For warming up, let me start this post with some quotations. First, a snippet from an online article titled "Discovery of Electron Spin":
"The discovery note in Naturwissenschaften is dated Saturday, 17 October 1925. One day earlier, Ehrenfest had written to Lorentz to make an appointment for the coming Monday to discuss a "very witty idea" of two of his graduate students. When Lorentz pointed out that the idea of a spinning electron would be incompatible with classical electrodynamics, Uhlenbeck asked Ehrenfest not to submit the paper. Ehrenfest replied that he had already sent off their note, and he added: "You are both young enough to be able to afford a stupidity!"
Ehrenfest's encouraging response to his students’ ideas contrasted sharply with that of Wolfgang Pauli. As it turned out, Ralph Kronig, a young Columbia University PhD who had spent two years studying in Europe, had come up with the idea of electron spin several months before Uhlenbeck and Goudsmit. He had put it before Pauli for his reactions, who had ridiculed it, saying that "it is indeed very clever but of course has nothing to do with reality." Kronig did not publish his ideas on spin. No wonder that Uhlenbeck would later refer to the "luck and privilege to be students of Paul Ehrenfest."
A. Pais, in Physics Today (December 1989)
M.J. Klein, in Physics in the Making (North-Holland, Amsterdam, 1989)"
So, we know that the birth of the spin concept was not an easy one. Ideas that would revolutionize physics were initially dismissed, sometimes with sharp words, and only managed to take root under the shelter of intellectual bravery and a bit of recklessness. But here we are, nearly a hundred years later. Spin is no longer a child; it has matured into a cornerstone of quantum mechanics. How is it faring in its adulthood?
To answer this question, let us turn to another fascinating quotation, this time from the comprehensive Wikipedia article "Spinor." In the section titled "Attempts at intuitive understanding," we find the following:
"Nonetheless, the concept is generally considered notoriously difficult to understand, as illustrated by Michael Atiyah's statement that is recounted by Dirac's biographer Graham Farmelo:
No one fully understands spinors. Their algebra is formally understood but their general significance is mysterious. In some sense, they describe the "square root" of geometry and, just as understanding the square root of −1 took centuries, the same might be true of spinors."
Centuries? Do we really have to wait that long? Can we afford to? And what is it, exactly, that remains a mystery? Despite all the successes of quantum theory and the remarkable applications of spin physics—from magnetic resonance imaging to quantum computing—why is it that some of the brightest minds in physics and mathematics are still uneasy? Are they just inveterate malcontents, destined to grumble in the face of progress? Or is there a deeper puzzle lurking beneath the surface, one that defies our current frameworks?
I think the answer depends on the level of curiosity, which varies greatly among individuals. It has nothing to do with optimism or pessimism but rather with an insatiable desire to dig deeper. And so, with this in mind, let us make an inventory of what we have learned so far. Let’s take stock of the journey that has brought us here and explore the questions still waiting for answers.
Our starting point was a "vector space". Why vectors? Well, we
have to start with "something". Vectors is a reasonable starting point.
Not excessively simple and not excessively complicated. Why 3
dimensions? This is a good question. There must be a reason for three
dimensions. One possible way of dealing with this question would be:
"because organic life is possible only in 3D". But that would be an
answer that begs other questions. How could some intelligence, that
creates it all, know in advance what is possible and what not? By being
able to see into the future, even if only vaguely? Perhaps, but that
would lead us into speculations with no end in view. So, let us stay
with an empirical fact - we live in 3D space.
Then we added an Euclidean metric to V. Another empirical fact. Perhaps on a large scale the geometry is non-Euclidean, but infinitesimally is Euclidean enough. So, let us start with the Euclidean flat space and see where we can go along this path. That is what we are doing now.
Then we endowed V with "orientation". This is more iffy. Why this orientation and not the opposite? It hurts our love feelings towards some "perfect symmetry". It hurts badly. Yes, there is an empirical fact - we live in a universe with broken parity. But it hurts. So, we look for possible remedy. Perhaps our universe is a two-sided surface, a boundary separating two higher dimensions? On one side of this surface there is one parity, on the other side there is an opposite parity? Perhaps the surface is not necessarily of zero thickness? Perhaps, occasionally, the two sides can "communicate" somehow? That is for the future. For now let us accept the fact: we live in a 3D space with a preferred orientation.
