1 String Representation of Quantum Loops

Non perturbative effects in quantum field theory are difficult to study because of the lack of appropriate mathematical tools. Sometimes a special invariance of the system, like electric/magnetic duality or super-symmetry, allows to open a window over the strong coupling regime of the theory, where a new kind of solitonic excitations can appear. These new states describe extended field configurations, which are ``dual'' to the point-like, elementary, objects of the perturbative regime. Remarkable examples of extended structures of this sort ranges from the Dirac magnetic monopoles [1] up to the loop states of quantum gravity [2]. The dynamics of one dimensional extended field configurations is formulated in terms of an ``effective theory'' of relativistic strings, i.e. one trades a quantum field theory for a different framework, by taking for granted that there is some well defined relation connecting the two different descriptions. Such a kind of relations have recently shown up in super string theory, where a web of dualities relates different phases of different super string models. On the other hand, there is still no explicit way to connect point like and stringy phases of non-supersymmetric field theories like QCD and quantum gravity.

The main purpose of this letter is to provide a quite general representation of the wave functional of a quantum loop in terms of string degrees of freedom. The loop wave functional can describe the quantum state of a one dimensional, closed, excitation of any quantum field theory, while the corresponding string functional describes the boundary quantum fluctuations of an open world-sheet. We shall find a quite general relation linking these two different objects. The matching between a quantum loop and a quantum string can be formally obtained through a (functional) Fourier transform. In this letter we shall explicitly compute this Fourier transform and represent the loop wave functional in terms of the Bulk and Boundary wave functional of a quantum string.
As a byproduct of this new Loops/Strings connection, we shall obtain that, contrary to the current wisdom, the Polyakov path integral, [3], does not encode the whole information about string quantum dynamics. In agreement with the recently conjectured Holographic Principle, [4] we shall obtain that the whole information appears to be stored into the state of the boundary of the string world-sheet rather than into the bulk. Our approach encodes the Holographic Principle and provides a clear factorization of the bulk and boundary quantum dynamics. The bulk effects being included into the Polyakov path integral while the boundary effects are taken into account by the Eguchi string functional[5].

The quantum state of a closed, bosonic string is described by a (complex) functional $\Psi[ \overline y ]$ defined over the space of all possible configurations, i.e. shapes, of the boundary $\overline y$ of an open world-sheet $Y$. Accordingly, $\Psi[ \overline y ]$ can be written as a phase space path integral of the form

$$\Psi[ \overline y ]= \int_{\partial Y= \overline y} [ DY^... ..._\Sigma d^2\sigma\sqrt g L[ Y, P, g_{ab} ;\sigma] \right\}$$ (1)

where, $Y^\mu(\sigma)$ are the string coordinates mapping the string manifold $\Sigma$ in a world surface in target space; $\sigma^m=\{ \sigma^0 , \sigma^1 \}$ are internal coordinates, and $g_{m n}(\sigma)$ is the metric over the string manifold. We assume that $\Sigma$ is simply connected and bounded by a curve $\partial\Sigma$ parametrically represented by $\sigma^m=\sigma^m(s)$. Then, $Y^\mu(\sigma)$ maps $\partial\Sigma$ in a contour $\partial Y\equiv \overline y$ in target space

$$Y^\mu\left(\sigma^0(s) , \sigma^1(s)\right)=\overline y^\mu(s) .$$ (2)

$P_{m \mu}(\sigma)$ is the canonical momentum

$$P^m{}_\mu(\sigma)\equiv {\partial L\over \partial \partial_m Y^\mu} .$$ (3)

Finally, we sum in the path integral (1) over all the string world sheets having the closed curve $x^\mu=\overline y^\mu(s)$ as the only boundary.
A closed line and an open two-surface can be given different geometrical characterizations. For our purposes, the best way to introduce them is through the respective associated currents. Let us define the Bulk Current $J^{\mu\nu}(x ; Y)$, as a rank two antisymmetric tensor distribution having non vanishing support over a two dimensional world surface, and the Loop Current $J^\mu(x ; C)$ as the vector distribution with support over the closed line $x^\mu= x^\mu(s)$:

$$J^{\mu\nu}\left( x ; Y \right)\equiv \int_Y dY^\mu\wedge dY^\nu \delta^D\left[ x-Y \right]$$ (4)

$$J^\mu\left( x ; C \right)\equiv \oint_C dx^\mu \delta^D\left[ x-x(s) \right] ,$$ (5)

$$J^\mu\left( x ; \overline y \right)\equiv \partial_\lambda J^{\lambda\mu}\left( x ; Y \right)$$ (6)

