Nakanishi–Kugo–Ojima quantization of general relativity in Heisenberg picture

Abstract

The Chern–Weil topological theory is applied to a classical formulation of general relativity in four-dimensional spacetime. Einstein–Hilbert gravitational action is shown to be invariant with respect to a novel translation (co-translation) operator up to the total derivative; thus, a topological invariant of a second Chern class exists owing to Chern–Weil theory. Using topological insight, fundamental forms can be introduced as a principal bundle of the spacetime manifold. Canonical quantization of general relativity is performed in a Heisenberg picture using the Nakanishi–Kugo–Ojima formalism in which a complete set of quantum Lagrangian and BRST transformations including auxiliary and ghost fields is provided in a self-consistent manner. An appropriate Hilbert space and physical states are introduced into the theory, and the positivity of these physical states and the unitarity of the transition matrix are ensured according to the Kugo–Ojima theorem. The nonrenormalizability of quantum gravity is reconsidered under the formulation proposed herein.

Appendix

1.1 A Co-translation operator and co-Poincaré symmetry

In this Appendix, two remarks related to co-translation symmetry are given based on previous studies [27, 38] by the author. They are essentially important to construct the Chern–Weil gravitational theory introduced in this study.

Suppose $P_a$ be a generator of translation group $T^4$ on four-dimensional local Lorentz manifold $\mathcal{M}$. Translation operator $\delta _T$ acts on vierbein and spin forms, respectively, as [26]

$$\begin{aligned} \delta _T{{\mathfrak {e}}}^a=\mathrm{d}\xi ^a+{{\mathfrak {w}}}^a_{~{\circ }}\xi ^{\circ }=\mathrm{d}_{{\mathfrak {w}}}\xi ^a&\mathrm{and}&\delta _T{{\mathfrak {w}}}^{ab}=0, \end{aligned}$$

where $\xi ^a$ is a local vector to characterize the translation. Zanelli showed the Einstein–Hilbert Lagrangian in a three-dimensional spacetime had a translation invariant owing to an accidental opportunity in the three dimensional space [26]. The co-translation is introduced to realize an extension of a translation symmetry in the Einstein–Hilbert Lagrangian in a four-dimensional spacetime.

A generator of the co-translation is defined using the translation and contraction operators as follows:

Definition A.1

(co-translation): Co-translation generator $P_{ab}$ is defined as

$$\begin{aligned} P_{ab}= & {} P_a\iota _b, \end{aligned}$$

where $\iota _\bullet$ is a contraction operator with respect to local vector $\xi ^\bullet$ in ${T\mathcal{M}}$, whose representation is provided using the trivial basis as

$$\begin{aligned} \iota _{a}= & {} \iota _{\xi ^a},~~\xi ^a=\eta ^{ab}{{\mathcal {E}}}^{\mu }_b\partial _\mu . \end{aligned}$$

The co-translation operator is represented as $\delta _{CT}=\xi ^a\times \delta _T\;\iota _a$, where “$\times$” is an antisymmetric tensor product of the two vectors such that

$$\begin{aligned} \xi ^a\times \xi ^b=\xi ^a\otimes \xi ^b-\xi ^b\otimes \xi ^a=-\xi ^b\times \xi ^a \end{aligned}$$

A contraction is a map $\iota :\Omega ^p({T^*\mathcal{M}}) \rightarrow \Omega ^{p-1}({T^*\mathcal{M}})$. One can obtain a relation,

$$\begin{aligned} \iota _{a}{{\mathfrak {e}}}^b={{\mathcal {E}}}^{\mu }_a{{\mathcal {E}}}_\nu ^b\delta ^\nu _\mu =\delta ^b_a,&\mathrm{and}&\delta _{CT}=\xi ^{\circ }\times \delta _T\left( \iota _{\circ }\;{{\mathfrak {e}}}^a\right) = \xi ^{\circ }\times \left( \delta _T\delta ^a_{\circ }\right) =\xi ^a, \end{aligned}$$

where $\delta _T\delta ^a_b=\delta ^a_b$ is used owing to its translation invariance in ${T\mathcal{M}}$. Above-mentioned relation is independent of the choice of bases in ${T\mathcal{M}}$. A projection manifold and a projection bundle can be introduced using a contraction operator as follows: let $\mathcal{M}_{\perp a}\subset \mathcal{M}$ be a three-dimensional submanifold of $\mathcal{M}$, such that $\iota _a{{\mathfrak {a}}}=0$ for ${{\mathfrak {a}}}$ with $\Omega ^1(\mathcal{M}_{\perp a})\ni {{\mathfrak {a}}}\ne 0$. The trivial frame bundles in $T\mathcal{M}_{\perp a}$ and $T^*\mathcal{M}_{\perp a}$ are regarded as sub-bundles of $T\mathcal{M}$ and $T^*\mathcal{M}$, respectively. If $\mathcal{M}$ has Poincaré symmetry ISO(1, 3), the submanifold $\mathcal{M}_{\perp a}$ has the ISO(1, 2) or the ISO(3) symmetry. Therefore, connection ${{\mathfrak {A}}}_a$ and curvature ${\mathfrak {F}}_a=\mathrm{d}{{\mathfrak {A}}}_a+{{\mathfrak {A}}}_a\wedge {{\mathfrak {A}}}_a$ are provided in $\mathcal{M}_{\perp a}$ as sub-bundles of ${{\mathfrak {A}}}$ and ${\mathfrak {F}}$ on $\mathcal{M}$, respectively. Using an equivalence relation $\sim _a$ such as ${{\mathfrak {a}}}\sim _a{{\mathfrak {b}}}\Leftrightarrow \iota _a{{\mathfrak {a}}}=\iota _a{{\mathfrak {b}}}$, quotient bundles ${\widetilde{{{\mathfrak {A}}}}}_a={{\mathfrak {A}}}/{\sim _a}$ and ${\widetilde{{\mathfrak {F}}}}_a={\mathfrak {F}}/{\sim _a}$ can be introduced, where ${{\mathfrak {a}}}$ and ${{\mathfrak {b}}}$ are any p-forms on $T^*\mathcal{M}$. It is clear that ${{\mathfrak {A}}}_a\simeq {\widetilde{{{\mathfrak {A}}}}}_a\subset {{\mathfrak {A}}}$ and ${\mathfrak {F}}_a\simeq {\widetilde{{\mathfrak {F}}}}_a\subset {\mathfrak {F}}$.

