Spin Chronicles Part 44: Archimedes Principle

 This is a continuation of "Spin Chronicles Part 43: Feeding GNS doll", where we have treated the GNS construction as if it was a lovely sweet doll. We played with it, and we have discussed three different ways of looking at it. The third method, called there Feed 3 is the most satisfactory, and in the following we will use it for the discussion of "time" emerging from space a'la Connes-Rovelli and Heller-Sasin. Do not worry now, we will get there soon, and all will be clear.

What if there is always a little bit of imaginary space and imaginary time?
Like a boat that floats on the surface, but is always embedded in water.
Perhaps there is a kind of  "Archimedes Principle" here? 

Why do I connect GNS to spinors? To answer this question let us first take a look how spinors are defined when we want to connect them to geometry, so that, for instance, they can be used also in curved spacetimes. There are several standard ways of doing so. There is no problem with getting the Clifford algebra and spin group inside it. In General Relativity it will be the Clifford algebra of the tangent space at a given point. But then what? One way is: take an irreducible representation of this algebra. Call vectors of this representation "spinors". Problem solved. Yes and no. There is unanswered question: where do I get this representation from? Which one? To ease this uneasiness we give the representation space a name "Clifford module". We are dealing now with named objects, we feel better.

The second approach is: introduce "spinor structure". We postulate its existence together with the covariant 2:1 map from spin frames to orthonormal frames. But where does it come from?  The answer is: "somehow, does it really matter?". And then it is added: for some spacetimes there may exist several inequivalent spin structures! 

The third approach is: take a minimal left ideal in the Clifford algebra, perhaps take two or four of them - you will fit different spinors to different Fermions. Isn't it nice? Even better, the right action will shuffle the ideals. Left action corresponds to spacetime rotations, right action corresponds to "internal symmetry operations". It can even fit the Standard Model. For all practical purposes it can even work, but the question remains: which ideal and why this and not another one? I am not completely happy with this. We are onto something, but what is it, this "something"?

GNS construction associates representations to "states". If we follow this philosophy, spacetime (or just "space") at a given point may be in some "state". It fits my intuition. Once We have "state", we have the associated representation. Pure states lead to irreducible representations. Mixed states lead to reducible ones (we will discuss this issue in the next post). This opens my memory bank. Pure states are "extreme points" of the convex set of all states. They are at the boundary. It is hard to get dynamically exactly to the boundary. Rather, when we experimentally attempt to get exactly pure state, we will obtain "almost pure" state, never "exactly pure". Pure states is an extreme idealization. No state is in the Nature is exactly pure. This opens a whole new perspective with possible physical consequences. I like it. It leads to another idea: we think that our space is "real", time is "real". Imaginary space and imaginary time are simply mathematical tricks. But are they? What if "impurities" ("defects")  are important? What if there is always a little bit of imaginary space and imaginary time? Like a boat that floats on the surface, but is always embedded in water. Perhaps there is a kind of  "Archimedes Principle" here?  In recent years Peter Woit develops a similar vision.

That is just "talk". Next post will be just "math". We will discuss "states dominated by other states", representations, cyclic vectors and irreducibility. 

The Spin Chronicles (Part 3): Spin frames

What’s a Spin Axis, Anyway?

This post was inspired by a question from Bjab, who asked: "What do you mean by ‘spin axis’?" Well, buckle up because I’m about to explain that! I’m not entirely thrilled with the answer I have right now, but hey—it works (kind of). Hopefully, in the near future, I'll come up with something that satisfies both Bjab and my restless curiosity. For now we have to be happy with the standard concept of the spin structure - representing the universe as a set of interlocked rotations - described below.

Spin structure

Where Do Spinors Live?

Imagine a strange and mysterious land—let’s call it V—a two-dimensional complex vector space. It’s not just any space, though. It has "structure." Picture it equipped with a hermitian non-degenerate scalar product that we’ll denote by $(u,v)$, and a bilinear antisymmetric non-degenerate form we’ll call $\epsilon$. Sounds fancy, right? These two should be related in a specific way, and here’s how:

There exists a special kind of basis for this space, which we’ll call “special orthonormal.” It satisfies:

1. $(e_1,e_1) = (e_2,e_2) = 1$,
2. $(e_1, e_2) = 0$,
3. $ϵ(e_1,e_2)=1.$

“Special” indeed. The first natural question is: Can this magical basis even exist?