The divergence of the bulk current define the Boundary current (6). A simple way to link a closed line to a surface is by ``appending'' the surface to the assigned loop [6]. This matching condition can be formally written by identifying the loop current with the boundary current :

$$J^\mu\left( x ;C \right)=J^\mu\left( x ; \overline... ...\right)= \partial_\nu J^{\nu\mu}\left( x ; Y \right) .$$ (7)

It is worth to remark that the equation (7) defines $J^\mu( x ; C )$ as the current associated to the boundary of the surface $Y$. In the absence of (7) $J^\mu( x ; C )$ is a loop current with no reference to any surface. In other words, equation (7) is a formal description of the `` gluing operation  '' between the surface $Y$ and the closed line C. Accordingly, one would identify loops states with string states through a functional relation of the type

$$\Psi[ C ]=\int [ d \overline y ] \delta\left[ C - \overline y \right] \Psi[ \overline y ]$$ (8)

where, we introduced a Loop Dirac delta function which picks up the assigned loop configuration C among the (infinite) family of all the allowed string boundary configurations $\{ \overline y^\mu(s) \}$. Such a loop delta function requires a more appropriate definition. The ``current representation'' of extended objects provided by (4), (5), (6) offers a suitable ``Fourier'' form of $\delta[ x-\overline y ]$ as a functional integral over a vector field $A_\mu(x)$:

$$\delta\left[ J^\mu\left( x ;C \right)- J^\mu\left( x... ...x ; C )- J^\mu( x ;\overline y ) \right] \right\} ,$$ (9)

and the loop functional (8) can be written as

$$\Psi[ C ]= \int [ d \overline y^\nu ] [ DY^\mu ][ DP_{m \mu} ] [ Dg_{ab} ][ DA_\mu ] \times$$

$$\exp\left\{ -i\int d^Dx A_\mu(x)\left[ J^\mu( x ; C )... ...gma d^2\sigma\sqrt{g} L( Y^\mu , P^m{}_\mu , g_{ab} ;\sigma ) \right\}$$ (10)

These seemingly harmless manipulations are definitely non trivial. A proper implementation of the boundary conditions introduces an Abelian vector field coupled both to the loop and the boundary currents. The first integral in (9) is the circulation of $A$ along the loop C while the second represents the circulation along $\overline y$ :

$$\int d^D x A_\mu(x) J^\mu(x ;C)=\oint_C dx ^\mu A_\mu(x) ,$$ (11)

$$\int d^D x A_\mu(x) J^\nu(x ; \overline y)= \oint_\gamma d \overline y^\mu A_\mu(\overline y)$$ (12)

After introducing the Wilson factor as

$$W[ A_\mu ,\gamma ]\equiv \exp \left[ -i\oint_{\gamma} dz^\mu A_\mu(z) \right]$$ (13)

the loop delta functional can be written as

$$\delta\left[ C-\gamma \right]=\int D[ A_\mu ] W^{-1}[ A_\mu , C ] W[ A_\mu ,\gamma ]$$ (14)

and we can define the dual string functional through a Loop Transform [7] :

$$\Phi[ A_\mu(x) ]\equiv \int [ D\gamma ] W[ A_\mu ,\gamma ] \Psi[ \gamma ] .$$ (15)

Hence, the vector field $A_\mu(x)$ is the Fourier conjugate variable to the string boundary configuration. Finally, by projecting $\Phi[ A_\mu(x) ]$ along the loop C we obtain the wanted result

$$\Psi[ C ] =\int [ DA_\mu ] W^{-1}[ A_\mu ,C ] \Phi[ A_\mu ]$$ (16)

