Lie Sphere Geometry Part 2: unoriented circles

 

Process philosophy, rooted in thinkers like Alfred North Whitehead and Charles Sanders Peirce, posits that reality is fundamentally dynamic—composed of processes, events, and relations rather than static objects. In this view, becoming precedes being; change and interaction are primary, while "things" are derivative patterns of stability within flux. Category theory aligns with this perspective by emphasizing arrows (morphisms) over objects, treating objects as mere nodes in a web of transformations. Peirce’s semiotics and process-oriented metaphysics anticipated this, framing reality as a continuum of signs and relations, where meaning and existence arise from dynamic interplay rather than fixed substances. Nature, then, is best understood as a network of interdependent processes.

Becoming precedes being

In Parts 19-23 we have discussed circles and conformal maps. Circles are 1D objects, and they have been discussed in two dimensions. But circles may be static or dynamic. A dynamical circle can be expanding or contracting. The dynamical state of the circle can be symbolized by an arrow showing the direction of its rotation with respect to its center. Then, choosing anticlockwise orientation, we draw a perpendicular to this arrow. It will point either towards the center, or outside of the center. State of contraction or expansion. We call them "oriented circles". Points are circles of zero radius. You can not give two orientations to a point. So points are somewhat special. Straight lines can be considered as circles of infinite radius. Lines can carry arrows much like circles. Again there are two areas, one on one side of an infinite straight line, another area on the other side. It may tell us how our line "moves" in one of the directions perpendicular to the line.

Our aim are, in fact, spheres rather than circles - thus one dimension higher. But we will tart with circles, as they are intuitive for our 3D perception. Only after we are done with oriented circles, we will move to oriented spheres.

We first start with unoriented circles. But we want to treat circles, lines, and points in a unified framework.  The first step will be to move from drawing circles on the flat plane to drawing them on a sphere - which is a plane plus a point at infinity. We know that stereographic projection maps the plane into the sphere, and maps circles and lines into circles, preserving the angles between them. We will skip the stereographic projection step, and start directly with drawing our circles on the unit sphere S² in R³.

Unoriented circles

Unoriented circles are simple to deal with. First we define S²:

S² = {x∈R³:  x² = 1}

Then, for every m∈S², and for every r∈[0,π] define

S_r(m) = {x∈S²:  x·m = cos r}.

Then S_r(m) is a circle on S² with center at m. For r=0 it reduces to one point x = m. For r = π it reduces to the opposite point x = -m. For r = π/2 we have a great circle. The parameter r can be considered as a distance between x and m measured along the shortest path (the great circle) connecting x and m. The maximal distance between two points on S² is π.

Let us discard here points and consider only nonpoint spheres. Let me quote (simplified for our needs) Proposition 5.5, p. 120,  from G.R. Jensen et al., Surfaces in Classical Geometries, Springer 2016:

Proposition 5.5. If Σ denotes the set of all nonpoint circles in S², then

Σ = (S² ⨉ (0,π))/ℤ₂,

where ℤ₂ acts on S² ⨉ (0,π) by

-1(m,r) = (-m, π - r).

G.R. Jensen et al., Surfaces in Classical Geometries, p. 121

Proof: Can you see it?

Lie Sphere Geometry Part 1: The Music of the Spheres

 

Circles and spheres. Spheres and circles. My mind is spinning. Circles seems to be more free than spheres. Two circles can entangle each other without touching. Spheres can't do such things. But sphere have inside and outside. Circles do not know about such concepts. There are also planes. They can be thought as spheres of infinite radius. Planes have two sides, but neither of them deserves to be called outside. Or both can be called such. Straight lines can be considered as circles of infinite radius. Points are spheres of zero radius - poor degenerate beings. So far talking about space I was dealing with points and linear concepts - like multivectors. It is time to get beyond linearity, and we will start our new adventure. It will be about the music of spheres. We will delve into the Lie sphere geometry. Wikipedia has an article on this subject. You can find some relevant references there.   Wikipedia has also an article on Music of the Spheres. There we read:

The musica universalis (literally universal music), also called music of the spheres or harmony of the spheres, is a philosophical concept that regards proportions in the movements of celestial bodies—the Sun, Moon, and planets—as a form of music. The theory, originating in ancient Greece, was a tenet of Pythagoreanism, and was later developed by 16th-century astronomer Johannes Kepler. Kepler did not believe this "music" to be audible, but felt that it could nevertheless be heard by the soul.

The music of the spheres

My soul (if I have one) is mathematical. So that is how we will be proceeding - through the math, I am only learning this subject. I wanted to learn it long ago, having a deep feeling of its importance, but I was always postponing it - for later. And now it comes to the fruit. Whatever I learn, I will write about it. Little by little, step by step. At first it will be just geometry. With time algebra (geometric algebra) will join in. Spinors will reappear in new clothes. So, for a while, we leave aside Spin Chronicles. For those curious I will start with the approach described in the paper "Lie Sphere Geometry in Hilbert Spaces" by Walter Benz, Result. Math. 40(2001), 9-38. It is a real poetry. 

In the article on Musica Universalis Wikipedia quotes William Shakespeare:

Sit, Jessica. Look how the floor of heaven

Is thick inlaid with patines of bright gold:

There's not the smallest orb which thou behold'st

But in his motion like an angel sings,

Still quiring to the young-eyed cherubins;

Such harmony is in immortal souls;

But whilst this muddy vesture of decay

Doth grossly close it in, we cannot hear it. 

