Jul 5, 2012 - refer to it as the Bohr-Sommerfeld-Heisenberg quantization [4]. ... The modern version of Bohr-Sommerfeld quantization rules requires that for.
arXiv:1207.1477v1 [math.SG] 5 Jul 2012
Bohr-Sommerfeld-Heisenberg Quantization of the 2-dimensional Harmonic Oscillator Richard Cushman and Jędrzej Śniatycki Abstract We perform a Bohr-Sommerfeld-Heisenberg quantization of the 2dimensional harmonic oscillator, and obtain a reducible unitary representation of SU(2). Each energy level carries an irreducible unitary representation. This leads to a decomposition of the representation of SU(2), obtained by quantization of the harmonic oscillator, into direct sum of irreducible unitary representations. Classical reduction of the energy level, corresponding to an irreducible unitary representation, gives the corresponding coadjoint orbit. However, the representation obtained by Bohr-Sommerfeld-Heisenberg quantization of the coadjoint orbit gives a unitary representation of SU(2) that is the direct sum of two irreducible irreducible representations.
1
Introduction
In the framework of geometric quantization we have formulated a quantization theory based on the Bohr-Sommerfeld quantization rules [1], [9]. The aim of this theory is to provide an alternative approach to quantization of a completely integrable system that incorporates the Bohr-Sommerfeld spectrum of the defining set of commuting dynamical variables of the system. The resulting theory closely resembles Heisenberg’s matrix theory and we refer to it as the Bohr-Sommerfeld-Heisenberg quantization [4]. In this paper we give a detailed treatment of Bohr-Sommerfeld-Heisenberg quantization of the 2-dimensional harmonic oscillator. We show that this approach to quantization yields all the usual results including the quantization representation of U(2). We also reduce of oscillator symmetry of the 1
2-dimensional harmonic oscillator, quantize the reduced space, and discuss commutation of quantization and reduction in this context. Let P be a smooth manifold with symplectic form ω. Consider a completely integrable system (f1 , ..., fn , P, ω), where n is equal to 21 dim P , and f1 , ..., fn are Poisson commuting functions that are independent on an open dense subset U of P . If all integral curves of the Hamiltonian vector fields of f1 , ..., fn are closed, then the joint level sets of f1 , ..., fn form a singular foliation of (P, ω) by n-dimensional Lagrangian tori. The restriction of this singular foliation to the open dense subset U of P is a regular foliation of (U, ω|U ). Each torus T of the foliation has a neighbourhood W in P such that the restriction of ω to W is exact, see [2, appendix D]. In other words, ω|W = dθW . The modern version of Bohr-Sommerfeld quantization rules requires that for each generator Γi of the fundamental group of the n-torus T, we have Z θW = mi h for i = 1, ..., n, (1) Γi
where mi is an integer and h is Planck’s constant. It is easy to verify that the Bohr-Sommerfeld conditions are independent of the choice of the form θW satisfying dθW = ω|W . Intrinsically, the Bohr-Sommerfeld conditions are equivalent to the requirement that the connection on the prequantization line bundle L when restricted to T has trivial holonomy group [7]. Let S be the collection of all tori satisfying the Bohr-Sommerfeld condition. We refer to S as the Bohr-Sommerfeld set of the integrable system (f1 , . . . , fn , P, ω). Since the curvature form of L is symplectic, it follows that the complement of S is open in P . Hence, the representation space H of geometric quantization of an integrable system consists of distribution sections of L supported on the Bohr Sommerfeld set S. Since these distributional sections are covariantly constant along the leaves of the foliation by tori, it follows that each n-torus T ∈ S corresponds to a 1-dimensional subspace HT of H. We choose a inner product ( | ) on H so that the family {HT T ∈ S} consists of mutually orthogonal subspaces. In Bohr-Sommerfeld quantization, one assigns to each n-tuple of Poisson commuting constants of motion f = (f1 , ..., fn ) on P an n-tuple (Qf1 , . . . , Qfn ) of commuting quantum operators Qfk for 1 ≤ k ≤ n such that for each ntorus T ∈ S, the corresponding 1-dimensional space HT of the representation 2
space (H, ( | )) is an eigenspace for each Qfk for 1 ≤ k ≤ n with eigenvalue fk|T . For any smooth function F ∈ C ∞ (Rn ), the composition F (f1 , ..., fn ) is quantizable. The operator QF (f1 ,...,fn ) acts on each HT by multiplication by F (f1 , ..., fn )|T . We assume that Pnthere exist global action-angle variables (Ai , ϕi ) on U such that ω = d( |U i=1 Ai dϕi ). Therefore, we can replace θW by the 1-form Pn i=1 Ai dϕi in equation (1) and rewrite the Bohr sommerfeld conditions as Z Ai dϕi = mi h for each i = 1, ..., n. (2) Γi
Since the actions Ai are independent of the angle variables, we can perform the integration and obtain Ai = mi ~ for each i = 1, ..., n.
(3)
where ~ = h/2π. For each multi-index m = (m1 , ..., mn ), we denote by Tm the torus satisfying the Bohr-Sommerfeld conditions with integers m1 , ..., mn on the right hand side of equation (3). Let SU be the restriction of the Bohr-Sommerfeld set S to the open neighbourhood U of P on which the functions f1 , ..., fn are independent. Consider a subspace HU of H given by the direct sum of 1-dimensional subspaces HTm corresponding to Bohr-Sommerfeld tori Tm ∈ SU . Let em be a basis vector of HTm . Each em is a joint eigenvector of the commuting operators (QA1 , . . . , QAn ) corresponding to the eigenvalue (m1 ~, . . . , mn ~). The vectors (em ) form an orthogonal basis in HU . Thus, (em | em0 ) = 0 if m 6= m0 .
(4)
For each i = 1, ..., n, introduce an operator ai on HU such that ai e(m1 ,..,,mi−1 ,mi ,mi+1 ,...,mn ) = e(m1 ,...mi−1 ,mi −1,mi+1 ,...,mn ).
(5)
In other words, the operator ai shifts the joint eigenspace of (QA1 , . . . , QAn ) corresponding to the eigenvalue (m1 ~, . . . , mn ~) to the joint eigenspace of (QA1 , . . . , QAn ) corresponding to the eigenvalue (m1 ~, . . . , mi−1 ~, (mi − 1)~, mi+1 ~, . . . , mn ~). Let a†i be the adjoint of ai . Equations (4) and (5) yield a†i e(m1 ,..,,mi−1 ,mi ,mi+1 ,...,mn ) = e(m1 ,...mi−1 ,mi +1,mi+1 ,...,mn ) 3
(6)
We refer to the operators ai and a†i as shifting operators.1 For every i = 1, ..., n, we have [ai , QAj ] = ~ ai δij , (7) where δij is Kronecker’s symbol that is equal to 1 if i = j and vanishes if i 6= j. Taking the adjoint, of the preceding equation, we get [a†i , QAi ] = −~a†i δij . The operators ai and a†i are well defined in the Hilbert space H. In [4], we have interpreted them as the result of quantization of apropriate functions on P . We did this as follows. We look for smooth complex-valued functions hi on P satisfying the Poisson bracket relations {hj , Ai } = iδij hj .
(8)
The Dirac quantization condition (9) [Qf , Qh ] = −i~ Q{f,h}
(9)
implies that we may interpret the operator aj as the quantum operator corresponding to hj . In other words, we set aj = Qhj . This choice is consistent with equation (8) because (7) yields [Qhj , QAi ] = −i~ Q{hj ,Ai } = −i~ Qiδij hj = δij ~ Qhj . Since ω|U = and Ai is
Pn
i=1
dAi ∧ dϕi , it follows that the Poisson bracket of eiϕj
{eiϕj , Ai } = XAi eiϕj =
∂ iϕj e ∂ϕi
= iδij eiϕj .
(10)
Comparing equations (8) and (10) we see that we may make the following identification ai = Qeiϕi and a†i = Qe−iϕi for i = 1, ..., n. Clearly, the functions hj = eiϕj are not uniquely defined by equation (8). We can multiply them by arbitrary functions that commute with all actions A1 , . . . , An . Hence, there is a choice involved. We shall use this freedom of choice to satisfy consistency requirements on the boundary of U in P . 1
In representation theory, shifting operators are called ladder operators. The corresponding operators in quantum field theory are called the creation and annihilation operators.
