Modified symplectic structures in cotangent bundles of lie groups

Modified symplectic structures in cotangent bundles of lie groups

F.J. Vanhecke; C. Sigaud; A.R. da Silva

Instituto de Física, Instituto de Matemática, UFRJ, Rio de Janeiro, Brazil

ABSTRACT

In earlier work [1], we studied an extension of the canonical symplectic structure in the cotangent bundle of an affine space Q = R^N, by additional terms implying the Poisson non-commutativity of both configuration and momentum variables. In this article, we claim that such an extension can be done consistently when Q is a Lie group G.

Keywords: Symplectic Mechanics; Noncommutative Configuration Space.

1. INTRODUCTION

As applied to physics, noncommutative geometry is understood mainly in two ways. The first one is the spectral triple approach of A.Connes [2] with the Dirac operator playing a central role in unifying, through the universal action principle, gravitation with the standard model of fundamental interactions. The second one is the quantum field theory on noncommutative spaces [3] with the Moyal product as main ingredient. Besides these, a proposition by several authors [4,5] was made to generalise quantum mechanics in such a way that the operators corresponding to space coordinates no longer commute: [

^k,^ℓ] ≠ 0 . This was implemented by an extension of the Poisson structure on the cotangent space such that the brackets satisfy {x^k, x^ℓ} ≠ 0. Upon quantisation, the corresponding operators should then also be noncommutative. A particle moving in an affine space A^N, has its configuration, in a fixed reference frame, given by an element {x^k} of the translation group: Q = R^N with cotangent bundle T^*(Q) = R^N × R^N. In [1], we examined such an extension of the canonical symplectic two-form ω₀ = dxⁱ ∧ dp_i → Ω = ω₀ + ω_F + ω_B:

This extension is form-invariant under a change of the reference frame lifted to the cotangent bundle:

For a general configuration space Q, a diffeomorphism ø:xⁱ → x^'i øⁱ(x), when lifted to T^*(Q), becomes

In general B'^kℓ is function of both variables {p', x'} and no intrinsic meaning can be given to the particular form of the extension Ω in equation (1.1).

In this work, we show that such an extension is achieved when Q = G is a Lie group. This is possible because the cotangent bundle T^*(G) has two distinguished trivialisations, the left- and right trivialisations [7] implemented respectively by the bases of the left- and right invariant differential forms.

In section 2., inspired by the rigid body motion, we use the left trivialisation with left invariant or body-coordinates and construct a left invariant two-form. In the case of constant F_ij and B^kℓ fields the ω_F term arises from a symplectic one-cocycle, as introduced by Souriau [8,9], and ω_B will be automatically left invariant. The constructed two-form Ω is obviously closed but the non degeneracy condition leads in general to a constrained Hamiltonian system. This is examined in more detail for SU(2) in section 3.. Final considerations are made in section 4.. Some elements of Lie algebra cohomology [9, 10] are recalled in the ^appendix.

2. THE PHASE SPACE {₀≡ T^*(G), ω₀}

Let {g^α, α = 1,2, ..., N} be coordinates of a group element g ∈ G. Natural or holonomic coordinates of points (g, p_g) ∈ T^*(G) are obtained using the basis {dg^µ} of the cotangent space (G). They are given by (g^α, p_µ)_hol, where p_g = p_µ dg^µ. Given a pair of dual bases {e_α} of the Lie algebra

T_e(G) and {ε^α} of its dual ^*, the differential and pull-back of the left- and right translations (L_g, R_g) define left- and right invariant vector fields and one forms: (g) L_g*|ee_α , (g)

_|e e_α , (g)

_-1|gε^α , (g)

_-1|gε^α. With canonical group coordinates, in terms of L^α_β(g, h) ∂(g h)^α/∂g^β and R^α_β (g, h) ∂(h g)^α/∂g^β, they are explicitely given by:

These bases implement canonical trivialisations of the tangent and cotangent bundle. For the cotangent bundle, which is the arena of symplectic or Hamiltonian formalism, we have a left and a right trivialisation:

They can be viewed as a change of coordinates of a point (g, p_g) in T^*(G):

In rigid body theory, the coordinates of the left trivialisation are the "body" coordinates, whence the subscript (,)_B. The right trivialisation yields "space" coordinates with subscript (,)_S. Both are related through the coadjoint representation of G in ^*:

