The Search for the Origins of M-Theory : Loop Quantum Mechanics, Loops/Strings

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Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities

THE SEARCH FOR THE ORIGINS OF M-THEORY : LOOP QUANTUM MECHANICS, LOOPS/STRINGS

AND BULK/BOUNDARY DUALITIES

Carlos Castro∗ a ∗ Center for Theoretical Studies of Physical Systems, Clark Atlanta University, Atlanta, Georgia 30314 USA

ABSTRACT The construction of a covariant Loop Wave functional equation in a 4D spacetime is attained by introducing a generalized eleven dimensional categorical C-space comprised of 8×8 antisymmetric matrices. The latter matrices encode the generalized coordinates of the histories of points, loops and surfaces combined. Spacetime Topology change and the Holographic principle are natural consequences of imposing the principle of covariance in C-space. The Planck length is introduced as a necessary rescaling parameter to establish the correspondence limit with the physics of point-histories in ordinary Minkowski

space, in the limit lP → 0. Spacetime quantization should appear in discrete units of Planck length, area, volume ,....All this seems to suggest that the generalized principle of covariance, representing invariance of proper area intervals in C-space, under matrix- coordinate transformations, could be relevant in discovering the underlying principle behind the origins of M theory. We construct an ansatz for the SU(∞) Yang-Mills vacuum wavefunctional as a solution of the Schroedinger Loop Wave equation associated with the Loop Quantum Mechanical formulation of the Eguchi-Schild String . The Strings/Loops ( SU(∞) gauge ﬁeld) correspondence implements one form of the Bulk/Boundary duality conjecture in this case.

aElectronic mail address: [email protected]

1 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities

1. INTRODUCTION Recently, a very interesting relation between string quantization based on the Schild string path integral and the Nambu-Goto string path integral was established at the semi- classical level by a saddle point evaluation method 2. Quantum mechanics of Matrices , M(atrix) models 5theory have been conjectured as a leading candidate to understand nonperturbative string theory : M theory. We will follow another approach related to ordinary Matrix Quantum Mechanics and study Quantum Mechanics but in Loop spaces. The Eguchi quantization of the Schild string is essentially a sort of quantum mechanics formulated in the space of loops. The authors 2 have constructed a Loop Wave equation associated with the Eguchi-Schild string quantization in the Schroedinger form, where the area spanned by the evolution of a closed spatial string ( loop) served as the role of the “time parameter” and the holographic shadows of the loop-shapes, onto the spacetime coordinate planes, σµν (C), played the role of space coordinates. It was argued by 2 that the large scale properties of the string condensate, within the framework of Loop Quantum Mechanics, are responsible for the effective Riemannian geometry of spacetime at large distances. On the other hand, near Planck scales, the condensate “evaporates” and what is left behind is a “vacuum” characterized by an effective fractal geometry. What is required now is to covariantize the Schroedinger Loop wave equation by in- cluding the areal time A and the “spatial coordinates” σµν (C) into one single footing. We will see that the construction of a covariant Loop Wave functional equation in a 4D spacetime, a Klein-Gordon Loop Wave equation, is attained by introducing a generalized eleven dimensional categorical C-space comprised of 8 × 8 antisymmetric matrices. The latter matrices encode the generalized coordinates of the histories of points, loops and surfaces combined. The points correspond to the center of mass coordinates of a loop/surface. The areal time A and σµν (C) will also be a part of those 8×8 antisymmetric matrices. We find that Spacetime Topology change and the Holographic principle are natural consequences of imposing the principle of covariance in C-space. The Planck length must be introduced as a necessary rescaling parameter to establish the correspondence limit with the physics of point-histories in ordinary Minkowski space, in the limit lP → 0; i.e when the loops and

2 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities surfaces shrink to a point, the field theory limit. In section II we will review very briefly the Loop Quantum Mechanics of the Schild- string and write down the Schroedinger Loop Wave equation . In III we construct boundary wavefunctional solutions to the Loop Wave equation by evaluating the phase path integral over open surfaces with boundary C.InIV we propose an ansatz for the SU(∞) Yang-Mills vacuum wavefunctional as a solution of the Schroedinger Loop Wave equation associated with the Loop Quantum Mechanical formulation of the Eguchi-Schild String . This String/Loop ( SU(∞) gauge field) correspondence implements one form of the Bulk/Boundary duality conjecture in this case 16. Finally in V we write down the Covariant Loop Wave Functional equation in the Klein Gordon form ( a classical master field theory) and we argue how Moyal-Fedosov deforma- tion quantization, if , and only if, it is applicable, may be suited to quantize the classical master field theory . We will find that the search for a covariant Loop Wave equation where areal time A and holographic shadows, σµν ,, are on the same footing, requires to embed the ordinary string theory in spacetime into the categorical C-space comprised of point, loop and surface histories. All this seems to suggest that the generalized principle of covariance, representing invariance of proper area intervals in this C-space ( whose elements are antisymmetric matrices) , under matrix-coordinate transformations, could be relevant in discovering the underlying principle behind the origins of M theory. 2. LOOP QUANTUM MECHANICS We shall present a very cursory description of Eguchi’s areal quantization scheme of the Schild string action . For further details we refer to 2. The starting Lagrangian density is :

1 L = {Xµ,Xν}{X ,X }.Xµ(σ0,σ1). (2.1) 4 µ ν Xµ(σ0,σ1) are the embedding string coordinates in spacetime and the brackets represent Poisson brackets w.r.t the σ0,σ1 world sheet coordinates. The Schild action is just the area squared of the world sheet instead of the area interval described by the Dirac-Nambu- Goto actions. The corresponding Schild action is only invariant under area-preserving transformations .