4
2
The classical theory
We now describe the symplectic geometry of the 2-dimensional harmonic oscillator and then reduce the S 1 oscillator symmetry, which is generated by its motion. The configuration space of the 2-dimensional harmonic oscillator is R2 with coordinates x = (x1 , x2 ). The phase space is T ∗ R2 = R4 with coordinates (x, y) = (x1 , x2 , y1 , y2 ). On T ∗ R2 the canonical 1-form is Θ = y1 dx1 + y2 dx2 = hy, dxi. The symplectic form on T ∗ R2 is the closed nondegenerate 2-form ω = dΘ = dy1 ∧ dx1 + dy2 ∧ dx2 whose matrix representation is T dx 0 I2 dx . (11) ω= −I2 0 dy dy The structure matrix W(x, y) = on C ∞ (T ∗ R2 ) is −I0 2 I02 .
{xi , xj } {yi , xj }
{xi , yj } {yi , yj }
of the Poisson bracket { , }
The Hamiltonian function of the 2-dimensional harmonic oscillator is E : T ∗ R2 → R : (x, y) 7→ 21 (x21 + y12 ) + 12 (x22 + y22 ).
(12)
∂ ∂ The corresponding Hamiltonian vector field XE = hX1 , ∂x i + hX2 , ∂y i can be computed using − dE = XE ω = hX2 , dxi − hX1 , dyi. We get
X1 =
∂E ∂y
and X2 = − ∂E . ∂x
(13)
Therefore the equations of motion of the harmonic oscillator are x˙ = y
and y˙ = −x.
(14)
The solution to the above equations is the one parameter family of transformations x (cos t) I2 −(sin t) I2 x E = (15) φt (x, y) = A(t) y (sin t) I2 (cos t) I2 y This defines an S 1 action, called the oscillator symmetry, on T ∗ R2 , which is a map from R to Sp(4, R) that sends t to the 4 × 4 symplectic matrix A(t), which is periodic of period 2π. Since E is constant along the integral curves of XE , the manifold 3 E −1 (e) = {(x, y) ∈ R4 x2 + y 2 = 2e, e > 0} = S√ , 2e
5
(16)
which is a 3-sphere of radius
√
2e, is invariant under the flow of XE .
The configuration space R2 is invariant under the S 1 action S 1 × R2 → t − sin t R : (t, x) 7→ Rt x, where Rt is the matrix cos . This map lifts to a sin t cos t 1 ∗ 2 symplectic action Ψt of S on phase space T R that sends (x, y) to Ψt (x, y) = (Rt x, Rt y). The infinitesimal generator of this action is 2
Y (x, y) =
d Φt (x, y) = (−x2 , x1 , −y2 , y1 ). dt t=0
(17)
The vector field Y is Hamiltonian corresponding to the Hamiltonian function L(x, y) = hy, (x2 , −x1 )i = x1 y2 − x2 y1 ,
(18)
that is, Y = XL . L is readily recognized as the angular momentum. The integral curve of the vector field XL on R4 starting at (x, y) = (x1 , x2 , y1 , y2 ) is s 7→
ϕLs (x, y)
=
cos s sin s 0 0 − sin s cos s 0 0 x , 0 0 cos s sin s y 0 0 − sin s cos s
which is periodic of period 2π. The Hamiltonian of the harmonic oscillator is an integral of XL . The S 1 symmetry Ψ of the harmonic oscillator is called the angular momentum symmetry. Consider the completely integrable system (E, L, R4 , ω) where E (12) and L (18) are the energy and angular momentum of the 2-dimensional harmonic oscillator. The Hamiltonian vector fields XE = y1 ∂x∂ 1 + y2 ∂x∂ 2 − x1 ∂y∂ 1 − x2 ∂y∂ 2 and XL = −x2 ∂x∂ 1 + x1 ∂x∂ 2 − y2 ∂y∂ 1 + y1 ∂y∂ 2 corresponding to the Hamiltonians E and L, respectively, define the generalized distribution D : R4 → T R4 : (x, y) 7→ D(x,y) = span{XE (x, y), XL (x, y)}.
(19)
Before we can investigate the geometry of D (19) we will need some very detailed geometric information about the 2-torus action L Φ : T 2 × R4 → R4 : (t1 , t2 ), (x, y) 7→ ϕE (20) t1 ◦ ϕt2 (x, y) L generated by the 2π periodic flows ϕE t1 and ϕt2 of the vector fields XE and XL , respectively. Using invariant theory we find the space V = R4 /T 2 of orbits
6
of the T 2 -action Φ (20) as follows. The algebra of T 2 -invariant polynomials on R4 is generated by σ1 = 21 (y12 + x21 + y22 + x22 ) and σ2 = x1 y2 − x2 y1 . Let σ3 = 21 (y12 + x21 − y22 − x22 ) and σ4 = x1 y1 + x2 y2 . Then σ12 = σ22 + σ32 + σ42 ≥ σ22 . From the fact that σ1 ≥ 0, we get σ1 ≥ |σ2 |. Thus if σ1 = e ≥ 0, then −e ≤ ` = σ2 ≤ e. So the T 2 orbit space V is the semialgebraic variety {(e, `) ∈ R2 |`| ≤ e}. As a differential space V has a differential structure T2
equal to the space C ∞ (R4 ) of smooth T 2 -invariant functions on R4 , that is, the space of smooth functions in σ1 and σ2 . Note that the T 2 -orbit map π : R4 → V ⊆ R2 : (x, y) 7→ σ1 (x, y), σ2 (x, y) is just the energy momentum mapping EM : R4 → R2 : (x, y) 7→ E(x, y), L(x, y) of the 2-dimensional harmonic oscillator. Abstractly, the Whitney stratification of V is given by V2 = V \ Vsing ⊇ V1 = Vsing \ (Vsing )sing ⊇ V0 = (Vsing )sing ⊇ ∅, where Wsing is the semialgebraic variety of singular points of the semialgebraic variety W . In our situation V2 = {(e, `) ∈ R2 |`| < e}, V1 = {(e, `) ∈ R2 0 < |`| = e}, and V0 = {(e, `) ∈ R2 e = ` = 0}. Then {Vj , j = 0, 1, 2} is the Whitney stratification of V . For j = 0, 1, 2 let Uj = EM−1 (Vj ). It is straightforward to see that the 2 at (x, y) of the T 2 -action Φ (20) is isotropy group T(x,y) 2 T(x,y)
{e}, if (x, y) ∈ U2 S 1 if (x, y) ∈ U1 = 2 T , if (x, y) ∈ U0 .
Thus {Uj , j = 0, 1, 2} is the stratification of R4 by orbit type, which is equal to T 2 -symmetry type where all the isotropy groups are equal, since T 2 is abelian. Because Vj is the image of Uj under the energy momentum map, we see that Uj is the subset of R4 where the rank of DEM is j. Since ∗ ω ] (x, y) : T(x,y) R4 → T(x,y) R4 : XE (x, y) 7→ dE(x, y),
is bijective, it follows that dim D(x,y) = dim span{dE(x, y), dL(x, y)} = rank DEM(x, y).
7
(21)
Therefore Uj is the union of j-tori for j = 0, 1, 2 each of which is an energymomentum level set of EM. We turn to investigating the generalized distribution D (19). Since ω(XE , XL ) = {E, L}, we see that for every (x, y) ∈ R4 we have ω(x, y)D(x,y) = 0. Thus every subspace D(x,y) in the distribution D is an ω-isotropic subspace of the symplectic vector space T(x,y) R4 , ω(x, y) . If (x, y) ∈ Uj then dim D(x,y) = j. So if (x, y) ∈ U2 , then D(x,y) is a Lagrangian subspace of T(x,y) R4 , ω(x, y) . If u(x,y) , v(x,y) ∈ D(x,y) , then thinking of u(x,y) and v(x,y) as vector fields on T R4 we obtain [u(x,y) , v(x,y) ] = 0, because their flows commute. Thus D is an involutive generalized distribution, which is called the energymomentum polarization of the symplectic manifold (R4 , ω). The leaf (= integral manifold) L(x,y) of the distribution D through the point (x, y) ∈ R4 2 . In particular we have is diffeomorphic to T 2 /T(x,y) L(x,y)
T 2 , if (x, y) ∈ U2 T = S 1 , if (x, y) ∈ U1 = 0 T = pt, if (x, y) ∈ U0 . 1
Thus D is a generalized toric distribution. Note that the leaves of D, each of which is an energy-momentum level set of EM, foliate the strata Uj for j = 0, 1, 2. Thus the space R4 /D of leaves of D is the T 2 orbit space V . e
T2 S1 pt
` Figure 1. The bifurcation diagram of the energy momentum map.
