1. Introduction
There have thus far been a lot of studies on conformal factor or the conformal mode of the metric tensor field. The conformal factor is in essence a ghost mode in quantum field theory, which leads to negative energy at classical level and violates the unitarity at quantum level. Thus, understanding the conformal factor might give us some clues for problems associated with ghost fields such as the massive ghost in higher-derivative gravities.
The study of the conformal factor has been done mainly for three physical reasons. The first reason is that in the Euclidean approach of general relativity, the Einstein–Hilbert action is unbounded from below because of the conformal factor. The pragmatic approach for this issue is that the integration over the conformal factor is rotated in the complex plane so as to make the integral formally convergent [1].
The second reason comes from the well-known cosmological constant problem [2]. For instance, starting with a theory with a metric gμν and a scalar field φ, one can always construct a new gravitational theory with a new metric ${\tilde{g}}_{\mu \nu }$ and a new scalar field $\tilde{\phi }$ by performing a Weyl transformation ${\tilde{g}}_{\mu \nu }={\rho }^{2}{g}_{\mu \nu }$ and $\tilde{\phi }={\rho }^{-1}\phi $ where ρ is nothing but the conformal factor. It can be conjectured that some dynamical mechanism might give us an appropriate vacuum expectation value for the conformal factor in such a way that the cosmological constant is adjusted to be almost zero [3–8].
The final reason is related to the problem of the degenerate metric [9–12]. This problem is one of motivations behind the present work, so let us explain it in more detail. In classical gravitational theories, the metric, or equivalently the vierbein, is usually postulated to be non-degenerate in order to guarantee the existence of its inverse metric or inverse vierbein. On the other hand, in quantum gravity, the degenerate metric is expected to arise since the change of the space-time topology occurs through such a degenerate metric [13, 14].
It might then be useful to calculate the effective potential for the conformal factor of the metric since the non-vanishing vacuum expectation value of the conformal factor means that we have a non-degenerate metric at low energies even if there is a degenerate metric at high energies. Actually, this calculation has been performed so far in [9–12], but there seems to be some confusion, in particular, when we start with a classical matter action in a curved background and calculate the one-loop effective potential for the conformal factor from quantized matter fields, we arrive at two different effective potentials. The origin of the two different effective potentials lies in the fact that we have two different background metrics in the cutoff regularization. In this article, we would like to clarify which effective potential is correct if we would like to understand the problem of the degenerate metric.
The outline of this article is as follows: in the next section we review calculations of the effective potential for the conformal factor of the metric from a quantized scalar field. In section 3 , we carry out similar calculations in case of a quantized spinor field. In section 4 , on the basis of global GL(4) symmetry, we prove that the effective potential for the conformal factor must take the form of ${V}_{\mathrm{eff}}(g,\varphi )\,=\sqrt{-g}V(\varphi )$. The final section is devoted to a conclusion.
2. Bosonic scalar fields
We begin by reviewing calculations of the effective potential for the conformal factor of the metric from both quantized scalar and spinor fields. First, let us start with an action of a massive real scalar field in a curved background1
$\begin{eqnarray}\begin{array}{rcl}{S}_{B}(\varphi ,{g}_{\mu \nu }) & = & -\displaystyle \frac{1}{2}\displaystyle \int {{\rm{d}}}^{4}x\sqrt{-g}({g}^{\mu \nu }{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi +{m}^{2}{\varphi }^{2})\\ & = & -\displaystyle \frac{1}{2}\displaystyle \int {{\rm{d}}}^{4}x\sqrt{-g}\varphi (-{{\rm{\Delta }}}_{g}+{m}^{2})\varphi ,\end{array}\end{eqnarray}$
where we have defined the d'Alembertian operator Δg as $\begin{eqnarray}{{\rm{\Delta }}}_{g}=\displaystyle \frac{1}{\sqrt{-g}}{\partial }_{\mu }(\sqrt{-g}{g}^{\mu \nu }{\partial }_{\nu }).\end{eqnarray}$
It is well-known that from this classical action one can derive the following effective action: $\begin{eqnarray}\begin{array}{rcl}{\rm{\Gamma }}({g}_{\mu \nu }) & = & \displaystyle \frac{{\rm{i}}}{2}\mathrm{trlog}(-{{\rm{\Delta }}}_{g}+{m}^{2})\\ & = & \displaystyle \int {{\rm{d}}}^{4}x\sqrt{-g}\left[-{\rm{\Lambda }}+\displaystyle \frac{{M}_{{Pl}}^{2}}{2}R+{ \mathcal O }({R}^{2})\right],\end{array}\end{eqnarray}$
with the Planck mass squared being defined by ${M}_{{Pl}}^{2}=\tfrac{1}{8\pi {G}_{N}}$ where GN is the Newton's constant. Then, with gμν = ρ2ημν this effective action gives rise to the effective potential for the constant conformal factor ρ: $\begin{eqnarray}{V}_{\mathrm{eff}}={\rm{\Lambda }}{\rho }^{4},\end{eqnarray}$
where the effective potential Veff is defined via Γ(gμν) = − ∫d4xVeff. This is the stardard and well-known result, which was obtained in the induced gravity [16].On the other hand, there is a different calculation of the effective potential for the conformal factor, which will be described in what follows. Provided that we define a conformal factor ρ(x) by1 ) can be written as5 ), i.e. $\varphi =\rho {\left(x\right)}^{-1}\bar{\varphi }$ is needed to make the kinetic term for the scalar field be in the canonical form. Also note that the two actions (1 ) and (6 ) are equivalent at least classically.
