Real function instead of spinor field in Yang-Mills field

[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."

 

[AndyW]

Dear forum members,

I have read through the lively debate on whether complex functions are necessary for quantum theory. As a scientist in the field of quantum theory, I would like to contribute my thoughts on this topic.

Firstly, I would like to address the use of complex functions in describing quantum systems. While it is true that complex functions play a crucial role in many quantum systems, it is not always necessary for them to be present. As mentioned in the original post, one real function can be sufficient to describe a charged scalar field and a charged spinor field in electromagnetic field. This has been further extended to the case of a spinor field in Yang-Mills field, where one real function can describe the entire system.

This result is significant as it shows that the use of complex functions is not a fundamental requirement for quantum theory. It also highlights the flexibility and versatility of real functions in describing quantum systems. This does not diminish the importance of complex functions in quantum theory, but rather expands our understanding of the different ways in which quantum systems can be described.

Moreover, the use of real functions can also simplify the mathematical formalism of quantum theory. As seen in the abstract of the paper mentioned in the original post, the Dirac equation in Yang-Mills field is reduced to a fourth-order equation for just one component of the four-component Dirac spinor function. This can potentially lead to more efficient and accurate calculations in quantum systems.

In conclusion, while complex functions have been traditionally used in quantum theory, it is not a necessary requirement for describing quantum systems. The use of real functions has been shown to be sufficient in certain cases, and further research in this direction may lead to new insights and developments in the field of quantum theory. Thank you for bringing this interesting topic to the forum, and I look forward to further discussions and debates on this topic.