Sp(1)-

Sp (1)-

SOFIANE BOUARROUDJ AND GUOWU MENG

ABSTRACT. 𝔰𝔬^∗(4n) Sp(1)-. Laplace-Runge-Lenz Sp(1)- . A. Weinstein. 𝔰𝔬^∗(4n) T^∗ℍ_∗ⁿ∕Sp(1). ( ℍ_∗ⁿ := ℍⁿ∖{0} Sp(1) T^∗ℍ_∗ⁿ Sp (1) ℍ_∗ⁿ.)

. Laplace-Runge-Lenz Weinstein.

Date: May 1, 2026.

The authors were supported by the Hong Hong Research Grants Council under RGC Project No. 16304014; SB was also supported by the grant NYUAD-065.

1.

. SO(4) SO(3). 𝔰𝔬(4,2) . I. A. Malkin V. I. Manko [1] 1966. ( 2 [2].) 𝔰𝔬(4,2) .

MICZ-. A. Barut G. Bornzin [3] . MICZ- MICZ- [4] [5] [6].

[7, 8] . [9]: Laplace-Runge-Lenz .

[10] U(1)- ( [11]). . Sp(1)- [12] U(1)-. Laplace-Runge-Lenz Sp(1)- Laplace-Runge-Lenz.

ℍ thesetofquaternions Hn(ℍ) theJordanalgebraofquaternionichermitian matricesofordern Im ℍ thesetofimaginaryquaternions i,j,k thestandardorthonormalbasis forIm ℍ such thatij = k ¯q thequaternionicconjugateofquaternionq,e.g., ¯k = − k Req 1(q+ ¯q) Im q 21(q− ¯q) ⌟ 2theinteriorproductofvectorswith forms ∧ thewedge productofforms d theexteriorderivative operator πX : T∗X → X thecotangentbundleprojection G acompactconnectedLiegroup 𝔤,𝔤∗ theLiealgebraof G and itsdual ξ an elementin𝔤 ⟨, ⟩ eithertheparing ofvectors withco-vectorsorinnerproduct ⟨|⟩ theinnerproductontheJordan algebraHn(ℍ ) Ada theadjointactionofa ∈ G on𝔤 P → X aprincipalG -bundle Θ a𝔤-valueddifferentialone- formon P that definesa principalconnection onP → X Ra therightaction onP bya ∈ G Xξ thevector fieldon P whichrepresentsthe infinitesimal rightaction onP by ξ ∈ 𝔤 F ahamiltonianG -space ℱ := P ×G F thequotientofP × F bytheactionofG Φ : F → 𝔤∗ theG -equivariantmoment map ℱ♯ Sternbergphasespace 𝒲 Weinstein’suniversalphasespace

2.

Sp(1)- n μ 𝔰𝔬^∗(4n) . 𝔰𝔬^∗(4n) H_n(ℍ) n [13]. ( [14] ).

u ∈ V := H_n(ℍ) L_u u u,v ∈ V S_uv = [L_u,L_v] + L_uv [L_u,L_v] : L_uL_v −L_vL_u uv L_uv L_u(v) u v. {uvw} S_uv(w).

[Suv,Szw ] = S{uvz}w − Sz{vuw} forany u,v,z,w inV .

S_uv . 𝔰𝔱𝔯 V . 𝔰𝔱𝔯 = 𝔰𝔲^∗(2n) ⊕ ℝ ℝ L_e — e.

𝔠𝔬 𝔰𝔱𝔯.

𝔠𝔬 = V ⊕ 𝔰𝔱𝔯⊕ V∗.

z V X_z :

[Suv,Xz] = X {uvz}

V ^∗ . V V ^∗ w W Y _w

[Suv,Yw] = − Y{vuw}.

[Xu,Yv] = 0, [Yu,Yv] = 0, [Xu,Yv] = − 2Suv forany u,v inV .

𝔠𝔬 = 𝔰𝔬^∗(4n) .

3.

Sternberg [15] U(1)- [10].

