6.8 Diffeomorphism invariance6 The Formalism6.6 Algebraic version (``loop representation'')

6.7 Other structures in tex2html_wrap_inline2484  

The recent developments in the mathematical foundations of the connection representation have increased the mathematical rigor of the theory, raising it to the standards of mathematical physics. This has been obtained at the price of introducing heavy mathematical tools, often unfamiliar to the average physicist, perhaps widening the language gap between the quantum gravity and the high energy physics community.

The reason for searching a mathematical-physics level of precision is that in quantum gravity one moves on a very unfamiliar terrain -quantum field theory on manifolds- where the experience accumulated in conventional quantum field theory is often useless and sometimes even misleading. Given the unlikelihood of finding direct experimental corroboration, the research can only aim, at least for the moment, at the goal of finding a consistent theory, with the correct limits in the regimes that we control experimentally. In these conditions, high mathematical rigor is the only assurance of the consistency of the theory. In the development of quantum field theory mathematical rigor could be very low because extremely accurate empirical verifications assured the physicists that ``the theory may be mathematically meaningless, but it is nevertheless physical correct, and therefore the theory must make sense even if we do not understand how.'' Here, such an indirect experimental reassurance is lacking and the claim that the theory is well founded can be based only on a solid mathematical control of the theory.

One may object that a rigorous definition of quantum gravity is a vain hope, given that we do not even have a rigorous definition of QED, presumably a much simpler theory. The objection is particularly valid from the point of view of a physicist who views gravity ``just as any other field theory like the ones we already understand''. But the (serious) difficulties of QED and of the other conventional field theories are ultraviolet. The physical hope supporting the quantum gravity research program is that the ultraviolet structure of a diffeomorphism invariant quantum field theory is profoundly different from the one of conventional theories. Indeed, recall that in a very precise sense there is no short distance limit in the theory; the theory naturally cuts itself off at the Planck scale, due to the very quantum discreteness of spacetime. Thus the hope that quantum gravity could be defined rigorously may be optimistic, but it is not ill founded.

After these comments, let me briefly mention some of the structures that have been explored in tex2html_wrap_inline2484 . First of all, the spin network states satisfy the Kauffman axioms of the tangle theoretical version of recoupling theory [130Jump To The Next Citation Point In The Article] (in the ``classical'' case A =-1) at all the points (in 3d space) where they meet. (This fact is often misunderstood: recoupling theory lives in 2d and is associated by Kauffman to knot theory by means of the usual projection of knots from 3d to 2d. Here, the Kauffman axioms are not satisfied at the intersections created by the 2d projection of the spin network, but only at the nodes in 3d. See [77Jump To The Next Citation Point In The Article] for a detailed discussion.) For instance, consider a 4-valent node of four links colored a, b, c, d . The color of the node is determined by expanding the 4-valent node into a trivalent tree; in this case, we have a single internal links. The expansion can be done in different ways (by pairing links differently). These are related to each other by the recoupling theorem of pg. 60 in Ref.\ [130Jump To The Next Citation Point In The Article]


where the quantities tex2html_wrap_inline2970 are su (2) six-j symbols (normalized as in [130Jump To The Next Citation Point In The Article]). Equation (29Popup Equation) follows just from the definitions given above. Recoupling theory provides a powerful computational tool in this context.

Since spin network states satisfy recoupling theory, they form a Temperley-Lieb algebra [130]. The scalar product (14Popup Equation) in tex2html_wrap_inline2484 is given also by the Temperley-Lieb trace of the spin networks, or, equivalently by the Kauffman brackets, or, equivalently, by the chromatic evaluation of the spin network. See Ref. [77Jump To The Next Citation Point In The Article] for an extensive discussion of these relations.

