3 Non-commutative Phase Space

To match the large-$N$ $SU(N)$ gauge theory with some appropriate brane model we have to bridge the gap between non-commuting Yang-Mills matrices and commuting brane coordinates. Since the world-trajectory of a $p$-brane is the target spacetime image $x^\mu=X^\mu( \sigma^0,\sigma^1,\dots , \sigma^p )$ of the world manifold $\Sigma : \sigma^m=( \sigma^0,\sigma^1,\dots \sigma^p )$, $X^\mu$ belonging to the algebra $\mathcal{A}$ of $C^\infty$ functions over $\Sigma$, to realize our program we must deform ${\mathcal{A}}$ to a non-commutative ``starred'' algebra by introducing a $\ast$-product. The general rule is to define the new product between two functions as (for a recent review see [19]):

$$f \ast g =f \,g + \hbar\, P_\hbar(\, f\ ,g \,) \ ,$$ (23)

where $P_\hbar(\, f\ ,g\, )$ is a bilinear map $P_\hbar : {\mathcal{A}}\times {\mathcal{A}}\rightarrow {\mathcal{A}}$. $\hbar$ is the deformation parameter which is often denoted by the same symbol as the Planck constant to stress the analogy with quantum mechanics, where classically commuting dynamical variables are replaced by non-commuting operators. In our case the role of deformation parameter is played by

$$\hbar\equiv {2\pi\over N} \ .$$ (24)

For our purposes, we select $\Sigma={\mathbb{R}}^{2n}$ and choose the Moyal product as the deformed $\ast$-product

$$f(\sigma) \ast g(\sigma) \equiv \exp\left[ i {\hbar\over ... ...n} \right] f(\sigma) g(\xi) \Biggr\rceil_{\xi=\sigma} \ ,$$ (25)

where $\omega^{mn}$ is a non-degenerate, antisymmetric matrix, which can be locally written as

$$\omega^{mn}=\left( \begin{array}{cc} {\mathbb{O}}_{ n\times... ...times n} & {\mathbb{O}}_{ n\times n} \end{array} \right) \ .$$ (26)

The Moyal product (25) takes a simple looking form in Fourier space

$$F(\sigma) \ast G(\sigma)=\int {d^{ 2n}\xi\over (2\pi)^{n}} \... ...over 2}+ \xi \right) G\left( {\sigma \over 2}- \xi \right) \ ,$$ (27)

where $F$ and $G$ are the Fourier transform of $f$ and $g$. Let us consider the Heisenberg algebra

$$\left[ \mbox{\textit{\bf {}K}}^m, \mbox{\textit{\bf {}P}}^l \right]=i \hbar \delta^{ml}\ ;$$ (28)

Weyl suggested, many years ago, how an operator ${\mbox{\textit{\bf {}O}}}_F( \mbox{\textit{\bf {}K}}, \mbox{\textit{\bf {}P}} )$ can be written as a sum of algebra elements as

$$\mbox{\textit{\bf {}O}}_F= {1\over (2\pi)^n}\int d^np d^nk F... ...xtit{\bf {}K}}^m + i k_l \mbox{\textit{\bf {}P}}^l \right) \ .$$ (29)

The Weyl map (29) can be inverted to associate functions, or more exactly symbols, to operators

$$F(\, q\ ,k \,)= \int {d^n\xi\over (2\pi)^n} \exp\left(-i k \... ...\right) \Biggr\vert q - \hbar {\xi \over 2 } \Biggr\rangle \ ;$$ (30)

moreover it translates the commutator between two operators ${\mbox{\boldmath {$U$}}}$, ${\mbox{\boldmath {$V$}}}$ into the Moyal Bracket between their corresponding symbols ${\mathcal{U}}(\sigma)$, ${\mathcal{V}}(\sigma)$

$${1\over i \hbar} \left[ {\mbox{\boldmath {$U$}}}, {\mbox{\... ...\ast {\mathcal{V}} - {\mathcal{V}} \ast {\mathcal{U}} \right)$$

and the quantum mechanical trace into an integral over Fourier space

$$(2\pi)^n \mathrm{\mathrm{Tr}}_{{\mathcal{H}}} \mbox{\textit... ...F( \, q\ ,k\,) \equiv \int d^{2n}\sigma\, F( \, \sigma\,) \ .$$ (31)

A concise but pedagogical introduction to the deformed differential calculus and its application to the theory of integrable system can be found in [20].

