4 Plücker coordinates and $M$-theory

One of the most enlightening features of the Eguchi approach is the formal correspondence it establishes between the quantum mechanics of point-particles and string loops. Such a relationship is summarized in the translation code displayed in Table 2.

Table: The Particle/String ``Dictionary''

Physical object	massive point-particle	$\longrightarrow$	non-vanishing tension string
Mathematical model	point $P$ in $R ^{(3)}$	$\longrightarrow$	space-like loop $C$ in $R ^{(4)}$
Topological meaning	boundary (=endpoint) of a line	$\longrightarrow$	boundary of an open surface
Coordinates	$\{ x ^{1} , x ^{9} , x ^{3} \}$	$\longrightarrow$	area element $\sigma ^{\mu \nu} [C] = \oint _{C} x ^{\mu} dx ^{\nu}$
Trajectory	$1$-parameter family of points $\{ \vec{x} (t)\}$	$\longrightarrow$	$\{ x ^{\mu} (s ; A) \}$ $1$-parameter family of loops
Evolution parameter	``time'' $t$	$\longrightarrow$	area $A$ of the string manifold
Translations generators	spatial shifts: $\frac{\partial}{\partial x ^{i}}$	$\longrightarrow$	shape deformations: $\frac{\delta}{\delta \sigma ^{\mu \nu} (s)}$
Evolution generator	time shifts: $\frac{\partial}{\partial t}$	$\longrightarrow$	proper area variations: $\frac{\partial}{\partial A}$
Topological dimension	particle trajectory $D=1$	$\longrightarrow$	string trajectory $D=2$
Distance	$(\vec{x} - \vec{x} _{0}) ^{2}$	$\longrightarrow$	$(\sigma ^{\mu \nu} [C] - \sigma ^{\mu \nu} [C _{0}]) ^{2}$
Linear Momentum	rate of change of spatial position	$\longrightarrow$	rate of change of string shape
Hamiltonian	time conjugate canonical variable	$\longrightarrow$	area conjugate canonical variable

Instrumental to this correspondence is the replacement of the canonical string coordinates $x ^{\mu} (s)$ with the reparametrization invariant area elements $\sigma ^{\mu \nu} [C]$. We shall refer to these areal variables as Plücker coordinates [9]. Surprising as it may appear at first sight, the new coordinates $\sigma ^{\mu \nu} [C]$ are just as ``natural'' as the old $x ^{\mu} (s)$ for the purpose of defining the string ``position''. A naive argument to support this claim goes as follows. In the Jacobi- type formulation of particle dynamics, the position of the physical object is provided by the instantaneous end-point of its own world-line

$$x ^{\mu} (P) \equiv x ^{\mu} (T) = \int _{-\infty} ^{T}... ...c{dx ^{\mu}}{d \tau} = \hbox{world-line end point} \quad .$$ (17)

In other words, the instantaneous position of the particle ix given by the line integral of the tangent vector up to the chosen final value $T$. Similarly, then, it seeps natural to define the ``string position'' as the surface integral of the tangent bi-vector up to the final boundary of the world-sheet

$$\sigma ^{\mu \nu} [C] = \int _{0} ^{s _{0}} d \sigma \in... ..._{\sigma} x ^{\nu} = \hbox{world-surface boundary} \quad ,$$ (18)

which is nothing but the loop area element appearing in Eq. (15). A geometric interpretation of the new ``matrix''-coordinate assignment to the loop $C$ is that they represent the areas of the loop projected shadows onto the coordinate planes. In this connection, note that the $\sigma$-tensor satisfies the constraint

$$\epsilon _{\lambda \mu \nu \rho} \sigma ^{\lambda \mu} \sigma ^{\nu \rho} = 0$$ (19)

which ensures that there is a one-to-one correspondence between a given set of areas $\sigma ^{\mu \nu}$ and a loop $C$. Thus, to summarize, the Plücker coordinates refer to the boundary of the world sheet, and enter the string wave functional as an appropriate set of position coordinates. Then, it is not surprising that in such a formulation homogeneity requires a timelike coordinate with area dimension as well⁴.

Finally, we note for the record, that the Plücker coordinates provide a formal correspondence between string theory [10], and a certain class of electromagnetic field configurations [11]. As a matter of fact, a classical gauge field theory of relativistic strings was proposed several years ago [12], but only recently it was extended to generic $p$-branes including their coupling to $p+1$-forms and gravity [13].