$\begin{eqnarray}{g}_{\mu \nu }=\rho {\left(x\right)}^{2}{\bar{g}}_{\mu \nu },\qquad \varphi =\rho {\left(x\right)}^{-1}\bar{\varphi },\end{eqnarray}$
where ${\bar{g}}_{\mu \nu }$ and $\bar{\varphi }$ are respectively a fixed fiducial metric and a fixed background, the action ( $\begin{eqnarray}\begin{array}{rcl}{S}_{B}^{{\prime} }(\bar{\varphi },{\bar{g}}_{\mu \nu },\rho ) & = & -\displaystyle \frac{1}{2}\displaystyle \int {{\rm{d}}}^{4}x\sqrt{-\bar{g}}({\bar{g}}^{\mu \nu }{\partial }_{\mu }\bar{\varphi }{\partial }_{\nu }\bar{\varphi }+{m}^{2}{\rho }^{2}{\bar{\varphi }}^{2}\\ & & -2{\bar{g}}^{\mu \nu }{\rho }^{-1}{\partial }_{\mu }\rho \bar{\varphi }{\partial }_{\nu }\bar{\varphi }+{\bar{g}}^{\mu \nu }{\rho }^{-2}{\partial }_{\mu }\rho {\partial }_{\nu }\rho {\bar{\varphi }}^{2})\\ & = & -\displaystyle \frac{1}{2}\displaystyle \int {{\rm{d}}}^{4}x\sqrt{-\bar{g}}\left({\bar{g}}^{\mu \nu }{\partial }_{\mu }\bar{\varphi }{\partial }_{\nu }\bar{\varphi }\right.\\ & & \left.+{m}^{2}{\rho }^{2}{\bar{\varphi }}^{2}+...\right),\end{array}\end{eqnarray}$
where the ellipses denote terms including the derivatives of ρ. These terms are irrelevant to our later argument since we consider only constant ρ. The latter definition in (Based on the action (6 ), let us evaluate a one-loop effective potential for a constant conformal factor ρ in a flat Minkowski background ${\bar{g}}_{\mu \nu }={\eta }_{\mu \nu }$. The partition function is defined as2
$\begin{eqnarray}\begin{array}{rcl}Z(\rho ) & = & \displaystyle \int { \mathcal D }\bar{\varphi }\,{{\rm{e}}}^{{\rm{i}}{S}_{B}^{{\prime} }}={[\det (-\square +{m}^{2}{\rho }^{2})]}^{-\tfrac{1}{2}}\\ & = & {{\rm{e}}}^{-\tfrac{1}{2}\mathrm{trlog}(-\,\square +{m}^{2}{\rho }^{2})},\end{array}\end{eqnarray}$
where = ημν∂μ∂ν. Then, a one-loop effective action is defined as $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{B}(\rho ) & = & \displaystyle \frac{{\rm{i}}}{2}\mathrm{trlog}(-\square +{m}^{2}{\rho }^{2})\\ & = & \displaystyle \frac{{\rm{i}}}{2}\displaystyle \int {{\rm{d}}}^{4}x\langle x| \mathrm{log}(-\square +{m}^{2}{\rho }^{2})| x\rangle \\ & = & \displaystyle \frac{{\rm{i}}}{2}\displaystyle \int {{\rm{d}}}^{4}x\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{4}p}{{\left(2\pi \right)}^{4}}\mathrm{log}({p}^{2}+{m}^{2}{\rho }^{2})\\ & \equiv & -\displaystyle \int {{\rm{d}}}^{4}x\,{V}_{\mathrm{eff}}^{(B)},\end{array}\end{eqnarray}$
where ${V}_{\mathrm{eff}}^{(B)}$ indicates the one-loop effective potential for the conformal factor ρ.The effective potential can be computed as follows:12 ) occurs for a nonzero ρ, thereby implying that the metric or the vierbein is non-degenerate and the inverse metric gμν or vierbein ${e}_{a}^{\mu }$ exists.