A. Weinstein Sternberg. Sternberg . . A. Weinstein [16] Weinstein.

3.1. S. Sternberg A. Weinstein. S. Sternberg [15] A. Weinstein [16]. Sternberg:

(i): G G- P → X Θ
(ii): G F Ω G- Φ: F →𝔤^∗. 𝔤 G.

- G Kirillov-Kostant-Souriau G.

Θ G- P → X 𝔤 P :

1) R_a⁻¹^∗Θ = Ad_aΘ a ∈ G 2) Θ(X_ξ) = ξ ξ ∈𝔤.

a ∈ G R_a⁻¹ a⁻¹ P Ad_a a 𝔤 X_ξ P ξ ∈𝔤 P f ∈ C^∞(P)

|| || (3.1) ℒX ξf ||p =-d|| f(p⋅exp(tξ)). dt t=0

[ℒ_{X_ξ},ℒ_{X_η}] = ℒ_{X_[ξ,η]} [X_ξ,X_η] = X_[ξ,η].

() G P G T^∗P G- ρ: T^∗P →𝔤^∗. π_P T^∗P → P ρ

⟨ρ(z),ξ⟩ = ⟨z,Xξ(πP(z))⟩.

: p ∈ P

𝔤 → TpP (3.2) ξ ↦→ X ξ(p)

T_p^∗P 𝔤^∗. ρ: T^∗P →𝔤^∗.

G- ρ (R_g)_∗(X_ξ) = X_{Ad_g⁻¹(ξ)} Ad_g^∗(α) = α∘ Ad_g⁻¹ α ∈𝔤^∗. ρ d⟨ρ,ξ⟩ = − $ˆX$ _ξ ⌟ ω_P. ω_P T^∗P $ˆX$ _ξ T^∗P X_ξ.

∂Xi z = pi(z)dxi|πP(z), ωP = dpi ∧dxi, X ξ = Xiξ ∂-i, Xˆξ = Xiξ-∂i − pi-ξj-∂ ∂x ∂x ∂x ∂pj

⟨ρ,ξ⟩ = p_iX_ξⁱ. xⁱ P .

ρ Φ G-

ψ : T∗P × F → 𝔤∗.

(x,y) −ρ(x) + Φ(y).

P → X G- ρ|_{T_p^∗P}: T_p^∗P →𝔤^∗ p ∈ P ρ: T^∗P →𝔤^∗ . ψ . ψ⁻¹(0) T^∗P ×F . 0 ∈𝔤^∗ G ψ⁻¹(0) 1 [17] ω ψ⁻¹(0)∕G π^∗ω = ι^∗(ω_P + Ω) ω_P P π ι :

ψ− 1(0) T∗P × F πι ψ− 1(0)∕G

(ψ⁻¹(0)∕G,ω) Weinstein 𝒲. .

“” Θ P → X . p ∈ P x p X ( Θ) T_xX → T_pP .

T∗P T∗X ππPX P X

X P . $P˜$

∗ T X π P X X

G- T^∗P → $˜P$ F G-

∗ ˜ αΘ : T P × F → P × F.

Weinstein α_Θ ψ⁻¹(0) G

-- αΘ : ψ−1(0)∕G → P˜×G F.

ω_Θ ℱ^♯ := $˜P$ ×_GF α_Θ^∗(ω_Θ) = ω. (ℱ^♯,ω_Θ) Sternberg Θ [15]. Sternberg Weinstein 𝒲.

3.2. Sternberg Sp(1)-. Sp(1)- ( ) ( ) n ≥ 2 ( ) μ ≥ 0. n μ Sternberg ℱ_μ^♯ [12]:

(i)

G Sp(1) ( SU(2)) G-

n ℍ ∗ → 𝒞1 † (3.3) Z ↦→ nZZ

¯ (3.4) Θ = Im(Z-⋅d2Z-). |Z|

𝒞₁ H_n(ℍ) n. H_n(ℍ) 𝒞₁ H_n(ℍ). 𝒞₁ H_n(ℍ) 𝒞₁ [12] .

