Unoriented and oriented conformal completion of 1+1 dimensional spacetime

What we are doing here is playing in the sandbox of a conformally completed 1+1-dimensional spacetime. 

Duality and triality in action

The term “conformal compactification” is often used, but it is somewhat misleading. More accurately, what we perform is a completion: we extend spacetime by admitting infinity itself. In the flat ${}^{}$R1,1 version of spacetime, infinity lingers only as a possibility—an unrealized horizon. But once we complete it into its $S^1\times S^1$ form, infinity ceases to be merely potential: it becomes actual, tangible, part of the fabric.

Now, there are two distinct ways this completion may unfold: oriented and unoriented. In the unoriented version, ${\mathbb{R}}^{1,1}$ is completed by sewing on a single circle at infinity, upon which endless worldlines of the flat spacetime find their closure—though each closes, as a rule,  at a different point of the same circle. In the oriented version, however, we grant a deeper polarity: two circles are added, dividing the worldlines of the future from those of the past. Each kind closes upon its own eternal loop. Yet, such generosity comes with a price—an entire second copy of flat spacetime appears in the completion.

Metaphysically, this picture gains resonance. It reflects duality, echoing matter and antimatter. And beyond duality, it hints at triality: space, its mirror antispace, and the twin circles at infinity—mediators, thresholds, silent connectors between the two realms.

This essay continues the thread begun in Sunday special – projective cone. We pick up precisely where we left off there. 

Identifying points x and λx, with λ>0, on the null cone of R^1,1 we obtain the oriented Grassmannian. To obtain the unoriented one, we additionally identify x with -x. The oriented Grassmannian is  nothing else but the torus S¹⨉S¹. What is then the unoriented one. What becomes of the torus when we identify its opposite points? Different things can happen, depending on what we call "opposite". One way of doing it is by taking the quotient of S¹⨉S¹ by the action of Z₂ defined by

(z,w)⟼(1/z,-w),     |z| = |w| = 1.

This is described in details, for instance, in Conrad's online lecture note "Quotients by group actions", p.4, and following. We obtain the famous unorientable Klein bottle. But we have a different action of Z2 on the torus - our action is

(z,w)⟼(-z,-w).                (1)

Note. Well, we do not have (z,w), we have

(x¹)² + (x²)² = (x³)² + (x⁴)² = 1.                   (2)     

But we can always set z = x¹ + i x², w = x³ + i x⁴. Then |z|= |w| =1, and changing the sign of x = (x¹,x²,x³,x⁴) is the same as changing the signs of z and w.

Proposition 1. The quotient of the torus S¹⨉S¹ by the action of Z₂ defined by (z,w)⟼(-z,-w) is again a torus. The diffeomorphism of the quotient space onto the torus induced by the map

φ(z,w) = (zw,w/z).                (3)

Proof. The map φ is smooth on S¹⨉S¹, and φ(z,w) = φ(-z,-w). Moreover, for any u,v in S¹⨉S¹, the equation φ(z,w)= (u,v) has exactly two solutions for (z,w):

1) z = u^½v-½, w = u^½v^½,
2) z = -u^½v-½, w = -u^½v^½.                (4)

Therefore φ maps the torus onto itself with the required kernel. QED.

Exercise 1. Make sure that you completely understand the proof above.

So torus has the very special property that it can double cover (and even n-cover) itself.

The above special feature holds for the null cone in R^2,2, but does not hold in R^4,2, where we have S³⨉S¹ instead of S¹⨉S¹. For R^4,2 taking the quotient by an analogous Z₂ produces a topologically different manifold.

Parametrization of the unoriented Grassmannian

For the oriented Grassmannian we used Eq. (2) to set

x¹ = cos(α), x² = sin(α),
x³ = cos(β), x⁴ = sin(β) ,               (5)

to define uniquely the angles α and β of a point on the torus. However this parametrization distinguishes the points x and -x, which become identical under the equivalence relation defined by the action (1) of Z₂. Can we find a parametrization of the new torus, obtained by this identification? To find this new parametrization we will use Eq. (3) and Proposition 1.

If we set z = e^iα, w = e^iβ, then u = zw = e^i(α+β), v = w/z =e^i(β-α).  If we define then

φ = (β + α) mod 2π,
ψ = (β - α) mod 2π,                (6),

then (φ,ψ) give us the required parametrization. What we need for defining a point on the torus are cos and sin of φ and ψ. Using (5) and trigonometric formulas for cos and sin of a sum and of difference, we find:

cos(φ) = x¹x³ - x²x⁴,
sin(φ) = x²x³ + x¹x⁴,
cos(ψ)  = x¹x³ + x²x⁴,
sin(ψ) = x²x³ - x¹x⁴.

The required condition sin²+cos² = 1 follows then automatically from (2).

Exercise 2. Verify the last statement.

