# Relation between Dirac spinors, quaternions, and bicomplex numbers

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

• all can be expressed by unitary 4x4 matrices so they seem to represent 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 this questions is not discussed anywhere in relation to 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 natural question if physically meaningful Dirac equation can be expressed using quaternions instead of Dirac spinor? Or what physical consequences (like alternative physics) it would lead to?

• You might want to read Quaternionic Quantum Mechanics and Quantum Fields by Stephen L. Adler: books.google.co.uk/books?id=irt7nOFaR3sC – J.G. Jul 26 '18 at 12:45
• 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?" – iSeeker Jul 29 '18 at 18:35
• 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. – iSeeker Aug 1 '18 at 17:47