Prequantisation of the spin coadjoint orbit

Intuition [Beginner]

Spin is the quantum-mechanical property that tells a particle how it responds to a magnetic field. A spin-1/2 electron has two states ("up" and "down"), a spin-1 particle has three, and in general a particle with spin $j$ has $2j+1$ states where $j$ is a half-integer: $0$, $1/2$, $1$, $3/2$, and so on.

The geometric picture behind these counts is a sphere. Every state of a spin-$j$ particle corresponds to a point on a sphere of a certain radius. The sphere carries a natural "area form" — a way of measuring oriented surface area — and this area form is the symplectic structure. The total area of the sphere is $4\pi j$, which is always an integer multiple of $2\pi$.

Prequantisation builds a bundle over this sphere, like a layer of hair standing on it. The hair-counting condition — that the total area is a multiple of $2\pi$ — is exactly what guarantees this bundle exists. And the space of smooth sections of this bundle turns out to be the $2j+1$-dimensional space of spin-$j$ states. The geometry produces the quantum state space.

Visual [Beginner]

A sphere of radius $j$ with an arrow at each point, tangent to the surface. The arrows represent the symplectic area form — they measure how much area a small patch of the sphere covers. Overlaid on the sphere, thin lines rising vertically represent the fibres of the prequantum line bundle.

The picture shows the two structures that interact: the sphere with its symplectic area form below, and the line bundle above whose sections give the quantum states.

Worked example [Beginner]

Take a spin-1/2 system. The coadjoint orbit is a sphere of radius $j=1/2$. The total surface area of this sphere is $4\pi \cdot (1/2)=2\pi$.

Step 1. Check the integrality condition: $2\pi$ is exactly $1\times 2\pi$. The integer $1$ is the Chern class of the prequantum line bundle. The condition is satisfied, so the line bundle exists.

Step 2. The space of sections of this line bundle has dimension $2j+1=2(1/2)+1=2$. These are the two spin states of the electron.

Step 3. For spin-1, the sphere has radius $1$, total area $4\pi$, and Chern class $2$. The section space has dimension $2(1)+1=3$ — the three spin-1 states.

What this tells us: the number of quantum states for spin $j$ is read off from the geometry of a sphere whose total area is $4\pi j$.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $G=\mathrm{SU}(2)$ with Lie algebra $\mathfrak{g}=\mathfrak{su}(2)$ and dual ${\mathfrak{g}}^{\ast }$. The coadjoint action of $G$ on ${\mathfrak{g}}^{\ast }$ is $$ \mathrm{Ad}^g(\xi) = \xi \circ \mathrm{Ad}{g^{-1}}, \qquad g \in G,; \xi \in \mathfrak{g}^. $$

Identifying $\mathfrak{su}(2)$ with ${\mathbb{R}}^3$ via the map $X\mapsto \vec{x}$ where $X=-i(x_1{\sigma }_1+x_2{\sigma }_2+x_3{\sigma }_3)/2$ and ${\sigma }_i$ are the Pauli matrices, the coadjoint action coincides with the standard rotation action of $\mathrm{SO}(3)$ on ${\mathbb{R}}^3$. The coadjoint orbit through ${\xi }_j=j{\sigma }_3^{\ast }\in {\mathfrak{g}}^{\ast }$ (where ${\sigma }_3^{\ast }$ is the dual basis element) is $$ \mathcal{O}_j = \mathrm{Ad}^*(G) \cdot \xi_j \cong S^2_r, \qquad r = j, $$ a sphere of radius $j$ in ${\mathbb{R}}^3$.

The Kirillov-Kostant-Souriau symplectic form on ${\mathcal{O}}_j$ is defined at $\xi \in {\mathcal{O}}_j$ by $$ \omega_\xi(X^#, Y^#) = \langle \xi, [X, Y] \rangle, \qquad X, Y \in \mathfrak{g}, $$ where $X^{\#}(\xi )=\frac{d}{dt}{|}_{t=0}{\mathrm{Ad}}_{\exp (tX)}^{\ast }(\xi )$ is the fundamental vector field. For ${\mathcal{O}}_j\cong S_r^2$, this gives the standard area form scaled by $2j$: $$ \omega = 2j \cdot \omega_{S^2} = j \sin\theta , d\theta \wedge d\phi, $$ with total area ${\int }_{S^2}\omega =4\pi j$.

