Complex numbers and physical meaning

[muscaria]

I don't see how this contradicts my statement that you can do with just one real wave function in quantum theory, at least in the very important cases of the Klein-Gordon and Dirac equations. Should you do that? This is a different question.

Apologies, I didn't mean to suggest it contradicted your statement, just that it seemed to me Urs had established the required structure for wavefunctions:

Hence the question comes down to what choices of rings R there is such that U(1) canonically maps to their multiplicative groups of invertible elements.

 

[Urs Schreiber]

.I have much reading to do then!

You are in luck, what you need to know is sruveyed at the PF-Insight-article The Covariant Phase Space

A quick example of a phase space that is not a cotangent bundle is the 2-sphere with its canonical symplectic form. This is the phase space for internal spin degrees of freedom. See at geometric quantization of the 2-sphere.

(Now of course the 2-sphere happens to admit a complex structure, so this is not an example for a phase space not admitting a complex polarization, just an example of a phase space that is not a cotangent bundle.)

 

[muscaria]

I guess this choice does "affect the underlying issue", as you can choose the gauge in such a way that the wave function will become real everywhere (at least locally).

I didn't really appreciate or think about what you were saying, sorry about that, I'll try to pay attention more carefully in the future and "listen". Unfortunately I don't have access to my University portal and couldn't find Schrodinger's paper you mention in free view..

So if you end up with a real wavefunction, the gauge potential coupled to the charge must then be identical to the mechanical momenta along the streamlines (integral curves/thinking quantum Hamilton Jacobi formalism) traced out by every initial infinitesimal volume of the wave, right? If one is to describe any real non-stationary wavefunction that is. So that the gauge potential effectively serves as a velocity field for the wave?

It's just I'm finding it a bit hard because usually the vector potential couples to the current and simply curves the path taken by the current without changing its magnitude, but you need current in the first place for this to take effect. Or is the idea to find a condition for a dynamic scalar potential field which somehow constrains the phase to 0 everywhere, and the vector potential then serves as a mechanical momentum field for the wave?

Thinking again in H-J/de Broglie-Bohm terms, a real wavefunction would give the following scenario for some free space H in the non-relativistic limit:
Taking $$i\partial_t\Psi = \left[\frac{(\textbf{p}-\textbf{eA})^2}{2m} -e\phi\right]\Psi$$
and $$\Psi=Re^{i\theta}$$
where I'll take the phase to be constant over space and time after our calculation. Otherwise we end up simply with a continuity equation and no quantum H-J equation and we lose potential insight into a possible relation existing between the gauge potential and the quantum potential which would yield an effectively real wavefunction. So doing that the usual pair of equations of continuity and QHJ: $$\nabla\cdot(R^2\textbf{v})+\frac{\partial R^2}{\partial t}$$ $$\frac{\partial\theta}{\partial t}+\frac{(\nabla\theta -e\textbf{A})^2}{2m} -e\phi + Q =0$$ reduce to
$$\nabla\cdot\left(R^2\frac{\textbf{A}}{m}\right)=\frac{\partial R^2}{\partial t}$$ $$\frac{(e\textbf{A})^2}{2m} -e\phi + Q =0.$$
Is the last equation the gauge condition which gives rise to a real wavefunction? I'm not sure if what I've done is wrong, but if it isn't then it would seem like the quantum potential $Q\propto\frac{\nabla^2R}{R}$, which depends on the form of the wave, and the electromagnetic potentials are constrained, giving rise to a situation where the form of the wave constrains the E-M field potentials which in turn dictates the current of the wave from the role taken on by the vector potential as a velocity field. Hmm...seems like a non-linear gauge condition in any case, with some effective feedback between wave and EM field?
I'm guessing carrying this out relativistically may lead to a more symmetric gauge condition, which must be what Schrodinger did for the K-G field? Any thoughts? Also, does this have any significant implications for the divergence of A? Is this complete rubbish? :p

 

[muscaria]

You are in luck

It's nice to know I'll never get bored! Thanks for the links, much appreciated.
EDIT: Just scanned through the notes.. and find the "you are in luck, what you need to know..." amusing given the amount of thought and digestion that's going to be needed in order to understand what you are describing! I have even more reading to do.

 

[Urs Schreiber]

even more reading to do.

Sorry if it comes across looking like a lot of work. The key fact I meant to point to is quite simple: What a phase space of a Lagrangian field theory is, generally, is simply the space of all classical solutions, equipped with the canonical (pre-)symplectic 2-form, and the reduced phase space is the symplectic space obtained from this by quotienting out flows in the kernel of the presymplectic form. This has in general no reason to be equivalent to a cotangent bundle, even if in many popular examples it is.

