# If a flagpole visualizes a Pauli spinor, what visualizes a Dirac spinor?

@AndrewSteane writes in his textbook that a Pauli spinor is a flagpole - with length, direction, flag orientation - and a sign.

A Dirac spinor is a more complicated object. What is the most intuitive visualization of a Dirac spinor?

Update: Steane proposes two flagpoles.

The 8 parameters of a Dirac spinor are 1 magnitude, 3 flagpole angles, 3 boost parameters (such as vx vy vz), and 1 fraction specifying particle and antiparticle components.

Does it make sense to contract the latter 4 numbers into a second flagpole? How can one relate the two descriptions?

• A flagpole and an antimatter flagpole ?
– JEB
Commented Jun 29, 2023 at 1:17
• This would be possible: but how does one relate the two poles to the boost parameters? Commented Jun 29, 2023 at 10:40
• Why not 2 spinning flagpoles with opposite chirality Commented Jul 2, 2023 at 21:09
• @Craig Steane proposes this as well. But then, how are matter and antimatter flagpoles related to the boost parameters? Commented Jul 3, 2023 at 5:20
• For what its worth I think it's not a great analogy. Nothing will give you intuition for the various spinors except for working with them. Commented Jul 3, 2023 at 7:05

Andrew Steane also writes, a Dirac spinor is essentially a pair of two Pauli spinors (see An introduction to spinors, chapter "VI. Dirac spinors and particle physics"):

Such a pair is called a bispinor or Dirac spinor. It can conveniently be written as a $$4$$-component complex vector, in the form $$\Psi=\begin{pmatrix}\phi_R \\ \chi_L\end{pmatrix} \tag{78}$$ where it is understood that each entry is a $$2$$-component spinor, $$\phi_R$$ being right-handed and $$\chi_L$$ left-handed.

Therefore I would visualize a Dirac spinor by a pair of two flagpoles (each with length, direction, flag orientation, and sign).

• He's talking about Weyl spinors. I don't think a Dirac spinor can be interpreted as two Pauli spinors (as I understand the term). Commented Jun 28, 2023 at 23:12
• Thank you for the reference! Steane says that a spinor is a flagpole and a sign. But how can a flagpole be right-handed or left-handed? Commented Jun 29, 2023 at 6:27
• @Thomas Fritsch How does one relate the two poles to the boost parameters and to the spin orientation and also to the particle.antparticle ratio? Commented Jul 1, 2023 at 6:32
• @KlausK I don't know. Sorry. Commented Jul 1, 2023 at 6:38