (4.8) is because η MN is 'diagonal' in the sense that its off-diagonal, bose-fermi and fermi-bose, matrix elements vanish, i.e., η MN = 0 when |M | = |N |, so that |M | = |N | = |M | · |N | in front of η MN . Note also that theη NM introduced here is just the transposed metric η T NM = η MN . So we have Noting the d'Alembert's equations of motion for X M , Nakanishi constructed the conserved currents 6 He showed from the equal-time commutation relations (ETCR) derived in the Einstein gravity theory that their charge operators generate the following transformations on all the fields Φ, gravity and matter fields φ j as well as the OSp(8|8) ghost-'fields' where the Nakanishi transformation δ MN N is an OSp rotation for the OSp(8|8) ghost-fields X L given by (4.14) This transformation, in particular, gives for the coordinatex ρ which is nonvanishing only when X M or/and X N is b µ . And the Nakanishi transformation of the gravity/matter fields φ j is given by Therefore, if either X M or X N equals b µ , the δ MN transformation is just the GC transformation with transformation parameter ε ρ (x) → E MN ρ = −κ(δ MN Nx ρ ) for the gravity/matter fields φ j , 17) and, for the OSp(8|8) ghost-fields X L , the GC transformation as scalar fields plus an OSp rotation: Note here that the 'transformation parameter' E MN ρ may now be fermionic when |M | + |N | = 1. We have therefore put the factors G j ρ and T jµ ρ linear in φ j behind the parameter E MN ρ in Eqs. (4.17) and (4.18) to avoid the sign factor (−1) |φ j |(|M|+|N |) .
The choral invariance of our total Lagrangian L = L cl + L GF+FP is now clear; the gravity/matter fields receives just a special GC transformation with parameter ε ρ (x) = E MN ρ in Eq. (4.17) and so the L cl part is invariant. The gauge fixing Lagrangian L GF+FP is also clearly invariant since in this form of the Lagrangian (4.6), the OSp-vector field components X M including the coordinatex ρ are treated as scalar fields and hence E µν is clearly µν tensor and hg µν E µν is manifestly a GC scalar density, and, moreover, E µν written in the form (4.6) is manifestly invariant under (global) OSp(8|8) rotation. Note that this invariance is made manifest by making the mere parameter coordinate x µ transforms as if being a field both under the GC transformation and the OSp rotation; actually, those two transformations onx µ cancels each other and the coordinate x µ remains intact under δ MN as any non-field parameters should be: indeed, Eq. (4.13) indicates for Φ =x µ , where the first term is the OSp rotation and the second term is the GC transformation ofx µ regarded as a 'scalar field'. Let us now compute the Noether currents corresponding to these choral symmetries in our general higher derivative GC invariant system. We shall show that the Noether currents coincides with the Nakanishi's simple form (4.10) aside from the divergence of an antisymmetric tensor.
To do this systematically, we devise a local version of the choral symmetry transformation (4.13), or (4.17) and (4.18). We multiply them by a local graded transformation parameter ε NM (x) from the left so that it reduces to the original OSp(8|8) transformation in the global limit ε NM (x) → ε NM :const.; namely, we define the transformation, for gravity/matter fields φ j , (4.20) We take our parameter ε NM Grassmann even or odd according to |N | + |M | = 0 or 1, respectively, so that the product (ε NM E MN ρ ) always becomes an ordinary bosonic 'parameter' and hence can be moved to anywhere without worrying about sign changes. Note, however, that, in order to obtain correctly the Noether current corresponding to the OSp transformation δ MN in Eq. (4.13), or Eqs. (4.17) and (4.18), we have to factor out the parameter ε NM from the left since it is multiplied from the left here. However, the general procedure explained in the previous sections to derive Noether current in the higher derivative theories, which we follow now, have placed the transformation parameter at the most right end, and the troublesome point is that the transformation parameter for the OSp transformation is the graded one ε NM but not the bosonic product (ε NM E MN ρ ). It is necessary to move those graded quantities separately and freely to apply the general procedure to this case, although the transformation parameter ε NM has eventually to be factored out from the left. The best way to forget about the bothering sign factors appearing in changing