Lifting the left multiplication in G to the cotangent bundle yields a group action: _a:T^*(G) → T^*(G):x = (g, p_g)→ y = (ag, p^'_ag = _^-1|ag p_g). In body coordinates: (_a)_B:(g^α,)_B→ ((ag)^α,)_B. The pull-back of the cotangent projection κ:T^*(G) → G:x (g, p_g) → g, acting on the {ε^α(g)} yield _a invariant one forms on T^*(G) : 〈 (x)| = (κ(x)) and the differentials of the left invariant functions on T^*(G) also yield _a invariant one forms on T^*(G). Together they provide a left invariant basis of the cotangent space at x = (g^α,)_B ∈ T^*(G):

Its dual basis in the tangent space T_x(T^*(G)) is given by

The canonical Liouville one-form 〈θ₀| = p_α 〈dg^α| and its associated symplectic two-form ω₀ = -dθ₀ = 〈dg^α|∧ 〈dp_α|, are obtained as:

The Hamiltonian vector field associated to a function A(g,π^L) on phase space ₀≡ T^*(G), is defined by: ı_Xω₀ = 〈dA|. Its components are:

With ı_Yω₀ = 〈dB| , the Poisson bracket of dynamical variables: {A, B}₀ω₀(X, Y), is obtained explicitely in (g^α,) variables as:

In particular, the basic Poisson brackets are:

The flow of a particular observable, the Hamiltonian H(g, π^L), determines the time evolution of any observable A(g, π^L) by the equation: dA/dt = {A, H}₀. We assume a Hamiltonian is of the form H(g, π^L) = K(π^L) + V(g).

Here, as in rigid body mechanics, the kinetic energy is given by

where I^αβ is the inverse of a constant, positive definite, inertia tensor I_µn in the "body" frame. The potential energy is a function V defined on the group manifold. The Euler equations of motion read:

The first of these equations (2.11) relates the angular momentum with the angular velocity in the body frame :

while the second (2.12) takes the classical form

An example of V(g) is given by a gravitational potential energy as follows. Let L = e_α L^α be a constant vector in (the position of the centre of mass in the body frame) and γ = γ_αε^α a constant vector in ^* (the gravitational force in the space fixed frame). The potential energy is defined as:

where ( | ) denotes the canonical pairing between and its dual ^*. To compute 〈dV|〉 we use the representation of the Maurer-Cartan form:

D(g^-1) dD(g) = D'(g^-1dg)

where D is any representation D of G, with derived representation D' of . In particular, dAd(g) = Ad(g) ad(e_µ) (g) and dK(g) = K(g) k(e_µ) (g). This yields:

where Γ(g) K(g^-1) γ is the variable gravitational force in the body-fixed frame. Using the above formulae to compute dK(g^-1), we obtain:

Equation (2.14) reads:

Together with (2.13),

the equations (2.17) and (2.18) form the so-called Euler-Poisson system.

3. MODIFIED SYMPLECTIC STRUCTURE ON T^*(G)

In ^{appendix A} it is shown that, if Θ = Θ_αβε^α ∧ ε^α ∈ Λ² (^*), obeys the cocycle condition (A.1), then Θ_L(g) (1/2) Θ_αβ (g) ∧ (g) is a closed left-invariant two-form on G. Including this closed two-form in the canonical two-form, one obtains another symplectic two-form on T^*(G), which, furthermore, is _a invariant. So we define:

The Poisson brackets are also modified and (2.8), (2.9) become:

In particular, the fundamental brackets are:

The modified symplectic structure induces an additional interaction and the Euler equations become:

The relation between the velocity in the body frame and the angular momentum (2.13) is maintained: = I_µν , while the second (2.14) takes the interaction into account:

For a semisimple Lie algebra , we have Θ_αβ = - ξ_µ f^µ _αβ and we may define a modified Liouville one-form:

and the symplectic two-form reads

This means that such that {g^α, p'_µ = p_µ + ξ_β L^β_µ (g^-1; g)} are Darboux coordinates:

In (g^α, ) coordinates, the Hamiltonian reads

and the Euler equations read:

which, obviously are equivalent to (3.4) and (3.12).

