Weaving a classical geometry with quantum threads

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Weaving a classical geometry with quantum threads

Abhay Ashtekar†, Carlo Rovelli‡, and Lee Smolin†

† Department of Physics, Syracuse University, Syracuse, NY 13244-1130, U.S.A.

‡ Department of Physics, University of Pittsburgh, Pittsburgh, PA 15260; and Dipartimento di Fisica, Universita di Trento, Italia

Abstract

Results that illuminate the physical interpretation of states of nonperturbative quantum gravity are obtained using the recently introduced loop variables. It is shown that: i) While local operators such as the metric at a point may not be well-defined, there do exist non-local operators, such as the area of a given 2-surface, which can be regulated diffeomorphism invariantly and which are finite without renormalization; ii) there exist quantum states which approximate a given flat geometry at large scales, but such states exhibit a discrete structure at the Planck scale.

It is by now generally accepted that perturbative approaches to quantum gravity fail because they assume that space-time geometry can be approximated by a smooth continuum at all scales. What is needed are non-perturbative approaches which can predict –rather than assume– what the true nature of the micro-structure of this geometry is. In such an approach, background fields such as a classical metric or a connection cannot play a fundamental role; quantum theory must be formulated in a diffeomorphism invariant fashion. An important task in these programs is then to introduce techniques needed to describe geometry and to “explain” from first principles how smooth geometries can arise on macroscopic scales.

Over the past five years, two avenues have been pursued to test if quantum general relativity can exist non-perturbatively: the first is based on numerical simulations, while the second is based on canonical quantization. This letter concerns the second approach. While the canonical approach itself was introduced by Dirac in the late fifties, the recent work departs from the early treatment in two important ways: i) it is based on a new canonically conjugate pair, the configuration variable being a connection; and, ii) it uses a new representation in which quantum states arise as suitable functions on the space of closed loops on a (spatial) 3-manifold. The new ingredients have led to technical as well as conceptual simplifications which, in turn, have led to a variety of new results. In particular, these methods have opened up new bridges between quantum gravity and other areas in mathematics and physics such as knot theory, Chern-Simons theory, and Yang-Mills theory.

The purpose of this letter is to report on the picture of quantum geometry that arises from the use of the loop variables. To explore geometry non-perturbatively, we must first introduce operators that carry the metric information and regulate them in such a way that the final operators do not depend on any background structure introduced in the regularization. We will show that such operators do exist and that they are finite without renormalization. (Furthermore, some of the operators are invariant under 3-dimensional diffeomorphisms and have a well-defined action on diffeomorphism invariant states which, in the loop representation, depend only on the knot class of the loop.) Using these operators, we seek non-perturbative states which can approximate any given flat classical geometry up terms $𝑂({𝑙}_{𝑝}\mathrm{/}𝐿)$ where ${𝑙}_{𝑃}$ is the Planck length and $𝐿$ is a macroscopic length scale, lengths being defined by the given geometry. We find that such states do exist but that they exhibit a discrete structure at the Planck scale ${𝑙}_{𝑃}$. Such a result was anticipated on general grounds since the 30’s. Indeed, there exist a number of quantum gravity programs that begin by postulating discrete structures at the Planck scale and then attempt to recover from it the known macroscopic physics. The key difference is that, in our approach, discreteness is the output of the framework rather than input. The input is only general relativity and quantum mechanics.

In this letter, we will only sketch the main ideas involved; details are described in referenced works.

Introduction

Let us begin with the classical phase space. The configuration variable ${𝐴}_{𝑖}^{𝑎}$ is a complex, SU(2)-connection and its conjugate momentum, ${\widetilde{𝐸}}_{𝑖}^{𝑎}$ –the mathematical analog of the electric field in Yang-Mills theory– is a triad with density weight one. (Throughout we will let $𝑎,𝑏,\mathrm{.}\mathrm{.}\mathrm{.}$ denote the spatial indices and $𝑖,𝑗,\mathrm{.}\mathrm{.}\mathrm{.},$ the internal indices. A tilde over a letter will denote a density weight 1.) The first step is the introduction of loop variables which are manifestly SU(2)-gauge invariant functions on the phase space. The configuration variables are the Wilson loops: Given a closed loop $𝛾$ on the 3-manifold $\mathrm{\Sigma }$, we set