We collect all the geometric information about the leaves of the generalized distribution D in the bifurcation diagram of the energy momentum 8
map of the 2-dimensional harmonic oscillator above. The range of this map is indicated by the shaded region. We now find action-angle coordinates for the 2-dimensional harmonic oscillator. Let x1 =
−1 √ (r 2 1
x2 =
√1 (−r1 2
cos ϑ1 + r2 cos ϑ2 ) sin ϑ1 + r2 sin ϑ2 )
y1 =
√1 (r1 2
y2 =
√1 (−r1 2
sin ϑ1 + r2 sin ϑ2 ) cos ϑ1 + r2 cos ϑ2 ).
(22)
A computation shows that E(r, ϑ) = 21 (r12 + r22 ) and L(r, ϑ) = 21 (r12 − r22 ) and that the change of coordinates (22) pulls back the symplectic form ω = dy1 ∧dx1 +dy2 ∧dx2 to the symplectic form Ω = d( 21 r12 )∧d ϑ1 +d( 21 r22 )∧d ϑ2 . Let A1 = 21 r12 = 12 E(r, ϑ) + L(r, ϑ) ≥ 0 and A2 = 12 r22 =
1 2
E(r, ϑ) − L(r, ϑ) ≥ 0.
Then (A1 , A2 , ϑ1 , ϑ2 ) with A1 > 0, and A2 > 0 and symplectic form Ω = dA1 ∧ d ϑ1 + dA2 ∧ d ϑ2 are real analytic action-angle coordinates for the 2dimensional harmonic oscillator. These coordinates extend real analytically to the closed domain A1 ≥ 0 and A2 ≥ 0. Let R4 have coordinates (ξ, η) = (r1 cos ϑ1 , r1 sin ϑ1 , r2 cos ϑ2 , r2 sin ϑ2 ) with symplectic form dη1 ∧ dξ1 + dη2 ∧ dξ2 = r1 dr1 ∧ d ϑ1 + r1 dr2 ∧ d ϑ2 . For j = 1, 2 let zj = ξj + i ηj = rj ei ϑj . Then R4 becomes C2 with coordinates 1 (z1 , z2 ). The flow ϕE t of the vector field XE becomes the complex S -action ϕ : S 1 × C2 → C2 : t, (z1 , z2 ) 7→ (eit z1 , eit z2 ). The algebra of S 1 -invariant polynomials on R4 is generated by π1 = Re z 1 z2 = ξ1 ξ2 + η1 η2
π2 = Im z 1 z2 = ξ1 η2 − ξ2 η1
π3 = =
1 2 1 2
(z1 z 1 − z2 z 2 ) (ξ12
+
η12
−
ξ22
π4 = −
η22 )
=
1 2 1 2
(z1 z 1 + z2 z 1 ) (ξ12 + η12 + ξ22 + η22 )
subject to the relation π12 + π22 = |z 1 z2 |2 = (z1 z 1 )(z2 z 2 ) = (π4 + π3 )(π4 − π3 ) = π42 − π32 , π4 ≥ 0. Note that L expressed in the (ξ, η) coordinates is π3 . 9
Using the Poisson bracket on C ∞ (R4 ) we get the structure matrix W (π) = ({πi , πj }) of the Poisson bracket on R4 with coordinates (π1 , . . . , π4 ), namely {πi , πj } π1 π2 π3 π4
π1 0 −2π3 2π2 0
π2 2π3 0 −2π1 0
π3 −2π2 2π1 0 0
π4 0 0 0 0
Table 1. The Poisson bracket on C ∞ (R4 ).
3
The Hopf fibration and reduction
In this section we discuss the Hopf fibration, which is the reduction map of the harmonic oscillator symmetry of the 2-dimensional harmonic oscillator. 3 = {(ξ, η) ∈ R4 ξ12 + η12 + ξ22 + η22 = 2e} be the 3-sphere of radius Let S√ 2e 2e and let Se2 = {π ∈ R3 π12 + π22 + π32 = e2 } be the 2-sphere of radius e. The map
√
3 ρ : S√ ⊆ R4 → Se2 ⊆ R3 : ζ = (ξ, η) 7→ π = (π1 (ζ), π2 (ζ), π3 (ζ)) 2e
(23)
is called the Hopf fibration. Let π ∈ Se2 . Then ρ−1 (π) is a great circle on 3 S√ . We prove this as follows. 2e Case 1. π ∈ Se2 \ {(0, 0, −e)}. Suppose (ξ, η) ∈ ρ−1 (π). Since ξ12 + η12 + ξ22 + η22 = 2e and ξ12 + η12 − ξ22 − η22 = 2π3 it follows that ξ12 + η12 = e + π3 > 0. Therefore we may solve the linear equations ξ1 η1 ξ2 π1 = (24) −η1 ξ1 η2 π2 to obtain π1 ξ1 − π2 η1 − (e + π3 )ξ2 = 0 π2 ξ1 + π1 η1 − (e + π3 )η2 = 0. The above equations define a 2-plane Ππ in R4 , since ππ12
−π2 −(e+π3 ) 0 π1 0 −(e+π3 )
3 has rank 2. Hence ρ−1 (π) ⊆ Ππ ∩ S√ . Reversing the argument shows that 2e π 3 −1 −1 Π ∩ S√2e ⊆ ρ (π). Thus ρ (π) is a great circle.
10
Case 2. π = (0, 0, −e). Then ξ12 + η12 = 0 which implies ξ1 = η1 = 0. Thus ρ−1 (π) = {(0, ξ2 , 0, η2 ) ∈ R4 ξ22 + η22 = 2e},
(25)
3 which is a circle. Because ρ−1 (π) = S√ ∩ {ξ1 = η1 = 0}, it follows that 2e −1 ρ (π) is a great circle.
Each fiber of the Hopf fibration is an integral curve of the harmonic oscillator vector field XE of energy e. Thus the space E −1 (e)/S 1 of orbits of the harmonic oscillator of energy e is Se2 . In other words, the Hopf fibration ρ (23) is the reduction map of the S 1 symmetry of the harmonic oscillator generated by the vector field XE . Consider a smooth function K : R4 → R, which is invariant under the flow of XE . Then K is an integral of XE , that is, £XE K = 0. Moreover, there is a e : R4 → R such that K(ξ, η) = K(π e 1 (ξ, η), π2 (ξ, η), π3 (ξ, η), smooth function K π4 (ξ, η)). Since {πj , π4 } = 0 for j = 1, . . . , 4, we obtain e = π˙ j = {πj , K}
3 3 X X e e ∂K ∂K e × π)j εjkl {πj , πk } =2 πl = 2(∇K ∂π ∂π k k k=1 l=1 k=1
3 X
for j = 1, 2, 3 and π˙ 4 = 0. e to E −1 (e) and define K e e (π1 , π2 , π3 ) = K(π e 1 , π2 , π3 , Restrict the function K e e × π) is satisfied by integral e). Set π = (π1 , π2 , π3 ) ∈ R3 . Then π˙ = 2(∇K 3 curves of a vector field X on R defined by e e × π). X(π) = 2(∇K
(26)
The 2-sphere Se2 is invariant under the flow of the vector field X, because e e (π) × πi = 0. £X hπ, πi = 2hπ, πi ˙ = 4hπ, ∇K The structure matrix of the Poisson bracket { , } on C ∞ (R3 ) is
W (π) = ({πj , πk }) =
0 −2 π3 −π2
11
−π3 0 π1
π2 −π1 0
(27)
Since ker W (π) = span{π} and Tπ Se2 = span{π}⊥ , the matrix W (π)|Tπ Se2 is invertible. On Se2 define a symplectic form ωe (π)(u, v) = h(W (π)T )−1 u, vi, where u, v ∈ Tπ Se2 . Let y ∈ Tπ Se2 . Since W (π)T y = 2π × y = z we get π × z = π × (2π × y) = 2π × (π × y) = 2(π hπ, yi − y hπ, πi) = −2y hπ, πi = −2e2 y, which implies y = (W (π)T )−1 z = − 2e12 π × z. Therefore ωe (π)(u, v) = − 2e12 hπ × u, vi = − 2e12 hπ, u × vi.
(28)
The vector field X (26) on (Se2 , ωe ) is Hamiltonian with Hamiltonian e e , because function K e e × π) × ui = − 12 hπ × (∇K e e × π), ui ωe (π)(X(π), u) = − e12 hπ, (∇K e e e hπ, πi − πh∇K e e , πi, ui = −h∇K e e , ui = − 12 h∇K e
e e (π)u, = − dK
where u, v ∈ Tw Se2 .