Sit, Jessica. Look how the floor of heaven

And indeed heaven outside, soul inside, and yet they are one. Like in Lie sphere geometry that will come.

Spin Chronicles Part 51: Tomita's thermal flow and Mirror symmetry

 We continue from Part 50.

Relation to "thermal time"

J - Mirror symmetry of  the time flow

To relate this purely mathematical construction to the concept of "thermal time" related to physics, we need to direct our attention to quantum dynamics. To not complicate our discussion we will use Planck units. In quantum dynamics, in the Schrodinger picture, time evolution of state vectors is described by unitary operators

U(t) = e^-itH,                     (1)

where H is the Hamiltonian (energy operator). Comparing U(t) with U_ρ(s)  given by Eq. (5) we see that both expressions agree if we set s=t and

H = -log ρ.                         (2)

This is a self-adjoint, and positive (Why?) operator, unbounded if H is infinite-dimensional (Why?), and we can interpret U_ρ(s) as the operators defining the time evolution of a quantum system.

In order to understand why it is called "thermal flow" let us calculate the expectation value <E>_ρ of the energy represented by the operator H. We have

<E>_ρ = Tr(ρ H) = -Tr(ρ log ρ).

But this is exactly the expression for von Neumann entropy of the state ρ! Therefore it is the entropy of the state that drives the evolution defined by Tomita's modular flow. Calling it a "thermal flow" is thus justified.

Remark. Usually, for instance in Ref. [1], there is a different explanation, based on the particular example of a Gibbs thermal equilibrium state. But, I think, the explanation above serve its purpose as well.

Let us consider now the induced evolution U_ρ(s) on the space of Hilbert-Schmidt operators B₂(H), defined by Eq. in Part 50

U_ρ(t)| a > = | ρ^itaρ^-it >, a∈B₂(H), t∈R.      (3)

We rewrite it using ρ^it =  U(t) = e^-itH:

U_ρ(t)| a > = | e^-itHaρ^itH >.                           (4)

Since U_ρ(t) is a group of unitary operators on B₂(H), representing the time evolution there, we can write

U_ρ(t) = e^-itℋ,

where ℋ is the Hamilton's operator on B₂(H). Differentiating (4) with respect to t at t=0 we then obtain

ℋ| a > = | Ha - aH > = | [H,a] >.

It is now easy to find eigenvectors and eigenvalues of  ℋ. From the spectral decomposition of :

ρ = ∑_n p_n P_n, p_n>0,  ∑_n p_n = 1,

and Eq. (1), we get

U(t) e_n = e^{it log p_n} e_n.

Therefore, with e_mn = |e_m)(e_n|, we have

U(t) e_mn = e^{it(log p_m-log p_n)} e_mn.

Differentiating with respect to t at t=0 we get

ℋ e_mn = -(log p_m - log p_n) e_mn.         (5)

Thus e_mn are eigenvectors of ℋ, and the corresponding eigenvalues of ℋ are differences of eigenvalues of of H. In particular the spectrum of ℋ is symmetric with respect to the eigenvalue 0.

The "ground state"  Ω_ρ

We have defined Ω_ρ as

 Ω_ρ = √ρ = ∑_n (p_n)^½ P_n.

But P_n = |e_n)(e_n| = e_nn. Therefore

 Ω_ρ = ∑_n (p_n)^½ e_nn.                     (6)

From (5) we see that ℋ vanishes not only on  Ω_ρ  (which is a stationary state for U_ρ(t)), but it vanishes also on each e_nn participating in the sum (6).

Mirror symmetry

The anti-unitary operator J provides a symmetry between the two "algebras of observables" π(A), acting on B₂(H) from the left, and π'(A) acting from the right. Left and right actions commute. In fact  π(A) and π'(A) are commutants of each other, while

π(A)∩π'(A) = C1

If the intersection of a von Neumann algebra with its commutant (the center of the algebra) is trivial, we call the algebra a factor. So π(A) and π'(A) are factors with Jπ(A)J=π'(A). We can easily get (How?)

JU_ρ(t)J = U_ρ(-t),

thus

JℋJ = -ℋ.

The operator J reverses the "flow of time". It reverses the "energy" sign.

In quantum theory commuting observables are interpreted as representing two "compatible measurements". Measuring one observable does not "disturb" the measurement of another observable. Here we have two quantum "worlds". One represented by the algebra π(A), the other represented by π'(A). Measurements of one world do not disturb measurements of the other world. The two worlds have opposite "time arrows" and opposite "energy spectra". The ground state  Ω_ρ is kind of situated perfectly in the middle. The "energy" can be added to this ground state, or subtracted from it.

The two worlds have opposite "time arrows" and opposite "energy spectra".

We have already seen a similar arrangement while studying the Clifford algebra of space Cl(V). The algebra acts on itself by left and by right actions. The two actions commute. The left regular representation is reducible. It has a commutant, which is the right regular representation. Elements of the algebra are multivectors. The spin group acts on multivectors simultaneously from the left and from the right, when spinor (here a vector in H) rotates by 180 degrees, a multivector (here an element of B₂(H)) rotates by 360 degrees. Going from H to B₂(H) some information is lost.

References

[1] A. Connes, C. Rovelli, "Von Neumann algebra automorphisms and time-thermodynamics relation in generally covariant quantum theories", Class. Quantum Grav. 11 (1994) 2899 .