\documentstyle[aps]{revtex}
\begin{document} %\draft
\twocolumn[\hsize\textwidth\columnwidth\hsize\csname
@twocolumnfalse\endcsname
\title{General Relativity in terms of Dirac Eigenvalues}
\author{Giovanni Landi${}^1$, Carlo Rovelli${}^2$}
\address{\it
${}^1$ Dipartimento di Scienze Matematiche, Universit\`a di Trieste,
I-34127, Trieste, Europe \\ [A
${}^1$ INFN, Sezione di Napoli, I-80125 Napoli, Europe \\
${}^2$ Physics Department, University of Pittsburgh,
Pittsburgh
Pa 15260, USA \\
${}^2$ Center for Gravity and Geometry, Penn State
University, State College Pa 16801, USA
}
\maketitle
\begin{abstract}
The eigenvalues of the Dirac operator on a curved spacetime are
diffeomorphism-invariant functions of the geometry. They
form an infinite set of ``observables'' for general relativity.
Recent work of Chamseddine and Connes suggests that they can be taken
as variables for an invariant description of the gravitational
field's dynamics. We compute the Poisson brackets of these eigenvalues
and find them in terms of the energy-momentum of the eigenspinors and the
propagator of the linearized Einstein equations. We show that the
eigenspinors' energy-momentum is the Jacobian matrix of the change of
coordinates from metric to eigenvalues. We also consider a minor
modification of the spectral action, which eliminates the disturbing
huge cosmological term and derive its equations of motion. These are
satisfied if the energy momentum of the trans Planckian
eigenspinors scale linearly with the eigenvalue; we argue that this
requirement approximates the Einstein equations.
\end{abstract}
\pacs{PACS: 04.20.Cv, 04.20.Fy, 04.60.-m, 02.40.Ky, \hskip3cm\today}
\vskip1cm
]
\section{Introduction}
One of the important lessons that we learn from general relativity is that
fundamental physics is invariant under diffeomorphisms: there is no
fixed nondynamical structure with respect to which location or motion
could be defined. A fully diffeomorphism-invariant description of the
geometry has consequently long been searched in general relativity;
but so far without much success. As emphasized by many authors, such
a description would be precious for quantum gravity \cite{diff}.
Recently, Alain Connes' intriguing attempt of using the particle physics
standard model for unraveling a microscopic noncommutative structure of
spacetime \cite{alainb,daniel3,landi}, has generated --in a sense as a
side product-- the remarkable idea that the curved-spacetime Dirac
operator $D$ codes the full information about spacetime geometry
in a way that can be used for describing the dynamics of the latter.
Indeed, not only the geometry can be reconstructed from the (normed)
algebra generated by (the inverse of) $D$ and the smooth functions on
the manifold, but the Einstein-Hilbert action itself is approximated by
the trace of a simple function of $D$ \cite{alain}. In its simplest reading,
this result suggests the possibility of taking the eigenvalues
$\lambda^{n}$ of the Dirac operator as ``dynamical variables'' for
general relativity. Indeed, these form an infinite family of fully
four-dimensional diffeomorphism invariant observables, precisely the
kind of object that was long searched in relativity. This approach has
limitations. The most serious of these are that so far it works for
Euclidean general relativity only (see \cite{eli} for some attempts to
overcome this problem), and somewhat unrealistic predictions for the
bare couplings \cite{daniel2}. However, it definitely opens
a new window on the study of the dynamics of spacetime.
In order to use these ideas in the classical or in the quantum theory,
one must translate structures from the metric variables to the
$\lambda^{n}$ variables. In particular, one needs information on the
constraints that the $\lambda^{n}$'s satisfy if they correspond to a
smooth geometry, and on their Poisson brackets. The difficulty is
that the dependence of the $\lambda^{n}$'s on the geometry is defined
in a very implicit manner, and these quantities seem too hard to
compute.
In this letter, we show that these difficulties can be circumvented.
Following some earlier results in \cite{roberto} (valid only for the non
4d-invariant eigenvalues of the fixed-time Weyl operator),
we derive here an expression for the
Poisson brackets of the Dirac eigenvalues. Rather surprisingly, this
expression turns out to be given in terms of the energy-momentum tensors
of the Dirac eigenspinors. These tensors form the matrix elements of the
Jacobian matrix of the change of coordinates between metric and
eigenvalues. The Poisson brackets are quadratic in these tensors, with a
kernel given by the propagator of the linearized Einstein equations.