The co-translation acts on the fundamental forms as follows:

$$\begin{aligned} \delta _{CT}\left( {{\mathfrak {S}}}_{ab}\right)= & {} \frac{1}{2}\epsilon _{abcd}\left( \xi ^c\times \mathrm{d}_{{{\mathfrak {w}}}}\xi ^d-\xi ^\mathrm{d}\times \mathrm{d}_{{{\mathfrak {w}}}}\xi ^c,\right) \end{aligned}$$

(52)

$$\begin{aligned} \delta _{CT} {{\mathfrak {w}}}^{ab}= & {} 0. \end{aligned}$$

(53)

Remark A.2

(Remark 3.2 in Ref. [27]) The Einstein–Hilbert Lagrangian given in (11) with $\Lambda _c=0$ is invariant under the co-Poincaré transformation up to a total derivative.

Proof

The gravitational Lagrangian with $\Lambda _c=0$ is considered. The invariance under the local SO(1, 3) symmetry of the Einstein–Hilbert Lagrangian is trivial from its definition. A co-translation operator $\delta _{CT}$ induces a transformation on the Einstein–Hilbert Lagrangian ${{\mathfrak {L}}}_G$ as

$$\begin{aligned} (\hbar \kappa )\delta _{CT}{{\mathfrak {L}}}_G= & {} \delta _{CT}\left( \frac{1}{2} \epsilon _{{\circ \circ }{\circ \circ }}{\mathfrak {R}}^{\circ \circ }\wedge {{\mathfrak {e}}}^{\circ }\wedge {{\mathfrak {e}}}^{\circ }\right) =\epsilon _{{\circ \circ }{\circ \circ }}{\mathfrak {R}}^{\circ \circ }\wedge \left( \xi ^{\circ }\times \mathrm{d}_{{{\mathfrak {w}}}}\xi ^{\circ }\right) \nonumber \\= & {} \mathrm{d}_{{{\mathfrak {w}}}}\left\{ \epsilon _{{\circ \circ }{\circ \circ }}{\mathfrak {R}}^{\circ \circ }(\xi ^{\circ }\times \xi ^{\circ }) \right\} , \end{aligned}$$

(54)

where (52), (53) and the Bianchi identity $\mathrm{d}_{{{\mathfrak {w}}}}{\mathfrak {R}}=0$ are used. Owing to the definition of the covariant derivative (2), $\mathrm{d}_{{{\mathfrak {w}}}}\{\cdots \}=\mathrm{d}\{\cdots \}$ is maintained in (54); thus, the co-translation operator only induces a total derivative term on the Einstein–Hilbert Lagrangian without the cosmological constant. $\square$

Remark A.3

(Theorem 4.2 in Ref. [27] and Remark 2.1 in Ref. [38]) The Einstein–Hilbert Lagrangian without the cosmological constant is defined using the curvature of the co-Poincaré bundle as (24). It has a topological invariant as the second Chern class $c_2({\mathfrak {F}}_{c\mathrm {P}}):$

$$\begin{aligned} {{\mathfrak {L}}}= \frac{8\pi ^2}{\kappa }c_2({\mathfrak {F}}_{c\mathrm {P}})\in {\mathbb {R}}\otimes H^4(\mathcal{M},{\mathbb {Z}}). \end{aligned}$$

Proof

The right-hand side of (24) is expressed using the structure constants as

$$\begin{aligned} \mathrm {Tr}\left[ {\mathfrak {F}}_{c\mathrm {P}}\wedge {\mathfrak {F}}_{c\mathrm {P}}\right]:= & {} \frac{1}{2} {{\mathcal {F}}}^K_{~IJ}\;{\mathfrak {F}}_{c\mathrm {P}}^I\wedge {\mathfrak {F}}_{c\mathrm {P}}^J\;T_K. \end{aligned}$$

Owing to the definition of the co-Poincaré curvature (23) and the Lie algebra of the co-Poincaré symmetry, a direct calculation with a trivial basis yields

$$\begin{aligned} \mathrm {Tr}\left[ {\mathfrak {F}}_{c\mathrm {P}}\wedge {\mathfrak {F}}_{c\mathrm {P}}\right]= & {} \mathrm {Tr}\left[ {\mathfrak {R}}^{ac}\wedge \mathrm{d}_{{\mathfrak {w}}}\left( P_{a}^{b}\;{{\mathfrak {S}}}_{cb}\right) \right] = \mathrm{d}_{{\mathfrak {w}}}\left( \mathrm {Tr}\left[ P_{a}^{b}\right] {\mathfrak {R}}^{ac}\wedge {{\mathfrak {S}}}_{cb}\right) = -\mathrm{d}_{{\mathfrak {w}}} \left( P_{{\circ }}^{{\circ }} \;{\mathfrak {R}}\wedge {{\mathfrak {S}}}\right) , \end{aligned}$$

where Bianchi identity $\mathrm{d}_{{\mathfrak {w}}}{\mathfrak {R}}=0$ and co-translation invariance $P\;({\mathfrak {R}})=0$ are used. For the last equality, we note that $\mathrm {Tr}\left[ {\mathfrak {R}}^{ac}\wedge {{\mathfrak {S}}}_{cb}\right] =-\delta ^a_b({\mathfrak {R}}\wedge {{\mathfrak {S}}})$. The Einstein–Hilbert gravitational Lagrangian is co-translation-invariant up to total derivative as given as Remark A.2; thus, P is removed from the last expression and it is embedded in a five-dimensional line bundle of ${\mathscr {M}}_5:={\mathscr {M}}\otimes {\mathbb {R}}$, whose boundary is given as $\partial {\mathscr {M}}_5=\Sigma$. Consequently,