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The Magic of C²: Relax, It’s All Possible

Yes, it exists! Here’s why. Let’s choose any basis in V. Using this, we can identify V with $C^2$ (our familiar two-dimensional complex space). In $C^2$, the scalar product $(u,v)$ is defined as:

$$(u,v)=u^{\mathrm{1\ast }}{v}^1+u^{\mathrm{2\ast }}{v}^2$$

where the star (*) represents complex conjugation. As for $\epsilon$, we define it as:

$$ϵ(u,v)=u^1{v}^2-u^2{v}^1$$

Now, using a standard basis $e_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$, $e_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$, it’s easy to check that our special orthonormal basis conditions hold. So, crisis averted! We don’t need to worry about whether this basis exists—it does. Phew.

Oh, and here’s a fun fact: if $(e_1, e_2)$ is a special orthonormal basis, then $(-e_1, -e_2)$ is also a special orthonormal basis. Isn’t symmetry beautiful?

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Back to the Mysterious World of V

So, we’ve got our scalar product, our fancy $\epsilon$, and we’re cruising in V—home to the enigmatic spinors. But are there other “special orthonormal” bases? And if so, how many?

Let’s say $e'_1, e'_2$ is another basis. There exists some matrix $A$ such that:

$$e_i^{\mathrm{\prime }}=e_j{A}_i^j(i,j=1,2)$$

(Yes, that’s a summation over “j”.)

Now, here’s the neat part: $e'_1, e'_2$ will also be special orthonormal if and only if the matrix $A$ is unitary with determinant 1. Matrices like this form a group called SU(2).

By the way, SU(2) is isomorphic to the group of unit quaternions. I won’t dive into that now, but it’s a pretty cool connection between algebra and geometry.

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The Spinor Connection

Let’s denote by O(V) the space of all these special orthonormal bases. The group SU(2) acts on O(V) from the right, both transitively and freely. The elements of O(V) are what we call spinor bases, and the elements of V are simply spinors. If $u$ is a spinor, and you change the basis using an SU(2) matrix $A$, the components of the spinor change according to this formula:

$$u^{\mathrm{\prime }i}=A_j^i{u}^j$$

Got it? Good! Now, let’s connect this with something real, like the lab—where actual experiments happen (sometimes with coffee stains on the data sheets).

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What About Our Lab?

To bridge this to the physical world, we need some more algebra (don’t worry, just a little more). Quaternions are often used for handling 3D rotations, but since we’re sticking with matrices, here’s what we do in quantum mechanics:

For each point $x = (x^1, x^2, x^3)$ in $R^3$, associate the following 2x2 Hermitian matrix, $\sigma(x)$:

$$\sigma (\mathrm{<b>x</b>})=\left(\begin{array}{cc}{x}^3& x^1-i{x}^2\\ x^1+i{x}^2& -x^3\end{array}\right)$$

Now, for any SU(2) matrix $A$, it turns out:

$$A\sigma (\mathrm{<b>x</b>})A^{\ast }=\sigma (R(A)\mathrm{<b>x</b>})$$

where $R(A)$ is a 3x3 real rotation matrix, an element of the special orthogonal group SO(3). In fact, the map from $A\to R(A)is\; a\; homomorphism.\; And\; every\; element\; of\; SO(3)\; can\; be\; generated\; this\; way.\; The\; catch?\; If$$A$ and $B$ are two SU(2) elements, then $R(A) = R(B)$ if and only if $B = \pm A$.

Hence, SU(2) is a double cover of SO(3). It’s like the universe giving you two chances to rotate.

Note: With quaternions it would look more "natural". To each x we would associate a pure imaginary quaternion 

q(x) = xi+yj+zk, 

and for each unit quaternion u, we would have 

uq(x)u* = q(R(u)x), 

where R(u) is in SO(3). 

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The Spin Structure: So What About That Spin Axis?

Let’s bring this back to the real world—our lab, where we have Cartesian coordinates. Any two right-handed orthogonal systems (say, different setups in the same room) are related by a unique rotation matrix from SO(3). The set of all such frames forms a space O(L), which SO(3) acts on.

Now, here’s the kicker: nature somehow maps each element of O(V) (the spinor world) to an element of O(L) (our lab’s coordinate system) in a way that respects the rotations. We call this map the spin structure.

In other words, when you rotate an element of O(V) by an SU(2) matrix A, the corresponding lab element rotates, too, By and SO(3) matrix R(A). It’s like a cosmic interlock between spinors and real-world rotations.

Note (for experts): This is a baby version of a spin structure, good for a 3D flat space or, more generally, for a space with a distant parallelism. But this baby version is all that we need for now.

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What’s Next?

All this brings us closer to answering the question: what do I mean by a spin axis? Stay tuned, because we’ll tackle that mystery in the next post! We will also take under the loop the spin structure - the most important point here.

P.S. I am experimenting with the mathematical notation on this blog. Sometimes I mess up, then correct what can be corrected.

P.S. I have uploaded the pdf version of this post to academia.edu.

P.S. 18-10-24 9:33 Igor Bayak (1.1.6)