# A Complex numbers in QM

#### Demystifier

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#### Peter Hearty

That book's been on my reading list for some time. One reason, although not a major one, is because of the complex numbers stuff. I presume something else must be used to accommodate the extra degrees of freedom that complex numbers gives, either via multiple simultaneous equations, or perhaps Clifford algebras.

Perhaps I should have said "why complex numbers are used", rather than "needed".

#### Demystifier

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Presumably one can correct it by saying they are needed up to isomorphism.
Then classical Maxwell equations also need complex numbers up to isomorphism.

#### atyy

Then classical Maxwell equations also need complex numbers up to isomorphism.
In how many space + time dimensions?

#### PeroK

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In my view the question over whether to admit the use the complex numbers doesn't have the physical or mathematical significance that some people like to think it has. You could avoid calling them "numbers" or you could find a way to conduct the mathematics without anything that has the mathematical properties of complex numbers. But, really, why worry about complex numbers in particular?

Moreover, in a formal development of numbers it is the real numbers where a lot of the problems lie. And, especially, if you try to justify the real numbers as something physical. For example, a typical real number is indescribable, in the sense that it requires an infinite amount of information to quantify it. If you do something like take the position, $x$, of a particle at some time, $t$, where $x, t$ are real numbers, then you have something quite hard to physically justify. There is no way to write down a typical real number.

And yet, in all the physics I've some across measurement values are assumed to be real numbers. A formal justification of this would not be easy in my opinion.

Why, then, does no one try to do physics based on rational numbers and countable sets? I imagine that it is possible. But, I suspect that the sacrifice in losing calculus and much else besides is too much. Whereas, the sacrifice in losing the complex numbers is perhaps more manageable.

#### atyy

Why, then, does no one try to do physics based on rational numbers and countable sets? I imagine that it is possible. But, I suspect that the sacrifice in losing calculus and much else besides is too much. Whereas, the sacrifice in losing the complex numbers is perhaps more manageable.
The biggest obstacle is the chiral lattice fermion problem. Even if the chiral fermion problem is solved, one would still have to discretize the vector space, so it wouldn't be a complete solution. But for the present, we even fail at space discretization because of the chiral lattice fermion problem.

#### vanhees71

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Of course, you can express anything in real numbers what you can express with complex numbers, but why should you do so? It's more cumbersome at best. If you wish you can do all of analysis with natural numbers too; after all the real numbers are built step by step from the natural numbers via first algebraic then topological "completion"; but then it becomes a real nuisance ;-))).

#### A. Neumaier

in all the physics I've some across measurement values are assumed to be real numbers. A formal justification of this would not be easy in my opinion.
In electrical engineering, using complex observables is common. Of course, you can reduce everything complex to real by doubling the number of variables but often this is inefficient, and one loses all nice properties of complex numbers (analytic continuation, etc.) that are very important in the applications. Reducing to rationals is not possible, for you may measure the diagonal of a rectangle by measuring its sides and applying the theorem of Pythagoras, or the area of a circle by measuring its radius - and you have irrationals and even transcendentals....

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#### PeroK

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In electrical engineering, using complex observables is common. Of course, you can reduce everything complex to real by doubling the number of variables but often this is inefficient, and one loses all nice properties of complex numbers (analytic continuation, etc.) that are vvery important in the applications. Reducing to rationals is not possible, for you may measure the diagonal of a rectangle by measuring its sides and applying the theorem of Pythagoras, or the area of a circle by measuring its radius - and you have irrationals and enven transcendentals....
... but, the point is that triangles, circles and all of Euclidean geometry are a mathematical abstraction. You could argue that these are just as much abstract mathematical tools as the complex numbers. In a way, it's not important whether physics works without real numbers or not. They are in many ways harder to justify physically then the complex numbers. My question is simply: why pick on the complex numbers for removal from QM?

#### A. Neumaier

... but, the point is that triangles, circles and all of Euclidean geometry are a mathematical abstraction. You could argue that these are just as much abstract mathematical tools as the complex numbers. In a way, it's not important whether physics works without real numbers or not. They are in many ways harder to justify physically then the complex numbers. My question is simply: why pick on the complex numbers for removal from QM?
Measurements are also mathematical abstractions.

In simple cases of everyday life, we see that a pointer is close to a particular bar of a scale, and translate that into a measurement value by an abstract process of estimating the distance to the next two bars, figuring out the label that these bars should have, from the few labels given, and then proceeding to linear interpolation. The result is our claimed measurement result.

To measure the (complex-valued) refractive index of an optical material, say, is a much more elaborate process. Scientific measurements are quite complex - some literally, most others even when they do not use complex numbers!

#### A. Neumaier

There is no way to write down a typical real number.
Here are some typical real numbers, written down in some unambiguous way: $0$, $1$, $-1$, $2/3$, $\sqrt{2}$, $\pi$, $(1+\sqrt{5})/2$, $1.05$, $3\cdot 10^5$, $10^{-12}$, etc.

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#### vanhees71

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In how many space + time dimensions?
In (1+3) of course. The Riemann-Silberstein notation is indeed sometimes a very elegant and manifestly covariant description of Maxwell's equations, using the group isomorphism between the proper orthochronous Lorentz group and $\text{SO}(3,\mathbb{C})$.

#### PeroK

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Here are some typical real numbers, written down in some unambiguous way: $0$, $1$, $-1$, $2/3$, $\sqrt{2}$, $\pi$, $(1+\sqrt{5})/2$, $1.05$, $3\cdot 10^5$, $10^{-12}$, etc.
Those are very atypical real numbers. Even the set of algebraic numbers is countable. To add to that we have a few special transcendental numbers and that's it.

