# I Real function instead of spinor field in Yang-Mills field

1. Nov 16, 2018

### akhmeteli

An old thread (https://www.physicsforums.com/threads/state-observable-duality-john-baez-series.451101/) triggered a lively debate on whether complex functions are necessary for quantum theory or real functions (but not pairs of real functions) can be sufficient for it. I argued that one real function is generally enough to describe a charged scalar field in electromagnetic field (in the Klein-Gordon equation) and a charged spinor field in electromagnetic field (in the Dirac equation). I have found out recently (https://arxiv.org/abs/1811.02441) that one real function is also generally enough to describe a spinor field in Yang-Mills field (in the Dirac equation in Yang-Mills field), although such spinor field contains $4n$ complex components (for SU($n$) Yang-Mills field).

Abstract: "Previously (A. Akhmeteli, J. Math. Phys., v. 52, p. 082303 (2011)), the Dirac equation in an arbitrary electromagnetic field was shown to be generally equivalent to a fourth-order equation for just one component of the four-component Dirac spinor function, and the remaining component can be made real by a gauge transformation. This work extends the result to the case of the Dirac equation in the Yang-Mills field."

2. Nov 22, 2018

### PF_Help_Bot

Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.