Complex numbers and physical meaning

[The Count]

I have to say that I am a bit confused with the use of complex numbers. I know that:
1. They have been created by mathematicians to solve the "real"ly unsolved equation of x^2=-1.
2. They are used in many aspects of physics, like waves and quantum theory, with terrific correspondence to the maths and experimental results.
3. But when it comes to physical meaning, there is a gap. Tricks like squaring to find the probability width in order to get rid of the unwanted "imaginary" part, or just using the "real" part.

I tried to find a physical explanation but with no lack. Specifically I would appreciate a physical correspondence in two aspects.

A. The waveparticle solution of Schroedinger Equation.
B. The Unit operator of time (evolution).

Thanks for your time.

 

[gonadas91]

I would say this is more a philosophical question, rather than scientific one, but interesting anyway! From my point of view, the complex numbers arise in too many parts of physics just as the only formalism needed to develop a theory. To explain it better, if you would try to develop some physical theories without them, they will end up being non-sense. The complex plane is an extension of the real numbers set, where more combinations of numbers are possible, and therefore, the possibilities of making a physical theory grow.

In QM for example, all operators algebra is defined in complex algebraic form: Observable quantities are always real, and we know that hermitian operators have real eigenvalues always. If we would try to develop the theory just with real symmetric matrices, our "movements" would be very limited. The wave equation (Schrodinger) is a differential equation for a complex function, which by itself doesn't have any physical meaning because its complex. It is the magnitude square of this function. Why the wavefunction is complex arises from the definition of the hamiltonian operator: \hat{H}=i\hbar\partial_{t}
which is hermitian. This is a fascinating question and I hope some more people will join!

 

[bhobba]

II tried to find a physical explanation but with no lack. Specifically I would appreciate a physical correspondence in two aspects.

There is a deep reason complex numbers are necessary in QM:
http://www.scottaaronson.com/democritus/lec9.html

Just to show all roads lead to Rome its also fundamental to Schroedinger's equation:
http://arxiv.org/pdf/1204.0653.pdf

The Unit operator of time (evolution).

Almost forgot to mention complex numbers are required for a very important theorem - called Wigners Theorem - see attached file

Thanks
Bill

 

[The Count]

Thanks for your time.

Excuse me if I didn't understand correctly but again I received reasons and necessity of using complex numbers and not physical explanation!

I think, I have already stated that they seem a necessity and they help with the maths. I am not questioning that, nor I am searching for a different way, I am just confused with their physical correspondence. In every lecture I have watched or read, they just overpass the physical justification and they just prove the maths. As far as I am concerned there is an obligation as physics teacher to give a physical meaning of everything I am using. If not then there is a gap, and it is not a philosophical query. It is a deep gap in our understanding of the maths and equations that we are using.

We cannot use a mathematical tool without explaining it's physical correspondence. The addition is putting together, multiplication is growing by a number, division is splitting, derivative of space in time is the velocity, the second derivative is the acceleration, the curvature of the gravitational field is the mass, ω and k are the circular frequency and the wave number respectively, and so on. What is the meaning of an imaginary number in physics? Not its squared product, nor its "real" part.

I am trying to find out if there is any explanation out there that I am missing, and until now, nobody has given me a satisfying answer. I hoped somebody would enlighten me.

 

[PeroK]

Thanks for your time.

Excuse me if I didn't understand correctly but again I received reasons and necessity of using complex numbers and not physical explanation!

I think, I have already stated that they seem a necessity and they help with the maths. I am not questioning that, nor I am searching for a different way, I am just confused with their physical correspondence. In every lecture I have watched or read, they just overpass the physical justification and they just prove the maths. As far as I am concerned there is an obligation as physics teacher to give a physical meaning of everything I am using. If not then there is a gap, and it is not a philosophical query. It is a deep gap in our understanding of the maths and equations that we are using.

We cannot use a mathematical tool without explaining it's physical correspondence. The addition is putting together, multiplication is growing by a number, division is splitting, derivative of space in time is the velocity, the second derivative is the acceleration, the curvature of the gravitational field is the mass, ω and k are the circular frequency and the wave number respectively, and so on. What is the meaning of an imaginary number in physics? Not its squared product, nor its "real" part.

I am trying to find out if there is any explanation out there that I am missing, and until now, nobody has given me a satisfying answer. I hoped somebody would enlighten me.

I don't know that much QM, but I can give you a mathematical perspective.

First, the term "imaginary" has no significance. Also, that complex numbers are called "numbers" has no significance. They could have been called, perhaps, "multiplicative vectors". They are a mathematical tool. A lot of people seem to get worked up about complex numbers and their physical "existence", while not worrying about the physical existence of an $n \times n$ matrix, for example.

