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 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 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: {f1, f2} = −{f2, f1}

{f1, αf2 + βf3} = α{f1, f2} + β{f1, f3}

{f1, f2f3} = {f1, f2}f3 + f2{f1, f3} Leibnitz Rule

{f1, {f2, f3}}} + {f3, {f1, f2}}} + {f2, {f1, f3}}} = 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 = {H, f }. This vector field defines the time evolution of the Hamiltonian f˙ = XH f = {H, f } .

This means that for conserved quantities {H, f1} = {H, f2} = 0

From the Jacobi identity f1 and f2 are conserved quantities then {f1, f2} is also conserved. A Poisson bracket can be degenerate: this happens when functions f (M) exist such that {f , g} = 0 for all g. However this is not the case for those defining a . These functions are known as the center of the .

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 {f1, f2}(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 ω = −Σi

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 5 / 28 Poisson manifolds: R3 example

A 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:

{xi , xj } = 2ijk xk

{x1, x2} = 2x3, {x1, x3} = −2x2, {x2, x3} = 2x1

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 6 / 28 Symplectic manifolds and Hamiltonian

The symplectic manifold (M, ω) 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 ω or dH = iXH Ω(XH , •). A symplectic transformation γ is a bijection acting on points m ∈ M in the phase space γ : M 7→ M. Consider the transformation: ∗ (γ ω)m(V , W ) = ωγ(m)(γ∗V , γ∗W ). Whenever γ∗ω = ω the transformation is symplectic. Infinitesimal transformations are specified by the vector field X on M such that LX ω = 0 where LX = d ◦ iX + iX d is the . Poission brackets transform as γ{f , g} = {γf , γg}.

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 ω. Now taking the lie derivative w.r.t this vector field we can see that

LXH ω = d ◦ (iXH ω) + iXH dω. Using dω = 0 and iXH ω = −dH substituting 2 this in the Lie derivative becomes LXH ω = −d H = 0

2n The simplest example of a symplectic manifold is R . This has cannonical coordinates (pi , qi ) and Poisson brackets {qi , qj } = 0 {pi , pj } = 0 and {pi , qj } = 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 ∈ G by the adjoint action X → (Ad g)(X ) = gXg −1 where g ∈ 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 ∈ G, Ξ ∈ G∗, X ∈ G These actions can be taken infinitesimally whereby the Lie Algebra acts on itsself and its dual: ad X (Y ) = [X , Y ], X , Y ∈ G ad∗ X Ξ(Y ) = −Ξ([X , Y ]), X , Y ∈ G Ξ ∈ 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 ∈ , |α|2 + |β|2 = 1 β α∗ C

Thesu ˜ (2) Lie algebra can be written as:

n  ib c + id o su˜ (2) = b, c, d ∈ ' 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 ∈ g ' g ∈ R . To find the coadjoint orbit of SU(2) one must find the points 0 ∗ 3 0 −1 F ∈ g ' g ∈ R such that F = gFg where g ∈ 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 ∈ ' 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 ∈ R we have r ∈ 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 ∈ G and h ∈ F(G∗) then Ad∗gh(Ξ) = h(Ad∗g −1(Ξ)) Consider again the function on h(G∗). The differential dh ∈ 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 ∈ 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. Kostant -Kirillov bracket ∗ If one considers h1, h2 ∈ F(G ) then the Kostant -Kirillov bracket is defined by:

{h1, h2}(Ξ) = Ξ([dh1, dh2])

This is antisymmetric and satisfies the Jacobi identity. This can be rewritten by letting h1(Ξ) = Ξ(X ) and h2(Ξ) = Ξ(Y ) where X , Y ∈ G and dh1 = X , dh2 = Y are elements of the Lie algebra bracket.

{Ξ(X ), Ξ(Y )} = Ξ([X , Y ])

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 16 / 28 Symplectic 2 form

This can then be used to define the symplectic 2-form on the coadjoint orbit. For any two tangent vectors at the point Ξ of the orbit, ∗ ∗ VX = ad X Ξ and VY = ad Y Ξ, define Kostant -Kirillov 2-form

ωK (VX , Vy ) = Ξ([X , Y ])

This form is closed and non-degenerate on any G-orbit.

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 17 / 28 Bracket on the SU(2) coadjoint orbit

Now we look at the SU(2) example again. In this case the Kostant -Kirillov bracket is the Poisson bracket on the sphere representing the coadjoint orbit. For a given radius r this is defined by the brackets {θ, θ} = 0, {φ, φ} = 0 and {φ, θ} = −{θ, φ}. 3 To continue from this point we look again at the Poisson brackets on R again:

{xi , xj } = 2ijk xk

{x1, x2} = 2x3, {x1, x3} = −2x2, {x2, x3} = 2x1

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 18 / 28 Bracket on the SU(2) coadjoint orbit

We now evaluate { x1 , x } using the Leibnitz rule and translate into x2 3 spherical polar coordinates:

x1 1 x1 1 x1 { , x3} = {x1, x3} − 2 {x2, x3} = (−2x2) − 2 (2x1) x2 x2 x2 x2 x2 2 x1 2 2 =2(−1 − 2 ) = −2(1 + tan(φ) ) = − 2 x2 cos(φ)

From this we can use x1 = f (φ) = tan(φ) and x = f (θ) = r cos(θ) x2 3 The nontrivial bracket {θ, φ} can be determined using the relation ∂f ∂g {f (φ), f (θ)} = ∂φ ∂θ {φ, θ}. 1 2 Hence {tan(φ), r cos(θ)} = −( cos2(φ) )rsin(θ){φ, θ} = − cos(φ)2 . 2 From this one can obtain {φ, θ} = r sin(θ) .

