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\title{Thermostring Quantization. Interpretation of Strings as Particles at Finite
Temperature}
\author{Zahid Zakir) \\
%EndAName
Institute of Noosphere, 167a B.Ipak Yuli, 700187 Tashkent, Uzbekistan
}
\maketitle

\begin{abstract}
Temperature paths of a point particle in the space - temperature manyfold
can be represented as strings with length L=1/kT (thermostrings). During the
time evolution the thermostring swept any surface in the
space-time-temperature manyfold. The thermostring is closed, it's points can
be rearranged and the charge is to be distributed along the length. Some
predictions of this method for statistical mechanics and string theory are
discussed. It is shown that superstrings IIA and heterotic strings can be
interpreted as thermostrings at Planck temperature.
\end{abstract}

\thispagestyle{empty}

\newpage

\section{Introduction}

The well known analogy between quantum mechanics and quantum statistical
mechanics of particles under replacements $\Delta t\rightarrow -i\Delta
\beta ,$ $\Delta \beta \rightarrow i\Delta t,$ commonly used as formal
procedure when only one of $t-$ , or $\beta -$evolutions are considered \cite
{1}. Here $\Delta t-$ is the time interval, $\Delta \beta -$ is the
(inverse) temperature evolution parameter's interval (''cold'') $0\leq
\Delta \beta \leq 1/kT$, $T-$ is the temperature, $k-$ is the Bolzman's
constant. But when we consider $\beta -$evolution of particles in {\it %
combination} with $t-$evolution, this analogy turns into some $(t-\beta )-$ 
{\it symmetry.} Here the inverse temperature parameter $\beta $ can be
treated as a new geometrical degree of freedom in addition to the space and
time. So, as a result, we have some space-time-temperature manyfold with $%
D=d+2$ dimensions, where $d$ is the dimension of space with coordinates $%
{\bf r}$. We shall introduce in this manyfold {\it the} {\it thermostring
representation of density matrix }or{\it \ the} {\it thermostring
quantization}.{\it \ }Reparametrizational symmetry $t\rightarrow t^{\prime
}(t)$ then can be naturally extended into two-dimensional
reparametrizational symmetry $t\rightarrow t^{\prime }(t,\beta ),$ $\beta
\rightarrow \beta ^{\prime }(t,\beta )$ \cite{2}. For nonrelativistic
particles the thermostring quantization leads to some reformulation of the
statistical mechanics. The thermostring quantization of relativistic
particles at the Planck temperature leads to the string formalism which is
identical with theories of closed strings (see also my preceeding paper \cite
{3}). Here the ralativistically invariance of the Planck temperature $T_p$
is very important and this fact allowed us treated the temperature degree of
freedom as one of geometrical dimensions of physical manyfold. The
thermostring quantization makes it possible a new physical interpretation of
superstrings IIA and heterotic strings in terms of point particles.

\section{Thermomechanics of particles}

The density matrix for a nonrelativistic particle at the finite temperature $%
kT=1/\Delta \beta ,$ can be represented as:

\begin{equation}
\begin{array}{c}
\rho ( 
{\bf r},{\bf r}_0;\Delta \beta )=\sum \exp (-E_i\Delta \beta )\cdot \psi _i(%
{\bf r})\cdot \psi _i^{*}({\bf r}_0)= \\  \\ 
=\sum \psi _i({\bf r},\beta )\cdot \psi _i^{*}({\bf r}_0,\beta _0)=\rho (%
{\bf r},\beta ;{\bf r}_0,\beta _0) 
\end{array}
\end{equation}

where $\Delta \beta =\beta -\beta _0,$ $E_i$- is the energy of particle, $%
{\bf r}$ are $d$-dimensional space coordinates. Here wave functions $\psi _i(%
{\bf r},\beta )$ are pure states of the particle with the Hamiltonian $H$ in 
$({\bf r},\beta )$-manyfold:

\begin{equation}
\begin{array}{c}
\psi _i( 
{\bf r},\beta )=\exp (-H\beta )\cdot \psi _i({\bf r}),\psi _i^{*}({\bf r}%
,\beta )=\psi _i^{*}({\bf r})\cdot \exp (H\beta ) \\  \\ 
\int \psi _i^{*}({\bf r},\beta )\cdot \psi _j({\bf r},\beta )\cdot d{\bf r}%
=\delta _{ij} 
\end{array}
\end{equation}

