# 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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"The complex numbers in QM"

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