A Polyakov fermionic string as a quantum state of

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Dec 18, 1997 - connection solves the Wheeler-De. Witt equation of Einstein gravitation theory written in terms of the Ashtekar-Sen variables. 0 1997 Elsevier ...
18 December 1997 PHYSICS

ELSEVIER

LETTERS

B

Physics Letters B 415 (1997) 231-235

A Polyakov fermionic string as a quantum state of Einstein theory of gravitation Luiz CL. Botelho Uniuersidade Federal do Park Departamento de Fisica, Campus lJniuersit&io do Guam& 66.075-900 Bel&m-Par& Brazil Received 29 January 1997; revised 14 April 1997 Editor: H. Georgi

Abstract We show that a Polyakov ferrnionic string functional integral in the presence of the external spatial Ashtekar-Sen connection solves the Wheeler-De Witt equation of Einstein gravitation theory written in terms of the Ashtekar-Sen variables. 0 1997 Elsevier Science B.V.

In recent years a new quantization of Einstein gravitation theory has been pursued by several authors, which seems appropriate for writing explicit solutions of the Wheeler-De Witt equation. It makes use of the so called SU(2)-Ashtekat-Sen connection as dynamical variable (see Refs. [1,2]) which has the geometrical meaning of being the projected spin connection on the space-time (three-dimensional) boundary [3]. A linear wave equation for this new quantum gravity dynamical variable was derived which supports a Wilson Loop solution [2]. In this letter, following our previous studies in this subject [4], we consider a new solution for the above mentioned equation defined by a Polyakov fermionic string functional integral [S]. Let us start our analysis by considering the following Polyakov string functional integral in the presence of a SU(2) connection All]=/W&AB],“p[ l),ij]P~[ X”]

i i=

0370.2693/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. PII SO370-2693(97)01153-2

[ qJ~(o,o)~;(o,27r)] 1

232

L.C.L. Botelho/

Physics Letters B 415 (19971 231-235

tI I

X

(,a)

(1)

The open string surface { XF( [,a),~ = 1,2,3} is immersed in the space-time (three-dimensional) boundary and does not possess holes and handles. The string surface parameter domain is taken to be the half-strip quanR&, = {( 5. IT), - x I 5 I + =;O < CT5 297) without loss of generality. The Polyakov two-dimensional tum ravity (string) metric is denoted by { gAB([,~)} and satisfies the trivial topological condition j?“,d~J~“dv n-. The two-dimensional intrinsic fermions Dirac fields belong to a complex SU(2) x( f gNg)Xt,o-)=2 fundamental representation and are denoted by {$i( (,a); $i( 5.a)) with the subscript A associated to the two-dimensional string (Euclidean) Lorentz Group SO(2) and the superscript i associated to the SU(2) group index. The interaction of the Polyakov string and the SU(2) three-dimensional Ashtekar-Sen connection is given by the explicit interaction of the SU(2) connection flux and the intrinsic fermion current as in the SU(2) QCD gauge theory. The functional measures in the Polyakov string functional integral are the well-known De-Witt covariant functional measures with boundary terms (we take the string boundary X,)

S(5-5’)S(a-a’)S~“(E)

(44

233

L.C.L. Botelho / Physics Letters B 415 (19971231-235

with h = det( h,,) h,,

= (~AX%X,)(

(4b)

C?(+)

where 6,“‘(O) is a regularized form of the singular term S”‘(O) (see Ref. [5] for details). The evaluation of the fermionic functional integral average in Eq. (3) is straightforward, since in two-dimensional QCD @U(2)) one can use the Roskies gauge decoupling fermion gauge [8,9] and thus, the ultra-violet limit implied by Eq. (4) leads that the average of the fermion currents in Eq. (3) is effectively defined by Fermion free fields (asymptotic freedom). It yields terms of the form

y&wY

‘La

where the UV regularized

(5)

form of the fermion propagator

SYB(((,a);( &7~))(“)=

used to obtain Eq. (5) is given by

i(Y1)(5-t’+a)+i(y2)(u-u’+a)gij

(6)

(t-~‘+a)2+(a-ua’+a)2

By absorbing the two-dimensional UV infinity a --z 0 in the bare model coupling constant e’s,“)(O), we can follow the argument of Refs. [2,4] to conclude that Eq. (5) vanishes identically as a consequence of being a contraction of the antisymmetric ( ~,zJ) tensor F,,( A(X*( (,cr))) and the ( p,v) symmetric tensor in front of the above mentioned tensor in Eq. (5). It is worth pointing out that we have used the Polyakov conformal gauge g,,(

[,a)

= eq(:.0)6AB

(7)

in the above calculations in order to factorize the metric field dependence of the fermionic propagator under analysis. Another important observation to be made is that proposed Polyakov string quantum gravity state Eq. (11)) contains the usual Wilson loop quantum gravity state celebrated in the literature [l-3] as a simple overall factor. In order to show this claim it is enough to integrate the fermions fields in the string path integral to obtain the result

(8) where the two-dimensional Eq. (8) is given explicitly %[X”(

&a)1

QCD external SU(2) gauge field entering in the fermion by the 2D surface induced SU(2) gauge field

= (A:(

X”( &o))h,)(d,XJY

5,~))

functional

determinant

in

(9)

Note that the appearance of the Wilson loop functional in Eq. (8) is nothing more than the (boundary) fermion propagator associated to our fermion boundary current in Eq. (1) projected on the spatial loop 1,(o) =x~(o,~)(x~(o,o) =xP) Let us comment the results presented in this letter differ significantly from those of Ref. [4] since here we have not considered the theory of self-avoiding string neither the restrictive Ashtekar-Sen connection boundary condition (J~Fp”(Axx) = 0); both conditions necessary to obtain the validity of the results presented in this reference.

