# Global geometry of physical systems and its quantum manifestations

This article appeared in issue 2017-2 of Periodiek, the magazine of the FMF student association.

Looking at geometric properties of physical systems from a global point
of view reveals a rich geometric structure which can be described using
ideas from **differential geometry** and **algebraic topology** [1].
These geometric properties are relevant to physical systems as
fundamental as the hydrogen atom and, even though they arise in the
context of classical mechanics, they have significant implications for
the corresponding **quantum** system.

To describe the main ideas that come into play in this research
direction I will focus on the example of the **spherical pendulum**. At
first sight, the spherical pendulum is not a very intriguing system. It
is **integrable**, that is, the differential equations describing its
motion can be solved in terms of known (but non-elementary)
functions.^{1} And you could reasonably ask, if we can find all motions,
then don’t we know everything about the system? Well, not quite. Looking
at individual motions is like looking through a microscope and may lead
to missing the whole picture. There are questions related to **global
properties** which can be answered only if we take a step back and look
at the system from a more abstract point of view.

The integrability of the spherical pendulum is not just a happy
accident. It has a much deeper reason, related to the existence of two
conserved quantities,^{2} the total **energy**

$$E = \frac{1}{2}\left( \dot{\theta}^{2} + \sin^2\theta,\dot{\phi}^{2} \right) + \cos\theta,$$

and the (vertical component of the) **angular momentum**

$$L = \sin^2\theta\ \dot{\phi}.$$

Here we are using spherical coordinates where
$\theta \in \left\lbrack 0,\pi \right\rbrack$ is the inclination angle,
$\phi \in \left\lbrack 0,2\pi \right\rbrack$ is the azimuthal angle, and
a dot denotes the corresponding time derivative. A very famous theorem
in classical mechanics, the Liouville-Arnold theorem, states that the
existence of these integrals (together with some technical conditions)
implies that the system is integrable. Moreover, the same theorem allows
us to make a first step toward a global description of an integrable
system. Applied to the spherical pendulum, it states that for most
choices of values of $(L,E)$ the points
$(\theta,\phi,\dot{\theta},\dot{\phi})$ in phase space with the given
$\left( L,E \right)$ form a **torus** $T^{2}$ (Figure 1). Consequently,
all motions with these values $(L,E)$ lie on the same torus.

There are some exceptions to having tori. The most relevant one for this
discussion is the **unstable equilibrium**, where the spherical pendulum
is motionless at its topmost point with $\theta = 0$, implying
$(L = 0,E = 1)$. The set of all points in phase space with
$\left( L = 0,E = 1 \right)$ form a **pinched torus** (Figure 2). The
**bifurcation diagram** (Figure 3) shows the values $(L,E)$
corresponding to tori (gray area), the exceptional value
$P = (L = 0,E = 1)$ corresponding to the pinched torus, and other
exceptional cases. This diagram and the associated information is
something that one cannot easily deduce by staring at complicated
algebraic expressions of the motions. Here, the general theory provides
powerful tools to reach these conclusions through qualitative arguments.

The next question from a geometric point of view, and a very natural one
for a mathematician to ask, is how all the tori fit together. To
understand this question, consider the well-known example of the
**cylinder** and the **Möbius band** (Figure 4). Even though a cylinder
and a Möbius band locally look identical, globally they are very
different. Both can be obtained by considering a rectangle, and gluing
two opposite sides of the rectangle in different ways.^{3} A consequence
of gluing the sides differently is that the cylinder is the Cartesian
product of a circle and an interval, while the Möbius band is
non-orientable and cannot be written as a Cartesian product.

Back to the spherical pendulum, consider a loop $L$ in the gray area in
Figure 3. Each point on the loop corresponds to a $T^{2}$ in phase space
and taking all these tori together gives a **bundle of tori over**
$\mathbf{L}$. Such bundles can be classified through a **gluing map**:
start with the Cartesian product $\lbrack 0,1\rbrack \times T^{2}$ and
then glue the tori at the endpoints of $\lbrack 0,1\rbrack$ to obtain
the required bundle (Figure 5). The gluing is given by a $2 \times 2$
integer matrix with determinant $\pm 1$, called **monodromy matrix**.
The main observation is that the bundle of tori is a Cartesian product
$L \times T^{2}$ if and only if the monodromy matrix is identity. In the
spherical pendulum, it has been shown that the monodromy matrix for a
loop $L$ going once around $P$ is

$$\begin{pmatrix} 1 & 1 \ 0 & 1 \ \end{pmatrix},$$

implying that in this case the bundle is not a Cartesian product and the
tori are glued with a “twist”. We say that the spherical pendulum has
**non-trivial monodromy** [2].

These are few of the things that one can say about the global geometry
of the spherical pendulum and there are many more interesting
connections that one can make with other deep geometric ideas. For
example, we have recently shown that the reason for non-trivial
monodromy is that a **Hopf fibration** is hidden in the neighborhood of
the unstable equilibrium [3].

The non-trivial geometry of the spherical pendulum manifests in the
corresponding quantum system. Here one can define the **joint spectrum**
of the two quantum operators that correspond to the energy and the
angular momentum (Figure 6). You can see in the joint spectrum that if
you take a small cell and you move it along a loop going around $P$ you
end up with a different cell than what you started with, a manifestation
of **quantum monodromy**. And if you pay close attention to the
difference between the sides of the initial and final cells you can even
read the monodromy matrix from this picture! Quantum monodromy has been
found in a wide variety of fundamental atomic and molecular systems, for
example, the hydrogen atom [4,5], and is important for assigning
quantum numbers to experimentally observed spectra.

The study of the global geometric properties of physical systems, both at the classical and quantum levels, reveals intriguing geometric structures and explains the structure of the corresponding joint quantum spectra. Here we have only touched on this active field of research in which the group of Dynamical Systems, Geometry, and Mathematical Physics at the JBI is deeply involved.

## References

[1] Global aspects of classical integrable systems, 2nd edition. R. H.
Cushman and L. M. Bates. Birkhäuser (2015). https://doi.org/10.1007/978-3-0348-0918-4

[2] On global action-angle coordinates. J. J. Duistermaat.
Communications on Pure and Applied Mathematics, 33(6):687–706 (1980). https://doi.org/10.1002/cpa.3160330602

[3] Monodromy of Hamiltonian systems with complexity 1 torus actions.
K. Efstathiou and N. Martynchuk. Journal of Geometry and Physics,
115:104-115 (2017). https://doi.org/10.1016/j.geomphys.2016.05.014

[4] Monodromy in the hydrogen atom in crossed fields. R. H. Cushman
and D. A. Sadovskií. Physica D: Nonlinear Phenomena, 142(1-2):166–196
(2000). https://doi.org/10.1016/S0167-2789(00)00053-1

[5] Defect in the joint spectrum of hydrogen due to monodromy. H.
Dullin and H. Waalkens. Arxiv preprint (2016). https://arxiv.org/abs/1612.00823

Notice the contrast with chaotic systems where it is not possible to solve the equations of motion. ↩︎

And there is an even deeper reason: the conservation of the energy and the angular momentum is related to symmetries of the system through Noether’s theorem. ↩︎

A very similar example is given by the torus and the Klein bottle where the rectangle is replaced by a cylinder and the two circles at the ends of the cylinder are glued in different ways. ↩︎