# Signed Coordinate Invariance, invariant lagrangians and manifolds, the time
problem in quantum cosmology, quantum space time, spacetimes and
antispacetimes.

Eduardo Guendelman<EMAIL_ADDRESS>Department of Physics, Ben-Gurion
University of the Negev, Beer-Sheva, Israel.
Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, 60438
Frankfurt am Main, Germany.
Bahamas Advanced Study Institute and Conferences, 4A Ocean Heights, Hill View
Circle, Stella Maris, Long Island, The Bahamas.

###### Abstract

Standard general coordinate invariance for the volume element is extended to
general coordinate transformations that have a negative jacobian. This is
possible by introducing a non Riemannian Measure of integration, which
transforms according to the jacobian of the coordinate transformation, not the
absolute value of the jacobian of the coordinate transformation as it is the
case with $\sqrt{-g}$. For some signed general transformations a change of
boundary conditions is involved in GR and restoring the original limits of
integrations can restore symmetry of the action. The discussion totally
changes when a non Riemannian Measure of integration is introduced, in the
case of signed general coordinate transformations, and even when we have,
boundaries, since the modified measure is constructed out of scalar fields
becomes also the integration manifold, Since the measure fields that define
the non metric measure are scalars. if a signed general coordinate
transformations is considered, there is no change in the measure fields,
indicating strict invariance of both lagrangian density and integration
manifold, implying boundary terms for manifolds with boundaries in coordinate
space to be irrelevant . This analysis can be applied to give a framework
where in certain scenarios, like t can give something similar to the Linde´s
Universe multiplication from first principles, Linde assumes that together
with the observed universe there is an additional one with analogous matter
and gravity content but with opposite action. The Linde model also has a
(restricted) version of the signed general coordinate invariance (without
affecting Boundary Conditions) . We consistently formulate the non Riemannian
measure theory extension of General Relativity, that could be related to
Linde´s model, although there are some fundamental differences with Linde´s
scenario. like there is here a local formulation. The assumption of a
coordinate independent measure of integration appears also useful in the study
of Baby Universe Creation. The formulation provides also a new natural way to
address the problem of time in Quantum Cosmology, a proposal for a quantum
space time and identification of space times and anti space times states in
the gravitational theory, which in fact relates very much to the model of
Linde, which can be interpreted now as a non local coexistence of space time
and anti spacetime.

##  I Introduction

In an interesting paper, Linde formulated a model that claims to resolve the
cosmological constant problem Lindemultiplication . This requires the
existence of two universes, each with its own set of coordinates $x^{\mu}$ and
$y^{\mu}$ containing matter and gravity components that mirror each other, but
with the corresponding actions having opposite signs, like spacetime and an
antispacetime, as in

$S=\int
d^{4}xd^{4}y{\sqrt{-g(x)}}{\sqrt{-\bar{g}(y)}}(\frac{M_{P}^{2}}{16\pi}R(x)+L(\phi(x))-\frac{M_{P}^{2}}{16\pi}R(y)-L((\bar{\phi}(y))$
(1)

$R(x)$ is defined in terms of a $g_{\mu\nu}$ metric, while $R(y)$ is defined
in terms of a $\bar{g}_{\mu\nu}$ metric and where $L(x)$ and $L(y)$ have
exactly the same functional form with respect to their corresponding mirror
fields, like for a scalar field $\phi(x)$, there will be a potential
$V(\phi(x))$, same with kinetic terms, etc. that are appearing in $L(x)$ ,
while in $L(y)$ there will be corresponding field $\bar{\phi}(y)$ with a
potential $V(\bar{\phi}(y))$, then in $L(y)$ the metric $\bar{g}_{\mu\nu}$
appears instead of the metric $g_{\mu\nu}$, then the theory is obviously
invariant under $V\rightarrow V+constant$ Lindemultiplication . This non local
coexistece of a spacetime and an antispacetime was shown by Linde to have
remarkable properies concerning its behavior with respect to the cosmological
constant problem. As we will see , we can think of this as having regions
where the measure of integration can change sign, an effect that must take
place at the same time as we double the space time, to realize Linde´s ideas.
The model by Linde is however non local, so this is an aspect that is not
desirable, we will explore how models inspired by modified measures (non
Riemannian) avoids this.

Another area where our formalism can help is the more complicated process of
quantum creation of inflationary bubbles. In the most popular paradigm for the
early universe, it is postulated that the universe suffers a period of
exponential expansion called “inflation” introduced by Guth Guth  ,
Starobinsky starobinsky  , Linde Linde  , ALBRECHTANDSTEINHARDT , (see also
the books early-univ ; primordial  and references therein). The possibility of
a ”local” version of inflation, that is an inflationary bubble, was studied in
BLAUGUENDELMANGUTH  a process which was later generalized so as to include the
quantum creation of such bubbles in FARHIGUVEN , for an alternative treatment
of creation of baby universes see FischlerPolchinski . For a popular review on
the subject of Baby Universes and further references see ZEEYA  . An
outstanding feature of FARHIGUVEN  is that in their euclidean tunneling
solution there are regions where the measure defining the volume element must
be negative. However this seems hard to understand standard Riemannian measure
$\sqrt{-g}$ cannot behave in this way, nevertheless FARHIGUVEN  gives very
good reasons that this should be the correct behavior of the measure in the
euclidean tunneling solution. For other approaches to the quantum creation of
a baby universe see FischlerPolchinski .

