Motion Captured – Deformable Cats Jumping Through Wilson Loops

February 1, 2015

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This month, at great expense and effort, we have for the first time in this series, not a simulation but a real, live experiment. Performed at 298.15 K, 101.325 kPa, the apparatus (my 4 year old female huskat: “Ket Nicholson”) was placed at a height of 1m above the vertical at an angle of $\pi$ radians ($\pm \pi/4$) to the standard orientation. The experiment was carried out in a terrestrial gravitational field, with the influence of nearby satellites neglected.1601147_10151989472086859_1951154934_n

The most fundamental (verified) theory of nature, the Standard Model of particle physics, is an example of a gauge theory. This is a wedding of quantum mechanics, special relativity and three unusual field theories. The three field theories are named after their fundamental gauge groups ( $SU(3)$, $SU(2)$ and $U(1)$ ). Though this is the context in which most students of physics usually meet gauge theory, gauge theory itself is an idea and a mathematical framework that has a completely independent existence. Mathematicians will talk about fiber bundles and principle sections to describe the same things as physicists, who tend to use fewer words and more indices. An interesting and illuminating application of gauge theory was given by Wilczek and Shapere in 1987 who considered the motion of deformable bodies from a gauge theory perspective (also see the excellent lecture notes of David Tong here).

Take Ket, our reluctant experimental subject, though she was dropped in mid air, without anything to push against, she managed to spin around and land successfully on her feet. This is impossible of course – Ket has no torques acting on her, therefore angular momentum is conserved, as she was dropped from a stationary start (more or less) she cannot, according to the usual theory, change her orientation. Of the many differences between Ket and a rigid rod, the crucial one is that Ket is deformable, she can change shape in mid air. Mathematically we have that angular momentum $L$ is the product of the Intertia tensor, $I$, and the angular velocity $\omega$.
\[\mathbf{L} = I \mathbf{ \omega }\]
(the angular equivalent to Newton’s $\mathbf{p} = m \mathbf{v}$ ). Differentiating with respect to time:
\[\dot{\mathbf{ L} } = \dot{I} \mathbf{\omega} + I \dot{\mathbf{\omega}} = 0\]
The shape of a cat or a rigid rod, is encoded mathematically in the inertia tensor which is something like the angular equivalent of mass, it tells us how much something is going to spin for a given amount of torque, in the same way as mass tells us how much something is going to move for a give amount of force. The inertia tensor of a group of point masses, rotating about an axis $P$ is given by
\[I_P = \sum_i m_i r_i^2\]
So we see that by moving the constituent masses around we can change it. This is exactly what Ket can do that the rigid rod can’t. In the angular momentum equation above $\dot{I} \neq 0$, so in order to balance $\dot{\omega}\neq 0$ i.e. there is a net rotation.

This is clear and seems to have nothing at all to do with particle physics. To see the connection consider the “configuration space” $C$ of a deformable body, like a cat or more simply: a rod with a hinge in the middle and a motor that allows us to open and close it. You could take this as a highly simplified model of a swimming clam. $C$ is the space where every point corresponds to a possible shape of the object. Hard to picture for a cat but for the rod with the hinge, using the angle in the middle as a coordinate, this space is just a circle. Now some of the shapes in $C$ will be related by a rotation; for the rod with hinge angles $\theta$ and $2\pi – \theta$ give equivalent shapes. Therefore we consider a smaller space “shape space”, $S$ where shapes that can be drawn on top of each other after a rotation are identified. For the mathematically inclined, for the rod $S = C / \mathbb{Z}_2$ and for the cat $S = C / SO(3)$, the $C$ here is the complex space containing all possible arrangements of the cat. We now consider closed paths in $S$ – which is our new, geometric, way to say deformations of the object which start and finish in the same shape.

The mathematicians like the following language, the shape space $S$ is what they call the base space, the space of all possible rotations of the object is what they call the fibre $F$. The whole space gives the fibre bundle $C = S \times F$. Making definitions often makes things more complicated but looking at the diagram below we can have a simple picture in mind throughout all of this
ribbon
$C$ is the ribbon or closed strip, $S$ is a circle and $F$ is a line based at a point on $S$. The falling cat traces a path like the red line through $C$ as she falls, going through a sequence of shapes, returning to the original one but reoriented, or in geometric language, at a different point on $F$.