Spacetime and anti-spacetime

 Here is the graphics representing the Minkowski space (yellow) and Minkowski anti-space (blue) embedded in the double cover of the conformal compactification. They are separated by (double cover of) the conformal infinity. Can light travel between spacetime and anti-spacetime? I am going to find an answer to this fascinating question.

And from the yellow side:

To be continued.....

P.S.1 Currently I am recalculating everything and finding really annoying misprints in my old papers. They drive me crazy!  Yesterday I have started an exchange with a well known mathematician in Novosibirsk, one of the authors of a Springer monograph about Riemannian geometry and geodesics,  also interested in conformal infinity. He tried to read my old papers and wasn't very happy, so I am preparing a CLEARLY WRITTEN DOCUMENT FOR HIM. Being very careful this time.

P.S.2.My new friend, mentioned in P.S.1, send me today his new paper - and introduction to Segal's chronometric theory (in Russian). There are some points when we agree (topology) and some when he disagrees with me (differentiable structure). We have to agree on everything before the publication of the paper. So I am now busy so that I do not have time to eat!

P.S.3. In his paper I have found new to me realization of the universal covering of the group U(2) in terms of quaternions. How could I not see it all by myself?! So finally we have some good use of quaternions in physics!!! Progress!

P.S.4. We now have snow in the sothern France. Yesterday is also snowing. Looks like we have a climate change? A while ago one of our intertnet cables got disconnected from the pole. Last time it happened on February 24 last year. It was the time of the start of the Russian "denazification" of Ukraine. Is it going to be a sign of a new decisive Russian offensive?

P.S.5. Valera (first name of my Russian friend, mathematician) sends me a fourth version of his 21 pages long paper - after his midnight!. It becomes more and more clear (the paper). His approach is through the unitary group  U(2), while I prefer to play with isotropic subspaces in R^6. Now even one result from my Quantum Fractals book is being quoted as useful, though it is concentrated on somewhat different issues.

P.S.6. I have to learn about "manifolds with corners". Perhaps the conformal infinity has what is called a "corner"? A corner of the universe? (Ark Fleet Ship B is being used  there). I don't know yet for sure.

P.S.7. Probably the best book on smooth manifolds: John M. Lee, Introduction to Smooth Manifolds, 2nd edition, Springer 2012. Table of content can be perused here. Includes concise appendices on linear algebra, on topology, sections on categories and functors, manifolds with corners, lot of examples.

.P.S.8. While working on the final version of one of the papers (in order to make the reviewers and the editor happy), and it will be sent, updated,  to the publisher today or tomorrow, so I will be able to return to the problem of light travelling along the conformal infinity on the boundary between space and anti-space) I have found (by chance of course) a very interesting paper that opens for me some new doors I was looking for a long time:

"Explicit construction of a time superoperator for quantum unstable systems"

November 2001 Chaos Solitons & Fractals 12:2591-2601

DOI: 10.1016/S0960-0779(01)00074-1

Gonzalo E Ordonez, Tomio Petrosky, Evgueni Karpov, I. Prigogine

Abstract: 

A time superoperator T conjugate to the Liouville superoperator LH=[H,] is constructed for a quantum system with one excited state or unstable particle. While there is no time operator conjugate to the Hamiltonian in the wave function space due to the positivity of energy, T may exist in the density matrix space as the spectrum of LH covers all the real axis. This is the first example of an observable that can only be formulated in the Liouville–von Neumann space of density matrices. In our example the expectation value of T gives the lifetime of the unstable particle. Once the time superoperator is obtained it is easy to define an entropy superoperator.

P.S.9. A mathematically more precise paper on the same fascinating subject:

Adolfo R. Ordonez, "Rigged Hilbert Spaces associated with Misra-Prigogine-Courbage Theory of Irreversibility", published in ; Physsica A 252 (1998) 362-376

P.S.10. In fact Prigogine's idea of considering "superoperators" fits perfectly the subject of this note. In finite-dimensional case the space in which superoperators act is H⊗H*, where H* is the dual Hilbert space (anti-space), and the spectrum of super-Hamiltonians made of ordinary Hamiltonians consists of the differences of the eigenvalues, thus is always symmetric: if a1-a2 is positive then a2-a1 is negative.

P.S.11. In fact I was playing with "superoperators" probably before the term was made famous by Ilya Prigogine. Here is a Remark I have made in my second published paper (that became a part of  my PhD Thesis) :

:

P.S.12 And today (Thursday January 26) we will have a close encounter: "In fact, this is one of the closest approaches by a known near-Earth object ever recorded"

 "The asteroid will be closest to Earth at 4:17 p.m. EST (2117 GMT) that day. At that point, it will be about 2,200 miles (3,600 kilometers) above the planet's surface, according to NASA."

P.S.13. I am still trying to find an aesthetically satisfactory way to justify mathematically the two images from this post. I am getting closer and closer.