Symplectic Geometry and Quantum Mechanics (original) (raw)
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Symplectic geometry, wigner-weyl-moyal calculus, and quantum mechanics, in phase space ; Part 3
2006
He regularly visits his alma mater, the University of Paris 6. De Gosson has done extensive research in the area of symplectic geometry, the combinatorial theory of the Maslov index, the theory of the metaplectic group, partial differential equations, and mathematical physics. He obtained his Ph.D. in 1978 on the subject of microlocal analysis at the University of Nice under the supervision of J. Chazarain, and his Habilitation in 1992 at the University Pierre et Marie Curie in Paris, under the mentorship of the late Jean Leray (Collège de France). Notation Our notation is as standard (and simple) as conflicting usages in the mathematical and physical literature allow. Number sets R (resp. C) is the set of all real (resp. complex) numbers. We denote respectively by
An Introduction to Symplectic Maps and Generalizations of the Toda Lattice
Congressus numerantium, 1970
The integer N may be fixed as finite or infinite, or N could be chosen to be an element from a finite cyclic group to invoke periodicity in the lattice. Then , for a given lattice particle, mj, we have that: j + N ≡ j for j ∈ Z+ ∪ {0}. The system we describe is called the Toda lattice [5] where ...
Maslov index and symplectic sturm theorems
Functional Analysis and Its Applications, 1998
In this section we construct transversally oriented hypersurfaces in the Lagrange-Grassmann manifold that determine a one-dimensional cocycle coinciding with the Maslov index. Let us consider a symplectic vector space (R 2n, w). Denote by An the manifold of all Lagrangian subspaces in (R 2n, w), which we call the Lagrange-Grassmann manifold. We choose Darboux coordinates (P, q) = (Px,...,pn, ql,..., qn) in (R 2n, w), w = ~ dpi A dqi = dp A dq. The plane p = 0 is called the q-plane and the plane q = 0 the p-plane. Let us consider the set X C An of all Lagrangian planes transversal to the plane p = 0. The set X is open and dense in An (a chart of An) that is diffeomorphic to R n(n+l)/2. A Lagrangian plane belonging to X can be identified with a symmetric matrix, namely, to a matrix A, the plane q = Ap corresponds. Definition [2]. The set of all Lagrangian planes that are not tran.qversal to a given Lagrangian plane is called the train of this plane. In the chart X, the train of the p-plane is given by the equation det A = 0. We identify each element of the tangent space at a point of a vector space, as well as each translationinvariant vector fields on the vector space, with an element of the vector space. In our case, we identify translation-invariant vector fields in the chart X with matrices. Let us consider a hypersurface in X specified by the equation (LA,"" LA,~ det)(A) = 0, where A1, ..., Ak are positive-definite matrices (0 < k < n), L. is the derivative along the vector field v, and det is the determinant.
Journal De Mathematiques Pures Et Appliquees, 2009
Using the ideas of Keller, Maslov introduced in the mid-1960's an index for Lagrangian loops, whose definition was clarified by Arnold. Leray extended Arnold results by defining an index depending on two paths of Lagrangian planes with transversal endpoints. We show that the combinatorial and topological properties of Leray's index suffice to recover all Lagrangian and symplectic intersection indices commonly used in symplectic geometry and its applications to Hamiltonian and quantum mechanics. As a by-product we obtain a new simple formula for the Hörmander index, and a definition of the Conley–Zehnder index for symplectic paths with arbitrary endpoints. Our definition leads to a formula for the Conley–Zehnder index of a product of two paths.Utilisant les idées de Keller, Maslov introduisit au milieu des années 1960 un indice pour les lacets lagrangiens ; Arnold clarifia par la suite la définition de Maslov. Leray étendit les résultats de Arnold en définissant un indice dépendant de deux chemins lagrangiens dont les extrêmités sont transversales. Nous montrons que les propriétés combinatoires et topologiques qui caractérisent l'indice de Leray sont suffisantes pour retrouver tous les indices d'intersection lagrangiens et symplectiques communément utilisés en géométrie symplectique, et ses applications à la mécanique hamiltonienne et quantique. Nous obtenons en outre une nouvelle formule simple pour l'indice de Hörmander, ainsi qu'une définition de l'indice de Conley–Zehnder pour les chemins symplectiques sans condition de transversalité. Notre définition permet en outre de démontrer une formula pour l'indice de Conley–Zehnder du produit des deux chemins symplectiques.
PoS (CORFU2011) 060 Quantization of the symplectic groupoid
2011
Quantization of Poisson manifolds has been the main motivation for introducing the concept of symplectic groupoid, see for instance in [1 2]. A symplectic groupoid is a Lie groupoid S with a compatible symplectic form ΩS. This compatibility means that the graph of the groupoid multiplication is a lagrangian submanifold of S× S× S, where S means S endowed with− ΩS.
Symplectic Structures and Quantum Mechanics
Modern Physics Letters B, 1996
Canonical coordinates for the Schrödinger equation are introduced, making more transparent its Hamiltonian structure. It is shown that the Schrödinger equation, considered as a classical field theory, shares with Liouville completely integrable field theories the existence of a recursion operator which allows for the infinitely many conserved functionals pairwise commuting with respect to the corresponding Poisson bracket.
The real symplectic groups in quantum mechanics and optics
Pramana, 1995
We present a utilitarian review of the family of matrix groups Sp(2n,), in a form suited to various applications both in optics and quantum mechanics. We contrast these groups and their geometry with the much more familiar Euclidean and unitary geometries. Both the properties of finite group elements and of the Lie algebra are studied, and special attention is paid to the socalled unitary metaplectic representation of Sp(2n,). Global decomposition theorems, interesting subgroups and their generators are described. Turning to n-mode quantum systems, we define and study their variance matrices in general states, the implications of the Heisenberg uncertainty principles, and develop a U (n)-invariant squeezing criterion. The particular properties of Wigner distributions and Gaussian pure state wavefunctions under Sp(2n,) action are delineated.