Abstract.We want to give an overview of our model of quantum
gravity which we developed since 2012 based on the canonical
quantization of a globally hyperbolic spacetime of dimension \(n+1\), \(n\ge 3\).
A unified quantum theory incorporating the four fundamental forces of nature is
one of the major open problems in physics. The Standard Model combines
electromagnetism, the strong force and the weak force, but ignores gravity. The
quantization of gravity is therefore a necessary first step to achieve a unified
quantum theory.
General relativity is a Lagrangian theory, i.e., the Einstein equations are derived
as the Euler-Lagrange equation of the Einstein-Hilbert functional
where \(N=N^{n+1}\), \(n\ge 3\), is a
globally hyperbolic Lorentzian manifold, \(\bar R\) the scalar curvature and \(\Lam \) a
cosmological constant. We also omitted the integration density in the
integral. In order to apply a Hamiltonian description of general relativity, one
usually defines a time function \(x^0\) and considers the foliation of \(N\) given by the
slices
\begin{equation} M(t)=\{x^0=t\}. \end{equation}
We may, without loss of generality, assume that the spacetime
metric splits
where the naming refers to
the given foliation. For the tangential Einstein equations one can define
equivalent Hamilton equations due to the groundbreaking paper by Arnowitt,
Deser and Misner [1]. The normal Einstein equations can be expressed
by the so-called Hamilton condition
where \(\mc H\) is the Hamiltonian used in
defining the Hamilton equations. In the canonical quantization of gravity the
Hamiltonian is transformed to a partial differential operator of hyperbolic type \(\hat {\mc H}\)
and the possible quantum solutions of gravity are supposed to satisfy
the so-called Wheeler-DeWitt equation
in an appropriate setting, i.e.,
only the Hamilton condition \(\mathrm {(\ref {E:1.6})}\) has been quantized, or equivalently, the
normal Einstein equation, while the tangential Einstein equations have been
ignored.
In [2] we solved the equation \(\mathrm {(\ref {E:1.7})}\) in a fiber bundle \(E\) with base space \(\socc \),
the elements of which are the positive definite symmetric tensors of order two,
the Riemannian metrics in \(\socc \). The hyperbolic operator \(\hat {\mc H}\) is then expressed
in the form
where \(\D \) is the Laplacian of the DeWitt metric given in the
fibers, \(R\) the scalar curvature of the metrics \(g_{ij}(x)\in F(x)\), and \(\f \) is defined by
where
\(\rho _{ij}\) is a fixed metric in \(\so \) such that instead of densities we are considering
functions.
The Wheeler-DeWitt equation only represents the quantization of the normal
Einstein equations and ignores the tangential Einstein equations. In order to
quantize the full Einstein equations we incorporated the Hamilton condition into
the right-hand side of the Hamilton equations to obtain a scalar evolution
equation such that the Hamilton equations and this scalar evolution equation
are equivalent to the full Einstein equations, cf. [6, Theorem 1.3.4, p.
12]. For the quantization of this evolution equation we defined the base
space of the fiber bundle \(E\) to be the Cauchy hypersurface \((\socc ,\bar \s _{ij})\) of the quantized
spacetime, where \(\bar \s _{ij}\) is the induced metric. We also choose the metric \(\rho _{ij}\) in \(\mathrm {(\ref {E:1.11})}\) to be
equal to \(\bar \s _{ij}\). The result of this quantization was a hyperbolic equation in
\(E\).
The fibers \(F(x)\) over \(x\in \so \) are Riemannian metrics \(g_{ij}(x)\) if external fields are excluded. In an
appropriate local trivialization we obtained a coordinate system \((\xi ^a)\), \(0\le a \le m\),
and the metric \(\s _{ij}\) belongs to the hypersurface or
subbundle
\begin{equation}\nt M=\{t=1\}\su E. \end{equation}
The solutions \(u\) then depend on the variables \((t,\s _{ij},x)\), where \(\s _{ij}\) does not
depend on \(t\) and \(t\) not on \(x\). We refer to \(t\) as quantum time and \(x,\s _{ij}\) as spatial
variables.
