Strings versus Anti Strings in the inversion invariant or proper volume
formulation

E.I. Guendelman

Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva, Israel

Frankfurt Institute for Advanced Studies, Giersch Science Center, Campus
Riedberg, Frankfurt am Main, Germany

Bahamas Advanced Studies Institute and Conferences, 4A Ocean Heights, Hill
View Circle, Stella Maris, Long Island, The Bahamas

E-mail<EMAIL_ADDRESS>

###### Abstract

The specific model studied is in the context of the modified measure
formulation the string or branes where tension appear as an additional
dynamical degree of freedom . We then consider the signed reparametrization
invariant volume element formulation of dynamical strings and branes and find
that the dynamical tension can produce positive tensions or negative tensions,
corresponding exactly to strings and branes and anti strings and anti branes
respectively. The anti strings are realized when a scalar time that defines
the modified measure runs in the opposite direction to the world sheet time.
For strings with positive tension, both times run in the same direction. The
situation resembles the situation in Relativistic Quantum Mechanics with
positive and negative energies, proper time of particles running forward with
respect of coordinate time , while for anti particles proper tome runs
opposite of coordinate time. An example where string anti string pair creation
takes place in analogy to the pair creation in an external electric field in
QED background field, this time in the presence of a background scalar field
that couples to the strings and locally changes the tension, the tension
field.

##  I Introduction

String Theories have been considered by many physicists for some time as the
leading candidate for the theory everything, including gravity, the
explanation of all the known particles that we know and all of their known
interactions (and probably more) stringtheory . According to some, one
unpleasant feature of string theory as usually formulated is that it has a
dimension full parameter, in fact, its fundamental parameter , which is the
tension of the string. This is when formulated the most familiar way. The
consideration of the string tension as a dynamical variable, using the
modified measures formalism, which was previously used for a certain class of
modified gravity theories under the name of Two Measures Theories or Non
Riemannian Measures Theories, see for example d ; b ; Hehl ; GKatz ; DE ;
MODDM ; Cordero ; Hidden . It is also interesting to mention that the modified
measure approach has also been used to construct braneworld scenarios
modified_measures_branes  and that Modified Measure Theories could be the
effective Theories of Causal fermion systems MMandCFS  Most recently, we have
shown that modified measure gravity theories can be be formulated using an
invariant volume element under Signed General Coordinate Invariance, that is
for general coordinate transformations with negative jacobian signedGR , which
could provide for a framework for the quantum creation of a baby universe as
formulated by Farhi, Guth and Guven FARHIGUVENGUTH  which requires negatives
measures for space time and for Linde´s Universe multiplication scenario
Lindemultiplication , which also involves negative measures for space time in
a parallel universe, which requires the measure of integration to become
negative at certain regions of the tunneling solution.

When applying these principles to string theory, this leads to the modified
measure approach to string theory, where rather than to put the string tension
by hand it appears dynamically.

This approach has been studied in various previous works a ; c ; supermod ;
cnish ; T1 ; T2 ; T3 ; cosmologyandwarped . See also the treatment by Townsend
and collaborators for dynamical string tension xx ; xxx , cosmologyandwarped ,
which does not involve changing integration measures, so it cannot be used to
achieve the goals presented in this paper however.

Here we will see that this modified volume element can play a similar role to
the proper time, which for antiparticles runs in the opposite direction to
laboratory time and how introducing appropriate background fields, string anti
string pair creation could be possible.

##  II The Modified Measure String Theory

The standard world sheet string sigma-model action using a world sheet metric
is pol1 , pol2 , pol3 ,

$S_{sigma-model}=-T\int
d^{2}\sigma\frac{1}{2}\sqrt{-\gamma}\gamma^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}g_{\mu\nu}.$
(1)

Here $\gamma^{ab}$ is the intrinsic Riemannian metric on the 2-dimensional
string worldsheet and $\gamma=det(\gamma_{ab})$; $g_{\mu\nu}$ denotes the
Riemannian metric on the embedding spacetime. $T$ is a string tension, a
dimension full scale introduced into the theory by hand.

