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Superficially Dirac spinor resp. Dirac gamma matrices and quaternions and bicomplex numbers seem to be very similar objects.

  • all can be expressed by unitary 4x4 matrices, so they seem to represent a kind of rotation in 4D space.
  • all can be expressed as 2x2 matrix of complex numbers

Questions:

  • So, are Dirac spinors just a subset of quaternions or not?
  • Or what is the relation, and what are the distinctions?
  • And what are the physical consequences?

I wonder why these questions are not discussed anywhere in relation to the Dirac equation, since this is the first question I would have. Since the Schrödinger equation is expressed using complex numbers, and the Dirac equation is its 4D version, I would naturally think first about quaternions as the 4D analog of complex numbers.

This also leads to the natural question if a physically meaningful Dirac equation can be expressed using quaternions instead of a Dirac spinor? Or what physical consequences (like alternative physics) it would lead to?

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    $\begingroup$ You might want to read Quaternionic Quantum Mechanics and Quantum Fields by Stephen L. Adler: books.google.co.uk/books?id=irt7nOFaR3sC $\endgroup$
    – J.G.
    Commented Jul 26, 2018 at 12:45
  • $\begingroup$ There's an easily accessible discussion in arxiv.org/pdf/1003.0075.pdf which touches briefly on your question: "In Dirac formalism, an electron, for instance, is described by a four component wavefunction (spinors). In the present formalism the particle is described by a quaternion wavefunction having also four components. Here, one component is a scalar and the three components are vector components. We may however ask the question that: are the two formalisms equivalent to each other?" $\endgroup$
    – iSeeker
    Commented Jul 29, 2018 at 18:35
  • $\begingroup$ Further to previous comment, I've just come across "Spinors in Spacetime Algebra and Euclidean 4-Space", available via researchgate.net/profile/Garret_Sobczyk which has sections on "Geometric spinors", "Quaternion spinors" and "Classical Dirac spinors" from a mathematical/geometric algebraic perspective. $\endgroup$
    – iSeeker
    Commented Aug 1, 2018 at 17:47
  • $\begingroup$ I'll put this as a comment because it doesn't really answer the question, but still feels relevant: If you multiply the Pauli matrices describing non-relativistic spin with $i$, you get a representation of the quaternions. $\endgroup$
    – paulina
    Commented Apr 24 at 9:24

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Since four anticommuting variables are required, but quaternions only provide three, a 2×2 quaternion matrix is required for the Dirac equation (the physical reason is the spin 1/2 requires two field components). See, f.ex., https://www.vttoth.com/CMS/physics-notes/165-quaternions-and-the-dirac-equation

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No they are NOT similar. Dirac gamma matrices have 16 degrees of freedom, while quaternions and bicomplex numbers have 4 degrees of freedom. Algebraically, they are totally different animals.

Let's count the degrees of freedom in detail:

  • Dirac algebra $Cl(1,3)$ (restricted to the real algebra): one scalar 1, 4 vectors $\gamma_\mu (\mu = 0,1,2,3)$, 6 bi-vectors $\gamma_{\mu\nu}$, 4 3-vectors $\gamma_{\mu\nu\kappa}$, and one 4-vector $\gamma_0\gamma_1\gamma_2\gamma_3$. In total 16 degrees of freedom.
  • Quaternions: scalar 1 plus $i$, $j$, and $k$. In total 4 degrees of freedom.
  • Bicomplex numbers: 2 degrees of freedom for each of the two complex numbers, therefore in total 4 degrees of freedom.

The only way to link Dirac algebra $Cl(1,3)$ with quaternions is via the isomorphism (also noted by the answer of @Francisco R. Villatoro): $$ Cl(1,3) \sim M_2(H) $$ where $M_2(H)$ stands for the $2\times2$ matrices of quaternions which have 4*4 = 16 degrees of freedom that match with Dirac algebra $Cl(1,3)$. In the $M_2(H)$ representation, a Dirac spinor is a bi-quaternion (column of 2 quaternions) with $4+4=8$ degrees of freedom.

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The person who seems to have delved into this most deeply is Dennis Morris and although his writing style is very individual his ideas are interesting and very thought provoking. See his: The Quaternionic Dirac Equation and his bools on quaternions such as Non-commutative Differentiation and the Commutator. The special features that come out of his writing that I haven't found elsewhere is the modelling of Electro-weak theory which emerges quite naturally from his discussion of Left and Right Chiral quaternions.

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