Lie Sphere Geometry Part 6: The Gospel of Q

I am kind of a Platonist. But I'm not the only one!

We continue from Part 5. Let us first recall the definitions.

We consider a 6-dimensional real vector space V with a scalar product of signature (4,2). We select an orthonormal basis in this space, e_i (i=0,...,5) , so that

(e₀,e₀)=(e₁,e₁,)=(e₂,e₂)=(e₃,e₃)=1, (e₄,e₄)=(e₅,e₅)= -1.

with vectors e_i being orthogonal to each other:

(e_i,e_j) = 0 for i≠j.

We call this space, with a selected basis,  R^4,2. A general vector x of V  is then written as

x = x⁰e₀+...+x⁵e₅.

We define Q as the set of all  isotropic (i.e. (u,u)=0 for any nonzero vector u on the line) lines through the origin 0 in V. Alternatively Q can be written as (see Part 5):

Q = {[u]∈P(V): (u,u) = 0},

where P(V) is the real projective space of V: P(V)=(V∖{0})/∼, with x∼y if and only if there exists  a real λ such that y=λx. We now want to prove the following proposition:

Proposition. The following formula defines an explicit isomorphism between the space of oriented spheres (discussed in Lie Sphere Geometry Part 4: oriented spheres) and Q:

(m,t) ⟼ x(m,t) = [m+cos(t)e₄+sin(t)e₅].                    (1)

Proof. Let us start with analyzing the meaning of the formula (1). On the left hand side we have m and t. Together they define an oriented sphere as explained in Part 4.  But in Part 4 we had coordinates x¹,...,x⁴, while here we have coordinates x⁰,...,x⁵. How should we understand (1)? On the right hand side we see m+cos(t)e₄+sin(t)e₅. This suggests that m in (1) should be understood as m=m⁰e₀+m¹e₁+m²e₂+m³e₃.  So our S³ in Part 4 is now in the space of coordinates x⁰,...,x³ instead of x¹,...,x⁴ as it was previously.  Since m is assumed to be on the unit sphere S³, we have (m,m)=1. We also have (m,e₄)=(m,e₅)=0.

Then we find that

(x(m,t), x(m,t)) = (m,m)-sin²(t)-cos²(t)=1-1=0,

so that x(m,t) is an isotropic vector in V. It is certainly a non-zero vector (Why?), so it defines a point of Q.

Next, we remember from Part 4 that (m, t) and (-m, t+π mod 2π) define the same oriented sphere. Do they define the same point of Q? We easily see, from the properties of sin and cos functions, that x(-m,t+π mod 2π) = -x(m,t). Therefore indeed (1) defines a well defined map from the set of oriented spheres  (S³⨉S¹)/Z₂ into Q.

It remains to be shown that the map is one-to one and onto. Let us first check that the map is one-to-one. Suppose [x(m,t)]=[x(m',t')]. That means x(m',t')=λx(m,t), or, explicitly

λ(m+cos(t)e₄+sin(t)e₅)=(m'+cos(t')e₄+sin(t')e₅) .

Since 1=cos²(t)+sin²(t) = cos²(t')+sin²(t')=1, it follows that λ=±1. If λ=1, then t'=t and m'=m. If λ=-1, then t'=t+π mod 2π, and m'=-m. Thus (m,t) and (m',t') define the same oriented sphere. So our map is injective.

We will show now that it is also surjective. The simplest way of doing this is by explicitly calculating the inverse map from Q to (S³⨉S¹)/Z₂. Let's do it. We take any point [u] in Q. Then (u,u)=0 and u≠0. That means

(u⁰)²+(u¹)²+(u²)²+(u³)²-(u⁴)²-(u⁵)² = 0,

or

(u⁰)²+(u¹)²+(u²)²+(u³)² = (u⁴)²+(u⁵)² .

But u≠0, therefore  (u⁴)²+(u⁵)² > 0. Set u'=λu, where λ = ( (u⁴)²+(u⁵)² )^-½ . Then [u']=[u]. But now (u'⁴)²+(u^'5)²= 1, therefore there exists a unique t in [0,2π) such that u'⁴=cos(t), u'⁵=sin(t). Set m=u'⁰e₀+u'¹e₁+u'²e₂.+u'³e₃. Then  m fulfills (m,m) =1, so for each point  [u] in Q there is oriented sphere (m,t) for which x(m,t)=[u].  QED.

The mystery of zero-radius spheres

The space (S³⨉S¹) looks like a homogeneous space - none of its points is better than any other. And (S³⨉S¹)/Z₂ or isomorphic to it Q doesn't look any worse. In fact, we will see later on that both are homogeneous spaces under the action of SO(4,2). And yet among the oriented spheres in S³ there are those with zero radius and no orientation. The same applies to our experiences in the 3D world we live in. How can these two different views  be reconciled? That is a bugging question. Here is my attempt to answer it, and details I hope to give in future posts.

