6.7 Other structures in 6 The Formalism6.5 The representation

6.6 Algebraic version (``loop representation'') and differential version (``connection representation'') of the formalism, and their equivalence

Imagine we want to quantize the one dimensional harmonic oscillator. We can consider the Hilbert space of square integrable functions tex2html_wrap_inline2914 on the real line, and express the momentum and the hamiltonian as differential operators. Denote the eigenstates of the hamiltonian as tex2html_wrap_inline2916 . It is well known that the theory can be expressed entirely in algebraic form in terms of the states tex2html_wrap_inline2918 . In doing so, all elementary operators are algebraic: tex2html_wrap_inline2920, tex2html_wrap_inline2922 . Similarly, in quantum gravity we can directly construct the quantum theory in the spin-network (or loop) basis, without ever mentioning functionals of the connections. This representation of the theory is denoted the ``loop representation''.

A section of the first paper on loop quantum gravity by Rovelli and Smolin [184Jump To The Next Citation Point In The Article] was devoted to a detailed study of ``transformation theory'' (in the sense of Dirac) on the state space of quantum gravity, and in particular on the relations between the loop states


and the states tex2html_wrap_inline2754 giving the amplitude for a connection field configuration A, and defined by


Here tex2html_wrap_inline2724 are ``eigenstates of the connection operator'', or, more precisely (since the operator corresponding to the connection is ill defined in the theory) the generalized states that satisfy


However, at the time of [184Jump To The Next Citation Point In The Article] the lack of a scalar product made transformation theory quite involved.

On the other hand, the introduction of the scalar product (14Popup Equation) gives a rigorous meaning to the loop transform. In fact, we can write, for every spin network S, and every state tex2html_wrap_inline2754


This equation defines a unitary mapping between the two presentations of tex2html_wrap_inline2484 : the ``loop representation'', in which one works in terms of the basis tex2html_wrap_inline2936 ; and the ``connection representation'', in which one uses wave functionals tex2html_wrap_inline2754 .

The development of the connection representation followed a winding path through tex2html_wrap_inline2496 -algebraic [12Jump To The Next Citation Point In The Article] and measure theoretical [14Jump To The Next Citation Point In The Article, 16Jump To The Next Citation Point In The Article, 15Jump To The Next Citation Point In The Article] methods. The work of Ashtekar, Isham, Lewandowski, Marolf, Mourao and Thiemann has finally put the connection representation on a firm ground, and, indirectly has much clarified the mathematics underlying the original loop approach. In the course of this development, doubts were raised about the precise relations between the connection and the loop formalisms. Today, the complete equivalence of these two approaches (always suspected) has been firmly established. In particular, the work of Roberto DePietri [76] has proven the unitary equivalence of the two formalisms. For a recent discussion see also [139Jump To The Next Citation Point In The Article].

6.7 Other structures in 6 The Formalism6.5 The representation

image Loop Quantum Gravity
Carlo Rovelli
© Max-Planck-Gesellschaft. ISSN 1433-8351
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