Graduate Complex Numbers Not Necessary in QM: Explained

[atyy]

I think another version of this question is why does QM use amplitudes rather than dealing in probabilities directly?

A lot of work in SIC-POVMs and quantum information has it that this comes from QM being a probability theory with multiple sample spaces related via the uncertainty principle (rather than a single space like Kolmogorov Probability theory).

Within a single QM sample space you have the normal law of total probability relating the outcomes of two random variables:
$$P(B_j) = \sum^{m}_{i} P(B_j|A_i)P(A_i)$$
just as in Kolmogorov probability.

However between two of QM's sample spaces the law of total probability gets modified by an additional term:
$$P(B_j) = \sum^{m}_{i} P(B_j|A_i)P(A_i) + \sum_{k<m} 2cos\left(\theta_k\right)\sqrt{P(A_k)P(B_j|A_k)P(A_m)P(B_j|A_m)}$$
with $\theta_k$ measuring the angle between sample spaces (or "contexts" in Quantum Information language), i.e. a measure of how much Bayesian updating within one sample space updates the probability distributions in another. Complex number amplitudes are then just an alternate more compact way of encoding these Probabilities and the angles of interference between their contexts. However you could if you wanted use purely real numbers and deal with probabilities directly.

Another way of saying it is that QM's use of multiple sample spaces introduces the concept of the relation between these spaces. This is expressed as interference between their probability distributions as measured by the angle between the spaces. Thus complex numbers represent a geometric element to the probabilities in QM that isn't present in Kolmogorov probability.

Is this the same reason, or a different one, from the reason that complex numbers are used in classical electrodynamics (see @Demystifier's post #13 and @vanhees71's post #14)?

 

[DarMM]

Is this the same reason, or a different one, from the reason that complex numbers are used in classical electrodynamics (see @Demystifier's post #13 and @vanhees71's post #14)?

Yes, I think. The physical reasons are different, but the basic reason of having a more compact algebraic expression for geometric relations holds I think. Ultimately in electromagnetism its a way of encoding a bivector and its Hodge dual into a single complex spinor (a special case of using Clifford algebras to simplify geometric quantities) with this spinor being in a rep of the double cover of the component of $SO(1,3)$ as @vanhees71 said.

EDIT: In general I think complex numbers in physics are often a way of "off loading" some of the geometry (e.g. overlap angles) into algebra.

 

[haushofer]

Historically, when did people realize the wavefunction needs to be complex/have 2 real degrees of freedom and one real degree of freedom (i.e. a real scalar function) does not suffice? Was it in the introduction of Heisenberg's commutation relations?

 

[A. Neumaier]

Historically, when did people realize the wavefunction needs to be complex and one real degree of freedom (i.e. a real scalar function) does not suffice?

The Schrödinger equation at its inception already contained a factor $i$. It is built into quantum mechanics quite independent of any interpretation. The canonical commutation relation also involves $i$.

 

[A. Neumaier]

why does QM use amplitudes rather than dealing in probabilities directly?

You gave a complex reason :-)

The real reason is that amplitudes satisfy a simple differential equations, probabilities don't. Knowing all probabilities at a fixed time is not even enough to determine the future probabilities, since probabilities lack the phase information at each point in configuration space.

 

[DarMM]

You gave a complex reason :-)

The real reason is that amplitudes satisfy a simple differential equations, probabilities don't. Knowing all probabilities at a fixed time is not even enough to determine the future probabilities, since probabilities lack the phase information at each point in configuration space.

The expression in my post is actually equivalent to the Schrodinger equation in the case where $B$ is $A$ at a latter time. Certainly knowing the probabilities isn't enough, but the interference phases (angles $\theta$) are present in that relation.

What you're saying isn't incompatible with what I'm saying. The angles arise from the existence of relations between multiple sample spaces. However the evolution equations are cumbersome when dealing with the probabilities and angles directly, where as combining them into an complex amplitude is an alternate simpler expression of them that gives nicer differential equations, i.e. simplicity is the reason for using complex numbers as you said. I expressed this as "alternate more compact", one could say "simpler" as you have.