The bulk of the entry that I pointed to goes beyond this, it explains how to prequantize such "covariant phase spaces". For just the picture of the covariant phase space as such a good original article to look at is [Zuckerman 87] and a review is in [Khavkine 14].

 

[Urs Schreiber]

you should explain where and why his published article was wrong.

Before being able to be wrong one has to be precise about what one is talking about. The argument you point to is the simple observation that for every particular section of a complex line bundle there is a local trivialization where it is given by a real function. Sure. Notice however that:

1. other sections than the particular chosen one that the gauge is adapted to will not have such a real representation;
2. if we speak about Klein-Gordon or Dirac fields, then these are classical fields that still need to be quantized. They look like quantum states, but they are not, they are the input of "second quantization". An actual wave function will be a complex-value function on the space of all such KG/Dirac fields (an element in the corresponding Fock space, say) and clearly these are again complex valued functions.

But of course to just arrive at the conclusion that it is false that quantum physics may work with just real valued wave functions only, you don't need to detour into relativistic wave equations. Just open any basic QM textbook and check that it all breaks down in p. 3 if you insist in replacing all complex numbers by real numbers.

 

[akhmeteli]

I didn't really appreciate or think about what you were saying, sorry about that, I'll try to pay attention more carefully in the future and "listen". Unfortunately I don't have access to my University portal and couldn't find Schrodinger's paper you mention in free view..

So if you end up with a real wavefunction, the gauge potential coupled to the charge must then be identical to the mechanical momenta along the streamlines (integral curves/thinking quantum Hamilton Jacobi formalism) traced out by every initial infinitesimal volume of the wave, right? If one is to describe any real non-stationary wavefunction that is. So that the gauge potential effectively serves as a velocity field for the wave?

It's just I'm finding it a bit hard because usually the vector potential couples to the current and simply curves the path taken by the current without changing its magnitude, but you need current in the first place for this to take effect. Or is the idea to find a condition for a dynamic scalar potential field which somehow constrains the phase to 0 everywhere, and the vector potential then serves as a mechanical momentum field for the wave?

Thinking again in H-J/de Broglie-Bohm terms, a real wavefunction would give the following scenario for some free space H in the non-relativistic limit:
Taking $$i\partial_t\Psi = \left[\frac{(\textbf{p}-\textbf{eA})^2}{2m} -e\phi\right]\Psi$$
and $$\Psi=Re^{i\theta}$$
where I'll take the phase to be constant over space and time after our calculation. Otherwise we end up simply with a continuity equation and no quantum H-J equation and we lose potential insight into a possible relation existing between the gauge potential and the quantum potential which would yield an effectively real wavefunction. So doing that the usual pair of equations of continuity and QHJ: $$\nabla\cdot(R^2\textbf{v})+\frac{\partial R^2}{\partial t}$$ $$\frac{\partial\theta}{\partial t}+\frac{(\nabla\theta -e\textbf{A})^2}{2m} -e\phi + Q =0$$ reduce to
$$\nabla\cdot\left(R^2\frac{\textbf{A}}{m}\right)=\frac{\partial R^2}{\partial t}$$ $$\frac{(e\textbf{A})^2}{2m} -e\phi + Q =0.$$
Is the last equation the gauge condition which gives rise to a real wavefunction? I'm not sure if what I've done is wrong, but if it isn't then it would seem like the quantum potential $Q\propto\frac{\nabla^2R}{R}$, which depends on the form of the wave, and the electromagnetic potentials are constrained, giving rise to a situation where the form of the wave constrains the E-M field potentials which in turn dictates the current of the wave from the role taken on by the vector potential as a velocity field. Hmm...seems like a non-linear gauge condition in any case, with some effective feedback between wave and EM field?
I'm guessing carrying this out relativistically may lead to a more symmetric gauge condition, which must be what Schrodinger did for the K-G field? Any thoughts? Also, does this have any significant implications for the divergence of A? Is this complete rubbish? :p

Thank you very much for your interest.

As for Scroedinger's article, I believe I reproduce much of its contents in my articles:
http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf
(Journal of Mathematical Physics, beginning of Section II)
http://akhmeteli.org/akh-prepr-ws-ijqi2.pdf
(International Journal of Quantum Information, beginning of Section 3. You may be also interested in my paraphrase of the Dirac's 1951 article (that inspired the Schroedinger's 1952 article) in Section 2. Another interesting thing there - the Lagrangian of Eq. (14) (derived by other people) - the matter field is real there, but the physics is the same as with the standard Klein-Gordon-Maxwell Lagrangian)
http://link.springer.com/content/pdf/10.1140/epjc/s10052-013-2371-4.pdf
(European Physical Journal C, beginning of Section 2)
However, when you have a chance, you might wish to read the beautiful but almost forgotten Schroedinger's article.