$𝑇[𝛾]=\frac{1}{2}\text{TrP exp}\left(𝐺{\oint }_{𝛾}{𝐴}_{𝑎}𝑑{𝑙}^{𝑎}\right),$

where $𝐺$ is Newton’s constant. (Throughout, we use the 2-dimensional representation of the gauge group to evaluate traces.) Variables with momentum dependence are constructed by inserting ${𝐸}_{𝑖}^{𝑎}$ at various points on the loop before taking the trace. Thus, for example, the loop variable quadratic in momenta is given by:

${𝑇}^{𝑎{𝑎}^{\mathrm{\prime }}}[𝛾](𝑦,{𝑦}^{\mathrm{\prime }})=\text{Tr}[\left(\text{P exp}\left(𝐺{\int }_{𝑦}^{{𝑦}^{\mathrm{\prime }}}{𝐴}_{𝑎}𝑑{𝑙}^{𝑎}\right)\right){𝐸}^{𝑎}(𝑦)\left(\text{P exp}\left(𝐺{\int }_{{𝑦}^{\mathrm{\prime }}}^{𝑦}{𝐴}_{𝑎}𝑑{𝑙}^{𝑎}\right)\right){𝐸}^{{𝑎}^{\mathrm{\prime }}}({𝑦}^{\mathrm{\prime }})],$

where $𝑦$ and ${𝑦}^{\mathrm{\prime }}$ are any two points on the loop $𝛾$. Note that in the limit when the loop $𝛾$ shrinks to a single point $𝑥$, ${𝑇}^{𝑎{𝑎}^{\mathrm{\prime }}}[𝛾]$ tends to ${𝐸}_{𝑖}^{𝑎}(𝑥){𝐸}_{𝑖}^{{𝑎}^{\mathrm{\prime }}}(𝑥)$, which, when ${\widetilde{𝐸}}_{𝑖}^{𝑎}$ is invertible, is related to the metric ${𝑞}_{𝑎𝑏}$ by $\text{det}𝑞(𝑥){𝑞}^{𝑎{𝑎}^{\mathrm{\prime }}}(𝑥)={𝐸}_{𝑖}^{𝑎}(𝑥){𝐸}_{𝑖}^{{𝑎}^{\mathrm{\prime }}}(𝑥)$. Thus, one can recover the metric from the loop variable ${𝑇}^{𝑎{𝑎}^{\mathrm{\prime }}}$.

In quantum theory, states are represented by suitably regular functions $\mathrm{\Psi }[𝛼]$ of loops satisfying certain algebraic conditions and quantum operators corresponding to the loop variables are defined in such a way that the Poisson algebra of the loop variables is mirrored in the commutator algebra in the usual fashion. For example, the action of the loop operator $\widehat{𝑇}[𝛾]$ is given by:

$\widehat{𝑇}[𝛾]\circ \mathrm{\Psi }[𝛼]=\mathrm{\Psi }[𝛼\mathrm{\#}𝛾]+\mathrm{\Psi }[𝛼\mathrm{\#}{𝛾}^{-1}],$

where $𝛼\mathrm{\#}𝛾$ is an "eye glass loop" which is equal to $𝛼\circ 𝜂\circ 𝛾\circ {𝜂}^{-1}$ for an arbitrary segment $𝜂$ joining $𝛼$and $𝛾$ and the same segment is used in both terms. Similarly, the action of higher-order loop operators such as ${\widehat{𝑇}}^{𝑎{𝑎}^{\mathrm{\prime }}}(𝑦,{𝑦}^{\mathrm{\prime }})$ simply involves gluing, breaking, and re-routing of loops.

It is tempting to try to define the local metric operator as the limit of ${\widehat{𝑇}}^{𝑎𝑏}$. However, the resulting operator has to be regulated and then renormalized–it involves products of ${\widetilde{𝐸}}_{𝑖}^{𝑎}$ and ${\widetilde{𝐸}}_{𝑖}^{𝑏}$ evaluated at the same point–and, because of the density weights involved, the renormalized operator carries an imprint of the background structure used in this procedure. This is because the renormalization procedure changes the density weight as it must replace a product of delta functions by a single delta function and, on a manifold, delta functions are densities. (In Minkowskian field theories, there is a preferred background metric and the only ambiguity in defining analogous operators is that of a multiplicative renormalization constant.) This appears to be a general feature of diffeomorphism invariant theories and it obstructs the introduction of meaningful local operator-valued distributions carrying geometric information.

Fortunately, however, there do exist non-local operators carrying the same information. We now sketch the construction of two of these.

