Graduate Complex Numbers Not Necessary in QM: Explained

[Demystifier]

I don't think this is true.

Why?

 

[stevendaryl]

How about the following statement? For any real number $x$ there is a language $L$ in which $x$ is definable. However, there is no language $L$ such that any real number $x$ is definable in $L$.

Well, I think that's trivially true. Given any real $r$ between 0 and 1, you can add a function symbol $f$ and add infinitely many axioms saying $f(n) = r_n$. Then within this theory, the number $r$ is definable.

 

[A. Neumaier]

Why?

Well, I think that's trivially true. Given any real $r$ between 0 and 1, you can add a function symbol $f$ and add infinitely many axioms saying $f(n) = r_n$. Then within this theory, the number $r$ is definable.

No. The problem is that you cannot ''give'' undefinable reals!

Given any real r makes r an anonymous real, never a particular one. It is just the conventional way of expressing that what follows has a formal variable r quantified over with the all quantor. Thus nothing is actually defined.

 

[stevendaryl]

No. The problem is that you cannot ''give'' undefinable reals!

In mathematical logic, one is allowed to consider theories with a non-computable collection of axioms. For example, the true theory of arithmetic. We can't actually write down such a collection, but it exists (in the same sense that any abstract mathematical objects exist). So for every real $r$, there exists (as a mathematical object) a theory that defines $r$ uniquely. We can't write it down, but that's a different matter.

 

[A. Neumaier]

In mathematical logic, one is allowed to consider theories with a non-computable collection of axioms. For example, the true theory of arithmetic. We can't actually write down such a collection, but it exists (in the same sense that any abstract mathematical objects exist). So for every real $r$, there exists (as a mathematical object) a theory that defines $r$ uniquely. We can't write it down, but that's a different matter.

Where is one allowed to do that? Not in first order logic, which we discuss here. In strange logics, strange things may of course happen.

 

[stevendaryl]

Where is one allowed to do that? Not in first order logic, which we discuss here. In strange logics, strange things may of course happen.

I am talking about first-order logic. In mathematical logic, one can study theories where the set of axioms are noncomputable.

 

[A. Neumaier]

I am talking about first-order logic. In mathematical logic, one can study theories where the set of axioms are noncomputable.

Please give a reference where this is done and leads to significant results. In this case one doesn't even know what the axioms are...

 

[stevendaryl]

Please give a reference where this is done and leads to significant results. In this case one doesn't even know what the axioms are...

Well, the most important non-axiomatizable theory is the theory of true arithmetic. You define the language of arithmetic, which is typically:

• constant symbol $0$
• unary function symbol $S(x)$
• two binary function symbols $+$ and $\times$
• one relation symbol $=$

You can, in set theory, define an interpretation of these symbols in terms of the finite ordinals, and then you can define the theory of true arithmetic as the set of formulas in this language that are true under this interpretation.

It's a noncomputable set of formulas, but it's definable in ZFC. (and much weaker theories)

 

[A. Neumaier]

Well, the most important non-axiomatizable theory is the theory of true arithmetic. You define the language of arithmetic, which is typically:

• constant symbol $0$
• unary function symbol $S(x)$
• two binary function symbols $+$ and $\times$
• one relation symbol $=$

You can, in set theory, define an interpretation of these symbols in terms of the finite ordinals, and then you can define the theory of true arithmetic as the set of formulas in this language that are true under this interpretation.

It's a noncomputable set of formulas, but it's definable in ZFC. (and much weaker theories)

''true arithmetic'' is not a theory in first order logic, but ''the set of all sentences in the language of first-order arithmetic that are true'' in the standard model of the natural numbers (itself not a first order logic notion).

Yes, it is a noncomputable set of formulas, but not a set of axioms of some first order theory. It is a nonaxiomatizable theory (i.e., not a first order logic theory), as you correctly said.