Thus, the energy-momentum tensors of the Dirac eigenspinors turn out
to be the key tool for analyzing the representation of spacetime
geometry in terms of Dirac eigenvalues.
We study also the Chamseddine-Connes spectral action. In the form
presented in \cite{alain} this is quite unrealistic as a pure gravity
action, because of a huge cosmological constant term that implies that
the geometries for which the action approximates the Einstein-Hilbert
action are {\em not} solutions of the theory. However, a very small
modification of the action eliminates the cosmological constant term.
We derive the equations of motion directly from the (modified) spectral
action. We argue that they amount to the requirement that the energy
momenta of the high mass eigenspinors scale linearly with the mass,
and that this requirement approximates the vacuum Einstein equations.
These results suggest that --even independently from its application
to the standard model-- the Chamseddine-Connes {\em gravitational}
theory can be viewed as a manageable gravitational theory by itself
(see also \cite{fr,gianni}), possibly with powerful applications to
classical and quantum gravity. It reproduces general relativity at
low energies; it is formulated in terms of fully diffeomorphism
invariant variables; and, of course, it prompts fascinating extensions of
the very notion of geometry.
\section{GR in terms of eigenvalues}
Consider Euclidean general relativity (GR) on a compact 4d (spin-) manifold
without boundary. We work in terms of the tetrad field $E_{\mu}^{I}(x)$. Here
$\mu=1\ldots 4$ are spacetime indices and $I=1\ldots 4$ are internal
Euclidean indices, raised and lowered by the Euclidean metric
$\delta_{IJ}$. The metric field is given by $g_{\mu\nu}(x)=
E_{\mu}^{I}(x)E_{\nu\, I}(x)$, and is used to raise and lower
spacetime indices. The spin connection $\omega_{\mu}^{I}{}_{J}$ is
defined by the equation $\partial_{[\mu}E_{\nu]}^{I} =
\omega_{[\mu}^{I}{}_{J} E_{\nu]}^{J}$. The dynamics is defined by the
action $S[E]=\frac{1}{16\pi G} \int d^{4} x \sqrt{g} R$, where $g$ and
$R$ are the determinant and the Ricci scalar of the metric.
In spite of a curiously widespread popular belief of the contrary,
phase space is a covariant notion: the covariant definition of the
phase space is as the space of the solutions of the equations of
motion, modulo gauge transformations \cite{phase}. In the theory
considered, the gauge transformations are given by 4d diffeomorphisms
and by the local internal rotations of the tetrad field. Thus, the
phase space $\Gamma$ of GR is the space of the tetrad fields $E$ that
solve the equation of motion (Einstein equations), modulo internal
rotations and diffeomorphisms. $\Gamma$ can be identified with the space of the
gauge orbits on the constraint surface and with the space of the Ricci
flat 4-geometries.
We shall use the following notation. We denote the space of smooth
tetrad fields as $\cal E$; the space of the orbits of the gauge groups
--diffeomorphisms and local orthogonal tetrad rotations-- in $\cal E$
as $\cal G$ (these are ``4-geometries''); and the space of the orbits
that satisfy the Einstein equation as $\Gamma$ (these are the Ricci-flat
4-geometries, which form the phase space of GR). We call functions
on $\Gamma$ ``observables''. Observables correspond to functions on
the constraint surface that commute with {\em all\/} the constraints.
We now define an infinite family of such observables. Consider spinor
fields $\psi$ on the manifold and the curved Dirac operator
\begin{equation}
D = \imath \gamma^{I} E^{\mu}_{I} \left(\partial_{\mu}+\omega_{\mu\,
JK}\gamma^{J} \gamma^{K}\right),
\end{equation}
which acts on them. Here $\gamma$ are the (Euclidean) hermitian Dirac
matrices. For each given field $E$, the Dirac operator is
a self-adjoint operator on the Hilbert space of spinor fields with
scalar product
\begin{equation}
(\psi,\phi)=\int d^{4}x \sqrt{g}\ \overline{\psi(x)} \phi(x).