In the papers [3, 5] we could express \(u\) as a product of eigenfunctions
\begin{equation} u=w\hat v v, \end{equation}
where \(w=w(t)\) is the
temporal eigenfunction, \(\hat v=\hat v(\s _{ij}(x))\) can be identified with an eigenfunction of the Laplacian
of the symmetric space
where \(\bar \s _{ij}\) is the fixed induced metric of \(\so \). The
eigenfunctions \(\hat v\) represent the elementary gravitons corresponding to the degrees of
freedom in choosing the entries of Riemannian metrics with determinants equal to
one. These are all the degrees of freedom available because of the coordinate
system invariance: For any smooth Riemannian metric there exists an atlas such
that the determinant of the metric is equal to one, cf. [6, Lemma 3.2.1, p. 74]. The
function \(v\) is an eigenfunction of an essentially self-adjoint differential operator in
\(\so \).
At first, the temporal eigenfunctions \(w\) were only the solutions of an ODE. Later, in
[5, Section 5] we proved that they were the eigenfunctions of an essentially
self-adjoint differential operator in \(\R []_+\), provided \(n\) is sufficiently large and \(\Lam <0\) and
the Cauchy hypersurface \((\socc ,\bar \s _{ij})\) is either a space of constant curvature like \(\R [n]\)
and \(\Hh [n]\) or a metric product of the form
\begin{equation} \bar \s =\de \otimes g \end{equation}
is a metric product; \(\de \) is the
standard Euclidean metric and \(g\) a Riemannian metric in \(M_0\), cf. [5, Section
5].
But in [6, Chapter 4.2] we were able to prove this property for arbitrary \(n\ge 3\) and \(\Lam <0\) and,
in case \(n=3\), even for \(\Lam >0\) by introducing an additional scalar fields map in the action
functional, i.e., a map
\begin{equation} \F :N\ra \R [k], \end{equation}
where \(N=I\times \so \) is the original spacetime which is to be
quantized. Let \((\bar g_{\al \bet })\) be the Lorentzian metric in \(N\), the scalar field Lagrangian is
defined by
There are two ways how to treat \(\mathrm {(\ref {E:1.27})}\) as an eigenvalue equation: First, the
cosmological constant \(\Lam \), or better \(-\Lam \) can be looked at as an implicit eigenvalue, or
secondly, if we consider \(\Lam <0\) to be fixed, we could try to solve the eigenvalue problem
In this case the corresponding eigenfunction \(w\) would be a
solution of \(\mathrm {(\ref {E:1.27})}\), i.e., it would be a temporal eigenfunction of our model of
quantum gravity. We solved the implicit as well as the explicit eigenvalue
problem in [6, Chapter 4] by choosing \(k\) in \(\mathrm {(\ref {E:1.28})}\) sufficiently large such that
\(\bar \mu <0\).
Since \(\mu _0\) is in general positive, unless we choose \(\abs {\theta _0}\) large which is not always possible or
desirable, we considered the orthogonally equivalent function
and \(\mu _0>0\). Then, there exists \(0<t_0<1\) and positive constants \(p,c_1,c_2\) such that \(u\) does
not vanish in the interval \((0,t_0]\) and can be estimates by
Here, we adapted the wording slightly to reflect the present assumptions, cf. [6,
Theorem 4.2.4, p. 118].
If we combine gravity with the forces of the Standard Model then we cannot
quantize the full Einstein equations but only the normal Einstein equation, i.e.,
the Hamilton condition. As a result we obtain the Wheeler-DeWitt equation which
again can be solved by a product of spatial and temporal eigenfunctions or
eigendistributions. In this case the temporal eigenfunction equation has the form,
after using the same ansatz as before,
Comparing this equation with
equation \(\mathrm {(\ref {E:1.34})}\) there are two differences: First, the term \(\mu _0\) does not depend on \(\abs {\theta _0}\)
since we
had to choose \(\theta _0=0\), and secondly, the exponent of \(t\) on the right-side is \(-\frac 23\). The first
difference implies that only by requiring \(k\) to be large we could enforce \(\bar \mu <0\)
and the negative exponent that the estimate \(\mathrm {(\ref {E:1.38})}\) is slightly worse, but still
good enough for our purpose. Indeed, we proved in [6, Theorem 5.5.5, p.
