By Ruark A.E.

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A submanifold Pˆ → P is called coisotropic iﬀ T ⊥ Pˆ ⊂ T Pˆ . Since ω is non degenerate we have dim Tp Pˆ + dim Tp⊥ Pˆ = dim Tp P , hence dim Tp⊥ Pˆ = codim Pˆ . This means that for coisotropic embeddings i : Pˆ → P the kernel7 of the pulled-back symplectic form ω ˆ := i∗ ω on Pˆ has the maximal ˆ possible number of dimensions, namely codim P . Definition 4. A constrained system Pˆ → P is said to be of ﬁrst class iﬀ Pˆ is a coisotropic submanifold of (P, ω). From now on we consider only ﬁrst class constraints.

L´eon van Howe: Sur le probl`eme des relations entre les transformations unitaires de la m´ecanique quantique et les transformations canoniques de la m´ecanique classique. Acad. Roy. Belg. Bull. (Cl. ) 37, 610–620 (1951) Lectures on Loop Quantum Gravity Thomas Thiemann MPI f¨ ur Gravitationsphysik, Albert-Einstein-Institut, Am M¨ uhlenberg 1, 14476 Golm, Germany Abstract. Quantum General Relativity (QGR), sometimes called Loop Quantum Gravity, has matured over the past ﬁfteen years to a mathematically rigorous candidate quantum ﬁeld theory of the gravitational ﬁeld.

E. up to ﬁnite multiplicity, on H. e. that there is no Fquant with Fquant ⊂ Fquant ⊂ Fpol (+, {, }). Note that the choice of Fquant is generally far from unique. e. the polynomials of at most quadratic order, we could choose Fquant = Fpol(∞,1) , the polynomials of at most linear order in momenta with coeﬃcients which are arbitrary polynomials in q. A general element in Fpol(∞,1) has the form f (q, p) = g(q) + h(q) p (59) where g, h are arbitrary polynomials with real coeﬃcients. The Poisson bracket of two such functions is {f1 , f2 } = {g1 + h1 p, g2 + h2 p} = g3 + h3 p , (60) g3 = g1 h2 − g2 h1 (61) where and h3 = h1 h2 − h1 h2 .