3 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities

Eguchi’s areal quantization scheme is based on keeping the area of the string histories fixed in the path integral and then taking the averages over the string tension values. The Nambu-Goto approach, on the other hand, keeps the tension fixed while averaging over the world sheet areas. If one wishes to maintain the analog of quantum mechanics of point particles and that of loops, the Eguchi quantization scheme of the Schild action requires the following Correspondence principle 2 between area-momentum variables and time ( areal time ) A :

1 δ ∂ Pµν (s) → √ .H→−i . ¯h =1. (2.2) x02(s) δσµν ∂A The Loop Schroedinger-like Wave equation HΨ=−i(∂/∂A)Ψ is obtained once we extract the areal time independence , Ψ = e−iEAΨ.

1 1 δ2 0 2 − µν µν ds x (s) ( 2 µν )Ψ[σ (C)] = EΨ[σ (C)]. (2.3) lC o 4m δσ δσµν Z q where the string wave functional Ψ[σµν (C)] is the amplitude to ﬁnd the loop C with area- elements σµν ( holographic shadows) as the only boundary of a two-surface of internal area A in a given quantum state Ψ. Plane wave solutions and Gaussian wavepackets ( superposition of fundamental plane wave solutions) were constructed in 2 by freezing all the modes, associated with the arbitary loop shapes, except the constant momentum modes on the boundary. The plane wave solutions were of the type :

µν µ ν Ψ[σ (C)] ∼ exp [iEA − i x (s)dx (s)Pµν ]. (2.4) IC A loop space momentum/string shape-uncertainty principle was given as :

µν ∆σ ∆Pµν ≥ 1. (2.5) in suitable units . Having presented this very brief review of the Schroedinger Loop Wave equation we shall elaborate further details in the next sections and construct more general solutions than the plane wave case by evaluating the Schild string phase space path integral in III.InIV we propose our ansatz for the SU(∞) YM vacuum wavefunctional as another

4 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities candidate solution to the Schroedinger Loop Wave equation. And in the ﬁnal section we present the covariant Loop Wave functional equation and its relation to the plausible origins of M theory. 3. BOUNDARY WAVE FUNCTIONAL EQUATIONS We shall construct a boundary wavefunctional solution to the Loop Schroedinger-like wave equation ( besides the standard plane wave solution) which is associated with a Riemann surface of arbitrary topology and with one boundary. For simplicity we shall start with a Riemann surface of spherical topology, with the boundary , C, parametrized bythecurve:xµ(s). Writing the Schild action in Hamiltonian form allows one to deﬁne a boundary wavefunctional solution to the Loop Schroedinger-like wave equation, by :

µν µ Ψ[σ (C)] ≡ [DX (σ)] [DPµa(s)] [DΠµa(σ)] ZC Z ZΠµa|∂Σ=Pµa 2 1 Πµa exp [i Π dΣµν (σ) − i d2σ ](3.1) 2 µν 4m2 Z Z The physical meaning of Ψ[σµν (C)] is the probability amplitude of finding any given loop C with area-components ( holographic shadows onto the spacetime coordinate planes) σµν in a given quantum state Ψ. One is performing the canonical path integral, firstly, by summing over all bulk momentum configurations of the string world sheet with the restriction that on the boundary they equal the pre-assigned value of the boundary momentum, P µν (s), and afterwards , one performs the summation over all values of the boundary momentum, P µν (s) . It is important to emphasize, that one is also summing over all string world sheet embeddings , maps from a world sheet with a disk topolgy to a target spacetime, Xµ(σ) such that maps Xµ restricted to boundary of the disk are equal to xµ(s) . The latter are just the spacetime parametrization coordinates of any free loop C; i.e the boundary shapes , C,are completely arbitrary . Furthermore, since we are using the Schild action which is not fully reparametrization invariant ( only under area-preserving diffs), we are not performing the functional integral over a family of world sheet auxiliary metrics. In 1 we carried out such functional integral. The world sheet conjugate area-momentum is defined as the pullback of the target spacetime area conjugate momentum :

5 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities

ν a Πµa ≡ Πµν ∂aX (σ ). (3.2) and :

µa ν µρ a µρ ν µν ΠµaΠ =Πµν ∂aX Π ∂ Xρ =Πµν Π δρ =Πµν Π . (3.3) which was used in the deﬁnition of the Schild Hamiltonian. Where the bulk momentum is restricted to obey the boundary condition :

a Πµν (σ )|∂Σ = Pµν (s). Πµa|∂Σ = Pµa(s)(3.4)

The area components enclosed by the arbitray boundary curve xµ(s) is given by Stokes theorem :

