The Spin Chronicles (Part 20):Group action on the sphere and the spin idea

 In Part 18 we have met the Klein absolute, which, for the case of the 2D (x,y) plane, happens to be the null cone (ζ₀)²- (ζ₁)² - (ζ₂)² - (ζ₃)² = 0 in the 4D Minkowski space. As it is usual in projective geometry we remove the origin ζ=0 from this null cone - we do not need it. 

Projective geometry is about light rays

Let us recall that we have added two extra dimensions x₀ and x₃. We added x₃ for the stereographic projection, and then we added x₀ to get rid of denominators and, instead of doing the stereographic projection we employed the projective space.

Our null cone  ζ² = 0 is invariant under the action ζ⟼g ζ τ(g) of the Clifford group G of the Clifford algebra Cl(V). The action is linear. We already know that g ζ τ(g) = Λ(g)ζ, where Λ(g) is a Lorentz transformation. We have calculated explicitly these transformations for typical one-parameter subgroups of G and obtain either rotations or Lorentz boosts. Now, since the action is linear, it defines the action on equivalence classes of the equivalence relation "∼" defining the projection (see Part 18). If ζ=λη, λ>0, then Λ(g)ζ = λΛ(g)η. Therefore g acts on the equivalence classes of "∼". Let us take a closer look at the structure of the set of these equivalence classes. What do we get? We rewrite ζ² = 0 as

(ζ₁)² + (ζ₂)² + (ζ₃)² = (ζ₀)².

Since we have removed the origin from the null cone, ζ₀ must be non zero.

Exercise 1. Why?

Thus either ζ₀ > 0, or ζ₀ < 0. We choose λ = 1/|ζ₀| and obtain a unique point in the equivalence class of
ζ with the zero coordinate equal +1 or -1. Let us concentrate on the case +1. We obtain the equation
(ζ₁)² + (ζ₂)² + (ζ₃)² =1 for this unique point. It represents a point on the two-dimensional sphere. It stereographically projects onto a point in (x,y) plane except for the South Pole of the sphere with ζ₃ = -1.

Let us return to our embedding formula from Part 18.

After some thinking, and prompted by Bjab's observation I have changed the embedding there to get rid of unnecessary minuses. It became

ζ₀ = (1+x·x)/2,

ζ₁ = x₁,

ζ₂ = x₂,

ζ₃ = (1-x·x)/2.

Now ζ₀ + ζ₃ = 1. If we have any point ζ' on the cone with ζ'₀ + ζ'₃ > 0, we can always rescale it by a unique positive λ to get ζ₀ + ζ₃ = 1, and then read the coordinates from ζ (the rescaled ζ') the two coordinates on the plane. The line with ζ₀ + ζ₃ = 0 form an equivalence class defining one point on the sphere - its South Pole, the Infinite Point on the plane.

Exercise 2. If ζ₀ + ζ₃ = 0 on the null cone, then, necessarily ζ₁ = ζ₂ = 0. Why?

Now we can return to the action of the Clifford group on Minkowski space, its null cone, the sphere, and the plane. We have a distinguished point on the sphere - its South Pole. Thus we can extract a special subgroup of transformations, namely those that do not move that distinguished point. These transformations will transform the (x,y) plane into itself - they will be easy to interpret. But there will be also other transformations that move the infinity point into some other point. Inverse of such a transformation will move a point on the plane into Infinity Point on the sphere, producing a singularity on the plane. We will analyze all this in the next post.

Where Have All the Spinors Gone?

We’ve got "spinors" boldly declared in the title of this series, yet somewhere along the way, they’ve managed to slip out of sight. This is unacceptable! Spinors should be front and center, the main act, not some backstage crew hiding in the shadows. So, just to keep them from becoming the forgotten middle child of mathematical objects, let’s bring them back into focus.

Recently, I received a seven-page paper from V.V. Varlamov. The topic? Clifford algebras and spinors, inspired by Rozenfeld's work on non-Euclidean geometry. I opened it eagerly, confident that by page seven, I’d emerge enlightened, ready to declare, "At last, I understand spinors!"

I did not.

The math in this paper was so advanced, it left me feeling like a freshman who wandered into a graduate-level seminar by mistake. My head began to spin—not in the cool quantum way, but in the “I need aspirin” way. The formulas weren’t just over my head; they were in orbit. It became clear that to keep my sanity intact, I needed to ditch the "follow the paper" approach and start thinking in my way.

I needed to ditch the "follow the paper" approach and start thinking in my way

So, here’s the mental road trip I embarked on:

We’ve got our trusty geometric Clifford algebra of space. It’s an elegant, eight-dimensional creature, cozy with bi-quaternions and complexified Minkowski space. Lovely. What’s more, this algebra isn’t just lounging around—it acts. And it acts on itself, no less. How?