Since L is a function which is invariant under the flow ϕE t of XE on E (e), it induces the function π e3 = π3 |Se2 on the reduced phase space Se2 . We look at the completely integrable reduced system (e π3 , Se2 , ωe ). −1
The flow of the reduced vector field Xπe3 (π) = 2(e3 × π) on Se2 ⊆ R3 is
ϕπte3 (π) = e2tE3 π = 0
where π ∈ Se2 and E3 =1 0
−1 0 0
cos 2t sin 2t 0
− sin 2t 0 cos 2t 0π, 0 1
0 0 0
. Note that ϕπte3 is periodic of period π.
Therefore an integral curve of Xπe3 on Se2 is either a circle or a point. For |`| < e we have π3−1 (`) is the circle {(π1 , π2 , `) ∈ Se2 π12 + π22 = e2 − `2 }; while when ` = ±e, we see that π −1 (`) is the point (0, 0, ±e). Introduce spherical coordinates π1 = e sin θ cos 2ψ,
π2 = e sin θ sin 2ψ, and π3 = e cos θ
with 0 ≤ θ ≤ π and 0 ≤ ψ ≤ π. Therefore the reduced symplectic form on Se2 is ωe = − 1e (e2 sin θ) dθ ∧ dψ, 12
1 volSe2 . Here volSe2 is the standard volume 2-form − 1e hπ, u × vi that is, ωe = 2e R 2 2 2 on Se , where π ∈ Se and u, v ∈ Tπ Se . Note that S 2 volSe2 = 4πe2 . e
Following our convention, for i = 1, 2, 3, we denote by π ei the push-forward to Se2 of the invariant function πi on R3 . We now explain why the functions π ei satisfy commutation relations for su(2). Consider the Poisson algebra A = (C ∞ (R3 ), { , }R3 , ·), where R3 has coordinates (π1 , π2, π3 ), the Poisson 0 2π −2π bracket { , }R3 has structure matrix W (π) = ({πj , πk }) =−2π3 03 2π12 , 2π2
−2π1
0
and · is pointwise multiplication of functions. Since C = π12 + π22 + π32 is a Casimir of A, that is, {C, πi }R3 = 0 for i = 1, 2, 3, it generates a Poisson ideal I of A. Observe that I = {f ∈ C ∞ (R3 ) f |Se2 = 0}, where Se2 = C −1 (e). Therefore B = (C ∞ (Se2 ) = C ∞ (R3 )/I, { , }Se2 , ·) is a Poisson algebra, where {f , g}Se2 = {f, g}R3 |Se2 and f 7→ f is the natural projection map of C ∞ (R3 ) onto C ∞ (Se2 ). Note that f = f |Se2 , because every smooth function on R3 , which vanishes on Se2 , is a multiple of C by a smooth function on R3. Consequently, the structure matrix W of the Pois0 2e π −2e π son bracket { , }Se2 is −2eπ3 03 2eπ12 , where π ei = πi |Se2 . In other words, 2e π2 −2e π1 0 P3 ek , which are the bracket relations for su(2), as de{e πi , π ej }Se2 = 2 k=1 εijk π sired. To see why the reduced phase space Se2 is symplectomorphic to a coadjoint orbit of SU(2) we refer the reader to the appendix. The Hamiltonian vector field Xπ˜3 is Xπ˜3
∂ , ∂ψ
because
ωe = e sin θ dθ = − d(e cos θ) = − de π3 = − da.
Note that the flow of Xπ˜3 on Se2 is periodic of period π. On Se2 \ {(0, 0, ±e)} we have da ∧ dψ = de π3 ∧ dψ = d(e cos θ) ∧ dψ = e sin θ dψ ∧ dθ = ωe . Thus (˜ π3 , ψ) are real analytic action-angle coordinates on Se2 \ {(0, 0, ±e)}. For later purposes we now show the inverse image under the harmonic oscillator reduction map ρ (23) of the reduced level set π e3−1 (`), which is the circle {(π1 , π2 , `) ∈ Se2 π12 + π22 = e2 − `2 }, when |`| < e, or the point (0, 0, sgn `), when |`| = e, is the 2-torus Ta21 ,a2 labeled by the actions A1 = a1 = 21 (e+`) and A2 = a2 = 12 (e−`), when |`| < e, or the 1-torus Ta21 ,0 , where 2 a1 = 2e when ` = e or the 1-torus T0,a , where a2 = 2e when −` = e. Suppose 2 13
that |`| < e. Because ξ12 + ξ22 = e + ` > 0 we can solve the linear equations 1 1 π1 ξ2 ξ1 η1 to obtain ξ2 = e+` = (ξ1 π1 −η1 π2 ) and η2 = e+` (η1 π1 +ξ1 π2 ). π2 η2 −η1 ξ1 −1 −1 Thus ρ (e π3 (`)) is equal to
ξ1 ,
1 (ξ π e+` 1 1
− η1 π2 ), η1 ,
1 (η π e+` 1 1
+ ξ1 π2 ) ∈ R4 π12 + π22 = e2 − `2 & ξ12 + ξ22 = e + ` .
1 Clearly ρ−1 (e π3−1 (`)) is contained in A−1 (a1 ) = A−1 1 ( 2 (e + `)). Since
ξ22 + η22 =
1 (ξ 2 (e+`)2 1
+ η12 )(π12 + π22 ) =
1 (e (e+`)2
+ `)(e2 − `2 ) = e − `,
it follows that ρ−1 (e π3−1 (`)) is contained in A−1 (a2 ) = A−1 ( 12 (e − `)). Therefore ρ−1 (e π3−1 (`)) = A−1 (a1 ) ∩ A−1 (a2 ) = Ta21 ,a2 . The first equality holds since ρ−1 (e π3−1 (`)) is diffeomorphic to a 2-torus. Suppose that |`| = e, then 2 2 2 2 −1 ξ1 + η1 = 0, if −` = e if ` = e . So ρ (e π3−1 (`)) is the circle ξξ12 ++ ηη12 == 2e, ., that is, 2 2 ξ2 + η2 = 0, if ` = e. 2e, if −` = e. 2 2 2 2 the 1-torus Ta1 =e,0 if ` = e, or the 1-torus T0,a2 =e if −` = e.
4
Quantization of the 2d harmonic oscillator
Following the general procedure outlined in the introduction and detailed in [4], in this section we describe the Bohr-Sommerfeld-Heisenberg quantization of the 2-dimensional harmonic oscillator in the energy-momentum representation. In §2 we have shown that action-angle coordinates are (A1 , A2 , ϑ1 , ϑ2 ), where A1 = 21 (E + L) ≥ 0 and A2 = 21 (E − L) ≥ 0.
(29)
Bohr Sommerfeld conditions applied to these action-angle variables are I I A1 dϑ1 = mh and A2 dϑ2 = nh, (30) where m and n are integers, and h is Planck’s constant. Since the actions are independent of the angles, we can integrate equation (30) to get A1 = m~ and A2 = n~, where ~ is Planck’s constant divided by 2π. According to Bohr and Sommerfeld, equations (30) give the joint spectrum {(m~, n~} of the quantum operators QA1 and QA2 corresponding to the dynamical variables A1 and A2 , 14
respectively. Since the actions A1 and A2 are nonnegative, the joint spectrum of the quantum operators (QA1 , QA2 ) is the quarter lattice {(m~, n~) ∈ h(Z≥0 × Z≥0 )}. Let H be the Hilbert space of states of the quantized harmonic oscillator. We assume that the family {em,n ∈ H | m ≥ 0, n ≥ 0} of vectors is an orthogonal basis in H. For each basis vector em,n of H, the 1-dimensional subspace of H spanned by em,n corresponds to a 2-torus labeled by the actions (A1 , A2 ) whose values are ~(m, n). 6a1 s4
s
s
s
s
s2
s
s
s
s
s3 s1 s
s s
s
1
s s
s
2
s s
s
3
s s
s -
4
a2
Figure 2. The quantum lattice for the joint spectrum of the quantum operators (QA1 , QA2 ) of the 2-dimensional harmonic oscillator. The axes are in units of ~.
For each (m, n) in the first quadrant of Z2 , we note that the vector em,n in H satisfies QE em,n = QA1 em,n + QA2 em,n = (m + n)~em,n and QL em,n = QA1 em,n − QA2 em,n = (m − n)~em,n . Then (m + n)~, (m − n)~ ) with (m, n) ∈ Z≥0 × Z≥0 is the joint spectrum of the quantum operators (QE , QL ). Since QE and QL commute, they span a 2-dimensional abelian Lie algebra t . For every nonnegative integer N let HN be the subspace of H spanned by 2
15
L the vectors emn where m + n = N . Then H = N ≥0 HN is an orthogonal direct sum decomposition of H into N +1-dimensional t2 -invariant subspaces. Because every basis vector of HN is an eigenvector of QL , this representation of t2 is reducible. Following the general theory outlined in the introduction, for j = 1, 2 define the shifting operators aj and (aj )† by a1 em,n = em−1,n , (a1 )† em,n = em+1,n , a2 em,n = em,n−1 , (a2 )† em,n = em,n+1 .