$\begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{eff}}^{(B)} & = & -\displaystyle \frac{{\rm{i}}}{2}\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{4}p}{{\left(2\pi \right)}^{4}}\mathrm{log}({p}^{2}+{m}^{2}{\rho }^{2})\\ & = & \displaystyle \frac{1}{2}\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{4}{p}_{E}}{{\left(2\pi \right)}^{4}}\mathrm{log}({p}_{E}^{2}+{m}^{2}{\rho }^{2})\\ & = & \displaystyle \frac{1}{32{\pi }^{2}}{\displaystyle \int }_{0}^{{{\rm{\Lambda }}}^{2}}{{\rm{d}}{p}}_{E}^{2}\,{p}_{E}^{2}\mathrm{log}({p}_{E}^{2}+{m}^{2}{\rho }^{2}),\end{array}\end{eqnarray}$
where we have performed the Wick rotation p0 = ip4 and defined ${p}_{E}^{2}={p}_{1}^{2}+{p}_{2}^{2}+{p}_{3}^{2}+{p}_{4}^{2}$. By using the mathematical formula $\begin{eqnarray}\begin{array}{rcl}{\displaystyle \int }_{0}^{{{\rm{\Lambda }}}^{2}}{\rm{d}}x\,x\mathrm{log}(x+c) & = & \displaystyle \frac{1}{2}({{\rm{\Lambda }}}^{4}-{c}^{2})\mathrm{log}({{\rm{\Lambda }}}^{2}+c)\\ & & -\displaystyle \frac{1}{4}{{\rm{\Lambda }}}^{4}+\displaystyle \frac{1}{2}c{{\rm{\Lambda }}}^{2}+\displaystyle \frac{1}{2}{c}^{2}\mathrm{log}c,\end{array}\end{eqnarray}$
where c is a constant, the one-loop effective potential for the conformal factor takes the form $\begin{eqnarray}{V}_{\mathrm{eff}}^{(B)}=\displaystyle \frac{1}{32{\pi }^{2}}\left[\displaystyle \frac{1}{2}{m}^{4}{\rho }^{4}\left(\mathrm{log}\displaystyle \frac{{m}^{2}{\rho }^{2}}{{{\rm{\Lambda }}}^{2}}-\displaystyle \frac{1}{2}\right)+{m}^{2}{{\rm{\Lambda }}}^{2}{\rho }^{2}\right],\end{eqnarray}$
where the constant terms and the terms including ${ \mathcal O }(\tfrac{1}{{{\rm{\Lambda }}}^{2}})$ are subtracted. Adding suitable counterterms of the form Λ2ρ2 and ${\rho }^{4}\mathrm{log}{\rm{\Lambda }}$, one can obtain the renormalized one-loop effective potential for the conformal factor $\begin{eqnarray}{V}_{\mathrm{eff}}^{(B)}=\displaystyle \frac{1}{64{\pi }^{2}}{m}^{4}{\rho }^{4}\left(\mathrm{log}\displaystyle \frac{{m}^{2}{\rho }^{2}}{{\mu }^{2}}-\displaystyle \frac{1}{2}\right),\end{eqnarray}$
where μ is a renormalization mass [10]. From this effective potential, we can see that the minimum of the renormalized one-loop effective potential (Here an important remark is in order. In deriving the effective potential (12 ), by means of the metric ${\bar{g}}^{\mu \nu }$, we have adopted a momentum cutoff, ${\bar{g}}^{\mu \nu }{p}_{E\mu }{p}_{E\nu }={\delta }^{\mu \nu }{p}_{E\mu }{p}_{E\nu }\lt {{\rm{\Lambda }}}^{2}$. On the other hand, if we use a different momentum cutoff by using the metric gμν, i.e. gμνpEμpEν = ρ−2δμνpEμpEν < Λ2, a similar calculation provides us with a different one-loop effective potential12 ) but (13 ) that coincides with equation (4 ), which is the standard and well-known result.