Now instead of using the measure $\sqrt{-\gamma}$ , on the 2-dimensional
world-sheet, in the framework of this theory two additional world sheet scalar
fields $\varphi^{i}(i=1,2)$ are considered. A new measure density is
introduced:

$\Phi(\varphi)=\frac{1}{2}\epsilon_{ij}\epsilon^{ab}\partial_{a}\varphi^{i}\partial_{b}\varphi^{j}.$
(2)

There are no limitations on employing any other measure of integration
different than $\sqrt{-\gamma}$. The only restriction is that it must be a
density under arbitrary diffeomorphisms (reparametrizations) on the underlying
spacetime manifold. The modified-measure theory is an example of such a
theory.

Then the modified bosonic string action is (as formulated first in a  and
latter discussed and generalized also in c

$S=-\int
d^{2}\sigma\Phi(\varphi)(\frac{1}{2}\gamma^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}g_{\mu\nu}-\frac{\epsilon^{ab}}{2\sqrt{-\gamma}}F_{ab}(A)),$
(3)

where $F_{ab}$ is the field-strength of an auxiliary Abelian gauge field
$A_{a}$: $F_{ab}=\partial_{a}A_{b}-\partial_{b}A_{a}$.

It is important to notice that the action (LABEL:eq:5) is invariant under
conformal transformations of the internal metric combined with a
diffeomorphism of the measure fields,

$\gamma_{ab}\rightarrow j\gamma_{ab},$ (4)

$\varphi^{i}\rightarrow\varphi^{{}^{\prime}i}=\varphi^{{}^{\prime}i}(\varphi^{i})$
(5)

such that

$\Phi\rightarrow\Phi^{{}^{\prime}}=j\Phi$ (6)

Here $j$ is the jacobian of the diffeomorphim in the internal measure fields
which can be an arbitrary function of the world sheet space time coordinates,
so this can called indeed a local conformal symmetry.

To check that the new action is consistent with the sigma-model one, let us
derive the equations of motion of the action (3).

The variation with respect to $\varphi^{i}$ leads to the following equations
of motion:

$\epsilon^{ab}\partial_{b}\varphi^{i}\partial_{a}(\gamma^{cd}\partial_{c}X^{\mu}\partial_{d}X^{\nu}g_{\mu\nu}-\frac{\epsilon^{cd}}{\sqrt{-\gamma}}F_{cd})=0.$
(7)

since $det(\epsilon^{ab}\partial_{b}\varphi^{i})=\Phi$, assuming a non
degenerate case ($\Phi\neq 0$), we obtain,

$\gamma^{cd}\partial_{c}X^{\mu}\partial_{d}X^{\nu}g_{\mu\nu}-\frac{\epsilon^{cd}}{\sqrt{-\gamma}}F_{cd}=M=const.$
(8)

The equations of motion with respect to $\gamma^{ab}$ are

$T_{ab}=\partial_{a}X^{\mu}\partial_{b}X^{\nu}g_{\mu\nu}-\frac{1}{2}\gamma_{ab}\frac{\epsilon^{cd}}{\sqrt{-\gamma}}F_{cd}=0.$
(9)

One can see that these equations are the same as in the sigma-model
formulation . Taking the trace of (9) we get that $M=0$. By solving
$\frac{\epsilon^{cd}}{\sqrt{-\gamma}}F_{cd}$ from (8) (with $M=0$) we obtain
the standard string eqs.

The emergence of the string tension is obtained by varying the action with
respect to $A_{a}$:

$\epsilon^{ab}\partial_{b}(\frac{\Phi(\varphi)}{\sqrt{-\gamma}})=0.$ (10)

Then by integrating and comparing it with the standard action it is seen that

$\frac{\Phi(\varphi)}{\sqrt{-\gamma}}=T.$ (11)

That is how the string tension $T$ is derived as a world sheet constant of
integration opposite to the standard equation (1) where the tension is put ad
hoc. Let us stress that the modified measure string theory action does not
have any ad hoc fundamental scale parameters. associated with it. This can be
generalized to incorporate super symmetry, see for example c , cnish ,
supermod  , T1 . For other mechanisms for dynamical string tension generation
from added string world sheet fields, see for example xx  and xxx . However
the fact that this string tension generation is a world sheet effect and not a
universal uniform string tension generation effect for all strings has not
been sufficiently emphasized before.

Notice that Each String in its own world sheet determines its own tension.
Therefore the tension is not universal for all strings.