Here we will address Linde´s scenario and tentativelly also the much more
complicated baby universe creation in the framework of the metric independent
non Riemannian measures, which has been used for the construction of modified
gravity theories Refs.TMT-orig-1 -TMT-orig-3  (see also Refs.TMT-recent-1-a
-TMT-recent-2 ), in some instances we have included the standard measure as
well, where the standard Riemannian integration measure might also contain a
Weyl-scale symmetry preserving $R^{2}$-term TMT-orig-3 . Some applications
have been, (i) $D=4$-dimensional models of gravity and matter fields
containing the new measure of integration appear to be promising candidates
for resolution of the dark energy and dark matter problems, the fifth force
problem, and a natural mechanism for spontaneous breakdown of global Weyl-
scale symmetry TMT-orig-1 -TMT-recent-2 , (ii) To study in Ref.susy-break  of
modified supergravity models with an alternative non-Riemannian volume form on
the space-time manifold, non singular emergent models leading to inflation and
then decaying into a dark enery and dark matter phase ourquintessence  \-
ourquintessencewithEDE ,(iv) Gravity-Assisted Emergent Higgs Mechanism in the
Post-Inflationary Epoch, Gravityassisted , (v) To study of reparametrization
invariant theories of extended objects (strings and branes) based on employing
of a modified non-Riemannian world-sheet/world-volume integration measure
mstring , nishino-rajpoot , leads to dynamically induced variable string/brane
tension and to string models of non-abelian confinement, interesting
consequences from the modified measures spectrum mstringspectrum , and
construction of new braneworld scenarios mstringbranes , . Modifed Measures
Theories have been discussed as effective theories for causal fermion theories
MMT .

We will see here how in this framework one can construct a General coordinate
invariant which has extended general coordinate transformations that includes
also transformations with a negative jacobian, this is possible by introducing
a non Riemannian Measure of integration, which transforms according to the
jacobian of the coordinate transformation, not the absolute value of the
jacobian of the coordinate transformation as it is the case with $\sqrt{-g}$.
This analysis can be applied to give a framework for the Farhi, Guven and Guth
treatment of the Quantum creation of a baby Universe, since here the new
theory certainly allows for negative measures of integration required by
Linde´s universe multiplication and the Farhi, Guth, Guven tunneling solutions
where the measure of integration also can be negative.

##  II  General Relativity and other theories use a Riemannian volume element
that is not invariant under signed general coordinate transformations

The action of GR, and other theories that use the standard Riemannian volume
element $d^{4}x{\sqrt{-g}}$ is of the form,

$S=\int d^{4}x{\sqrt{-g}}L$ (2)

where $L$ is a generally coordinate invariant lagrangian. Now notice that
under a general coordinate transformation,

$d^{4}x\rightarrow Jd^{4}x$

, while

$\sqrt{-g}\rightarrow\mid J\mid^{-1}\sqrt{-g}$

where $J$ is the jacobian of the transformation and $\mid J\mid$ is the
absolute value of the transformation. Therefore
$d^{4}x{\sqrt{-g}}\rightarrow\frac{J}{\mid J\mid}d^{4}x{\sqrt{-g}}$, so
invariance is achieved only for $J=\mid J\mid$, that is if $J>0$, that is
signed general coordinate transformations are excluded.

One could argue that when taking the square root of the determinant of the
metric one may choose the negative solution when it suit us, but this would be
an arbitrary procedure if no specific rule is given to choose the positive or
the negative root. We choose instead to declare that $\sqrt{-g}$ is always
positive and replace it in the measure by something else whose sign is well
defined.

###  II.1 Invariance of the action with non invariant lagrangian density
(integrand) and compensating non invariant manifold of integration?

If conditions are optimal, the non invariance of the lagrangian density
(integrand) , which includes the measure, in a signed coordinate
transformation, could be compensated by the non invariance of the manifold of
integration. For example, in a time reversal transformation, the integrand
will change sign, but , if there no obstructions, and we can then change the
limits of integrations, the exchange of the limits of integrations will
involve an additional exchange of signs that can compensate for the sign
change in the integrand.

If the manifold has boundaries in coordinate space, introducing coordinate
transformations that change the boundaries involve compensationg terms at the
boundaries, even when the transformations are infinitesimal, so at this point
the invariance of the action becomes complicated and problematic, in
particular in cosmology where the universe may have a beginning in time.

As we discuss in the next sections, where we will discuss a possible
realization of the invariance under signed general coordinate transformations
in the context of the non local Linde Universe multiplication model, without
invoking a change in the manifold of integration, but rather transforming the
field variables, and then in the following section a return to a local theory
by the use of the modified measure formalism.

##  III Invariance of signed general coordinate transformations in the non
local Linde Universe multiplication model, or spacetime antispace model

The Linde non local model can offer a limited way out to obtain signed general
coordinate invariance, so we can allow general coordinate transformations
where the jacobian of the transformation, say in the $x$ space , is negative,
but still we do not consider the possibility that it could change from
positive to negative.

As long as the jacobian is uniformly negative over all space, invariance will
be achieved if the same transformation is performed both in the $x$ and $y$
coordinates In these cases, we do not invoke any transformation or change in
the manifold of integration, which we have argued is a questionable operation.

As we will see in the next section, the use of Metric Independent Non-
Riemannian Volume-Forms and Volume elements allows us to resolve this issue
with no such restrictions on possible changes of signs of the jacobian of the
coordinate transformation in different regions of space time and without
invoking non local actions.