A wave function is a complex function of a two real variables $x, t$. That means that the wave function is two dimensional. So, you can write:

$\Psi (x, t) = \Psi_1(x, t) + i \Psi_2 (x, t)$

If you put this into the Schrodinger equation, you can generate a pair of simultaneous differential equations:

$\hbar \frac{\partial \Psi_1}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi_2}{\partial x^2} + V \Psi_2$

$-\hbar \frac{\partial \Psi_2}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi_1}{\partial x^2} + V \Psi_1$

This pair of equations is mathematically equivalent to the Schrodinger equation and is devoid of complex numbers. Perhaps your question, therefore, boils down to the physical significance of the wave function comprising two interdependent (real) wave functions.

 

[my2cts]

The reason is invariance under time reversal. The consequence is that there are always two degenerate solutions that are 90 degrees out of phase. This twofold degeneracy is conveniently described with complex numbers.

 

[bhobba]

Excuse me if I didn't understand correctly but again I received reasons and necessity of using complex numbers and not physical explanation!

Since physics is written in the language of math I am not sure how you separate a math explanation from a physical one. Wanting an explanation on your terms is not a good idea because nature may not oblige. Its better to accept nature as is. If that means the explanation is what you call mathematical rather than physical then that's the way it is.

Anyway here is another take on the issue:
http://www.scottaaronson.com/papers/island.pdf

See point 2.

Thanks
Bill

 

[bhobba]

Perhaps your question, therefore, boils down to the physical significance of the wave function comprising two interdependent (real) wave functions.

You are of course correct.

Such questions really are - why is QM most elegantly expressed using complex numbers.

Looking at it that way it's quite reasonable the answer is mathematical rather than physical - not that its easy to separate the two.

Thanks
Bill

 

[PeroK]

You are of course correct.

Such questions really are - why is QM most elegantly expressed using comple numbers.

Looking at it that way it's quite reasonable the answer is mathematical rather than physical - not that its easy to separate the two.

Thanks
Bill

The most elegant thing that complex numbers do, IMO, is the way they unify the exponential function with the simple harmonic functions (sin and cos). Looking at the second question (B) in the OP, the "imaginary" exponential function is an elegant way to express two simple harmonics 90° out of phase.

 

[sandy stone]

Speaking strictly as a layman, this is how I visualize it: The probability amplitudes that are calculated in QM consist of an absolute value multiplied by a phase angle. When probability amplitudes are added, as in a superposition, they must be added vectorially, taking phase angle into account, before being squared to find an actual physical probability. Complex numbers enter the picture because they are a traditional, convenient way to express the phase angle in terms of its x and y components, the cosine and sine. Senior posters please correct me.

 

[zonde]

What is the meaning of an imaginary number in physics?

I would like to give shorter version of what PeroK said that does not look like a math:
Imaginary number corresponds to two real quantities that are interdependent in a certain way in respect to final effect.

 

[muscaria]

This is a slightly reversed take on it.. In analogy to the situation of analytical mechanics where one requires stipulation of the generalised coordinates and generalised velocities such that the total set of these quantities constitutes the independent dynamical variables, the phase θ(x, t) and the amplitude R(x, t) of the wavefunction must be specified at any given time for every point in space. In effect, the phase and the amplitude serve as the independent dynamical variables for quantum mechanical systems. This is clearly evident in view of the Schrodinger equation being a first order complex partial differential equation with respect to time. At first sight, one may perceive a significant structural difference between the mechanics of classical and quantum systems given that the former seemingly evolve in time according to a set of n second order Euler-Lagrange equations which require specification of quantities which are zeroth order and first order in time. However, the dynamics of classical mechanical systems occur in a phase space formed by the 2n coordinates (q, p) which are zeroth order in time ”positional” type coordinates, analogous
to the situation for quantum mechanical systems. In an identical fashion to the classical domain where the canonical momenta encode the flow of particles and are determined by the normal vectors of surfaces of constant action, quantum mechanical flow of probability is determined by local phase differences and directed normal to surfaces of constant phase. Accordingly, the requirement of specifying generalised velocity variables in the Euler-Lagrange formalism which are first order in time, translates to specifying a distribution of the phase of the wavefunction over space in the quantum realm, a quantity which is zeroth order in time. This is in complete analogy with the passage from the Lagrangian to Hamiltonian formalism of mechanics. So given that the S.E is first order in time and Newtonian mechanics is governed by a second order equation, 2 zeroth order dynamical variables are required: one quantity representing the "initial coordinates" i.e. the amplitude function and another quantity representing the "initial" velocities i.e. the phase function.