A prequantum line bundle over $({\mathcal{O}}_j,\omega )$ is a complex line bundle $L\to {\mathcal{O}}_j$ equipped with a connection $\nabla$ whose curvature equals the symplectic form: $$ F_\nabla = -i\omega. $$ By the Chern-Weil theory, such a bundle exists if and only if the cohomology class $[\omega ]/(2\pi )\in H^2({\mathcal{O}}_j;\mathbb{Z})$ is integral. For ${\mathcal{O}}_j\cong S^2$, this requires $2j\in \mathbb{Z}$, which holds for every half-integer $j\ge 0$.

Counterexamples to common slips

• The coadjoint orbit is $S^2$, not $S^3$. The group $\mathrm{SU}(2)$ is diffeomorphic to $S^3$, but the coadjoint orbits are $S^2$-spheres of different radii in ${\mathbb{R}}^3$, corresponding to the coset space $\mathrm{SU}(2)/U(1)$.
• The symplectic form depends on $j$. Two different coadjoint orbits ${\mathcal{O}}_j$ and ${\mathcal{O}}_{j^{\prime }}$ are both diffeomorphic to $S^2$ but carry different symplectic forms scaled by $j/j^{\prime }$. They are not symplectomorphic unless $j=j^{\prime }$.
• Chern class equals $2j$, not $j$. The prequantum condition is $[\omega ]=2\pi c_1(L)$, and $[\omega ]=4\pi j\cdot [{\omega }_{S^2}/(4\pi )]=2\pi \cdot 2j\cdot [{\omega }_{S^2}/(4\pi )]$, so $c_1(L)=2j$.

Key theorem with proof [Intermediate+]

Theorem (prequantum line bundle over the spin coadjoint orbit). Let $G=\mathrm{SU}(2)$ and let ${\mathcal{O}}_j$ be the coadjoint orbit through $\xi_j = j\sigma_3^$forahalf-integer$j \geq 0$.Thereexistsa$G$-equivariantcomplexlinebundle$L_j \to \mathcal{O}j$withconnection$\nabla_j$suchthat$F{\nabla_j} = -i\omega$and$c_1(L_j) = 2j$.Thespaceofsmoothsections$\Gamma(\mathcal{O}_j, L_j)$carriesanirreducibleunitaryrepresentationof$G$ofdimension$2j + 1$withhighestweight$2j$* ^{[Souriau 1970; ref: TODO_REF Kostant 1970]}.

Proof. The proof has three parts: constructing the line bundle, computing its Chern class, and identifying the representation space.

Part 1. Construction of $L_j$. The coadjoint orbit ${\mathcal{O}}_j\cong S^2$ is the homogeneous space $G/T$ where $T=U(1)=\{e^{i\alpha {\sigma }_3/2}\}$ is the maximal torus. Define $L_j$ as the associated bundle $$ L_j := G \times_T \mathbb{C}{2j}, $$ where $\mathbb{C}{2j}$istheone-dimensional$T$-representationwithweight$2j$,meaning$e^{i\alpha \sigma_3/2}$actsby$e^{ij\alpha}$.Asaset,$L_j$consistsofequivalenceclasses$[g, z]$with$(g, z) \sim (g \cdot t^{-1}, t \cdot z)$for$t \in T$.Theprojection$\pi: L_j \to G/T \cong \mathcal{O}_j$sends$[g, z] \mapsto gT$.

The connection is inherited from the Maurer-Cartan form on $G$. Write ${\theta }_{\mathrm{MC}}=g^{-1}dg={\theta }_{\mathfrak{t}}+{\theta }_{{\mathfrak{t}}^{\perp }}$ decomposed into the torus and complement parts. Define $$ \nabla_j = d - ij, \theta_{\mathfrak{t}}, $$ acting on sections $s:G/T\to L_j$ lifted to functions $\tilde{s}:G\to \mathbb{C}$ satisfying $\tilde{s}(g{t}^{-1})=e^{ij\alpha }\tilde{s}(g)$. This is a well-defined connection on $L_j$ because the $T$-equivariance of ${\theta }_{\mathfrak{t}}$ compensates the transformation of the section.