The density matrix $\rho $ in $({\bf r},\beta )-$manyfold is the transition
amplitude for $\beta -$evolution of wave functions $\psi _i({\bf r},\beta )$:

\begin{equation}
\psi ({\bf r},\beta )=\int d{\bf r}_0\cdot \rho ({\bf r},\beta ;{\bf r}%
_0,\beta _0)\cdot \psi _i({\bf r}_0,\beta _0) 
\end{equation}

with property:

\begin{equation}
\rho ({\bf r},\beta ;{\bf r}_0,\beta _0)=\int d{\bf r}_1\cdot \rho ({\bf r}%
,\beta ;{\bf r}_1,\beta _1)\cdot \rho ({\bf r}_1,\beta _1;{\bf r}_0,\beta
_0) 
\end{equation}

and with path integral representation \cite{1}:

\begin{equation}
\rho ({\bf r}_n,\beta _n;{\bf r}_0,\beta _0)={\int }D{\bf r}(\beta )\cdot
\exp (-\frac m2{\int\limits_{\beta _0}^{\beta _n}}d\beta \cdot (\partial 
{\bf r}/\partial \beta )^2) 
\end{equation}

The partition function defined as:

\begin{equation}
Z(\Delta \beta )=\int d{\bf r}\cdot \rho ({\bf r},\beta ;{\bf r}_0,\beta
_0)\mid _{{\bf r}(\beta )={\bf r}(\beta _0)} 
\end{equation}

In this representation of the density matrix:

a) States of particles are described in the space-temperature manyfold $(%
{\bf r},\beta )$, i.e. the temperature degree of freedom is treated as a
geometrical dimension of manyfold as well as space and time. Distances along
this axis are proportional to the Planck constant $\hbar $ and inverse
proportional to the temperature $T$;

b) Mixed states for the statistical ensemble of point particles in $({\bf r}%
,t)-$manyfold are described by the density matrix $\rho ({\bf r},{\bf r}%
_0;\beta )$, which after factorization $\Delta \beta =\beta -\beta _0$ may
be represented as a transition amplitude $\rho ({\bf r},\beta ;{\bf r}%
_0,\beta _0)$ for ''pure'' states $\psi _i({\bf r},\beta )$ in $({\bf r}%
,\beta )$- manyfold;

c) The temperature parameter of evolution $\beta $ is nonlimited $-\infty
\leq \beta \leq \infty ,$ but intervals $\Delta \beta $ are restricted by
condition $0\leq \Delta \beta \leq 1/kT$, so temperature paths of particles
have limited length;

d) Only closed temperature paths with conditions ${\bf r}(\beta _n)={\bf r}%
(\beta _0)$ contribute to the partition function and for their coordinates
we have periodicity conditions ${\bf r}(\beta )={\bf r}(\beta +1/kT)$.

In the Feynman's path integral representation of the density matrix \cite{1}
exists any asymmetry between time and temperature degrees of freedom,
because the temperature evolution is described by the path integral, whereas
the time evolution is described by the differential equations. Further we
shall consider another method of summation of probability of temperature
paths, in which both temperature and time evolutions are described in the
same manner.