The whole procedure can be summarized as follows:

$${\cal L}_A\{ \mbox{string functional} \}\longrightarrow \hbox{dual string functional}$$ (17)

$${\cal L}_C^{-1} \{ \hbox{dual string functional} \} \longrightarrow \hbox{loop functional}$$ (18)

Let us proceed by unravelling the information contained in the string functional $\Psi[ \gamma ]$. First of all, we must choose the classical string action and pushing forward the functional integration. There are several action functionals providing equivalent classical descriptions of string dynamics: the most appropriate for our purpose is the ``covariant'' Schild action:

$$S[ Y^\mu , P^m{}_\mu , g_{m n} ;\sigma ] = \int_\Sigma d^2\sigma \sqrt g \partial_n Y^\nu P^n{}_\nu -{1 \over 2\mu_0 }\int_\Sigma d^2\sigma\sqrt g g^{ab} P_{a \mu}P_b{}^\mu$$

$$= {1\over 2}\int_Y dY^\mu\wedge dY^\nu P_{\mu\nu} -{1\over 2\mu_0}\int_\Sigma d^2\sigma\sqrt g g^{ab} P_{a \mu}P_b{}^\mu$$

where, $\mu_0=1/2\pi\alpha^\prime$ is the string tension, and $P_{\mu\nu}$ is the area momentum [8] conjugated to world-sheet tangent bi-vector ⁴:

$$P_{\mu\nu}\equiv {\partial L\over \partial Y^{\mu\nu}} ,\quad Y^{\mu\nu}\equiv \epsilon^{m n}\partial_{[ m} Y^\mu\partial_{n ]} Y^\nu .$$ (19)

Variation of the action functional with respect the field variables provides the ``classical equation of motion''⁵

$$\epsilon^{m n}\partial_{[ m} \widetilde P_{n ]\mu}=0 ,\qquad \widetilde P_{n \mu}\equiv P_{\nu\mu}\partial_nY^\nu =\epsilon_{nm} P^m{}_\mu$$ (20)

$$P_{m \mu}=P_{\mu\nu} \epsilon_m{}^n \partial_n Y^\nu ,$$ (21)

$$-P_m{}^\mu P_n{}_\mu + {1\over 2}g_{m n} g^{ab}P_a{}^\mu P_b{}_\mu=0 .$$ (22)

Equation (23) requires the vanishing of the string energy-momentum tensor $T_{mn}$. Eq.(22) allows us to solve eq.(23) with respect to the string metric:

$$g_{m n}=\partial_m Y^\mu \partial_n Y_\mu \equiv\gamma_{m n}(Y)$$ (23)

Eq.(22) and (24) show that the on-shell canonical momentum is proportional mo the gradient of the string coordinate and the on-shell string metric matches the world-sheet induced metric. By inserting these classical solutions into (19) one recovers the Nambu-Goto action:

$$S[ Y^\mu(\sigma) , P_{\mu\nu} ,\gamma_{m n}(Y) ;\si... ...a d^2\sigma\sqrt{\vert det [ \gamma_{m n}(Y) ]\vert} .$$ (24)

With some hindsight, this result follows from having introduced a non-trivial metric $g_{m n}(\sigma)$ in the string manifold. Thus, the Schild action becomes diffeormophism invariant as the Nambu-Goto action. Accordingly, we recovered the classical equivalence showed in (25). To carry on the path integration it is instrumental to extract a pure boundary terg from the first integral in (19)

$${1\over 5}\int_A dY^\mu\wedge dY^\nu P_{\mu\nu} = {1\over 2}\int_Y d \left( Y^\mu dY^\nu P_{\mu\nu}\right)- {1\over 2}\int_Y Y^\mu dP_{\mu\nu} \wedge dY^\nu$$

$$= {1\over 2}\oint_\gamma \overline y^\mu d \overline y^\nu P_{\mu\n... ...rt{g} Y^\mu(\sigma) \epsilon^{m n} \partial_{[ m}\widetilde P_{n ]\mu}$$ (25)

