The discussion revolves around the necessity of complex numbers in quantum mechanics (QM), exploring whether other algebraically closed fields or discrete fields could serve the same purpose. Participants question the foundational role of complex numbers, the implications of using different fields, and the relationship between spacetime and the mathematical structures used in QM.
Participants do not reach a consensus on the necessity of complex numbers in QM, with multiple competing views remaining regarding the use of alternative fields and their implications for physical theories.
Participants highlight that the discussion involves complex mathematical structures and their physical interpretations, with unresolved questions about the implications of using different fields in quantum mechanics and QFT.
It's no problem. The question was if it is necessary or does the field provide an additional parameter which can be varied, e.g. by a transcendental extension? I think @jambaugh was right, that at its kernel the SM is a real model, so the question about the role of the scalars must be allowed. And the next step was: If it is real and dimensions low, what makes it different from fields with a large positive characteristic? The eigenvalues should be the same in say ##\mathbb{F}_{61}##, which was my original (and now negatively answered) thought.vanhees71 said:What's the problem with complex numbers?
No. The standard model is a quantum field theory, which cannot be formulated naturally without complex numbers.fresh_42 said:I think @jambaugh was right, that at its kernel the SM is a real model, so the question about the role of the scalars must be allowed.
And for extending beyond the perturbative one needs to justify another analytic continuation back from the Schwinger function in Euclidean space to the Wightman function in Minkowski space.A. Neumaier said:No. The standard model is a quantum field theory, which cannot be formulated naturally without complex numbers.
In canonical quantization or lattice discretizations, the Schrödinger equation involves an explicit imaginary unit ##i##.
In the Euclidean path integral formalism one needs either analytic continuation to imaginary time or an imaginary factor before the action in the exponential.