Symplectic geometry Daniel Bryan University of Hamburg May 5, 2020 Daniel Bryan (Uni-hamburg) Short title May 5, 2020 1 / 28 Overview 1 Introduction to symplectic geometry Poisson and symplectic manifolds Poisson bracket and Hamiltonian Poisson brackets, coordinates and symplectic form Symplectic form and Hamiltonian 2 Coadjoint Orbits and SU(2) example Coadjoint orbits SU(2) Example Kostant-Kirillov symplectic structure 3 Hamiltonian reduction Locally Hamiltonian actions of Lie algebra Poissonian action of Lie group Daniel Bryan (Uni-hamburg) Short title May 5, 2020 2 / 28 Poisson and symplectic manifolds We consider a phase space M and differentiable functions on them. An algebra of these functions F(M) can be defined. The Poisson bracket can be defined as a bilinear antisymmetric derivation of the algebra : F(M) × F(M) 7! F(M). The Poisson brackets obey the following conditions: Antisymmetry: ff1; f2g = −{f2; f1g ff1; αf2 + βf3g = αff1; f2g + βff1; f3g ff1; f2f3g = ff1; f2gf3 + f2ff1; f3g Leibnitz Rule ff1; ff2; f3ggg + ff3; ff1; f2ggg + ff2; ff1; f3ggg = 0 Jacobi Identity Daniel Bryan (Uni-hamburg) Short title May 5, 2020 3 / 28 Poisson bracket and Hamiltonian One can associate a vector field with the Hamiltonian H of the system XH f = fH; f g. This vector field defines the time evolution of the Hamiltonian f_ = XH f = fH; f g . This means that for conserved quantities fH; f1g = fH; f2g = 0 From the Jacobi identity f1 and f2 are conserved quantities then ff1; f2g is also conserved. A Poisson bracket can be degenerate: this happens when functions f (M) exist such that ff ; gg = 0 for all g. However this is not the case for those defining a symplectic manifold. These functions are known as the center of the Poisson Algebra. Daniel Bryan (Uni-hamburg) Short title May 5, 2020 4 / 28 Poisson brackets, coordinates, and symplectic form In coordinates the Poisson brackets can be written as ij @f1 @f1 ff1; f2g(x) = Σij P (x) @xi @xj using the bilinearity and Leibnitz rule. The function Pij (x) = −Pji (x). is jk ks ij js ik The Jacobi identity becomes P @s P + P @s P + P @s P = 0 Assume M is even dimensional. Then Pij (x) can be invertible meaning that the center of the algebra is trivial such that ij ia −1 bj @s P = −P @s (P )abP . This can be inserted into the Jacobi Triple product where it becomes: P is ja kb −1 −1 −1 s;a;b P P P (@s (P )ab + @b(P )sa + @a(P )bs ) = 0 Pij is invertible this is equivalent to −1 −1 −1 @s (P )ab + @b(P )sa + @a(P )bs = 0 ij i i This can be interpreted as the closed 2-form ! = −Σi<j P dx ^ dx this is d! = 0. Daniel Bryan (Uni-hamburg) Short title May 5, 2020 5 / 28 Poisson manifolds: R3 example A Poisson manifold is a manifold on which a Poisson bracket can be defined. A Poisson manifold is more general than a symplectic manifold as it contains degenerate Poisson brackets from which a symplectic structure cannot be constructed. For example odd dimensional manifolds can be Poisson but are not 3 symplectic. Consider R for which the Poisson brackets are defined as: fxi ; xj g = 2ijk xk fx1; x2g = 2x3; fx1; x3g = −2x2; fx2; x3g = 2x1 Daniel Bryan (Uni-hamburg) Short title May 5, 2020 6 / 28 Symplectic manifolds and Hamiltonian The symplectic manifold (M;!) is defined as a manifold with a non-degenerate closed 2 -form described previously which is globally defined on the manifold. The manifold has a Hamiltonian vector field XH defined as dH = iXH ! or dH = iXH Ω(XH ; •). A symplectic transformation γ is a bijection acting on points m 2 M in the phase space γ : M 7! M. Consider the transformation: ∗ (γ !)m(V ; W ) = !