(ii)

𝔤^∗ 𝔤 := Im ℍ ⟨,⟩ i j k 𝔤. G

¯ 2 (3.5) F := {ξ ∈ Imℍ | ξξ = μ }

Ω_μ Kirillov-Kostant-Souriau

⟨ξ,dξ ∧dξ⟩ (3.6) Ω μ = --2|ξ|2---.

ξ = ξ¹i + ξ²j + ξ³k F :

1 2 3 2 3 1 3 1 2 (3.7) {ξ ,ξ } = ξ , {ξ ,ξ } = ξ , {ξ ,ξ } = ξ .

(: ij = k.)

(iii)

G-

Φ : F → Im ℍ (3.8) ξ ↦→ 2ξ

( G F Im ℍ .) Φ G 2 (3.8)

d⟨Φ,η⟩ = X η ⌟ Ωμ.

X_η F F η X_η(ξ) = (ξ,[η,ξ]) ∈ T_ξF .

3.3. Weinstein Sp(1)-. Sp(1) ℍ_∗ⁿ (Z,q) ∈ ℍ_∗ⁿ × Sp(1) Z [q] — Z 1 × 1- [q]. Sp (1) T^∗ℍ_∗ⁿ . Tℍ_∗ⁿ ℍ Tℍ_∗ⁿ Z W . ℍⁿ:

⟨U,V ⟩ = Re(U†V),

T^∗ℍ_∗ⁿ Tℍ_∗ⁿ Tℍ_∗ⁿ : U,V ∈ ℍⁿ

{⟨U,Z ⟩,⟨V,W ⟩} = ⟨U,V ⟩, {⟨U,Z ⟩,⟨V,Z⟩} = {⟨U,W ⟩,⟨V,W ⟩} = 0.

T^∗ℍ_∗ⁿ Tℍ_∗ⁿ 𝔤^∗ 𝔤 ρ Tℍ_∗ⁿ 𝔤 (Z,W) −Im(W^†Z). ψ: Tℍ_∗ⁿ × F →𝔤

† (3.9) ψ (Z, W,ξ) = Im(W Z) + 2ξ.

g (Z,W,ξ) (Z ⋅ g⁻¹,W ⋅ g⁻¹,gξg⁻¹).

ψ $ψ˜$ : Tℍ_∗ⁿ ×𝔤 →𝔤 : $˜ψ(Z,W, ξ) = Im (W †Z )+ 2ξ.$ $˜ψ$ ⁻¹(0) ξ = − $1 2$ Im(W^†Z). G-. .

† † {⟨i,W Z ⟩,⟨j,W Z ⟩} = {⟨W i,Z⟩,⟨W j,Z ⟩} = {⟨W i,Z⟩,⟨W j,Z ⟩}− < i ↔ j > = − {⟨W i,Z ⟩,⟨W, Zj⟩}− < i ↔ j > = − ⟨W i,Zj⟩†− < i ↔ j > = − 2⟨k,W Z⟩.

ξ¹ = − $1 2$ ⟨i,W^†Z⟩ ξ² = − $1 2$ ⟨j,W^†Z⟩ ξ³ = − $1 2$ ⟨k,W^†Z⟩ {ξ¹,ξ²} = ξ³ (3.7) .

Sp(1)- Tℍ_∗ⁿ ×𝔤 → Tℍ_∗ⁿ $ψ˜$ ⁻¹(0) Sp(1)- $ψ˜$ ⁻¹(0) Tℍ_∗ⁿ. ψ⁻¹(0) Sp(1)- $ψ˜$ ⁻¹(0)

{ } −1 1 † 𝒲 μ := ψ (0)∕Sp(1) = Sp(1)⋅(Z,W, ξ) | ξ = − 2Im(W Z),|ξ| = μ

Tℍ_∗ⁿ∕Sp(1). 𝒲_μ

{Sp(1) ⋅(Z,W ) | |Im(W †Z)| = 2μ}

Tℍ_∗ⁿ∕Sp(1). Sternberg ℱ_μ^♯ Weinstein 𝒲_μ Sternberg Tℍ_∗ⁿ∕Sp(1).