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 19 / 28 Kostant -Kirillov bracket for SU(2)

We can now find the Kostant -Kirillov bracket for SU(2) by considering: 1 ∂f µν ∂g {f , g} = − 2 ∂xµ ω (x) ∂xν . In this case we have:

2 ! µν 0 r sin(θ) ω = 2 − r sin(θ) 0

The Kostant -Kirillov bracket in this case reads: ω = r sin(θ)dθ ∧ dφ.

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 20 / 28 Hamiltonian reduction

Hamiltonian reduction gives a general formalism of physical symmetries in terms of a Hamiltonian which enables indentification of the associated Lie algebras. We consider the action of a Lie group on a symplectic manifold. Considering a point m ∈ M we know that the action of g ∈ G is symplectic if g ⇒ gm is symplectic. This also means that for the Poisson brackets: {f1(gm), f2(gm)} = {f1, f2}(gm) To look at the action of the Lie algebra X ∈ G consider the 1 -parameter group g t = e(tX ). In the limit t → 0 the action of the Lie group on the functions on the d −tX symplectic manifold is: X · f (m) = dt f (e · m)|t=0 From this we get a representation of functions on the Lie algebra: X · (Y · f ) − Y · (X · f ) = [X , Y ]f

Furthermore we have X · f = −LX ·mf where L is the Lie derivative.

The symplecticity condition is : {X · f1 f2} + {f1, X · f2} = X ·{f1, f2}.

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 21 / 28 Locally Hamiltonian actions of the lie algebra

Consider again the Lie group acting on M with a symplectic diffeomorphism. Propose that the action of one-parameter subgroup of G is locally Hamiltonian such that there exists a function HX on M such that:

X · f = {HX , f } Proof:

Consider again the condition {X · f1 f2} + {f1, X · f2} = X ·{f1, f2}. Take f1 = f and f2 = h then the identity becomes: {X · f h} + {f , X · h} − X ·{f , h} = 0 We then use the cannonical Darboux coordinates and expand these brackets: X [(∂pj Xqi − ∂pi Xqj )∂qi f ∂qj h + (∂pj Xpi + ∂qi Xqj )∂pi f ∂qj h i,j

−(∂qj Xqi + ∂pi Xpj )∂qi f ∂pj h + (∂qj Xpi − ∂qi Xpj )∂pi f ∂pj h]

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 22 / 28 Locally Hamiltonian actions of the lie algebra

One can compare this to the 1-form ΩX = −iX ·mω P This can be written as ΩX = i (Xqi dpi − Xpi dqi ) One can notice that the condition dΩX = 0 is satisfied by the previous condition.

This means that there exists locally a function HX such that ΩX = dHX . ∂HX ∂HX Therefore one can write Xq = and Xp = − i ∂pi i ∂qi Remembering that the action of the Lie algebra on the point m as P pi qi X · m = i (X ) ∂pi + (X ) ∂qi this then proves: X · f = P − ∂HX ∂f + ∂HX ∂f = {H , f } i ∂qi ∂pi ∂pi ∂qi X

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 23 / 28 Hamiltonian

If we know there is an invariant 1-form α such that ω = dα

As α is invariant the Lie derivative is vanishing LX α = 0.

Consider LX α = (iX d + diX )α = ω(X , ·) + dα(X ) = 0

From dHX = −iX ω we have dHX − dα(X ) = 0

Therefore we obtain HX = α(X · m).

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 24 / 28 Poissonian action of Lie group

The relation X · f = {HX , f } can be substituted into the Jacobi Identity: X · (Y · f ) − Y · (X · f ) = [X , Y ]f .

This identity then becomes X · (Y · f ) − Y · (X · f ) = {{HX , HY }, f }}

Substituting X · f = {HX , f } into the remaining relations in the above equation one obtains: {H[X ,Y ] − {HX , HY }, f }} = 0

This does not prove in general that H[X ,Y ] = {HX , HY } because constants can also commute with a function. In the special case this is satisfied is called Poissonian if the Hamiltonians are also globally defined and depend linearly on X .

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 25 / 28 Conclusions

We have introduced symplectic geometry by going through Poisson and symplectic manifolds and its Hamiltonian formulation. We have defined coadjoint orbits and the Kostant-Kirillov symplectic structure. We have seen how this works for the SU(2) example by computing its coadjoint orbit and Kostant -Kirillov symplectic structure. We have seen how Hamiltonian reduction works by expressing the action of a 1-parameter subgroup of the Lie algebra in terms of the Hamiltonian.

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 26 / 28 References

(1) O.Babelon, D. Bernard, M. Talon , Laboratoire de Physique Theorique et Hautes Energies, Universites Paris VI–VII, Service de Physique Theorique de Saclay, Gif-sur-Yvette, 2003 Introduction to Classical Integrable Systems, chapter 14

(2) J. Maes, Master Thesis, Department of Mathematics, Utrecht University, April 4th, 2011 An Introduction to the Orbit Method

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 27 / 28 The End

Daniel Bryan (Uni-hamburg) Short title May 5, 2020 28 / 28