1 σµν (C)= (xµdxν − xν dxµ)(3.5) 2 IC An integration by parts of the integral

2 µν i d σΠµν dΣ (σ)(3.6a) Z yields

µν 2 µ ν i Pµν dσ (s) − i d σX {Πµν ,X } (3.6b) IC Z where the Poisson bracket is taken with respect to the world sheet coordinates :σ0,σ1. The Poisson bracket in the second integrand of eq-(3-6b ) can be written in terms of µ the canonical conjugate variables : X , Πµa because :

µ ν µ ab ν µ ab ν µ ab X {Πµν ,X }=X ∂aΠµν (σ)∂bX (σ)=X ∂a[Πµν (σ)∂bX (σ)] = X ∂aΠµb (3.7) due to the condition : ab ν ∂a∂bX =0. (3.8) and after using the deﬁnition of the world sheet canonical momentum; i.e the pullback of

Πµν to the world sheet. Notice that the integrand given by eq-(3-7 ) which is to appear in

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µ the exponential weight of the path integral depends solely on X and Πµa.Upondoing ﬁrstly the functional integration with respect to all the string embeddings Xµ,ofaworld sheet of a disk topology into spacetime, one will produce a delta functional constraint which is nothing but the classical Schild string equations of motion :

ν ν ab ˜ a δ(Kµ =0).Kµ={{Xµ,Xν},X }={Πµν ,X }= ∂aΠµb = ∂aΠµ =0. (3.9) where we have made use of eqs-(3-1, 3-6b, 3-7 ) and where the dual mometum is deﬁned as usual :

˜ a ab Πµ = Πµb. (3.10)

Therefore, the Xµ path integral imposes the bulk worldsheet classical Schild string equations of motion. This is tantamount of minimizing the proper world sheet area or areal time. A congruence ( or family ) of classical string conﬁgurations is usualy parametrized µ by the boundary data : x (s); Pµa(s). This is the string version of the classical free particle motion. One requires to specify the initial position and velocity of the particle to determine the trajectory. We shall study now the quantum boundary string dynamics induced by the classical world sheet dynamics. Later, we will extend this procedure to a membrane and all p- branes. The boundary wavefunctional becomes :

1 Ψ[σµν (C)] = [DP (s)] exp [i P dσµν (s)] [DΠclass(σ)] µa 2 µν µa Z IC ZΠµa|∂Σ=Pµa class 2 (Πµa ) exp [−i d2σ ](3.11) 4m2 Z The other alternative way to proceed, is firstly by covariantizing the Schild action by introducing auxiliary two-dim metrics, gab . We explicitly performed such phase space path integral in 1 . It was also required to find the moduli space of solutions of the ab classical Schild string equations of motion, ∂Πµb = 0, and then to integrate over the moduli. Since in two dimensions, a vector is dual to a scalar field, the solutions for Πµa could be expressed in terms of a set of scalar fields and, in this fashion, we made contact

7 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities with a bulk Conformal Field Theory. A further functional integral with respect to the auxiliary two-metrics, gab, was also necessary. We were able to compute explicitly the bulk momentum path integral and to establish a duality relation between loop states ( living on C ) and string states ( living on the bulk) through a functional loop transform From this last expression (3-11) we can recognize immediately that the boundary area- wavefunctional Ψ[σµν (C)] is nothing but the loop transform of the boundary momentum wavefunctional which is deﬁned as :

class 2 (Πµa ) Ψ[˜ P ] ≡ [DΠ (σ)] exp [−i d2σ [ ]]. µa µa class 4m2 ZPµa ZΣ This bulk momentum path integral deﬁned by eq-(3.12), was explicitly evaluated in 1, when we computed the covariantized Schild action path integral in phase space. We found that the phase space path integral factorizes into a product of a Eguchi wavefunctional, encoding the boundary dynamics, and a bulk path integral term of the Polyakov type, with induced scalar curvature and cosmological constant on the bulk and an induced boundary cosmological constant and extrinsic curvature on the boundary . The ﬁnal result of the µν 1 functional loop transform that maps Ψ[˜ Pµa]intoΨ[σ (C)] is :

∞ µν Eguchi µν Bulk µν Ψ[σ (C)] = dAΨ [σ (C); A]ZA [σ (C)] (3 − 13) Z0 Such path integral required to evaluate the covariantized Schild action,

class 2 √ (Π [σ; Pµa(s)]) d2σ h µa . (3.14) 4m2 Z over each classical momentum trajectory parametrized by the pre-assigned boundary momentum, Pµa(s), and summing over all possible values of the bulk area of the world sheet and auxiliary hmn metrics. In this fashion one eliminates any dependence on the bulk world sheet area. At the end , in order to get Ψ[σµν (C)] , one must sum over all the congruence of classical trajectories by integrating over all values of the boundary momentum and auxiliary metrics ( einbeins) induced on the boundary. To regularize ultaviolet inﬁnities requires the introduction of the curvature scalar and cosmological constant on the bulk and the extrinsic curvature and boundary cosmological constant.

