I Necessity of Complex Numbers in Quantum Mechanics

[vanhees71]

What's the problem with complex numbers? Of course, you can decompose everything in real numbers and work with real quantities only, but why do you want to do this? To the contrary, usually one uses complex exponential functions in purely real theories like electromagnetics instead of sines and cosines, simply because it's easier to handle.

 

[fresh_42]

What's the problem with complex numbers?

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.

 

[A. Neumaier]

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.

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.

 

[Tendex]

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.

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.