 

[A. Neumaier]

However the evolution equations are cumbersome when dealing with the probabilities and angles directly, where as combining them into an complex amplitude is an alternate simpler expression of them that gives nicer differential equations, i.e. simplicity is the reason for using complex numbers as you said.

Has anyone written down the evolution equations in terms of probability equations and angles? They must be ugly and impossible to motivate (without resorting to amplitudes), and they would have never been popular. Without complex numbers, the evolution equations probably would not even have been discovered...

 

[DarMM]

Has anyone written down the evolution equations in terms of probability equations and angles? They must be ugly and impossible to motivate (without resorting to amplitudes), and they would have never been popular. Without complex numbers, the evolution equations probably would not even have been discovered...

I've seen them, it's not too hard to derive, but I don't think anybody would use them. What I'm saying is that the existence of multiple sample spaces produces the multiple interference angles present in QM, a feature missing from Kolmogorov probability with its single sample space. However the resulting geometry of meshed sample spaces is cumbersome to deal with directly in terms of the probabilities and angles, hence the amplitude formalism.

Another thing is the constraints related to ensuring probabilities always sum to $1$ over exclusive outcomes imposes the structure of a Hilbert space on the amplitudes.

Adán Cabello's papers on the exclusivity principle focus on this where he shows requiring probabilities to be consistent across contexts implies they have relations between each other equivalent to them coming from (squares of) inner products on a Hilbert space. The Hilbert space being complex then occurs from requiring local tomography.

So I should say the specific form of the relation I posted is important as well, without the root you'd have a real Hilbert space.

Summing up, because I've blathered a bit:
The probabilities in QM involve multiple sample spaces that mesh together in a way that ensures consistency with probabilities summing to $1$ across contexts and consistency with local tomography*. This implies all probabilities and all relations between sample spaces can be encoded in a complex Hilbert space. Thus the use of complex numbers in QM.

*Local tomography is also basically imposed by special relativity, real Hilbert space QM has global degrees of freedom

 

[haushofer]

The Schrödinger equation at its inception already contained a factor $i$. It is built into quantum mechanics quite independent of any interpretation. The canonical commutation relation also involves $i$.

But what convinced Schrodinger then that the wavefunction can't be real? What's wrong with real wavefunctions and real operators?

I.e. what exactly becomes inconsistent if you try to build a quantum theory with a real wavefunction (1 real component)?

 

[DarMM]

But what convinced Schrodinger then that the wavefunction can't be real? What's wrong with real wavefunctions and real operators?

I.e. what exactly becomes inconsistent if you try to build a quantum theory with a real wavefunction (1 real component)?

It has global degrees of freedom inconsistent with special relativity.

 

[A. Neumaier]

But what convinced Schrodinger then that the wavefunction can't be real? What's wrong with real wavefunctions and real operators?

I.e. what exactly becomes inconsistent if you try to build a quantum theory with a real wavefunction (1 real component)?

The stationary states whose energies gave the connection to the older quantum theory from spectroscopy have complex phases. With real wave functions one can handle static issues only.

 

[Fra]

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.

I fullly agree with this stance.

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.

This is exactly what i am trying to do. Ie. to take representability and computability seriously. That means countable sets of distinguishable events are in the starting points. Real numbers can be thought of as an "approximation" that actually makes the math easier when you reach high complexity, but one must not forget that once you trace things back to LOW complexity (think big bang and primordal observers) the whole continuum mathematics are invalid as basis for physics IMO. This way of reconstructing measures will also automatically solve renormalization problems, that are really created simply because the limits are take and their orders are lost.

But there is not much published on this, and the mainstream paradigms also heavily rests of analysis and real numbers. It also requires a new understanding of symmetries, as the continuuum symmetries in this light may need to be reunderstood as approximations of complex systems rather than as fundamental mathematical truths. This complicates a lot of things in how we think of mathematics toolboxes of physics. I am personally convinced its the way to go though.