What we see, what we experience, is a projection, like a shadow of a higher-dimensional structure on a lower-dimensional wall. A sphere becomes a point, but the sphere itself is just a shadow. Change the position of the lamp, and the sphere will become a point and the point will become a sphere.

Now, formally it looks like a reasonable answer, but after some thinking the next question comes to mind: how can we do it in practice? Can we? Or is it just mathematics that has nothing to do with the physical world?

I think we can, and, I think, it has to do a lot with physics. Special Relativity deals with inertial frames. But there are no truly inertial frames in nature. Lorentz transformations are transformations between inertial frames. It is an idealization valid as long as accelerations can be neglected. This is our overwhelming experience. We do not have much experience with accelerations. Well, fighter jet pilots have some, but still their experiences are local and short, not more than second's long 10 Gs. That's next to nothing on the Universe scale. What happens inside elementary particles? We have no idea. Their internal structure is still a pure guess. We have models that answer some questions, but leave other questions open. I will approach this subject from the side of mathematics. The physical interpretation and possible application of the mathematical formulas is a hard and demanding work. And, anyway, what we do here is just playing with one particular model, a model that, perhaps, catches some of the properties of Reality, but leaves aside other important properties.

So, in the next post we study group action on Q. And after that I will propose a way how to get rid of taking quotient by Z₂, which bothers me aesthetically.  Why must we take this quotient? To take another shadow for reality? As you see I am kind of a Platonist. But I'm not the only one!

P.S. 13-04-25 1:40

In the comment Igor BayakApril 12, 2025 at 10:02 PM Igor Bayak wrote:

"the product of spheres loses its orientability as a result of factorization."

Let us see that this statement is, in our case, false. We have two spheres S¹ and S³. For  S¹  let us take the group of all complex numbers c with modulus one: |c|=1.

For S³.we take the group of unitary 2x2 matrices of determinant one - the group SU(2). They can be parametrized as 

  x₁+ iy₁, x₂+iy₂

-(x₂-iy₂), x₁-iy₁

with the determinant given by  (x₁)²+(y₁)²+(x₂)²+(y₂)² = 1.

This is the equation of  S³. So we have our S¹ and S³ realized as Lie groups. One is a group of complex numbers, the second one a group of 2x2 complex matrices.

We tak the product U(1) x SU(2). It is the group of pairs (c,A), where c is in U(1) and A is in U(2).

Now we define the map, let us call it  f from U(1) x SU(2) into the group of all unitary 2x2  matrces:

f(c,A) = cA.

It is clear that if A is in SU(2), and c in U(1), the product of A by the complex number c is a unitary matrix. Moreover f is a group homomorphism from U(1) x SU(2) into U(2). And every unitary matrix can be represented as such a product. Therefore f is surjective

We easily see that f(-c,-A)=f(c,A). Indeed f(c,A)=f(c',A') if and only if c=c' and A=A' or c=-c' and A=-A'. 

In other words 

U(2) =(S³⨉S¹)/Z₂ 

where the factorization is by identifying the opposite points simultaneously.

But U(2) is a Lie group, it has 4 parameters. Its Lie algebra is 4-dimensional. Thus we have four linearly independent vector fields on U(2) depending smoothly on the point. It follows that taking the quotient by Z₂ , in this case, produces again an orientable manifold, contrary to what Igor was thinking,

P.S. 13-04-25 17:30 Reading Plato Apology, 21 b,c,d:

... whereas when I do not know, neither do I think I know; so I am likely to be wiser than he to this small extent, that I do not think I know what I do not know.

"Consider that I tell you this because I would inform you about the origin of the slander. When I heard of this reply I asked myself: "Whatever does the god mean? What is his riddle? I am very conscious that I am not wise at all; what then does he mean by saying that I am the wisest? For surely he does not lie; it is not legitimate for him to do so." For a long time I was at a loss as to his meaning; then I very reluctantly turned to some such investigation as this: I went to one of those reputed wise, thinking that there, if anywhere, I could refute the oracle and say to it: "This man is wiser than I, but you said I was." Then, when I examined this man—there is no need for me to tell you his name, he was one of our public men—my experience was something like this: I thought that he appeared wise to many people and especially to himself, but he was not. I then tried to show him that he thought himself wise, but that he was not. As a result he came to dislike me, and so did many of the bystanders. So I withdrew and thought to myself: "I am wiser than this man; it is likely that neither of us knows anything worthwhile, but he thinks he knows something when he does not, whereas when I do not know, neither do I think I know; so I am likely to be wiser than he to this small extent, that I do not think I know what I do not know." After this I approached another man, one of those thought to be wiser than he, and I thought the same thing, and so I came to be disliked both by him and by many others."