\label{product}
\end{equation}
where the bar indicates complex conjugation, and the scalar
product in spinor space is the natural one in $C^{4}$. Therefore, $D$
admits a complete set of real eigenvalues and eigenfunctions
(``eigenspinors''). Since the manifold is compact, the spectrum is
discrete. We write
\begin{equation}
D \psi^{n} = \lambda^{n} \psi^{n}.
\end{equation}
Here and below, $n=0,1,2 ... $ is an index, not an exponent. We
convene to label the eigenvalues so that $\lambda^{n}$ is non
decreasing in $n$, namely $\lambda^n\leq\lambda^{n+1}$ (each eigenvalue is
repeated according to its multiplicity). In order to emphasize the dependence
of Dirac operator, eigenvalues and eigenspinors on the tetrad field, we use
also the notation
\begin{equation}
D[E]\ \psi^{n}[E] = \lambda^{n}[E]\ \psi^{n}[E]
\end{equation}
where the dependence on the tetrad is indicated explicitly.
$\lambda^{n}[E]$ defines a discrete family of real
functions on the space $\cal E$ of the tetrad fields
\begin{eqnarray}
\lambda^{n}:\ \ {\cal E} & \longrightarrow & R \nonumber \\
E & \longmapsto & \lambda^{n}[E].
\end{eqnarray}
Equivalently, they define a function $\lambda$ from $\cal E$ into
the space of infinite sequences $R^\infty$
\begin{eqnarray}
\lambda:\ \ {\cal E} & \longrightarrow & R^\infty \nonumber \\
E & \longmapsto & \{\lambda^{n}[E]\}.
\end{eqnarray}
Since we have chosen to order the $\lambda^n$'s in non-decreasing
order, the image of $\cal E$ under this map, which we denote as
$\lambda({\cal E})$ is entirely contained in the cone $\lambda^n \leq
\lambda^{n+1}$ of $R^\infty$.
Now, for every $n$, the function $\lambda^{n}$ is invariant under 4d
diffeomorphisms and under internal rotations of the tetrad field.
Therefore the functions $\lambda^{n}$ are well defined functions on
$\cal G$. In particular, they are well defined on $\Gamma$: they are
observables of GR.
Two metric fields with the same $\lambda^{n}$'s are called
``isospectral''. Isometric (that is, gauge equivalent) $E$ fields are
isospectral, but the converse might fail to be true \cite{drum,roberto}.
Therefore $\lambda$ may not be injective even if restricted to $\cal G$.
The $\lambda^n$'s may fail to coordinatize $\cal G$. They may also
fail to coordinatize $\Gamma$. However, they presumably ``almost do it''.
In the following, we explore the idea of analyzing GR in terms of the set
of observables $\lambda^{n}[E]$.
\section{Symplectic structure}
The phase space $\Gamma$ is a symplectic manifold and a Poisson brackets
structure is defined on the physical observables. We now study the
Poisson brackets $\{\lambda^{n}, \lambda^{m}\}$.
To this purpose, we first construct the symplectic structure on $\Gamma$. This
can be written in covariant form following Ref~\cite{abhay}. A vector
field $X$ on $\Gamma$ can be written as a differential operator
\begin{equation}
X = \int d^{4}x \ X_{\mu}^{I}(x)[E]\ \ \frac{\delta}{\delta
E_{\mu}^{I}(x)}
\end{equation}
where $X_{\mu}^{I}(x)[E]$ is any solution of the Einstein equations
{\it linearized\/} over the background $E$. The symplectic two-form
$\Omega$ of GR is given by
\begin{equation}
\Omega(X,Y) = \frac{1}{32\pi G} \int_{\Sigma}d^{3}\sigma\ n_{\rho}
[
X_{\mu}^{I}\ \overleftarrow{\overrightarrow{\nabla}}{}_{\tau}\ Y_{\nu}^{J}
] \epsilon^{\tau}_{IJ\upsilon} \epsilon^{\upsilon\rho\mu\nu}
\label{ome}
\end{equation}
where $[X_{\mu}^{I}\ \overleftarrow{\overrightarrow{\nabla}}{}_{\tau}\
Y_{\nu}^{J}] \equiv [X_{\mu}^{I}\ \nabla_{\tau}\ Y_{\nu}^{J} -
Y_{\mu}^{I}\ \nabla_{\tau}\ X_{\nu}^{J}]$; from now on we put $32\pi G=1$.