/Fredrik

 

[PeterDonis]

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.

It has all the properties that can be described by a countable first-order axiomatization. But does that include all of the properties of the real numbers that are important for constructing physical models using, for example, calculus?

The particular property I'm thinking might be problematic is the least upper bound property: that every bounded set of real numbers has a least upper bound that is also a real number. But does every bounded set of definable real numbers have a least upper bound that is also a definable real number?

 

[A. Neumaier]

The particular property I'm thinking might be problematic is the least upper bound property: that every bounded set of real numbers has a least upper bound that is also a real number. But does every bounded set of definable real numbers have a least upper bound that is also a definable real number?

Yes. Defining $\sqrt{2}:=\sup\{x\in Q\mid x^2<2\}$ is a valid definition of a particular real number.

More generally, if $S$ is a set of definable real numbers then $\sup S$ gives a definition of its supremum, hence the latter is also definable.

 

[PeterDonis]

if SSS is a set of definable real numbers then $\sup S$ gives a definition of its supremum, hence the latter is also definable.

That's what I thought, but then I started reading about things like this...

https://en.wikipedia.org/wiki/Specker_sequence

...which is a sequence of computable real numbers whose supremum is not computable. But "computable" would seem to be a stricter notion than "definable", since the latter does not require that you actually know how to compute the supremum (for instance), only that you know how to define it using a countable set of axioms. So I guess that's the difference.

 

[A. Neumaier]

That's what I thought, but then I started reading about things like this...

https://en.wikipedia.org/wiki/Specker_sequence

...which is a sequence of computable real numbers whose supremum is not computable. But "computable" would seem to be a stricter notion than "definable", since the latter does not require that you actually know how to compute the supremum (for instance), only that you know how to define it using a countable set of axioms. So I guess that's the difference.

The computable numbers are meant in the sense of constructive mathematics, numbers defined in such a way that the definition implies an algorithm for computing arbitrarily close rational numbers. Unlike definable reals, computable reals do not form a countable model for the real numbers.

Being algorithmic is a much stronger condition than ''using a countable set of axioms''!

 

[DarMM]

This is very interesting! What can one not do with the definables? Can all of analysis be built atop them?

 

[PeroK]

Yes. Defining $\sqrt{2}:=\sup\{x\in Q\mid x^2<2\}$ is a valid definition of a particular real number.

More generally, if $S$ is a set of definable real numbers then $\sup S$ gives a definition of its supremum, hence the latter is also definable.

Let $S$ be a set of rational numbers. All rational numbers are definable. Therefore $\sup S$ is definable. But, every real number is the supremum of a set of rational numbers. Hence, every real number is definable.

In particular the set of all sets of rational numbers is uncountable. So you'll have to careful about how you define a set.

Something like the set of all rational sequences comes easily enough, from a finite set of axioms.

 

[PeterDonis]

Unlike definable reals, computable reals do not form a countable model for the real numbers.

This would seem to imply that there are more computable reals than definable reals. That doesn't seem right, since we can define numbers that are not computable.

 

[PeroK]

This is very interesting! What can one not do with the definables? Can all of analysis be built atop them?

If it can I'll eat my real analysis book.

 

[PeterDonis]

Let $S$ be a set of rational numbers. All rational numbers are definable. Therefore $\sup S$ is definable.

Are all sets of rational numbers definable? The set of all sets of rational numbers is uncountable.

 

[PeterDonis]

if $S$ is a set of definable real numbers then $\sup S$ gives a definition of its supremum, hence the latter is also definable.

In this light of the response I just gave to @PeroK, there is a missing step in this argument which in fact is invalid. The supremum of a definable set of definable numbers is definable; but not all sets of definable numbers are definable sets; they can't all be, because the set of all sets of definable numbers is uncountable.

 

[PeroK]

Are all sets of rational numbers definable? The set of all sets of rational numbers is uncountable.