##  IV Metric Independent Non-Riemannian Volume-Forms and Volume elements
invariant under signed general coordinate transformations

One can define a metric independent measure from a totally anti symmetric
tensor gauge field, for example

$\Phi(A)=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\partial_{\mu}A_{\nu\kappa\lambda}\quad,$
(3)

Then, under a general coordinate transformation

$\Phi(A)\rightarrow J^{-1}\Phi(A)$

. . Therefore $d^{4}x\Phi(A)\rightarrow d^{4}x\Phi(A)$, so invariance is
achieved regardless of the sign of $J$.

##  V Theory using Metric Independent Non-Riemannian Volume-Forms

First we review our previous papers where we have considered the action of the
general form involving two independent non-metric integration measure
densities generalizing the model analyzed in quintess  is given by

$S=\int d^{4}x\,\Phi_{1}(A)\Bigl{[}R+L^{(1)}\Bigr{]}+\int
d^{4}x\,\Phi_{2}(B)\Bigl{[}L^{(2)}+\epsilon
R^{2}+\frac{\Phi(H)}{\sqrt{-g}}\Bigr{]}\;.$ (4)

Here the following definitions are used:

  * •

The quantities $\Phi_{1}(A)$ and $\Phi_{2}(B)$ are two densities and these are
independent non-metric volume-forms defined in terms of field-strengths of two
auxiliary 3-index antisymmetric tensor gauge fields

$\Phi_{1}(A)=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\partial_{\mu}A_{\nu\kappa\lambda}\quad,\quad\Phi_{2}(B)=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\partial_{\mu}B_{\nu\kappa\lambda}\;.$
(5)

The density $\Phi(H)$ denotes the dual field strength of a third auxiliary
3-index antisymmetric tensor

$\Phi(H)=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\partial_{\mu}H_{\nu\kappa\lambda}\;.$
(6)

  * •

The scalar curvature $R=g^{\mu\nu}R_{\mu\nu}(\Gamma)$ and the Ricci tensor
$R_{\mu\nu}(\Gamma)$ are defined in the first-order (Palatini) formalism, in
which the affine connection $\Gamma^{\mu}_{\nu\lambda}$ is a priori
independent of the metric $g_{\mu\nu}$. Let us recall that $R+R^{2}$ gravity
within the second order formalism was originally developed in starobinsky .

  * •

The two different Lagrangians $L^{(1,2)}$ correspond to two matter field
Lagrangians

On the other hand, the variation of (4) w.r.t. auxiliary tensors
$A_{\mu\nu\lambda}$, $B_{\mu\nu\lambda}$ and $H_{\mu\nu\lambda}$ becomes

$\partial_{\mu}\Bigl{[}R+L^{(1)}\Bigr{]}=0\quad,\quad\partial_{\mu}\Bigl{[}L^{(2)}+\epsilon
R^{2}+\frac{\Phi(H)}{\sqrt{-g}}\Bigr{]}=0\quad,\quad\partial_{\mu}\Bigl{(}\frac{\Phi_{2}(B)}{\sqrt{-g}}\Bigr{)}=0\;,$
(7)

whose solutions are

$\frac{\Phi_{2}(B)}{\sqrt{-g}}\equiv\chi_{2}={\rm
const}\;\;,\;\;R+L^{(1)}=-M_{1}={\rm const}\;\;,\;\;L^{(2)}+\epsilon
R^{2}+\frac{\Phi(H)}{\sqrt{-g}}=-M_{2}={\rm const}\;.$ (8)

Here the parameters $M_{1}$ and $M_{2}$ are arbitrary dimensionful and the
quantity $\chi_{2}$ corresponds to an arbitrary dimensionless integration
constant.

The resulting theory is called a Two Measure Theory, due to the presence of
the Two measures $\Phi_{1}(A)$ and $\Phi_{2}(A)$. But for the purpose of this
paper this is two general, since we want to restrict to a theory that will
give us ordinary General Relativity, and we want to keep the general
coordinate invariance under signed general coordinate invariance.

For obtaining GR dynamics, we can restrict to one measure, so let us take

$\Phi_{1}(A)=\Phi_{2}(B)=\Omega$

also to make some contact for example with FARHIGUVEN  , where an additional
set of four fields is introduced, we express $\Phi$ in terms of four scalar
fields

$\Omega=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\varepsilon^{abcd}\partial_{\mu}\varphi_{a}\partial_{\nu}\varphi_{b}\partial_{\kappa}\varphi_{c}\partial_{\lambda}\varphi_{d}\quad$
(9)

(one has to point out that in the earlier formulations of modified measures
theories we used the 4 scalar field representation for the measure, see TMT-
orig-1  , ) The mapping of the four scalars to the coordinates $x^{\mu}$ may
be topologically non trivial, as in FARHIGUVEN  and this multivaluedness could
be of use to obtain Linde´s Universe multiplication as well. Finally, we have
to correct the equation

$\frac{\Phi_{2}(B)}{\sqrt{-g}}\equiv\chi_{2}={\rm const}.$ (10)

for another equation the will be invariant under signed general coordinate
invariant transformations, which will be

$\frac{\Omega^{2}}{(-g)}\equiv\chi=K^{2}={\rm const}>0.$ (11)

without loss of generality we define $K$ to be positive. The resulting action
that replaces (5) is,