4. THE CLOSED TWO FORM ω_L

Configuration space coordinates which do not Poisson commute, are obtained through the addition of a left-invariant and closed two-form to (3.1):

With the notation S_{αβ ≡} ( f^µ_αβ - Θ_αβ), we wite ω_L in matrix form:

The degeneracy of (ω_L) is examined comsidering the equation

In the bases (2.4), (2.5): X^α 〈|X〉 , X_µ |X〉 and (4.4) reads:

where we introduced the matrices, linear in the momenta:

They are mutually transposed and the products Φ S = S Ψ , ϒ Φ = Ψ ϒ are antisymmetric. The fundamental equation (4.4), defining Hamiltonian vector fields, has a solution if Φ and Ψ have inverses, i.e. if

The matrices ϒ Φ^-1 = Ψ^-1ϒ and Φ^-1S = S Ψ^-1 are then also antisymmetric. The Hamiltonian vector fields are obtained as:

The Poisson brackets between the basic dynamical variables are:

For a Hamiltonian H = K + V, the equations of motion are:

Since Φ, Ψ are linear in π^L, Δ is a polynomial in π^L of degree at most equal to N, the dimension of the Lie group. It defines an algebraic variety in ^*:

and its complement

_Δ

^*\Π₁ defines a manifold

with symplectic structure given by ω_L, restricted to . If it happens that Π₁ itself is an algebraic manifold, an imbedded submanifold is obtained:

with imbedding in

₀

G×^*: j₁:₁

₀. The system is then constrained to ₁ and we may look for solutions of (4.4) restricted to ₁. Such solutions may exist if further conditions are imposed on the Hamiltonian. To proceed systematically, we follow the algorithm of Gotay, Nester and Hinds [11]. To keep things simple, this will be done in the next section for the semi-simple group SU(2).

5. A CASE STUDY: SU(2)

The dynamical variables are functions on

₀

SU(2) × su(2)^*. A basis {e_α} of the Lie algebra su(2) may be chosen such that its structure constants are the Kronecker symbols [e_α,e_β] = e_µ ε^µ_αβ. The Killing metric η_αβ ε^µ_αν ε^ν_βµ= -2 δ_αβ, provides an isomorphism between su(2) and su(2)^*. The metric δ_αβ with inverse δ^µν will be freely used to raise or to lower indices. Θ_L is written in terms of a magnetic field ξ_µ as Θ_αβ = -ξ_κε^κ_αβ and any antisymmetric ϒ can be written in terms of τ^λ, a dual magnetic field in momentum space, as Y^µν = τ^λε_λ^µν. Defining

+ ξ_κ, ω_L reads:

The fundamental equation (4.4): ı_{|X 〉} ω_L = 〈dH| becomes:

X^αε^κ_αβ - X_β = H_β, X^ν + X_µτ^λε_λµν = H^ν

where H_β (∂H/∂g^α) L^α_β(g, e) , H^ν (∂H/∂ ). The matrices (4.6) are given explicitely by Φ _α^ν C₁δ_α^ν+ τ_απ' ^ν and Ψ^µ_β C₁δ^µ_β+ π^'µτ_β, where C₁ (1-π'·τ). They obey Φ_α^ν (δ_ν^β -τ_νπ^'β) = C₁δ_α^β and Ψ^µ_β (δ^β_ν- π^'βτ_ν) = C₁δ^µ_ν. It follows that (4.5) implies:

5.1. The non degenerate case

The determinant of the matrices Φ and Ψ is given by Δ = (C₁)². Obviously the plane Π₁ {(g,π^L)|(1-π'·τ) = 0} is an algebraic manifold in ^*. Its complement Δ

^*\Π₁ defines a manifold

G × _Δ with symplectic structure ω_L, retricted to . On , Φ and Ψ have inverses:

For a Hamiltonian H = K(π^L) + V(g), the Hamiltonian vector fields are read off from (5.2) and (5.3) with ensuing equations of motion:

For a purely kinetic Hamiltonian, we obtain:

5.2. The degenerate case

The equation C₁≡ (1 - π' · τ) = 0 defines a two dimensional plane Π₁ in su(2)^*≅ R³. The primary constrained manifold, defined by ₁

SU(2) × Π₁, is imbedded in ₀

SU(2) × su(2)^*. On ₁, the closed two-form ω_L is degenerate and the pairing of π' 𕏤 su(2)^* with τ ∈ su(2) equals 1. So |τ 〉 ≠ 0 and, without loss of generality, we take {τ^α} = {0,0, τ}. In what follows, greek indices {α,β,µ,ν,...} shall vary in {1,2,3}, while latin indices {a, b, m, n,...} assume only the values {1,2}. The imbedding is given by:

with its differential or push-forward:

The pull-back transforms forms on

₀ into forms on ₁:

In particular the pull-back of ω_L to the five dimensional manifold ₁ is

The restriction of ω_L to ₁, not to be confused with its pull-back, is denoted by ω_{L| 1}ω_L° j₁. In matrix representation:

Let (T

₀)_{| 1} {(x,X) ∈ T

₀ | x ∈ ₁} be the subbundle of T

₀ restricted to ₁. Following the GNH algorithm [11], we look for a vector field |X〉 in (T

₀)_{| 1}, tangent to ₁ and solution of ı_{|X 〉} ω_{L| 1} = 〈dH| ° j₁. Explicitely :

Two independent null vectors of ω_{L| 1}, solution of ı_|Z〉 ω_{L| 1} = 0, are given by:

Consistency requires {〈dH|Z^a〉 = 0} for (a = 1,2) and = 1/τ.

These two equations define a secondary constrained manifold

₂⊂ ₁, on which a particular solution of (??) is

where C₂₃≡ (∂K/∂)- (∂K/∂)-〈dV|〉. The general solution |X_G〉 of (??), , still contains two arbitrary functions ζ₁ and ζ₂:

This vector must be tangent to

₁ and ₂. This leads to three equations

If these three equations determine or not the two arbitrary functions ζ₁ and ζ₂ , will depend on the kinetic energy K(π^L) and on the particular form of the potential V(g). If they do so, the system will have a solution. If not, they will define a tertiary constraint manifold ₃ and the analysis must proceed.

6. CONCLUSIONS

In this work, we analysed the consistency of a modification of the symplectic two-form on the cotangent bundle of a group manifold. This was done in order to obtain classical, i.e. Poisson, noncommuting configuration (group) coordinates. This was achieved in the non degenerate case, with the closed two-form ω_L which is then symplectic. We do not address here the general quantization problem of such a system and refer e.g. to [12] for a general review on quantization methods. It should be stressed that, whatever the quantisation scheme, any such obtained framework has little to do with non commutative geometry, either in the sense of A.Connes or as a quantum field theory on non-commutative spaces.

(Received on 10 April, 2008)

The one-cocycle σ is called symplectic if Σ(u, v) 〈σ(u)|v〉 is antisymmetric, Σ(u, v) = - Σ(v, u) or Σ_[αµ]σ_α,µ= - σ_µ,α . Any antisymmetric Θ defined in terms of θ ∈ C¹(,^*,k) as Θ_[αβ] = θ_α,β is actually a 2-cochain on with values in R and trivial representation: Θ ∈ C²(,R,0). Furthermore, when θ ∈ Z¹(,^*,k), Θ is a 2-cocycle of Z²(,R,0):

In general let Θ = Θ_αβε^α∧^β∈ Λ²(^*), obey the cocycle condition (A.1). Acting with yields the left-invariant two form:

Using the cocycle relation and the Maurer-Cartan structure equations, it is seen that Θ_L(g) is a closed left-invariant two-form on G.

When is semisimple, Θ is exact. Indeed, the Whitehead lemmas state that H¹(, R, 0) = 0 and H²(, R, 0) = 0. In particular, Θ ∈ B²(,R,0) is a coboundary and there exists an element ξ of C¹(, R, 0) ≡ ^* such that Θ(u, v) = (δ₁(ξ))(u, v) = - ξ([u, v]) or

The constant vector ξ ∈ T^*() is the analogue of a magnetic field in the abelian case G ≡ R³.

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appendix

with values in acts with the coadjoint representation ,→ 〈θ(,

A one-cochain θ on with values in ^*, on which acts with the coadjoint representation k, θ ∈ C¹ (,^*,k), is a linear map θ: → ^*:u → θ (u). Its components are θ_α,µ 〈θ(e_µ)|e_α〉. It is a one-cocycle, θ ∈ Z¹(,^*,k), if its coboundary, (δ₁θ) (u, v) k(u)θ(v)-k(v)θ(u)-θ([u, v]), vanishes.

Publication Dates

• Publication in this collection
  15 June 2009
• Date of issue
  Mar 2009

History

• Received
  10 Apr 2008