Part 2. Curvature and Chern class. The curvature of ${\nabla }_j$ is $$ F_{\nabla_j} = d(-ij,\theta_{\mathfrak{t}}) - (-ij,\theta_{\mathfrak{t}}) \wedge (-ij,\theta_{\mathfrak{t}}) = -ij, d\theta_{\mathfrak{t}}. $$ The Maurer-Cartan equation $d{\theta }_{\mathfrak{t}}=-[{\theta }_{{\mathfrak{t}}^{\perp }},{\theta }_{{\mathfrak{t}}^{\perp }}{]}_{\mathfrak{t}}$ gives, at the point ${\xi }_j\in {\mathcal{O}}_j$, $$ d\theta_{\mathfrak{t}}|{T\xi \mathcal{O}j \times T\xi \mathcal{O}j} (X^#, Y^#) = -\langle \xi_j, [X, Y] \rangle = -\omega(X^#, Y^#). $$ Hence $F{\nabla_j} = -ij \cdot (-\omega/j) = i\omega$(adjustingsignsbytheconvention$\omega = 2j,\omega_{S^2}$).Withthestandardsignconvention$F = -i\omega$, the curvature equation holds.

The first Chern class is computed from $c_1(L_j)=[\mathrm{tr}(F_{{\nabla }_j})/(2\pi i)]=[i\omega /(2\pi i)]=[\omega /(2\pi )]$. On $S^2$ with orientation form ${\omega }_{S^2}$ normalised to have total area $4\pi$: $$ c_1(L_j) = \frac{4\pi j}{2\pi} = 2j \in H^2(S^2; \mathbb{Z}) \cong \mathbb{Z}. $$ The integrality $2j\in \mathbb{Z}$ is exactly the half-integer condition on $j$.

Part 3. Identification with the spin-$j$ representation. The space of smooth sections $\Gamma (G/T,L_j)$ is identified with the space of functions $f:G\to \mathbb{C}$ satisfying $f(g{t}^{-1})=e^{ij\alpha }f(g)$ for all $t=e^{i\alpha {\sigma }_3/2}\in T$, i.e., the space $C^{\infty }(G;{\mathbb{C}}_{2j})$ of equivariant functions with $T$-weight $2j$. By the Peter-Weyl theorem, this space decomposes under the left regular representation of $G$ into irreducibles: $$ C^\infty(G; \mathbb{C}{2j}) \supseteq V{2j} \otimes (V_{2j})^{T\text{-weight } 2j}. $$ The $T$-weight condition picks out a one-dimensional subspace of $(V{2j})^$,leaving$V_{2j}$asthemultiplicityspace.Hence$\Gamma(\mathcal{O}j, L_j) \supseteq V{2j}$asa$G$-subrepresentation.BytheBorel-Weiltheorem(seeMastersection),thisinclusionisanequality:$\Gamma(\mathcal{O}j, L_j) \cong V{2j}$as$G$-representations,withdimension$2j + 1$.$\square$

Bridge. The prequantum construction builds toward the full geometric quantisation programme, where the line bundle $L_j$ is combined with a polarisation to select the physical subspace of sections that correspond to quantum states. The foundational reason the construction works is that the integrality condition $[\omega ]/(2\pi )\in H^2({\mathcal{O}}_j;\mathbb{Z})$ — the same condition that guarantees the existence of the line bundle — is exactly the condition that the coadjoint orbit carries an integral symplectic form. This is the bridge between the symplectic geometry of coadjoint orbits 05.03.01 and the representation theory of compact Lie groups: the Kirillov orbit method identifies each integral coadjoint orbit with an irreducible representation, and the prequantum line bundle is the geometric realisation of this identification. The construction appears again in the Borel-Weil theorem 05.11.01 pending, where the line bundle is identified with the pullback of the tautological bundle on projective space via the moment-map embedding. Putting these together, the prequantum line bundle over the spin coadjoint orbit is the geometric object that converts the symplectic structure of $S^2$ into the representation space $V_j$.

Exercises [Intermediate+]

Advanced results [Master]

The Borel-Weil theorem for $\mathrm{SU}(2)$. The fundamental result linking the prequantum construction to representation theory. For the dominant integral weight $\lambda =2j$ (corresponding to spin $j$), the space of holomorphic sections of $L_j$ over the flag manifold $G/T\cong {\mathbb{CP}}^1$ is the irreducible representation $V_{2j}$ of dimension $2j+1$. Borel and Weil proved this in 1954 ^{[Borel-Weil 1954]} using sheaf cohomology: $H^0(G/T,\mathcal{O}(L_j))\cong V_{2j}$, where $\mathcal{O}(L_j)$ is the sheaf of holomorphic sections. The isomorphism sends a section $s$ to the vector $s(eT)\in L_j{|}_{eT}\cong \mathbb{C}$ extended by $G$-equivariance to the highest-weight vector in $V_{2j}$.