 

[stevendaryl]

''true arithmetic'' is not a theory in first order logic,

Yes, it is. In the study of mathematical logic, a "theory" is a set of formulas closed under logical implication.

 

[vanhees71]

The intuitionistic reals behave mathematically very different from the reals taught in any analysis course.

In intuitionistic math, most concepts from ZFC ramify into several meaningful nonequivalent ones, depending on which intuitionistic version of the axioms one starts with (all of which would become equivalent if the axiom of choice were assumed in addition). Thus one has to be very careful to know which version of the reals one is talking about.

I never understood, what intuitionistic math is good for beyond the fact that it might be an intellectually interesting game of thought ;-).

 

[A. Neumaier]

I never understood, what intuitionistic math is good for beyond the fact that it might be an intellectually interesting game of thought ;-).

Well, it shows the extent to which things can be made fully constructive. Thus it gives insight into the structure of mathematical reasoning. A physicist doesn't need it, of course...

 

[stevendaryl]

I never understood, what intuitionistic math is good for beyond the fact that it might be an intellectually interesting game of thought ;-).

I spent a good number of years studying intuitionistic and constructive mathematics. I think it's interesting, but I'm not convinced that anything worthwhile comes from it.

An interesting fact about intuitionistic mathematics is the isomorphism between intuitionistic proofs and computer programs. Intuitionistically, if you prove a statement of the form

$\forall x \exists y: \phi(x,y)$

you can extract a program (expressed as a lambda-calculus expression) that given any $x$ returns a $y$ satisfying $\phi(x,y)$.

Every proposition in constructive logic (I'm a little hazy about the exact distinction between constructive and intuitionistic) corresponds to a type, in the computer-science sense, and the proofs of those propositions correspond to mathematical objects of that type. So for example:

$A \wedge B$ corresponds to the set of ordered pairs $(a,b)$ where $a$ is a proof of $A$ and $b$ is a proof of $b$.
$A \rightarrow B$ corresponds to the set of functions which given a proof of $A$ returns a proof of $B$.
$A \vee B$ corresponds to the disjoint union of proofs of $A$ and proofs of $B$.
etc. (quantifiers correspond to product types and tagged unions).

I think it's all very interesting, and it gives some insight into logic and programming and the connection between them. But ultimately, I don't see the whole endeavor as being tremendously useful.

 

[stevendaryl]

The biggest difference between constructive and classical logic is the extent to which it is possible to prove that something exists without being able to give an example. You can't do that in constructive logic. So a proof that there exists a nonmeasurable set doesn't go through. However, you can recover most of classical mathematics by doing a "double negation". In almost all cases where statement $A$ is provable classically, $\neg \neg A$ is provable constructively. A double-negation is not equivalent to the original statement in constructive logic.

 

[A. Neumaier]

In almost all cases where statement A is provable classically, ¬¬A is provable constructively.

In all cases, this is provable in the intuitionistic setting.

This shows that there is no loss of quality in assuming classical logic. One can only gain, never inherit a contradiction that is not already there on the intuitionistic level.

This is the reason why intuitionistic logic is irrelevant in practice.

 

[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.

I've been thinking about this a bit more and I'm not sure it is correct. Quaternionic quantum mechanics for example can be given simple differential equations. Also the differential equations are a dynamical feature where as the presence of complex numbers is a Kinematical feature of quantum theory.

I think the reason for complex numbers is the fact of QM being a multiple sample space probability theory obeying local tomography. Conservation of probability in such a generalized probability model implies those simple evolution equations.

 

[A. Neumaier]

I think the reason for complex numbers is the fact of QM being a multiple sample space probability theory obeying local tomography. Conservation of probability in such a generalized probability model implies those simple evolution equations.

This cannot have been the reason in the early days of QM when there was no notion of multiple sample space.

The true reason is that the Schrödinger equation already contains $i$ (and needs it for unitary evolution with a Hermitian generator), and it worked exceedingly well, so nothing more complicated was superior (or even competitive).