(ii) 1 H_n(ℍ) Tℍ_∗ⁿ∕Sp(1) 𝒲_μ Sternberg ℱ_μ^♯.

4.

Sp(1)- n μ. Sternberg ℱ_μ^♯ T^∗𝒞₁. V := H_n(ℍ) T^∗𝒞₁ T𝒞₁ ℱ_μ^♯ T𝒞₁ . x π ( V )

T𝒞 1 TV ιxπτtVV V

ι τ_V t V . x π ℱ_μ^♯ → T𝒞₁ x π. H_n(ℍ) ⟨∣⟩ u ∈ H_n(ℍ) ⟨x∣u⟩ ℱ_μ^♯.

1. H_n(ℍ) Sternberg ℱ_μ^♯ Y _u 𝒴_u := ⟨x∣u⟩ X_e 𝒳_e := ⟨x∣π²⟩ + $-μ2-- ⟨e|x⟩$ . (e_α) H_n(ℍ) ℒ_u L_u :

∑ (4.1) 2- ℒ2eα = ℒ2e + 𝒳e𝒴e − μ2. n α

( ) X_e Y _u .

. Sternberg Weinstein

-- −1 ♯ ˜ (4.2) αΘ : 𝒲μ := ψ (0)∕Sp(1) → ℱμ := P ×Sp(1) F,

Weinstein 𝒲_μ .

(i).

Sp(1) ⋅ (Z,W,ξ) 𝒲_μ 𝒴_u 𝒳_e 1.

𝒴 ∘α- (Sp(1)⋅(Z,W, ξ)) = ⟨Z, uZ⟩, 𝒳 ∘ α-(Sp(1)⋅(Z, W,ξ)) = 1 |W |2. u Θ e Θ 4

(ii).

u v V := H_n(ℍ) Sp(1)-

( 1 ||{ 𝒳u := 4⟨W,uW ⟩ (4.3) 𝒴v := ⟨Z, vZ⟩ ||( 1 𝒮uv := 2⟨W,(u⋅v)Z ⟩

Tℍ_∗ⁿ. u ⋅ v u v. u v z w V :

( ||{𝒳u, 𝒳v } = 0, {𝒴u,𝒴v} = 0, {𝒳u, 𝒴v} = − 2𝒮uv, ||{ {𝒮uv, 𝒳z} = 𝒳 {uvz}, {𝒮uv,𝒴z} = − 𝒴{vuz}, |||| ( {𝒮uv, 𝒮zw} = 𝒮{uvz}w − 𝒮z{vuw}.

T^∗ℍ_∗ⁿ∕Sp(1) H_n(ℍ).

(iii).

e V (e_α) V ℒ _u = 𝒮 _eu u ∈ V .

∑ (4.4) -2 ℒ 2eα = ℒ 2e + 𝒳e 𝒴e − μ2. n α

Proof.

(i).

α_Θ Θ. Z ∈ ℍ_∗ⁿ x = nZZ^†. (x,ẋ) 𝒞₁ x (Z,Ż) Z ℍ_∗ⁿ. (3.4)

† † † n(Z˙Z + ZZ˙ ) = ˙x, Im (Z ˙Z) = 0.

1 trx˙ ˙Z = n|Z|2(x˙Z − -2- Z).

Θ- T^∗ℍ_∗ⁿ → T^∗𝒞₁ (Z,⟨W, ⟩) (x,⟨π∣⟩) x = nZZ^†

1 ⟨ tr ˙x ⟩ (4.5) ⟨π | ˙x⟩ = n|Z-|2 W, ˙xZ − -2-Z .

(x,ux) ∈ T_x𝒞₁

1 ⟨ tr(ux) ⟩ ⟨π | ux⟩ = n-|Z-|2 W,(ux)Z − --2---Z ux istheJordan productofu withx 1 ⟨ † † † ⟩ = 2|Z|2 W, uZZ Z + ZZ uZ − Retr(uZZ )Z 1 (4.6) = 2 ⟨W, uZ⟩.