 

[Urs Schreiber]

Here is one way to see it:

First of all, a careful inspection shows that action functionals in physics have to be regarded as taking values not in the real numbers, as sometimes assumed, but in the real numbers modulo the integers. (This is discussed in the "PF-Insights"-article titled Higher prequantum geometry II: The global action functional.)

The first implication of this is that the covariant phase space of a field theory carries a principal bundle for the circle group U(1) = R/Z. (This is the content of Higher prequantum geometry IV: The covariant phase space. For more background see also the exposition at fiber bundles in physics).

Now quantum wave functions are the sections of a linearization of this U(1)-principal bundle on phase space. Such a linearization is given by a choice of ring R and a group homomorphism U(1) -> GL_1(R) from the circle group to the group of invertible elements of R.

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.

The choice R = C the complex numbers is the canonical such choice.

 

[muscaria]

After Urs's post the following may seem a bit trivial :p, but may contribute in some way. The canonical equations of motion show that the Hamiltonian function serves as a generating function for time translations of the independent dynamical variables. The simplectic structure of these $2n$ equations and their appearance in pairs, can be cast in the form of a single set $n$ if the paired conjugate variables are replaced by complex variables: $$ c_k = \frac{1}{\sqrt{2}} (q_k +ip_k) $$ $$ c_k^* = \frac{1}{\sqrt{2}} (q_k -ip_k) $$. With this transformation, we now have $n$ complex canonical equations: $$ i\frac{d}{dt}c_k =\frac{\partial{H}}{\partial{c_k^*}} $$. Come to think of it, is this actually why we call it "phase space"?

 

[Urs Schreiber]

The simplectic structure of these $2n$ equations and their appearance in pairs, can be cast in the form of a single set $n$ if the paired conjugate variables are replaced by complex variables:

What this is pointing out is that the phase space $\mathbb{R}^{2n}$ has, apart from its standard "real polarization" also a complex polarization (see at polarization). However, not every phase space admits a complex polarization (locally it does, but globally there may be obstructions). So, while complex polarizations are intimately related to the complex phase of wave functions, the occasional existence of complex polarizations does not seem to really answer the question as to why, fundamentally, wave functions have complex phases, I would think.

Come to think of it, is this actually why we call it "phase space"?

Yeah, this is something one should wonder about at least once in a life. I once wondered about this, too. But the history was different. The short answer is that Boltzmann thaught of a point in phase space as a specifying "phase of the motion" of a gas in a box, and that's where the term more or less comes from. But the long answer is much longer, much more convoluted, in fact somewhat weird, also entertaining, and in any case a good reminder for anyone who believes that the scientific community as a whole operates in a rational manner: see

• David Nolte, The tangled tale of phase space, PhysicsToday (April 2010) (pdf)

 

[muscaria]

Thank you Urs for a very informative reply! When you say

However, not every phase space admits a complex polarization (locally it does, but globally there may be obstructions).

do you intend that there are situations which occur with gauge field singularities or something? Preventing analysis of the connection between different complex polarizations at different points of the cotangent bundle? Apologies for my lack of knowledge in differential forms..

 

[Urs Schreiber]

of the cotangent bundle

First of all, phase spaces need in general not be cotangent bundles. Even cotangent bundles in general only carry almost complex structure, not necessarily complex structure.

 

[akhmeteli]

Thanks for your time.

Excuse me if I didn't understand correctly but again I received reasons and necessity of using complex numbers and not physical explanation!

I think, I have already stated that they seem a necessity and they help with the maths. I am not questioning that, nor I am searching for a different way, I am just confused with their physical correspondence.

Actually, you can question that (please see https://www.physicsforums.com/threads/why-are-qm-wave-functions-complex.683821/#post-4337679 )

 

[Urs Schreiber]

you can question that

There does not seem to be any room to question that quantum states in general have complex phases. That you may find one with a real phase is not proof to the contrary.

And this is precisely due to the obvious physical interpretation that the OP is after: a space-dependent complex phase of a wave function is the way that momentum is encoded by the wave function. A wave function of a free particle of fixed momentum in flat space is nothing but a complex phase, linearly varying in space, and this is at the very heart of the nature of quantum physics, the uncertainty relation, and all those things that were experimentally verified and understood 100 years back.

 

[Khashishi]

There are two philosophical stances you can take here.
1. Numbers are tools we use in various models to describe nature.
2. Numbers are real things that we discover.
I choose to accept 1. Complex numbers are just a different and more general tool than real numbers to use in our models. And there are plenty of models that use complex numbers. Statement 2 raises all sorts of questions: what kind of numbers are "real"? Just whole numbers? Real numbers? I don't think these questions can be answered. But statement 1 is easy to come to grips with.