##  III Elementary review of Particles and Antiparticles and Pair Creation in
QED

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

$m\frac{d^{2}z^{\mu}}{d\tau^{2}}=e\frac{dz_{\nu}}{d\tau}F^{\mu\nu}$ (12)

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

$m\frac{d^{2}z^{\mu}}{d\tau^{2}}=-e\frac{dz_{\nu}}{d\tau}F^{\mu\nu}$ (13)

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.

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.

###  III.1  Pair Creation in a Strong Uniform Electric Field in QED,
importance of different directions of time and proper time for identifying
particles and antiparticles

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

$\displaystyle F^{\mu}_{\ \ \nu}=\left[\begin{array}[]{llll}0&E&0&0\\\
E&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{array}\right]$ (18)

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

$m\frac{d^{2}x^{\mu}}{d\tau^{2}}=eF^{\mu}_{\ \ \nu}\frac{dx^{\nu}}{d\tau}$
(19)

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

$\displaystyle\exp[\frac{e}{m}F^{\alpha}_{\ \
\beta}\tau]^{\mu}_{\nu}=\delta^{\mu}_{\nu}+\frac{e}{m}\tau
E\Delta^{\mu}_{\nu}+\frac{1}{2}(\frac{e}{m}\tau
E\Delta^{\mu}_{\nu})^{2}+\cdots$ (20)

where

$\displaystyle\Delta=\left[\begin{array}[]{llll}0&1&0&0\\\ 1&0&0&0\\\
0&0&0&0\\\ 0&0&0&0\end{array}\right]$ (25)

Separating even and odd power in Eq. 20 we have

$\displaystyle u^{0}$ $\displaystyle=$
$\displaystyle\cosh\left(\frac{eE\tau}{m}\right)u^{0}(0)+\sinh\left(\frac{eE\tau}{m}\right)u^{1}(0)$
$\displaystyle u^{1}$ $\displaystyle=$
$\displaystyle\sinh\left(\frac{eE\tau}{m}\right)u^{0}(0)+\cosh\left(\frac{eE\tau}{m}\right)u^{1}(0)$
(26)

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

$\displaystyle x^{0}$ $\displaystyle=$
$\displaystyle\frac{m}{eE}\left\\{\sinh\left(\frac{eE\tau}{m}\right)u^{0}(0)+\cosh\left(\frac{eE\tau}{m}\right)u^{1}(0)\right\\}$
$\displaystyle x^{1}$ $\displaystyle=$
$\displaystyle\frac{m}{eE}\left\\{\cosh\left(\frac{eE\tau}{m}\right)u^{0}(0)+\sinh\left(\frac{eE\tau}{m}\right)u^{1}(0)\right\\}$
(27)

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

$\displaystyle x^{0}$ $\displaystyle=$
$\displaystyle\frac{m}{eE}\sinh\left(\frac{eE\tau}{m}\right)$ $\displaystyle
x^{1}$ $\displaystyle=$
$\displaystyle\frac{m}{eE}\cosh\left(\frac{eE\tau}{m}\right)$ (28)

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

$\displaystyle x^{0}$ $\displaystyle=$
$\displaystyle-\frac{m}{eE}\sinh\left(\frac{eE\tau}{m}\right)$ $\displaystyle
x^{1}$ $\displaystyle=$
$\displaystyle-\frac{m}{eE}\cosh\left(\frac{eE\tau}{m}\right)$ (29)

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

$(x^{1})^{2}-(x^{0})^{2}=\left(\frac{m}{eE}\right)^{2}$ (30)

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. 30 becomes a circle

$(x^{1})^{2}+t_{E}^{2}=\left(\frac{m}{eE}\right)^{2}$ (31)

This tunneling solution can now interpolate between the anti-particle and
particle solutions. (see Fig. 2). In the imaginary time region, the action,
$S=-iS_{E}$ where $S_{E}$ is given by

$S_{E}=\int
dt_{E}\left\\{m\sqrt{1+\left(\frac{dx}{dt_{E}}\right)^{2}}-eEx\right\\}$ (32)

which is the analytic continuation of Eq. LABEL:action and Eq.
LABEL:actionE&M. 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 $\Delta
x=\frac{m}{eE}$ which has a physical interpretation is manifest in writing it
as

$W=eE\Delta x=2mc^{2}$ (33)

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.