$\begin{eqnarray}{\tilde{V}}_{\mathrm{eff}}^{(B)}=\displaystyle \frac{1}{64{\pi }^{2}}{m}^{4}{\rho }^{4}\left(\mathrm{log}\displaystyle \frac{{m}^{2}}{{\mu }^{2}}-\displaystyle \frac{1}{2}\right).\end{eqnarray}$
It is worthwhile to stress that it is not the effective potential (To understand the difference between the two effective potentials, let us notice that the actions (1 ) and (6 ) are invariant under a fake Weyl transformation14 ). However, we easily find that the former cutoff regularization breaks this fake symmetry while the latter cutoff regularization preserves it.
$\begin{eqnarray}{\bar{g}}_{\mu \nu }\to \omega {\left(x\right)}^{2}{\bar{g}}_{\mu \nu },\qquad \bar{\varphi }\to \omega {\left(x\right)}^{-1}\bar{\varphi },\qquad \rho \to \omega {\left(x\right)}^{-1}\rho ,\end{eqnarray}$
since both gμν and φ are trivially invariant under (Normally, without anomalies an effective action should possess the same contents of symmetries as its classical action even if the symmetry is a fake symmetry, as in (14 ).3 In fact, we can verify that the effective action corresponding to the effective potential (13 ) is invariant under the fake Weyl transformation (14 ). This can be exhibited by rewriting the conformal factor ρ4 as $\sqrt{-g}$:14 ) as well as the general coordinate transformation since it is expressed in terms of the metric gμν, which is trivially invariant under the fake Weyl transformation (14 ).
$\begin{eqnarray}\begin{array}{rcl}\tilde{{\rm{\Gamma }}}({g}_{\mu \nu }) & \equiv & -\int {{\rm{d}}}^{4}x{\tilde{V}}_{\mathrm{eff}}^{(B)}=-\displaystyle \frac{1}{64{\pi }^{2}}{m}^{4}\\ & & \times \,\int {{\rm{d}}}^{4}x\sqrt{-g}\left(\mathrm{log}\displaystyle \frac{{m}^{2}}{{\mu }^{2}}-\displaystyle \frac{1}{2}\right),\end{array}\end{eqnarray}$
where $\sqrt{-g}={\rho }^{4}$ was used. This effective action is certainly invariant under the fake Weyl transformation (On the other hand, an effective action corresponding to the effective potential (12 ) is not invariant under the fake Weyl transformation (14 ). Of course, we can rewrite the effective action as an expression which is invariant under the fake Weyl transformation (14 ), but then the general coordinate symmetry is violated since
$\begin{eqnarray}\begin{array}{rcl}{\rm{\Gamma }}({g}_{\mu \nu }) & \equiv & -\int {{\rm{d}}}^{4}{{xV}}_{\mathrm{eff}}^{(B)}=-\displaystyle \frac{1}{64{\pi }^{2}}{m}^{4}\\ & & \times \,\int {{\rm{d}}}^{4}x\sqrt{-g}\left(\mathrm{log}\displaystyle \frac{{m}^{2}{\left(-g\right)}^{\tfrac{1}{4}}}{{\mu }^{2}}-\displaystyle \frac{1}{2}\right),\end{array}\end{eqnarray}$
where the presence of the factor ${(-g)}^{\tfrac{1}{4}}$ breaks the general coordinate symmetry.In this sense, we can regard not ${V}_{\mathrm{eff}}^{(B)}$ but ${\tilde{V}}_{\mathrm{eff}}^{(B)}$ as a physically plausible effective potential for the classical theory defined by the action (1 ). To put it differently, the effective potential ${V}_{\mathrm{eff}}^{(B)}$ defines a quantum theory which is different from the classical theory (1 ) in the sense that ρ no longer has the meaning of the conformal factor, but a mere scalar field, which might be called dilaton since it is created from symmetry breaking of the fake Weyl symmetry. This observation will be verified from a proof mentioned in section 4 .