Non-local Operators

Note first that, given a smooth 1-form ${𝜔}_{𝑎}$ on $\mathrm{\Sigma }$, we can define a function $𝑄[𝜔]$ on the classical phase space which carries the metric information:

$𝑄[𝜔]:={\int }_{\mathrm{\Sigma }}{𝑑}^3𝑥({\widetilde{𝐸}}_{𝑖}^{𝑎}{𝜔}_{𝑎}{\widetilde{𝐸}}_{𝑖}^{{𝑎}^{\mathrm{\prime }}}{𝜔}_{{𝑎}^{\mathrm{\prime }}}{)}^{\frac{1}{2}},$

where the integral on the right is well-defined because the integrand is a density of weight 1. When the triads are smooth, we can reconstruct the density-weighted inverse metric from the knowledge of $𝑄[𝜔]$ (for all $𝜔$). In terms of the classical loop variable ${𝑇}^{𝑎{𝑎}^{\mathrm{\prime }}}$, this function can be re-expressed as:

$𝑄[𝜔]={\lim }_{𝜖\to 0}\int {𝑑}^3𝑥{(\int {𝑑}^3𝑦\int {𝑑}^3{𝑦}^{\mathrm{\prime }}{𝑓}_{𝜖}(𝑥,𝑦){𝑓}_{𝜖}(𝑥,{𝑦}^{\mathrm{\prime }}){𝑇}^{𝑎{𝑎}^{\mathrm{\prime }}}[{𝛾}_{𝑦,{𝑦}^{\mathrm{\prime }}}](𝑦,{𝑦}^{\mathrm{\prime }}){𝜔}_{𝑎}(𝑦){𝜔}_{{𝑎}^{\mathrm{\prime }}}({𝑦}^{\mathrm{\prime }}))}^{\frac{1}{2}},$

where ${𝑓}_{𝜖}(𝑥,𝑦)$ is a smearing function, a density of weight 1 in $𝑥$, which tends to ${𝛿}^3(𝑥,𝑦)$ as $𝜖$ tends to zero and where ${𝛾}_{𝑦,{𝑦}^{\mathrm{\prime }}}$ is an arbitrarily defined smooth loop that passes through points $𝑦$ and ${𝑦}^{\mathrm{\prime }}$, such that it goes smoothly to a point as ${𝑦}^{\mathrm{\prime }}\to 𝑦$. Expression of $𝑄[𝜔]$ is well-suited for translation to quantum theory: We can define the quantum operator $\widehat{𝑄}[𝜔]$ simply by replacing ${𝑇}^{𝑎{𝑎}^{\mathrm{\prime }}}$ in with the loop operator ${\widehat{𝑇}}^{𝑎{𝑎}^{\mathrm{\prime }}}$ and taking the limit $𝜖\to 0$ in the action of the operator on states. The resulting operator is well-defined – it carries no memory of the additional structure used in the construction of the smearing functions – and finite without any renormalization. The resulting action of the operator on states is quite simple, if $𝛼$ is a nonintersecting loop it is

$\widehat{𝑄}[𝜔]\circ \mathrm{\Psi }[𝛼]=(\sqrt{6}{𝑙}_{𝑃}^2{\oint }_{𝛼}𝑑𝑠\mathrm{\mid }{𝛼}^{𝑎}{𝜔}_{𝑎}(𝛼(𝑠))\mathrm{\mid })\mathrm{\Psi }(𝛼)\mathrm{.}$

Thus, the operator acts simply by multiplication. Hence the loop representation is well-suited to find states in which the 3-geometry –rather than its time evolution– is sharp.

The second class of operators corresponds to the area of 2-surfaces. Note first that, given a smooth 2-surface $𝑆$ in $\mathrm{\Sigma }$, its area ${𝐴}_{𝑆}$ is a function on the classical phase space. We first express it using the classical loop variables. Let us divide the surface $𝑆$ into a large number $𝑁$ of area elements ${𝑆}_{𝐼}$, $𝐼=1,\mathrm{2...}𝑁$, and set ${𝐴}_{𝐼}^{𝑎𝑝𝑝𝑟}$ to be