$$\begin{aligned} -\int _{\Sigma }\mathrm {Tr}\left[ {\mathfrak {F}}_{c\mathrm {P}}\wedge {\mathfrak {F}}_{c\mathrm {P}}\right]= & {} \int _{{\mathscr {M}}_5}\mathrm{d}_{{\mathfrak {w}}}\left( {\mathfrak {R}}\wedge {{\mathfrak {S}}}\right) = \int _{{\mathscr {M}}_5}\mathrm{d}\left( {\mathfrak {R}}\wedge {{\mathfrak {S}}}\right) = \int _{\partial {\mathscr {M}}_5=\Sigma }{\mathfrak {R}}\wedge {{\mathfrak {S}}}. \end{aligned}$$

A surface term is eliminated owing to a boundary condition. We note that

$$\begin{aligned} \mathrm{d}_{{\mathfrak {w}}}\left( {\mathfrak {R}}\wedge {{\mathfrak {S}}}\right) = \mathrm{d}\left( {\mathfrak {R}}\wedge {{\mathfrak {S}}}\right) +{{\mathfrak {w}}}\wedge {\mathfrak {R}}\wedge {{\mathfrak {S}}}-{\mathfrak {R}}\wedge {{\mathfrak {S}}}\wedge {{\mathfrak {w}}}=\mathrm{d}\left( {\mathfrak {R}}\wedge {{\mathfrak {S}}}\right) , \end{aligned}$$

owing to the definition of a covariant derivative.

The second Chern class with respect to the co-Poincaré curvature is obtained as

$$\begin{aligned} c_2({\mathfrak {F}}_{c\mathrm {P}}):= & {} \frac{1}{8\pi ^2}\left( \mathrm {Tr}\left[ {\mathfrak {F}}_{c\mathrm {P}}\right] ^2 -\mathrm {Tr}\left[ {\mathfrak {F}}_{c\mathrm {P}}\wedge {\mathfrak {F}}_{c\mathrm {P}}\right] \right) , \end{aligned}$$

and it has a homology class as $c_2({\mathfrak {F}}_{c\mathrm {P}})\in H^4(\mathcal{M},{\mathbb {R}})\subset \mathrm {Im}\;H^4(\mathcal{M},{\mathbb {Z}})$ owing to the Chern–Weil theory. The co-Poincaré curvature ${\mathfrak {F}}_{c\mathrm {P}}$ is traceless; thus, the remark is maintained. $\square$

This result suggests that appropriate fundamental forms of the symplectic geometry for general relativity can be identified as $({{\mathfrak {w}}},{{\mathfrak {S}}})$. This choice of the canonical pair of general relativity is equivalent to that proposed by Kanatchikov [81] on the basis of the de Donder–Weyl Hamiltonian theory.

1.2 B Proof of nilpotency

The coordinate vectors are fundamental vectors on $T{\mathscr {M}}$. Nilpotent can be confirmed as

$$\begin{aligned}&\delta _{\mathrm {BRST}}\left[ \delta _{\mathrm {BRST}}\left[ x^{\mu }\right] \right] =\delta _{\mathrm {BRST}}\left[ \chi ^\mu \right] =\delta _{\mathrm {BRST}}\left[ g^{\mu \nu }\chi _\nu \right] ,\\&\quad = g^{\mu \rho }\left( \partial _\rho \chi ^\nu \right) \chi _\nu + g^{\rho \nu }\left( \partial _\rho \chi ^\mu \right) \chi _\nu -g^{\mu \nu }g_{\nu \rho }\left( \partial ^\rho \chi ^\sigma + \partial ^\sigma \chi ^\rho \right) \chi _\sigma ,\\&\quad = \left( \partial ^\mu \chi ^\nu \right) \chi _\nu + \left( \partial ^\nu \chi ^\mu \right) \chi _\nu -\left( \partial ^\mu \chi ^\sigma \right) \chi _\sigma - \left( \partial ^\sigma \chi ^\mu \right) \chi _\sigma =0. \end{aligned}$$

Starting from the BRST transformation of the metric tensor (29), nilpotent is provided as

$$\begin{aligned} \delta _{\mathrm {BRST}}\left[ \delta _{\mathrm {BRST}}\left[ g_{\mu \nu }\right] \right]= & {} \delta _{\mathrm {BRST}}\left[ -g_{\mu \rho }\partial _\nu \chi ^\rho \right] +\delta _{\mathrm {BRST}}\left[ \mu \leftrightarrow \nu \right] ,\\= & {} \left\{ -\delta _{\mathrm {BRST}}\left[ g_{\mu \rho }\right] \partial _\nu \chi ^\rho +g_{\mu \rho }\left( \partial _\nu \delta _{\mathrm {BRST}}\left[ x^\sigma \right] \right) \partial _\sigma \chi ^\rho \right\} +\left\{ \mu \leftrightarrow \nu \right\} ,\\= & {} \left\{ g_{\mu \sigma }\left( \partial _\rho \chi ^\sigma \right) \partial _\nu \chi ^\rho + g_{\mu \rho }\left( \partial _\nu \chi ^\sigma \right) \partial _\sigma \chi ^\rho \right\} +\left\{ \mu \leftrightarrow \nu \right\} =0, \end{aligned}$$

where anticommutativity of the ghost filed is used.