Once we have our starting point, we turn on the Clifford algebra machine. It rewards us with the complex geometric *-algebra A,
isomorphic to complex quaternions, it rewards us with three
involutions, with the group isomorphic to the group SL(2,C) of special
relativity, and with the group isomorphic to SU(2), usually employed in
the study of simple spinors. But there are no spinors yet.
In quantum physics spinors transform under the irreducible
representation of SU(2). We were not discussing group representations so
far, but we were discussing the representation of A.
Once we have a representation of A, we have also a representation of any
group contained inside the algebra. And, in quantum theory, to deal
with spinors we really need the algebra, not only the group.
To relate the Clifford algebra to quantum theory the method of
searching for minimal ideals has been invented by algebraists. Usually
just one left ideal is picked up and it is shown that this enables us to do
all standard tricks with spinors, write equations, add interactions,
etc.
We already know how to construct these ideals. We need to choose a Hermitian idempotent p. Each such nontrivial p is of the form p=(1+n)/2, where n is a unit vector in V, a direction in 3D space. Algebraists define then the ideal generated by p as Ap = {up: u∈A}. I have chosen another, equivalent, way, a way that looks as a solution of a (right-sided) eigenvalue problem:
Ap = {u∈A: up=u} = {u∈A: un=u}
Note: We can also look for the eigenspace belonging to the eigenvalue 0 of p. This would give us a complementary left ideal. It can also be obtained as the eigenvalue 1 subspace for p'=(1-n)/2, corresponding to the choice of the opposite direction.
We consider p as an operator acting on A from the right. Since p2=p, eigenvalues of p are 0 and 1, and we are looking for a subspace belonging to the eigenvalue 1. Similarly for n. Since n2=1, n has eigenvalues +1 or -1, and we are looking for a subspace In belonging to the eigenvalue +1.
What can be the meaning of selecting just one such ideal?
Perhaps it is with selecting a point in space to set up a reference
frame there, as Alain Cagnati suggested? We select a reference direction
in space to be able to quantify what we measure? This defines our
two-dimensional reference Hilbert space. Or it is like choosing a
certain perspective, so that we can map the 3D house on a 2D canvas, as
suggested by Anna?
So we select a reference direction, set up our axes e1,e2,e3 in V, and E1,E2 in In, and produce our 2D complex Hilbert space with a basis E1,E2. We get then Pauli matrices as representing left action of the basis vectors of V. Standard quantum theory of spin is reproduced within In.
Another choice of n would then give us
hopefully equivalent description. Except what "equivalent" is is not
completely clear. It needs to be clarified. We will come to this point
later on.
Given a projection p=(1+n)/2 we can act with it from the left or from the right (or both ways at once). Given two projections p=(1+n)/2, q=(1+m)/2, we can ask two questions at the same time: find all u in A, satisfying simultaneously both equations:
1. up = u
2. qu = u.
We can interpret 1. as setting a reference direction to be n, and interpret 2. as finding all spin states with spin direction m. This is intersection of one left ideal with one right ideal. It is always one-dimensional.
Exercise 1. Prove this last statement.
What if we do not want to select a reference direction n? We can ask 2. without asking 1. We get a 2-dimensional (complex) right ideal. Strange. We are dealing now with two-dimensional complex subspace of a 4-dimensional complex space A. 2D subspaces suggest using bi-vectors. Bi-vectors are elements of a Grassmann algebra or Clifford algebra. Which suggests using the Clifford algebra of the Clifford algebra. Why not. Since A is real 8-dimensional, this would be Cl(8) - the beloved Clifford algebra of many. But that is just dreaming.
In the next post I will use another toy, going from pure
algebraists to C* and von-Neumann "non-commutativists". They use so called
GNS construction for playing with reducible and irreducible
representations of *algebras. This will give us another perspective.