γ(m)(γ∗V ; γ∗W ). Whenever γ∗! = ! the transformation is symplectic. Infinitesimal transformations are specified by the vector field X on M such that LX ! = 0 where LX = d ◦ iX + iX d is the Lie derivative. Poission brackets transform as γff ; gg = fγf ; γgg. Daniel Bryan (Uni-hamburg) Short title May 5, 2020 7 / 28 Symplectic form and Hamiltonian Any Hamiltonian flow is equivalent to a symplectic transformation : Proof This can be seen by considering dH = −iXH !. Now taking the lie derivative w.r.t this vector field we can see that LXH ! = d ◦ (iXH !) + iXH d!. Using d! = 0 and iXH ! = −dH substituting 2 this in the Lie derivative becomes LXH ! = −d H = 0 2n The simplest example of a symplectic manifold is R . This has cannonical coordinates (pi ; qi ) and Poisson brackets fqi ; qj g = 0 fpi ; pj g = 0 and fpi ; qj g = δij . P @H @H The Hamiltonian vector field in this case is X = − @p + @q H i @qi i @pi i Daniel Bryan (Uni-hamburg) Short title May 5, 2020 8 / 28 Coadjoint orbits Coadjoint orbits are important as they are used to find irreducible unitary representations of Lie algebras and physically help to find and quantize the internal degrees of freedom of a physical system. Consider a connected Lie group G and an associated Lie algebra G. The group G acts on X 2 G by the adjoint action X ! (Ad g)(X ) = gXg −1 where g 2 G The co-adjoint action of G on the dual G∗ of the Lie Algebra is: (Ad∗gΞ)(X ) = Ξ(Ad g −1(X )); g 2 G; Ξ 2 G∗; X 2 G These actions can be taken infinitesimally whereby the Lie Algebra acts on itsself and its dual: ad X (Y ) = [X ; Y ]; X ; Y 2 G ad∗ X Ξ(Y ) = −Ξ([X ; Y ]); X ; Y 2 G Ξ 2 G∗ Daniel Bryan (Uni-hamburg) Short title May 5, 2020 9 / 28 SU(2) example Consider the Lie group SU(2), which can be written in terms of the matricies: n α −β∗ o SU(2) = α; β 2 ; jαj2 + jβj2 = 1 β α∗ C Thesu ~ (2) Lie algebra can be written as: n ib c + id o su~ (2) = b; c; d 2 ' 3 −c + id −ib R R Daniel Bryan (Uni-hamburg) Short title May 5, 2020 10 / 28 SU(2) finding coadjoint orbits To find the coadjoint orbit in this case it is important to notice that there is an isomorphism between the Lie algebra and its dual: ∗ 3 F 2 g ' g 2 R . To find the coadjoint orbit of SU(2) one must find the points 0 ∗ 3 0 −1 F 2 g ' g 2 R such that F = gFg where g 2 G. This means we are looking for a conjugation invariant function Q(F ) = Q(F 0) = Q(gFg −1) such that the function Q(F ) = c defines coadjoint orbits F for different constants. Daniel Bryan (Uni-hamburg) Short title May 5, 2020 11 / 28 SU(2) finding coadjoint orbits It is known that the Lie algebrasu ~ (2) is semi-simple and has a trace which can also be used to define an invariant inner product < X ; Y >= tr(XY ). This acts a function tr :su ~ (2) ! C and by considering the cyclicity tr[gFg −1] = tr[F ]. However from the matricies it is obvious that tr[F ] = 0 and hence doesn't give information about coadjoint orbits. However we can try tr[F 2]. This is also conjugation invariant tr[gFg −1gFg −1] = tr[gF 2g −1] = tr[g −1gF 2] = tr[F 2] Daniel Bryan (Uni-hamburg) Short title May 5, 2020 12 / 28 SU(2) orbits Remembering the formsu ~ (2) Lie algebra : n ib c + id o su~ (2) = b; c; d 2 ' 3 −c + id −ib R R Now we take Tr[F 2]: n ib c + id ib c + id o Tr[F 2] = Tr · = −c + id −ib −c + id −ib n −b2 − c2 − d2 ::: o Tr ::: −b2 − c2 − d2 So the trace finally becomes Tr[F 2] = −2(b2 + c2 + d2). Daniel Bryan (Uni-hamburg) Short title May 5, 2020 13 / 28 SU(2) orbits 2 1 2 We can now define r = − 2 Tr[F ] r 2 = b2 + c2 + d2. 2 As b; c; d 2 R we have r 2 R 2 3 Therefore we can interpret r as the radius of a sphere S ⊂ R . 3 Therefore the coadjoint orbits of SU(2) are the set of spheres in R with all possible radii r and the point at the origin r = 0 . Higher order traces Tr[F N ] can be checked and give the same coadjoint orbits so these are all the coadjoint orbits of SU(2). Daniel Bryan (Uni-hamburg) Short title May 5, 2020 14 / 28 Functions on G∗ Consider the space of functions F(G∗) The coadjoint action of G on G∗ induces an action on the functions of G∗. Consider g 2 G and h 2 F(G∗) then Ad∗gh(Ξ) = h(Ad∗g −1(Ξ)) Consider again the function on h(G∗). The differential dh 2 G. This holds because G∗ is a vector space and hence dh is a linear form on G∗ as G∗∗ ∼ G. From this one can also write δΞ 2 G∗ and hence it becomes possible to expand the function as : h(Ξ + δΞ) = h(Ξ + δΞ) = h(Ξ) + δΞ(dh) + O((δΞ)2) Daniel Bryan (Uni-hamburg) Short title May 5, 2020 15 / 28 Kostant -Kirillov symplectic structure The Kostant -Kirillov symplectic structure refers to a symplectic structure on the coadjoint orbit of a Lie Algebra again this is defined by a Poisson bracket and symplectic 2-form.