Since the ghost field has two parts, nilpotent is checked separately. First, nilpotent of the $\chi _\mu$ is shown as

$$\begin{aligned} \delta _{\mathrm {BRST}}\left[ \delta _{\mathrm {BRST}}\left[ \chi _\mu \right] \right]= & {} \delta _{\mathrm {BRST}}\left[ \delta _{\mathrm {BRST}}\left[ g_{\mu \nu }\chi ^\nu \right] \right] =\delta _{\mathrm {BRST}}\left[ \delta _{\mathrm {BRST}}\left[ g_{\mu \nu }\right] \right] \chi ^\nu =0, \end{aligned}$$

where $\delta _{\mathrm {BRST}}[\chi ^\mu ]=0$ and nilpotent of the metric tensor are used. Direct calculation from (27) gives the same result, too. The second part becomes

$$\begin{aligned} \delta _{\mathrm {BRST}}\left[ \delta _{\mathrm {BRST}}\left[ \chi ^a_{~b}\right] \right]= & {} \delta _{\mathrm {BRST}}\left[ \chi ^a_{~c}\chi ^{c}_{~b}\right] = \chi ^a_{~c_2}\chi ^{c_2}_{~~c_1}\chi ^{c_1}_{~~b}-\chi ^a_{~c_1}\chi ^{c_1}_{~~c_2}\chi ^{c_2}_{~~b}=0, \end{aligned}$$

due to anticommutativity of the ghost field.

Next, a tensor $\partial _{\mu }\chi ^{\nu }$ is also nilpotent as

$$\begin{aligned}&\delta _{\mathrm {BRST}}\left[ \delta _{\mathrm {BRST}}\left[ \partial _{\mu }\chi ^{\nu }\right] \right] = -\delta _{\mathrm {BRST}}\left[ \partial _{\mu }\chi ^{\rho }\partial _{\rho }\chi ^{\nu }\right] \\&\quad =-\partial _{\mu }\chi ^{\rho _1}\partial _{\rho _1}\chi ^{\rho _2}\partial _{\rho _2}\chi ^{\nu } +\partial _{\mu }\chi ^{\rho _1}\partial _{\rho _1}\chi ^{\rho _2}\partial _{\rho _2}\chi ^{\nu }=~0. \end{aligned}$$

Nilpotent of the vierbein form is provided as

$$\begin{aligned} \delta _{\mathrm {BRST}}\left[ \delta _{\mathrm {BRST}}\left[ {{\mathfrak {e}}}^a\right] \right]= & {} \delta _{\mathrm {BRST}}\left[ {{\mathfrak {e}}}^b\chi ^a_{~b}\right] =~ {{\mathfrak {e}}}^{b_1}\chi ^{b_2}_{~~b_1}\chi ^a_{~b_2}+{{\mathfrak {e}}}^{b_2}\chi ^a_{~b_1}\chi ^{b_1}_{~~b_2}=0, \end{aligned}$$

due to anticommutativity of the ghost field.

One can trace the same calculation as a case of the vierbein form due to $\delta _{\mathrm {BRST}}\left[ \mathrm{d}\chi ^{ab}\right] =0$. Detailed calculations are omitted here.

The volume form is global scalar, and their BRST transformation is expected to vanish, which can be confirmed as

$$\begin{aligned} \delta _{\mathrm {BRST}}\left[ {\mathfrak {v}}\right]= & {} \frac{1}{4!}\epsilon _{{\circ \circ \circ \circ }} \delta _{\mathrm {BRST}}\left[ {{\mathfrak {e}}}^{\circ }\wedge {{\mathfrak {e}}}^{\circ }\wedge {{\mathfrak {e}}}^{\circ }\wedge {{\mathfrak {e}}}^{\circ }\right] \\= & {} \frac{1}{3!}\epsilon _{a_1{\circ }{\circ \circ }} \delta _{\mathrm {BRST}}\left[ \chi ^{a_1}_{~~a_2}{{\mathfrak {e}}}^{a_2}\wedge {{\mathfrak {e}}}^{\circ }\wedge {{\mathfrak {e}}}^{\circ }\wedge {{\mathfrak {e}}}^{\circ }\right] ~=~0, \end{aligned}$$

due to ${{\mathfrak {e}}}^{\circ }\wedge {{\mathfrak {e}}}^{\circ }\wedge {{\mathfrak {e}}}^{\circ }\wedge {{\mathfrak {e}}}^{\circ }\propto \epsilon ^{{\circ \circ \circ \circ }}$ and $\chi ^{a_1}_{~~a_2}=0$ when $a_1=a_2$.

The BRST transformation of the surface form is provided as

$$\begin{aligned} \delta _{\mathrm {BRST}}\left[ {{\mathfrak {S}}}_{ab}\right]= & {} \frac{1}{2}\epsilon _{abc_1c_2}\delta _{\mathrm {BRST}}\left[ {{\mathfrak {e}}}^{c_1}\wedge {{\mathfrak {e}}}^{c_2}\right] =\epsilon _{abc_1c_2}\chi ^{c_1}_{~c_3}{{\mathfrak {e}}}^{c_3}\wedge {{\mathfrak {e}}}^{c_2}~ \left( =~{{\mathfrak {c}}}^{\circ }\wedge {\overline{{{\mathfrak {e}}}}}_{ab{\circ }}\right) . \end{aligned}$$

Applying the BRST-transformation on it again, one can get

$$\begin{aligned}&\delta _{\mathrm {BRST}}\left[ \delta _{\mathrm {BRST}}\left[ {{\mathfrak {S}}}_{ab}\right] \right] ~=~ \epsilon _{abc_1c_2}\delta _{\mathrm {BRST}}\left[ \chi ^{c_1}_{~c_3}{{\mathfrak {e}}}^{c_3}\wedge {{\mathfrak {e}}}^{c_2}\right] ,\\&\quad =\epsilon _{abc_1c_2}\Bigl \{ \chi ^{c_1}_{~c_4}\chi ^{c_4}_{~c_3}{{\mathfrak {e}}}^{c_3}\wedge {{\mathfrak {e}}}^{c_2}- \chi ^{c_1}_{~c_3}\chi ^{c_3}_{~c_4}{{\mathfrak {e}}}^{c_4}\wedge {{\mathfrak {e}}}^{c_2}- \chi ^{c_1}_{~c_3}\chi ^{c_2}_{~c_4}{{\mathfrak {e}}}^{c_3}\wedge {{\mathfrak {e}}}^{c_4} \Bigr \}=0, \end{aligned}$$

because first term is the same as the second term and the third term is symmetric with $c_1$ and $c_2$ exchange.