I confess that your post is a bit difficult for me: your point of view and (physical) language seem unusual. Furthermore, I was not much interested in the non-relativistic case, as the relativistic case is much more realistic.

Indeed, the 4-potential of electromagnetic field in the unitary gauge (where the scalar wave function is real) has the same direction as the current of the matter field in every point for the Klein-Gordon-Maxwell system, but I suspect this is not so for the Dirac-Maxwell system.

You mention de Broglie-Bohm, and I show that the electromagnetic field in the unitary gauge plays the role of the guiding field of dBB.

Maybe I'll try to answer your specific questions later.

 

[akhmeteli]

Before being able to be wrong one has to be precise about what one is talking about. The argument you point to is the simple observation that for every particular section of a complex line bundle there is a local trivialization where it is given by a real function. Sure. Notice however that:

1. other sections than the particular chosen one that the gauge is adapted to will not have such a real representation;

1. Yes, but those "other sections" are not necessary to describe the physics of this system, so you can do just with real wave functions (the other sections give the same physics)

2. [*]if we speak about Klein-Gordon or Dirac fields, then these are classical fields that still need to be quantized. They look like quantum states, but they are not, they are the input of "second quantization". An actual wave function will be a complex-value function on the space of all such KG/Dirac fields (an element in the corresponding Fock space, say) and clearly these are again complex valued functions.

3. I guess this paragraph of yours hints that you need more sophisticated arguments than those of your previous post to prove that real functions are not enough. And I am not sure your latest argument is waterproof either. Classical fields in 3+1 dimensions can have the same unitary evolution as quantum field theories (please see, e.g., Section 4 of my article http://link.springer.com/content/pdf/10.1140/epjc/s10052-013-2371-4.pdf in European Physical Journal C, where I use other people's results)

But of course to just arrive at the conclusion that it is false that quantum physics may work with just real valued wave functions only, you don't need to detour into relativistic wave equations. Just open any basic QM textbook and check that it all breaks down in p. 3 if you insist in replacing all complex numbers by real numbers.

I don't quite see how "it all breaks down in p. 3". It may be more difficult to describe wave functions of non-relativistic quantum physics using only real functions, but we don't need it, as the relativistic wave equations are more realistic.

Let me summarize. You can do just with real functions at least in some general and important cases, so one can indeed at least question the absolute necessity of complex functions in quantum theory.

 

[akhmeteli]

So if you end up with a real wavefunction, the gauge potential coupled to the charge must then be identical to the mechanical momenta along the streamlines (integral curves/thinking quantum Hamilton Jacobi formalism) traced out by every initial infinitesimal volume of the wave, right? If one is to describe any real non-stationary wavefunction that is. So that the gauge potential effectively serves as a velocity field for the wave?

Looks that way for Klein-Gordon-Maxwell, not sure about Dirac-Maxwell, probably not.

[QUOTE="muscaria]It's just I'm finding it a bit hard because usually the vector potential couples to the current and simply curves the path taken by the current without changing its magnitude, but you need current in the first place for this to take effect.[/QUOTE]
Let me note that current arises in the Maxwell equations as well, not just in the Klein-Gordon equation.
[QUOTE="muscaria]Or is the idea to find a condition for a dynamic scalar potential field which somehow constrains the phase to 0 everywhere, and the vector potential then serves as a mechanical momentum field for the wave?[/QUOTE]
I am not sure I quite understand this phrase. Scalar potential does not seem to define the phase uniquely: for example, you can change the phase by a constant without changing the scalar potential. Even the expression "scalar potential" is a bit strange in this context - I guess you mean a component of the 4-vector potential.