 

[akhmeteli]

There does not seem to be any room to question that quantum states in general have complex phases. That you may find one with a real phase is not proof to the contrary.

It seems that Schroedinger had a different opinion on this issue. In his article (Nature, v.169, p.538(1952)) he wrote, discussing the Klein-Gordon-Maxwell electrodynamics: "That the wave function ... can be made real by a change of gauge is but a truism, though it contradicts the widespread belief about 'charged' fields requiring complex representation." And he did not have in mind a replacement of a complex function by two real ones. Of course, even Schroedinger could be wrong, but then, I guess, you should explain where and why his published article was wrong. Let me note that wave function's phase is not gauge-independent, so it does not have any definite sense until you've fixed the gauge. Let me add that you can make any scalar wave function real by a gauge transform (at least, locally). Moreover, you can use just one real function instead of the Dirac spinor function, as I showed in my article in the Journal of Mathematical Physics (http://akhmeteli.org/wp-content/uploads/2011/08/JMAPAQ528082303_1.pdf)

And this is precisely due to the obvious physical interpretation that the OP is after: a space-dependent complex phase of a wave function is the way that momentum is encoded by the wave function. A wave function of a free particle of fixed momentum in flat space is nothing but a complex phase, linearly varying in space, and this is at the very heart of the nature of quantum physics, the uncertainty relation, and all those things that were experimentally verified and understood 100 years back.

There is no contradiction between what I wrote above and "the uncertainty relation, and all those things that were experimentally verified and understood 100 years back." I use the standard Klein-Gordon and Dirac equations of quantum mechanics.

 

[muscaria]

First of all, phase spaces need in general not be cotangent bundles.

Ok..I have much reading to do then! Are a set of n coordinates ("momenta" usually but given arbitrariness under canonical transformations, I'll just say n coordinates) taking on values in the cotangent spaces not a requirement to ensure an invariant differential form of the action?: $$dA=\sum_i p_i dq_i$$

 

[muscaria]

It seems that Schroedinger had a different opinion on this issue. In his article (Nature, v.169, p.538(1952)) he wrote, discussing the Klein-Gordon-Maxwell electrodynamics: "That the wave function ... can be made real by a change of gauge is but a truism, though it contradicts the widespread belief about 'charged' fields requiring complex representation." And he did not have in mind a replacement of a complex function by two real ones.

Was the point not considered in Urs's earlier post, that action functionals take on values in the real numbers modulo the integers (conjugate action-angle variables pairs). The point being that action twists around from point to point and flow occurs along directions of non-zero action/phase twist? This is aside from the fact that one needs to also specify a second scalar function i.e amplitude function, serving as an initial or boundary "particle distribution".

 

[muscaria]

Let me note that wave function's phase is not gauge-independent, so it does not have any definite sense until you've fixed the gauge.

I'm guessing a choice of gauge was presumed and doesn't seem (at first sight anyway) to affect the underlying issue, the gauge field just curves the space over which flow (canonical momenta) occurs and simply needs to be accounted for. As an analogy, we could still ask about the structure of line elements in Minkowski space without needing to specify the general Riemannian geometry of spacetime.

 

[muscaria]

There are two philosophical stances you can take here.
1. Numbers are tools we use in various models to describe nature.
2. Numbers are real things that we discover.
I choose to accept 1. Complex numbers are just a different and more general tool than real numbers to use in our models. And there are plenty of models that use complex numbers. Statement 2 raises all sorts of questions: what kind of numbers are "real"? Just whole numbers? Real numbers? I don't think these questions can be answered. But statement 1 is easy to come to grips with.

I'm unsure of whether the OP was looking for a reason for the use of Complex numbers as resulting from a required mathematical structure underpinning a physical theory or whether he was looking for some direct correspondence of complex numbers to physical reality. I usually presume people mean the former when they ask a question of this type. Any theory, be it physical or mathematical, remains an abstraction of reality and the idea that it is in direct correspondence with reality or that it $\textit{is}$ the reality, seems to me anyway like a rather limited and incoherent view. So.. yeah, guessing the OP's question is more geared towards the former.

