Jones Calculus

A quaternion valued wave equation $\Psi_{tt} = D^2 \Psi$ can be solved as usual with a d’Alembert solution $\Psi(t) = \cos(D t) \Psi(0) + \sin(D t) D^{-1} \Psi'(0)$ . We can write this more generally as $e^{\beta D t} (u(0) - \beta v(0))$ where $\beta$ is a unit space quaternion where $\psi(0)=u(0) - \beta v(0)$ is the initial wave. Now, $\exp(\beta x) = \cos(x) + \beta \sin(x)$ holds for any space unit quaternion $\beta$ . Unlike in the complex case, we have now an entire $2$ -sphere which can be used as a choice for $\beta$ . If $u(0)$ and $v(0)$ are real, then we stay in the plane spanned by $1$ and $\beta$ . If $u(0)$ and $v(0)$ are in different plane, then the wave will evolve inside a larger part of the quaternion algebra.

Also as before, the wave equation has not be put in artificially. It appears when letting the system move freely in its symmetry. In the limit of deformation we are given an anti-symmetric matrix $B= \beta (b+b^*)$ and get a unitary evolution $\exp(i B t)$ . As we have used Pauli matrices to represent the quaternion algebra on $C^2$ , a wave is now given as a pair $(\psi(t),\phi(t))$ of complex waves. Using pairs of complex vectors is nothing new in physics. It is the Jones calculus named after Robert Clark Jones (1916-2004) who developed this picture in 1941. Jones was a Harvard graduate who obtained his PhD in 1941 and after some postdoc time at Bell Labs, worked until 1982 at the Polaroid Corporation.

Why would a photography company emply a physisists dealing with quaternion valued waves? The Jones calculus deals with polarization of light. It applies if the electromagnetic waves F =(E,B) have a particular form where E,B are both in a plane and perpendicular to each other. Remember that light is described by a 2-form F=dA which has in 4 dimensions B(4,2)=6 components, three electric and three magnetic components. The Maxwell equations dF=0, d* F=0 are then in a Lorentz gauge d^*A=0 equivalent to a wave equation L A =0, where L is the Laplacian in the Lorentz space. Now, if light has a polarized form, one can describe it with a complex two vector $\Psi=(u,v)$ rather than by giving the 6 components (E,B) of the electromagnetic field. How is this applied? Sun light arrives unpolarized but when scattering at a surface, it catches an amount of polarization. Polarized sunglasses filter out part of this light reducing the glare of reflected light. The effect is also used in LCD technology or for glasses worn in 3D movies. It can not only be used for light, but in radio wave technology, polarization can be used to “double book” frequency channels. And for radar waves, using polarized radar waves can help to avoid seeing rain drops. Even nature has made use of it. Octopi or cuttlefish are able to see polarization patterns. See the encylopedia entry for more. Mathematically the relation with quaternion is no suprise because the linear fibre of a 1-form A(x) at a point is 4-dimensional. Describing the motion of the electromagnetic field potential A (which satisfies the wave equation) is therefore equivalent to a quaternion valued field.

We have to stress however that the connection between a quaternion valued quantum mechanics and wave motion of the electromagnetic field is mostly a mathematical one. First of all, we work in a discrete setup over an arbitrary finite simplicial complex. We don’t even have to take the de Rham complex: any elliptic complex D=d+d^* as discribed in a discrete Atiyah-Singer setup will do. The Maxwell equations even don’t need to be 1 forms. If $E \oplus F=\oplus E_k + \oplus F_k$ is the arena of vector spaces on which D:E \to F, F \to E$ acts, then one can see for a given $j \in D_k$ the equations $dF=0,d^*F=j$ as the Maxwell equation in that space. For $F=dA$ and gauge $d^*A=0$ , the Maxwell equations reduce to the Poisson equation $D^2 A=j$ which in the case of an absense of “current” j gives the wave equation $D^2 A=0$ meaning that $A$ is a harmonic k-form. Now, in a classical de Rham setup on a simplicial complex G, A is just an anti-symmetric function on k-dimensional simplices of the complex. Still, in this setup, when describing light on a space of k-forms, it is given by real valued functions. If we Lax deform the elliptic complex, then the exterior derivatives become complex but still, the harmonic forms do not change because the Laplacian does not change. Also note that we don’t incorporate time into the simplicial complex (yet). Time evolution is given by an external real quantity leading to a differential equation. The wave equation $u_{tt}=Lu$ can be described as a Schrödinger equation $u_t = i Du$ . We have seen that when placing three complex evolutions together that we can get a quaternion valued evolution. But the waves in that evolution have little to do with the just described Maxwell equations in vacuum, which just describes harmonic functions in the elliptic complex.