• From the left: $L(u)v=uv$
• From the right: $R(u)v = vu$
  

Now, here’s the juicy part: when you let a Clifford group element $g$ act on something from the left, $g \cdot x$, it drags you out of the real Minkowski space. (Thanks a lot, $g$.) To reel things back in, you need to hit it from the right with something like $\tau(g)$. This leads to a beautiful commuting relationship:

$$L(u)R(w)=R(w)L(u)\mathrm{.}$$

This is clean, elegant math. But what does it have to do with the quantum physics of spin? Here’s where my mind started making some leaps—and possibly doing a few backflips.

Left, Right, and Quantum Duality

In quantum theory, there’s always this duality: the "observer" versus the "system under observation." For spins, we’ve got the laboratory frame with its neat axes, along with a physicist who’s busy assigning complex spin vectors in Hilbert space. And then we have the spin itself—precessing, pirouetting, doing its quantum dance.

In my mental model, I associate the left action with the quantum system itself and the right action with the observer. Or maybe it’s the other way around? I’ll admit, this part’s still a work in progress. The details need ironing out, like a wrinkly shirt you’re not sure is clean or dirty, but you’re wearing it anyway.

Enter the Hairy Ball Theorem

Somehow, all these thoughts led me to the Hairy Ball Theorem. Yes, that theorem from topology—the one that proves you can’t comb a hairy ball flat without creating a cowlick. If you’re wondering what this has to do with spinors, quantum dualities, or Clifford algebra, congratulations—you’re just as confused as I am.

But don’t worry, this will all (hopefully) become clearer in future posts. For now, I’ll leave you with this cliffhanger: Can spinors help us avoid cowlicks in quantum mechanics?

Rudolf Haag and the Interpretation of Quantum Mechanics

 This note is a continuation of Why Algebra? Rudolf Haag 

Haag’s Assertions

Concerning the interpretation of quantum mechanics Haag noted that 

“to this day, there remains some uneasiness about its status, some disagreement concerning its interpretation. This is not restricted to crackpots. Different camps of eminent scientists advance widely different opinions.” 

Cranks and Crackpots

It is good, at this point, to remember the advice of the philosopher, Bertrand Russell, namely that when experts do not agree on a given subject, then “no opinion can be regarded as certain by a non-expert”. Then Haag defines his position, and I will concentrate on just two points here:

1. The subject of physics is ‘nature’ and, whatever this means precisely, it is something beyond and apart from human knowledge.

In other words, physics should be formulated in such a way that it concerns objective phenomena. All twists, so popular, trying to make physics subjective, invoking the necessity of an “observer”, and so on, do not belong to physics! They result from arbitrarily assumed metaphysical prejudices that affect rationality of thinking.

Then we have the second assertion:

B. Individual knowledge is gained by observation, typically by experiment. In describing the set up and the result of an experiment we are bound by limits emphasized by Bohr: ‘We must be able to tell our friends what we have done and what we have learned.’ Bohr concludes from this that in the description of both the arrangement and the result we are bound to use “the language of classical physics’’.

Thus Haag takes a rational point of view. The language of physics, one that physicists should use when talking about Nature, should be a classical language that deals with real, material objects and their placement in space time.

Popper’s “Propensity”

Further in his paper he notes that many issues in quantum theory are still open, in particular the issue of whether the correct description should be deterministic or not. Instead of using the term “probability”, that requires repeated experiments, Haag opts for the Popperian term “propensity”, that characterizes an objective property associated with a given situation. The fact that “propensity” per se cannot be observed should not worry us, because:

The use of concepts in the theory which are not directly amenable to observation is neither forbidden nor unusual. It seems unavoidable.”

Then he addresses the quantum mechanical superposition principle and superselection rules:

Strict superselection rules forbid (coherent) linear combinations of states which differ in some charge quantum number (electric, baryonic, leptonic,…).

This alone, at least for me, dismisses the “Schrödinger cat paradox”; the paradox exists only because it is based on arbitrary assumptions. 

The fact that there IS a paradox tells us that these assumptions are wrong. But, for some reason, most physicists simply ignore this fact.

The Problem of the Observer

The concept of an “observer” was, for Haag, and also for me, causing a serious problem, as it meant that we forcefully limited the scope of physics, which was/is unnecessary. Instead of introducing an “observer” that is external to the physical system, it is better to introduce the concept of an event – as a fundamental concept. Things happen, with or without observers. Reality is Real on its own terms. Another fundamental concept that we should not be afraid of is irreversibility at a fundamental level. All these corrections should lead to:

…a self consistent theory in which all relevant objects are included as parts of the physical system.