-- (4.7) αΘ(G ⋅(Z,W, ξ)) = G⋅(Z,nZZ †,π,ξ).

-- † † 𝒴u ∘ αΘ(G ⋅(Z, W,ξ)) = ⟨x | u⟩ = ⟨nZZ | u⟩ = Re tr (ZZ u) = ⟨Z,uZ ⟩

-- μ2 nμ2 𝒳e ∘αΘ(G ⋅(Z,W,ξ)) = ⟨x | π2⟩+ ⟨e | x⟩-= ⟨π | πx⟩ + trx 1 μ2 = - ⟨W, πZ ⟩+ --2- using Eq.(4.6) 2 |Z | 2 = n-⟨π | (ZW †)+⟩+ μ-2. 2 |Z |

(ZW^†)₊ = $1 2$ (ZW^† + WZ^†). (x,(ZW^†)₊) 𝒞₁ x. (4.5)

⟨ ⟩ 𝒳 ∘α- (G ⋅(Z,W, ξ)) = -1--- W, (ZW †) Z − tr-(ZW--†)+-Z + -μ2- e Θ 2|Z |2 ⟨ + 2 ⟩ |Z|2 -1--- † ⟨W,-Z⟩- -μ2- = 2|Z |2 W, (ZW )+Z − 2 Z + |Z |2 -1---⟨ † ⟩ --1-- 2 μ2-- = 2|Z |2 W,(ZW )+Z − 4|Z|2⟨W, Z⟩ + |Z|2 1 ( 2 2 † 2 2) μ2 = 4|Z-|2- |W ||Z| + Re(W Z) − ⟨W, Z⟩ + |Z|2 1 ( ) μ2 = 4|Z-|2- |W |2|Z|2 + (Im(W †Z))2 + |Z-|2- 1 ( ) μ2 = ---2- |W |2|Z|2 − |Im (W †Z )|2 +---2 41|Z | |Z| = 4|W |2.

(ii).

{𝒳 _u, 𝒳 _v} = 0 {𝒴 _u, 𝒴 _v} = 0.

1 {𝒳u,𝒴v} = 4 {⟨W, uW ⟩,⟨Z,vZ ⟩} = {⟨W, uW ⟩,⟨Z, vZ⟩} = − ⟨uW,vZ ⟩ = − 2𝒮uv

{𝒮uv,𝒮zw } = 1 {⟨W, (u⋅v)Z⟩,⟨W,(z ⋅w )Z⟩} 41 = 4 {⟨W, (u⋅v)Z⟩,⟨W,(z ⋅w )Z⟩}− < (u⋅v) ↔ (z ⋅w) > 1 † 1 † = − 4⟨(z ⋅w) W,(u ⋅v)Z ⟩+ 4⟨(u⋅v) W,(z ⋅w )Z ⟩ = 1⟨W, [u ⋅v,z ⋅w]Z⟩ 4 = 𝒮 {uvz}w − 𝒮z {vuw}

{uvz} = $1 2$ (u ⋅ v ⋅ z + z ⋅ v ⋅ u) {vuw} = $1 2$ (v ⋅ u ⋅ w + w ⋅ u ⋅ v).

1 1 {𝒮uv,𝒳z } = 8 {⟨W, (u⋅v)Z⟩,⟨W,zW ⟩} = 4 {⟨W,(u ⋅v)Z ⟩,⟨W, zW ⟩} 1 1 = 4⟨W, (u ⋅v⋅z)W ⟩ = 4 ⟨W, (z ⋅v⋅u)W ⟩ = 1⟨W, (z ⋅v⋅u +u ⋅v⋅z)W ⟩ 8 = 𝒳 {uvz}.

1 {𝒮uv,𝒴z} = -{⟨W,(u ⋅v)Z ⟩,⟨Z,zZ ⟩} = {⟨W,(u ⋅v)Z ⟩,⟨Z,zZ ⟩} = −2⟨Z,(z ⋅u⋅v)Z ⟩ = − ⟨Z,(v⋅u ⋅z)Z⟩ 1 = − 2⟨Z,(z ⋅u ⋅v+ v ⋅u⋅z)Z⟩ = − 𝒴{vuz}.

