If I had a hammer - the mystery of light

If I had a hammer
I'd hammer in the morning
I'd hammer in the evening
All over this land

etc.

So sung Pete Seeger,

and after him Johnny Cash with JuneCarter, Trini Lopez, Peter, Paul and Mary, and others.

But I got a hammer

And I've got a bell
And I've got a song to sing
All over this land

It is the song about the light and its mystery. And twistors and (later) also bi-twistors. And I am crazy about it all. 

In fact I do have a hammer - the projective variety of isotropic lines in H^2,2. 

It occurred to me this morning that infinity is not a static concept. Like the horizon it moves when the observer, or rather the consciousness moves. It is a dynamical "thing". Yet it is objective, because consciousness is also objective. But that is for later.

For now let's start with this (one hour long) quite an interesting discussion: Einstein and the Quantum: Entanglement and Emergence

Einstein-Rosen bridge is the keyword. The bridge between the universe and its dual anti-universe. The double cover and the dynamical infinity layer between the two. Made of trapped light. 

Once upon a time I even wrote a paper featuring this bridge. But as my many other projects it has not been finished, waiting for "better times". I realized yesterday that I do not even know where the degeneracy of the metric on the bridge comes from? Does it come from breaking the light-cone (conformal) structure, or from the scale factor? Will have to look at it again. 

By the way: around 40 minutes into the discussion it is said that it follows from quantum theory that the amount of information in the universe is "conserved". I have no idea where this idea came from? Apparently these physicists think that they know what they are talking about. I don't.

H^2,2 deserves a poem. It deserves a symphony. I am going to use my algebraic hammer and my complex projective bell to do just that. No more postponing for "later". Increasing the number of dimensions up to countable infinity and even further than that.

From the abstract of my paper: "Hanson-Regge gravitational instanton and Einstein-Rose bridge.We argue that a region of space-time with vanishing vierbein but smooth principal connection can be, in principle, detected by scattering experiments."

The paper features just one bridge. But if there is one, there may be infinitely many of them. And there are. Unstable gravity waves that can be read and utilized by consciousness units. If anything is objective (and there are more than one of such "things"), consciousness certainly is.

And what is this that I am so crazy about? This H^2,2 ? It is a four-dimensional (2+2=4) complex vector space endowed with a Hermitian scalar product of signature (2,2). Why four? For instance because of Jung's love of "quaternio", See for instance the thesis "COUNTING TO FOUR: ASSESSING THE QUATERNITY OF C.G. JUNG IN THE LIGHT OF LACAN AND SOPHIOLOGY":

"Though we will see many similarities between the two psychoanalysts, ultimately I will argue that the quadrilaterals of Lacan operate in the opposite direction from Jung’s quaternities. They work to break open heuristic structures that, when closed in on themselves, result in dualisms and even monisms (in the case of Jungian quaternities, into a kind of monism of consciousness). "

Or better: Jung subconsciously knew about twistors? Collective unconscious?

Elements of this space are sometimes called "twistors" (when we double the dimension we get bi- twistors, containing twistors and anti-twistors, like in the Dirac equation). Twistors are spinors for the conformal group (rather than for the Lorentz group as it is for spinors).

 

"... As a student, I love 

this subject from childhood and decided to stay with it forever. ..." 

From  the "Einstein and the Quantum: Entanglement and Emergence" video:

00:43:40,880 --> 00:43:46,240

again that the information in a system

00:43:43,358 --> 00:43:48,639

any system is stored far away

...

00:45:30,480 --> 00:45:36,240

wherever the boundaries of space are

....

00:45:33,358 --> 00:45:39,358

now the space have a boundary who knows?

At the dynamical conformal infinity! On the "bridge".

Back to H^2,2

We denote by (u,v) the Hermitian scalar product, assumed to be linear in, antilinear in v. A basis e₁, e₂, e₃, e₄ in is said to be orthonormal if (e_i,e_j) = G_ij, where G is the diagonal matrix (-1,-1,+1,+1).

We denote by Q the set of all non-zero isotropic vectors:

Q = {u ∈ H^2,2: (u,u) = 0, u ≠ 0}

Q is a real manifold of (real) dimension 7. Why seven? 4 complex = 8 real. The condition (u,u) = 0 is one real condition, since (u,u) is real. 8-1 = 7.

Let C be the field of complex numbers, and let C* be the multiplicative group of complex numbers different from zero. Le R⁺ be the multiplicative group of positive real numbers, let U(1) be the circle group - the group of complex numbers of modulus 1. Then we have C* = R⁺ x U(1).

