beware, some prefactors may still need harmonizingβ¦
: Lagrangians and Action functionals + Geometric Quantization
,
,
=
prequantum 1-bundle = , regular, = lift of to
=
,
,
,
,
, ,
,
, ,
,
,
,
,
,
,
, , ,
, ,
,
,
, , ,
, ,
, , ,
, , ,
,
, , ,
The 2-sphere $S^2$ is naturally equipped with the structure of a symplectic manifold with symplectic form its canonical volume form, up to a constant factor. As such, one can consider its geometric quantization.
From the point of view of physics, this yields a quantum mechanical system that describes a rigid rotor or spinor in 3-dimenensional Euclidean space (and just the rotation/spin degrees of freedom, not the translational degrees of freedom). Accordingly, the coresponding Hilbert-space of quantum states carries a natural Hamiltonian action by the angular momentum quantum operators that generate the rotation of the sphere.
More abstractly in mathematics, the 2-sphere is naturally identified with a coadjoint orbit of the special unitary Lie algebra/special unitary group $\mathfrak{su}(2)/SU(2)$ (and hence of the spin group $Spin(3)$, by one of the exceptional isomorphisms). As such the geometric quantization of the 2-sphere is a special case of the general orbit method for producing irreducible representations of suitable Lie groups by geometric quantization. In this case it identifies the quantization of the Hamiltonian action of $SU(2)$ on $S^2$ in terms of quantum operators acting on the space of quantum states with an irreducible representation of $SU(2) \simeq Spin(3)$.
The following walks through all the ingredients and steps of the geometric quantization of the 2-spere. Several of the ingredients/steps described are not specific to the 2-sphere but apply generally in geometric quantization, or more specifically in KΓ€hler polarization quantization, but are spelled out again for completeness. Similarly, much of the discussion is really about the complex geometry of the Riemann sphere.
We introduce first the 2-sphere with its canonical structure of a symplectic manifold and KΓ€hler manifold.
For all of the following it is useful to choose an adapted atlas of coordinate charts on the 2-sphere, namely that induced by regarding it as the Riemann sphere.
Choose any point $x_{\infty} \in S^2$. Then the complement $S^2 - \{x_\infty\}$ is diffeomorphic to the plane $\mathbb{R}^2$, which in turn we regard as the complex plane. Hence
is one coordinate chart on the 2-sphere. Accordingly, removing the antipodal point
yields another coordinate chart
We denote the canonical complex coordinate function on $U_{+/-}$ by $z_{+/-}$, respectively. In terms of these the canonical identification of the two coordinate charts on their common overlap
is given by
Hence on $S^2 - \{x_0, x_\infty\}$ the two coordinate functions are complex inverses of each other
This defines an atlas by two open charts which defines the 2-sphere as a smooth manifold. Since both patches are complex planes and the transition function is a holomorphic function this moreover gives $S^2$ the structure of a complex manifold. As such it is called the Riemann sphere.
In the following we often just write β$z$β for β$z_+$β or β$z_-$β when the chart $U_+$ or $U_-$ is understood.
By the canonical embedding
into the 3-dimensional Cartesian space as the unit 2-sphere given by
the 2-sphere inherits a Riemannian metric as the pullback of the canonical metric $\sum_{i = 1} ^3 d x_i \otimes d x_i$ on $\mathbb{R}^3$.
In the coordinate charts of def. , the canonical functions
have the expressions
$x_1 = \frac{1}{2} \frac{z_+ + \overline{z}_+}{1 - {\vert z_+\vert}^2}$
$x_2 = -\frac{i}{2} \frac{z_+ - \overline{z}_+}{1 - {\vert z_+\vert}^2}$
$x_3 = 1 - \frac{1}{1 + {\vert z_+\vert}^2}$.
The canonical Riemannian metric on $S^2$ induces the volume form $\omega_1 \in \Omega^2(S^2)$ given in the coordinate charts of def. by
More generally, for $S^2 \hookrightarrow \mathbb{R}^3$ embedded as the 2-sphere of radius $k \in (0,\infty)$, the corresponding volume form is
Equipped with this Riemannian metric for $k \in (0,\infty)$ and with the complex manifold structure of def. the 2-sphere is a KΓ€hler manifold with
KΓ€hler potential on $U_\pm$ given by
KΓ€hler form on $U_\pm$ given by
The above KΓ€hler form indeed has the same coordinate expression on both patches $U_\pm$: on $U_+ \cap U_- \simeq (S^2 - \{x_0, x_\infty\})$ we have
In the following we leave the subscript $(-)_k$ implicit and write just β$\omega$β and β$K$β etc, the dependency on the parameter $k$ being understood.