%we restore physical units below.)
Both sides of (\ref{ome}) are
functions of $E$, namely scalar functions on $\Gamma$; this $E$ is
used to transform internal indices into spacetime indices. Here
$\Sigma:\sigma\longmapsto x(\sigma)$ is an arbitrary three-dimensional
``ADM'' surface, and $n_{\rho}$ is the normal one-form to this surface.
The coefficients of the symplectic form can be written as
\begin{equation}
\Omega^{\mu\nu}_{IJ}(x,y) = \! \int_{\Sigma}\! d^{3}\sigma\ n_{\rho}
[\delta(x,x(\sigma)) \overleftarrow{\overrightarrow{\nabla}}_{\tau}
\delta(y, x(\sigma)) ] \epsilon^{\tau}_{IJ\upsilon}
\epsilon^{\upsilon\rho\mu\nu} .
\label{omega}
\end{equation}
The Poisson bracket between two functions $f$ and $g$ on $\Gamma$ is
given by
\begin{equation}
\{f,g\}= \int d^{4}x\int d^{4}y\ \ P_{\mu\nu}^{IJ}(x,y) \
\frac{\delta f}{\delta E_{\mu}^{I}(x)}\
\frac{\delta g}{\delta E_{\mu}^{I}(y)}.
\label{pp}
\end{equation}
where $P_{\mu\nu}^{IJ}(x,y)$ is the inverse of the symplectic form
matrix. It is defined by
\begin{equation}
\int d^{4}y\ P_{\mu\nu}^{IJ}(x,y)\ \Omega^{\nu\rho}_{JK}(y,z)=
\delta(x,z)\ \delta_{\mu}^{\rho}\ \delta^{I}_{K}.
\end{equation}
Since the symplectic form is degenerate on the space of the fields
(it is non-degenerate only when restricted to the space
of equivalent classes of gauge-equivalent fields), we can only invert
it on this space by fixing a gauge. Let us assume this has been done.
More precisely, integrating the last equation against a vector field
$F_{\rho}^{K}(z)$ that satisfies the linearized Einstein equations
over $E$, we have
\begin{eqnarray}
&\int d^{4}y \int d^{4}z\ P_{\mu\nu}^{IJ}(x,y)\
\Omega^{\nu\rho}_{JK}(y,z) \
F_{\rho}^{K}(z) = & \nonumber \\
& \ \ \ = \int d^{4}z \ \delta(x,z)\
\delta_{\mu}^{\rho}\ \delta^{I}_{K} \ F_{\rho}^{K}(z),&
\end{eqnarray}
Integrating over the delta functions, and using (\ref{omega}), we have
\begin{eqnarray}
&\int_{\Sigma} d^{3}\sigma \ n_{\rho} [
P^{IJ}_{\mu\nu}(x,x(\sigma)) \overleftarrow{\overrightarrow{\nabla}}_{\rho}
F^{K}_{\tau}(x(\sigma)) ]
\epsilon^{\rho}_{JK\upsilon}\epsilon^{\upsilon\nu\tau\sigma} = &
\nonumber \\
& \ \ \ \ \ = F_{\mu}^{I}(x).&
\label{p}
\end{eqnarray}
This equation, where $F$ is any solution of the linearized equations,
defines $P$. But this equation is precisely the definition of the
propagator of the linearized Einstein equations over the background
$E$ (in the chosen gauge). For instance, let us chose the surface
$\Sigma$ as $x^{4}=0$ and fix the gauge with
\begin{equation}
X^{4}_{4} = 1, \ \
X^{4}_{a} = 0, \ \
X^{i}_{4} = 1, \ \
X^{i}_{a} = 0.
\end{equation}
where $a=1,2,3$ and $i=1,2,3$. Then equation (\ref{p}) becomes
\begin{equation}
F^{i}_{a}(\vec x, t)=
\int d^3\vec y\ (P^{ib}_{aj}(\vec x, t; \vec y, 0)
\overleftarrow{\overrightarrow{\nabla}}_{0} F^{j}_{b}(\vec y, 0)),
\end{equation}
where we have used the notation $\vec x = (x^1, x^2, x^3)$ and
$t=x^4$, and the propagator can be easily recognized.