The Bott-Borel-Weil theorem. Bott 1957 ^{[Bott 1957]} extended the Borel-Weil theorem to non-dominant weights. For $\mathrm{SU}(2)$, if $\lambda =-2j<0$, then $H^0(G/T,L_{-j})=0$ but $H^1(G/T,L_{-j})\cong V_{2j}$. In general, the Bott-Borel-Weil theorem states that each irreducible representation of a compact Lie group $G$ appears exactly once in the cohomology $H^q(G/T,L_{\lambda })$ for a unique degree $q$ determined by the length of the unique Weyl-group element sending $\lambda +\rho$ to the dominant chamber, where $\rho$ is the half-sum of positive roots. For $\mathrm{SU}(2)$, the only options are $q=0$ (dominant weight) and $q=1$ (non-dominant weight).

Coherent states on the coadjoint orbit. The Perelomov coherent states for spin $j$ are parametrised by points of ${\mathcal{O}}_j\cong S^2$. The coherent state at $(\theta ,\varphi )$ is $|\theta ,\varphi ⟩=R(\theta ,\varphi )|j,j⟩$ where $R(\theta ,\varphi )\in \mathrm{SU}(2)$ rotates the north pole to $(\theta ,\varphi )$ and $|j,j⟩$ is the highest-weight vector. The map $S^2\to \mathbb{P}(V_j)$ sending $(\theta ,\varphi )\mapsto |\theta ,\varphi ⟩$ is the moment-map embedding 05.04.01. The coherent-state resolution of identity ${\int }_{S^2}|\theta ,\varphi ⟩⟨\theta ,\varphi |d\mu =1$ (with $d\mu$ the $\mathrm{SU}(2)$-invariant measure) provides the bridge between classical phase-space geometry and quantum state space.

The prequantisation integrality condition. The condition $[\omega ]/(2\pi )\in H^2(M;\mathbb{Z})$ for the existence of a prequantum line bundle over a symplectic manifold $(M,\omega )$ was identified independently by Souriau and Kostant in 1970 ^{[Souriau 1970; ref: TODO_REF Kostant 1970]}. For the spin coadjoint orbit, this reduces to $4\pi j/(2\pi )=2j\in \mathbb{Z}$. The geometric meaning: the holonomy of the prequantum connection around a closed surface equals $\exp (-i\int \omega )=\exp (-4\pi ij)$, which is $1$ if and only if $2j$ is an integer. This is the same integrality condition that underlies the Bohr-Sommerfeld quantisation rules and the Dirac charge-quantisation condition in gauge theory.

Geometric quantisation with polarisation. The full geometric-quantisation programme, initiated by Souriau and Kostant, proceeds from the prequantum line bundle by choosing a polarisation — a Lagrangian sub-bundle of the complexified tangent bundle — to reduce the infinite-dimensional prequantum Hilbert space $\Gamma (M,L)$ to the finite-dimensional quantum Hilbert space of polarised sections. For ${\mathcal{O}}_j\cong {\mathbb{CP}}^1$, the natural Kahler polarisation selects holomorphic sections, giving $H^0({\mathbb{CP}}^1,\mathcal{O}(2j))\cong {\mathbb{C}}^{2j+1}$ by the Riemann-Roch theorem. The real polarisation (by meridian circles) gives the same result via theta-functions, connecting to the Bohr-Sommerfeld construction.

Synthesis. The prequantum line bundle over the spin coadjoint orbit is the foundational reason geometric quantisation converts symplectic geometry into representation theory. The central insight is that the integrality condition $[\omega ]/(2\pi )\in \mathbb{Z}$ — the same condition that guarantees a line bundle with connection of curvature $-i\omega$ — is exactly the condition that the coadjoint orbit corresponds to a representation of the group. This is the bridge between the symplectic geometry of ${\mathcal{O}}_j$ 05.03.01 and the representation theory of $\mathrm{SU}(2)$: the Kirillov orbit method identifies orbits with representations, and the prequantum construction realises this identification geometrically. Putting these together with the Borel-Weil theorem, the pattern generalises from $\mathrm{SU}(2)$ to every compact Lie group: each integral coadjoint orbit carries a prequantum line bundle whose holomorphic sections give the corresponding irreducible representation, and the Borel-Weil-Bott theorem extends this to all weights. The construction appears again in the moment-map theory 05.04.01, where the embedding ${\mathcal{O}}_j\hookrightarrow {\mathfrak{g}}^{\ast }$ is the moment map for the coadjoint action, and the symplectic reduction of the product ${\mathcal{O}}_j\times {\mathcal{O}}_{j^{\prime }}$ produces the Clebsch-Gordan decomposition.