3. Fermionic spinor fields
Before doing so, for the sake of completeness and recent works [11, 12], let us consider the case of a massive Dirac spinor in a curved background. We will find the similar problem, i.e. the existence of two different effective potentials depending on the regularization procedure. The action of the massive Dirac spinor is given in a curved background:
$\begin{eqnarray}{S}_{F}({\rm{\Psi }},{e}_{\mu }^{a})=\int {{\rm{d}}}^{4}x\,e({\rm{i}}\bar{{\rm{\Psi }}}{e}_{a}^{\mu }{\gamma }^{a}{D}_{\mu }{\rm{\Psi }}-m\bar{{\rm{\Psi }}}{\rm{\Psi }}),\end{eqnarray}$
where the covariant derivative is defined as Dμ$\Psi$ = (∂μ + ωμ)$\Psi$ with the spin connection ${\omega }_{\mu }^{{ab}}$.As in equation (5 ), when we define a conformal factor ρ(x) by17 ) can be written as
$\begin{eqnarray}{e}_{\mu }^{a}=\rho (x){\bar{e}}_{\mu }^{a},\qquad {\rm{\Psi }}=\rho {\left(x\right)}^{-\tfrac{3}{2}}\psi ,\end{eqnarray}$
the action ( $\begin{eqnarray}{S}_{F}^{{\prime} }(\psi ,{\bar{e}}_{\mu }^{a},\rho )=\int {{\rm{d}}}^{4}x\,\bar{e}[\bar{\psi }({\rm{i}}\rlap{/}{D}-m\rho )\psi +...],\end{eqnarray}$
where the ellipses again denote terms including derivatives of ρ.Following a similar argument to the case of the scalar field, we find that a one-loop effective action is given by
$\begin{eqnarray}\begin{array}{rcl}{{\rm{\Gamma }}}_{F}(\rho ) & = & -{\rm{i}}\mathrm{tr}\,\mathrm{log}({\rm{i}}\rlap{/}{D}-m\rho )\\ & = & -{\rm{i}}\displaystyle \int {{\rm{d}}}^{4}x\langle x| \mathrm{tr}\,\mathrm{log}({\rm{i}}\rlap{/}{D}-m\rho )| x\rangle \\ & = & -{\rm{i}}\displaystyle \int {{\rm{d}}}^{4}x\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{4}p}{{\left(2\pi \right)}^{4}}\mathrm{trlog}(\rlap{/}{p}-m\rho )\\ & \equiv & -\displaystyle \int {{\rm{d}}}^{4}x\,{V}_{\mathrm{eff}}^{(F)},\end{array}\end{eqnarray}$
where we have set ωμ = 0 in the third equality.This effective potential can be computed as follows:
$\begin{eqnarray}\begin{array}{rcl}{V}_{\mathrm{eff}}^{(F)} & = & {\rm{i}}\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{4}p}{{\left(2\pi \right)}^{4}}\mathrm{tr}\,\mathrm{log}(\rlap{/}{p}-m\rho )\\ & = & {\rm{i}}\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{4}p}{{\left(2\pi \right)}^{4}}\mathrm{tr}\,\mathrm{log}{\gamma }^{5}(\rlap{/}{p}-m\rho ){\gamma }^{5}\\ & = & {\rm{i}}\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{4}p}{{\left(2\pi \right)}^{4}}\mathrm{tr}\,\mathrm{log}(-\rlap{/}{p}-m\rho )\\ & = & \displaystyle \frac{1}{2}{\rm{i}}\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{4}p}{{\left(2\pi \right)}^{4}}\mathrm{tr}\,[\mathrm{log}(\rlap{/}{p}-m\rho )+\mathrm{log}(-\rlap{/}{p}-m\rho )]\\ & = & 2{\rm{i}}\displaystyle \int \displaystyle \frac{{{\rm{d}}}^{4}p}{{\left(2\pi \right)}^{4}}\mathrm{log}({p}^{2}+{m}^{2}{\rho }^{2}),\end{array}\end{eqnarray}$
where we have used Clifford algebra, {γa, γb} = − 2ηab. This effective potential has the same form as that of the scalar field, so we can arrive at the renormalized one-loop effective potential for the conformal factor $\begin{eqnarray}{V}_{\mathrm{eff}}^{(F)}=-\displaystyle \frac{4}{64{\pi }^{2}}{m}^{4}{\rho }^{4}\left(\mathrm{log}\displaystyle \frac{{m}^{2}{\rho }^{2}}{{\mu }^{2}}-\displaystyle \frac{1}{2}\right).\end{eqnarray}$
In the above derivation, we have used the cutoff regularization given by ${\bar{g}}^{\mu \nu }{p}_{E\mu }{p}_{E\nu }={\delta }^{\mu \nu }{p}_{E\mu }{p}_{E\nu }\lt {{\rm{\Lambda }}}^{2}$.As in the scalar field, if we use a different momentum cutoff by using the metric gμν, i.e., gμνpEμpEν =ρ−2δμνpEμpEν < Λ2, we can obtain the one-loop effective potential
$\begin{eqnarray}{\tilde{V}}_{\mathrm{eff}}^{(F)}=-\displaystyle \frac{4}{64{\pi }^{2}}{m}^{4}{\rho }^{4}\left(\mathrm{log}\displaystyle \frac{{m}^{2}}{{\mu }^{2}}-\displaystyle \frac{1}{2}\right).\end{eqnarray}$