#### Demystifier

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But for the present, we even fail at space discretization because of the chiral lattice fermion problem.
I don't think that it is such a big problem. The Wilson method removes the doublers by a rather simple method. Essentially, one adds to the Lagrangian a discretized version of
$$a\partial^{\mu}\bar{\psi} \partial_{\mu}\psi$$
where $a$ is the lattice spacing. Sure, it violates the chiral symmetry, but so what? Lattice violates also the Lorentz, the rotational and the translational symmetry, and yet nobody gets too excited about it.

#### Demystifier

2018 Award
Here are some typical real numbers, written down in some unambiguous way: $0$, $1$, $-1$, $2/3$, $\sqrt{2}$, $\pi$, $(1+\sqrt{5})/2$, $1.05$, $3\cdot 10^5$, $10^{-12}$, etc.
The set of all "typical" real numbers is a set of measure zero.

#### PeroK

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I just checked and even the set of computable numbers is countable. I guess the interesting question is whether you could do physics using the computable numbers instead of the real numbers?

#### A. Neumaier

Those are very atypical real numbers. Even the set of algebraic numbers is countable. To add to that we have a few special transcendental numbers and that's it.
I don't share your notion of typicality. The real numbers I gave are very typical when viewed in terms of the usage in the literature on mathematics, physics, and engineering.
I just checked and even the set of computable numbers is countable. I guess the interesting question is whether you could do physics using the computable numbers instead of the real numbers?
The set of definable real numbers (or any other mathematical objects) is countable and has all the properties of the reals (or the objects in question). It gives a countable model of them. This is called Skolem's paradox, but is not a real paradox. It has the consequence that one may restrict without loss of generality to the definable numbers. The ones I gave are typical examples, though one can easily give more complicated ones, e.g., ''the smallest prime with $10^{10}$ decimal digits''.

#### PeroK

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I don't share your notion of typicality. The real numbers I gave are very typical when viewed in terms of the usage in the literature on mathematics, physics, and engineering.

The set of definable real numbers (or any other mathematical objects) is countable and has all the properties of the reals (or the objects in question). It gives a countable model of them. This is called Skolem's paradox, but is not a real paradox. It has the consequence that one may restrict without loss of generality to the definable numbers. The ones I gave are typical examples, though one can easily give more complicated ones, e.g., ''the smallest prime with $10^{10}$ decimal digits''.
That's fascinating. But, from what I understand Skolem's paradox does not mean that we can treat the reals as though they are countable. And assume that analysis and calculus can be done unimpaired by issues of uncountablilty.

#### A. Neumaier

That's fascinating. But, from what I understand Skolem's paradox does not mean that we can treat the reals as though they are countable. And assume that analysis and calculus can be done unimpaired by issues of uncountability.
Reals are intrinsically uncountable by Cantor's diagonal argument. But they are meta-countable (as objects talked about on the metalevel), and that is what counts on the level of usage.

#### vanhees71

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The set of definable real numbers (or any other mathematical objects) is countable and has all the properties of the reals (or the objects in question). It gives a countable model of them. This is called Skolem's paradox, but is not a real paradox. It has the consequence that one may restrict without loss of generality to the definable numbers. The ones I gave are typical examples, though one can easily give more complicated ones, e.g., ''the smallest prime with $10^{10}$ decimal digits''.
The definable real numbers are not contable, as famously proven by Cantor. The real numbers are defined as the unique Archimedean ordered complete field of numbers (via Dedekind cuts or equivalence classes of Cauchy series with the standard topology).

#### A. Neumaier

The definable real numbers are not contable, as famously proven by Cantor. The real numbers are defined as the unique Archimedean ordered complete field of numbers (via Dedekind cuts or equivalence classes of Cauchy series with the standard topology).
The reals are not countable by Cantor's diagonalization argument. But most real numbers are not definable: Each definable real number is defined by a finite formula, and these formulas form a countable set. Thus the definable reals from a countable set only.

#### vanhees71

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This is a very strange interpretation of "definable". I thought the modern definition of the real numbers is as I said above precisely for the reason not to be dependent on the necessarily only finite (and thus incomplete in the sense of the mathematical real numbers) "formulas". Of course, you cannot write a number like $\pi$ in terms of the usual decimal places, but it's still a well defined real number (e.g., by defining $\pi/2$ as the smallest positive solution of $\cos(\pi/2)=0$, which of course you cannot solve exactly with a finite algorithm or notation).

#### A. Neumaier

This is a very strange interpretation of "definable". I thought the modern definition of the real numbers is as I said above precisely for the reason not to be dependent on the necessarily only finite (and thus incomplete in the sense of the mathematical real numbers) "formulas". Of course, you cannot write a number like $\pi$ in terms of the usual decimal places, but it's still a well defined real number (e.g., by defining $\pi/2$ as the smallest positive solution of $\cos(\pi/2)=0$, which of course you cannot solve exactly with a finite algorithm or notation).
My usage of definable real numbers is standard; see the link!

There is a difference between defining the set of real numbers and defining individual real numbers. To use a number it must be defined as a particular number from the set of all possible numbers. $\pi$ is a definable real number, defined by a finite formula, e.g. $\pi=\int_{-1}^1 (1-x^2)^{-1/2}dx$. On the other hand, only countably many among the uncountably many real numbers are definable. This includes all algebraic numbers and many transcendentals, such as $e$ and $\pi$.

"Complex numbers in QM"

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