Next, we need the functional derivative of the eigenvalues
with respect to the metric.
The variation of $\lambda^{n}$ for a variation of $E$ can be computed
using standard techniques for first order variations of eigenvalues;
(standard time-independent quantum-mechanics perturbation theory). For
a self-adjoint operator $D$ depending on a parameter $v$ and whose
eigenvalues are nondegenerate, we have
\begin{equation}
\frac {d\lambda^{n}}{d v} = (\psi ^{n}| \frac{d}{dv } D(v)|\psi^{n}).
\end{equation}
In our situation, for generic metrics with nondegenerate eigenvalues
we have that
\begin{eqnarray}
\frac{\delta \lambda^{n}}{\delta E_{\mu}^{I}(x)} & = &
(\psi^{n}|\frac{\delta}{\delta E_{\mu}^{I}(x)}D|\psi^{n}) \\
& = &
\int \sqrt{g}\ \bar\psi^{n} \frac{\delta}{\delta E_{\mu}^{I}(x)}
D\psi^{n}
\nonumber
\\ & = &
\frac{\delta}{\delta E_{\mu}^{I}(x)} \int \sqrt{g}\ \bar\psi^{n}
D\psi^{n} - \int \frac{\delta\sqrt{g}}{\delta E_{\mu}^{I}(x)}
\bar\psi^{n} D \psi^{n}
\nonumber
\\ & = &
\frac{\delta}{\delta E_{\mu}^{I}(x)} \int \sqrt{g}\ \bar\psi^{n}
D\psi^{n} - \int \frac{\delta\sqrt{g}}{\delta E_{\mu}^{I}(x)}
\bar\psi^{n} \lambda^n \psi^{n}
\nonumber
\\ & = &
\frac{\delta}{\delta E_{\mu}^{I}(x)} \int \sqrt{g}\
(\bar\psi^{n} D\psi^{n} - \lambda^n \bar\psi^{n}\psi^n)
\nonumber
\\
& = & T^{n}{}^{\mu}_{I}(x).
\label{t}
\end{eqnarray}
Now, $T^{n}{}^{\mu}_{I}(x)$ is nothing but the usual energy-momentum
tensor of the spinor field $\psi^{n}$ in tetrad notation (see for
instance \cite{stanley}). Indeed, the usual Dirac energy-momentum
tensor is defined by
\begin{equation}
T^{\mu}_{I}(x)\equiv\frac{\delta}{\delta E_{\mu}^{I}(x)}S_{\rm Dirac},
\end{equation}
where $S_{\rm Dirac}=\int \sqrt{g}\ (\bar
\psi D \psi - \lambda\bar\psi\psi)$ is the usual
Dirac action of a spinor with mass $\lambda$.
We have shown that the energy-momentum tensor of the
eigenspinors
gives the Jacobian matrix of the transformation from $E$ to $\lambda$;
namely it gives the variation of the eigenvalues for a small
change in the geometry. This fact suggests that we can study the map $\lambda$
locally by studying the space of the eigenspinor's energy-momenta.
By combining (\ref{pp},\ref{p}) and (\ref{t}) we obtain our main result:
\begin{equation}
\{\lambda^{n},\lambda^{m}\}= %{\textstyle 32\pi G}
\int\!\! d^{4}x\!\!
\int\!\! d^{4}y \ T^{[n}{}^{\mu}_{I}(x)\ P_{\mu \nu}^{IJ}(x, y) \
T^{m]}{}^{\nu}_{J}(y)
\label{main}
\end{equation}
which gives the Poisson bracket of two eigenvalues of the Dirac operator
in terms of the energy-momentum tensor of the two corresponding
eigenspinors and of the propagator of the linearized Einstein
equations. The r.h.s. does not depend on the gauge chosen for $P$.