the order of graded quantities, is to adopt a convention similar to the so-called 'implicit grading' [29]. We take as a natural order of those graded quantities, ε NM the first, E MN ρ second and the other graded quantities like X M , G J ρ and T jmu ρ third. Initially, these quantities appear in this natural order, since the product factor (ε NM E MN ρ ) appearing in Eq. (4.20) is bosonic and can be placed on the most left in any case. Then from this natural order we freely move those factors separately anywhere without writing any sign factors. Implicit grading scheme means that the correct sign factors should be recovered when necessary; that is, in any terms containing those graded quantities, the necessary sign factor can be found by counting how many times of changing order are necessary to bring those factors into the natural order. We adopt hereafter this implicit grading scheme. We should note that the GC transformation part of this transformation (4.20) now takes exactly the same form as the GC transformation (2.5) with (bosonic) transformation parameter ε ρ (x) ≡ ε NM E MN ρ : for all the fields Φ j = (φ j , X M ), and so the total action is still invariant, meaning that the total Lagrangian transforms as a scalar density: δL = ∂ µ (Lε µ ) = ∂ µ (L ε NM E MN ρ ). As for the rest OSp(8|8) rotation part of the OSp coordinate 'fields' X L , however, the L GF+FP in Eq. (4.6) is no longer invariant under the rotation with x-dependent parameter ε MN . As shown explicitly in the Appendix A, we can immediately find the change of L GF+FP as Eq. (A.6): where note that we have already used 'implicit grading' at the last equality. Following this implicit grading scheme, we can write the change of our total Lagrangian L = L cl + L GF+FP under our transformation (4.20) in the form Now we can rewrite our transformation in the same form as the general gauge transformation (2.5), which contains the zero-th and first order differentiation of the transformation parameter ε NM ; that is, it is unifiedly given for gravity/matter and ghost fields Φ I = (φ j , X L ) in the form where the coefficients G IMN and T IMNµ are given as So we can now follow the general discussions presented in the previous two sections to derive the Noether currents in this system. We should also note that the Lagrangian change δL is now given by Eq. (4.24) in place of Eq. (2.8). Then, we see that the previous δL-identities following from the coefficients of n-th order derivatives ∂ µ1···µn ε ρ of the transformation parameter ε ρ , Eqs. (2.12) and (2.14) now also hold with understanding that the coefficients G ρ and T µ ρ of the gauge transformation (2.5) there are now replaced by the present transformation's ones: That is, by making the replacement 27) and taking account of the form of δL given in Eq. (4.24), we find the identities as coefficients of n-th order derivatives ∂ µ1···µn ε NM with n = 0, 1 and n ≥ 2, respectively, N n=0 L ;µ1···µn where K k MN ν1···ν k µ now reads where F MN νµ is the antisymmetric tensor (as an ambiguity term of OSp Noether current) given by for k = 0, 1, 2, · · · . (4.35) The Eq. (4.33) shows that the Noether current of OSp(8|8) symmetry takes the form on-shell, so that the charge can be given by the Nakanishi's simple form as symmetry generators for any local operators. 7 Finally, recall that we are adopting implicit grading. Then, the factor E MN µ or its derivatives contained in G IMN or T IMN µ in the definitions of the current J NMµ , (4.32), the field strengthF MN νµ , (4.34), andK k+1 , (4.35), should be placed at the most left, and if they are kept at the place as written, then the grading sign factor should be put necessary for bringing them there from the most left. Fortunately, however, the sign factors actually turn out to be unnecessary here. This is because the other graded quantities to jump over when bringing the factor E MN µ to the most left are essentially only the bosonic Lagrangian L as a whole. For instance, the Noether current (4.32) contains the terms (4.38) In the second term, it is the Lagrangian L itself for E MN µ to jump over. In the first term, on the other hand, L ;µα1···αn I = ∂L/∂ µα1···αn φ I is fermionic when the field φ I is a fermion. But, it is immediately followed by the factor G I ρ = ∂ ρ φ I in the G IMN = G I ρ E MN ρ , so that the net factor L ;µα1···αn I G I ρ in front of E MN ρ is always bosonic, carrying the same statistics as the original L.