I think the problem is that you need a new definition of a set. For example, the set $\{1/2, 3/4, 7/8 \dots \}$ is a specific set of definable numbers. But the set $\{a_1, a_2, \dots \}$ where the $a_n$ are arbitrary, unspecified definable numbers won't have the property being claimed.

There are too many of these sets, so something would need to be done about that.

 

[PeterDonis]

the set $\{a_1, a_2, \dots \}$ where the $a_n$ are arbitrary, unspecified definable numbers won't have the property being claimed

Yes, which is why I think that you have found a missing premise in the argument @A. Neumaier was making that, per my post #52, is actually invalid; so I'm not convinced that the definable real numbers have all of the necessary properties of the full set of real numbers to allow real analysis to be "built" on just the definable real numbers.

 

[SSequence]

That's what I thought, but then I started reading about things like this...

https://en.wikipedia.org/wiki/Specker_sequence

...which is a sequence of computable real numbers whose supremum is not computable. But "computable" would seem to be a stricter notion than "definable", since the latter does not require that you actually know how to compute the supremum (for instance), only that you know how to define it using a countable set of axioms. So I guess that's the difference.

I am hardly an expert on this, but based on what I have read on various points or sources, here are few things that might be helpful:
(1) For example, I think the things like specker sequence example might just be an artifact of requiring decimal expansions to be computable. When one switches or changes the definitions even computable analysis goes quite far.

(2) Outside of pure comptuable mathematics, a vast majority of mathematics can be done in very weak systems. One often cited system is ACA₀ (one of systems of reverse mathematics) ... but I am not really familiar with details.
But for example, things like specker sequence example will hardly be a problem when we use arithmetic sets (even with decimal expansion definition) ... which are very small but natural collection of subsets of natural numbers.

But even outside of reverse mathematics, there are good number of revisionist approaches that have been applied successfully (there are number of people who have worked on this successfully).

 

[haushofer]

The stationary states whose energies gave the connection to the older quantum theory from spectroscopy has complex phases. With real wave functions one can handle static issues only.

Do you have a reference for that? I've read a few QM-books and find it odd that somehow this question is not or barely treated :)

 

[haushofer]

It has global degrees of freedom inconsistent with special relativity.

Do you have a reference or can you elaborate?

 

[DarMM]

Do you have a reference or can you elaborate?

Here are two I like:
https://arxiv.org/abs/1611.09029
http://www.dima.unige.it/microlocal/wp-content/uploads/2016/12/OPPIO.pdf

You can see that only the complex case doesn't contradict Poincaré symmetry.

It's also the only case that allows local tomography, i.e. the statistics of the state is recoverable from local measurements. This means for example that the state of a two particle system can be recovered from measurements on both particles individually:
https://arxiv.org/abs/1202.4513

 

[vanhees71]

Historically, when did people realize the wavefunction needs to be complex/have 2 real degrees of freedom and one real degree of freedom (i.e. a real scalar function) does not suffice? Was it in the introduction of Heisenberg's commutation relations?

The commutation relations are due to Born. Always Born is forgotten although without him Heisenberg didn't even know what he was doing at Helgoland. The formulation of matrix mechanics is mostly achieved by Born and Jordan (the latter even quantizing the electromagnetic field already in the famous 2nd paper by Born, Jordan, and Heisenberg quite a while before Dirac).

 

[DarMM]

Summing this up (@atyy this is a bit more accurate than my previous post):

QM is a probability theory with multiple sample spaces, it then has angles $\theta_j$ that represent relations between these sample spaces. The probabilities and interference angles can (at first glance) be combined into vectors that are either real, complex or quaternionic which results in a much simpler Hilbert space formalism, where as dealing with probabilities and interference angles directly is cumbersome.

However on closer inspection the quaternionic case has "too many" angles resulting in the possibility of one sample space interfering with another in such a way as to force its probabilities to exceed $1$. Or another way of looking at it, the quaternionic uncertainty relations can imply uncertainties that break unitarity.

The real case then turns out to break Poincaré symmetry.

Thus the only multiple sample space consist probability theory is one whose angles can be encoded in complex vectors, i.e. QM.