Planck constant is Lorentz invariant?

[keji8341]

I don't know if it is "universal", but this much is correct. A spherically symmetric wave can indeed be expanded as an infinite sum of plane waves. I don't know what the result of an infinite sum of Doppler shifts would be.

Any references? Please.

 

[Dale]

I would start here, particularly the introductory paragraph.
http://en.wikipedia.org/wiki/Fourier_optics

 

[keji8341]

I would start here, particularly the introductory paragraph.
http://en.wikipedia.org/wiki/Fourier_optics

I did not find the theorem as you mentioned: "A spherically symmetric wave can indeed be expanded as an infinite sum of plane waves. "

Be specific, Please.

 

[Dale]

Did you not even read the first paragraph? "In Fourier optics, by contrast, the wave is regarded as a superposition of plane waves which are not related to any identifiable sources; instead they are the natural modes of the propagation medium itself."

If you want something more in depth you can get a textbook:
http://www.wiley.com/WileyCDA/WileyTitle/productCd-0780334116.html

Or Google "Fourier optics" or "plane wave spectrum".

 

[keji8341]

Did you not even read the first paragraph? "In Fourier optics, by contrast, the wave is regarded as a superposition of plane waves which are not related to any identifiable sources; instead they are the natural modes of the propagation medium itself." ...

Thanks. Yes, I read these words:

“In Fourier optics, by contrast, the wave is regarded as a superposition of plane waves which are not related to any identifiable sources; instead they are the natural modes of the propagation medium itself.”

Did you get your conclusion from the above words? It seems to me that, the above ambiguous words cannot result in your conclusion:

"A spherically symmetric wave can indeed be expanded as an infinite sum of plane waves."

Probably that is your own work, related to your intellectual property. If so, I won’t ask.

I don’t have much math. But to my knowledge, whether a function f(x) can be represented as a Fourier integral depends on the property which the function has. Here is a theorem I got from
http://mathworld.wolfram.com/FourierTransform.html

A function f(x) has a Fourier transform if
1. The integral of absolute f(x) exists.
2. There are a finite number of discontinuities.
3. The function has bounded variation. A sufficient weaker condition is fulfillment of the Lipschitz condition.

I think the field produced by a point source has a singularity. I am not sure if the first and the third math conditions can be satisfied.

 

[Dale]

“In Fourier optics, by contrast, the wave is regarded as a superposition of plane waves which are not related to any identifiable sources; instead they are the natural modes of the propagation medium itself.”

Did you get your conclusion from the above words? It seems to me that, the above ambiguous words cannot result in your conclusion:

"A spherically symmetric wave can indeed be expanded as an infinite sum of plane waves."

Why not? They are saying the same thing.

 

[PhilDSP]

A function f(x) has a Fourier transform if
1. The integral of absolute f(x) exists.
2. There are a finite number of discontinuities.
3. The function has bounded variation. A sufficient weaker condition is fulfillment of the Lipschitz condition.

I think the field produced by a point source has a singularity. I am not sure if the first and the third math conditions can be satisfied.

Probably even more basic considerations are what domain you are interested into evaluate or define the Fourier Transform. For the field at a single point you only need a one dimensional Fourier Transform and at that point the transform (and wave equation) will be the same whether the wave is a plane wave or whether it is spherical.

It's only when you wish to evaluate a spatially extended region that you will see a difference between values for a plane wave or spherical wave originating from a particular point. In that case you'll need to define and evaluate a Fourier Transform in at least 2 dimensions. You will see a phase variation across the extended region which will be visible in the FT.

 

[keji8341]

Probably even more basic considerations are what domain you are interested into evaluate or define the Fourier Transform. For the field at a single point you only need a one dimensional Fourier Transform and at that point the transform (and wave equation) will be the same whether the wave is a plane wave or whether it is spherical.

It's only when you wish to evaluate a spatially extended region that you will see a difference between values for a plane wave or spherical wave originating from a particular point. In that case you'll need to define and evaluate a Fourier Transform in at least 2 dimensions. You will see a phase variation across the extended region which will be visible in the FT.