We define quotient manifolds:

Q' = Q/R⁺

Q'' = Q/C*

Q'' is 5-dimensional (because orbits of C* in Q are two-dimensional). Q' is 6-dimensional (because orbits of R⁺ in Q are one-dimensional). Q'' will be the manifold of all (compact) light rays. What is Q' ? At the moment I have no idea.  Extra U(1) gauge freedom? Photon's polarization? It would be weird. Mathematically all is clear: Q' is a U(1) principal fiber bundle over Q''. But what would be the use of it in physics? If any ....

I am not 100% sure if all of the above is correct. Perhaps I am wrong. Then I will retract and fix it. In old good times internet user Bjab was kindly catching and fixing almost all my mistakes. But that is a history now - now I am responsible for all, and have to catch and fix it all by myself.  Normality in life. Interests come and go.

Notice that for a while I am adapting stuff from my paper "Some comments on projective quadrics subordinate to pseudo--Hermitian spaces".

Why is the space of all light rays 5-dimensional? Cut spacetime by the hyperplane t=0. Every light ray intersects this hyperplane in exactly one point. This gives us three coordinates. But then we have to specify which direction on the heavenly sphere the light goes to. This adds additional two (angular) coordinates. Together 3+2=5.

Kaironic fields are, in fact, fields on Q'', unlike other fields used in physics. I didn't know that when I was creating the theory of Kairons (it began there: Anticausal Currents for Massless Particles, A. Jadczyk, Preprint Wroclaw-491, Dec 1979. 11p, never published as a paper). Kaironic fields on Q' would describe charged kairons. Massless charged "particles"? These are not yet experimentally discovered (at least officially, in mainstream science). Little is also known theoretically about such strange fairy-tales creatures.

From "our" perspective light is something that interacts with our retina, the quanta. From a different perspective, that of a block universe compactified, higher density perspective, light ray, the whole ray, with its all "history" is just a point in a 5-dimensional "space". The whole history is a point. Another history - another point. Choosing between different histories is just selecting different points. Our life is just a point in the manifold of all possible lives. Applied mathematics - that's what it is. Consciousness is light trapped by gravity. Approximately.

We denote by P the canonical projection P":Q→Q''. The projection P" can be implemented in two steps: first taking the quotient with respect to R⁺  to obtain Q', then quotienting Q' by U(1) to obtain Q''. We denote the corresponding projections by P' and π respectively. Thus we have

P" = π∘P'

Q'' = Q'/U(1)

Q' and Q'' are compact.

In order to discuss the topology we will need a "split'. Choosing a split in , Q' itself will split. It will split into the product S³xS³. Choosing another "split" - it will split into S³xS³ differently. But what is this split? In fact we will need to operators: ":split" and "flip".

Notation:  For the ease of notation in the following we will use the letter V to denote H^2,2. If A is a linear operator on V, we denote by A* its adjoint with respect to the indefinite scalar product (u,v) in V.

Definition: A split is a direct sum de3composition V_- ⊕ V₊, where V_- are two two-dimensional subspaces of V, with the scalar product (u,v) positively defined on V₊ and negatively defined on V_-.   

If e₁, e₂, e₃, e₄ is an orthonormal basis in V, then it defines a split, where V_- is the subspace spanned by  e₁, e₂,  and V₊ is spanned by e₃, e₄. Let S(V) be the set of all splits. Every split determines a unique operator S defined by Su=-u on V_- and Su=u on V₊. Then S=S* and S²=1. Moreover the scalar product (u,v)_S  defined as

(u,v)_S = (u,Sv) = (Su,v) .

is positive definite on V. Thus the set S(V) of all splits can be equivalently defined as the set of all such operators. 

Definition: Give a split S, we define a flip as a linear operator F on V such that F=F*, F²=1, and anticommuting with S: SF+FS=0. Evidently such an F exchanges positive and negative eigensubspaces of S. If e₁, e₂, e₃, e₄ is an orthonormal basis and if S is the canonical split defined by this bases, then, written in a bloc matrix form, with 2x2 block matrices we have 

S =

-1 0

0 1

and the canonical flip can be defined by the block matrix

F =

0 1

1 0.

Notation: From now on I will us TeX notation "^" and "_" for superscripts and subscripts respectively.

The topology of Q' =P'(Q) and Q'' = P"(Q)

It will be convenient to introduce an orthonormal basis e_i (i=1,2,3,4). Le u be in Q. Then the condition u is Q implies

|u_1|^2 + |u_2|^2-|u_3|^2-|u_4|^2 = 0

or 

|u_1|^2 + |u_2|^2 = |u_3|^2-|u_4|^2 >0

The last inequality follows from the fact that we have excluded the zero vector from Q. Thus let

r^2 = |u_1|^2 + |u_2|^2 = |u_3|^2-|u_4|^2 , r>0

We can then replace u by u/r, and we have P'(u)=P'(u/r) - the same point of Q'. After that replacement we have 

|u_1|^2 + |u_2|^2 = |u_3|^2-|u_4|^2 = 1.