𝒮 _uv 𝒳 _z 𝒴 _w Tℍ_∗ⁿ ( T^∗ℍ_∗ⁿ) Sp(1) Sp(1) T^∗ℍ_∗ⁿ T^∗ℍ_∗ⁿ∕Sp(1) H_n(ℍ).

(iii).

ℒ _u = $1 2$ ⟨W,uZ⟩

2∑ ℒ 2 = -1-∑ ⟨W, eαZ⟩2 = n∑ ⟨(ZW †)+ | eα ⟩2 n α eα 2n α 2 α n- † † 1 ( † )2 = 2 ⟨(ZW )+ | (ZW )+⟩ = 2tr (ZW )+ = 1tr (ZW † + W Z †)2 8 = 1tr(Z(W †ZW †)+ (W Z †W )Z † + |W |2ZZ † + |Z |2W W †) 81 † 2 † 2 1 2 2 = 8((W Z) + (Z W ) )+ 4|W ||Z| -1 † † 2 -1 † † 2 1 2 2 = 16(W Z − Z W ) + 16(W Z + Z W ) + 4|W ||Z| = − 1|Im(W †Z )|2 + 1⟨W, Z⟩2 + 1|W |2|Z|2 42 2 4 4 = − μ + ℒe + 𝒳e 𝒴e.

□

1. . (ii) 𝔰𝔬^∗(4n) T^∗ℍ_∗ⁿ∕Sp(1) . Sternberg T^∗ℍ_∗ⁿ∕Sp(1) (i) 1. 1 (iii) .

4.1. 1 Sp(1)- n μ.

4.2. Sternberg T^∗ℍ_∗ⁿ∕Sp(1) Sternberg T^∗ℍ_∗ⁿ∕Sp(1). Sp(1) T^∗ℍ_∗ⁿ Sp(1) n 6 [12].

4.3. [9] 1 Sp(1)- T𝒞₁

H = 1𝒳e-− -1- 2𝒴e 𝒴e

Laplace-Runge-Lenz

( ) 𝒜 = 1 𝒳 − 𝒴 𝒳e- + 𝒴u- u 2 u u𝒴e 𝒴e

𝒳_u 𝒴_u X_u Y _u 1.

1⟨x|π2⟩ -μ2 1 H = 2 r + 2r2 − r,

1.1 [12]. r = $trx- n$ .

5.

1 S_uv X_z Y _w

𝒮u,v, 𝒳z, 𝒴w

. ℒ_u 𝒮_ue ℒ_u,v $1 2$ (𝒮_uv + 𝒮_vu).

1: Laplace-Runge-Lenz. [10].

1. e_α H_n(ℍ). α β. 1 :

(i): 𝒳_{e_α}ℒ_{e_α} = n𝒳_eℒ_e 𝒴_{e_α}ℒ_{e_α} = n𝒴_eℒ_e
(ii): 4 nℒ_{e_α,u}ℒ_{e_α} = −𝒳_u𝒴_e + 𝒳_e𝒴_u
(iii): 𝒳_{e_α}² = n𝒳_e² 𝒴_{e_α}² = n𝒴_e²
(iv): 2 nℒ_{e_α,u}𝒳_{e_α} = −𝒳_uℒ_e + ℒ_u𝒳_e 2 nℒ_{e_α,u}𝒴_{e_α} = 𝒴_uℒ_e −ℒ_u𝒴_e
(v): 𝒳_{e_α}𝒴_{e_α} = n(ℒ_e² + μ²),
(vi): $4- n3$ ℒ_{e_α,e_β}² = 𝒳_e𝒴_e −ℒ_e² + $n−2 n$ μ².

2. e_α H_n(ℍ) L² = 1 2 ∑ _α,βℒ_{e_α,e_β}² A² = −1 + ∑ _α𝒜_{e_α}². H

( 2 n2(n − 1) 2) (n )2 2 (5.1) − 2H L − ---4----μ = 2- (n− 1− A ).

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