##  IV Extension to signed reparametrization invariant Modified Measure volume
element as the proper volume of Strings

In signedGR , we discussed the generalization of general coordinate
transformations to signed general coordinate transformations and found the
modified volume which is invariant under signed general coordinate
transformations.

For the case of strings now notice that under a signed general
reparametrization transformation of the world sheet, that could for example
change the direction of time,

$d^{2}\sigma\rightarrow Jd^{2}\sigma$

, while

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

where $J$ is the jacobian of the transformation and $\mid J\mid$ is the
absolute value of the jacobian of the transformation. Therefore

$d^{2}\sigma{\sqrt{-\gamma}}\rightarrow\frac{J}{\mid
J\mid}d^{2}\sigma{\sqrt{-\gamma}}$

,

so invariance of the volume element is achieved only for $J=\mid J\mid$, that
is if $J>0$, that is signed general coordinate transformations are excluded if
we want to define signed invariant reparametrization volume element.

Notice that we are not discussing invariance of the action here, which is a
different matter, since under a signed reparametrization transformation the
time direction may change for example. Then the limits of integration change
and when restoring back the limits of integration we obtain anothe minus sign
and the action restores its invariance.

However what we want to know is, if we now can have a measure that does not
change sign, even if we change for example the direction of the coordinate
time, which will be the corresponding generalization of the proper time, which
continues to increase as the particle goes backwards in time and now in the
case of the strings, the modified volume can be monotonically increasing even
as the strings go backwards in time, Furthermore, this will imply that there
are effects that cannot be described in the standard formalism, like string
anti string pair creation, which would require the generalization of the
proper time, which continues to increase as the string goes backwards in time
.

Indeed , replacing measure $\sqrt{-\gamma}$ by the modified measure $\Phi$
provides us with such measure since $\Phi$ transforms according to $J^{-1}$
instead of $\mid J\mid^{-1}$. So we can have now a positive defined measure
for strings propagating backwards in the string sheet time in exact analogy to
the QED case.

Notice that

$\Phi(\varphi)d^{2}\sigma=d\varphi_{1}d\varphi_{2}$

So the integration domain in the $\varphi_{1}\varphi_{2}$ space is invariant
under these signed reparametrizations, since $\varphi_{1}\varphi_{2}$ are both
scalars and do not transform under inversions . So we can take the time of the
string, say $\sigma_{1}$ to go in the opposite direction to that of the
coordinate time, thus defining an anti string, the same way as the
antiparticle was defined when the proper time and the coordinate time ran in
different directions. $\varphi_{2}$ could be taken as the spacial coordinate
of the string, periodic for a closed sting for example, and will not play a
role in the discussion on the difference between strings and anti strings.
Generically, the consideration of the $\varphi_{1}\varphi_{2}$ space as
opposed to the $\sigma_{1}\sigma_{2}$ space enlarges the space of possible
configurations and solutions and allows for example the consideration of pair
creation or pair annihilation , which requires non trial mappings between the
$\sigma_{1}\sigma_{2}$ space and the $\sigma_{1}\sigma_{2}$ spaces .
Furthermore if we want the full action density to be invariant under signed
reparametrizations, in (LABEL:Action_Mod_Measure_String1), we have to correct
the term

$\Phi(\varphi)\frac{\epsilon^{ab}}{2\sqrt{-\gamma}}F_{ab}(A)$

, since $\Phi(\varphi)$ and $\sqrt{-\gamma}$ differ by a sign under signed
reparametrization transformation, To cure this, we replace this by

$\Phi(\varphi)^{2}\frac{\epsilon^{ab}}{2(-\gamma)}F_{ab}(A)$

,

obtaining the action invariant under signed reparametrization invariant, Where
both the domain of integration and the lagrangian density are invariant under
signed reparametrizations transformations.