We are now ready to formulate the alleged relationship between the quenched model (22) and membrane model: the symbol of the matrix $\mbox{\textit{\bf {}A}}^{(\mathrm{q})}_j$ is proportional to the $2n$-brane coordinate $X^j( \sigma^1 , \dots , \sigma^{2n} )$. Going through the steps discussed above the action $S^{\mathrm{BFSS}}$ transforms into its symbol $W^{\mathrm{BFSS}}$:

$$S^{\mathrm{BFSS}}\rightarrow W^{\mathrm{BFSS}}= {N V_H\over ... ...}}\ast \left\{ A^i, A^n \right\}_{\mathrm{MB}}}{4} \right] \ .$$ (32)

The action (32) is manifestly Lorentz non-covariant, as it is expected (the covariant, supersymmetric, higher dimensional version of the action (32) is discussed in [21]). The adopted quenching scheme explicitly breaks the equivalence between spacelike and timelike coordinates. Accordingly, our final result takes a typical ``non-relativistic'' look.
Up to now we have not fixed the Fourier space dimension $n$. To give $A^i$ the meaning of embedding function, we have to choose $2n\le D-1$, where $D$ is the target spacetime dimension. To match QCD in four spacetime dimensions we set $n=1$. In this case

$$\omega^{mn}=\epsilon^{mn}=\left( \begin{array}{cc} 0 & 1\\ -1 & 0 \end{array} \right)$$ (33)

and we rescale the Yang-Mills charge and field⁶as

$${N \over g_0{}^2} \longmapsto {1 \over g^2_{\mathrm{YM}}}$$ (34)

$$A^i \longmapsto V^{-2/3}_H\, X^i \ .$$ (35)

Since the ``glue'' is supposed to be confined inside an hadronic size volume $V_H$, we can assign to $g_{\mathrm{YM}}$ the standard value at the confinement scale

$${g^2_{\mathrm{YM}}\over 4\pi} \simeq 0.18 \ .$$ (36)

Finally, if $N \gg 1$ the Moyal bracket can be approximated by the Poisson bracket

$$\left\{ X^i, X^j \right\}_{\mathrm{MB}} \longmapsto \left\{ X^i, X^j \right\}_{\mathrm{PB}}$$

and (32) takes the form [22]

$$S^{p=2}_{\mathrm{NR}}= \int dt d^2\sigma \left[ {1\over 2 } ... ...athrm{PB}} \left\{ X^i, X^j \right\}_{\mathrm{PB}} \right] \ ,$$ (37)

where $\mu _{0}$ and $\alpha$ are defined by

$$\mu_0\equiv { V_H^{-1/3}\over 2\pi g^2_{\mathrm{YM}} } \quad... ...ad \alpha\equiv { V_H^{-5/3}\over 2\pi g^2_{\mathrm{YM}} } \ .$$

Moreover from the definition of the Poisson bracket

$$\left\{ X^i, X^j \right\}_{\mathrm{PB}} \equiv \epsilon^{mn}\partial_m X^i \partial_n X^j$$

we can compute

$$\left\{ X^i\ , X^j \right\}_{\mathrm{PB}}\left\{ X_i\ , X_j ... ...partial_m \, X^k \, \partial_n \, X_k \,\right)\equiv 2 \gamma$$

so that the action (37) can be rewritten as

$$S^{p=2}_{\mathrm{NR}}= \int dt d^2\sigma \left[ {1\over 2 } ... ...partial_m \, X^k \, \partial_n \, X_k \,\right)\, \right] \ .$$ (38)