In this case as well, the physically plausible effective potential is given by ${\tilde{V}}_{\mathrm{eff}}^{(F)}$, but not ${V}_{\mathrm{eff}}^{(F)}$.As a final comment, it is straightforward to calculate the effective potential for the conformal factor where matter fields are taken to be the abelian electromagnetic field or the nonabelian Yang–Mills field. However, these fields are invariant under the Weyl transformation, or equivalently a local scale transformation, so we have no effective potential for the conformal factor. This situation remains unchanged even if there is a conformal anomaly since the conformal anomaly is expressed in terms of the Riemannian tensors.
4. Effective potential from GL(4) symmetry
In the previous two sections, we have reached a somewhat strange conclusion: although we started with two theories which are classically equivalent to each other, we have two different quantum theories irrespective of any anomalies. The source of this strange conclusion is obvious: the metric field plays a dual role, one of which is a dynamical variable and the other is a geometrical object. The geometrical role enters in a theory in defining the momentum squared. Actually, with the presence of a real scalar field φ, there is an ambiguity in the choice of the metric, that is, the original metric gμν or a generalized metric where the metric gμν is multiplied by some function of a scalar field, that is, f(φ)gμν. In the previous examples, the choice of the original metric gμν produces an effective potential which is quartic in the conformal factor whereas that of the modified metric ${\bar{g}}_{\mu \nu }={\rho }^{-2}{g}_{\mu \nu }$ leads to a Coleman–Weinberg-like effective potential.
In section 2 , we insisted that the physically plausible effective potential must take the form Veff = Cρ4 with a certain constant C if we would like to regard ρ as the conformal factor of the metric. To support our opinion, in this section, we will prove that the effective potential for the conformal factor of the metric in the presence of a scalar field φ necessarily takes the form of $\sqrt{-g}V(\varphi )$ on the basis of global GL(4) symmetry.4
For this aim, we assume that the vacuum is translationally invariant, by which all fields are constants in space-time. For such constant fields, the general coordinate transformation is reduced to a global GL(4) transformation
$\begin{eqnarray}{x}^{\mu }\to {x}^{{\prime} \mu }={\left({M}^{-1}\right)}^{\mu }{\,}_{\nu }{x}^{\nu },\end{eqnarray}$
where Mμ ν is a constant 4 × 4 matrix obeying $\det M\ne 0$. Under the GL(4) transformation, the constant fields and the Lagrangian density transform as $\begin{eqnarray}\begin{array}{rcl}{g}_{\mu \nu } & \to & {g}_{\mu \nu }^{{\prime} }={g}_{\alpha \beta }{M}^{\alpha }{\,}_{\mu }{M}^{\beta }{\,}_{\nu },\qquad {\varphi }^{{\prime} }\to \varphi ,\\ { \mathcal L }(g,\varphi ) & \to & {{ \mathcal L }}^{{\prime} }({g}^{{\prime} },{\varphi }^{{\prime} })=\det \,M\cdot { \mathcal L }(g,\varphi ).\end{array}\end{eqnarray}$
Here, note that the scalar field φ is invariant under the GL(4) transformation.With the infinitesimal GL(4) transformation, ${M}^{\mu }{\,}_{\nu }\,={\delta }_{\nu }^{\mu }+\delta {M}^{\mu }{\,}_{\nu }$ with ∣δMμ ν∣ ≪ 1, the constant fields and the Lagrangian density transform as
$\begin{eqnarray}\delta {g}_{\mu \nu }=\delta {M}_{\mu \nu }+\delta {M}_{\nu \mu },\qquad \delta \varphi =0,\qquad \delta { \mathcal L }=\mathrm{tr}\delta M\cdot { \mathcal L },\end{eqnarray}$
where $\mathrm{tr}\delta M\equiv {g}^{\mu \nu }\delta {M}_{\mu \nu }$. Note that the last transformation implies that the Lagrangian density indeed transforms as a density under the GL(4) transformation.Given the constant fields, under the infinitesimal GL(4) transformation, the Lagrangian density transforms as26 ). Then, this equation is simply solved to be