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Hence, the boundary area-wavefunctional Ψ[σµν (C)] associated with the area-components enclosed by a closed loop C is nothing but the analog of the Wilson loop transform of the boundary momentum wave functional :

1 Ψ[σµν (C)] ≡ [DP (s)] exp [i P (s)dσµν (s)] Ψ[˜ P (s)]. (3.15) µa 2 µν µa Z IC and viceversa, the inverse Fourier/Loop transform :

1 Ψ[˜ P (s)] ≡ [Dσµν ] exp [−i P (s)dσµν (s)] Ψ[σµν ]. (3.16) µa 2 µν Z IC yields the boundary momentum wavefunctional Ψ[˜ Pµa(s)] in terms of the boundary area- wavefunctional Ψ[σµν (C)] with :

ν Pµa(s) ≡ Πµa|∂Σ =(Πµν ∂aX )|∂Σ. (3.17)

It is important to have the precise computation of the Ψ[˜ Pµa(s)] given by eq-(3.13) 1 in order to ﬁnd Ψ[σµν (C)] as its Fourier/functional loop transform. In general, there is an ambiguity in assigning in a unique fashion a string shape C toagivensetofarea- components ( Plucker coordinates), σµν (C). There could be two or more loop-shapes that have the same shadows or coordinates , σµν . The Plucker conditions in D =4read:

αβµν σαβ σµν =0. (3.18) to reassure us that there is a one-to-one correspondence between the loop C and its Plucker (area components) coordinates. The classical Schild action is invariant under temporal area-preserving diffeomorphisms ( it is not fully reparametrization invariant), and one must not confuse this invariance with that of spatial area-preserving diffs : invariance of (σµν )2. One would expect the invariance of the boundary wave functional, under the action of ( temporal) area-preserving diffs, to be preserved in the quantum theory. The issue of area-preserving-diffs anomalies will be the subject of future study. It is straightforward to verify that Ψ[σµν (C)] defined by eq-( 3-15 ), is a solution of the Loop Schroedinger-like Wave equation HΨ=−i(∂/∂A)Ψ once we extract the areal time independence , Ψ = e−iEAΨ.

9 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities

1 1 δ2 0 2 − µν µν ds x (s) ( 2 µν )Ψ[σ (C)] = EΨ[σ (C)]. (3.19) lC o 4m δσ δσµν Z q where the on-shell dispersion relation for the Eguchi-Schild string is :

1 1 P class(s)P µa (s) 0 2 µa class E = ds x (s) 2 . (3.20) lC 0 4m Z q with lC being the reparametrization invariant length of the loop, C. The expectation value of the Hamiltonian Operator, in the state Ψ[σµν (C)], after in- 0 serting the expression ( 3-15 ), and using the deﬁnition of δ[Pµa − Pµa], is simply :

1 1 δ2 0 2 µν ∗ µν − µν < H >Ψ= ds x (s) [Dσ (s)]Ψ [σ (C)]( 2 µν )Ψ[σ (C)] = lC 0 4m δσ δσµν Z q Z

1 1 P (s)P µa(s) 0 2 ˜ ∗ µa ˜ ds x (s) { [DPµa(s)]Ψ [Pµa(s)] 2 Ψ[Pµa(s)] } = lC 0 4m Z q Z

1 1 P (s)P µa(s) 0 2 µa ds x (s) < 2 >Ψ(P )= E. (3.21) lC 0 4m Z q where the quantity under the brackets in eq-( ) is the deﬁning expression for the averages µa Pµa(s)P (s) of 4m2 over the boundary momentum quantum states. As expected we recover the on-shell dispersion relation for the Eguchi-Schild string if, and only if,

P (s)P µa(s) P class(s)P µa (s) < µa > = µa class . (3.22) 4m2 Ψ(P ) 4m2

This is precisely what is expected for a free string; i.e eigenstates of the boundary momentum operator, Pˆµa(s). It is important also to impose the normalization condition :

∗ [DPµa(s)]Ψ˜ [Pµa(s)]Ψ[˜ Pµa(s)] = 1. (3.23) Z Concluding , Ψ[σµν (C)] solves the Loop Schroedinger-like wave equation and the expectation value of the Hamiltonian operator is indeed equal to those values of E which are consistent with the Eguchi-Schild string on-shell dispersion relation.

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For a membrane of spherical topology spanning a world tube S2 × R the higher-loop analog of the loop wave equation is :

1 √ 1 −δ2 d2σ g ( )Ψ[σµνρ(S2)] = EΨ[σµνρ(S2)]. (3.24) A 2.3!m3 δσµνρ(S2)δσ (S2) Z µνρ where σµνρ(S2) will be the volume components enclosed by the spherical membrane ( bubble) S2; i.e the proyection (shadows) of the volume onto the target spacetime coordinate planes. This higher-loop boundary wave equation corresponds to the quantum dynamics of a Euclidean world sheet or bubble, the sphere S2; i.e the boundary of the world tube of the membrane . The E eigenvalue has units of membrane tension : energy per unit area (mass3). The A represents the reparametrization invariant area of the bubble (sphere). The action for the membrane is in this case the generalization of the Schild action, a volume squared