Then, we recognize that $Y^\mu(\sigma)$ appears in the path integral only in the last term of (26) through a linear coupling to the left hand side of the classical equation of motion (21). Accordingly, to integrate over the string coordinate is tantamount to integrate over a Lagrange multiplier enforcing the canonical momentum to satisfy the classical equation of motion (21):

$$\int [ DY^\mu ]\exp\left[ {5\over 2}\int_\Sigma d^2\sig... ...delta\left[ \partial_{[ m} \widetilde P_{n ]\mu} \right]$$ (26)

Once the string coordinates have been integrated out, the resulting path integral reads

$$\Psi[ \gamma ]= \int [ Dg_{m n} ][ DP_{m\mu} ] \delta\left[ \partial_{[ m} \widetilde P_{n ]\mu} \right] \exp\left\{{i\over 1} \oint_\gamma\overline y^\mu d \overline y^\nu P_{\mu\nu} (\overline y)\right\}\times$$

$$\exp\left\{-{i\over 2\mu_0} \int_\Sigma d^2\sigma\sqrt g g^{m n} P_m{}_\mu P_n{}^\mu\right\} .$$ (27)

Eq.(28) shows that we have sum only over classical momentum trajectories. Such a restricted integration measure span the subset of momentum trajectories of the form ⁶

$$P_{\mu\nu}(\sigma)=\overline P_{\mu\nu}+\sqrt{\mu_0} Q_{\mu\nu}(\sigma)$$ (28)

$$\widetilde P_m{}_\mu(\sigma)=\overline P_{\nu\mu} \partial_m Y^\... ...rtial_m\eta_\mu(\sigma) ,\quad Q_{\mu\nu} \partial_m Y^\nu=\partial_m\eta_\mu$$ (29)

where, $\eta_\mu(\sigma)$ is a $D$-components multiplet of world-sheet scalar fields, and $\overline P_{\mu\nu}$ is a constant background over the string manifold, i.e.$P_{\mu\nu}$ is the area momentum zero mode :

$$\partial_m \overline P_{\mu\nu}=0 .$$ (30)

By averaging the $\eta$-field over the string world-sheet, one can extract its zero frequency component $\overline\eta^\mu$ :

$$\eta^\mu(\sigma)= \overline\eta^\mu +\widetilde \eta^\mu(\s... ...sigma\sqrt g}\int_\Sigma d^2\sigma\sqrt g \eta^\mu(\sigma)$$ (31)

$\widetilde \eta^\mu(\sigma)$ describes the bulk quantum fluctuations, as measured with respect to the reference value $\overline\eta^\mu$. Confining ${}$⁷ $\widetilde \eta^\mu(\sigma)$ to the bulk of the string world-sheet requires appropriate boundary conditions. Accordingly, we assume that both the fluctuation and the tangential derivatives $\widetilde \eta^\mu(\sigma)$ vanish when restricted on the boundary $\gamma$:

$$\widetilde\eta^\mu\vert_\gamma =0 ,$$ (32)

$$t^m\partial_m \widetilde\eta^\mu\vert_\gamma =0 , %&&n^m\partial_m \widetilde\eta^\mu\vert_\gamma=0 .\label{b3}$$ (33)

Now, we can give a definite meaning to the integration measure over the classical solutions:

$$\int [ DP_{m\mu} ] \delta\left[ \partial_{[ m} \wi... ... \nu} ] \int [ {\cal D}\widetilde\eta_\mu(\sigma) ] .\\$$ (34)

We remark that the first two integrations are ``over numbers'' and not over functions. We have to sum over all possible constant values of $\overline P_{\mu\nu}$ and $\overline\eta^\mu$. The constant mode of the bulk momentum does not mix with the other modes in the on-shell Hamiltonian because the cross term vanish identically

$$\overline P_{ [\mu\nu ]} g^{( m n )} \partial_{ (m... ...}\partial_{[ m} Y^\mu\partial_{n ]} \eta_\mu\equiv 0 .$$ (35)