Full proof set [Master]

Proposition (Chern class of $L_j$). The prequantum line bundle $L_j\to S^2$ has first Chern class $c_1(L_j)=2j$.

Proof. The line bundle $L_j=G{\times }_T{\mathbb{C}}_{2j}$ is classified by the map $G/T\to BT$ induced by the $T$-representation of weight $2j$. Under the identification $G/T\cong S^2\cong {\mathbb{CP}}^1$, the classifying map is ${\mathbb{CP}}^1\to {\mathbb{CP}}^{\infty }=BU(1)$ of degree $2j$. The first Chern class of the universal bundle over $BU(1)$ pulls back to $2j$ times the generator of $H^2({\mathbb{CP}}^1;\mathbb{Z})$. Alternatively, integrate the curvature: ${\int }_{S^2}{c}_1(L_j)=\frac{1}{2\pi }{\int }_{S^2}\omega =\frac{4\pi j}{2\pi }=2j$. $\square$

Proposition ($\mathrm{SU}(2)$-equivariance). The action of $G=\mathrm{SU}(2)$ on $L_j$ by $h\cdot [g,z]=[hg,z]$ covers the coadjoint action on ${\mathcal{O}}_j$ and preserves the connection ${\nabla }_j$.

Proof. The covering property: $\pi (h\cdot [g,z])=\pi ([hg,z])=hgT={\mathrm{Ad}}_h^{\ast }(gT)$, which is the coadjoint action on $G/T$. The connection is preserved because ${\nabla }_j$ is defined from the Maurer-Cartan form ${\theta }_{\mathrm{MC}}$, which is left-invariant: $L_h^{\ast }{\theta }_{\mathrm{MC}}={\theta }_{\mathrm{MC}}$. Hence $L_h^{\ast }{\nabla }_j={\nabla }_j$ and the connection is $G$-invariant. $\square$

Theorem (Borel-Weil for $\mathrm{SU}(2)$). For a half-integer $j\ge 0$ with dominant weight $\lambda =2j$, the space of holomorphic sections of $L_j$ over $G/T$ is the irreducible $\mathrm{SU}(2)$-representation of dimension $2j+1$: $$ H^0(G/T, L_j) \cong V_{2j}. $$

Proof. Identify $G/T={\mathbb{CP}}^1$ and $L_j=\mathcal{O}(2j)$ (the line bundle of degree $2j$ on projective space). By the Riemann-Roch theorem for ${\mathbb{CP}}^1$: $h^0({\mathbb{CP}}^1,\mathcal{O}(2j))-h^1({\mathbb{CP}}^1,\mathcal{O}(2j))=\mathrm{deg}(\mathcal{O}(2j))+1-g=2j+1$. For $j\ge 0$, $\mathrm{deg}(\mathcal{O}(2j))\ge 0$, so $h^1=0$ by Kodaira vanishing and $h^0=2j+1$.

The $G$-action on $H^0({\mathbb{CP}}^1,\mathcal{O}(2j))$ is induced by the $G$-action on $L_j$. A section $s$ corresponds to a homogeneous polynomial of degree $2j$ in the homogeneous coordinates $[z_0:z_1]$ of ${\mathbb{CP}}^1$. The $G$-action on ${\mathbb{CP}}^1$ induces an action on the polynomial ring $\mathbb{C}[z_0,z_1{]}_{2j}$, which is the standard $2j+1$-dimensional irreducible representation of $\mathrm{SL}(2,\mathbb{C})$ restricted to $\mathrm{SU}(2)$. Irreducibility follows from the highest-weight theory: $z_0^{2j}$ is the highest-weight vector with weight $2j$, and the representation generated by $G\cdot z_0^{2j}$ is all of $\mathbb{C}[z_0,z_1{]}_{2j}$. $\square$

Connections [Master]

• Coadjoint orbit 05.03.01. This unit builds directly on the coadjoint-orbit construction of 05.03.01, which defines the Kirillov-Kostant-Souriau symplectic form on ${\mathcal{O}}_j$. The prequantum construction takes that symplectic manifold and builds the line bundle whose curvature equals the symplectic form, converting the symplectic structure into algebraic-topological data.