The BRST transformation of ${{\mathfrak {c}}}^a$ is given by

$$\begin{aligned} \delta _{\mathrm {BRST}}\left[ {{\mathfrak {c}}}^a\right]= & {} \delta _{\mathrm {BRST}}\left[ \chi ^a_{~b}~{{\mathcal {E}}}^b_\mu ~\mathrm{d}x^\mu \right] ,\\= & {} \chi ^a_{~b_1}\chi ^{b_1}_{~b_2}{{\mathcal {E}}}^{b_2}_\mu \mathrm{d}x^\mu - \chi ^a_{~b_1}~{{\mathcal {E}}}^{b_2}_\mu \chi ^{b_2}_{~b_1}\mathrm{d}x^\mu +\chi ^a_{~b}~\left( \partial _\mu \chi ^\nu \right) {{\mathcal {E}}}^b_\nu ~\mathrm{d}x^\mu -\chi ^a_{~b}~{{\mathcal {E}}}^b_\mu \mathrm{d}\chi ^\mu =0, \end{aligned}$$

Nilpotent of other forms is trivial, and the proof is omitted here.

The quantum Lagrangian must be the BRST-null. Gauge-fixing and Fadeef–Popov Lagrangians are constructed to satisfy the BRST-null condition in Sect. 6-2. Nilpotent of only the gravitational Lagrangian is given here. The BRST transformation of the gravitational Lagrangian is provided as

$$\begin{aligned} \delta _{\mathrm {BRST}}\left[ {{\mathfrak {L}}}_G\right]= & {} \frac{1}{2}\delta _{\mathrm {BRST}}\left[ \left( \mathrm{d}{{\mathfrak {w}}}^{\circ \circ }+{c_{gr}}\;{{\mathfrak {w}}}^{\circ }_{~\star }\wedge {{\mathfrak {w}}}^{\star {\circ }}\right) \wedge {{{\mathfrak {S}}}}_{\circ \circ }-\frac{\Lambda }{3!}{\mathfrak {v}}\right] . \end{aligned}$$

The BRST transformation for the volume form is vanished by itself. For the derivative term,

$$\begin{aligned} \delta _{\mathrm {BRST}}\left[ \mathrm{d}{{\mathfrak {w}}}^{\circ \circ }\wedge {{{\mathfrak {S}}}}_{\circ \circ }\right]= & {} \epsilon _{abc_2c_3}\chi _{~c_1}^{b}\mathrm{d}{{\mathfrak {w}}}^{ac_1}\wedge {{\mathfrak {e}}}^{c_2}\wedge {{\mathfrak {e}}}^{c_3}+ \epsilon _{abc_2c_3}{{\mathfrak {w}}}^{ac_1}\wedge \mathrm{d}\chi _{~c_1}^{b}\wedge {{\mathfrak {e}}}^{c_2}\wedge {{\mathfrak {e}}}^{c_3}\\&+\epsilon _{abc_1c_2}\chi ^{c_1}_{~c_3}~\mathrm{d}{{\mathfrak {w}}}^{ab}\wedge {{\mathfrak {e}}}^{c_3}\wedge {{\mathfrak {e}}}^{c_2}\\= & {} 2~{{\mathfrak {w}}}^{ac_1}\wedge \mathrm{d}\chi ^{b}_{~c_1}\wedge {{\mathfrak {S}}}_{ab}, \end{aligned}$$

where first- and third-terms are cancelled each other. Remnant term is transformed as

$$\begin{aligned}&\delta _{\mathrm {BRST}}\left[ {{\mathfrak {w}}}^{\circ }_{~\star }\wedge {{\mathfrak {w}}}^{\star {\circ }}\wedge {{{\mathfrak {S}}}}_{\circ \circ }\right] \\&\quad =\epsilon _{abc_2c_3}\chi ^{c_2}_{~c_4}{{\mathfrak {w}}}^{ac_1}\wedge {{\mathfrak {w}}}_{c_1}^{~~b}\wedge {{\mathfrak {e}}}^{c_4}\wedge {{\mathfrak {e}}}^{c_3}+ \epsilon _{abc_3c_4}\chi ^{c_2}_{~c_1}{{\mathfrak {w}}}^{ac_1}\wedge {{\mathfrak {w}}}_{c_2}^{~~b}\wedge {{\mathfrak {e}}}^{c_3}\wedge {{\mathfrak {e}}}^{c_4}\\&\qquad +\epsilon _{abc_3c_4}\chi ^{b}_{~c_2}{{\mathfrak {w}}}^{ac_1}\wedge {{\mathfrak {w}}}_{c_1}^{~~c2}\wedge {{\mathfrak {e}}}^{c_3} \wedge {{\mathfrak {e}}}^{c_4}-2{{c_{gr}}}^{-1}{{\mathfrak {w}}}^{ac_1}\wedge \mathrm{d}\chi ^b_{~c_1}\wedge {{\mathfrak {S}}}_{ab},\\&\quad =-2{{c_{gr}}}^{-1}{{\mathfrak {w}}}^{ac_1}\wedge \mathrm{d}\chi ^b_{~c_1}\wedge {{\mathfrak {S}}}_{ab}. \end{aligned}$$

In the r.h.s of the first line, the second term is zero as itself, and first- and third-terms are cancelled each other. Therefore, one can confirm $\delta _{\mathrm {BRST}}\left[ {{\mathfrak {L}}}_G\right] =0$ after summing up all terms.

If we use a following remake, we can give simpler proofs for above forms.