Of course, one could think of the Schrödinger equation of a single particle with spin as being quaternionic, but this does not extend nicely to the multiparticle case. Quaternionic quantum mechanics, p-adic quantum mechanics, and other exotics were never found interesting - except by the few who tried it and reported their meager results.

 

[DarMM]

I don't disagree with what you wrote, but that seems to be more the reason why historically QM has complex numbers. I more answering "Why couldn't it have been another field?" or "What's wrong with using another field?"

Similar to somebody asking why General Relativity has no Torsion. One can of course answer that Einstein didn't use torsion and there was never any need for it. However you can also explain what might be wrong with such an approach.

I'm basically driving at why the other approaches didn't work and QM must be complex.

 

[Fra]

This cannot have been the reason in the early days of QM when there was no notion of multiple sample space.

The true reason is that the Schrödinger equation already contains $i$ (and needs it for unitary evolution with a Hermitian generator), and it worked exceedingly well, so nothing more complicated was superior (or even competitive).

The historical reason is for things is interesting on its own, but if we seek reasons involving a deeper understanding of QM i think the notion of "multiple sample spaces" has the potential to be a better framework for deeper understanding. Then it not argument that this insight was lacking almost a century ago.

I now understand you want to avoid completely the "statistical" or "probabilistic" interpretation of things in the thermal interpretation, and in that process i can see why one sees no value in "multiple P-spaces", but I see another way. We can attach the statistical foundation to subjective statistics of an agent, which is also a bit related to underdetermined systems and reasoning upon incomlpete information. Here the notion of multiple samples spaces which are related makes excellent sense IMO, as one can motivate them from the perspective of compressed sensing, for final encoding in memory records - from which probabilistic expectations of future follows.

So I would insist we ask ourselfs, WHY multiple sample spaces? When one is required to discard information, also due to limited storage, the question is WHICH information can be throw away? (or which information about its environment can an electron throw again) with minimal consequences? Here one may evolve a recoding into conjugate samples spaces or otherwise any alternative coding, and each transformations may have benefits from survival perspective dependin how the environment acts.

In this evolutionary picture, I see the "multiple sample space" mental picture as the one allowing for more natural insight.

/Fredrik

 

[akhmeteli]

I don't disagree with what you wrote, but that seems to be more the reason why historically QM has complex numbers. I more answering "Why couldn't it have been another field?" or "What's wrong with using another field?"

Similar to somebody asking why General Relativity has no Torsion. One can of course answer that Einstein didn't use torsion and there was never any need for it. However you can also explain what might be wrong with such an approach.

I basically driving at why the other approaches didn't work and QM must be complex.

As I tried to explain in this thread (post 78, https://www.physicsforums.com/threads/complex-numbers-in-qm.966895/page-4#post-6141369), it is not quite obvious that "QM must be complex": for example, you can get the same physics as that described by the Schrödinger equation or the Klein-Gordon equation using just one real wave function (all you need to do for that is to perform a gauge transform). The same is true for the Dirac equation (although this is a bit more difficult to demonstrate).

 

[A. Neumaier]

it is not quite obvious that "QM must be complex"

Simplicity forces complex numbers. Of course, one can rewrite things equivalently in a more cumbersome conceptual setting, but what's that good for??

 

[akhmeteli]

it is not quite obvious that "QM must be complex"

Simplicity forces complex numbers.

Your statement may be too strong, as it seems to be equally applicable to classical electrodynamics, where complex numbers certainly make life much easier. But that would suggest that the status of complex numbers in QM is the same as in classical electrodynamics. Is this something you would agree with?

Of course, one can rewrite things equivalently in a more cumbersome conceptual setting, but what's that good for??