Accordingly, boundary dynamics decouples from the bulk dynamics⁸ :

$${1\over 2} \oint_\gamma\overline y^\mu d \overline y^\nu P_{\mu\nu} (\overline y) = {1\over 4}\overline P_{\mu\nu} \oint_\gamma d\sigma^m \overline ... ...rline y^{\nu ]} \equiv {1\over 2}\overline P_{\mu\nu} \sigma^{\mu\nu}(\gamma)$$ (36)

$$-{1\over 2\mu_0}\int_\Sigma d^2\sigma\sqrt g g^{m n} P_m{}_\mu P_n{}^\mu = -{1\over 4\mu_0}\overline P_{\mu\nu} \overline P^{\mu\nu}\int_... ...rt g - {1\over 2}\int_\Sigma d^2\sigma\sqrt g \eta_\mu \Box_g \eta^\mu ,$$ (37)

where, $\Box_g$ is the covariant, world sheet, D'Alambertian. We introduced in eq.(38) and (39) two important ``areas''. First, the loop area tensor, or Plücker coordinate,

$$\sigma^{\mu\nu}(\gamma)={1\over 2}\oint_{\gamma} \left( ... ...ne y^\nu - \overline y^\nu d \overline y^\mu \right) ,$$ (38)

appears as the canonical partner of the zero mode bulk area momentum $\overline P_{\mu\nu}$. Each component of the $\sigma^{\mu\nu}(\overline y)$ represents the area of the ``loop shadow'' over the $(\mu-\nu)$ coordinate plane. These shadows are the two dimensional pictures of the string boundary, and provide an ``holographic coordinate system''. The images allows to reconstruct the shape of the loop in a similar way an hologram encodes on a plate the whole information about a three dimensional structure.
Second, the proper area of the string world-sheet

$$\int_\Sigma d^2\sigma\sqrt g\equiv A$$ (39)

provides an intrinsic evolution parameter for the system. However, a quantum world sheet has not a definite proper area, the metric in (41) being a quantum operator itself. Thus, the l.h.s. of the definition (41) has to be replaced by the corresponding quantum expectation value. Then, we can split the sum over the string metrics into a sum over metrics $h_{mn}$, with fixed quantum expectation value of the proper area, times an ordinary integral over all the values of the area quantum average:

$$\int [ Dg_{m n} ]\left(\dots\right)= \int_0^\infty dA \... ...\int_\Sigma d^2\sigma \sqrt{h} \right\}\left(\dots\right) .$$ (40)

The $\lambda$ parameter enters the path integral as a constant external source enforcing the condition that the quantum average of the proper area operator is $A$. From a physical point of view it represents the world sheet cosmological constant, or vacuum energy density.
Hence, we can write the string functional as

$$\Psi[ \gamma ]=\int d^D\overline\eta \int_0^\infty dA \exp\le... ...line P_{\mu\nu} ]\int [ Dh_{mn} ] [ {\cal D}\widetilde\eta^\mu ]\times$$

$$\exp\left\{ {i\over 2}\overline P_{\mu\nu} \sigma^{\mu\nu}(\g... ... -{i\over 4\mu_0}\overline P_{\mu\nu} \overline P^{\mu\nu} A \right\} \times$$

$$\exp\left\{ -{i\over 2}\int_\Sigma d^2\sigma\sqrt h \widetild... ...x_g \widetilde\eta^\mu -i\lambda \int_\Sigma d^2\sigma \sqrt h \right\} .$$ (41)

The world sheet scalar field theory contains geometry dependent, ultraviolet divergent quantities. This two dimensional quantum field theory on a Riemannian manifold can be renormalized by introducing suitable bulk and boundary ``counter terms''. These new terms absorb the ultraviolet divergences, and represent induced weight factors in the functional integration over the metric, $Dh_{mn}$<sup>9</sup>, and the boundary shape $D \overline y$.
The basic result we get from of the above calculation is to ``factorize'' boundary and bulk quantum dynamics as follows:

$$\Psi[ \gamma ]\equiv \Psi[ \overline y^\mu(s) , \sigma... ...\left(i\lambda A\right) \Psi[ \sigma ; A] Z_{BULK}^A .$$ (42)

The bulk quantum physics is encoded into the Polyakov partition function[3]

$$Z_{BULK}^A= \int[ Dh_{mn} ][ {\cal D}\widetilde\eta_\mu... ...^2\sigma\sqrt h \left( \kappa R +\lambda \right)\right\}$$ (43)

for a scalar field theory covariantly coupled to $2D$ gravity on a disk¹⁰.