If both of two fields, $\alpha$ and $\beta$, are nilpotent, $\alpha \beta$ is also nilpotent.

Proof

If a field X is nilpotent, signatures of the Leibniz rule satisfy $\epsilon _{X}=-\epsilon _{\delta X}$ due to $\delta _{\mathrm {BRST}}[\delta _{\mathrm {BRST}}[X]]=0$ and (28), where $\epsilon _{X}$ ($\epsilon _{\delta X}$) is a signature of X ($\delta _{\mathrm {BRST}}[X]$), respectively. Therefore,

$$\begin{aligned} \delta _{\mathrm {BRST}}\left[ \delta _{\mathrm {BRST}}\left[ \alpha \beta \right] \right]= & {} \epsilon _{\alpha }\delta _{\mathrm {BRST}}\left[ \alpha \right] \delta _{\mathrm {BRST}}\left[ \beta \right] + \epsilon _{\delta \alpha }\delta _{\mathrm {BRST}}\left[ a\right] \delta _{\mathrm {BRST}}\left[ \beta \right] =0. \end{aligned}$$

$\square$

1.3 C Equations of motion

From the classical Lagrangian form, the torsionless condition and the Einstein equation are obtained as equations of motion by requiring a stationary condition for variation of action. The same procedure can be used to obtain Euler–Lagrange equations from the quantum Lagrangian. Here, the quantum Lagrangian is summarized as

$$\begin{aligned}&{{{\mathfrak {L}}}_{QG}}={{\mathfrak {L}}}_G+{{{\mathfrak {L}}}_{GF}}+{{{\mathfrak {L}}}_{FP}},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(31)\\&\quad \left\{ \begin{array}{llc} {{\mathfrak {L}}}_G&{}=&{}~~~~~~\frac{1}{2}\left( {\mathfrak {R}}^{\circ \circ }-\frac{\Lambda }{3!}{\overline{{{\mathfrak {S}}}}}^{\circ \circ }\right) \wedge {{{\mathfrak {S}}}}_{\circ \circ },~~~~~~~~~~~~~~~~~~~~~~(11)\\ {{{\mathfrak {L}}}_{GF}}&{}=&{} -\frac{1}{2}\left( \mathrm{d}{{\mathfrak {b}}}^{\circ \circ }+\alpha {{\mathfrak {b}}}^{\circ }_{~\star }\wedge {{\mathfrak {b}}}^{\star {\circ }} \right) \wedge {{\mathfrak {S}}}_{\circ \circ },~~~~~~~~~~~~~~~~~~~~(32)\\ {{{\mathfrak {L}}}_{FP}}&{}=&{} -\frac{i}{2}\left( d{\tilde{{{\mathfrak {c}}}}}^{\circ \circ }+ \alpha {\tilde{{{\mathfrak {c}}}}}^{{\circ }}_{~\star }\wedge {{\mathfrak {b}}}^{\star {\circ }}\right) \wedge {{\mathfrak {c}}}^\star \wedge {\overline{{{\mathfrak {e}}}}}_{\star {\circ \circ }},~~~~~~~~~~~~~~~(33) \end{array} \right. \end{aligned}$$

Euler–Lagrange equations are obtained as follows:

$$\begin{aligned} {{\mathfrak {T}}}^a= & {} \mathrm{d}{{\mathfrak {e}}}^a+{c_{gr}}\;{{\mathfrak {w}}}^a_{~{\circ }}\wedge {{\mathfrak {e}}}^{\circ }=0. \end{aligned}$$

This is the torsionless condition in the same form as the classical Lagrangian.

$$\begin{aligned}&\frac{1}{2}\overline{\left( {\mathfrak {R}}^{\circ \circ }\wedge {{\mathfrak {e}}}^{\circ }\right) }_a-\Lambda {\mathfrak {V}}_a -\frac{1}{2}\left( \mathrm{d}{{\mathfrak {b}}}^{\circ \circ }+\alpha {{\mathfrak {b}}}^{\circ }_{~\star }\wedge {{\mathfrak {b}}}^{\star {\circ }} \right) \wedge {\overline{{{\mathfrak {e}}}}}_{a{\circ \circ }}\nonumber \\&\quad -\frac{i}{2}\left( d{\tilde{{{\mathfrak {c}}}}}^{\circ \circ }+\alpha {\tilde{{{\mathfrak {c}}}}}^{\circ }_{~\star }\wedge {{\mathfrak {b}}}^{\star {\circ }}\right) \wedge {\overline{{{\mathfrak {c}}}}}_{a{\circ \circ }} =0, \end{aligned}$$

(55)

where ${\mathfrak {V}}_a=\epsilon _{ab_1b_2b_3}{{\mathfrak {e}}}^{b_1}\wedge {{\mathfrak {e}}}^{b_2}\wedge {{\mathfrak {e}}}^{b_3}/3!$. The first two terms give the Einstein equation without matter fields. Third- and fourth-terms newly appeared from the gauge-fixing and Faddeev–Popov Lagrangian forms.

$$\begin{aligned} \mathrm{d}{{\mathfrak {S}}}_{ab}-\alpha \left( {{\mathfrak {b}}}^{\circ }_a\wedge {{\mathfrak {S}}}_{{\circ }b}-i{\tilde{{{\mathfrak {c}}}}}^{\circ }_a\wedge {{\mathfrak {c}}}^{\circ }\wedge {\overline{{{\mathfrak {e}}}}}_{{\circ \circ }b} \right)= & {} 0. \end{aligned}$$

We note that ${{\mathfrak {b}}}$ and $\delta _{{\mathfrak {b}}}$ are anticommute each other, and the variation operator is applied from the left. When the Landau gauge is used, the de Donder gauge-fixing condition $\mathrm{d}{{\mathfrak {S}}}_{ab}=0$ is obtained.