We will deal with the problematic of time elsewhere. Just to state now that describing a space time with a finite simplicial complex does not seem to work. It migth be beautiful and interesting to describe finite discrete space times but one can hardly solve the Kepler problem with it. Mathematically close to the Einstein equations is to describe simplicial complexes with a fixed number of simplices which have maximal or minimal Euler characteristic among all complexes. Anyway, describing physics with waves evolving on finite geometries is appealing because the mathematics of its quantum mechanics is identical to the mathematics of the quantum mechanics in the continuum, just that everything is finite dimensional. Yes there are certain parts of quantum mechanics which appear needing infinite dimensions but if one is interested in the PDE’s, the Schroedinger respectivly the wave equation on such a space there are many interesting problems already in finite dimensions. The question how fast waves travel is also iteresting in the nonlinear Lax set-up. See This HCRP project from 2016 of Annie Rak. In principle the mathematics of PDE’s on simplicial complexes (which are actually ordinary differential equations) has more resemblence with the real thing because if one numerically computes any PDE using a finite element method, one essentially does this.

Here is a photograph showing Robert Clark Jones:

Source: Emilio Segrè Visual Archives.

There are other places in physics where complex vector-valued fields appear. In quantum mechanics it appears from SU(2) symmetries, two level systems, isospin or weak isospin. Essentially everywhere, where two quantities can be exchanged, the SU(2) symmetry appears. A quaternion valued field is also an example of a non-abelian gauge field. In that case, one is interested (without matter) in the Lagrangian $|F|^2/2$ with $F=dA+A \wedge A$ , where $A$ is the connection 1-form. Summing the Lagranging over space gives the functional. One is interested then in critical points. The satisfy $d_A^* F=0, d_A F=0$ meaning that they are “harmonic” similarly as in the abelian case, where harmonic functions are critical points of the quadratic Lagrangian. There are differences however. In the Yang-Mills case, one looks at SU(2) meaning that the fields are quaternions of length 1. When we look at the Lax (or asymptotically for large t, the Schrödinger evolution) of quaternion valued fields $\psi(t)$ , then for exach fixed simplex x, the field value $\psi(t,x)$ is a quaternion, not necessarily a unit quaternion.

[Remark. A naive idea put forward in the “particle and primes allegory” is to see a particle realized if it has an integer value. The particles and prime allegory draws a striking similarity between structures in the standard model and combinatorics of primes in associative complete division algebras. The later is pure mathematics. As there are symmetry groups acting on the primes, it is natural to look at the equivalence classes. The symmetry groups in the division algebras are U(1) and SU(2) but there is also a natural SU(3) action due to the exchange of the space generators i,j,k in the quaternion algebra. This symmetry does not act linearly on the apace, but it produces an other (naturally called strong) equivalence relation. The weak (SU(2)) and strong equivalence relations combined lead to pictures of Mesons and Baryons among the Hadrons while the U(1) symmetry naturally leads to pictures of Electron-Positron pairs and Neutrini in the Lepton case. The nomenclature essentially pairs the particle structure seen in the standard model with the prime structure in the division algebras. As expected, the analogy does not go very far. The fundamental theorem of algebra for quaternions leads to some particle processes like pair creation and annihilation and recombination but not all. It does not explain for example a transition from a Hadron to a Lepton. The set-up also leads naturally to charges with values 1/3 or 2/3 but not all. Also, number theory has entered physics in many places, it is not clear why “integers” should appear at all in a quantum field theory. What was mentioned in the particles and primes allegory is the possibility to see particles only realized at a simplex x, if the field value is an integer there. As in a non-linear integrable Hamiltonian system like the Lax evolution soliton solutions are likely to appear and so, if the wave takes some integer value p at some time t and position x, it will at a later time have that value p at a different position. The particle has traveled. But as during the time it has jumped from one vertex to an other, it can have changed to a gauge equivalent particle. If the integer value is not prime, it decomposes as a product of primes. Taking a situation where space is a product of other spaces allows to model particle interactions. One can then ask why a particle like an electron modeled by some non-real prime is so stable and why if we model an electron-positron pair by a 4k+1 prime, the position of the electron and positron are different. A Fock space analogy is to view space as an element in the strong ring, where every part is a particle. Still the mathematics is the same, we have a geometric space G with a Dirac operator D. Time evolution is obtained by letting D go in its symmetry group.]

Jones calculus

Author: oliverknill