These operators correspond to quantum operators aj = Qeiϑj and (aj )† = Qe−iϑj . However, the functions e±iϑj do not extend smoothly to the origin (0, 0) in R2 ⊆ R4 with coordinates (ξj , ηj ). On the other hand, the complex conjugate coordinate functions zj = ξj + i ηj = rj eiϑj and z j = ξj − i ηj = rj e−iϑj
for j = 1, 2,
are smooth on all of R4 . Moreover, they satisfy the required Poisson bracket relations: {zj , zk } = {z j , z k } = 0 and {zj , z k } = −2i δjk . Therefore, we may introduce new quantum operators Qzj and Qzj , where 2m~ em−1,n , when m ≥ 1 0, when m = 0 p = Qz1 em,n = 2(m + 1)~ em+1,n √ 2n~ em,n−1 , when n ≥ 1 = 0, when n = 0 p = Qz2 em,n = 2(n + 1)~ em,n+1 .
Qz1 em,n = Q†z1 em,n Qz2 em,n Q†z2 em,n
√
(31)
Now consider polynomials π1 = Re z 1 z2 , π2 = Im z 1 z2 , π2 and π3 = (z 1 z1 − z 2 z2 ) = L, which are invariant under the S 1 action (t, z) 7→ eit z, generated by the flow of the Hamiltonian vector field XE . Look at the corresponding quantum operators Qπ1 , Qπ2 , and Qπ3 . Then 1 2
Q2π1 em,n = Qz1 z2 em,n + Qz1 z2 em,n = Qz1 Qz2 em,n + Qz1 Qz2 em,n p p = Qz 1 2(n + 1)~ em,n+1 ) + Qz1 ( 2(m + 1)~ em,n−1 p p = 2~ m(n + 1) em−1,n+1 + (m + 1)nem+1,n−1 ,
16
p √ Similarly, Q2iπ2 em,n = 2~ m(n + 1) em−1,n+1 − m + 1, n em+1,n−1 and Q2π3 em,n = 2(m − n)~ em,n . The following calculation shows that the quantum operator Qπ1 on the Hilbert space (H, ( | )) is self adjoint. Q†2π1 em,n = Q†z1 z2 em,n + Q†z1 z2 em,n = Qz† 2 Q†z1 em,n + Q†z2 Qz† 1 em,n
= Q†z2 Qz1 em,n + Q†z2 Qz1 em,n p √ = 2(m + 1)~ Qz2 em+1,n + 2m~ Qz2 em−1,n p p = 2 (m + 1)n~ em+1,n−1 + 2 m(n + 1)~ em−1,n+1 = Q2π1 em,n . Similarly, the linear operators Qπk for k=2, 3, 4 are also self adjoint. We now show that the operators Qπ1 , Qπ2 , and Qπ3 satisfy the commutation relations [Qπ1 , Qπ2 ] = ~i (2Qπ3 ), [Qπ1 , Qπ3 ] = − ~i (2Qπ2 ), [Qπ2 , Qπ3 ] = ~i (2Qπ1 ), which define a Lie algebra that is isomorphic to su(2). We compute [Qπ1 , Qπ2 ]em,n = Qπ1 Qπ2 em,n − Qπ2 Qπ1 em,n p p m(n + 1) em−1,n+1 − (m + 1)nem+1,n−1 = Qπ1 ~i p p − Qπ2 ~ m(n + 1) em−1,n+1 + (m + 1)nem+1,n−1 p p p = ~i m(n + 1)~ (m − 1)(n + 2) em−2,n+2 + m(n + 1) em,n p p p − ~i m(n + 1)~ (m + 1)n em,n + (m + 2)(n − 1) em+2,n−2 p p p (m − 1)(n + 2) em−2,n+2 + m(n + 1) em,n − ~ m(n + 1) ~i p p p − ~ (m + 1)n ~i (m + 1)n em,n + (m + 2)(n − 1) em+2,n−2 = ~i (2~(m − n) em,n ) = ~i (2Qπ3 )em,n .
Similarly [Qπ1 , Qπ3 ]em,n = − ~i (2Qπ2 )em,n and [Qπ2 , Qπ3 ]em,n = ~i (2Qπ1 )em,n . We can rewrite these commutation relations as [ i~1 Qπ1 , i~1 Qπ2 ] =
1 (2Qπ3 ), i~
[ i~1 Qπ1 , i~1 Qπ3 ] = − i~1 (2Qπ2 ),
[ i~1 Qπ2 , i~1 Qπ3 ] =
1 (2Qπ1 ). i~
(32)
Thus we obtain the following representation of su(2) on H by skew hermitian operators µ e : su(2) → u(H, C) : i~1 Qπj 7→ µ e( i~1 Qπj ), 17
where µ e( i~1 Qπj ) is the skew hermitian operator H → H : em,n 7→
1 Q e . i~ πj m,n
A straightforward computation shows that the quantum operator QE commutes with the quantum operators Qπj for j = 1, 2, 3. Thus in addition to the bracket relations (32) for i~1 Qπj , j = 1, 2, 3 we have the relations [ i~1 QE , i~1 Qπj ] = 0,
for j = 1, 2, 3.
(33)
The bracket relations (32) and (33) define the Lie algebra u(2). Therefore we obtain the following representation of u(2) on the Hilbert space (H, ( | )) by linear skew hermitian maps ( 1 ( 1 Q , µ(QE ), E i~ i~ µ : u(2) → u(H, C) : → 7 1 1 Q , j = 1, 2, 3 µ e(Qπj ), j = 1, 2, 3, i~ πj i~ where µ( i~1 QE ) : H → H : em,n 7→
1 Q e . i~ E m,n
Recall that HN = spanC {emn ∈ H m + n = N }. Then HN is a complex (N + 1)-dimensional subspace of H, which is invariant under QL E , being its eigenspace corresponding to the eigenvalue N ~. Therefore H = N ≥0 HN is an orthogonal direct sum decomposition, because QE is self adjoint. Since p p Qπ1 em,n = ~ m(n + 1) em−1,n+1 + (m + 1)n em+1,n−1 p p m(n + 1) em−1,n+1 − (m + 1)n em+1,n−1 Qπ2 em,n = ~i Qπ3 em,n = ~(m − n)em,n
it follows that for j = 1, 2, 3 the linear skew hermitian quantum operator 1 Q maps HN into itself. Thus for j = 1, 2, 3 the linear skew hermitian i~ πj operators i~1 Qπj |HN define a representation of su(2) on HN , which is clearly irreducible. So we have decomposed the representation of u(2) on H into a sum of irreducible u(2) representations.
5
Quantization of the reduced system
For fixed e ≥ 0 we quantize the classical system (e π3 , Se2 , ωe ) obtained by reducing the harmonic oscillator symmetry of the Hamiltonian system (L, T ∗ R2 , ω), see §3. We want to understand the relation of its quantum spectrum to the quantum spectrum of the 2-dimensional harmonic oscillator. 18
Recall that (Se2 , ωe ) is symplectomorphic to a coadjoint orbit of SU(2). Moreover, functions π e1 , π e2 , π e3 correspond to evaluation of elements of the orbit on a basis of su(2), see appendix. Hence, quantization of the system (e π3 , Se2 , ωe ) should lead to a representation of SU(2). We assume that the symplectic manifold (Se2 , ωe ) is prequantizable. In other words, for q ∈ Z we have Z Z 1 1 qh = ωe = 2e volSe2 = 2e (4πe2 ) = 2πe, Se2
Se2
that is, 0 ≤ e = q~, where ~ = Se2 ,
h . 2π
Thus q is a fixed nonnegative integer.