As for the idea... It is the same as in Dirac-1951 and Schroedinger -1952: the Maxwell equations in a certain gauge can describe both the electromagnetic field and the matter field.
[QUOTE="muscaria]Thinking again in H-J/de Broglie-Bohm terms, a real wavefunction would give the following scenario for some free space H in the non-relativistic limit:
Taking $$i\partial_t\Psi = \left[\frac{(\textbf{p}-\textbf{eA})^2}{2m} -e\phi\right]\Psi$$
and $$\Psi=Re^{i\theta}$$
where I'll take the phase to be constant over space and time after our calculation. Otherwise we end up simply with a continuity equation and no quantum H-J equation and we lose potential insight into a possible relation existing between the gauge potential and the quantum potential which would yield an effectively real wavefunction. So doing that the usual pair of equations of continuity and QHJ: $$\nabla\cdot(R^2\textbf{v})+\frac{\partial R^2}{\partial t}$$ $$\frac{\partial\theta}{\partial t}+\frac{(\nabla\theta -e\textbf{A})^2}{2m} -e\phi + Q =0$$ reduce to
$$\nabla\cdot\left(R^2\frac{\textbf{A}}{m}\right)=\frac{\partial R^2}{\partial t}$$ $$\frac{(e\textbf{A})^2}{2m} -e\phi + Q =0.$$
Is the last equation the gauge condition which gives rise to a real wavefunction? I'm not sure if what I've done is wrong, but if it isn't then it would seem like the quantum potential $Q\propto\frac{\nabla^2R}{R}$, which depends on the form of the wave, and the electromagnetic potentials are constrained, giving rise to a situation where the form of the wave constrains the E-M field potentials which in turn dictates the current of the wave from the role taken on by the vector potential as a velocity field. Hmm...seems like a non-linear gauge condition in any case, with some effective feedback between wave and EM field?
I'm guessing carrying this out relativistically may lead to a more symmetric gauge condition, which must be what Schrodinger did for the K-G field? Any thoughts? Also, does this have any significant implications for the divergence of A? Is this complete rubbish? :p[/QUOTE]
I did not read this very carefully, but I don't like the idea of making the wave function real in a non-relativistic equation: on the one hand, it gives a gauge choice that is different from the choice required to make real the wave function in a relativistic wave equation, on the other hand, the relativistic equation is more realistic.

 

[The Count]

As for the idea... It is the same as in Dirac-1951 and Schroedinger -1952: the Maxwell equations in a certain gauge can describe both the electromagnetic field and the matter field.
[QUOTE="muscaria]Thinking again

Can you please give me more details about this?
What do you mean the "matter field"?
Where I can find bibliography on this subject?

 

[akhmeteli]

Can you please give me more details about this?
What do you mean the "matter field"?
Where I can find bibliography on this subject?

Thank you very much for your interest.
The matter field is the field describing massive particles, say, electrons/positrons, e.g., the Klein-Gordon field or the Dirac field.
For bibliography and details, please see my published articles quoted in post 37 in this thread. The IJQI paper cites Dirac-1951 and Schroedinger-1952 articles. A recent preprint http://arxiv.org/abs/1502.02351 shows how to rewrite the Dirac equation for one real wave function in a general case.
The most representative examples can be found in the EPJC article. For example, after a gauge transform, the matter field can be algebraically eliminated from scalar electrodynamics (Klein-Gordon-Maxwell electrodynamics), and the resulting equations define independent evolution of electromagnetic field. Thus, modified Maxwell equations in the unitary gauge (where the scalar matter field is real) describes both electromagnetic field and the matter field).

 

[anorlunda]

Richard Feynman's book QED The Strange Throery of Matter and Light used clever graphics to illustrate what the OP asks. At the very least, you see that if you simplify those little vectors to real numbers and add them up, you get the wrong answer. The vectors must have both magnitude and direction to get the right answer. That is what complex numbers do for us.

OP, if you want to learn more. I reccomend Feynman's book.

 

[The Count]

Thank you all for your time.
You gave me a lot of homework.
I really appreciate the interest that you showed to the question.

I have to say that finally I did not get an answer, but maybe there is not any (yet).
I must insist that there is no way that a mathematical tool used in physics not to have a physical interpretation.

The only close enough interpretation that seems to have a point to me is the relation of imaginary part with time.
I mentioned it in my first question, that the unitary matrix of the progression of time is imaginary.
I found it in wikipedia on Henri Poincaré and Minkowski 4 dimensional space.
I found a relation to some of your answers too.

I would appreciate if anybody could help...

 

[bhobba]

I have to say that finally I did not get an answer,

I thought you got many answers.

I must insist that there is no way that a mathematical tool used in physics not to have a physical interpretation.

That's been the whole issue here. You have decided what is just a mathematical tool and what is a physical explanation. Physics is written in the language of math. The math is the physical explanation.

Imaginary time is a trick associated with Wick rotation:
https://www.physicsforums.com/insights/useful-integrals-quantum-field-theory/

But of course you do not accept mathematical tricks as physical. That means many many things will perplex you.

Thanks
Bill

 

[The Count]

Thanks again for your interest in my stubbornness.
I want first to thank all of you once more for your time and let you know that it was appreciated.

I must repeat that you all helped me a lot to think, but, as I see it, I did not understood a clear physical interpretation to be given.
I may be wrong, but for me mathematical interpretation is not enough for physics. I cannot imagine relativity to leave the curvature of spacetime, without the interpretation of mass/energy.

In this point I would prefer to use Plank's quote for black body radiation.
"Six years I struggled with the problem of thermal equilibrium radiation and matter without success. I knew that this problem was fundamental to physics. I knew the formula that represents the energy distribution of the spectrum. A theoretical interpretation had to be given, at all costs, no matter how high."

Thanks for imaginary time link, I will study it and come back.