The necessity of complex numbers in quantum mechanics (QM) is a central topic of discussion, focusing on their role in modeling continuous spectra of observables such as position, momentum, and energy. Participants argue that while algebraically closed fields can theoretically replace complex numbers, the continuous nature of spacetime requires the use of the complex field ##\mathbb{C}##. The conversation also explores the implications of using transcendental extensions like ##\mathbb{C}(t)## and the potential for alternative mathematical frameworks, such as ring theory, to incorporate spacetime directly into quantum field theory (QFT). Ultimately, the consensus is that the continuous algebraically closed field ##\mathbb{C}## is essential for maintaining the integrity of QM and its observables.
PREREQUISITESPhysicists, mathematicians, and researchers interested in the foundations of quantum mechanics, quantum field theory, and the mathematical structures underlying physical theories.
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.