$$\begin{aligned} \left( d{\tilde{{{\mathfrak {c}}}}}^{\circ \circ }+ \alpha {\tilde{{{\mathfrak {c}}}}}^{{\circ }}_{~\star }\wedge {{\mathfrak {b}}}^{\star {\circ }}\right) \wedge {\overline{{{\mathfrak {e}}}}}_{a{\circ }{\circ }}= & {} 0, \end{aligned}$$

(56)

where the anticommutation among ${{\mathfrak {c}}}$, $\delta {{\mathfrak {c}}}$ and ${{\mathfrak {b}}}$ is used.

$$\begin{aligned} \epsilon _{ab{\circ \circ }}\left( \mathrm{d}\left( {{\mathfrak {c}}}^{\circ }\wedge {{\mathfrak {e}}}^{\circ }\right) -\alpha {{\mathfrak {b}}}^{\circ }_{~\star }\wedge {{\mathfrak {c}}}^\star \wedge {{\mathfrak {e}}}^{\circ }\right) = \epsilon _{ab{\circ \circ }}\left( \mathrm{d}{{\mathfrak {c}}}^{\circ }-\alpha {{\mathfrak {b}}}^{\circ }_{~\star }\wedge {{\mathfrak {c}}}^\star \right) \wedge {{\mathfrak {e}}}^{\circ }~=~0, \end{aligned}$$

where the de Donder condition is used.

The BRST transformation may give another set of equations, which must be consistent with above equations:

$$\begin{aligned} \delta _{\mathrm {BRST}}\left[ {{\mathfrak {T}}}^a\right]= & {} \chi ^a_{~{\circ }}~\mathrm{d}{{\mathfrak {e}}}^{\circ }+\mathrm{d}\chi ^a_{~{\circ }}\wedge {{\mathfrak {e}}}^{\circ }+{c_{gr}}\;\chi ^a_{~{\circ }}~{{\mathfrak {w}}}^{~{\circ }}_{\star }\wedge {{\mathfrak {e}}}^\star \\&+{c_{gr}}\;\chi _{\star }^{~{\circ }}~{{\mathfrak {w}}}^{a}_{~{\circ }}\wedge {{\mathfrak {e}}}^\star -\mathrm{d}\chi ^a_{~{\circ }}\wedge {{\mathfrak {e}}}^{\circ }+{c_{gr}}\;\chi ^{\circ }_{~\star }{{\mathfrak {w}}}^a_{~{\circ }}\wedge {{\mathfrak {e}}}^\star \\= & {} \chi ^a_{~{\circ }}~{{\mathfrak {T}}}^{\circ }~=~0. \end{aligned}$$

This is consistent with the torsion-less condition. The BRST transformation for the volume form is vanished, and last two terms are cancelled each other such as

$$\begin{aligned} \delta _{\mathrm {BRST}}\left[ i\left( d{\tilde{{{\mathfrak {c}}}}}^{\circ \circ }+\alpha {\tilde{{{\mathfrak {c}}}}}^{\circ }_{~\star } \wedge {{\mathfrak {b}}}^{\star {\circ }}\right) \wedge {\overline{{{\mathfrak {c}}}}}_{a{\circ \circ }}\right]= & {} -\left( \mathrm{d}{{\mathfrak {b}}}^{\circ \circ }+\alpha {{\mathfrak {b}}}^{\circ }_{~\star }\wedge {{\mathfrak {b}}}^{\star {\circ }}\right) \wedge {\overline{{{\mathfrak {e}}}}}_{a{\circ \circ }} \end{aligned}$$

Therefore, the BRST transformation of (55) is given by

$$\begin{aligned} 0= & {} \epsilon _{abc\bullet }\delta _{\mathrm {BRST}}\left[ \left( \mathrm{d}{{\mathfrak {w}}}^{ab} +{c_{gr}}\;{{\mathfrak {w}}}^a_{~{\circ }}\wedge {{\mathfrak {w}}}^{{\circ }b}\right) \wedge {{\mathfrak {e}}}^c\right] \\= & {} \epsilon _{abc\bullet }\Bigl \{ \chi ^b_{~{\circ }}\left( \mathrm{d}{{\mathfrak {w}}}^{a{\circ }} +{c_{gr}}\;{{\mathfrak {w}}}^a_{~\star }\wedge {{\mathfrak {w}}}^{\star {\circ }}\right) \wedge {{\mathfrak {e}}}^c +\left( \mathrm{d}{{\mathfrak {w}}}^{ab}+{c_{gr}}\;{{\mathfrak {w}}}^a_{~{\circ }}\wedge {{\mathfrak {w}}}^{{\circ }b}\right) \wedge {{\mathfrak {c}}}^c \Bigr \}. \end{aligned}$$

This is consistent with the Einstein equation.

$$\begin{aligned} \delta _{\mathrm {BRST}}\left[ \mathrm{d}{{\mathfrak {e}}}^a-\alpha \left( {{\mathfrak {b}}}^a_{~{\circ }}\wedge {{\mathfrak {e}}}^{\circ }-i{\tilde{{{\mathfrak {c}}}}}_{\circ }^a\wedge {{\mathfrak {c}}}^{\circ }\right) \right]= & {} \mathrm{d}{{\mathfrak {c}}}^a=0, \end{aligned}$$

where $\alpha$-terms are cancelled each other. The BRST transformation of (56) gives an equation of motion for ${{\mathfrak {b}}}$ in (55).

$$\begin{aligned} \epsilon _{ab{\circ \circ }}\delta _{\mathrm {BRST}}\left[ \mathrm{d}\left( {{\mathfrak {c}}}^{\circ }\wedge {{\mathfrak {e}}}^{\circ }\right) -\alpha {{\mathfrak {b}}}^{\circ }_{~\star }\wedge {{\mathfrak {c}}}^\star \wedge {{\mathfrak {e}}}^{\circ }\right]= & {} 0, \end{aligned}$$

where $\epsilon _{ab{\circ \circ }}{{\mathfrak {c}}}^{\circ }\wedge {{\mathfrak {c}}}^{\circ }=0$ is used. This is not an equation, but an identity.