$S=\int d^{4}x\,\Omega\Bigl{[}R+L\Bigr{]}+\int
d^{4}x\,\Omega^{2}\Bigl{[}\frac{\Phi(H)}{{(-g)}}\Bigr{]}\;.$ (12)

the density $\Phi(H)$ remains defined eq. (6) so the integration obtained from
the variation of the $H$ gauge field is eq. (11) now. The solution of eq. (11)
are

$\frac{\Omega}{\sqrt{(-g)}}=\pm{K}.$ (13)

where the sign in (13) will be dynamically determined, as opposed to the Linde
scenario, where the negative measure, associated with the $y$ universe always
exists. The non locality in the Linde´s scenario could be replaced by the
introduction of a new set of coordinates in a way such that the $\varphi_{a}$
scalars have a relation to the coordinates $x$ to these scalars may be multi
valued, What is no doubt is similar is the existence of positive and negative
measures of integration in both theories. In both cases it is related to
signed general coordinate transformation, although in the Linde model the sign
has to be uniform over space time and in the modified measure case this
restriction does not exist, in fact the sign of the jacobian of the
transformation can vary from one region of space time to the other in the non
Riemannian measure formulation,

Another possibility for a measure that would transform like the the jacobian
of the coordinate transformation, not the absolute value of the jacobian,
would be the determinant of the vierbein. This will detroy however (up to a
sign) the invariance of the theory under signed local Lorentz transformation
of the vierbeins. that is Lorentz transformations with negative determinants,
so, it is not a solution, rather we trade one asymmetry for another.

##  VI The invariant scalar integration manifold and invariant lagrangian
density

Notice that using the volume element converts the the integration over
coordinates in the action into integration over scalar fields, since

$\Phi d^{4}x=d\varphi_{1}d\varphi_{2}d\varphi_{3}d\varphi_{4}$

The integration manifold existing in the four scalar field manifold is in fact
completely unaffected by any coordinate transformation taking place in the $x$
space. The lagrangian density is also a scalar not affected by any coordinate
transformation, the theory formulated in this way does not require any
boundary terms if the boundaries are for example formulated in the scalar
field space.

In the case of (12) for example,

$S=\int d\varphi_{1}d\varphi_{2}d\varphi_{3}d\varphi_{4}L$

, where

$L=\Bigl{[}R+L\Bigr{]}+\,\Omega\Bigl{[}\frac{\Phi(H)}{{(-g)}}\Bigr{]}\;$

##  VII Gravitational Equations of motion

We start by considering the equation that results from the variation of the
degrees of freedom that define the measure $\Omega$, that is the scalar fields
$\varphi_{a}$, these are,

$A^{\mu}_{a}\partial_{\mu}(R+L+2\Omega\frac{\Phi(H)}{{(-g)}})=0$ (14)

where

$A^{\mu
a}=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\varepsilon^{abcd}\partial_{\nu}\varphi_{b}\partial_{\kappa}\varphi_{c}\partial_{\lambda}\varphi_{d}\quad$
(15)

Notice that the determinant of $A^{\mu a}$ is proportional to $\Omega^{3}$, so
if the measure is not vanishing, the matrix $A^{\mu a}$ is non singular and
therefore $\partial_{\mu}(R+L+2\Omega\frac{\Phi(H)}{{(-g)}})=0$, so that,

$R+L+2\Omega\frac{\Phi(H)}{{(-g)}}=M=constant$ (16)

The variation with respect to the metric $g^{\mu\nu}$, we obtain.

$\Omega(R_{\mu\nu}+\frac{\partial L}{\partial
g^{\mu\nu}})+g_{\mu\nu}\Omega^{2}\frac{\Phi(H)}{{(-g)}}=0$ (17)

solving $\Omega\frac{\Phi(H)}{{(-g)}}$ from (16) and inserting into (17), we
obtain,

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R+\frac{1}{2}Mg_{\mu\nu}+\frac{\partial
L}{\partial g^{\mu\nu}}-\frac{1}{2}g_{\mu\nu}L=0$ (18)

which gives exactly the form of Einstein equation with the canonical energy
momentum defined from $L$

$T_{\mu\nu}=g_{\mu\nu}L-2\frac{{\partial{}}}{{\partial{g^{\mu\nu}}}}L\;.$ (19)

The equations of motion of the connection (in the first order formalism)
implies that the connection is the Levi Civita connection. L can describe a
scalar field with the potential and the term $\frac{1}{2}M$ can be interpreted
as a shift of the scalar field potential by a constant or a floating
contribution to the cosmological constant- In FARHIGUVEN  the tunneling
solution is described by considering another embedding space in addition to
the standard $x^{\mu}$ space, the mapping between the two set of coordinates
is multi valued , so that for each point in the $x^{\mu}$ space there are many
points in the embedding space. In the modified measure approach to this
problem is best to consider the embedding space as the one defined by the four
scalar fields $\varphi_{a}$ that define the measure $\Omega$ (9). Therefore
the most fundamental space is the $\varphi_{a}$ , since only in this space we
can formulate the full description and solution of the problem. The
calculations in FARHIGUVEN  are consistent with GR, just extended to allow for
negative measures of integration, which is exactly what we have formulated
here in the context of a modified measure theory.