145]
Theorem 1.2.Let \(u\in \mc H_2\) satisfy the equation
\begin{equation}\lae {1.43} A_1u=-t^{-1}\frac \pa {\pa t}\big (t \pde ut\big )+t^{-2}\mu ^2 u +t^2 m_2^2 u=\lam t^{-\frac 23} u, \end{equation}
where the constants \(\mu , m_2\) and \(\lam \) are strictly
positive. Since \(\mu \) is especially important, let us emphasize that
and \(\mu _0>0\). Then,
for any small \(\e _0>0\), there exist \(0<t_0<1\) and positive constants \(p,c_1,c_2\) such that \(u\) does not
vanish in the interval \((0,t_0]\) and can be estimated by
The eigenvalue equations \(\mathrm {(\ref {E:1.36})}\) and \(\mathrm {(\ref {E:1.43})}\) in the Hilbert space \(\mc H_2\) can both be solved by
complete sequences of mutually orthogonal eigenfunctions \(u_i\) with corresponding
positive eigenvalues \(\lam _i\) of multiplicity one satisfying
and \(\mu _0>0\). Then, for any small \(\e _0>0\) there exists \(0<t_0<1\) and positive constants \(p,c_1,c_2\), such that \(w_i\) does
not vanish in the interval \((0,t_0]\) and can be estimates by
The eigenfunctions \(w_i\) in the previous theorem are the solutions of the original
temporal eigenfunctions equation and they are the eigenfunctions of a self-adjoint
operator in a Hilbert space. The \(u_i\) are the unitarily equivalent eigenfunctions of a
unitarily equivalent self-adjoint operator. In [7, Section 3] we proved that the
unitarily equivalent eigenfunctions
can be extended past the singularity by an
even reflection as sufficiently smooth functions provided the coefficient \(\mu ^2\) in \(\mathrm {(\ref {E:1.44})}\) is large
enough. More precisely, we proved:
Theorem 1.4.Let \(2\le m_0\in \N \) be arbitrary and assume
in \(\R []\), where we have to
replace \(t^q\) by \(\abs t^q\) for obvious reasons. Let us emphasize that the lower order coefficients
of the ODE exhibit a singularity in \(t=0\) but that both sides of the equation are
continuous in the interval \((-\un ,\un )\) and vanish in \(t=0\).
Here, the exponent \(q\) is any real number satisfying
\begin{equation} -2<q<2. \end{equation}
2. The equations of quantum gravity
The tangential Einstein equations are equivalent to the Hamilton equations and
the normal Einstein equation is equivalent to the Hamilton condition.
By quantizing the Hamilton condition we obtain the Wheeler-DeWitt
equation while ignoring the tangential Einstein equations. In order to
quantize the full Einstein equations we consider the second Hamilton
equations
On
the right-hand side of this evolution equation we then implement the
Hamilton condition \(H=0\) in the form
\begin{equation} p H=0, \end{equation}
where \(0\not =p\in \R []\) is an arbitrary real number to
be determined later. After the quantization of the modified evolution
equation \(\mathrm {(\ref {E:2.3})}\) we obtain the hyperbolic equation
The preceding equation is
evaluated at \((x,t,\s _{ij},\theta ^a)\), where \(x\in \so \), \(t\in \R []_+\), \(\s _{ij}\in M\) is the induced metric of a Cauchy hypersurface
of the quantized globally hyperbolic spacetime and \(\theta =\theta (x)\) is a coordinate in
the fiber \(\R [k]\). Let us recall that after quantization the components \(\F ^a\) of the
scalar field are equal to the coordinates \(\theta ^a\) in \(\R [k]\) such that
\begin{equation} \F ^a(x)=\theta ^a(x)\qq \A \, x \in \so \end{equation}
Since we only
introduced the scalar field in order to prove that the temporal "eigenfunctions"
are indeed eigenfunctions of a self-adjoint operator with a pure point
spectrum we can simplify the left-hand side of \(\mathrm {(\ref {E:4.2.47.4})}\) by choosing
\begin{equation} \theta ^a(x)=1\qq \A \, x \in \so , \;\A \, 1\le a\le k. \end{equation}
where \(u\) depends on \((x,t,\s _{ij},\theta ^a)\). The parameter \(p\in \R []\), \(p\not =0\), is not yet
specified.
As mentioned before the solution \(u\) should be a product of spatial and
temporal eigenfunctions. In order to ensure that the temporal eigenfunctions
are eigenfunctions of a self-adjoint operator we have to distinguish three
cases:
For a more detailed exposition we refer to [6, Chapter
4.2].