${𝐴}_{𝐼}^{𝑎𝑝𝑝𝑟}={({\int }_{{𝑆}_{𝐼}}{𝑑}^2{𝑆}_{𝑏𝑐}(𝑥){𝜖}^{𝑎𝑏𝑐}{\int }_{{𝑆}_{𝐼}}{𝑑}^2{𝑆}_{{𝑏}^{\mathrm{\prime }}{𝑐}^{\mathrm{\prime }}}^{\mathrm{\prime }}({𝑥}^{\mathrm{\prime }}){𝜖}^{{𝑎}^{\mathrm{\prime }}{𝑏}^{\mathrm{\prime }}{𝑐}^{\mathrm{\prime }}}{𝑇}^{𝑎{𝑎}^{\mathrm{\prime }}}[{𝛾}_{𝑥,{𝑥}^{\mathrm{\prime }}}](𝑥,{𝑥}^{\mathrm{\prime }}))}^{\frac{1}{2}},$

where ${𝜖}^{𝑎𝑏𝑐}$ is the (metric independent) Levi-Civita density of weight -1. ${𝐴}_{𝐼}^{𝑎𝑝𝑝𝑟}$ approximates the area function on the phase space defined by the surface elements ${𝑆}_{𝐼}$, the approximation becoming better as ${𝑆}_{𝐼}$ – and hence loops ${𝛾}_{𝑥,{𝑥}^{\mathrm{\prime }}}$ – shrink. Therefore, the total area ${𝐴}_{𝑆}$ associated with $𝑆$ is given by

${𝐴}_{𝑆}={\lim }_{𝑁\to \mathrm{\infty }}{𝐴}_{𝐼}^{𝑎𝑝𝑝𝑟}\mathrm{.}$

To obtain the quantum operator ${\widehat{𝐴}}_{𝑆}$, we simply replace ${𝑇}^{𝑎{𝑎}^{\mathrm{\prime }}}$ by the quantum loop operator ${\widehat{𝑇}}^{𝑎{𝑎}^{\mathrm{\prime }}}$. This somewhat indirect procedure is necessary because there is no well-defined operator that represents the metric or its area element at a point. Again, the operator ${\widehat{𝐴}}_{𝑆}$ is finite and its action is simple when evaluated on a nonintersecting loop $𝛼$:

${\widehat{𝐴}}_{𝑆}\circ \mathrm{\Psi }[𝛼]=\sqrt{6}{𝑙}_{𝑝}^2𝐼(𝑆,𝛼)\mathrm{\Psi }[𝛼],$

where $𝐼(𝑆,𝛼)$ is simply the unoriented intersection number between the 2-surface $𝑆$ and the loop $𝛼$. Thus, in essence, a loop $𝛼$ contribute one Planck unit of area to any surface it intersects. The area operator also acts simply by multiplication in the loop representation.

Because of the simple form of operators $\widehat{𝑄}[𝜔]$ and ${\widehat{𝐴}}_{𝑆}$, a large set of simultaneous eigenstates can be immediately constructed. There is one associated to every nonintersecting loop $𝛾$, which we will label ${\mathrm{\Psi }}_{𝛾}[𝛼]$ and call the characteristic state of $𝛾$. It is equal to one when evaluated on $𝛾$ and zero when $𝛼$ is any other nonintersecting loop. Its value on intersecting loops may be found by an explicit computation. We may note that the corresponding eigenvalues of area are then quantized in integer multiples of $\sqrt{6}{𝑙}_{𝑃}^2$. There are also eigenstates associated with intersecting loops; these are discussed in other referenced works.

Weave States

Let us now turn to the second of our main results. The goal here is to introduce loop states which approximate a given flat 3-metric ${ℎ}_{𝑎𝑏}$ on $\mathrm{\Sigma }$ on scales $𝐿$ large compared to ${𝑙}_{𝑝}$. (Note that the large scale limit is equivalent to the semi-classical limit since, in source-free, non-perturbative quantum general relativity, $\mathrm{\hslash }$ and $𝐺$ always occur in the combination $\mathrm{\hslash }𝐺={𝑙}_{𝑝}^2$.) The basic idea is to weave the classical metric out of quantum loops by spacing them so that (on average) precisely one line crosses every surface element whose area, as measured by the given ${ℎ}_{𝑎𝑏}$, is one Planck unit. Such loop states will be called weaves. Given a weave, one can obtain others by, e.g., adding small quantum fluctuations. We now present a concrete example of such a state.