One issue that should be addressed is that of the gauge fixing in the
$\varphi_{a}$ space. Indeed, we notice that the only thing where these fields
appear in the equations of motion is $\Omega$, but this quantity is invariant
under volume preserving diffeomorphisms of the fields $\varphi_{a}$,
$\varphi^{\prime}_{a}=\varphi^{\prime}_{a}(\varphi_{a})$ which satisfy

$\epsilon_{a_{1}a_{2}a_{3}a_{4}}\frac{\partial{\varphi^{\prime}}_{b_{1}}}{\partial\varphi_{a_{1}}}\frac{\partial{\varphi^{\prime}}_{b_{2}}}{\partial\varphi_{a_{2}}}\frac{\partial{\varphi^{\prime}}_{b_{3}}}{\partial\varphi_{a_{3}}}\frac{\partial{\varphi^{\prime}}_{b_{4}}}{\partial\varphi_{a_{4}}}=\epsilon_{b_{1}b_{2}b_{3}b_{4}}$
(20)

so the study of the best gauge for the $\varphi_{a}$ fields for further
comparison with the $x^{\mu}$ space could be a very important subject. Of
course when we say that the mapping between the $\varphi_{a}$ and the
$x^{\mu}$ spaces, we want to exclude multi valuedness due to volume preserving
diffeomorphisms of the fields $\varphi_{a}$, if for example different signs
for $\Omega$ are associated to the same point in $x^{\mu}$ space, it is clear
that there are at least two points in $\varphi_{a}$ space associated to one
point in $x^{\mu}$ space, and these two points in the $\varphi_{a}$ are not
related through a volume preserving diff. This could be an effect analogous to
the Universe Multiplication of Linde.

##  VIII Linde´s universe Multiplication and relation to a Brane Anti Brane
system and measure field multivaluedness instead of non locality

We can immediately see some similar features between the Linde universe
multiplication as described by eq. (1) and the modified measure theory, with
the measure assuming a positive or a negative value, as expressed by eq, (13),
instead of the obvious non locality of the Linde approch, the modified measure
approach can offer instead multi valued feature of the $\varphi_{a}$ space
with respect to the $x^{\mu}$ space. The double solution for the measure (13)
can be valid for the same coordinate $x^{\mu}$ , which may correspond however
to non unique values in the $\varphi_{a}$ space. The doubling of the measure
(13) has its correspondence in the signed reparametrization invariant
formulation of modified measure stringsandantistrings  and the corresponding
existence of strings and antistrings as well as branes and anti branes in such
fromulation.

##  IX Turning Manifolds with Boundaries into Manifolds without Boundaries,
the problem of time in Quantum Cosmology, Quantum Space Time, space times and
anti space times

As we will discuss in this section, these concepts can have interesting
applications to quantum cosmology: 1) the consideration of one of the measure
fields as a time, instead of the ordinary coordinate time can transform a
manifold with boundaries, as a universe with an origin of time, which is
difficult to handle, into a manifold which avoids these difficulties in by
considering that the fundamental manifold in terms of the measure fields does
not have boundaries and 2) the problem of time, associated with the coordinate
time, which produces vanishing Hamiltonians is avoided. Considering instead
one of the measure fields as time avoids this problem and lead us to a new
formulation of a quantum space time, 3) using the measure fields as the basis
to construct the quantum space time 4) defining space times and
antispacetimes, this is in fact related to 1).

Indeed, concerning the first point, in most cosmological models we assume that
the universe has an origin in time, this leads then to a manifold with
boundaries if we think the fundamental manifold is coordinate space and time.
A well known example of such a phenomena is the pair creation in QED, which we
review in the next subsection.

###  IX.1  Pair Creation in a Strong Uniform Electric Field in QED, as an
example of turning a manifold with boundary in time to a manifold without
boundary in proper time

From the ’Feynman’ perspective, negative energy waves propagating into the
past are physically realized as the antiparticles propagating into the future.
As he has shown Fyn:e-m  that from the classical equations of motion for a
particle in an external field can be written as

$m\frac{d^{2}z^{\mu}}{d\tau^{2}}=e\frac{dz_{\nu}}{d\tau}F^{\mu\nu}$ (21)

where $\tau$ is the proper time. If we note as Feynman has that if we allow
$\tau\rightarrow-\tau$ the equation becomes

$m\frac{d^{2}z^{\mu}}{d\tau^{2}}=-e\frac{dz_{\nu}}{d\tau}F^{\mu\nu}$ (22)

which is identical to the previous equation except that the particles charge
has changed. In other words, as far as its charge is concerned, it has become
the antiparticle. Thus, proper time running backward (i.e.,
$\tau\rightarrow-\tau$), while keeping the coordinate time unchanged. led to
the particle becoming an antiparticle. Of course we can take the equivalent,
but more suitable for our purposes transformation that we change the direction
of coordinate time, while requiring that the proper time remains unchanged.

