By Maurice A. de Gosson

This booklet deals a whole dialogue of thoughts and themes intervening within the mathematical remedy of quantum and semi-classical mechanics. It starts off with a really readable creation to symplectic geometry. Many issues also are of actual curiosity for natural mathematicians operating in geometry and topology.

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**Example text**

17. (i) Let ∆jk = (δjk )1≤j,k≤n (δjk = 0 if j = k, δjk = 1). Show that the matrices Xjk = ∆jk 0 Zjk = 0 −∆jk , Yjk = 1 0 2 ∆jk + ∆kj 0 0 1 0 2 0 ∆jk + ∆kj , 0 (1 ≤ j ≤ k ≤ n) form a basis of sp(n). (ii) Show, using (i) that every Z ∈ sp(n) can be written in the form [X, Y ] = XY − Y X with X, Y ∈ sp(n). One should be careful to note that the exponential mapping exp : sp(n) −→ Sp(n) is neither surjective nor injective. This is easily seen in the case n = 1. We claim that S = exp X with X ∈ sp(1) =⇒ Tr S ≥ −2.

I) The condition is necessary, taking for B the canonical symplectic bases. If conversely is the graph of a symmetric matrix M , then it is immediate to check that σ(z; z ) = 0 for all z ∈ . (ii) The intersection ∩ P consists of all (x, p) which satisfy both conditions Xx + P p = 0 and x = 0. 18). 19) follows from the trivial equivalence (x, p) ∈ S P ∩ P ⇐⇒ Bp = 0. 18 Chapter 1. 15 we can ﬁnd a symplectic basis B = {e1 , . . , en } ∪ {f1 , . . , fn } of (E, ω) and the spaces = Span {e1 , . .

The associated symmetric matrix M = D2 Q (the Hessian matrix of Q) has µ+ positive eigenvalues and µ− negative eigenvalues. We will call the diﬀerence µ+ − µ− the signature of the quadratic form Q and denote it by sign Q: sign Q = µ+ − µ− . 27. Let ( , , ) be an arbitrary triple of Lagrangian planes in a symplectic space (E, ω). The “Wall–Kashiwara index” (or: signature) of the triple ( , , ) is the signature of the quadratic form Q(z, z , z ) = ω(z, z ) + ω(z , z ) + ω(z , z) on ⊕ ⊕ . This signature is denoted by τ ( , , ).