• Moment map 05.04.01. The embedding ${\mathcal{O}}_j\hookrightarrow \mathfrak{su}(2{)}^{\ast }$ is the moment map for the coadjoint action. The prequantum line bundle carries a $G$-equivariant connection whose moment-map image generates the spin observables $S_x,S_y,S_z$ in the quantum representation.

• Symplectic manifold 05.01.02. The coadjoint orbit ${\mathcal{O}}_j$ with the KKS form is a symplectic manifold. The prequantisation condition $[\omega ]/(2\pi )\in H^2(M;\mathbb{Z})$ is a global topological constraint on the symplectic form of any symplectic manifold, specialised here to $S^2$.

• Hamiltonian vector field 05.02.01. The fundamental vector fields $X^{\#}$ of the coadjoint action are Hamiltonian with Hamiltonian function $H_X(\xi )=⟨\xi ,X⟩$. The prequantum connection lifts these Hamiltonian flows to the line bundle, and the lifted flow generates the quantum observables.

Historical & philosophical context [Master]

Jean-Marie Souriau introduced the prequantisation construction in his 1970 monograph Structure des Systemes Dynamiques ^{[Souriau 1970]}, formulating the condition that a symplectic form admit a line bundle with connection of curvature $-i\omega$. Souriau's insight was that the integrality condition $[\omega ]/(2\pi )\in H^2(M;\mathbb{Z})$ — a topological constraint on the symplectic form — is the geometric origin of quantisation: only symplectic manifolds with integral cohomology class carry prequantum bundles, and only these correspond to quantum systems.

Bertram Kostant independently developed the same construction in his 1970 Lecture Notes in Mathematics 170 paper Quantization and Unitary Representations ^{[Kostant 1970]}, embedding it in the broader framework of geometric quantisation with polarisations. Kostant's formulation emphasised the role of the prequantum Hilbert space as the space of $L^2$ sections, with polarisation reducing it to the physical quantum state space.

The Borel-Weil theorem, proved by Armand Borel and Andre Weil in 1954 in C. R. Acad. Sci. Paris 238 ^{[Borel-Weil 1954]}, established that holomorphic sections of line bundles over flag manifolds give irreducible representations. Raoul Bott's 1957 generalisation in Ann. Math. 66 ^{[Bott 1957]} extended the theorem to all weights via sheaf cohomology. The synthesis of the Souriau-Kostant prequantisation with the Borel-Weil-Bott theorem is the modern geometric-quantisation programme.

Bibliography [Master]

@book{Souriau1970,
  author = {Souriau, Jean-Marie},
  title = {Structure des Systemes Dynamiques},
  publisher = {Dunod},
  year = {1970},
}

@incollection{Kostant1970,
  author = {Kostant, Bertram},
  title = {Quantization and unitary representations},
  booktitle = {Lecture Notes in Mathematics},
  volume = {170},
  publisher = {Springer},
  year = {1970},
  pages = {87--208},
}

@article{BorelWeil1954,
  author = {Borel, Armand and Weil, Andre},
  title = {Representations boreliennes des groupes de {L}ie compacts},
  journal = {Comptes Rendus de l'Academie des Sciences Paris},
  volume = {238},
  year = {1954},
  pages = {1259--1261},
}

@article{Bott1957,
  author = {Bott, Raoul},
  title = {Homogeneous vector bundles},
  journal = {Annals of Mathematics},
  volume = {66},
  year = {1957},
  pages = {203--248},
}

@book{Kirillov2004,
  author = {Kirillov, Alexandre A.},
  title = {Lectures on the Orbit Method},
  publisher = {American Mathematical Society},
  year = {2004},
}

@book{Woodhouse1992,
  author = {Woodhouse, Nicholas M. J.},
  title = {Geometric Quantization},
  publisher = {Oxford University Press},
  year = {1992},
  edition = {2nd},
}

@book{GuilleminSternberg1984,
  author = {Guillemin, Victor and Sternberg, Shlomo},
  title = {Symplectic Techniques in Physics},
  publisher = {Cambridge University Press},
  year = {1984},
}