For the specific case of a falling cat, the paper of Montgomery analyses a simplified cat in a lot more detail than we can go into here (the paper can be found here, see especially page 18). The Kane-Scher model pictures a cat as two cylinders joined in the middle via a ball and socket joint with a no twist condition (cats can’t spin their two body halves 360 degrees independently), the two cylinders separately can be in any orientation in three dimensions so the configuration space is $SO(3) \times SO(3)$. Overall rotations of the combined cylinder system do not change the shape so the fibre is $SO(3)$. What we are saying here is: for every configuration of a cat, a particular arrangement of paws and whiskers and everything else, we choose a standard orientation and identify every configuration which can be rotated into the standard orientation with the same point in shape space. Enforcing constraints for staying connected and against twisting mean that (after lots of advanced group theory) Montgomery can show the shape space is $\mathbb{RP}^2$ the real projective plane in two dimensions.

For something that will remind you a lot of field theory if you know it, say that a cat at some standard orientation is specified by some coordinates $\mathbf{r}_i’$. In $C$ other points/configurations can be rotated into this one, i.e. there is some rotation matrix $R$ such that $\mathbf{r}_i(t) = R(t) \mathbf{r}_i'(t)$.
We define angular velocity as (see Tong’s notes if this is unclear!)
\[
\Omega = R^{-1} \dot{R}
\]
This is non-zero since the cat is changing shape and therefore $R$ is changing. We would like to write $\Omega$ in terms of shape space co-ordinates. In these coordinates a vector $x$ is a physical shape and its coordinates $x^A$ are the directions in which the shape can change e.g. angles between the legs or spinal twist. We put $\Omega = \Omega_A(x) \dot{x}^A$. If the shape changes an infinitesimal amount $x^A \rightarrow x^A +\delta x^A$ then a matrix, $\Omega_A(x)$, is associated to every direction in shape space. Now what happens if we want to change the reference shapes we used? Mathematically
\[
\mathbf{r}_i’ \rightarrow S(x) \mathbf{r}_i’
\]
Where $S(x)$ rotates the old standard configuration for the shape $x$ into the new one. With this change $R(t) \rightarrow R(t) S^{-1}(x)$. After a small bit of algebra using $\Omega = R^{-1} \dot{R} = \Omega_A(x) \dot{x}^A$, we have
\[
\Omega_A \rightarrow S \Omega_A S^{-1} + S \frac{dS}{dx^A}
\]
The physical situation doesn’t change, but the quantities $\Omega_A$, which depend on our arbitrary choices, do.

We can solve for $R$ in terms of $\Omega$,
\[
R = P \exp\left(\oint \Omega_A dx^A\right)
\]
This is just the usual solution to the differential equation, but, since the $R$’s are non-commuting we have to put them in a special order, which is what the $P$ accomplishes. Now it can be shown that under a redefinition of the “standard shapes”
\[
R(t) \rightarrow S(x) R(t) S^{-1}(x)
\]
operationally this says: go back to the original standard, do the rotation then rotate to the new standard. The equations above are all common in field theory. $\Omega_A$ behaves exactly like a gauge field. The equation above for $R$ is the same as for a Wilson Loop, or what mathematicians call a holonomy. It is measuring the space between the two red dots on the picture above. It is exactly what we wanted: how Ket rotates as she traces a path in shape space. This framework gives us a crucial constraint, if we ask something ambiguous we will get an answer that depends on our choice of standard orientations. We have a method of detecting unphysical questions. This is exactly what it means for a theory to be gauge invariant.

As well as cats, this formulation of gauge theory can be used to study parallel parking, bacterial swimming, polarized light, satellites and many other mechanical systems under the heading of control theory. Besides its possible usefulness, gauge theory in this context sheds new light on its application in particle physics. A gauge invariance is nothing more than a redundancy in the physical description that makes theories more symmetric and hence easier to write down and solve. Here we had different cat shapes and we arbitrarily chose a standard orientation for each one, physics doesn’t care where we draw our axes so the results had better not depend on it, this criteria led to the construction above that measurable quantities should be gauge invariant. Without the ability to rotate shapes we would have had to have some complicated rule about choosing axes for arbitrary shapes, introducing a redundancy led to a nicer and more natural looking theory but there was nothing fundamental about it. Recent insights from the study of duality imply that the same may be true in particle physics. Rather than finding the ultimate gauge group, so called “grand unification”, it may be that gauge theory emerges as nothing more than a way to write down a fundamental theory in terms of redundant variables that make the mathematical form easier to grasp.

Our experimental subject was indignant but unharmed, and considered herself well compensated for her invaluable assistance.