The first term in (38) is a straightforward generalization of the kinetic energy of a non-relativistic particle; the second term represents the ``potential energy'' associated to the elastic deformations of the membrane. The action (38) still displays a residual symmetry under area preserving diffeomorphisms, leaving only one dynamical degree of freedom describing transverse oscillations of the membrane surface<sup>7</sup>. If the action (38) has any chance to provide a membrane model of hadronic objects, then it must be able to provide at least the correct order of magnitude of hadronic masses. Our model does not take into account spin effects, therefore it is consistent to look for spherically symmetric configurations. Again this is a sort of quenching even if of a more geometric type. Infinite vibration modes of the brane, corresponding to local shape deformations, are frozen and the dynamics is reduced to the ``radial'' breathing mode alone. This kind of approximation, commonly called ``minisuperspace'' approximation, is currently adopted in Quantum Cosmology, where it amounts, in practice, to quantize a single scale factor (thereby selecting a class of cosmological models, for instance, the Friedman-Robertson-Walker spacetimes) while neglecting the quantum fluctuations of the full metric. The effect is to turn the exact, but intractable, Wheeler-DeWitt functional equation [23] into an ordinary quantum mechanical wave equation [24]. As a matter of fact, the various forms of the ``wave function of the universe'' that attempt to describe the quantum birth of the cosmos are obtained through this kind of approximation [25] or modern refinements of it [26]. This non-standard approximation scheme was applied to a relativistic membrane in the seminal paper by Collins and Tucker [27], and since then it has been used several times [28], including the case of self-gravitating objects [29].

Following [27], we parametrize the membrane coordinates as follows

$$X^1 \equiv R(t) \sin\theta\cos\phi$$

$$X^2 \equiv R(t) \sin\theta\sin\phi$$ (39)

$$X^3 \equiv R(t)\cos\theta$$

and the transverse, dynamical degree of freedom corresponds to $R$. The metric $\gamma_{ab}$ induced on the membrane by the embedding (39) is:

$$\gamma_{ab}= \mathrm{diag}\left( R^2(t), R^2(t) \sin^2\theta... ...,\quad \det\left( \gamma_{mn} \right)= R^4(t) \sin^2\theta \ .$$

The corresponding action turns out to be

$$S= \int L dt = \pi^2\int dt \left[ {1\over 2} \mu_0 \left( {dR\over dt} \right)^2 + {\alpha\over 4} R^4 \right] \ ;$$

accordingly the momentum conjugated to the only dynamical degree of freedom is

$$P_R\equiv {\partial L\over \partial ( dR/dt )}=\pi^2 \mu_0 {dR\over dt}$$

from which the Hamiltonian can be calculated as

$$H\equiv P_R {dR\over dt} - L ={1\over 2\pi^2 \mu_0} P_R^2 + {\alpha\pi^2\over 4} R^4 \ .$$

Then the action in Hamiltonian form is

$$S=\int dt \left[ P_R {dR\over dt} - \left( {1\over 2\pi^2 \mu_0}P_R^2 + {\alpha\over 4} R^4 \right) \right] \ .$$

The above results allow one to compute the hadronic mass spectrum from the spherical membrane Schrödinger equation

$$\left[ -{1\over 2\pi^2 \mu_0}{d^2\over dR^2} + {\alpha\pi^2\over 4} R^4 \right] \Psi( R )= M_n \, \Psi( R )$$ (40)

with the following boundary conditions:

$$\Psi( 0 ) = 0$$ (41)

$$\lim_{R\to\infty}\Psi( R ) = 0 .$$ (42)

The lowest mass eigenvalues can be evaluated numerically [30] or through WKB approximation [28]:

$$\left( 2\pi^2 \mu_0 M_n \right)^{1/2}={4\pi\over\beta(1/4, 3... ...YM}}^2 } \right)^{1/3} \left( n +{3\over 4} \right)^{2/3} \ .$$ (43)

where, $\beta$ is the Euler $\beta$-function: $\beta(1/4, 3/2)\equiv {\Gamma(1/4)\, \Gamma(3/2)\over\Gamma(7/4)}$. The WKB formula (43) gives a mass scale of the correct order of magnitude, $M_n \propto 4\pi\, g_{\mathrm{YM}}^{2/3}V_H^{-1/3 }\simeq 1 \mathrm{GeV}$. More sophisticated estimates of the glueball mass spectrum, including topological corrections [31], are not very different from the values given by (43). Thus, we conclude that the QCD membrane action (38) encodes, at least the dominant contribution, to the gluon bound states spectrum. Hopefully, an improvement of this result will come from an extension of the minisuperspace approximation along the line discussed in [32], where a new form of the p-brane propagator has been obtained.