We first discuss the complex line bundles on the 2-sphere
and then consider genuine prequantum line bundles
Hermitian complex line bundles are classified via their first Chern class (just as circle group-principal bundles) by second ordinary cohomology with integer coefficients, which for the 2-sphere is
By the clutching construction a line bundle given by two trivializing sections $\sigma_\pm$ on $U_{\pm}$, respectively, of the trivial line bundle on the coordinate patch $\mathbb{C}$ has class the winding number of the transition function
For $k \in \mathbb{Z} \simeq H^2(S^2, \mathbb{Z})$ we take the standard Cech cocycle representing this class to be that given on $U_+ \cap U_- \simeq S^2 - \{x_0, x_\infty\}$ by the transition function
By the discussion there, a spin structure on $S^2$ is equivalently a square root $\sqrt{\Omega^{1,0}} \in \mathbf{Line}_{\mathbb{C}}(S^2)$ of the canonical line bundle $\Omega^{1,0}$, which here is simply the holomorphic 1-form bundle.
The first Chern class of the canonical line bundle on the 2-sphere is
(up to a unique choice of sign, which determines the isomorphism on the right). Accordingly the 2-sphere has an essentially unique spin structure whose theta characteristic $\sqrt{\Omega^{1,0}}$ is the complex line bundle with unit first Chern class
The canonical section of the holomorphic 1-form bundle on $\mathbb{C}$ is simply the canonical 1-form $d z$ itself. By the coordinate charts of def. we have on $\mathbb{C} - \{0\}$
and so the transition function of the canonical bundle in this local trivialization is
This has winding number $2$. Therefore the first Chern class of the holomorphic 1-form bundle $\Omega^{1,0}$ is $c_1(\Omega^{1,0}) = \pm 2$ (the sign being an arbitrary convention, determined by the identification $H^2(S^2, \mathbb{Z}) \simeq \mathbb{Z}$).
And so it follows that there is a unique spin structure, namely given by choosing $\sqrt{\Omega^{1,0}}$ to be the line bundle on $S^2$ with first Chern class $\pm 1$.
We discuss equipping the above complex line bundles on $S^2$ with connections.
For $k \in H^2(S^2, \mathbb{Z}) \simeq \mathbb{Z} \hookrightarrow \mathbb{R}$, refine the Cech cocycle of def. to a Cech-Deligne cocycle with curvature differential 2-form the KΓ€hler form of prop.
by taking the connection 1-form on $U_{\pm}$ to be
We need to check that the connection 1-form satisfies on $U_+ \cap U_- \simeq (S^2 - \{x_0, x_\infty\})$ the transition law
To see this first notice that on $U_+ \cap U_- \simeq (\mathbb{C} - \{0\})$
With this it follows that on $U_+ \cap U_-$ indeed we have
Compare this invariance of the form of the connection to the invariance of the form of its curvature in remark above.
For $\epsilon \in (0,1)$ let $D_{(\epsilon, \infty)}$ be the disk of radius $\epsilon$ around $x_\infty \in S^2$. Then by the Stokes theorem with def. we have the integral equality
In the limit that $\epsilon \to 0$ this shows that for integral $k$ as above, $\omega$ has integral periods
This is essentially the relation that historically was first recognized by Paul Dirac in the context of what today is known as the Dirac charge quantization argument. In more modern language this expresses the defining homotopy pullback property of ordinary differential cohomology.
The KΓ€hler manifold structure on $S^2$ of prop. which lifts its symplectic structure given by the volume form canonically induces a polarization of the symplectic structure: the corresponding KΓ€hler polarization.
Given a prequantum line bundle $(L,\nabla)$ for $(S^2, \omega)$, a section $\Psi \in \Gamma(L)$ is polarized if for all $v \in \Gamma(T^{0,1}S^2)$ the covariant derivative of $\psi$ along $v$ vanishes
By prop. the connection $\nabla$ is what is called βwell adaptedβ in that in the complex coordinate patch $U_\pm \simeq \mathbb{C}$ the covariant derivative along anti-holomorphic vector fields is given simply by the ordinary anti-holomorphic derivative:
This means that:
The KΓ€hler-polarized sections are precisely the holomorphic sections.
Given a prequantum line bundle $(L,\omega)$ prequantizing $(S^2, \omega)$, the space of quantum states is the vector space of polarized sections of $L \otimes \sqrt{\Omega^{1,0}}$, the wave functions:
Here the tensor product with $\sqrt{\Omega^{1,0}}$ is the metaplectic correction, remark , which by prop. is the essentially unique line bundle of unit first Chern class.
