[What’s changed in the last ~100 years]
A recent Scientific American article reminded me that quantum spin underlies the stability of matter – without which there’d be no life. But the article prompted another dive into the “mathematical machinery” describing the quantum state of a single electron or a single photon.
The Stern–Gerlach experiment established that an atomic-scale system has intrinsic quantum properties. In particular, quantization of an electron’s (intrinsic) angular momentum – spin. As such, yet another quantum state with an observable in superposition. In this case, a superposition of up/down spin direction (along an axis).
Quantum superposition is a fundamental principle of quantum mechanics. … that every quantum state can be represented as a sum of two or more other distinct states. [Note reference to the diffusion equation in section on Hamiltonian evolution.]
Vectors and spinors and bispinors, oh my!
Here’s a historical recap of a seminal physics experiment 100 years ago – the Stern-Gerlach experiment . Regarding a property of elementary particles which underlies the stability of matter – the spatial orientation of [intrinsic] angular momentum: the concept of quantum spin (in 4-D Hilbert space). In this case for leptons, e.g., electrons .
• Scientific American > Quantum Physics > “100 Years Ago, a Quantum Experiment Explained Why We Don’t Fall through Our Chairs” by Davide Castelvecchi (February 8, 2022) – The basic concept of quantum spin provides an understanding of a vast range of physical phenomena.
Without quite realizing what they were seeing, [Otto] Stern and his fellow physicist and collaborator Walther Gerlach discovered quantum spin: an eternal rotational motion that is intrinsic to elementary particles …
As physicist Wolfgang Pauli would explain in 1927, spin is quite unlike other physical concepts such as velocities or electric fields. Like those quantities, the spin of an electron is often portrayed as an arrow, but it is an arrow that does not exist in our three dimensions of space. Instead it is found in a 4-D mathematical entity called a Hilbert space .
Unlike in modern experiments, the displacement of the beams was tiny – about 0.2 millimeter – and had to be spotted with a microscope.
It was only after modern quantum mechanics was founded, beginning in 1925, that physicists realized that the silver atom’s magnetism is produced not by the orbit of its outermost electron but by that electron’s intrinsic spin, which makes it act like a tiny bar magnet.
… to this day, physicists continue to argue about how to interpret the experiment [regarding “quantum superposition” and the measurement problem].
The characterization of an electron as a spinor / Dirac spinor (aka bispinor) is helpful. But following all the math is beyond my ken. Everyday (3D) analogies offer some insight.
Spinors and tensors are associated with continuous geometries of energy density, stress, flux. Kinked energy, as I’ve noted elsewhere. The mathematical framework implies higher dimensional (3-sphere) topology. Which limits an intuitive grasp. Mathematically, the well-defined formalism may be routine; but visualization elusive – the geometric significance still mysterious.
 Wiki notes that “the experiment was performed several years before Uhlenbeck and Goudsmit formulated their hypothesis of the existence of the electron spin.”
The Stern–Gerlach experiment has become a prototype for quantum measurement, demonstrating the observation of a single, real value (eigenvalue) of an initially unknown physical property.
 And, as also noted in Wiki’s article, the correspondence between an electron’s spin and angular momentum of the photon – photon polarization.
• Wiki > Spin (physics)
For photons, spin is the quantum-mechanical counterpart of the polarization of light; for electrons, the spin has no classical counterpart.
Spin is described mathematically as a vector for some particles such as photons, and as spinors and bispinors for other particles such as electrons.
 Wiki notes:
The significance of the concept of a Hilbert space was underlined with the realization that it offers one of the best mathematical formulations of quantum mechanics. In short, the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries of the system are unitary operators, and measurements are orthogonal projections.
The orbitals of an electron in a hydrogen atom are eigenfunctions of the energy.
 Mathematical formalism / representation and reality
Of course, there’s an interesting history even for numbers which were considered fictitious in some sense. These numbers changed our perspective on things, whether or not actually existing or exactly measurable in our everyday “real” landscape. Useful representations (as are infinite series). Numbers like zero. Transcendental numbers (e.g., π and e). Irrational numbers. Imaginary numbers (e.g., the square root of negative one).
Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to describe the intrinsic angular momentum, or “spin”, of the electron and other subatomic particles.
Spinors are characterized by the specific way in which they behave under rotations. They change in different ways depending not just on the overall final rotation, but the details of how that rotation was achieved (by a continuous path in the rotation group).
Regarding the history of irrational numbers, see also my March 16, 2017 comment re Effective Theory and Pi Day.
• Wiki > Dirac’s bra–ket notation
• Wiki > Spinor > Attempts at intuitive understanding
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.
• Wiki > Bispinor
In physics, and specifically in quantum field theory, a bispinor, is a mathematical construction that is used to describe some of the fundamental particles of nature, including quarks and electrons.
A bispinor is more or less “the same thing” as a Dirac spinor. … That is, the Dirac spinor is a bispinor in the Dirac convention. By contrast, the article below concentrates primarily on the Weyl, or chiral representation, is less focused on the Dirac equation, and more focused on the geometric structure, including the geometry of the Lorentz group.
Bispinors are so called because they are constructed out of two simpler component spinors, the Weyl spinors. Each of the two component spinors transform differently under the two distinct complex-conjugate spin-1/2 representations of the Lorentz group. This pairing is of fundamental importance, as it allows the represented particle to have a mass, carry a charge, and represent the flow of charge as a current, and perhaps most importantly, to carry angular momentum.