Finally, let us look at the Wheeler-DeWitt equation which we solved when we
quantized gravity combined with the forces of the Standard Model, cf. [4]. For our
purpose the reference [6, Chapter 5.4] is more suitable since, there, we also added
a scalar field map such that the combined Hamilton function has the form
where
the subscripts \(YM\), \(H\), \(D\) refer to the Yang-Mills, Higgs and Dirac fields and \(SM\) to the fields
of the Standard Model or to a corresponding subset of fields. The Hamilton
constraint
\begin{equation} \mc H=0 \end{equation}
will be quantized by first quantizing the Hamiltonians \(\mc H_G+\mc H_S\) in the fibers for
general metrics resulting in a hyperbolic operator
\begin{equation} \hat {\mc H}_G u+\hat {\mc H}_S u \end{equation}
will
be evaluated \((x,t,\de _{ij}, \bar \theta ^a)\), where \(\de _{ij}\) is the standard Euclidean metric in \(\so =\R [n]\), \(n=3\), and
\begin{equation} \bar \theta ^a(x)=1\qq \A \, 1\le a\le k. \end{equation}
The
Hamilton function \(\mc {\tilde H}_{SM}\), which represents spatial fields and is independent of \(t\), is
quantized in \((\so , \de _{ij})\) by the usual methods of Quantum Field Theory (QFT).
The Wheeler-DeWitt equation then has the form
where \(\al _N\) is a positive
coupling constant and where we also assume that \(u\) does not depend on
\(\theta ^a(x)\).
We then solve the Wheeler-DeWitt equation by using separation of variables. The
operator \(\hat {\mc H}_{SM}\) is acting only in the base space \(\so \), such that the spatial eigendistributions,
or approximate eigendistributions, \(\psi \) satisfying
can be derived by applying
standard methods of QFT.
The remaining operator in \(\mathrm {(\ref {E:2.5.4.6.1.1})}\) is acting only in the fibers, i.e., we can use the
eigenfunctions \(v=v(\s _{ij})\) of \(-\D _M\), which represent the elementary gravitons, satisfying
if \(n=3\), compare [6, equation (2.2.34), p. 49] and
\(\mathrm {(\ref {E:1.30})}\).
Hence, we make the ansatz
\begin{equation} u=wv\psi , \end{equation}
where \(w=w(t)\) only depends on \(t>0\). Then, combining \(\mathrm {(\ref {E:2.5.4.6.1.1})}\), \(\mathrm {(\ref {E:2.5.4.71.1})}\), \(\mathrm {(\ref {E:2.5.4.73.1})}\), \(\mathrm {(\ref {E:2.5.4.74.1})}\) and \(\mathrm {(\ref {E:2.5.4.75.1})}\)
we derive an ODE which must be solved by \(w\), namely,
then the left-hand side of \(\mathrm {(\ref {E:2.28})}\) is identical to
the left-hand side of equation \(\mathrm {(\ref {E:1.31})}\). However, on the right-hand side of these
equations we have different powers of \(t\) which will lead to slightly different
asymptotic estimates from above near the origin for the corresponding
solutions. In order to unify the approach we shall consider the temporal
equation
is negative if \(k\in \N \) is large enough. If in addition the cosmological
constant is also negative
\begin{equation} \Lam <0, \end{equation}
then \(\mathrm {(\ref {E:2.35})}\) can be looked at as an eigenvalue equation with
positive eigenvalues \(m_1\) in an appropriate Hilbert space. This eigenvalue
problem can be easily solved and in [7] we proved asymptotic estimates
near the singularity which allowed us to deduce that unitarily equivalent
eigenfunctions can be extended past the singularity as sufficiently smooth
functions.
3. The missing antimatter
In our model of quantum gravity the physical states are described by solutions of
a hyperbolic equation in a fiber bundle with base space \(\socc \) which is isometric to a
Cauchy hypersurface of the quantized spacetime. The solutions of the hyperbolic
equation can be expressed as a product of temporal and spatial eigenfunctions of
self-adjoint operators acting in appropriate Hilbert spaces. The coefficients of the
temporal eigenfunction equation as well as the corresponding eigenfunctions \(w_i\)
have a singularity in \(t=0\) similar to the big bang singularity of the quantized
spacetime.