The Bohr-Sommerfeld conditions for the integrable reduced system (e π3 , ωe ) in action-angle coordinates (a, ψ), see §3, are for p ∈ Z I Z π ph = a dψ = ` 2 dψ = 2π`. a−1 (`)
0
The [0, π] → Se2 : ψ 7→ √ second equality √ above follows because the curve e2 − `2 cos 2ψ, e2 − `2 sin 2ψ, ` parametrizes a−1 (`). Thus ` = p~. Since ` = π3 cos θp and | cos θp | ≤ 1, it follows that |`| ≤ e, which implies |p| ≤ q and cos θp = pq . So the Bohr-Sommerfeld set Seq for the reduced integrable system 2 2 (π3 |Sq~ , Sq~ , ωq~ ) with q ∈ Z≥0 is 2 ∈ Z with |p| ≤ q}. Seq = { q~ sin θp cos 2ψ, q~ sin θp sin 2ψ, p~ ∈ Sq~ The set Seq is a disjoint union of 2q − 1 circles of radius ( |p| < q and two points (0, 0, ±q~) when p = ±q.
p q 2 − p2 )~ when
e q which corresponds to the 1Let eep,q with |p| ≤ q be a vector in H 2 ψ ∈ [0, π]} in the Bohrtorus { q~ sin θp cos 2ψ, q~ sin θp sin 2ψ, p~ ∈ Sq~ e q so that he e Sommerfeld set Sq . Define an inner product h , i on H ep,q |e ep0 ,q0 i = 2 δ(p,q),(p0 ,q0 ) . From now on we use the notation π ej , j = 1, 2, 3 for πj |Sq~ . For each integer p with |p| ≤ q we have Qπe3 eep,q = q~ cos θp eep,q = p~ eep,q . e q = span{e So H epq |p| ≤ q} consists of eigenvectors of Qπe3 . 19
(34)
e so that We want to define shifting operators e aq and e a†q on H e aq eep,q = eep−1,q and e a†q eep,q = eep+1,q By general theory, we expect to identify the operators e aq and e a†q with the quantum operators Qeiψ and Qe−iψ , respectively. However, the functions e±iψ 2 are not single-valued on the complement of the poles in Sq~ . In order to get single-valued functions, we have take squares of e±iψ , namely the functions 2 e±2iψ . The functions e±iψ do not extend to smooth functions on Sq~ . However, the functions q e32 )e±2iψ = q~ sin θ cos 2ψ ± i q~ sin θ sin 2ψ = π e1 ± ie π2 π e± = ( (q~)2 − π do. Moreover, we have {e π± , π e3 } = {e π1 , π e3 } ± i {e π2 , π e3 } = 2e π2 ± 2ie π1 = ±2i(e π1 ± ie π2 ) = ±2i π e± . Under reduction the quantum operator i~1 QE corresponds to the reduced quantum operator i~1 Qπe4 and the quantum operator i~1 QL corresponds to the reduced quantum operator i~1 Qπe3 . As in the discussion of quantization of coadjoint in [4], we can define shifting operators Qπe± as follows. Set Qπe+ eep,q = bp eep−2,q and Q†πe+ eep,q = bp+2 eep+2,q and Qπe− eep,q = bp+2 eep+2,q and Q†πe− eep,q = bp eep,q , where bp ∈ R and b−q = 0 = bq+2 . Then Qπe+ Qπe− eep,q = bp+2 Qπe+ eep+2,q = b2p+2 eep,q and Qπe− Qπe+ eep,q = b2p eep,q . So [Qπe+ , Qπe− ]e ep,q = Qπe+ Qπe− eep,q − Qπe− Qπe+ eep,q = (b2p+2 − b2p )e ep,q . From {e π+ , π e− } = {e π1 + ie π2 , π e1 − ie π2 } = −2i {e π1 , π e2 } = −4i π e3 it follows that we should have [Qπe+ , Qπe− ] = −i~ Q{eπ+ ,eπ− } = −i~ Q−4i πe3 = −4~ Qπe3 . 20
(35)
Evaluating both sides of equation (35) on eep,q gives b2p+2 − b2p = −4~2 p, for every p = −q + 2k, where 0 ≤ k ≤ q and k ∈ Z. In particular we have −4~2 q = b2q+2 −b2q = −b2q , since bq+2 = 0 and −4~2 (−q) = b2−q+2 −b2−q = b2−q+2 , since b−q = 0. Now for every p = −q + 2k, where k ∈ {0, 1, 2, . . . , q} we have b2p = (b2p − b2p−2 ) + (b2p−2 − b2p−4 ) + · · · + (b2−q+2 − b2−q ) + b2−q = −4~2 (p − 2) − 4~2 (p − 4) + · · · + −4~2 (−q), = −4~2
k X `=1
(p − 2`),
since b−q = 0
since p = −q + 2k
2
= −4~ (pk − (k + 1)k) = 4~2 k(q + 1 − k) = ~2 (p + q)(q − p + 2).
(36)
Therefore when p ∈ {−q, −q + 2, −q + 4, . . . , q − 2, q} and 2k = p + q we have p Qπe+ eep,q = bp eep−2,q = 2~ k(q + 1 − k) eep−2,q p = ~ (p + q)(q − p + 2) eep−2,q and p Qπe− eep,q = bp+2 eep+2,q = 2~ (k + 1)(q − k) eep+2,q p = ~ (p + q + 2)(q − p) eep+2,q We now show that equation (35) holds. We compute (Qπe+ Qπe− − Qπe− Qπe+ )e ep,q = Qπe+ bp+2 eep+2,q − Qπe− bp eep−2,q
= (b2p+2 − b2p )e ep,q = 4~2 (k + 1)(q − k) − k(q + 1 − k) eep,q = −4~2 (−q + 2k)e ep,q = −4~2 p eep,q = −4~ Qπe3 eep,q .
π+ + π e− ) and π e2 = Since π e1 = 21 (e
1 (e π+ 2i
−π e− ), we get
Qπe1 eep,q = 12 Qπe+ eep,q + 12 Qπe− eep,q = 12 bp eep−2,q + 21 bp+2 eep+2,q and similarly Qπe2 eep,q =
1 2i
bp eep−2,q − 21
1 2i
bp+2 eep+2,q .
(37)
The following calculation shows that the linear operator Qπe1 is self adjoint eq. on H Q†πe1 eep,q = Q†πe+ eep,q + Q†πe− eep,q = bp+2 eep+2,q + bp eep−2,q = Qπe1 eep,q . Similarly, the operators Qπek for k = 2, 3, 4 are self adjoint. The operators Qπe1 , Qπe2 , and Qπe3 satisfy the commutation relations [Qπe1 , Qπe2 ] = −i~ (2Qπe3 ), [Qπe2 , Qπe3 ] = −i~ (2Qπe1 ), [Qπe3 , Qπe1 ] = i~ (−2Qπe2 ),
because [Qπe1 , Qπe2 ]e ep,q = Qπe1 Qπe2 eep,q − Qπe2 Qπe1 eep,q 1 = 2i bp Qπe1 eep−2,q − 2i1 bp+2 Qπe1 eep+2,q − 12 bp Qπe2 eep−2,q − 12 bp+2 Qπe2 eep+2,q eep−2,q + 21 bp eep,q ) = 2i1 bp ( 12 bm−2 e − 2i1 bp+2 ( 12 bp+2 eep+2,q + 12 bp+4 eep+4,q ) − 21 bp ( 2i1 bp−2 eep−2,q − 2i1 bp eep,q ) − 21 bp+2 ( 2i1 bp+2 eep−2,q − 2i1 bp+4 eep,q ) = 4i1 (2b2p − 2b2p+2 )e ep,q = −i~ (2p~)e ep,q = −i~(2Qπe3 )e ep,q .
Similarly, [Qπe2 , Qπe3 ]e ep,q = −i~ (2Qπe1 )e ep,q and [Qπe1 , Qπe3 ]e em,q = −i~ (−2Qπe2 )e ep,q . Therefore the skew hermitian operators bracket relations 1 1 [ i~ Qπe1 , i~ Qπe2 ] =
1 e3 ), i~ (2Qπ
1 1 [ i~ Qπe1 , i~ Qπe3 ] =
1 Qe1 , i~1 Qπe2 , i~ π
1 e2 ), i~ (−2Qπ
and
1 Qe3 i~ π
satisfy the
1 1 [ i~ Qπe2 , i~ Qπe3 ] =
,
1 e1 ) i~ (2Qπ
which defines the Lie algebra su(2). Compare with table 1. Next we construct a representation of su(2) by skew hermitian linear operators. The bracket relations show that for j = 1, 2, 3 the mapping e q , C) : µ eq+1 : su(2) → u(H
1 Qej i~ π
→µ eq+1 ( i~1 Qπej ),
eq → H e q : eep,q 7→ 1 Qπe eep,q is a faithful complex 2q + 1where µ eq+1 ( i~1 Qπej ) : H j i~ e dimensional representation of su(2) on Hq by skew hermitian linear operators. 22
@s @
@ @s @ @
−4
−3
6 e
s @s @ @
−2
s4
s
3
@s @ @
−1
s2
1 @c
s s
1
s s
s
s
-
2
3
4
`
Figure 3. The quantum lattice for the 2-dimensional harmonic oscillator after reducing the oscillator symmetry. The axes are in units of ~.