This would be exactly analogous to taking one of the four scalars as an
analogous of the proper time and the coordinate time, which would be another
entity,

We are now in a position where we can discuss the scattering of a particle in
an external field. Four possibilities are seen to exist: i) the scattering of
the particle by the field, ii) the creation of a particle-antiparticle pair.
iii) the annihilation particle-antiparticle pair iv) the scattering of an
antiparticle by the field. Below, we review one of them and in the following
section we draw conclusions concerning the dynamical string theory and the
analogy of the modified measure in the string case with the proper time in the
particle case. We shall present here a simplified version, by A.Vilenkin
Vilenkin , of the calculation of pair production first performed by Schwinger
Schwinger1  To begin our discussion, we consider a particle with charge $e$
and mass $m$ in a constant electric field. The general equation of a particle
in a field is most convenient written in terms of the Maxwell tensor
$F_{\mu\nu}$ where for a constant electric $\mathbf{E}$ in the $x-$direction,
$F^{01}=~{}-~{}E,\ \ F^{10}=E\ \ F^{0}_{\ 1}=E,\ \ F^{1}_{\ \ 0}=E$. More
explicitly, we have

$\displaystyle F^{\mu}_{\ \ \nu}=\left[\begin{array}[]{llll}0&E&0&0\\\
E&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}\right]$ (27)

The equation of motion for this particle in a constant electric field is

$m\frac{d^{2}x^{\mu}}{d\tau^{2}}=eF^{\mu}_{\ \ \nu}\frac{dx^{\nu}}{d\tau}$
(28)

The formal solution of for $u^{\mu}=\frac{dx^{\mu}}{d\tau}$ is
$u^{\mu}(\tau)=\exp[\frac{e}{m}F^{\alpha}_{\ \
\beta}\tau]^{\mu}_{\nu}u^{\nu}(0)$ The exponential can be expanded and we have

$\displaystyle\exp[\frac{e}{m}F^{\alpha}_{\ \
\beta}\tau]^{\mu}_{\nu}=\delta^{\mu}_{\nu}+\frac{e}{m}\tau
E\Delta^{\mu}_{\nu}+\frac{1}{2}(\frac{e}{m}\tau
E\Delta^{\mu}_{\nu})^{2}+\cdots$ (29)

where

$\displaystyle\Delta=\left[\begin{array}[]{llll}0&1&0&0\\\ 1&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\end{array}\right]$ (34)

Separating even and odd power in Eq. 29 we have

$\displaystyle u^{0}$ $\displaystyle=$
$\displaystyle\cosh\left(\frac{eE\tau}{m}\right)u^{0}(0)+\sinh\left(\frac{eE\tau}{m}\right)u^{1}(0)$
$\displaystyle u^{1}$ $\displaystyle=$
$\displaystyle\sinh\left(\frac{eE\tau}{m}\right)u^{0}(0)+\cosh\left(\frac{eE\tau}{m}\right)u^{1}(0)$
(35)

Integrating with respect to $\tau$ yields (where we have dropped arbitrary
constants of integration)

$\displaystyle x^{0}$ $\displaystyle=$
$\displaystyle\frac{m}{eE}\left\\{\sinh\left(\frac{eE\tau}{m}\right)u^{0}(0)+\cosh\left(\frac{eE\tau}{m}\right)u^{1}(0)\right\\}$
$\displaystyle x^{1}$ $\displaystyle=$
$\displaystyle\frac{m}{eE}\left\\{\cosh\left(\frac{eE\tau}{m}\right)u^{0}(0)+\sinh\left(\frac{eE\tau}{m}\right)u^{1}(0)\right\\}$
(36)

We choose the following boundary conditions $u^{1}(0)=0,\ \ u^{0}(0)=1$ which
leads to

$\displaystyle x^{0}$ $\displaystyle=$
$\displaystyle\frac{m}{eE}\sinh\left(\frac{eE\tau}{m}\right)$ $\displaystyle
x^{1}$ $\displaystyle=$
$\displaystyle\frac{m}{eE}\cosh\left(\frac{eE\tau}{m}\right)$ (37)

and for the boundary condition $u^{1}(0)=0,\ \ u^{0}(0)=-1$ leads to

$\displaystyle x^{0}$ $\displaystyle=$
$\displaystyle-\frac{m}{eE}\sinh\left(\frac{eE\tau}{m}\right)$ $\displaystyle
x^{1}$ $\displaystyle=$
$\displaystyle-\frac{m}{eE}\cosh\left(\frac{eE\tau}{m}\right)$ (38)

The solution given by Eq. IX.1 represents a particle solution while the
solution of Eq. IX.1 represents the anti-particle solution. Both solutions,
together satisfy

$(x^{1})^{2}-(x^{0})^{2}=\left(\frac{m}{eE}\right)^{2}$ (39)

At classical level, these solutions are distinct and one solution can not
evolved into the other. Thus, a particle couldn’t evolve to an anti-particle.
However, the semi-classical approximation which consists of considering the
classical equations of motion but with imaginary time. Then inserting
$t=-it_{E}$ we obtain that the hyperbola of Eq. 39 becomes a circle

$(x^{1})^{2}+t_{E}^{2}=\left(\frac{m}{eE}\right)^{2}$ (40)

This tunneling solution can now interpolate between the anti-particle and
particle solutions. In the imaginary time region, the action, $S=-iS_{E}$
where $S_{E}$ is given by

$S_{E}=\int
dt_{E}\left\\{m\sqrt{1+\left(\frac{dx}{dt_{E}}\right)^{2}}-eEx\right\\}$ (41)

Introduction the angular variable $\theta$ where $x=\frac{m}{eE}\cos\theta,\ \
t_{E}=\frac{m}{eE}\sin\theta$ we obtain that $S_{E}=\pi m^{2}/eE$. Since the
probability is given (up to pre factors) by $\exp{(-S_{E})}=\exp(-\frac{\pi
m^{2}}{eE})$ for $eE>0$. Notice that the distance of the particle and
antiparticle at the moment of creation is $\Delta x=\frac{2m}{eE}$ which has a
physical interpretation is manifest in writing it as

$W=eE\Delta x=2mc^{2}$ (42)

where we have restored the ’c’ to make its physical meaning clearer. Thus we
can create a pair of particle-antiparticle in a constant electric field by
performing work, $W$ in a distance $\Delta x$ equal to the the sum of the rest
masses of the particles, i.e. $2mc^{2}$. In the case for an electric field
that doesn’t extend through all of space, still it must be extended enough to
perform the work equal to the sum rest masses of the two particles in order to
create a pair.