Finally, if the transformation between the ``coordinates''
$E_{\mu}^{I}(x)$ and the ``coordinates'' $\lambda^{n}$ is locally
invertible on the phase space $\Gamma$, we can write the symplectic
form directly in terms of the $\lambda^{n}$'s as
\begin{equation}
\Omega=\Omega_{mn}\ d\lambda^{n} \wedge d\lambda^{m},
\end{equation}
where a sum over indices is understood, and
where $\Omega_{mn}$ is defined by
\begin{equation}
\Omega_{mn}\ T^{n}{}^{\mu}_{I}(x)\ T^{m}{}^{\nu}_{J}(y)=
\Omega^{\mu\nu}_{IJ}(x,y).
\end{equation}
Indeed, let $d E_{\mu}^{I}(x)$ be a (basis) one-form on $\Gamma$,
namely the infinitesimal difference between two solutions of Einstein
equations, namely a solution of the Einstein equations linearized
over $E$. We have then
\begin{eqnarray}
\Omega & = & \int d^{4}x \int d^{4}y \
\Omega^{\mu\nu}_{IJ}(x,y)\ d E_{\mu}^{I}(x) \wedge d E_{\nu}^{J}(y)
\nonumber \\
& = & \int d^{4}x \int d^{4}y
\Omega_{mn}\ T^{n}{}^{\mu}_{I}(x)\ d E_{\mu}^{I}(x) \wedge
T^{m}{}^{\nu}_{J}(y)\ d E_{\nu}^{J}(y)
\nonumber \\
& = & \Omega_{mn} \
d\lambda^{n} \wedge d\lambda^{m}.
\end{eqnarray}
An explicit evaluation of the matrix $\Omega_{nm}$ would be of great interest.
\section{Action}
As shown in \cite{alain}, the gravitational action can be expressed as
\begin{equation}
S = Tr[\chi(D)]
\label{action1}
\end{equation}
in natural units $\hbar=G=c=1$. Here $\chi$ is a smooth monotonic
function on $R^{+}$ such that
\begin{eqnarray}
\chi(x) &=& 1,\ \
{\rm for}\ x < 1-\delta, \nonumber \\
\chi(x) &=& 0,\ \
{\rm for}\ x> 1 + \delta.
\end{eqnarray}
where $\delta<<<1$. Equivalently, $S$ is the number of $\lambda^n$'s
smaller than the Planck mass, which is 1 in natural units.
The action (\ref{action1}) approximates the Einstein-Hilbert action
with a large cosmological term for ``large-scale'' metrics with small
curvature (with respect to the Planck scale). This can be seen as follows.
Let $N[E]$ be the integer such that $\lambda^{N}[E]$ is the largest
eigenvalue of $D[E]$ smaller than the Planck mass $M_{P}={1\over
L_{P}}=1$. A moment of reflection shows then that we have
\begin{equation}
S[E]=N[E]
\label{sn}
\end{equation}
by definition. For large $n$, the growth of the eigenvalues of the Dirac
operator is given by the Weyl formula;
\begin{equation}
\lambda^n = V^{-{1\over 4}}\ n^{{1\over 4}} + \ldots
\label{qua}
\end{equation}
where $V$ is the volume and we are neglecting factors of the
order of unity. The next term in this asymptotic
approximation can be obtained from the fact that the Dixmier trace
(the logarithmic divergence of the trace) of $D^{-2}$ is the
Einstein-Hilbert action \cite{alainb,alain,landi}; therefore
\begin{equation}
(\lambda_{n})^{-2} = V^{{1\over2}}\ n^{-{1\over 2}} + \int\sqrt{g}R\
{n^{-1}} + \ldots
\end{equation}
Under our assumptions on the geometry, at the Planck scale we are in
asymptotic regime: the first term dominates over the second, and the
remaining terms are negligible. Writing the last equation for $n=N$
and using (\ref{sn}), we have
\begin{equation}
1 = V^{{1\over2}}\ S^{-{1\over 2}} + \int\sqrt{g}R\ S^{-1} + \ldots
\end{equation}
Solving for $S$ we obtain
\begin{equation}
S = \frac{1}{L_P^4} V + \frac{1}{L_P^2} \int\sqrt{g}R +
\ldots. \end{equation}
which shows that the action (\ref{action1}) is dominated by the
Einstein-Hilbert action with a cosmological term. In the last
equation, we have explicitly reinserted the Planck length $L_P$.