Now that we have established the connection between areal quantization and the path integral formulation of quantum strings, it seems almost compelling to ponder about the relationship, if any, between the $\sigma ^{\mu \nu} [C]$ matrix coordinates and the matrix coordinates which, presumably, lie at the heart of the $M$-Theory formulation of superstrings. Since the general framework of $M$-Theory is yet to be discovered, it seems reasonable to focus on a specific matrix model recently proposed for Type $IIB$ superstrings⁵[15]. The dynamics of this model is encoded into a simple Yang-Mills type action

$$S _{\mathrm{IKKT}} = - \frac{\alpha}{4} \mathrm{Tr} ... ... + \beta \mathrm{Tr} \mathbb{I} + \hbox{fermionic part}$$ (20)

where the $A ^{\mu}$ variables are represented by $N \times N$ hermitian matrices, and $\mathbb{I}$ is the unit matrix. The novelty of this formulation is that it identifies the ordinary spacetime coordinates with the eigenvalues of the non-commuting Yang-Mills matrices. In such a framework, the emergence of classical spacetime occurs in the large-$N$ limit, i.e., when the commutator goes into a Poisson bracket⁶ and the matrix trace operation turns into a double continuous sum over the row and column indices, which amounts to a two dimensional invariant integration. Put briefly,

$$\lim _{N \rightarrow \infty} \lq\lq \mathrm{Tr}'' \rightarrow - i \int d\tau d\sigma \sqrt{\gamma}$$ (21)

$$- i \lim _{N \rightarrow \infty} [ A ^{\mu} , A ^{\nu} ] \rightarrow \{ x ^{\mu} , x ^{\nu} \}$$ (22)

$$\sqrt \gamma \{ x ^{\mu} , x ^{\nu} \} \equiv \dot{x} ^{\mu \nu} \equiv \partial _{\tau} x ^{\mu} \wedge \partial _{\sigma} x ^{\nu} \quad .$$ (23)

What interests us is that, in such a classical limit, the IKKT action (20) turns into the Schild action in Eq. (13) once we make the identifications

$$\alpha \longleftrightarrow - m ^{2} \quad , \qquad \beta ... ... N (\tau , \sigma) \longleftrightarrow \sqrt{\gamma} \quad ,$$ (24)

while the trace of the Yang-Mills commutator turns into the oriented surface element

$$\lim _{N \rightarrow \infty} \mathrm{Tr} \left[ A ^{\m... ... ^{\nu} \equiv \sigma ^{\mu \nu} (\partial \Sigma ) \quad .$$ (25)

This formal relationship can be clarified by considering a definite case. As an example let us consider a static $D$-string configuration both in the classical Schild formulation and in the corresponding matrix description. It is straightforward to prove that a length $L$ static string stretched along the $x ^{1}$ direction, i.e.

$$x ^{\mu} = \tau T \delta ^{\mu 0} + \frac{L \sigma}{2 \p... ...\mu 1} \quad , \qquad 0 \le \tau \le 1 \quad , \qquad 0 \le \sigma \le 2 \pi$$ (26)

$$ x ^{\mu} = 0 \quad , \qquad \mu \ne 0,1$$ (27)

solves the classical equations of motion

$$\{ x _{\mu} , \{x ^{\mu} , x ^{\nu} \} \} = 0 \quad .$$ (28)

During a time lapse $T$ the string sweeps a time-like world-sheet in the $(0,1)$-plane characterized by an area tensor

$$\sigma ^{\mu \nu} (L , T) = \int _{0} ^{1} d \tau \int _... ...} = T L \delta ^{0 [ \mu} \delta ^{\nu ] 1} \quad .$$ (29)

Eq.(29) gives both the area and the orientation of the rectangular loop which is the boundary of the string world-sheet. The corresponding matrix solution, on the other hand, must satisfy the equation

$$[ A _{\mu} , [ A ^{\mu} , A ^{\nu} ] ] = 0 \quad .$$ (30)

Consider, then, two hermitian, $N \times N$ matrices $\hat{q}$, $\hat{p}$ with an approximate $c$-number commutation relation $${ [ \hat{q} , \hat{p} ]= i}$$, when $N \gg 1$. Then, a solution of the classical equation of motion (30), corresponding to a solitonic state in string theory, can be written as

$$A ^{\mu} = T \delta ^{\mu 0} \hat{q} + \frac{L}{2 \pi} \delta ^{\mu 1} \hat p \quad .$$ (31)

In the large-$N$ limit we find

$$- i [A ^{\mu} , A ^{\nu}] = - i \frac{L T}{2 \pi} \del... ...ox \frac{L T}{2 \pi} \delta ^{0 [ \mu} \delta ^{\nu ] 1}$$ (32)

and

$$- i \mathrm{Tr} [ A ^{\mu} , A ^{\nu} ] \approx \in... ... x ^{\mu} , x ^{\nu} \} = \sigma ^{\mu \nu} (L , T) \quad .$$ (33)

These results, specific as they are, point to a deeper connection between the loop space description of string dynamics and matrix models of superstrings which, in our opinion, deserves a more detailed investigation. Presently, we shall limit ourselves to take a closer look at the functional quantum mechanics of string loops with an eye on its implications about the structure of spacetime in the short distance regime.