$S=-\int
d^{2}\sigma\Phi(\varphi)(\frac{1}{2}\gamma^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}g_{\mu\nu}-\Phi(\varphi)\frac{\epsilon^{ab}}{2(-\gamma)}F_{ab}(A)),$
(34)

now the equation of motion that comes from the variation of the internal gauge
field is ,

$\epsilon^{ab}\partial_{b}(\frac{\Phi^{2}(\varphi)}{(-\gamma))})=0.$ (35)

which is integrated to,

$\frac{\Phi^{2}(\varphi)}{(-\gamma)}=T^{2}$ (36)

where without loosing generality we can take $T$ positive. The above equation
implies that positive tensions are accompanied with negative ones,

$\frac{\Phi(\varphi)}{\sqrt{-\gamma}}=\pm T.$ (37)

the negative tension strings we associate with the ¨anti-strings¨. We can see
that the results obtained from the signed reparametrization invariant action
are (34) to those obtained in the previous section, except for the possibility
that there are positive and negative tensions. To start with, the equation
that replaces (7) is

$\epsilon^{ab}\partial_{b}\varphi^{i}\partial_{a}(-\gamma^{cd}\partial_{c}X^{\mu}\partial_{d}X^{\nu}g_{\mu\nu}+2\Phi\frac{\epsilon^{cd}}{(-\gamma)}F_{cd})=0.$
(38)

which again can be integrated, assuming non singular measure $\Phi$ ,

$-\gamma^{cd}\partial_{c}X^{\mu}\partial_{d}X^{\nu}g_{\mu\nu}+\frac{\Phi\epsilon^{cd}}{(-\gamma)}F_{cd}=M.$
(39)

The variation with respect to $\gamma^{ab}$ gives now, for $\Phi\neq 0$ as

$T_{ab}=\partial_{a}X^{\mu}\partial_{b}X^{\nu}g_{\mu\nu}-\Phi\gamma_{ab}\frac{\epsilon^{cd}}{(-\gamma)}F_{cd}=0.$
(40)

Taking the trace of (39) and comparing with ( 40) we obtain $M=0$ then solving
$\frac{\Phi\epsilon^{cd}}{(-\gamma)}F_{cd}$, from 39 now for $M=0$ and
inserting in (40) we recover the standard Polyakov equation for a string, but
now we have two species of strings, with positive and negative tensions.

Notice however that from a ¨practical¨ point of view, it is not necessary to
work with an action density that is completely invariant under the signed
reparametrizations, in particular, we can still work with the original action
3, which avoids quadratic equations for the tension, but still allows for
positive and negative tensions, just as in the modified version that has
complete signed reparametrization invariance and is simple to manipulate,
since everything becomes linear. We now consider possible ways to scatter
strings from an external field so that the tension may change, still in the
most simple formalism.

## Introducing a New Background Field, The Tension Field

Schwinger Julian Schwinger  had an important insight and understood that all
the information concerning a field theory can be studied by understanding how
it reacts to sources of different types. Different types of background bulk
fields, like the dilaton field, the two index antisymmetric gauge field have
been discussed in the text book by Polchinski for example Polchinski  . Here ,
instead of the traditional background fields usually considered in
conventional string theory, one may consider another scalar field that induces
currents in the string world sheet and since the current couples to the world
sheet gauge fields, this produces a dynamical tension controlled by the
external scalar field as shown at the classical level in Ansoldi . In the next
two subsections we will study how this comes about in two steps, first we
introduce world sheet currents that couple to the internal gauge fields in
Strings and Branes and second we consider a coupling to an external scalar
field by defining a world sheet current that is induced by such external
scalar field.and then coupling this current to the internal gauge fields of
the Strings . After this is done, we argue that such external field could lead
to string anti string pair creation, in analogy to the QED process of electron
anti electron pair production. So the Tension Field probes exactly the string
- anti string structure of the modified measure string theory.

### Introducing world sheet currents that couple to the internal gauge fields
and locally change the tension

If to the action of the string we add a coupling to a world-sheet current
$j^{a}$, i.e. a term

$S_{\mathrm{current}}=\int d^{2}\sigma A_{a}j^{a},$ (41)

then the variation of the total action with respect to $A_{a}$ gives

$\epsilon^{ab}\partial_{a}\left(\frac{\Phi}{\sqrt{-\gamma}}\right)=j^{b}.$
(42)

We thus see indeed that, in this case, the dynamical character of the brane is
crucial here.