The presence of the huge Planck-mass cosmological term is a bit devastating
because the solutions of the equations of motions have Planck-scale
Ricci scalar, and therefore they are {\em all\/} out of the regime for which
the approximation taken is valid!
However, the cosmological term can be canceled easily by replacing $\chi(x)$
with $\tilde\chi(x)$ defined by
\begin{equation}
\tilde\chi(x)=\chi(x)-\epsilon^{4} \chi(\epsilon x)
\label{epsilon}
\end{equation}
for a small number $\epsilon$. The additional term cancels exactly
the cosmological term, leaving only the Einstein Hilbert action, with
corrections which are small for low curvature geometries, which, {\em
now}, {\em are} solutions of the theory. In fact
\begin{eqnarray}
\tilde S &\equiv & Tr(\tilde{\chi}(D)) \nonumber \\
&=& Tr({\chi}(D)) -\epsilon^{4}
Tr({\chi}(\epsilon D)) \nonumber \\
& =& \frac{V}{L_{P}^4} + \frac{1}{L_{P}^2} \int\sqrt{g}R \nonumber \\
& & -\epsilon^{4}\left(\epsilon^{-4} \frac{V}{L_{P}^4}
+\epsilon^{-2} \frac{1}{L_{P}^2} \int\sqrt{g}R\right) + \ldots.
\nonumber \\ &=& \frac{1}{L_{P}^2}\ \int\sqrt{g} R + \ldots
\end{eqnarray}
If we write $S$ directly in terms of the observables $\lambda^{n}$, we
have the following expression for the action
\begin{equation}
\tilde S[\lambda] = \sum_n\ \tilde\chi(\lambda^n).
\label{action}
\end{equation}
One cannot vary the $\lambda^{n}$'s in this expression to
obtain (approximate) Einstein equations. These equations are obtained
minimizing (\ref{action}) on the surface $\lambda({\cal E})$, not on
the entire $R^{\infty}$. In other words, the $\lambda^{n}$'s are not
independent: there are relations among them. These relations code the
complexity of GR. The equations of motion are obtained by varying $S$
with respect to the tetrad field. They can be written as
\begin{equation}
0 = \frac{\delta \tilde S}{\delta E_{\mu}^{I}(x)}
= \sum_n\ \frac{\partial \tilde S}{\partial \lambda^{n}} \
\frac{\delta \lambda^{n}}{\delta E_{\mu}^{I}(x)}
= \sum_n\ \frac{d\tilde\chi(\lambda^n)}{d \lambda^{n}}\
T^{n}{}^{\mu}_{I}(x).
\label{ee}
\end{equation}
Let $f(x) = \frac{d}{dx}\tilde\chi(x)$. The equations of motion of the
theory are then
\begin{equation}
\sum_{n} f(\lambda^{n})\ T^{n}{}^{\mu}_{I}(x) = 0.
\end{equation}
We close by analyzing the content of these equations. $f(x)$ is a
function that vanishes everywhere except on two narrow peaks. A
positive peak (width $\delta$ and height $1/\delta$) around the Planck
mass 1; and a negative peak (width $\delta\epsilon$ and height
$\epsilon^{5}/\delta$) around the arbitrary large number
$s={1\over\epsilon}>>1$. The equation is therefore solved if above
the Planck mass ($n>>N$), the energy momentum tensor scales as
\begin{equation}
\rho(1)\ T^{N}{}^{\mu}_{I}(x) = s^{-4}\ \rho(s)\
T^{N(s)}{}^{\mu}_{I}(x),
\end{equation}
where $\rho(s)$ is the density of the eigenvalues at the scale $s$ and
$\lambda^{N(s)}=s$, because in this case the two terms from the two
peaks cancel. But from (\ref{qua}) we have that the density of
eigenvalues grows as $N^{3}$, and that $N(s)=s^{4}$. This yields
\begin{equation}
T^{n}{}^{\mu}_{I}(x) = \lambda^{n}\ \ T^{N}{}^{\mu}_{I}(x).
\label{scaling}
\end{equation}
for $n>>N$. (Recall that $\lambda^{N}=1$.)
In other words: {\it the equations of motion for the geometry are solved
when above the Planck mass the energy-momentum of the eigenspinors
scales as the mass}.