\section{Temperature paths as Thermostrings}

The density matrix as the $\beta $-evolution transition amplitude can be
represented through infinity set of intermediate states:

\begin{equation}
\begin{array}{c}
\rho ( 
{\bf r}_n,\beta _n;{\bf r}_0,\beta _0)=\lim _{n\to \infty }\int d{\bf r}%
(\beta _1)...d{\bf r}(\beta _{n-1})\cdot \psi _i({\bf r}_n,\beta _n)\cdot
\psi _i^{*}({\bf r}_{n-1},\beta _{n-1})\psi _i({\bf r}_{n-1},\beta
_{n-1})\cdot ... \\  \\ 
...\cdot \psi _i^{*}({\bf r}_1,\beta _1)\psi _i({\bf r}_1,\beta _1)\cdot
\psi _i^{*}({\bf r}_0,\beta _0) 
\end{array}
\end{equation}

We can transfer all $\psi _i({\bf r}_k,\beta _k)$ to the left hand side and
all $\psi _i^{*}({\bf r}_k,\beta _k)$ to the right hand side and then we can
compose wave functionals $\Psi _i,\Psi _i^{*}$ as product of infinity number 
$(n\rightarrow \infty )$ intermediate state wave functions of particles:

\begin{equation}
\begin{array}{c}
\rho ( 
{\bf r}_n,\beta _n;{\bf r}_0,\beta _0)=\int d{\bf r}(\beta _1)...d{\bf r}%
(\beta _{n-1})\cdot \{\psi _i({\bf r}_n,\beta _n)\cdot \psi _i({\bf r}%
_{n-1},\beta _{n-1})...\psi _i({\bf r}_1,\beta _1)\}\cdot \\  \\ 
\cdot \{\psi _i^{*}( 
{\bf r}_{n-1},\beta _{n-1})...\psi _i^{*}({\bf r}_1,\beta _1)\cdot \psi
_i^{*}({\bf r}_0,\beta _0)\}={\sum }{\int\limits_{{\bf r}(\beta _0)}^{{\bf r}%
(\beta _n)}}D{\bf r}(\beta )\cdot \Psi _i[{\bf r}(\beta );\beta _n,\beta
_0]\cdot \Psi _i^{*}[{\bf r}(\beta );\beta _n,\beta _0] \\  
\end{array}
\end{equation}

Here $D{\bf r}(\beta )$ is the path integration measure, the wave functional 
$\Psi $ describe the states of temperature path:

\begin{equation}
\begin{array}{c}
\Psi _i[ 
{\bf r}(\beta );\beta ,\beta _0]=\prod \psi _i({\bf r}_k,\beta _k) \\ \\ \int D%
{\bf r}(\beta )\cdot \Psi _i^{*}[{\bf r}(\beta );\beta _n,\beta _0]\cdot
\Psi _j[{\bf r}(\beta );\beta _n,\beta _0]=\delta _{ij} 
\end{array}
\end{equation}

We see that the temperature path of the point particle ($\beta -$world line)
can be described as some one dimensional physical object in
space-time-temperature manyfold with the wave functional $\Psi $ and further
we call this object as a {\it thermostring.}

In general case the wave functional $\Psi $ must be symmetrized under
permutations of points of the temperature path (thermostring). This
permutations depended on the type of statistics of particles in Gibbs
ensemble and they determine the type of statistics of thermostring (bosonic
or fermionic). In the ordinary string theory such permutations are
impossible since ordinary strings are treated as continuous one dimensional
objects in the physical space.

\section{The Time Evolution of Thermostrings}

The time evolution of the density matrix can be described by summing of all
surfaces which swept temperature path during it's time evolution. This
circumstance allow us to introduce the thermostring representation of the
quantum statistical mechanics of particles or {\it the thermostring
quantization} of particles at finite temperatures.

Taking into account time evolution formulas for wave functions $\psi _i({\bf %
r},\beta ):$

\begin{equation}
\psi _i({\bf r},\beta ,t)=\exp (-iHt)\cdot \psi _i({\bf r},\beta ), 
\end{equation}

we can obtain the time evolution expression for the wave functional $\Psi .$

In $({\bf r},\beta )-$ manyfold we have $(d+1)$-dimensional coordinates $%
{\bf q}$ with components $({\bf r},\beta ).$ Time derivatives of this
vectors are space-temperature velocities of particles and they are separated
into longitudinal and transverse to the temperature path components:

\begin{equation}
\begin{array}{c}
\partial 
{\bf q}/\partial t={\bf v=v}_{\perp }+{\bf v}_{\parallel } \\  \\ 
{\bf v}_{\perp }={\bf v}-{\bf k}\cdot ({\bf q}^{\prime }\cdot {\bf v}),{\bf q%
}^{\prime }=\partial {\bf q}/\partial \beta ,{\bf k}={\bf q}^{\prime }/{\bf q%
}^{\prime 2} 
\end{array}
\end{equation}