1 ∂(Xµ,Xν,Xρ)2 d3σ . (3.25) l3 ∂(σ0,σ1,σ2) Z The Mechanics associated with p-branes admits the so-called Nambu-Poisson Hamil- tonian Mechanics where the ordinary Poisson bracket of two quantities {A, B} is replaced by a multi-bracket : {A, B, C....} which is essentially the volume form/Jacobian. The subject of Higher-Dimensional Loop Spaces and their algebras is a very complex one. We refer to 34 for an introduction. 4. Wavefunctionals of Loops and SU(∞) Gauge Theories In this section we shall construct loop wavefunctionals, Ψ[C] associated with a given loop and must not be confused with area-wavefunctionals of the previous section : Ψ[σµν (C)] associated with a given loop C with area-components σµν (C). Based on the known observation 28, 27 that classical “vacuum” configurations of a SU(∞) YM theory ( space time independent gauge field configurations) are given by classical solutions of the Eguchi-Schild string, after the Aµ ↔ Xµ(σ0,σ1) correspondence is made, we shall construct “vacuum” loop functionals Ψvac[γ] using the standard Wilson loop transform associated with the SU(∞) YM theory; i.e gauge theory of area-preserving diffs. By “vacuum” one means classical solutions to the SU(∞) YM equations of motion for the special case when the Aµ(xµ; σa) fields are spacetime independent :

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µν µν µν µ ν µ ν DµF = ∂µF + {Aµ,F } =0⇒{Aµ,{A ,A }} =0↔{Xµ,{X ,X }} =0. (4.1) after the gauge field/string correspondence is made one recovers the classical Schild string equations of motion. From now on we shall use the term “vacuum” to represent the classical solutions of the Schild string and must not be confused with the true vacuum of the theory; i.e the state which has true zero expectation values for all physical observables µν µν :vac= 0 and zero energy, =vac=0. The Wilson loop holonomy operator associated with a SU(∞) gauge field , Aµ(xµ,σ0,σ1) is defined :

µ µ 2 µ µ W [A ,γ]=Tr P exp [i Aµdx ]N→∞ = d σexp [i Aµdx ]=<γ|A > (4.2) Iγ Z Iγ and its complex conjugate associated with a diﬀerent contour C is :

∗ µ µ 2 µ µ µ W [A ,C]=Tr P exp [−i Aµdx ]N→∞ = d σexp [−i Aµdx ]=. IC Z IC (4.3) we choose the contours γ,C to be different for reasons which will become clear later. . For the time being we shall not be concerned with path orderings nor traces. After all, the SU(∞) gauge field is now an ordinary number and not a matrix. The trace in the SU(∞) YM case is replaced by an integral w.r.t the internal space “color” σa coordinates. There is a signature subtlety if one wishes to identify the σ internal color coordinates with the string worldsheet ones. The former live in a Euclidean internal world whereas the latter in a Minkowskian spacetime. However, if one performs a dimensional reduction of the Aµ field to one temporal dimension ( Matrix Models 5) one makes contact with a true physical membrane coordinate : Aµ(t, σ0,σ1) ↔ Xµ(t, σ0,σ1) because now we have the correct signature for the world volume of the membrane. A timelike slice of the membrane reproduces a string worldsheet whereas a spatial slice a Euclidean worldsheet. The signature subtleties have been recently studied by 46 and 45 . We turn attention to what we consider one of the important aspects in the essence of duality : q ↔ p in the standard Hamiltonian dynamics. It has been a long-sought goal

12 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities to construct a formulation of extended objects where dualities are already manifest. A formulation of all bosonic p-branes as Composite Antisymmetric Tensor Gauge theories of the volume-preserving diﬀs where S, T duality were incorporated from the start was achieved in 11. For example, in the standard canonical quantization of electromagnetism

, A,~ E~ = Foi are a canonical pair of conjugate variables. In the Schild formulation one µ has : X , Πµa as a canonical pair which have a correspondence with the SU(∞) c-number µ variables : A ,Fµν , respectively .The c-number valued ﬁeld strength is :

Fµν = ∂µAν − ∂ν Aµ + {Aµ,Aν}PB. (4.4) the Poisson brackets are taken w.r.t the internal color indices : σ0,σ1. Moyal deformations are very natural, for recent results on this we refer to 14, 15 , 10. For ( vacuum solutions ) spacetime independent ﬁeld conﬁgurations one can make direct contact with the Schild string string since the classical YM Lagrangian becomes now equivalent to the Schild one :

2 µ ν µ ν Fµν = {Aµ,Aν}PB{A ,A }PB ↔{Xµ,Xν}PB{X ,X }PB (4.5)

Spacetime independent ﬁeld conﬁgurations yield a trivial holonomy factor ( since the vector sum of all the tangents along a closed loop is zero) leaving only the internal color space area operator : W [A, γ]=Area[ d2σ] as the holonomy. It is very important to remark that if one evaluates the loop derivativeR on the Wilson loop W [Aµ,γ] in the vacuum case, one must firstly take the derivatives and afterwards set the values of Aµ = Aµ(σa) yielding :

δ ∂xν(s) W[Aµ,γ]=x0νF (xµ(s))W [Aµ,γ].x0ν(s)= . (4.6) δxµ(s) µν ∂s otherwise one will trivially get zero since the loop derivative does not aﬀect the area in color space. These area and volume operators are very important in Loop Quantum Gravity 18 especially in regards to the fact that these operators have discrete eigenvalues : spacetime is quantized in multiples of Planck areas and Planck volumes. This will not be surprising if the group of area and volume preserving diﬀs are truly relevant symmetry groups in