Since by prop. this simply shifts the class of the prequantum bundle by one, and since in much of the traditional literature it is customary to ignore the metaplectic correction (which in the present case is indeed harmless), we will in the following regard $L \otimes \sqrt{\Omega^{1,0}}$ as the prequantum line bundle, by which we mean that we keep referring by β$k$β to the first Chern class of that line bundle on $S^2$ of which we consider sections etc. Strictly speaking the prequantum line bundle hence has class $k-1$ from now on, but this shift collides with all established literature and is not worth bothering with.
It is however maybe worth noticing that the shift of the class by +1 induced by the metaplectic correction can be regarded as lifting the possibly non-regular coadjoint orbits with $k \geq 0$ to the regular ones, with $k \geq 1$.
The dimension of the space of quantum states on the 2-sphere is
where $k \coloneqq c_1(L \otimes \sqrt{\Omega^{1,0}})$ is the first Chern class of the prequantum line bundle with metaplectic correction.
By prop. the polarized sections are precisely the holomorphic sections. By the construction in def. a holomorphic section of $L \otimes \sqrt{\Omega^{1,0}}$ is equivalently a pair of holomorphic functions $\Psi_\pm$ on $\mathbb{C}$ such that on $\mathbb{C} - \{0\}$ they are related by
A linear basis for the space of unconstrained holomorphbic functions on $\mathbb{Z}$ is of course given by the monomials $\{z^n\}$. Therefore a basis for the admissible pairs as above is given by pairs $\Psi_\pm(z_-) = z_-^{n_\pm}$ such that
hence such that
If $k \geq 0$ then this equation has $(k+1)$ solutions. If $k \lt 0$ it has no solution.
For $k = 2$ (by prop. the class of the canonical line bundle) the space of quantum states is 3-dimensional and canonically identified with the vector space underlying the special unitary Lie algebra $\mathfrak{su}(2)$.
For $k = 1$ (by prop. the class of the unique theta characteristic) it is 2-dimensional, being the space of states of a bare spinor in 3-dimensional Euclidean space. This is the standard model for a qbit. Below in example we see how the canonical quantum observables act on this space of states as the Pauli matrices.
We discuss the Hamiltonian action of the special unitary group/spin group $SU(2) \simeq Spin(3)$ on $(S^2, \omega)$.
First we state the abstract situation in terms of the orbit method, then we unwind what this implies in detail.
The Lie algebra $\mathfrak{su}(2)$ as a matrix Lie algebra is the sub Lie algebra on those matrices of the form
The standard basis elements of $\mathfrak{su}(2)$ given by the above presentation are
These are called the Pauli matrices.
The Pauli matrices satisfy the commutator relations
The maximal torus of $SU(2)$ is the circle group $U(1)$. In the above matrix group presentation this is naturally identified with the subgroup of matrices of the form
The coadjoint orbits of the coadjoint action of $SU(2)$ on $\mathfrak{su}(2)$ are equivalent to the subset of the above matrices with $x^2 + y^2 + z^2 = k^2$ for some $k \geq 0$.
These are regular coadjoint orbits for $k \gt 0$.
By the discussion at orbit method β The coadjoint orbit and the flag manifold it follows that
For $k \gt 0$ the corresponding coadjoint orbit and hence the symplectic 2-sphere is equivalent to the quotient/coset space
where the equivalence is induced by the map
Recall the canonical coordinate functions $x_1, x_2, x_2 \colon S^2 \to \mathbb{R}$ from prop. .
For $(L,\nabla)$ a prequantization of $(S,\omega)$ equipped with the above KΓ€hler polarization, the prequantum operators associated to the canonical coordinate functions
from prop. are quantum operators (respect the polarization) and in the canonical trivializaton of $L$ on $U_-$ their action on sections is that of the following vector fields:
For $k = 1$ these act as the Pauli matrices on the space of quantum states $\mathcal{H} \simeq \mathbb{C}^2$.
By the proof of prop. the space of quantum states is spanned by those wavefunctions which on $U_-$ are of the form
These are eigenvectors for $Q(x_3)$:
It is clear that $Q(x_1 - i x_2)$ acts as a lowering operator on these states. Notice that also $Q(x_1 + i x_2)$ indeed acts as a raising operator in that for the special value of $k = 1$ we have indeed
geometric quantization of the 2-sphere
For general references see at orbit method and at geometric quantization.
Reviews with an emphasis on the quantization of the 2-sphere include
section 3.1 of
section 7 of
Discussion in terms of Dirac induction is in
Last revised on February 9, 2017 at 05:31:26. See the history of this page for a list of all contributions to it.