This representation is can be decomposed into a sum of two irreducible representations as follows. Let e0 = H q
n M m=0
m
span{(Qπe+ ) eeq,q } =
n M m=0
span{e eq−2p,q }
and e1 = H q
n−1 M m=0
m
span{(Qπe+ ) eeq−1,q } =
n−1 M m=0
span{e eq−1−2p,q }.
e 0 and H e 1 are irreducible representations of su(2) of dimension q + 1 Then, H q q eq = H e0 ⊕ H e 1 . Thus, Bohr-Sommerfeldand q, respectively. Moeorever, H q q Heisenberg quantization of coadjoint orbits of SU(2) leads to a reducible su(2) representation by skew hermitian operators. Using the Campbell–Baker–Hausdorff formula and the fact that SU(2) is compact, we can exponentiate the representation µ eq+1 to a unitary represene tation Rq of SU(2) on Hq . For more details see [4].
23
6
Quantum reduction
We now discuss the explicit relation between the Bohr-Sommerfeld quantization of the 2-dimensional harmonic oscillator and the Bohr-Sommerfeld quantization of the reduced systems. Recall that the Bohr-Sommerfeld quantization of the 2-dimensional harmonic oscillator gives rise to the skew hermitian quantum operators p p Qπ1 em,n = ~ m(n + 1) em−1,n+1 + (m + 1)n em+1,n−1 p p Qπ2 em,n = ~i m(n + 1) em−1,n+1 − (m + 1)n em+1,n−1 Qπ3 emn = QL emn = (m − n)~ em,n (38) Qπ4 emn = QE em,n = (m + n)~ em,n (39) on the Hilbert space H = {em,n m ≥ 0 & n ≥ 0} with inner product h , i. Operators, Qπ1 , Qπ2 and Qπ3 generate a representation of su(2), which integrates to a unitary representation of SU(2) on H. We have seen that Qπ1 , Qπ2 and Qπ3 commute with Qπ4 = QE . Hence, fixing an eigenvalue e = q~ of QE , for some q = 0, 1, ..., we get an invariant subspace Hq of H carrying a unitary representation Uq of SU(2). It follows from the equations above that q = m + n, for m ≥ 0, n ≥ 0. Hence, Hq = span{em,n m + n = q, m ≥ 0 & n ≥ 0} =
q M m=0
span{em,q−m }
has dimension q + 1. We shall show that the representation Uq on Hq is irreducible. Consider the quantization of the reduced symplectic manifold (Se2 , ωe ) corresponding to e = q~. We get the quantum operators p p (p + q)(q − p + 2)e ep−2,q + (p + q + 2)(q − p)e ep+2,q p p = 2i1 ~ (p + q)(q − p + 2) ep−2,q − (p + q + 2)(q − p)e ep+2,q = p~ eep,q
Qπe1 eep,q = 12 ~ Qπe2 eep,q Qπe3 eep,q
e q = {e on the Hilbert space H ep,q |p| ≤ q} with inner product h , i.
To each em,n ∈ Hq , we associate eem−n,m+n ∈ H. By definition of Hq , we have m + n = q. Since −(m + n) ≤ m − n ≤ m + n, it follows that 24
e q . This yields a |m − n| ≤ m + n = q. Therefore, eem−n,m+n = eem−n,q ∈ H e q . The range of this map is linear map from Iq : Hq → H Iq
q M m=0
span{em,q−m } =
q M m=0
span{Iq em,q−m } =
q M m=0
e 0. span{e e2m−q,q } = H q
e 0 the restriction of Iq to codomain H e 0 . The map I 0 is Let Iq0 : Hq → H q q q bijective, because its inverse is given by eep,q 7→ e 1 (p+q), 1 (q−p) . Now 2
2
(Qπe4 ◦ Iq0 )em,n = Qπe4 eem−n,q = q~ eem−n,q = (m + n)~ Iq0 em,n = (Iq0 ◦ QE )em,n and similarly (Qπe3 ◦ Iq0 )em,n = (I 0q ◦ QL )em,n . Also (Qπe1 ◦ Iq0 )em,n = Qπe1 eep,q p p = 21 ~ Iq0 2m(2n + 2) em−1,n+1 + (2m + 2)2n em+1,n−1 = (Iq0 ◦ Qπ1 )em,n
and similarly (Qπe2 ◦ Iq0 )em,n = (Iq0 ◦ Qπ2 )em,n . Note that the quantum spectrum of the quantized reduced system on the horizontal line e = q in figure 3 corresponds to the diagonal line e = m + n in figure 1 of the quantum spectrum of the quantized 2-dimensional harmonic oscillator. e 0 conjugates the representation of su(2) generThe operator Iq0 : Hq → H q e 0 and the ated by the skew hermitian operators { i~1 Qπe1 , i~1 Qπe2 , i~1 Qπe3 } on H q representation on Hq generated by the skew hermitian operators { i~1 Qπ1 , i~1 Qπ2 , 1 e 0 is irreducible, it follows that QL }. Since, the representation of su(2) on H q i~ the representation of su(2) on Hq is irreducible. We can extend the representations of su(2) discussed here, to a representation of u(2) by adjoining the operator i~1 QE , which commutes with 1 Qπ1 , i~1 Qπ2 , and i~1 QL . We can decompose the i~ LHilbert space of our quantization of the harmonic oscillator as follows H = q≥0 Hq , where Hq = {em,n ∈ 0 0 e0 = L H e0 e0 H m + n = q}. Let H q≥0 q and I : H → H : em,n 7→ Im+n em,n . The preceding discussion gives a decomposition of the unitary representation of U(2), obtained by Bohr-Sommerfeld-Heisenberg quantization of the harmonic oscillator, into irreducible unitary representations of U(2), each of which occurs with multiplicity 1.
25
We have shown that in 2-dimensional harmonic oscillator, each irreducible component of the quantization representation of U(2) can be identified from the quantization of the corresponding coadjoint orbit reduced system using the inverse image of the momentum map J : R4 → su(2)∗ . Therefore, one could say that the Bohr-Sommerfeld-Heisenberg quantization of harmonic oscillator commutes with reduction.2 However, we have to remember that this notion of commutation of quantization and reduction is weaker than that used by Guillemin and Sternberg [6] because in our case the quantization of e 1 , which does not occur in Hq . the reduced space has a component H q
Appendix. Facts about SU(2) coadjoint orbits
7
∈ Gl(2, C) |α|2 + |β|2 = 1} on C2 The linear action of SU(2) = {U = αβ −β α with coordinates z = (z1 , z2 ) given by SU(2) × C2 → C2 : (U, z) 7→ U z has the following properties. 1) It preserves the 2-form Ω = − 2i dz1 ∧ dz 1 + dz2 ∧ dz 2 .
+ ix 2) It is Hamiltonian, because for every u = y +izix −y−iz ∈ su(2) = {u ∈ T gl(2, C) u + u = 0} the vector field X u = (iz)z1 + (−y + ix)z2 ∂z∂ 1 + (y − ix)z1 + (−iz)z2 ∂z∂ 2
satisfies X
u
− 2i
(iz)z1 + (−y + ix)z2 dz 1 + (y − ix)z1 + (−iz)z2 dz 2 = 21 ∂ x(z1 z 2 + z2 z 1 ) − iy(z1 z 2 − z2 z 2 ) + z(z1 z 1 − z2 z 2 )
Ω=
= ∂J u ,
where J u is the Hermitian form J u (z) = 21 z T (−iu)z =
1 2
x(z1 z 2 + z2 z 1 ) − iy(z1 z 2 − z2 z 2 ) + z(z1 z 1 − z2 z 2 ) .
(40)
From (40) it follows that the mapping su(2) → C ∞ (C2 ) : u 7→ J u is linear. 2
Various interpretations of the term “quantization commutes with reduction” are discussed in [8].
26
3) It has an SU(2)-momentum mapping J : C2 → su(2)∗ : z 7→ J(z), where J(z)u = J u (z) for every u ∈ su(2). The map J is SU(2)-coadjoint equivariant, because for every U ∈ SU(2) we have T J(U z)u = J u (U z) = − 2i (U z)T u(U z) = − 2i z T (U uU )z
= − 2i z T (AdU −1 u)z = J AdU −1 u (z) = (AdU −1 )T (J(z))u.