This is a unique parametrization of the points of Q' (with respect to the chosen orthonormal basis). It is now clear that Q' is topologically (but also as a smooth manifold) homeomorphic (and even diffeomorphic) to the product of two 3-spheres:

Q' ≈ S^3 x S^3

Then Q'' = (S^3 x S^3)/U(1)_{diag}

A kind of a Hopf fibration. The group U(1) (multiplication of u by complex numbers of modulus 1) acts on both S^3 simultaneously, so we wrote it as U(1)_{diag}. We notice that this product representation depends, in fact, on the split, and not on the orthonormal basis itself. Choosing a flip allows us to identify the two copies of S^3. But it is easier to see it using an orthonormal basis adapted to the split/flip pair.  

Back to the transcript from the World Science Festival video:

01:02:21,280 --> 01:02:25,119 people have been trying to put gravity

01:02:22,798 --> 01:02:27,440 and quantum mechanics together for a

01:02:25,119 --> 01:02:30,720 long time ...

01:02:40,559 --> 01:02:45,519 i very strongly believe what's going on

01:02:43,199 --> 01:02:48,639 is that quantum mechanics and gravity

01:02:45,519 --> 01:02:50,719 are simply so deeply connected so

01:02:53,280 --> 01:02:57,760 that trying to separate them into the

01:02:55,440 --> 01:02:59,280 classical theory of gravity and quantum

01:02:59,280 --> 01:03:03,839 will inevitably lead and then put them

01:03:01,679 --> 01:03:05,440 back together again separate them into

01:03:03,838 --> 01:03:06,880 these two things and then put them back

01:03:11,920 --> 01:03:16,240 except that quantum mechanics and

01:03:13,679 --> 01:03:18,159 gravity are almost the same thing

While quantization of gravity may be an absurd, physicists, who have nothing better to do do than to follow the mainstream, are organizing conferences and are all enthusiastic about it.

And what am I doing doing here, on this blog? These twistors... Is that quantum mechanics? Or is that gravity? Or is that just simple and pretty mathematics?

The video linked above starts with showing and putting together two Einstein's 1935 papers. On EPR "paradox" and on Einstein-Rosen bridge. In fact it should show and discuss also Einstein's (with Bergmann) 1938 paper "On a Generalization of Kaluza's Theory of Electricity". For a more recent discussion of the subject see "A Note On Einstein, Bergmann, and the Fifth Dimension", Edward Witten(Princeton, Inst. Advanced Study), Jan 30, 2014 7 pages Contribution to: Einstein Symposium 2005.

Perhaps an example first, and an image that should be kept in mind when happily playing with these abstract algebro-geometric structures.

Let e_i (i=1,2,3,4) be an orthonormal basis. Then u=e_1+e_3 is an isotropic vector. v=e_2+e_4 is another isotropic vector. Moreover u and v are orthogonal to each other - they span a two dimensional totally subspace. All vectors in this subspace are isotropic and mutually orthogonal. The vector u belongs to this subspace. Now consider the set of all totally isotropic subspaces to which u belongs. It is the set of all spacetime points on the light ray associated with u. Spacetime points are represented in our picture as totally isotropic subspaces of V. Light rays are represented by one-dimensional isotropic subspaces. Given light ray u, we can consider the "pencil" of all totally isotropic subspaces containing u. 

These are "events" on the null geodesic (aka light ray). Given a spacetime point - totally geodesic subspace of V, we can consider the set of all isotropic lines of this subspace. ("Point" is the plane itself, not its origin - thus not the zero vector of the plane)

These are light rays crossing a given space time point - the apex of the light cone. This is a kind of "sacred"  duality between points and lines of elementary Euclidean geometry. Point is an intersection of lines. Line is a collection of points.   

P.S.1. 13-02-23 13:20 Perhaps I will investigate closer the problem of massless charged particles (write a paper on this subject?). If nothing in nature prevents them from existing - they should exist. But if so, what do they do? 

Two papers on the subject that I am aware of:

Kurt Lechner, Electrodynamics of massless charged particles, JOURNAL OF MATHEMATICAL PHYSICS 56, 022901 (2015)

Ivan Morales, Bruno Neves, Zui Oporto, Olivier Piguet, Behaviour of Charged Spinning Massless Particles, https://arxiv.org/abs/1711.04127

see also further development here.

BTW the book by Kumar is really fascinating!

P.S.2. Since C²⊕ C² ≈ C² ⊗ C², we may be as well plying with an entangled pair of qubits. The universe in a pair of qubits? As above so below? Or as below so above? Are those the glimpses of  the physics of the future? Is that what quantum future is about?

Which takes me to https://arxiv.org/abs/1404.4978