Longitudinal components of velocity ${\bf v}_{\parallel }$ also have two
parts. The first part leads to the collective motion of the thermostring as
a whole object with synchronous displacements of all points of the
thermostring and in case of single thermostring this displacements can be
disregarded (as the zero mode). The second part of the longitudinal velocity
leads to permutations of points of thermostring in case of replacements of
neighboring points. This permutations do not contribute to the energy of
thermostring because of the indistinguishability of the points of
thermostring. In the string theory exclusion of ${\bf v}_{\parallel }$ from
the Lagrangian is one of difficulties of the theory \cite{4}, whereas in
case of thermostrings this is a natural and necessary procedure.

So, we have following time evolution expression for the wave functional:

\begin{equation}
\Psi _i[{\bf q}(\beta ,t),\beta _n,\beta _0;t]=\exp \{-\frac{i(t-t_0)}{%
\Delta \beta }{\int\limits_{\beta _0}^{\beta _n} }d\beta \cdot H({\bf v}%
_{\perp }^2)\}\cdot \Psi _i[{\bf q}(\beta );\beta _n,\beta _0;t_0] 
\end{equation}

This expression can be represented as a surface integral:

\begin{equation}
\Psi _i[{\bf q}(\beta ,t);\beta ,\beta _0;t]=P[{\bf q}(\beta
,t);t_n,t_0;\beta _n,\beta _0]\cdot \Psi _i[{\bf q}(\beta ,t);\beta _n,\beta
_0;t_0] 
\end{equation}

where

\begin{equation}
\begin{array}{c}
P[ 
{\bf q}(\beta ,t);t_n,t_0;\beta _n,\beta _0]=\int D{\bf q}(\beta ,t)\cdot
\exp \left( \frac i{\Delta \beta }{\int }d\beta dtL[{\bf v}_{\perp }(\beta
,t),{\bf q}(\beta ,t)]\right) = \\  \\ 
=\int D{\bf q}(\beta ,t)\cdot \exp (i\cdot S[{\bf q}(\beta ,t);t_n,t_0;\beta
_n,\beta _0)]) 
\end{array}
\end{equation}

Here $P[{\bf q}]$ is the propagator and $S[{\bf q}]$ is the action function
for thermostrings:

\begin{equation}
S[{\bf q}]=\frac 1{\Delta \beta }\int d\beta dtL[{\bf v}_{\perp }(\beta ,t),%
{\bf q}(\beta ,t)]=\frac m{2\Delta \beta }\int d\beta dt\cdot {\bf v}_{\perp
}^2 
\end{equation}

In case of the Gibbs ensemble of free relativistic particles we have
following action function for corresponding relativistic thermostrings:

\begin{equation}
S[x]=-\frac m{\Delta \beta }\int d\beta dt\cdot \sqrt{1-{\bf v}_{\perp }^2} 
\end{equation}

where $(d+2)$-vector $x^\mu $ have components $x^\mu ({\bf q,}t)=x^\mu ({\bf %
r,}\beta ,t{\bf ).}$ It can be shown \cite{4} that after introduction of
world sheet parameters $\tau ,\sigma $ and substitutions:

\begin{equation}
\begin{array}{c}
t=t(\tau ,\sigma ),d\beta =d\sigma 
\sqrt{x^{\prime 2}},\stackrel{.}{x}^\mu =\partial x^\mu /\partial \tau
,x^{\prime }=\partial x/\partial \sigma , \\  \\ 
dtd\sigma = 
\frac{\partial (t,\sigma )}{\partial (\tau ,\sigma )}d\tau d\sigma =%
\stackrel{.}{t}\cdot d\tau d\sigma , \\  \\ 
\partial {\bf q/}\partial t=\stackrel{.}{\bf q}/\stackrel{.}{t},\partial 
{\bf q/}\partial \sigma ={\bf q}^{\prime }-\stackrel{.}{\bf q}\cdot
(t^{\prime }/\stackrel{.}{t}), 
\end{array}
\end{equation}