Let me emphasize again that this is not about replacing complex numbers with pairs of real numbers. For example, for the Schrödinger equation or the Klein-Gordon equation we have just one real wave function instead of one complex function after the gauge transformation. For the Dirac equation, we can have just one real function instead of four complex components of the spinor wave function, and so on. For the Klein-Gordon-Maxwell electrodynamics you can algebraically eliminate the matter wave function altogether, which may be in sync with your emphasis on fields, rather than on particles. So the above seems to be a tentative answer to your question: "what's that good for??" and suggests that real numbers can even provide some conceptual simplicity.

 

[A. Neumaier]

the status of complex numbers in QM is the same as in classical electrodynamics. Is this something you would agree with?

Yes, it is a matter of convenience. In classical electrodynamics one uses complex numbers often, but there are many situations where they don't give an advantage and are not used.

In quantum mechanics avoiding complex numbers almost never gives an advantage, so it is hardly ever used. But the real formulation in terms of real and imaginary part is sometimes useful for the numerical solution of the Schrödinger equation as one can then use symplectic integrators.

that real numbers can even provide some conceptual simplicity.

No, since one usually needs all components of the spinor wave function and its transformation properties, which are ugly and inconvenient in your formulation.

 

[DarMM]

As I tried to explain in this thread (post 78, https://www.physicsforums.com/threads/complex-numbers-in-qm.966895/page-4#post-6141369), it is not quite obvious that "QM must be complex": for example, you can get the same physics as that described by the Schrödinger equation or the Klein-Gordon equation using just one real wave function (all you need to do for that is to perform a gauge transform). The same is true for the Dirac equation (although this is a bit more difficult to demonstrate).

This might work for a fragment of QM, but in general it will cause states to violate local tomography and thus relativity.

 

[akhmeteli]

the status of complex numbers in QM is the same as in classical electrodynamics. Is this something you would agree with?

Yes, it is a matter of convenience. In classical electrodynamics one uses complex numbers often, but there are many situations where they don't give an advantage and are not used.
In quantum mechanics avoiding complex numbers almost never gives an advantage, so it is hardly ever used.

So I conclude that complex numbers are actually not a "must" in QM.

But the real formulation in terms of real and imaginary part is sometimes useful for the numerical solution of the Schrödinger equation as one can then use symplectic integrators.

I cannot emphasize enough that what I describe has nothing to do with replacing complex numbers with pairs of real numbers.

that real numbers can even provide some conceptual simplicity.

No, since one usually needs all components of the spinor wave function and its transformation properties, which are ugly and inconvenient in your formulation.

So you don't think using one real function instead of four complex functions can provide conceptual simplicity. Let us agree to disagree about that.

 

[akhmeteli]

As I tried to explain in this thread (post 78, https://www.physicsforums.com/threads/complex-numbers-in-qm.966895/page-4#post-6141369), it is not quite obvious that "QM must be complex": for example, you can get the same physics as that described by the Schrödinger equation or the Klein-Gordon equation using just one real wave function (all you need to do for that is to perform a gauge transform). The same is true for the Dirac equation (although this is a bit more difficult to demonstrate).

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This might work for a fragment of QM

You could call the Dirac equation "a fragment of QM", but I would say it is quite a large part of QM.

, but in general it will cause states to violate local tomography and thus relativity.

Let me note that even if this is so, it suggests that one needs pretty sophisticated arguments to prove that complex numbers are a must for QM. Anyway, could you please give a reference to the proof?

 

[DarMM]

You could call the Dirac equation "a fragment of QM", but I would say it is quite a large part of QM.

The point is with the multiparticle case. Also the Dirac equation as a wavefunction equation is seriously limited. The Dirac equation conceived of as the equation for the wave function of a single relativistic spin-1/2 particle is indeed a small fragment. One runs into problems with existence of bound states, positivity of energy and many other features.

Let me note that even if this is so, it suggests that one needs pretty sophisticated arguments to prove that complex numbers are a must for QM. Anyway, could you please give a reference to the proof?

I gave two in post #58

 

[akhmeteli]

The point is with the multiparticle case.