If it was the latter then I would say numbers and theories are (in ref to your second statement) things we create, which have relevance within a certain domain of experience of reality, if any at all. A map is an abstraction of a city which helps guide you through a city, yet there is evidently no part of the map which corresponds to the city. If it's a bad map you notice incoherence which becomes clear when you are not getting the result you expected (maybe you've just walked into a river) and you need a new map things, if it's a good map it will guide you well, so well sometimes that you may even start to believe the map is in direct correspondence with the city. When you start to probe things a bit more and ask more and more subtle questions, you realize you're continuously having to remap how things appear to you and that it wasn't that the old good maps were "wrong", but simply that there were limits to their relevance of our experience of reality.
The question of a direct correspondence with reality of a number or theory seems, in my view, nonsensical in the first place given that they are abstractions of reality.

 

[sandy stone]

Clearly the theory is not reality, any more than the map is the terrain. However, it could be argued that the explicit purpose of the theory is to produce numbers that correspond directly with reality. In other words, to predict experimental results. I don't think it's nonsensical to try to gain an intuitive understanding of the process that produces those numbers by looking for a correspondence between mathematical objects and operations, and elements of what we suppose to be reality. If I look at lines on a map, I know they're not really streets, but they still help me understand how to find my way around the city. Of course, as the theory becomes more abstract the correspondence becomes less obvious, and at some point completely obscure. In the particular case of complex numbers in QM, though, I think an intuitive understanding is easily achieved, and for people like myself, helpful.

 

[muscaria]

However, it could be argued that the explicit purpose of the theory is to produce numbers that correspond directly with reality.

As I said, a good map isn't wrong! I never meant to suggest that there was no correspondence of theory to reality whatsoever, there evidently is but only in an abstract sense. My post was in response to Khashishi's 2 stance suggestion which seemed somewhat limited given that in my mind the OP's question was directed towards understanding why complex numbers serve as an appropriate representation for certain physical objects, which up to now has been the central focus of this thread. Although

1. Numbers are tools we use in various models to describe nature.

is perfectly valid, it seems the OP is asking for more than that. The second part of my post was directed towards the idea:

2. Numbers are real things that we discover.

Statement 2 raises all sorts of questions: what kind of numbers are "real"? Just whole numbers? Real numbers?

.
Which he in fact believed to be unanswerable, for reasons potentially along the lines that have just been brought up? You may be misinterpreting my point of view.

intuitive understanding of the process that produces those numbers

is in my eyes a key aspect of a physical theory but the process which produces the numbers does not correspond to the reality, it's a process which is the product of thought. Although the validity of a physical theory rests purely on empirical grounds, I would consider the ontological content of a theory as being just as important as its epistemology, believing it is an essential driving force in the abstractions of new forms of order in nature. However I don't believe these have a $\textit{direct}$ correspondence to reality as they will always remain abstract. Ultimately there's no more or less abstract in terms of correspondence of theory to reality, just abstractions which have more or less relevance to reality.

and elements of what we suppose to be reality.

These elements are abstractions of reality, just like the concept of a particle or any physical object is an abstraction from this totality.

 

[akhmeteli]

Was the point not considered in Urs's earlier post, that action functionals take on values in the real numbers modulo the integers (conjugate action-angle variables pairs). The point being that action twists around from point to point and flow occurs along directions of non-zero action/phase twist? This is aside from the fact that one needs to also specify a second scalar function i.e amplitude function, serving as an initial or boundary "particle distribution".

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.

 

[John Mathon]

Roger Penrose has spent quite a life basically pondering this issue. He invented Spinors and Twistors to essentially give "life" to the idea of complex dimensions. My simplistic interpretation of the need for complex numbers is that they reflect an underlying Riemann topology, i.e. that the complex calculations and results are simply a reflection of an underlying spinning nature. Somehow somewhere whatever is reality is spinning. Spinning is important because it is isomorphic to wave like motion. What we seem to be discovering like String theory seems to posit is that the underlying fabric of the universe has wavelike properties. Waves and complex numbers have a natural relationship. So, in part there is a convenience of using complex numbers in that it helps us mathematically to express the wave-like properties of things. The Schroedinger wave-function thus represents in one way of interpretation the spinning of something in something that is easily expressed using complex numbers. Penrose has come up with Twistors to help us imagine this. This may not be the ultimate description as there are still problems extending Twistors to all particles and to all physics but great strides have been made in the last 5 years that are stunning.

 

[akhmeteli]

I'm guessing a choice of gauge was presumed and doesn't seem (at first sight anyway) to affect the underlying issue, the gauge field just curves the space over which flow (canonical momenta) occurs and simply needs to be accounted for. As an analogy, we could still ask about the structure of line elements in Minkowski space without needing to specify the general Riemannian geometry of spacetime.

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). As for the presumption... One can presume this choice, but I don't have to accept any presumptions without a compelling reason. Again, I am saying that you can do with a real wave function, I am not saying that you should. However, in some cases this can be useful.

 

[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.