Using the given flat metric ${ℎ}_{𝑎𝑏}$, fix a cubical lattice on $\mathrm{\Sigma }$ with lattice spacing $𝑎$. At each lattice site $\mathbf{𝑛}$ we center a circle ${𝛾}_{\mathbf{𝑛}}$ of radius $𝑎$ and random orientation (where ${ℎ}_{𝑎𝑏}$ is again used in the construction, and we require also that the loops be, for simplicity, nonintersecting.) Denote the collection of these circles by $\mathrm{\Delta }$.

The state ${\mathrm{\Psi }}_{\mathrm{\Delta }}(𝛼)$ is, as we now indicate, a weave state with the required properties. To see if it reproduces on a scale $𝐿>>{𝑙}_{𝑝}$ the geometry determined by the classical metric ${ℎ}_{𝑎𝑏}$, let us introduce a 1-form ${𝜔}_{𝑎}$ which is slowly varying on the scale $𝐿$ and compare the value $𝑄[𝜔](ℎ)$ of the classical $𝑄[𝜔]$ evaluated at the metric ${ℎ}_{𝑎𝑏}$, with the action of the quantum operator $\widehat{𝑄}[𝜔]$ on ${\mathrm{\Psi }}_{\mathrm{\Delta }}[𝛼]$. A detailed calculation yields:

$\widehat{𝑄}[𝜔]\circ {\mathrm{\Psi }}_{\mathrm{\Delta }}[𝛼]=(\frac{2𝜋\sqrt{6}}{{𝑙}_{𝑝}}{\left(\frac{{𝑙}_{𝑝}}{𝑎}\right)}^2𝑄[𝑤](ℎ)+𝑂\left(\frac{𝑎}{𝐿}\right)){\mathrm{\Psi }}_{\mathrm{\Delta }}[𝛼]\mathrm{.}$

Thus, ${\mathrm{\Psi }}_{\mathrm{\Delta }}$ is an eigenstate of $\widehat{𝑄}[𝜔]$ and the corresponding eigenvalue is closely related to $𝑄[𝜔](ℎ)$. However, even to the leading order, the two are unequal unless the lattice separation $𝑎$ equals $(2𝜋\sqrt{6}{)}^{1\mathrm{/}2}{𝑙}_{𝑃}$.

The situation is the same for the area operators ${\widehat{𝐴}}_{𝑆}$. Let $𝑆$ be a 2-surface whose extrinsic curvature varies slowly on a scale $𝐿>>{𝑙}_{𝑃}$. The state ${\mathrm{\Psi }}_{\mathrm{\Delta }}$ is an eigenvector of ${\widehat{𝐴}}_{𝑆}$ with eigenvalue equal to the area ${𝐴}_{𝑆}(ℎ)$assigned to $𝑆$ by ${ℎ}_{𝑎𝑏}$ (to $𝑂({𝑙}_{𝑃}^2\mathrm{/}{𝐴}_{𝑆}(ℎ))$ when the lattice spacing $𝑎$ satisfies precisely the condition stated above.

Thus, the requirement that ${\mathrm{\Psi }}_{\mathrm{\Delta }}$ should approximate the classical metric ${ℎ}_{𝑎𝑏}$ on large scales $𝐿$ tells us something non-trivial about the short-distance structure of the multi-loop $\mathrm{\Delta }$: $𝑎$ is fixed to be the Planck length in units defined by ${ℎ}_{𝑎𝑏}$. Now, naively, one might have expected that the best approximation to the classical metric would occur in the continuum limit in which the lattice separation goes to zero. It is, perhaps, surprising that this does not occur. The reason is that the factors of the Planck length in $(6)$ and $(9)$ force each line of the weave to contribute a Planck unit to the various geometrical observables. Finally, note the structure of the argument: we begin with a classical metric, use it to define the scale $𝐿$, the notion of “slowly varying” as well as the structure of $\mathrm{\Delta }$ and find that the lattice separation $𝑎$ is forced to be the Planck length as measured by ${ℎ}_{𝑎𝑏}$. This is necessary because, while the theory depends on a dimensional scale ${𝑙}_{𝑝}$, it is only with respect to those quantum states that can be associated with classical metrics in this way that there can be a meaningful specification of what physical intervals have this length.

We conclude with three remarks.