this expression leads to the Nambu-Goto action for the relativistic
thermostring:

\begin{equation}
S[x]=-\gamma {\int }d\sigma d\tau \cdot \sqrt{(\stackrel{.}{x{\bf \cdot }}%
x^{\prime })^2-\stackrel{.}{x}^2\cdot x^{\prime 2}} 
\end{equation}

Here $\gamma =m/\Delta \beta ,$ $\sigma $ and $\tau $ are world sheet
coordinates of thermostring. We see that in the thermostring representation
of quantum statistical mechanics of particles there exist
reparametrizational symmetries $\sigma ^{\prime }=f(\sigma ,\tau ),$ $\tau
^{\prime }=\varphi (\sigma ,\tau ),$ as `well as in string theories. 

Another form of the action of relativistic particles leads to second form
for the action of thermostrings\cite{2}:

\begin{equation}
S[x,e]=-\frac 1{2\Delta \beta }\int d\beta dt\cdot (e^{-1}{\bf v}_{\perp
}^2-e\cdot m) 
\end{equation}

This action is fully relativistically invariant if $\Delta \beta $ and limits
of $\beta $-integration are invariants. In ordinary temperatures it is
impossible, but if we given the relativistically invariant Planck temperature
$T_p$ as limiting temperature of thermostring with $\Delta \beta =1/kT_p$, we
have invariant action function. In terms of world-sheet coordinates 
$x(\sigma ,\tau )$ and metrics $g(\sigma ,\tau )$ this action leads to the
Polyakov surface integral \cite{5} for the thermostrings time evolution 
amplitude:

\begin{equation}
P[x,g]=\int Dx(\sigma ,\tau )Dg(\sigma ,\tau )\cdot \exp (i\cdot S[x,g]) 
\end{equation}

Finally, we conclude that the statistical mechanics of relativistic particle
at finite temperature in thermostring representation is identical with the
closed string theory formalism.

\section{Statistical Mechanics in Thermostring Representation}

The thermostring representation of statistical mechanics may be useful for
solving of equilibrium and nonequilibrium statistical mechanics problems
because of applicability of rich string theory methods. A new surface
integral representation of Green functions of statistical mechanics and a
geometrical formulation can lead to some nontrivial consequences. One
example is the problem of doubling of operators in the thermofield dynamics,
which developed by H.Umezawa et.al. In the thermostring representation this
doubling is simply explained as consequence of existence of left and right
moved modes of closed thermostrings. The thermal algebra for this doubling
states represented as the Virasoro algebra for thermostring modes.

Another nontrivial prediction of the thermostring quantization can be an
appearance of Liouville modes and tachyons in case of some statistical
systems as additional fields or collective excitations. Liouville modes of
thermostrings in statistical mechanics appeare because of the dimension of
manyfold is less than the critical dimension of string manyfold (10 or 26) 
\cite{5}. Tachyons also can be appeared in thermostring quantization of some
statistical systems because of the absence of fermions or supersymmetry \cite
{2}.

\section{Thermostring interpretation of closed strings}

In the string theory strings are introduced as new fundamental one
dimensional objects in the physical space. Introduction of such nonlocal
structures into physics leads to many conceptual difficulties concerning
measurability of points of strings, intrinsic dynamics, relativistic
causality, etc. Some disconnections exist between the point structure and
the metrics of space-time manyfold and the nonlocal nature of strings, some
modes of which are treated as gravitons. Strings are {\it aliens} to the
geometry of the space-time and the reduction of strings to space-time
properties is a nontrivial problem. The M-Theory is a very successful
attempt in this direction and here strings are treated as any nonlocal
secondary geometrical structures in physical space-time. Here the
dimensionality of the space is increased to one, then some local objects
(particles and fields) which more fundamental than strings are introduced
and strings are constructed by any limiting process.