### How a world sheet current can naturally be induced by a bulk scalar field,
the Tension Field

Suppose that we have an external scalar field $\phi(x^{\mu})$ defined in the
bulk. From this field we can define the induced conserved world-sheet current

$j^{b}=e\partial_{\mu}\phi\frac{\partial
X^{\mu}}{\partial\sigma^{a}}\epsilon^{ab}\equiv
e\partial_{a}\phi\epsilon^{ab},$ (43)

where $e$ is some coupling constant. The interaction of this current with the
world sheet gauge field is also invariant under local gauge transformations in
the world sheet of the gauge fields $A_{a}\rightarrow
A_{a}+\partial_{a}\lambda$.

For this case, (42) can be integrated to obtain

$T=\frac{\Phi}{\sqrt{-\gamma}}=e\phi+T_{0},$ (44)

The constant of integration $T_{0}$ could represent the initial value of the
tension before the string enters the region where the tension field is
present. Notice that the interaction is metric independent since the internal
gauge field does not transform under the the conformal transformations. This
interaction does not therefore spoil the world sheet conformal transformation
invariance in the case the field $\phi$ does not transform under this
transformation. Therefore, one way to dynamically change the string tension
tension dynamically is through the introduction this new background field,
that changes the value of the the tension locally in the world sheet Ansoldi
. This background field was then called the ¨tension field¨ and used for many
different scenarios cosmologyandwarped , Escaping , the results of which have
been summarized in summary  and most recently for the construction of
braneworld scenarios in dynamical string tension theories Life  and
Lidhtlikeandbraneworld , and as we will discuss in the next section, this
tension field could provide a way to communicate between strings and anti
strings.

##  V Scattering of a dynamical tension string field leading to string anti
string pair creation in analogy with the analog QED process

The QED example explained before in this paper inspires an analogous model for
a possible scenario for a string-anti string creation process. Here instead of
the the external electric field we can have the tension field.

We specify the initial state according to the $\varphi_{1}$ time as the anti
string, where the time $\varphi_{1}$ and the world sheet coordinate time
$\sigma_{1}$ (running in the same direction as the coordinate time $t$) run in
opposite directions, so the tension is therefore negative. Assume $T_{0}$ that
could represent the initial value of the tension before the string enters the
region where the tension field is present is negative,

Then as the $\varphi_{1}$ time continues to increase, and the world sheet time
continues to decrease, we encounter a region where the tension field $\phi$ is
present , and over this region the tension field suffers a a positive jump
$e\Delta\phi$ which can be bigger than $-T_{0}$, turning the tension from
negative to positive. This is the analog of the condition for pair creation in
the QED case, that is that the electric field is extended id a region of space
with minimum extension given by $W=eE\Delta x=2mc^{2}$. This again, as in the
QED case, cannot happen classically, there should be a tunneling region that
connects the anti string solution going backwards in coordinate time with the
string solution going forward in coordinate time. Unfortunately we cannot yet
present full details as compared with the QED case, because on the much higher
complexity of the string dynamics as compared with the particle dynamics of
the QED problem, but we plan to present more details in a future publication.

In the case we were to use the full signed reparametrization formulation the
equation 44 has to be modified and if the coupling to the tension field is the
same, now becomes,

$T^{2}=\frac{\Phi^{2}}{(-\gamma)}=e\phi+T_{0},$ (45)

As we see, there is a difference with the simpler case studied before, since
the above equation involves the square of the dynamical tension, so there must
be a boundary for the evolution of the tension field giver by $e\phi+T_{0}=0$
and after this point the tension field must bounce in the opposite direction,
and if particle antiparticle is desired, when bouncing in the opposite
direction the opposite sign of the tension must be taken. If the motion of the
tension field is symmetric around the bounce, the string and anti string will
have exactly opposite tensions, something that is not necessary in the
simplified model which is not signed reparametrization invariant.

##  VI Discussion on Signed reparametrization invariant volume element for
Brane Theories. Analogy with Relativistic QM positive and negative energies.