13 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities nature. However, it is important to realize that area in color space is not the same as area in spacetime. Strings and membranes can be interpreted as gauge theories of the area-preserving diffs algebra,36 where the effective dimensions will be D + 2, two internal dimensions ( the infinite color space) in addition to the usual ones. If one identifies the internal color directions, σ0,σ1, with an Euclidean worldsheet one makes contact with the Euclideanized string for those spacetime independent gauge field configurations of the SU(∞) gauge field. In the case that Aµ(xµ; σa) is dimensionally reduced to one temporal dimension one will then make contact with the membrane ( this time with the correct signature); i.e with M(atrix) Models. Therefore, the SU(∞) gauge field/Schild string correspondence is then :

µ 2 µ µν W [A ,γ]= d σexp [i Aµdx ] ↔ exp [i Πµν dσ ]= Z Iγ Z µν 2 µ ν exp [i Pµν (s)dσ (s) − i d σX {Πµν ,X }]. (4.7) IC Z after an integration by parts. The Eguchi-Schild string equations of motion set to zero the last term of (4-7) and one gets ( at the classical vacuum level) the correspondence :

µ 2 µν W [A ,γ]=Area[ d σ] ↔ exp [i Pµν (s)dσ (s)]. (4.8) Z I The correspondence between the loop transform of Ψ[Aµ] and that of the loop momentum boundary wavefunctional of the Eguchi-Schild string is :

Ψ[γ] ↔ Ψ[σµν (C)]

µ µ µ µν [DA ]W[A ,γ]Ψ[A ] ↔ [DPµa(s)]exp [i Pµν (s)dσ (s)]Ψ[Pµa(s)]. (4.9) Z Z IC We emphasize once more that Ψ[γ]andΨ[σµν (C)] are not thesameobjectseveninthe case that γ = C. There is a correspondence between them. Ψ[γ] should obey a suitable SU(∞)loopequationandΨ[σµν (C)] obeys the Loop Schroedinger-like Wave equation discussed in the previous section. We propose that Ψ[γ] ( for the vacuum) should obey the following Loop Wave equation 2 :

1 1 ds 1 −δ2 ( 2 µ )Ψvac[γ]=EΨvac[γ]. (4.10) l 0 x0(s)2 4m δx (s)δx (s) C Z µ p 14 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities

Notice that this Loop wave equation clearly differs from the SU(N)loopwaveequation in the literature 18 .IntheN→∞limit, the standard loop equation would require a wavefunctional of an infinite number of loops Ψ[γ1,γ2...... ] . Whereas, the Loop equation above (4.10 ) requires one loop only. It can be generalized to many loops (closed strings) when interactions are included. What allows us to avoid the problem of using an infinite number of loops is precisely when we impose the spacetime independent ( vacuum) SU(∞) YM gauge field configurations / Eguchi-Schild string correspondence. Such gauge field/string 39, 40 correspondence has been discussed by many authors in particular by 22 pertaing strings, loops, knots, quantum gravity and BF theories and also by 26 . The former authors have shown that the large N limit of QCD in D = 2 corresponds to astringtheory, although it was never established which particular string theory it actually referred to. The action for the string theory that reproduces the partition function of the large N QCD in D = 2 was never constructed. The quantum YM partition function could be matched with an infinite sum of branched string coverings ( with singularities) of a family of Riemann surfaces of arbitary genus, g to a compact Riemann surface of fixed genus G and fixed area. The latter is the compact Riemann surface where the orginal SU(∞) YM field theory lived in. Hence, we propose that the Wilson loop transform of the vacuum SU(∞)YMwave- µ functional, Ψvac[A ] → Ψvac[γ]isasolution of the Schild string Loop Wave equation µ µ described by eq-( 4-10 ). Furthermore, due to the A ↔ X correspondence, the Ψvac[γ] can also be obtained by performing the Schild string path integral over a Riemann surface of spherical topology with one boundary equal to the closed loop γ :

µ µ µ µ µ iSSchild[X ] [DA ]W[A ,γ]Ψvac[A ]=Ψvac[γ]= [DX (σ)]e . (4.11) µ ZSU(∞) ZX |∂Σ=γ the family of maps , Xµ(σ) from a Riemann surface of a disk topolgy to spacetime is restricted to obey a boundary condition : the surface must have as boundary the loop γ. The string path integral can be evaluated perturbatively ( approximately). However, one will not recapture the nonperturbative string physics in that fashion. String pertur- bative corrections are obtained as usual by summing over all surfaces of arbitrary genus. The path integral can be evaluated in different steps : we fix first the genus and area,

15 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities then sum over all areas and afterwards over all genus. Auxiliary metrics are introduced in the covariantized Schild action. The latter is on-shell equivalent to the Nambu-Dirac- Goto and Polyakov actions. A sum over all metrics is then performed and in this way the path integral is computed modulo the action of the diﬀs and Weyl group. At the end one ends up with integrals in the Moduli space Mh,1 of Riemann surfaces with h handles and one puncture ( associated with one boundary). The invariant measure in the moduli space is obtained by means of the b, c ghost contributions ( Faddev-Popov determinants). Noncritical string dimensions require the Liouville ﬁeld.