1. Work is in progress on the linearization of the exact state space of the theory in a neighborhood of a weave state. There are preliminary indications that the resulting theory is isomorphic to the Fock space of linearized gravitons to order ${𝑙}_{𝑃}\mathrm{/}{𝜆}_{𝑔𝑟𝑎𝑣𝑖𝑡𝑜𝑛}$. Thus, the notion of gravitons may be physically meaningful only at long wavelengths. In this connection it is important to point out that as it is an eigenstate of the three geometry ${\mathrm{\Psi }}_{\mathrm{\Delta }}[𝛼]$ is not a candidate for the vacuum of the theory; however candidates for the vacuum may be constructed by dressing ${\mathrm{\Psi }}_{\mathrm{\Delta }}[𝛼]$ with an appropriate distribution of loops corresponding to the virtual gravitons.

2. The main results presented in this letter can be obtained also in the connection representation, in which case the characteristic states take the form ${\mathrm{\Psi }}_{𝛾}[𝐴]=\text{TrP exp}(𝐺{\oint }_{𝛾}{𝐴}_{𝑎}𝑑{𝛾}^{𝑎})$. It is the loop operators (rather than the loop states) that are essential to the argument; they provide us with a regularization procedure that respects diffeomorphism invariance.

3. However, if we wish to consider physical states which are annihilated by the quantum constraints the use of the loop representation seems unavoidable since, at the present stage, solutions to all quantum constraints have been obtained only in the loop picture. Consider the loop state ${\mathrm{\Psi }}_{𝑓𝑙𝑎𝑡}$ defined on nonintersecting loops $𝛼$ by:

${\mathrm{\Psi }}_{𝑓𝑙𝑎𝑡}[𝛼]=\{\begin{array}{ll}1& \text{if }𝐾(𝛼)=𝐾(\mathrm{\Delta })\\ 0& \text{otherwise},\end{array}$

where $𝐾(𝛼)$ is the knot class to which $𝛼$ belongs. Results show that ${\mathrm{\Psi }}_{𝑓𝑙𝑎𝑡}(𝛼)$ solves all quantum constraints; in the Dirac terminology, it is a physical state of non-perturbative quantum gravity. It is natural to interpret ${\mathrm{\Psi }}_{𝑓𝑙𝑎𝑡}$ as representing the flat 3-geometry at large scales: just as the loop $\mathrm{\Delta }$ corresponds to a flat metric ${ℎ}_{𝑎𝑏}$, the equivalence class $𝐾(\mathrm{\Delta })$ of loops should correspond to the equivalence class of all metrics related to ${ℎ}_{𝑎𝑏}$ by a diffeomorphism. In the same spirit then, it is tempting to conjecture that knot classes which are inequivalent to $𝐾(\mathrm{\Delta })$represent non-flat geometries.

Acknowledgments: This work was supported in part by the NSF grants PHY90-12099, PHY90-16733, and INT88-15209 and by research funds provided by Syracuse University.

References:

1. B.S. DeWitt, E. Myers, R. Harrington, and A. Kapulkin, Nucl. Phys.B. (Proc. Suppl.) 20, 744 (1991); M.E. Agishtein and A. A. Migdal, Princeton pre-print PUPT-1272 (1991).

2. A. Ashtekar, Non-perturbative canonical quantum gravity (Notes prepared in collaboration with R.Tate) (World Scientific, Singapore, 1991).

3. C. Rovelli, Class. & Quant. Grav. 8, 1613 (1991).

4. L. Smolin, Recent developments in nonperturbative quantum gravity Syracuse pre-print SU-GP-92/2-2, to appear in the Proceedings of the 1991 GIFT International Seminar on Theoretical Physics (World Scientific, Singapore, in press.), hepth preprint number 9202022.

5. A. Ashtekar Phys. Rev. Lett. 57, 2244 (1986), Phys. Rev.D36, 1587 (1987).

6. C. Rovelli and L. Smolin, Phys. Rev. Lett. 61, 1155 (1988); Nucl. Phys. B133, 80 (1990).

7. A. Ashtekar, C. Rovelli, and L. Smolin, in preparation.

8. See, for example, J. A. Wheeler, in "Battelle Rencontres 1967", eds. C. DeWitt, J. A. Wheeler, Benjamin 1968; B. Ferretti, Lettere al Nuovo Cimento, 40, 169 (1984).; M. Toller, International Journal of Theoretical Physics, 29, 963 (1990); T. Padmanbhan, Class. and Quantum Grav. 4, L107, (1987).