234

L.C.L. Botrlho/

Physics Letters B 415 (19971 231-235

At this point of our letter, the question of physical observables suitable to our string quantum gravity state Eq. (1) should be considered. We start our discussion on this very important question by calling attention that it remains a open problem to understand canonical quantum gravity in light of the Copenhagen school interpretation of quantum mechanics. In the Wheeler-De Witt (canonical) framework, there is no time parameter in the associated quantum gravity Schriidinger equation (the well-known Wheeler-De Witt equation (Eq. (2))). As a consequence, the operation of taking quantum system averages makes no sense physically for the observer. Note that there are no bound-states, currents, energy observables. etc. in the canonical Wheeler-De Witt quantum gravity framework. There is only, in principle, the “vacuum” state of the 3D geometry satisfying the homogeneous Wheeler-De Witt equation and that was the main reason for the search of new field parametrization in Einstein quantum gravitation theory. We remark that among these frameworks the Ashtekar-Sen parametrization is the most promising scheme devised until now, since it leads to a mapping of the 3D metric field to the well studied SU(2) gauge theory (the old Faraday line interpretation for fields) and making, thus, the original non-linear Wheeler-De Witt equation a linear wave equation in terms of these new variables [l-3]. However, some geometrical (non-physical) objects have been studied [9] and leading to the result that the Ashtekar-Sen-Smoling Wilson Loop associated to smooth loops are eigenstates of these geometrically operators with eigenvalues given by the entanglement index of these infinitely differentiable loops with the smooth surface and smooth volume which are fixed by an (unphysical) observer measuring area and volume in the 3D geometry. We emphasize the use of the word unphysical for this observer, since there is no clear space-time diffeomorphism invariance in these geometricals observations. Following these attempts to evaluate formal observables in order to get a better insight in this very difficult problem, we remark that our string quantum state may be useful to evaluate a kind of spatial gravitation propagator given by the following quantum state average (see appendix of Ref. [51)

This object has a formal meaning of describing the process of a “spatial graviton” propagation from the pure vacuum state (nothing) to our proposed string state equation (Eq. (1)) defined by the Ashtekar-Sen connection A,(x) and loop 1,(a ). Following the Copenhagen School interpretation. we substitute the metric operations below [2]

(6gg” 1(x)

=62,‘6A$+!iA;(x)

(11)

inside the Polyakov string path integral representing the non-trivial quantum state in Eq. (10). As a consequence, we can easily write the “3D graviton propagator” as a two-point Polyakov string scattering amplitude associated to our proposed string theory Eq. (I ). Studies of the possible relevance of these scattering amplitudes for quantum gravity will appear elsewhere. Finally the argument that another surface solution with a topology of a cylinder may be obtained by simply taking the Wilson loop of the Ashtekar-Sen connection along a one parameter family of closed loops in the spatial manifold, and integrate the resulting one-parameter family of numbers over the parameter is clearly wrong since this object is not defined as a functional over the surface vector position {X,( 5,~ )} and, thus, losing all meaning of a functional of the cylinder surface. The above cited construction is nothing more than a superposition of the Wilson loop solutions which still satisfies the Wheeler-De Witt equation written in terms of Ashtekar-Sen variables, since this Schrijdinger quantum gravity equation is linear in this SU(2) gauge field

L.C.L. Botelho/

Physics Letters B 415 (1997) 231-235

235

parametrization. As a consequence of these remarks, this kind of superposition loop solutions do not bring new features besides those already studied in Ref. [9]. Note that our proposed solution being a string theory opens the possibility of using all machinery of 2D-quantum field models [lo,1 I] to understand four-dimensional Einstein quantum gravity. This research was partially

supported by CNPq-Brazil

References [ll A. Ashtekar, in: Gravitation and Quantization, Les Houches, France. Vol. 57, Elsevier, Amsterdam, [2] T. Jacobson, L. Smoling, Nucl. Phys. B 299 (1988) 295. [31 R. Cappovilla, T. Jacobson. J. Dell, Phys. Rev. Lett. 63 (1989) 2325. [4] L.C.L. Botelho, Phys. Rev. D 52 (4 September 1996) 6941. 151 L.C.L. Botelho, J. Math. Phys. 30 (1989) 2160. 161 B. Durhuus, P. Olesen. J.L. Pertersen, Nucl. Phys. B 198 (1982) 157. [7] L.C.L. Botelho, Phys. Lett. B 152 (1985) 358. 181R. Roskies, in: Festschrift for Feza Gursey’s, 60th birthday. 1982. unpublished. 191 R. De Pietri, C. Rove& Phys. Rev. 54 D (1996) 2664. 1101 A.M. Polyakov, Gauge Field and Strings, Harvard Academic Publishers, 1987. [I 11 L.C.L. Botelho, Phys. Rev. D 35 (1987) 1515.

1992.