New Paths Towards Quantum Gravity (Lecture Notes in Physics, 807)

134 N. ReshetikhinThe Lagrangian submanifold corresponding to the result of the gluing should beL(M f ) ={x ∈ S((∂ M) ′ )| such that there exists y ∈ S(∂ M) 1with(y, s( f )(y), x) ∈ L(M)}. (6)Notice that ∂ M f = (∂ M) ′ by definition. This axiom is known as the gluingaxiom. In classical mechanics the gluing axiom is the composition of the evolutionat consecutive intervals of time. 1A boundary condition in the Hamiltonian formulation is a Lagrangian submanifoldL b (∂ M) in the symplectic manifold S(∂ M), assigned to the boundary ∂ M ofthe manifold M, L b (∂ M) ⊂ S(∂ M). It factorizes into the product of Lagrangiansubmanifolds corresponding to connected components of the boundary:L b ((∂ M) 1 ⊔ (∂ M) 2 ) = L b ((∂ M) 1 ) × L b ((∂ M) 2 ).Classical solutions with given boundary conditions are intersection points L b (∂ M)∩L(M).In order to glue classical solutions along the common boundary (compositionof classical trajectories in classical mechanics) let us assume that boundaryLagrangian submanifolds are fibers of Lagrangian fiber bundles. That is, we assumethat for each connected component (∂ M) i of the boundary a symplectic manifoldS((∂ M) i ) is given together with a Lagrangian fiber bundle π i : S((∂ M) i ) →B((∂ M) i ) over some base space B((∂ M) i ) with fibers defining the boundaryconditions.3.3.2 Hamiltonian Formulation of Local Lagrangian Field TheoryHere again, instead of giving general definitions we will give a few illustratingexamples.3.3.2.1 Classical Hamiltonian Mechanics1. Let H ∈ C ∞ (M) be the Hamiltonian function generating Hamiltonian dynamicson a symplectic manifold M. 2 Here is how such a system can be reformulated inthe framework of a Hamiltonian field theory.1 I am grateful to V. Fock for many illuminating discussions of Hamiltonian aspects of field theory,see also [23].2 Recall that Hamiltonian mechanics is a dynamical system on a symplectic manifold (M,ω)with trajectories being flow lines of the Hamiltonian vector field v H generated by a function H ∈C ∞ (M), v H = ω −1 (dH).Hereω −1 : T ∗ M → TM is the isomorphism induced by the symplecticstructure on M.