Next, tex2html_wrap_inline2484 admits a rigorous representation as an tex2html_wrap_inline2978 space, namely a space of square integrable functions. To obtain this representation, however, we have to extend the notion of connection, to a notion of ``distributional connection''. The space of the distributional connections is the closure of the space of smooth connection in a certain topology. Thus, distributional connections can be seen as limits of sequences of connections, in the same manner in which distributions can be seen as limits of sequences of functions. Usual distributions are defined as elements of the topological dual of certain spaces of functions. Here, there is no natural linear structure in the space of the connections, but there is a natural duality between connections and curves in M : A smooth connection A assigns a group element tex2html_wrap_inline2984 to every segment tex2html_wrap_inline2546 . The group elements satisfy certain properties. For instance if tex2html_wrap_inline2546 is the composition of the two segments tex2html_wrap_inline2990 and tex2html_wrap_inline2992, then tex2html_wrap_inline2994 .

A generalized connection tex2html_wrap_inline2996 is defined as a map that assigns an element of SU (2), which we denote as tex2html_wrap_inline3000 or tex2html_wrap_inline3002, to each (oriented) curve tex2html_wrap_inline2546 in M, satisfying the following requirements: i) tex2html_wrap_inline3008 ; and, ii) tex2html_wrap_inline3010, where tex2html_wrap_inline3012 is obtained from tex2html_wrap_inline2546 by reversing its orientation, tex2html_wrap_inline3016 denotes the composition of the two curves (obtained by connecting the end of tex2html_wrap_inline2990 with the beginning of tex2html_wrap_inline2992) and tex2html_wrap_inline3022 is the composition in SU (2). The space of such generalized connections is denoted tex2html_wrap_inline3026 . The cylindrical functions tex2html_wrap_inline3028, defined in section 6.3 as functions on the space of smooth connections, extend immediately to generalized connections


We can define a measure tex2html_wrap_inline2500 on the space of generalized connections tex2html_wrap_inline3026 by


In fact, one may show that (31Popup Equation) defines (by linearity and continuity) a well-defined absolutely continuous measure on tex2html_wrap_inline3026 . This is the Ashtekar-Lewandowski (or Ashtekar-Lewandowski-Baez) measure [14Jump To The Next Citation Point In The Article, 15Jump To The Next Citation Point In The Article, 16Jump To The Next Citation Point In The Article, 26Jump To The Next Citation Point In The Article]. Then, one can prove that tex2html_wrap_inline3036, under the natural isomorphism given by identifying cylindrical functions. It follows immediately that the transformation (19Popup Equation) between the connection representation and the ``old'' loop representation is given by


This is the loop transform formula that was derived heuristically in [184Jump To The Next Citation Point In The Article]; here it becomes rigorously defined.

Furthermore, tex2html_wrap_inline2484 can be seen as the projective limit of the projective family of the Hilbert spaces tex2html_wrap_inline3040, associated to each graph tex2html_wrap_inline2650 immersed in M . tex2html_wrap_inline3040 is defined as the space tex2html_wrap_inline3048, where n is the number of links in tex2html_wrap_inline2650 . The cylindrical function tex2html_wrap_inline3028 is naturally associated to the function f in tex2html_wrap_inline3040, and the projective structure is given by the natural map (13Popup Equation) [18Jump To The Next Citation Point In The Article, 149].

Finally, Ashtekar and Isham [12] have recovered the representation of the loop algebra by using C*-algebra representation theory: The space tex2html_wrap_inline3060, where tex2html_wrap_inline3062 is the group of local SU (2) transformations (which acts in the obvious way on generalized connections), is precisely the Gelfand spectrum of the abelian part of the loop algebra. One can show that this is a suitable norm closure of the space of smooth SU (2) connections over physical space, modulo gauge transformations.

Thus, a number of powerful mathematical tools are at hand for dealing with nonperturbative quantum gravity. These include: Penrose's spin network theory, SU (2) representation theory, Kauffman tangle theoretical recoupling theory, Temperley-Lieb algebras, Gelfand's tex2html_wrap_inline2496 algebra spectral representation theory, infinite dimensional measure theory and differential geometry over infinite dimensional spaces.

6.8 Diffeomorphism invariance6 The Formalism6.6 Algebraic version (``loop representation'')

image Loop Quantum Gravity
Carlo Rovelli
© Max-Planck-Gesellschaft. ISSN 1433-8351
Problems/Comments to livrev@aei-potsdam.mpg.de