1.4 D proof of (44)

The Proof of (44) can be given as follows: Shorthand notations are introduced to omit indices in this appendix for simplicity as follows:

$$\begin{aligned} {\widehat{{\mathfrak {Q}}}}_{{\mathfrak {b}}}=\frac{1}{2}\widehat{{\varvec{\beta }}}{\circ }\delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] ,&~~&{\widehat{{\mathfrak {Q}}}}_{{{\tilde{{{\mathfrak {c}}}}}}}=\frac{i}{4}\widehat{{\varvec{\chi }}}{\circ }\delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] -\frac{1}{4}\widehat{{\varvec{\beta }}}{\circ }\widehat{{\varvec{{\mathfrak {Q}}}}}, \end{aligned}$$

where $\widehat{{\varvec{{\mathfrak {Q}}}}}$ is defined in Eq. (41). By using these notions, the commutation relation can be represented as

$$\begin{aligned} -8i\left[ {\widehat{{\mathfrak {Q}}}}_{{{\tilde{{{\mathfrak {c}}}}}}},{\widehat{{\mathfrak {Q}}}}_{{\mathfrak {b}}}\right]= & {} \left[ \widehat{{\varvec{\beta }}}{\circ }\delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] , \widehat{{\varvec{\chi }}}{\circ }\delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] \right] +i \left[ \widehat{{\varvec{\beta }}}{\circ }\delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] , \widehat{{\varvec{\beta }}}{\circ }\widehat{{\varvec{{\mathfrak {Q}}}}} \right] ,\nonumber \\= & {} \widehat{{\varvec{\beta }}}\left[ \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] , \widehat{{\varvec{\chi }}}\right] \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] +\widehat{{\varvec{\chi }}} \left[ \widehat{{\varvec{\beta }}},\delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] \right] \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] \nonumber \\&+i\; \widehat{{\varvec{\beta }}}\left[ \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] ,\widehat{{\varvec{\beta }}}{\circ }\widehat{{\varvec{{\mathfrak {Q}}}}}\right] +i \left[ \widehat{{\varvec{\beta }}}, \widehat{{\varvec{\beta }}}{\circ }\widehat{{\varvec{{\mathfrak {Q}}}}} \right] \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] . \end{aligned}$$

(57)

where $[\widehat{{\varvec{\beta }}},\widehat{{\varvec{\chi }}}]=0$ is used. The first term of (57) becomes

$$\begin{aligned} \widehat{{\varvec{\beta }}}\left[ \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] , \widehat{{\varvec{\chi }}}\right] \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right]= & {} \widehat{{\varvec{\beta }}} \left( \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] \widehat{{\varvec{\chi }}}- \widehat{{\varvec{\chi }}}~\delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] \right) \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] , \\= & {} \widehat{{\varvec{\beta }}} \left( \delta _{\mathrm {BRST}}\left[ \left\{ \widehat{{\varvec{{\mathfrak {Q}}}}},\widehat{{\varvec{\chi }}}\right\} \right] +i\left[ \widehat{{\varvec{\beta }}},\widehat{{\varvec{{\mathfrak {Q}}}}}\right] \right) \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] ,\\= & {} i\widehat{{\varvec{\beta }}}~\left[ \widehat{{\varvec{\beta }}},\widehat{{\varvec{{\mathfrak {Q}}}}}\right] \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] =~4{\widehat{{\mathfrak {Q}}}}_{{\mathfrak {b}}}, \end{aligned}$$

where $\delta _{\mathrm {BRST}}\left[ \left\{ \widehat{{\varvec{{\mathfrak {Q}}}}},\widehat{{\varvec{\chi }}}\right\} \right] =0$ due to (40), and (38) are used. Note that $\widehat{{\varvec{\chi }}}$ and $\widehat{{\varvec{{\mathfrak {Q}}}}}$ have $\epsilon _X=-1$ in (28). The second term of (57) is zero, because

$$\begin{aligned} \left[ \widehat{{\varvec{\beta }}},\delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] \right]= & {} \delta _{\mathrm {BRST}}\left[ \left[ \widehat{{\varvec{\beta }}},\widehat{{\varvec{{\mathfrak {Q}}}}} \right] \right] =~0, \end{aligned}$$

due to (38). The third term is also zero as

$$\begin{aligned} i\widehat{{\varvec{\beta }}}\left[ \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] ,\widehat{{\varvec{\beta }}}{\circ }\widehat{{\varvec{{\mathfrak {Q}}}}}\right]= & {} i\widehat{{\varvec{\beta }}}\widehat{{\varvec{\beta }}} \left[ \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] ,\widehat{{\varvec{{\mathfrak {Q}}}}}\right] + i\widehat{{\varvec{\beta }}}~ \delta _{\mathrm {BRST}}\left[ \left[ \widehat{{\varvec{{\mathfrak {Q}}}}},\widehat{{\varvec{\beta }}}\right] \right] \widehat{{\varvec{{\mathfrak {Q}}}}}=~0, \end{aligned}$$

due to (38). A relation $\left[ \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] ,\widehat{{\varvec{{\mathfrak {Q}}}}}\right] =0$ can be confirmed by direct calculations. The last term of (57) becomes

$$\begin{aligned} i\left[ \widehat{{\varvec{\beta }}}, \widehat{{\varvec{\beta }}}{\circ }\widehat{{\varvec{{\mathfrak {Q}}}}} \right] \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right]= & {} i\widehat{{\varvec{\beta }}}~ \left[ \widehat{{\varvec{\beta }}},\widehat{{\varvec{{\mathfrak {Q}}}}}\right] \delta _{\mathrm {BRST}}\left[ \widehat{{\varvec{{\mathfrak {Q}}}}}\right] =~4{\widehat{{\mathfrak {Q}}}}_{{\mathfrak {b}}}. \end{aligned}$$