Symplectic Structures| Symplectic Structures |
Symplectic Vector Space
$ Def: A vector space
V with a non-degenerate antisymmetric rank-2 tensor Ω.
* Symmetry group: The Lie
group preserving this structure is the symplectic group Sp(V,
Ω) ⊂ GL(V).
* Subspaces: W < V is
Isotropic if W ⊂ W⊥:=
{null vectors of Ω|W}; Coisotropic (first class) if
W⊥ ⊂ W;
Weakly symplectic (second class) if W ∩
W⊥ = 0; Lagrangian if isotropic
and coisotropic, W = W⊥;
Otherwise, it is called mixed.
Canonical Transformation
> s.a. fock space; quantum states.
$ Def: Given a symplectic vector
space (V, Ω), a canonical transformation is a linear map A:
V → V that preserves Ω, i.e., Ω(A(X),
A(Y)) = Ω(X, Y), for all X, Y
∈ V, or At Ω A = Ω.
* Generating function: The function
H(X) = \(-{1\over2}\)X At Ω X .
* Point transformation:
A canonical transformation that does not mix coordinates and momenta.
@ Generating functions: Anselmi EPJC(16)-a1511
[in classical mechanics and quantum field theory];
Anselmi EPJC(16)-a1604 [reference formulas].
@ Special types: Brodlie JMP(04)qp [non-linear];
Dereli et al IJMPA(09)-a0904 [in 3D phase space with Nambu bracket].
@ Quantum: Bordner JMP(97);
Davis & Ghandour PLA(01) [and system equivalence];
Cervero & Rodríguez IJTP(02)qp/01 [and squeezing];
Lacki SHPMP(04) [in early quantum mechanics].
@ Generalizations: Cariñena et al JGM(13)-a1303 [canonoid transformations, and symmetries];
Kupsch RVMP(14) [for fermions];
Valtancoli JMP(15)-a1510 [with non-commutative Poisson brackets, minimal length].
Kähler Spaces and Manifolds / Structures
> s.a. complex structure.
$ Kähler space: A
symplectic vector space (V, Ω) with a complex structure J,
compatible with Ω in the sense that the rank-2 tensor ΩJ is
positive-definite and symmetric (a Kähler metric).
$ Kähler manifold:
A triple (M, Ω, J), with M a differentiable
manifold, Ω and J strongly compatible, and J integrable;
> s.a. Wikipedia page.
@ Kähler space: in Artin 57;
Flaherty 76 [in relativity];
in Helgason 78;
Calabi & Chen JDG(02) [space of Kähler metrics].
@ Kähler manifold: Hashimoto et al JMP(97) [hyperkähler metrics from Ashtekar variables];
Pedersen et al LMP(99) [quasi-Einstein Kähler metrics];
Chen JDG(00) [space of Kähler metrics];
Cortés m.DG/01-ln [special Kähler manifolds];
Ross & Thomas JDG(06) [constant curvature, obstructions];
Aazami & Maschler a1711 [from Lorentzian geometries].
@ Generalizations:
Gualtieri CMP(14) [generalized complex geometry];
> s.a. Hyperkähler Structure.
Generalizations > s.a. Contact
Manifold; Cosymplectic Structures;
types of symplectic structures.
* Higher-order
structures: A manifold is n-plectic if it is equipped
with a closed, non-degenerate form of degree n + 1;
> s.a. formulations of general relativity.
@ General references: Zumino MPLA(91) [Fermionic coordinates];
Nikolić qp/98 [non equal-time];
Vanhecke BJP(06)mp/05 [non-commutative configuration space];
Sergyeyev Sigma(07)mp/06 [weakly non-local];
> s.a. non-commutative physics; poisson
structures [Moyal and Nambu brackets]; Supermanifolds.
@ Higher-order structures: Rogers PhD(11)-a1106;
Sämann & Szabo RVMP(13)-a1211 [quantization of 2-plectic manifolds].
Related Concepts 
see symplectic manifolds; symplectic structures in physics.
main page
– abbreviations
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