From the point of view of our discussion here, we see that in the pair
creation process, the proper time proceeds from minus infinity to plus
infinity, while for this manifold the minimum coordinate time sits in
euclidean space at $t_{E}=-\frac{m}{eE}$, the coordinate manifold is a
manifold with boundaries, while the proper time manifold does not have
boundaries associated with it.

###  IX.2 Resolution of The problem of Time in Quantum Cosmology using one of
the measure fields as time

A related difficulty related to the use of coordinate time in quantum
cosmology is that the invariance under reparametrizations in the coordinate
time leads to the constraint that the Hamiltonian equals zero. In this case
the analogous of rge Schroedinger equation for Quantum gravity will tell us
that the wave function is time independent.

In the presence of this problem, the ¨time problem in Quantum Cosmology¨, many
proposals for alternative fields to replace the cosmic time were proposed, for
a review see PROBLEmoftimeinQC .

To choose the corresponding time among the four measure fields, we can use the
volume preserving diffeomorphisms (20) or the general coordinate invariance to
choose three of the measure scalars as the three spacial coordinates $x,y,z$,
the remaining scalar can be defined as the time like coordinate, which cas run
in the same direction of the coordinate time or against.

###  IX.3 Quantum Space Time

By considering a gauge (of the volume preserving diffeomorphisms or the
general coordinate invariance groups ) where three of the coordinates
$(t,x,y,z)$, say $(x,y,z)$ are set to be equal to three of the measure fields,
we define a procedure by means of which coordinates become dynamical , since
the measure fields are real dynamical variables and space is now represented
by the measure fields. Notice that these measure fields have non vanishing
canonically conjugated momenta, etc. so they represent the dynamical and
quantized space time (for some papers and a review of quantum space time see
quantumspacetime ).

One can see that the for the canonically conjugate momenta for the measure
fields, the system shows constraints that relate the canonically conjugate
momenta with the spacial derivatives of these measure gauge fields, indeed,

$\pi^{\varphi_{a}}=M\epsilon^{aa_{1}a_{2}a_{3}}\epsilon^{ijk}\partial_{i}\varphi_{a_{1}}\partial_{j}\varphi_{a_{2}}\partial_{k}\varphi_{a_{3}}$
(43)

Showing a constraint that relates the canonically conjugate momenta of the
measure fields with the spacial derivatives of these fields. The commutators
could be calculated not from the Poison brackets but from the Dirac brackets
DIRACBRACKETS , that are defined from the Poisson brackets of the constraints,
including the gauge fixing, like the ones we have mentioned. On the mass shell
$M$ is a constant, but off the mass shell it must be defined as
$M=R+L+2\Omega\frac{\Phi(H)}{{(-g)}}$. The quadratic term in $\Omega$ in the
action implies that the dependence of the action on the measure time
derivatives is not linear, which is good because lagrangian systems linear in
the time derivatives of some variable have some difficulties.

If the Dirac brackets between the measure fields can be non vanishing, even if
the Poisson brackets are zero, we could obtain a non commutative space time .

###  IX.4 Particles vs antiparticles, Strings vs anti Stings , Branes vs anti
Branes and Space Time vs anti Space Time

We have reviewed already the discussion of particles vs antiparticles due to
Feynman, which consisted of interpreting antiparticles as particles where the
proper time moves in the opposite direction of the coordinate time.

In the case of extended objects, which also can be formulated in terms of
modified measures, as we have mentioned before, and in particular, one can
advance the concepts of strings vs anti strings and that of branes vs anti
branes stringsandantistrings  . The antistrings are identified, in analogy
with the Feynman picture for antiparticles when the sign of the Riemannian
measure is opposite to the metric independent measure, defined in terms of
world sheet scalars, so the same rule should be applied with the
identification of the anti space time in comparison with the space time. To
see this more clearly, in the case of a string, we can use the volume
preserving diffeomorphisms to set one measure field to a spatial world sheet
variable, then the remaining measure field is a time like coordinate that can
go in the direction of the world sheet coordinate or the opposite, which is
the difference between strings and anti strings and similar for branes and
anti branes.

For a $4D$ space time (or we can say a space time filling brane), we use the
volume preserving diffeomorphisms to set a gauge where three measure fields
are set equal to three spatial coordinates, then the remaining measure field
is the analog of the proper time in the particle case, which can run in the
same direction or oposite to the coordinate time, defining a space time or an
anti space time.

##  X Discussion

We have discussed how general coordinate invariance is extended to general
coordinate transformations that have a negative jacobian, one way to achieve
this invariance is in the case the coordinate has a negative jacobian in all
spacetime.

In Linde´s scenario, as long as the jacobian is uniformly negative over all
space, invariance will be achieved if the same transformation is performed
both in the $x$ and $y$ coordinates. No modifications of integration manifolds
are required in this case,

A local realization of signed general coordinate transformation invariance is
possible by introducing a non Riemannian Measure of integration, which
transforms according to the jacobian of the coordinate transformation, not the
absolute value of the jacobian of the coordinate transformation as it is the
case with $\sqrt{-g}$. The theory studied here could provide a way to achieve
similar similar effects to that of Linde´s universe multiplication scenario,
but without non localities.