The thermostring quantization is not introduce new objects into physics, the
dimensionality of the space is lovered to one and the connection with string
theory is exactly without any complicated constructions. This is only a new
method of quantization at finite temperature of ordinary systems - point
particles and local fields, where thermostrings appeared as some nonlocal
mathematical structures in the configurational {\it space-- time- temperature%
} manyfold. The physical reason for nonlocality is the Gibbs ensemble
averaging and the length of thermostring is nothing but the
space-temperature world-line of the point particle. So, in the thermostring
representation we introduce into physics only a new (temperature) dimension
in addition to the space and time and we are representing the physical
manyfold as the space-time-temperature manyfold with more interesting and
nontrivial physics than in the space-time.

The interpretation of superstrings \cite{2} as thermostrings due to the
identity of formalisms can preserve all achievements of the string theory
formalism and in the same time it can exclude some conceptual difficulties
associated with the introduction into physics a nonlocal fundamental objects
with nonmeasurable intrinsic structure.

Thermostring interpretation of superstrings leads to following consequences:

a) Strings is not fundamental physical objects in the physical space, they
are point particles moved in the space with Planck temperature fluctuations
and they are described by the string formalism because of the need for Gibbs
ensemble averaging;

b) One of dimensions of the string theory manyfold is the (inverse)
temperature dimension and therefore the dimension of space is equal 8 (or
24) which together with time and temperature degrees of freedom combined
into critical dimension;

c) Only closed strings can be appeared at initial and final states of the
string theory amplitudes as observable physical states;

d) The charge of particles must be distributed along the length of
thermostring;

e) Interactions of strings with point particles must be excluded from
amplitudes of strings;

f) The Planck temperature $T_p$ as upper limit of temperatures can be
consider at large distances and small temperatures as very large quantity $%
T_p\rightarrow \infty $, and the thermostring with $\Delta \beta \rightarrow
0$ is reduced to the point particle.

Among superstring theories only theories of closed strings satisfy to this
conditions and we may conclude that only superstrings IIA (in case of
neutral particles) and heterotic strings (in case of charged particles) can
be interpreted as thermostrings. This means that the thermostring
interpretation selects only two theories from the family of superstring
theories .

Thermostring interactions we can describe simultaneously in particle,
statistical ensemble and string languiges. The factorization of one
statistical ensemble into two ensembles or the merging of two ensembles into
one is physically clearer and simpler procedure than cutting or gluing of
rigid strings in the string theory.

We can provide procedure of thermostring quantization of physical strings as
one dimensional objects in physical space and we shall obtain as a result a
theory of membranes. So, if we provide the thermostring quantization of
physical D-branes, we shall obtain a theory of (D+1)-branes, i.e. the
dimensionality of initial objects of the theory increase to one. In the
treatment of D-branes the thermostring representation can be combined with
M-Theory methods if we interprete one of it's 11 dimensions as temperature
degree of freedom. But in case of strings the thermostring interpretation
probably is more simple and natural way for understanding of the
modification of local theories at Planck distances and temperatures.

\begin{thebibliography}{9}
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Integrals.} McGRAW-HILL.

\bibitem{2}  Green M.B.,Schwarz J.H.,Witten E. (1987) {\it Superstring
Theory, }Cambridge U.Press.

\bibitem{3}  Israilov Z.Zakir, {\it On the Statistical Interpretation of
String Theories.} In Proc.Int.Conf.''Quantum Physics and The Universe'',
Tokyo, Aug.19-22, 1992; Vistas in Astronomy (1993) v.37 (1-4), p.277.

\bibitem{4}  Barbashov B.M., Nesterenko V.V. (1987) {\it Model of
Relativistic String in Hadron Physics} (in Russian), Nauka, M.

\bibitem{5}  Polyakov A.M. (1981) Phys.Lett. v.103B, p.207,211

\end{thebibliography}

\end{document}