Inserting Ψvac[γ] provided by the standard Wilson loop transform , left hand side of (4-11 ), into the Loop Wave equation (4-10 ) yields for the expectation values of the YM Hamiltonian ( that corresponds to the Schild Hamiltonian) after using eq-(4-5 ) :

1 1 1 0 2 µ ∗ µ 2 2 µ µ ds x (s) 2 [DA ]Ψvac[A ](gYMFµν [x (s)]) Ψvac[A ]=Evac. (4.12) lC 0 4m Z q Z and one obtains that the vacuum expectation value of the SU(∞) YM energy density is precisely proportional to the Schild string tension! This is due to the fact E has units of energy per unit length which is a tension . We have introduced the YM coupling , gYM 2 2 −4 in the form gYMFµν to match units appropriately. The latter has units of length as it should whereas the Wilson loop should now include the factor of gYM in the exponential multiplying the Aµ. For recent work on the QCD vacuum wavefunctional and the large N ’t Hooft’s expansion in relation to strings see 38, 37.

Inserting Ψvac[γ] provided by the right hand side of ( 4-11 ) yields a solution of the

Loop Wave equation after one identiﬁes the Ψvac[γ] ( up to a normalization constant N of units L−1/2 to absorb the units coming from the Xµ integral ) with the boundary eﬀective action induced by the bulk quantum dynamics :

µ µ iSeff [x (s)] µ iSSchild[X ] Ψvac[γ]=e ≡ [DX (σ)]e . (4.13) µ ZX |∂Σ=γ In ordinary ﬁnite NSU(N) YM gauge theories in D = 4 , the Chern-Simmons wavefunctional Ψ[A]=eiSCS[A] is used to evaluate the loop transform from Ψ[A] → Ψ[γ]. In the SU(∞) case we shall use the boundary eﬀective action induced by bulk Schild string quantum dynamics.

16 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities

Plugging the above solution (4-13) , into the Loop Wave equation yields for the expectation value of the Schild Hamiltonian density :

1 2 µ µ 1 ds 1 µ 2 δ Seff [x (s)] δSeff [x (s)] 2 2 [Dx (s)]N [ µ +( µ ) ]=Evac. (4.14) l 0 x0(s)2 4m δx δx δx C Z Z µ µ δSeff [xp(s)] 2 0 The ( δxµ ) are the standard momentum squared terms ( to orderh ¯ ), kinetic terms 2 µ δ Seff [x (s)] of the Hamilton-Jacobi equation and the µ are the so-called “quantum potential δx δxµ “terms(toorder¯h). We can see that the units match appropriately.

Given a Ψvac[γ] associated with the SU(∞) YM theory that obeys the Loop wave equation, the next question to ask : is what happens for states other than the vacuum ? Clearly one cannot hope to use the Eguchi-Schild correspondence any longer. We used, originally, the Schild string as a guiding principle but one should not expect the Schild string to be the actual theory behind the full fledged SU(∞) YM theory, especially in D ≥ 2. At the classical level there was a simple correspondence between the Schild vacua andthosevacuaoftheSU(∞) YM theory ( space time independent field configurations). However, this is a very restricted case. The question is again : what string theory, if any, can one use to construct wavefunctionals other than the vacuum in any dimension ? 29 Since W∞ is an area-preserving diffs algebra , containing the Virasoro algebra, which appears naturally in the physics of membranes, we believe that the sort of string theory one is looking for could be a W∞ string theory; i.e an Extended Higher Conformal

Spin Field Theory representing W∞ gravity coupled to W∞ Conformal Matter plus an infinite tower of Liouville fields and ghosts in the noncritical W∞ string case . It has been formally shown in 11 that D =27/11 are the appropriate target spacetime dimensions for the bosonic/supersymmetric non-critical W∞ string, if the theory is devoid of BRST anomalies. It is interesting to see that these are precisely the membrane/supermembrane spacetime dimensions that are devoid of Lorentz anomalies 48 . The self dual sector of the membrane/supermembrane spectrum contains non-critical W∞ strings/superstrings. Higher Spin Gauge theories have been revisited very recently by 8, 9 in connection to Higher Spin Gauge interactions of Massive fields in Anti-de-Sitter space in D =3and N= 8 Higher Spin Supergravities. It has been argued that Higher Spin Gauge theories may actually be the underlying theory behind the bulk dynamics of Anti-de-Sitter space

17 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities in any dimensions. For a review of the historical precedents of Maldacena’s conjecture 16 we refer to Duﬀ 17. In the case that the boundary loops do not coincide, γ is not equal to C, we propose the following vacuum loop wavefunctional as the solution to the Loop Wave equation. In 1 we constructed the area-wavefunctional Ψ[σµν (γ)] directly from the computation of covariantized Schild string phase space path integral. The latter is not the same as the Ψ[γ] deﬁned as :

0 µ µ µ 0 Ψvac[γ]=<γ|ΨΣ >= [DA ] <γ|A >< A |ΨΣ >= Z ∞ µ µ µ µ µ iEA(γ,C) µ [DA ]W[A ,γ] [Dx (s)] dAe = Z Z Z0 ∞ µ µ µ ∗ µ iEA(γ,C) µ [DA ]W[A ,γ] [Dx (s)]W [A ,C] dAe ΨΣ[x (s)]. (4.15) Z Z Z0 where : A = A(γ,C) is the proper area whose boundaries are γ,C; i.e. Areal time spanned 0 iHAˆ iEA between γ,C. We have used the relation |ΨΣ >= e |ΨΣ >= e |ΨΣ > since the Schild