9. See, for example, D. Finkelstein, Phys. Rev. 184 (1969) 1261; R. Penrose, in Quantum Theory and Beyond, ed. T. Bastin (Cambridge University Press,1971); G. 't Hooft, Recent Developments in Gravitation, eds. in M. Levy and S. Deser, (Plenum,1979); L. Bombelli, J. Lee, D. Meyer, and R. D. Sorkin, Phys. Rev. Lett. 60 (1988) 655; J. Barbour and L. Smolin, Syracuse Preprint SU-GP-92/2-4 (1992).

10. This definition is equivalent to the one given in other referenced works in which the state space is extended to functions on multiloops and $\widehat{𝑇}[𝛾]\circ \mathrm{\Psi }[𝛼]=\mathrm{\Psi }[𝛼\cup 𝛾]$.

11. In fact the finiteness of these operators is closely related to their background independence. Basically, a dependence on a power of the regulator must come together with background dependence, because the regulator scale is defined with respect to the background metric introduced in regularization. It is tempting to conjecture that any operator that is regulated in such a way that no dependence on the background survives in the limit in which the regulator is removed, must be finite in this limit.

12. The action of the operator on intersecting loops is known explicitly, but is slightly more complicated. See references for details.

13. Note that such states are not normally considered in ordinary quantum field theories based on Fock inner products. However, in the absence of a background metric the Fock inner product cannot be defined. Instead, quantum field theories on manifolds without background metrics may be based on representations of the sort introduced by A. Ashtekar and C.J. Isham (Class. & Quan. Grav., in press) within which the characteristic states of loops are normalizable.

14. J. Zegwaard, Utrecht preprint THU-91/13 (1991).

15. A. Ashtekar, C. Rovelli, and L. Smolin, "Gravitons and loops" Phys. Rev. D 44 (1991) 1740.

16. T. Jacobson and L. Smolin, Nucl.Phys. B299, 295 (1988).

I Asked ChatGPT: What is the paper about?

The paper titled "Weaving a classical geometry with quantum threads" by Abhay Ashtekar, Carlo Rovelli, and Lee Smolin explores the physical interpretation of states in nonperturbative quantum gravity using loop variables. Here are the key points and findings from the paper:

1. Objective and Context:

   • The paper addresses the limitations of perturbative approaches to quantum gravity, which assume a smooth space-time geometry at all scales. Instead, it proposes non-perturbative methods to uncover the true micro-structure of space-time.
   • The authors focus on canonical quantization, building on work from the late 1950s, but incorporating new techniques such as using a connection as the configuration variable and representing quantum states as functions on the space of closed loops.

2. Loop Variables:

   • The paper introduces loop variables, which are manifestly SU(2)-gauge invariant functions on the phase space. These variables allow for the description of geometry in a diffeomorphism-invariant manner.
   • Key loop variables include Wilson loops (which depend on the connection) and variables that include momenta (involving the triad fields).

3. Non-local Operators:

   • The authors demonstrate that while local operators like the metric at a point may not be well-defined, non-local operators such as the area of a surface can be diffeomorphism-invariant and finite without renormalization.
   • The construction of these non-local operators involves regulating and defining them in a way that they do not depend on any background structure used in the regularization process.

4. Quantum States and Discreteness at Planck Scale:

   • The paper finds that there exist quantum states that approximate a given flat classical geometry at large scales but exhibit a discrete structure at the Planck scale. This discreteness is not an input assumption but an outcome of the framework.
   • States are represented by functions of loops, and the quantum operators corresponding to loop variables mirror the classical Poisson algebra in the quantum commutator algebra.

5. Weave States:

   • The authors introduce the concept of "weave states," which are loop states that approximate a given classical metric on large scales. These states are constructed by spacing loops so that each loop contributes one Planck unit of area to surfaces it intersects.
   • The weave states help in understanding how classical geometry emerges from quantum states on macroscopic scales.

6. Key Results:

   • The paper shows that operators representing areas and volumes can be defined in a way that respects diffeomorphism invariance and are finite without requiring renormalization.
   • It also demonstrates that the notion of gravitons may only be meaningful at long wavelengths, as the weave states can approximate the geometry determined by classical metrics on large scales.

7. Conclusion:

   • The paper concludes that the discrete structure at the Planck scale emerges naturally from the non-perturbative quantization framework. The approach based on loop variables and weave states offers a promising avenue for understanding the quantum nature of geometry and gravity.

Overall, the paper presents significant advancements in the understanding of quantum gravity by introducing non-perturbative methods that bridge the gap between classical and quantum descriptions of space-time.