The $d$ dimensional extended object ($d=1$ corresponds to the string case) has
an action that provides for a dynamically generated brane tension

$S=-\int
d^{d+1}\sigma\Phi(\varphi)(\frac{1}{2}\gamma^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}g_{\mu\nu}-\frac{\epsilon^{a_{1}a_{2}.....a_{d+1}}}{2\sqrt{-\gamma}}F_{a_{1}a_{2}.....a_{d+1}}(A)),$
(46)

where $F_{a_{1}a_{2}.....a_{d+1}}(A)$ is a field stregth deriving from the
totally anti symmetric derivative of a totally antisymmetric potential with
$d$ indices. Also the measure is constructed from the derivatives of $d+1$
scalars. The variation with respect to those totally antisymmetric potential
with $d$ indices gives the equation that the derivative of
$\frac{\Phi(\varphi)}{\sqrt{-\gamma}}$ is a constant $T$, which we identify
with the tension. (46) is not however invariant under signed reparametrization
transformations, in order to do this $\sqrt{-\gamma}$ sould not appear, only
$-\gamma$, so the c signed invariant generalization would be,

$S=-\int
d^{d+1}\sigma\Phi(\varphi)(\frac{1}{2}\gamma^{ab}\partial_{a}X^{\mu}\partial_{b}X^{\nu}g_{\mu\nu}-\frac{\Phi(\varphi)\epsilon^{a_{1}a_{2}.....a_{d+1}}}{2(-\gamma)}F_{a_{1}a_{2}.....a_{d+1}}(A)),$
(47)

from the variation of the internal gauge fields and integration of the
resulting equation, we get now $\frac{\Phi^{2}(\varphi)}{(-\gamma)}=T^{2}$,
just as in the string case, which again leads to positive and negative
tensions, i.e. $\frac{\Phi(\varphi)}{\sqrt{-\gamma}}=\pm T$. As in the
modified measure string, it is not strictly necessary to work with 47, 46 also
works.

As in relativistic quantum mechanics RLQM  one may attempt to eliminate
negative solutions, which in the case of relativistic quantum mechanics are
negative energy solutions, that attempt, known as the Relativistic
Schroedinger equation fails (RLQM ) and such equation cannot even describe the
scattering by a square well potential. As pointed out by Feynman, this is
related to the fact that the space of positive energy solutions is not large
enough to represent an arbitrary initial condition. The correct interpretation
of the Klein paradox RLQM  must involve particles and antiparticles, the
alternative of eliminating the negative energy solutions does not work.

We introduced a new background field,the tension field, which is specially
designed to probe the string anti string dynamics of the modified measure
string theories. The process of string anti string creation can be studied
qualitatively for a simpler formulation that is not totally invariant under
signed reparametrizations. in this case the resulting pair of string anti and
string may not have exactly opposite tensions. In the signed reparametrization
invariant formulation, we find that zero tension squared is a special point
from which it must bounce back and transition from negative to positive values
is possible, in most likelihood this bounce could be symmetric, leading to
pair creation with exactly opposite tensions for the string and the anti
string.

Concerning simultaneous use of positive and negative brane tensions, we can
point out that this has been used extensively in brane world models, see for
example RS . The interpretation of the negative tension branes as antibranes
could be useful.

One interesting aspect of the dynamical strings and brane, in that due to the
influence of the tension field a part of the string or brane could be anti
string or anti brane while the rest could be string or brane, i.e., have
positive tension. One may have an anti brane bubble in the midst of a brane
environment, since the transition for brane to anti brane or the reverse due
to the tension field is local.

Finally, the tension field that locally changes the value of the string
tension has been introduced as a background field, so far just an external
field. To have a complete picture, more work should be done to determine it
dynamically Some ideas in this directions were discussed for a system
containing two types of strings with different integration constants $T_{1}$
and $T_{2}$ and imposing separately conformal invariance on the two string
types separately, which lead to an algebraic equation for the tension field,
that indicates the formation of a braneworld at large times Life ,
Lidhtlikeandbraneworld . An interesting possibility could be if those two
types of strings could be strings and anti strings generated by a pair
creation process like the one that we have discussed here, this may complete
the picture described in those paper, including the quantum formation of such
braneworlds.

A different approach to dynamical string tension and the possibility of
negative string tension has been discussed in Paul , the relation of the
approach of this paper is not straightforward, but could exist at the level of
potentially similar effects.

##  VII Acknowledgments

I want to thank Stefano Ansoldi Emil Nissimov, Svetlana Pacheva and Paul
Steinhardt for conversations on dynamical string tensions, the Foundational
Questions Institute (FQXi) and the COST actions Quantum Gravity Phenomenology
in the multi messenger approach, CA18108 , COST Gravitational waves, Black
Holes and Fundamental Physics, CA16104 and COST COSMOVERSE, Action CA21136 for
financial support.

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