Then Haag discusses different possible strategies that could lead to such a theory, and, in particular, notes:

In a series of papers Ph. Blanchard and A. Jadczyk have described a formalism which generates real events in the interaction of an atomic object with a macroscopic measuring device. The latter is idealized as a classical system in the sense that it is described by a commutative algebra which may include some discrete variables. The scheme, called “Event Enhanced Quantum Theory”, introduces irreversible decisions into the interaction process and yields a good phenomenological description of the quantum measurement process. (Italics mine)

As this concerns my own contribution, it is time for me to develop the comment of my teacher, one of the clearest and sharpest minds I ever knew in my life.

P.S.1 My wife likes it. SC looks a lot like FLA - Her childhood home. This is for Her.

Joan Baez, Hickory Winds

It's hard way to find out that trouble is real
In a far away city, with a far away feel
But it makes me feel better each time it begins
Callin' me home, hickory wind

P.S.2.12-06 13:00  I am trying to find out if the conformally compactified Minkowski space is orientable or not. Its double cover certainly is (it is S^1 x S^3), but its image by the covering map? I feel really shameful that I don't know it. Big holes in my working knowledge of differential geometry. Work, work, work and more work needed! So I work.

P.S.3. 13:30 It is (isomorphic to) a Lie group (namely U(2)), so it should be orientable. But then I don't understand anything at all.

P.S.4. Possibly related:

Suspect UFOs Are Biblical Time Machines
Diana Walsh Pasulka

P.S.5. The beginning of "Geometry of the conformally compactified Minkowski space"

With time it will be, little by little,  continuously improved, completed, changed, mutated. Slow evolution. Survival of the fittest, as they say.

I welcome all questions, comments, suggestions concerning these notes. 

P.S.6 From: M. Lipkind, "Definition of consciousness. Impossible and unnecessary? "

In: Fritz Albert Popp and Lev Belousov, "Integrative Biophysics", Springer 2003, p. 439

"Thus,  Protoconsciousness  can  be  imagined  as  current  awareness  by  a living  cell  of the  gap,  that being expressed as  non-congruence,  divergence, incompatibility, collision, conflict - between the ideal geometrical form and its physical realization. This means that such discrepancy between the Ideal Geometry and  Robust Physics is "felt" by the cell.  The  ideal geometry is species-specific, initially pre-existing,  and  pre-determined, while the robust physics is actually occurring and constantly fluctuating to adapt, to adjust, to fit,  to accommodate, to approximate to the ideal geometry.  Accordingly, the living process can be expressed as continuous dynamic approximation of the "real" physical form  to  its  geometrical "ideal".  Since this  approximation is felt  until  there  is  the  non-congruence  between  physical  and  geometrical (which  can  diminish  only  asymptotically,  i.e.  the  physical  will  never coincide  with  geometrical),  then  the  geometrical  feeling (protoconsciousness)  is  an  inalienable  part of any  living  entity.  Hence,  the Geometrical  Feeling  is  suggested  for  the  role  of  the  Protophenomenal Fundamental alongside physical fundamentals (Mass, Charge, Time/Space). As  to  the  deep  ontological  meaning  of the  concept  of "Geometrical Feeling",  it  could  be  analogized  with  the  "Universal  Grammar"  by  N. Chomsky (1988), which may be considered as "intrinsic part of the structure of matter ever since the Big Bang, or a necessary part of the eternal Platonic world oflogic and mathematics" (Hamad, 2001). "

P.S.7. From Christopher Langan "The Theory of Theories":

"For example, modern physics is bedeviled by paradoxes involving the origin and directionality of time, the collapse of the quantum wave function, quantum nonlocality, and the containment problem of cosmology.  Were someone to present a simple, elegant theory resolving these paradoxes without sacrificing the benefits of existing theories, the resolutions would carry more weight than any number of predictions.  Similarly, any theory and model conservatively resolving the self-inclusion paradoxes besetting the mathematical theory of sets, which underlies almost every other kind of mathematics, could demand acceptance on that basis alone.  Wherever there is an intractable scientific or mathematical paradox, there is dire need of a theory and model to resolve it.  

 If such a theory and model exist – and for the sake of human knowledge, they had better exist – they use a logical metalanguage with sufficient expressive power to characterize and analyze the limitations of science and mathematics, and are therefore philosophical and metamathematical in nature.  This is because no lower level of discourse is capable of uniting two disciplines that exclude each other‘s content as thoroughly as do science and mathematics.   

 Now here‘s the bottom line: such a theory and model do indeed exist "

I don't think so. Attempts to create such a beast - they exist.

P.S.8. In the pdf note I have made a few changes comparing to the previous version.. I have multiplied the factor in front of the matrix by "i", and the matrix by "-i". The whole matrix U(Z) is unchanged. The proof  (I will write it later today) will be somewhat more "elegant" after these "cosmetical" changes.

Where did I get the matrix U(Z) from? I am not sure. Reading, thinking, making errors, and correcting them. Until it works the way I envisaged it to work.

P.S.9. Uploaded a new version of the pdf note. The proof of the last lemma is not yet finished.