Identifying the basis 0
i
0
E1 = i 0 , E2 = 1
−1 0
0
i
, and E3 = 0 −i
of su(2) with the standard basis e1 , e2 , and e3 , respectively, of R3 , the SU(2)momentum mapping J becomes the map Re z1 z 2 Im z1 z 2 1 2 2 2 (|z1 | − |z2 | )
ρe : C2 → R3 : (z1 , z2 ) 7→
=
π1 π2 . π3
In particular πj (z) = J(z)Ej . Identifying C2 with coordinates (z1 , z2 ) = (ξ1 + iη1 , ξ2 + iη2 ) with R4 with coordinates (ξ1 , ξ2 , η1 , η2 ) and then restricting 3 = {(z1 , z2 ) ∈ C2 |z1 |2 + |z2 |2 = 2e > 0}, the map ρe becomes the to S√ 2e Hopf fibration 3 ⊆ R4 → Se2 ⊆ R3 : (ξ1 , ξ2 , η1 , η2 ) 7→ ρ : S√ 2e
1 2
ξ1 ξ2 + η1 η2 ξ1 η2 − ξ2 η1 , 2 2 2 2 (ξ1 + η1 ) − (ξ2 + η2 )
which is the reduction map of the harmonic oscillator symmetry. Because 3 SU(2) acts transitively on S√ and J is SU(2)-coadjoint equivariant, the 2e 2 image Se of the map ρ is an SU(2)-coadjoint orbit OJ(z0 ) through J(z). Therefore for every x ∈ Se2 π ej (x) = (πj |OJ(z) )(x) = AdTU −1 (J(z)) Ej for some U ∈ SU(2) such that x = AdTU −1 (J(z)). 4) It preserves the Hamiltonian E = SU(2) we have 1 2
T
E(U z) = z U
T
1 0
0 1
27
1 2
(z1 z 1 + z2 z 2 ), since for every U ∈
U z = 21 z T z = E(z).
The Hamiltonian vector field XE = X1 ∂z∂ 1 + X2 ∂z∂ 2 of E satisfies XE ∂E, that is, − 2i X1 dz 1 − 2i X2 dz 2 = 12 z1 dz 1 + 12 z2 z 2 .
Ω=
So XE (z) = iz. The flow of XE defines the S 1 action S 2 × C2 → C2 : (t, z) 7→ eit z, which commutes with the linear SU(2) action on C2 . The next discussion, analogous to the one used in [4], leads to an explicit expression for the standard symplectic form on the coadjoint orbit Se2 . Define the bijective R-linear mapping j : su(2) → R3 : u =
iz y + ix
−y + ix −iz
7→
x y , z
(41)
which identifies su(2) with R3 . For u, u0 ∈ su(2) a straightforward calculation shows that iz −y + ix iz 0 −y 0 + ix0 , y0 + ix0 = y + ix −iz −iz 0 i(xy 0 − yx0 )
= 2 (zx0 − xz 0 ) + i(yz 0 − zy0 )
−(zx0 − xz 0 ) + i(yz 0 − zy 0 ) . −i(xy 0 − yx0 )
In other words, j([u, u0 ]) = 2 j(u) × j(u0 ),
for every u, u0 ∈ su(2).
Using the bijective R-linear map i : so(3) → R3 :
0 z −y
−z 0 x
y −x 0
identify so(3) with R3 , we may rewrite (42) as
(42) 7→
x y z
to
j(adu u0 ) = 2 j(u) × j(u0−1 (j(u))j(u0 ), that is, j(adu )j −1 (j(u0−1 (j(u))j(u0 ), or equivalently, as j(adu )j −1 = 2 i−1 (j(u)),
for every u ∈ su(2).
(43)
Exponentiating (43) gives e2 i
−1 (j(u))
= ej(adu )j
So j(Adexp u )v = e2 i
−1
= j(eadu )j −1 = j(Adexp u )j −1 .
−1 (j(u))
j(v), 28
for every u, v ∈ su(2).
(44)
The Killing form on su(2) is k(u, u0 ) = 12 tr u(u)T . Explicitly, k(u, u0 ) = 12 tr
iz y + ix
= 21 tr
zz 0 + xx0 + yy 0 + iyx0 − ixy 0 ∗
−y + ix −iz
−iz −y − ix
y − ix iz ∗ zz 0 + xx0 + yy 0 − iyx0 + ixy 0
= xx0 + yy 0 + zz 0 = j(u), j(u0 ) , where ( , ) is the Euclidean inner product on R3 . In other words, k = j ∗ ( , ). The Killing form is Ad-invariant, because if U ∈ SU(2), then k(AdU u, AdU u0 ) = 12 tr (AdU u)(AdU u0 )T = 21 tr (U uU −1 )((U −1 )T (u0 )T (U )T ) = 21 tr U u(u0 )T U −1 = k(u, u0 ). Define the linear mapping k [ : su(2) → su(2)∗ by k [ (u)v = k(u, v) for every u, v ∈ su(2). Since k is nondegenerate, the map k [ is invertible with inverse k ] . The linear map k ] intertwines the coadjoint action of SU(2) on su(2)∗ with the adjoint action of SU(2) on su(2). In other words, for every u ∈ su(2) and every U ∈ SU(2) we have k ] AdTU −1 k [ (u) = AdU u. To see this we argue as follows. For every v ∈ su(2) we have AdTU −1 k [ (u) v = k [ (u)AdU −1 v = k(u, AdU −1 v) = k(AdU u, v), since k is Ad-invariant = k [ (AdU u)v, which proves the assertion. Thus we may identify SU(2) coadjoint orbits on su(2)∗ with SU(2) adjoint orbits on su(2). Let Ow be the SU(2) adjoint orbit through w ∈ su(2). Using (44) we see that j(Ow ) ⊆ Se2 , where e2 = k(w, w) > 0. From (44) it follows that j|Ow is locally an open mapping. Because Se2 is compact, we obtain j(Ow ) = Se2 . We now determine the standard symplectic form on Se2 . From [4] it follows that the standard symplectic form ωw on Ow is ωw (v)(−adv u, −adv u0 ) = −k(v, [u, u0 ]),
where v ∈ Ow and u, u0 ∈ su(2).
So −adv u, −adv u0 ∈ Tv Ow . Let ωe = j ∗ ωw . Then ωe is a symplectic form on Se2 where e2 = k(w, w) > 0. We now find an explicit expression for ωe . Let x = j(v), y = j(u), and y 0 = j(u0 ). Then x ∈ Se2 and x × y, x × y 0 ∈ Tx Se2 and ωe (x)(x × y, x × y 0 ) = 41 ωe (j(v)) 2j(v) × j(u), 2j(v) × j(u0 ) 29
= 14 ωe (j(v)) j(adv u0 ), j(adv u0 ) = 14 ωw (v) − adv u0 , −adv u0 , since ωe = j ∗ ωw and j is linear = − 14 k(v, [u, u0 ]) = − 14 j(v), 2j(u) × j(u0 ) 1 volSe2 . = − 21 (x, y × y 0 ) = 2e The last equality follows using spherical coordinates on Se2 .
References [1] N. Bohr, “On the constitution of atoms and molecules” (Part I), Philosophical Magazine, 26 (1913) 1-25. [2] R.H. Cushman and L.M. Bates, Global aspects of classical integrable systems, Birkhäuser, Basel, 1997. [3] R. Cushman and J.J. Duistermaat, “The quantum mechanical spherical pendulum”, Bull. AMS 19 (1988) 475–479. [4] R. Cushman and J. Śniatycki, Bohr-Sommerfeld-Heisenberg Theory in Geometric Quantization, submitted. [5] R. Cushman and J. Śniatycki, Bohr-Sommerfeld quantization of the spherical pendulum, in preparation, 2012. [6] V. Guillemin and S. Sternberg, “Geometric quantization and multiplicities of group representations”. Invent. Math. 67 (1982) 515-538. [7] J. Śniatycki, Geometric Quantization and Quantum Mechanics, Applied Mathematics Series 30 (1980) Springer Verlag, New York. [8] J. Śniatycki, Differential Geometry of Singular Spaces and Reduction of Symmetries, Cambridge University Press, to appear. [9] A. Sommerfeld, “Zur Theorie der Balmerschen Serie”, Sitzungberichte der Bayerischen Akademie der Wissenschaften (München), mathematischphysikalische Klasse, (1915) 425-458. [10] N.M.J. Woodhouse, Geometric quantization, 2nd edition, Oxford University Press, Oxford, UK, 1997.
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