# I The complex numbers in QM

#### PeterDonis

Mentor
The complex numbers are nothing else:$\mathbb{C}=\mathbb{R}/\langle x^2+1\rangle$.
Huh? Not all complex numbers satisfy $x^2 + 1 = 0$. Nor is $\mathbb{C}$ a quotient of $\mathbb{R}$ by anything.

#### fresh_42

Mentor
2018 Award
Huh? Not all complex numbers satisfy $x^2 + 1 = 0$. Nor is $\mathbb{C}$ a quotient of $\mathbb{R}$ by anything.
Same as not all tensors satisfy the EFE. But all complex equations written as polynomials modulo $x^2+1$ are the same equations written as complex numbers. And if we restrict ourselves to calculations modulo EFE, then EFE is within the calculations from the beginning.

#### A. Neumaier

Shouldn't we use $L^2(\mathbb{F})$ instead and work with functionals which already carry the coordinates not as variables but as scalars, since the existence of a stress-energy tensors affects the behavior of the coordinates? $\mathbb{F}$ would then be $\mathbb{C}(t,x,y,z)$, leading to a space where $L^2(\mathbb{R}^3,\mathbb{C})$ is only a subspace. An all-in-one environment, so to say.
These fields are not linearly ordered.
If I understood it correctly, a metric tensor $g$ can be considered as an element of $\mathbb{R}^4 \otimes (\mathbb{R}^4 \otimes \mathbb{R}^4)^*$,
A metric tensor is an element of $C^\infty(\mathbb{R}^4,\mathbb{R}^4 \otimes \mathbb{R}^4)$.
This is not a triple tensor product.
Hence every calculation done in this vector space, and I'm sure it can be turned into a Hilbert space, automatically respects GR.
For quantum gravity one needs diffeomorphism invariance of the equations of motions for quantum fields. This requires far more than defining a Hilbert space. Moreover, a Hilbert space structure requires the field to be that of the real or complex numbers; otherwise the norm is not well-defined.
I meant $L^2(\mathbb{F},\mathbb{F})$.
This space has not the structure even to accommodate the spectrum of a harmonic oscillator, let alone that of more complicated physics.
The original thought of this thread had been: If we only consider a few representations of the Lie algebra of $U(1)\times SU(2)\times SU(3)$ and of rather low dimensions, what do we need the complex numbers for? Any sufficiently large algebraic closed field of characteristic not two would lead to the same results. And if so, why not vary the scalar field instead of the groups ($E_8,SU(5)$) or the Lie algebras ($\mathbb{Z}_2$ grading) to extend the theory.
To have temporal evolution one needs 1-parameter subgroups parameterized by the real line. Thus nonzero characteristics is out. (There are a few papers trying to generalize quantum physics to the p-adic case, which has characteristic zero but a number theoretic flavor, but these attemtps get nowhere close to being useful. Note that a huge number of results in quantum mechanics are corroborated by experiments and would have to be recovered by any alternative. Lack of reproducing all these results to the verified accuracy rules out all little-informed tinkering with the basics.

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#### jambaugh

Gold Member
An additional observation. At its heart quantum mechanics is a real theory. The symmetry groups are parameterized by real numbers e.g. SU(N) and not SL(N,C). Observables must have real eigen-values.

However I wildly speculate that to a large extent the complex representations are necessitated by the intimate connection between the U(1) gauge of electromagnetism and our use of electromagnetic interactions as our principle instrument of observation and interaction between observers and systems.

That may be a chicken & egg regress (i.e. one could as easily say EM gauge is necessitated by the complex representations in QM) or it may be totally off. I haven't explored it very deeply.

I do believe there's been research on QM with different, especially non-abelian fields yielding additional gauge degrees of freedom. I know one early foray into quaternionic QM turn out to recover weak isospin (D.R. Finkelstein but i don't have the citation handy.) I suspect it is equivalent to standard model type gauge theory.

#### HomogenousCow

Is it that time of the year again?

#### vanhees71

Gold Member
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

Mentor
2018 Award
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.