I consider the case of multiple particles in my work http://link.springer.com/content/pdf/10.1140/epjc/s10052-013-2371-4.pdf (published in EPJC), section 4. The treatment there may be not comprehensive enough for your taste, but it is not obvious that it cannot be generalized.

Also the Dirac equation as a wavefunction equation is seriously limited. The Dirac equation conceived of as the equation for the wave function of a single relativistic spin-1/2 particle is indeed a small fragment. One runs into problems with existence of bound states, positivity of energy and many other features.

The Dirac equation does not include all QED, but "a small fragment" of QM? Come on:-) Let me also add that a similar result can be obtained for the Dirac equation in the Yang-Mills field (see the reference at https://www.physicsforums.com/threa...d-of-spinor-field-in-yang-mills-field.960244/)

I gave two in post #58

And I am not impressed...

The Moretti/Oppio work seems to be about Wigner elementary relativistic systems - I guess there is no external field there, so it is not quite realistic (see also my short discussion with Moretti at https://physics.stackexchange.com/q...le-to-the-electromagnetic-field/268971#268971 (after his answer)).

The work on "local tomography" that you quoted contains an awful lot of "fairly natural constraints", which constraints may be good for eliminating approaches using real numbers but it is not obvious why one has to accept them.

 

[DarMM]

The Dirac equation does not include all QED, but "a small fragment" of QM? Come on

I don't understand what this means. The Dirac equation appears in QED as an equation of motion for field operators, this is separate to its use as a single particle relativistic wave equation (just like the Klein-Gordon equation). As the latter it is severely limited and runs into problems. As the former it's not really related to the discussion as its complexity or reality is unrelated to the complexity/reality of the Hilbert space.

In fact I'm confused as to what you are talking about. Are you arguing for the field of real numbers in the equations of motion of the field operators or as the underlying field for the Hilbert space of states. These are separate issues.

The work on "local tomography" that you quoted contains an awful lot of "fairly natural constraints", which constraints may be good for eliminating approaches using real numbers but it is not obvious why one has to accept them.

Could you describe what you are talking about here?

 

[akhmeteli]

I don't understand what this means. The Dirac equation appears in QED as an equation of motion for field operators, this is separate to its use as a single particle relativistic wave equation (just like the Klein-Gordon equation). As the latter it is severally limited and runs into problems. As the former it's not really related to the discussion as its complexity or reality is unrelated to the complexity/reality of the Hilbert space.

I mean that, while the single-particle Dirac equation does not describe all the physics described by QED, it describes an awful lot of quantum phenomena. Yes, the Dirac equation has its share of problems, but I would say it is better than, e.g., the nonrelativistic Schrödinger equation or the Klein-Gordon equation.

In fact I'm confused as to what you are talking about. Are you arguing for the field of real numbers in the equations of motion of the field operators or as the underlying field for the Hilbert space of states. These are separate issues.

I consider the following question: are complex numbers a must for QM (remember that the title of the thread is "Complex numbers in QM")? To this end, I consider non-second-quantized single-particle equations (Schrödinger, Klein-Gordon, Dirac). Traditionally it was believed that they required complex wave functions. It is not well-known that, as Schrödinger showed, complex wave functions (or pairs of real wave functions) are not required for, say, the Klein-Gordon equation. It is not well-known that they are not required for the Dirac equation either. Are they required for multiple particles? I am not sure. So far I am mostly speaking about solutions of single-particle equations. I believed this was relevant to your question "I basically driving at why the other approaches didn't work and QM must be complex."

Could you describe what you are talking about here?

Say, Proposition 1 in https://arxiv.org/abs/1202.4513 has a lot of assumptions. As the authors try to further motivate them, it is not obvious one must accept them. Furthermore, they write in the abstract: "orthodox finite-dimensional complex quantum mechanics with superselection rules is the only non-signaling probabilistic theory in which..." So it looks like if one agrees with their assumptions, one gets a finite-dimensional quantum mechanics. I guess that does not even cover the quantum mechanics of the nonrelativistic Schrödinger equation.