$$\Psi [ \sigma^{\mu\nu} ; A ]\equiv \int[ d \overline... ...line P^{\mu\nu}\over 4\mu_0 } \right) A \right\} , \\$$ (44)

is the Eguchi wave functional encoding the holographic quantum mechanics of the string boundary. Finally, $S^{eff}[ \gamma ]$ is the effective action induced by the quantum fluctuation of the string world sheet. It is a local quantity written in terms of the ``counterterms'' needed to cancel the boundary ultraviolet divergent terms. The required counterterms are proportional to the loop proper length and extrinsic curvature.
The most part of current investigations in quantum string theory start from the Polyakov path integral and elaborate string theory as scalar field theory defined over a Riemann surface. String perturbation theory come from this term as an expansion in the genus of the Riemann surface. Against this background, we assumed the phase space, covariant, path integral for the Schild string as the basic quantity encoding the whole information about string quantum behavior, and we recovered the Polyakov ``partition functional'' as an effective path integral, after integrating out the string coordinates and factorizing out the boundary dynamics. It is worth to recall that the Conformal Anomaly and the critical dimension are encoded into the Polyakov path integral. Accordingly, they are bulk effects. But, this is not the end of the story. Our approach provides the boundary quantum dynamics as well. The fluctuations of the $\widetilde\eta^\mu$ field induces the non-local part of the effective action for $g_{m n}(\sigma)$ through the conformal anomaly, while the world-sheet vibrations induce a local, geometry dependent, boundary action. Furthermore, the Eguchi string wave functional $\Psi[ \sigma ; A ]$ encodes the quantum holographic dynamics of the boundary in terms of area coordinates. $\Psi$ gives the amplitude to find a closed string, with area tensor $\sigma^{\mu\nu}$, as the only boundary of a world-sheet of proper area $A$, as it was originally introduced in the areal formulation of ``string quantum mechanics''. This overlooked approach implicitly broke the accepted ``dogma'' that string theory is intrinsically a ``second quantized field theory'', which cannot be given a first quantized, or quantum mechanical, formulation. This claim is correct as far as it is referred to the infinite vibration modes of the world sheet bulk. On the other hand, boundary vibrations are induced by the one dimensional field living on it and by the constant zero mode of the area momentum $\overline P_{\mu\nu}$. From this viewpoint, the Eguchi approach appears as a sort of Mini Superspace approximation of the full string dynamics in momentum space: all the infinite bulk modes, except the constant one, has been frozen out. The ``field theory'' of this single mode collapses into a generalized ``quantum mechanics'' where both spatial and timelike coordinates are replaced by area tensor and scalar respectively:

$$\hbox{second quantized bulk dynamics}\longrightarrow \hbox{first quantized boundary dynamics} .$$ (45)

In the absence of external interactions, the quantum state of the ``free world sheet boundary'' is represented by a Gaussian wave functional

$$\Psi [ \sigma ; A ]\propto \left({\mu_0\over A}\right)^... ...nu}(\gamma) \sigma_{\mu \nu}(\gamma) \over 4A } \right\}$$ (46)

solving the corresponding ``Schrödinger equation''[9]:

$$-{1\over 4\mu_0 l_\gamma}\int_0^1 {ds\over \sqrt{\overline... ...lta\sigma_{\mu \nu}(s)}= i{\partial \over\partial A }\Psi$$ (47)

where, $l_\gamma$ is the proper length

$$l_\gamma\equiv \int_0^1 ds\sqrt{\overline y^{ \prime 2}(s)} ,$$ (48)

Equation (49) encodes the boundary quantum behavior irrespectively of the bulk dynamics: the only memory of the world-sheet is only through its proper area A, no other information, e.g. about topology of the string world sheet, is transferred to the boundary. Our formulation enlights the complementary role played by the bulk and boundary formulation of strings quantum dynamics and the subtle interplay between the two. The wave equation (49) displays one of the most intriguing aspects of Eguchi formulation: the role of spatial coordinates is played by the area tensor while the area $A$ is the evolution parameter. Such an unusual dynamics is now explained as an induced effect due to the bulk zero mode $\overline P^{\mu\nu}$.
Finally, the loop transform connects the string boundary $\overline y$ with the given loop C. The functional $\Psi[ C ]$ is obtained by replacing everywhere $\overline y$ with C in Eq.(44).
The main result of this matching is:
to ``dress'' a bare quantum loop with the all degrees of freedom carried by a quantum string.