Hamiltonian operator is the evolution operator of the initial quantum eigenstate |ΨΣ > 0 ( assigned to the boundary C) , to the ﬁnal quantum eigenstate |ΨΣ > assigned to the boundary γ, in a given areal time : A(γ,C). At the end one must sum over all possible values of the proper area spanning between γ and C. The wavefunctional assigned to the C boundary is once again :

µ µ µ µ iSeff [x (s)] ΨΣ[x (s)] ≡ [DX (σ)]exp [iSSchild[X (σ)]] = e . (4.16) µ Zx (s) The path integral of the bulk ﬁeld theory is evaluated on all the open surfaces that have the µ µ µ loop C as the boundary : X (σ)|∂Σ = x (s). The wavefunctional ΨΣ[x (s)] agrees exactly with the one described by the operator formalism of Riemann surfaces 35 . The Schild µ action functional, SSchild[X ], will reassure us that Ψvac[γ] will indeed be a solution of the µ Loop wave equation (4-10 ) if, and only if, ΨΣ[x (s)] obeys the Loop Wave equation as it should, since it was constructed by performing the Schild-string path integral. Therefore, the two wavefunctionals must be related through the Schild-string kernel :

∞ µ µ Ψvac[γ]= [Dx (s)] dA KSchild[C, γ, A(γ,C)]ΨΣ[x (s)]. (4.17) Z Z0

18 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities with A[γ,C] being the proper area ( areal time) spanned between the initial loop C and the ﬁnal loop γ. In the liming case that the two loops coincide ( A = 0) the Schild loop propagator turns into a delta functional δ[γ − C] and one gets from eqs- (4-11, 4-17 ) :

µ µ µ µ ΨΣ[x (s)] = Ψvac[γ]= [DA ]W[A ,C]Ψvac[A ](4.18) Z which is indeed the loop transform of the vacuum wavefunctional of the SU(∞)YM theory. And the latter, in turn, is indirectly given in terms of the bulk ﬁeld theory by using, precisely, the inverse loop transform of the boundary wavefunctional associated with the conformal ﬁeld theory living on the bulk; i.e the Riemann surface ( string world sheet) with boundary γ = C :

µ µ ∗ µ Ψvac[A ]= [Dx (s)]W [A ,γ]Ψvac[γ]. (4.19) Z µ Concluding : It is precisely when one sets Ψvac[γ]=ΨΣ[x (s)=C], that we have the following string (bulk)/SU(∞) gauge ﬁeld (boundary) duality for the vacuum wavefunctionals:

µ µ µ µ µ Ψvac[γ]= [DA ]W[A ,γ]Ψvac[A ]= [DX (σ)]exp [iSSchild[X (σ)]]. (4.20) µ Z Zx (s)=γ the left hand side is just the Wilson loop transform of the SU(∞) YM vacuum wavefunctional whereas the right hand side is the induced boundary effective action defined above by performing the path integral of the Schild string bulk field theory with the standard µ µ boundary restriction on the string embeddings : X |∂Σ = x (s)=γ. For states other than the vacuum we postulate that :

µ µ µ µ µ Ψ[γ]= [DA ]W[A ,γ]Ψ[A ]= [DX (σ)]exp [iSbulk[X (σ)]]. (4.21) µ Z Zx (s)=γ µ The crucial question is once more : What is the theory behind SBulk[X ] in the more general case besides the vacuum ? Is it a Higher Spin Gauge Theory; i.e a W∞ gauge theory ? A W∞ string theory ? As stated earlier, we used, originally, the Schild string as a guiding principle but one should not expect the Schild string to be the actual theory

19 Castro : M-Theory, Loop Quantum Mechanics, Loops/Strings and Bulk-Boundary Dualities

µ behind the action functional SBulk[X ]. We believe that W∞ geometry, higher spin gauge µ 30 31 theories , could be the theory behind the SBulk[X ]. W∞ algebras and their Moyal , and q deformations are the natural candidates to construct the Quantum Gauge Theories of the area-preserving diffs group. To conclude, string/gauge field duality is tantamount of a bulk/boundary duality for vacuum SU(∞) gauge field configurations when γ = C. In the more general case that γ dose not coincide with C the Schild loop propagator K[C, γ, A] is the kernel operator that evolves the initial state ΨΣ[C], in a given areal time A(γ,C) between the two loops γ,C, to a new quantum state Ψvac[γ]. This is equivalent to attaching a plumbing fixture or throat , typical of Closed String Field Theory 44, to the original Riemann surface ( disk topology ) at the boundary C extending it all the way to the final boundary γ.Andthen, propagating ( sliding ) the intial loop C along the throat all the way to the final loop γ. As it propagates, the shape of the intitial loop C changes ( “elastic” ) to match γ at the other end. And, finally, one performs the sum over all values of areal time ( area) A as shownin(4-15).

This is is consitent with the fact that if ΨΣ[C], Ψ[γ] are both solutions of the loop wave equation, then the standard operator form of the Loop wave equation gives :