$\begin{eqnarray}\delta { \mathcal L }=\mathrm{tr}\delta M\cdot { \mathcal L }=\displaystyle \frac{\partial { \mathcal L }}{\partial {g}_{\mu \nu }}(\delta {M}_{\mu \nu }+\delta {M}_{\nu \mu }),\end{eqnarray}$
where we have used δφ = 0 in equation ( $\begin{eqnarray}{ \mathcal L }=-\sqrt{-g}V(\varphi ),\end{eqnarray}$
where V(φ) is a function depending on only the scalar field φ. This proof does not rely on perturbation theory so the result holds even in the non-perturbative regime.5Now let us apply this result to the present situation where the scalar field φ corresponds to the conformal factor ρ. Let us recall that in the effective potential ρ is treated as a constant. With gμν = ρ4ημν, equation (28 ) leads to
$\begin{eqnarray}{V}_{\mathrm{eff}}(\rho ,\varphi )={\rho }^{4}V(\varphi ),\end{eqnarray}$
where we have defined the ‘effective potential', ${V}_{\mathrm{eff}}(\rho ,\varphi )\,=-{ \mathcal L }$. If V(φ) is positive definite, ρ = 0 is only the minimum of the effective potential, thereby meaning that the metric is degenerate at low energies and the inverse metric does not exist, i.e. ⟨gμν⟩ = 0. It is worthwhile to note that the effective potential never takes the form of $\begin{eqnarray}{V}_{\mathrm{eff}}(\rho ,\varphi )={\rho }^{4}(a\mathrm{log}\rho +b),\end{eqnarray}$
where a, b are constants depending on the constant ρ.Then, a natural question arises of whether the quantum effects of matter and gravitational fields do not always drive the vacuum expectation value of the metric to be non-degenerate. To answer this question, let us note that the key ingredient in our derivation of equation (28 ) is the presence of a global GL(4) symmetry for constant fields. It is known that in the theory of quantum gravity, GL(4) symmetry is spontaneously broken to the Lorentz symmetry, and consequently the exact massless graviton emerges as a Nambu–Goldstone particle associated with this spontaneous symmetry breakdown [22]. In fact, if ${M}_{\nu }^{\mu }$ belongs to a generator of the Lorentz group SO(1, 3), we have
$\begin{eqnarray}\delta {g}_{\mu \nu }=\delta {M}_{\mu \nu }+\delta {M}_{\nu \mu }=0,\end{eqnarray}$
since δMμν is an antisymmetric matrix in the case of the Lorentz group. In such a broken phase of the GL(4) symmetry, our derivation based on the GL(4) symmetry does not make sense so there could be a possibility that a nontrivial effective potential appears, which drives the vacuum expectation value of the metric or vierbein to be non-degenerate.5. Conclusion
In this article, we have studied the issue of the effective potential of the conformal factor of the metric generated by quantized matter fields. It has been known that there are two kinds of the effective potentials, one of which produces a non-vanishing vacuum expectation value (VEV) of the conformal factor and the other does not exhibit such a feature, that is, a vanishing VEV. If the former effective potential were true, we could show that we have a non-degenerate metric at low energies even if there is a degenerate metric, which is required for the change of the space-time topology at high energies.
Based on a global GL(4) symmetry, which exists in the formulation of effective potentials, we have proven that the effective potential for the conformal factor always takes the form of Veff (ρ, φ) = ρ4V(φ), but not ${V}_{\mathrm{eff}}(\rho ,\varphi )={\rho }^{4}(a\mathrm{log}\rho +b)$. This proof clearly implies that we cannot show the existence of the non-degenerate metric at low energies from quantum effects associated with matter fields. Moreover, our proof suggests that we might be able to show the existence of the non-degenerate metric in a quantum gravity where the global GL(4) symmetry is spontaneously broken to the Lorentz symmetry.