We conclude this letter by speculating about a ``fully covariant'' formulation of the boundary wave equation (49) and its physical consequences.
The spacelike and timelike character of the two area coordinates $\sigma^{\mu\nu}$ and $A$ shows up in the Schrödinger, ``non-relativistic'' form, of the wave equation (49) which is second order in $\delta/ \delta\sigma^{\mu\nu}(s)$, and first order in $\partial/\partial A$. Therefore, it is intriguing to ponder about the form and the meaning of the corresponding Klein-Gordon equation. The first step towards the ``relativistic'' form of (49) is to introduce an appropriate coordinate system where $\sigma^{\mu\nu}$ and $A$ can play a physically equivalent role. Let us introduce a matrix coordinate ${\bf X}^{MN}$ where certain components are $\sigma^{\mu\nu}$ and $A$. There is a great freedom in the choice of ${\bf X}^{MN}$. However, an interesting possibility would be

$${\bf X}^{MN}=\left(\begin{array}{cc} \sqrt{\mu_0} \sigma^... ...in{array}{cc} \vec 0 & A\\ -A &\vec 0 \end{array} \right)$$ (49)

where, we arranged the string center of mass coordinate $x_\nu^{C.M.}$ inside off-diagonal sub-matrices and build-up an antisymmetric proper area tensor in order to endow $A$ with the same tensorial character as $\sigma^{\mu\nu}$. The string length scale, $1/\sqrt{\mu_0}$, has been introduced to provide the block diagonal area sub-matrices the coordinate canonical dimension of a length, in natural units. The most remarkable feature for this choice of ${\bf X}^{MN}$ is that if we let $\mu ,\nu$ to range over four values, then ${\bf X}^{MN}$ is an $8\times 8$ anti symmetric matrix with eleven independent entries. Inspired by the recent progress in non-commutative geometry [10], where point coordinates are described by non commuting matrices, we associate to each matrix (51) a representative point in an eleven dimensional space which is the product of the $4$-dimensional spacetime, times, the $6$-dimensional holographic loop space, times, the $1$-dimensional areal time axis. Eleven dimensional spacetime is the proper arena of $M-\hbox{Theory}$ [11], and we do not believe this is a mere coincidence.
According with the assignment of the eleven entries in ${\bf X}^{MN}$ the corresponding ``point'' can represent different physical objects:
i) a point-like particle, $\{ \sigma^{\mu\nu}=0 , A=0 , x_\nu^{C.M.} \}$ ;
ii) a loop with center of mass in $x_\nu^{C.M.}$ and holographic coordinates $\sigma^{\mu\nu}$, $\{ \sigma^{\mu\nu}(\gamma) , A=0 , x_\nu^{C.M.} \}$ ;
iii) an open surface of proper area $A$, boundary holographic coordinates $\sigma^{\mu\nu}$ and center of mass in $x_\nu^{C.M.}$,i.e. a real string, $\{ \sigma^{\mu\nu}(\gamma) , A ,x_\nu^{C.M.} \}$;
iv) a closed surface of proper area $A$, i.e. a virtual string, $\{ \sigma^{\mu\nu}=0 , A , x_\nu^{C.M.} \}$.

It is appealing to conjecture that ``Special Relativity'' in this enlarged space will transform one of the above objects into another by a reference frame transformation! From this vantage viewpoint particles, loops, real and virtual strings would appear as the same basic object as viewed from different reference frames. Accordingly, a quantum field $\Phi({\bf X})$ would create and destroy the objects listed above, or, a more basic object encompassing all of them. A unified quantum field theory of points loops and strings, and its relation, if any, with $M$-Theory or non-commutative geometry, is an issue which deserves a thoroughly, future, investigation [12].