To understand the meaning of this scaling requirement, notice that
$T^{n}{}_{\mu}^{I}$ is formed by a term linear in the derivatives of
the spinor field and a term independent from these, which is a function
of $(\psi, E, \partial_{\mu} E)$ quadratic in $\psi$.
\begin{equation}
T^{n}{}_{\mu}^{I}=\bar\psi^{n}\gamma^{I}
{\scriptstyle \overleftarrow{\overrightarrow{\partial}}}_{\mu}\psi^{n} +
S^{n}{}_{\mu}^{I}[\psi, E, \partial E].
\end{equation}
If we expand the last term around a point of the manifold with local coordinates
$x$, covariance and dimensional analysis require that
\begin{equation}
S^{n}{}_{\mu}^{I} =
c_{0} \lambda^{n} E_{\mu}^{I}+c_{1}\ R_{\mu}^{I}+ c_{2}\
R\,E_{\mu}^{I}+O\left( \frac{1}{\lambda^{n}}\right).
\end{equation}
for some fixed expansion coefficients $c_{0}, c_{1}$ and $c_{2}$.
Here $R_{\mu}^{I}$ is the Ricci tensor. [To be convinced that terms
of this form do appear, consider the following.
\begin{eqnarray}
T^{n}{}_{\mu}^{I} &=& \bar\psi^{n}\gamma^{I}D_{\mu}\psi^{n} +\ldots
\nonumber \\
& = & (\lambda^{n})^{-1}\
\bar\psi^{n}
\gamma^{I}\gamma^{\nu}D_{\mu}D_{\nu}\psi^{n}+\ldots
\nonumber \\
& = & (\lambda^{n})^{-1}\
\bar\psi^{n} \gamma^{I}\gamma^{\nu}[D_{\mu},D_{\nu}]\psi^{n} +\ldots
\nonumber \\
& = & (\lambda^{n})^{-1}\
\bar\psi^{n} \gamma^{I}\gamma^{\nu}R_{\mu\nu}\psi^{n} +\ldots
\nonumber \\
&=& (\lambda^{n})^{-1}\
\bar\psi^{n}
\gamma^{I}\gamma^{\nu}R_{\mu\nu}^{JK}\gamma_{J}\gamma_{K}\psi^{n} +\ldots
\nonumber \\
& = &
Tr\ \ \gamma^{I}\gamma^{\nu}R_{\mu\nu}^{JK}\gamma_{J}\gamma_{K} +\ldots
\nonumber \\
& = &
R^{I}_{\mu} + \ldots\ ]
\end{eqnarray}
For sufficiently high $n$, the eigenspinors are locally plane waves in
local cartesian coordinates: doubling the mass just doubles the frequency.
If
\begin{equation}
\lambda^{m}=t\ \lambda^{n}
\label{lsca}
\end{equation}
Then $\partial_\mu\psi^{m}=t\ \partial_\mu\psi^{n}$.
It follows that in general the energy momentum scales as
\begin{eqnarray}
T^{m}{}_{\mu}^{I} &=& t
\left[\bar\psi^{n}\gamma^{I}\partial_{\mu}\psi^{n}-
\partial_{\mu}\bar\psi^{n}\gamma^{I}\psi^{n} + c_{0}
\lambda^{n}E_{\mu}^{I}\right] \nonumber \\
& & + \left[c_{1}\ R_{\mu}^{I}+ c_{2}\ R\,E_{\mu}^{I}\right] +
O\left(\frac{1}{\lambda^{n}}\right).
\end{eqnarray}
For large $\lambda^{n}$ we can disregard the last term, and (\ref{scaling})
requires that the second square bracket vanishes. Taking the trace we
have $R=0$, using which we conclude
\begin{equation}
R^{I}_{\mu} = 0
\end{equation}
which are the vacuum Einstein equations. Thus, the variation of the
(modified) Chamseddine-Connes action implies a scaling requirement on the
high mass eigenspinors energy momentum tensors, and this requirement,
in turn, yields vacuum Einstein equations at low scale.
\vskip 1cm
We thank Alain Connes, Roberto De Pietri, J\"urg Fr\"ohlich and Daniel
Kastler for suggestions and conversations. This work was supported by
the Italian MURST and by NSF grant PHY-5-3840400.
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\end{document}