in the quantization process we proved
in [7, Theorem 8, p. 21] that there exists a complete sequence of unitarily
equivalent temporal eigenfunctions \(\tilde u_i\) which solve the eigenfunction equation
in the
interval \((0,\un )\), where \(\tilde \mu , m_2\) and \(\lam _i\) are strictly positive and
\begin{equation} -2<q<2 \end{equation}
is a fixed exponent, such that the
solutions \(\tilde u_i\) can be evenly or oddly mirrored to the negative axis as sufficiently
smooth functions across the singularity provided the dimension \(k\) of the target
space of \(\F \) is sufficiently large. The equation \(\mathrm {(\ref {E:5.2})}\) is then valid in \(\R []\) such that both sides
smoothly vanish in \(t=0\). Moreover, we proved in [7, Lemma 6, p. 23] that
the eigenfunctions \(\tilde u_i\) vanish exponentially fast near infinity which is also
valid for the unitarily equivalent eigenfunctions \(w_i\), cf. [7, Corollary 3, p.
24].
The hyperbolic equation in the fiber bundle comprised second order differential
operators acting in the fibers as well as in the base space. The temporal
equation also defines a second order differential operator acting in the
fibers because the Riemannian metrics, which are part of the variables
after quantization, can be written in the form
where \(0<t<\un \) and the \(\s _{ij}(x)\), \(x\in \socc \), are
elements of a subbundle with fibers \(M(x)\) such that by fixing an arbitrary metric \(\bar \s _{ij}(x)\)
which is supposed to be the induced metric of a Cauchy hypersurface of
the quantized spacetime, we may assume, after choosing an appropriate
atlas depending on \(\bar \s _{ij}\), that each fiber \(M(x)\) is isometric to the symmetric space
cf. Section 1. The elementary gravitons are then eigenfunctions of the
Laplacian in \(X\). The corresponding eigenvalues are already incorporated in the
coefficient \(\tilde \mu ^2\) of the temporal differential operator such that we are allowed to
consider, besides the temporal operator, only spatial operators acting in
\(\so \).
Thus, we look at a quantum spacetime \(Q\) which can be written as a product
and at
self-adjoint operators \(H_0\) and \(H_1\) acting in appropriate Hilbert spaces such that the
remaining hyperbolic equation in \(Q\) can be expressed in the form
in which the equation \(\mathrm {(\ref {E:5.7})}\) is also
valid. Moreover, the equation \(\mathrm {(\ref {E:5.2})}\) is even valid in \(\R []\) across the singularity.
Hence, we have to face the question how to interpret this behaviour. If we
assume that \(Q_{-}\) has the same light cone as \(Q\), then the singularity in \(t=0\) lies in the
future of \(Q_-\) and since the mirrored eigenfunctions \(w_i(t)\) become unbounded if \(t\)
tends to zero, the singularity in \(t=0\) would be called a big crunch, i.e., \(Q_-\) would
end in a big crunch but the corresponding classical spacetime would not
start with a big bang in view of the results in [7, Lemma 6 & Corollary
3].
Hence, we have to assume that \(Q_-\) has the opposite time orientation, i.e., the
singularity in \(t=0\) is also a big bang for \(Q_-\). In [4] and [6, Chapter 5] we proved that we
may consider \(H_1\) to be a spatial self-adjoint operator defined by the fields of the
Standard Model. If \(H_1\) is invariant with respect to parity and charge conjugation
then, in view of the CPT theorem, we would conclude that at the big bang two
universes had been created with different time orientation one filled with matter
and the other with antimatter.
References
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Gravitation: an introduction to current research (Louis Witten, ed.), John Wiley, New
York, 1962, pp. 227–265.
[2] Claus Gerhardt, The quantization of gravity in globally hyperbolic spacetimes,
Adv. Theor. Math. Phys. 17 (2013), no. 6, 1357–1391, arXiv:1205.1427,doi:10.4310/ATMP.2013.v17.n6.a5.
[3] _________ , The quantization of gravity: Quantization of the Hamilton equations,
Universe 7 (2021), no. 4, 91, doi:10.3390/universe7040091.
[4] _________ , A unified quantization of gravity and other fundamental forces of nature,
Universe 8 (2022), no. 8, 404, doi:10.3390/universe8080404.
[5] _________ , The quantization of gravity: The quantization of the full Einstein
equations, Symmetry 15 (2023), no. 8, 1599, doi:10.3390/sym15081599.
[6] _________ , The Quantization of Gravity, 2nd ed., Fundamental Theories of Physics,
vol. 194, Springer, Cham, November 2024, doi:10.1007/978-3-031-67922-3.
[7] _________ , Extending the solutions and the equations of quantum gravity past the big
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