Conclusion
In this paper we considered the general GC invariant theory which contains arbitrarily high order derivative fields. We identified there the explicit expression for the energy-momentum as the Noether current corresponding to the rigid case of GC transformation, and have shown that a linear combination of the equations of motion can be rewritten into the form of Maxwell-type field equation which has the total energy-momentum as its source. This was done both for gauge unfixed classical system and for the gauge-fixed quantum system in the de Donder-Nakanishi gauge. The Maxwell-type field equation in the latter has formally an additional term coming from the gauge-fixing compared with that in the former, which was shown to take a BRS-exact form just like in the well-known Yang-Mills case.
By using the same technique, we derived similar expressions of Maxwelltype equations for the Noether currents for the IOSp(8|8) choral symmetries. It confirmed that the Nakanishi's original result persists to any GC invariant system. These results will be useful for proving the existence theorem [27] of graviton (photon) in any GC (local U (1)) gauge invariant system as far as the translation (global U (1)) symmetry is not spontaneously broken. We hope we can report on this matter near future.
The techniques used for the higher derivative systems in this paper will also be useful for studying more general problems. For instance, the basic problems such as the (non-)equivalence between canonical quantization and path-integral quantization may be discussed in a general fashion for the higher derivative systems. This is so since the present technique can easily be combined with Ostrogradsky's canonical formalism for higher derivative systems [30].

Acknowledgment
The author would like to thank Ichiro Oda for valuable discussions on Weyl invariant gravity theories and reading the early version of this manuscript. This work is supported in part by the JSPS KAKENHI Grant Number JP18K03659.
A OSp transformation of L GF+FP Note that the parameter ε NM and the metric η NM = η NM have the following statistics and transposition properties: Using these properties and also noting that the metric η MN is a usual cnumber (i.e., bosonic) quantity, we can rewrite the two terms in the OSp transformation (4.22) of the OSp coordinates X L as where X, η −1 and ε are regarded as a column vector and matrices in the final expressions. Thus the two terms reduce to the same expression and so the OSp transformation (4.22) is rewritten concisely into Now it is easy to see how the gauge-fixing Lagrangian L GF+FP = −κ −1gµν E µν changes under the OSp transformation (4.22). Noting the expression of E µν in Eq. (4.6), we find Now we note that the OSp transformation parameter ε NM must be graded antisymmetric ε NM = −(−) |N |·|M| ε MN (A. 5) in order for the Lagrangian to be global OSp invariant. Indeed, only the second term remains in the global transformation, and it vanishes if and only if ε NM is graded antisymmetric since ∂ µ X M ∂ ν X N , multiplied byg µν , is graded symmetric. The second term thus vanishes also here, and the remaining first term is rewritten into where, in going to the second line, we have exchanged the dummy indices N ↔ M and used the graded antisymmetry property of ε NM , (A.5).

B OSp(2|2)-invariant scalar field system
To see the property of the OSp symmetry, let us consider here the OSp-invariant system on flat Minkowski background in which scalar fields belongs to an OSp(2|2) vector representation.
where φ M is a 2 + 2-component OSp-vector whose first 2-components are bosons and the rest 2-components are fermions and the OSp(2|2) metric is given by Note that we have put an overall minus sign to our Lagrangian (B.1) in order to make it coincide with the convention of the gauge fixing Lagrangian L GF+FP in Eq. (4.6), although it is not physically important in any case since the OSp metric is neither positive-nor negative-definite. The infinitesimal OSp rotation is parametrized as Under this rotation, the quadratic kinetic Lagrangian (B.1) is transformed as with ε NM ≡ η NL ε L M . This can further be rewritten as So, if the transformation parameter ε NM is graded anti-symmetric, i.e., Then the ETCR is given by 8 10) The Noether current J MNµ for the OSp(2|2) transformation is defined by where we have used the graded-antisymmetry property of ε NM in going from the second to third lines. Thus we have so that the charge is given by This actually generates the original OSp rotation by ETCR's: The IOSp(2|2) algebra is confirmed as