This analysis may also be applied to give a framework for the Farhi, Guven and
Guth treatment of the Quantum creation of a baby Universe, which just use
Einstein´s equations, as we have also, for a false vacuum bubble and find that
the self consistent solution must have regions where the measure of
integration becomes negative, although GR itself does not provide for a well
defined mechanism to produce negative measures of integration.

This negative measure issue is manifested in a somewhat hidden way in the
analysis of the second reference in FischlerPolchinski , the problem is that
(see discussion around eqn. (59) in this paper) the Euclidean lapse changes
sign within the integration range. Notice that only the square of the lapse
function has meaning in GR and the lapse function itself would only appear in
the square root of the determinant of the determinant of the metric, assigning
an arbitrary sign to the lapse function corresponds to an arbitrary
prescription for the definition of the measure of integration, something we
avoid here by stating that the measure is an independent object independent of
the metric.

The modified measure theory we have described here does this and the sign in
(13) will be dynamically determined, matching the solutions of Farhi, Guven
and Guth. The Farhi, Guven and Guth also require an additional space which is
multivalued with respect to the $x$ space and regions with negative measure to
be able to describe the tunneling solutions and the modified measure and a non
trivial mapping from the measure fields to the $x$ space can serve this
purpose.

The Linde universe multiplication, with one universe associated with a
negative measure is a simpler application of these ideas, since it can be
formulated already at the classical level. Again we would rely on multi
valuedness of the mapping of the coordinate space to the 4 scalars measure
space and of course the possibility of a positive and a negative measure for
space time.

One could also go beyond General Relativity like theories, and generalize the
modified measure theory (4) in the following way,

$S=\int d^{4}x\,\Phi_{1}(A)\Bigl{[}R+L^{(1)}\Bigr{]}+\int
d^{4}x\,\Phi_{2}(B)\Bigl{[}L^{(2)}+\epsilon
R^{2}+\Phi_{2}(B)\frac{\Phi(H)}{(-g)}\Bigr{]}\;.$ (44)

this will be now also a signed general coordinate invariant theory. One can
again explore a representation of modified measures in terms of 4 scalar
fields, etc. This case will be more involved, requiring solving for two
measures, defining an Einstein frame, etc, following for example the work done
in quintess . This will be done in a future publication.

Going back to what we have already done in this paper, in the negative
$\Omega$ measure case, as defined in (9), this represents a different
orientation of the $\varphi_{a}$ space with respect to the $x^{\mu}$ space, it
is reminiscent to the interpretation by Feynman of antiparticles as negative
energy particles traveling backwards in time, which he also described as a
situation where laboratory time and the proper time of the particle run in
different direction, so one could think of $\Omega$ negative as anti space. In
this context the Farhi, Guth and Guven FARHIGUVEN  , which is a creation of a
baby universe that seems to be associated to pair creation of spaces and anti
spaces. It is interesting to note the analogous observations in Linde´s
universe multiplication scenario Lindemultiplication  concerning the particle
and antiparticles.

The non locality in the universe multiplication scenario shows in the
equations of motion in the form of space time averages of field variables over
the complete history of the Universe which appear in the equations of motion
as effective coupling constants. These integrals could be infrared
divergences, depending on the assumed history of the universe
Lindemultiplication . And assuming complete symmetry between the negative
measure universe and the positive measure it is possible to argue for a zero
effective cosmological constant . The modified measures approach in contrast
is local , however if the set of four scalars that define the measure is
introduced, and the mapping of the coordinates to the four scalars is multi
valued, this could give rise then to the appearance of non locality , although
there would be no reason for complete symmetry between positive and negative
measure regions and therefore unlike in Linde´s scenario, for an exact
cancellation of an exactly zero effective cosmological constant.
Alternatively, to get similar effects to the Linde model with our local
theory, one could assume entanglement between positive and negative measure
regions of space,

Finally, we have discussed applications of these concepts to quantum
cosmology: 1) the consideration of one of the measure fields as a time,
instead of the ordinary coordinate time can transform a manifold with
boundaries, which is difficult to handle into a manifold which avoids these
difficulties, when this happens, we also see that the correspondance between
measure fields and standard coordinates is multivalued, just as in the case of
particle antiparticle creation in QED, where there are no boundaries for the
proper time, but the coorniate time has a minimum value, 2) the problem of
time, associated with the coordinate time, which produces vanishing
Hamiltonians is avoided, 3) a possible new avenue to the construction of
quantum space time is proposed, 4) the identification of space times vs anti
space times, that can play an important role in quantum cosmology. The Linde
model appears as a non local coexistence of spaces and antispaces and
manifolds containing spaces and anti spaces can easily avoid boundaries that
would appear in the naive coordinate space as we mentioned before.

###### Acknowledgements.

I want to thank Stefano Ansoldi, Alex Kaganovich , Emil Nissimov and Svetlana
Pacheva for conversations, FQXi for great financial support for work on this
project at BASIC in Ocean Heights, Stella Maris, Long Island, Bahamas, and
CosmoVerse • COST Action CA21136 and very special thanks to CA18108 - Quantum
gravity phenomenology in the multi-messenger approach for financial support
and for invitation to the COST Action: CA18108 - Quantum gravity phenomenology
in the multi-messenger approach Final Annual Conference where this paper was
presented.

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