Thanks. I think you are talking about numerical computations about FT. Actually my question is a pure theoretical problem. The point source has a sigularity at r=0; the function to be transformed is ~expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave. I am not sure if it can be represented by uniform plane waves, although some scientists often firmly claim “The plane wave decomposition is mathematically universal.”

The point-source solution actually is closely related so-called "invariant Green's function" in classical electrodynamic.

 

[Dale]

You may be interested in this page:
http://lmb.informatik.uni-freiburg.de/papers/download/wa_report01_08.pdf

 

[PhilDSP]

Thanks. I think you are talking about numerical computations about FT. Actually my question is a pure theoretical problem. The point source has a sigularity at r=0; the function to be transformed is ~expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave. I am not sure if it can be represented by uniform plane waves, although some scientists often firmly claim “The plane wave decomposition is mathematically universal.”

The point-source solution actually is closely related so-called "invariant Green's function" in classical electrodynamic.

I'm not sure what you really mean by "mathematically universal". Sure, very often EM problems or SR problems are expressed and evaluated in terms of plane waves for simplicity. It seems that a spherical wave decomposition of a plane wave is much more commonly described than the converse. But even that is rather complicated and potentially fraught with technical problems such as in this description:

http://farside.ph.utexas.edu/teaching/jk1/lectures/node102.html

In short, it seems far more complicated to force a spherical wave solution to a problem described in terms of a plane wave than to re-describe the problem in terms of spherical waves.

By the term "evaluate" I mean evaluate analytically (not numerically). Usually to evaluate a FT numerically you need to use a Discrete Fourier Transform which at best only approximates a FT (Continuous Fourier Transform)

If you could rewrite your function so that it doesn't already have terms for frequency then it might be simple to determine the FT analytically. Can you somehow replace the frequency term with a partial derivative (involving dr/dt with a constant c) ?

 

[keji8341]

I'm not sure what you really mean by "mathematically universal". Sure, very often EM problems or SR problems are expressed and evaluated in terms of plane waves for simplicity. It seems that a spherical wave decomposition of a plane wave is much more commonly described than the converse. But even that is rather complicated and potentially fraught with technical problems such as in this description:

http://farside.ph.utexas.edu/teaching/jk1/lectures/node102.html

In short, it seems far more complicated to force a spherical wave solution to a problem described in terms of a plane wave than to re-describe the problem in terms of spherical waves.

By the term "evaluate" I mean evaluate analytically (not numerically). Usually to evaluate a FT numerically you need to use a Discrete Fourier Transform which at best only approximates a FT (Continuous Fourier Transform)

If you could rewrite your function so that it doesn't already have terms for frequency then it might be simple to determine the FT analytically. Can you somehow replace the frequency term with a partial derivative (involving dr/dt with a constant c) ?

1. "It seems that a spherical wave decomposition of a plane wave is much more commonly described than the converse. "
You are right. This is usually presented in textbook; for example, J. D. Jackson, Classical Electrodynamics, 3rd edition, (John Wiley & Sons, NJ, 1999), p. 471, Eq. (10.44) in Chapter 10.

The converse: A plane wave decomposition of spherical wave is given in the paper by MacPhie and Ke-Li Wu, “A Plane Wave Expansion of Spherical Wave Functions for Modal Analysis of Guided Wave Structures and Scatterers”, IEEE Trans. Antennas and Propagation 51, 2801 (2003). The spherical waves are analytical at r=0.

2. "Can you somehow replace the frequency term with a partial derivative (involving dr/dt with a constant c)?"
I think that's a different problem. My question: Can "expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave" be represented as a sum of uniform plane waves? I guess the Fourier image integral and the original function are actually not one-to-one correspondence in such a case. Like the 4d invariant Green's function, the outgoing wave, incoming wave, and outgoing+incoming are all corresponding to the same Fourier integral, just taking different contours. If that is true, using the plane-wane decomposition of a spherical wave to explain moving point-source Doppler effect is questionable.

 

[Dale]

I guess the Fourier image integral and the original function are actually not one-to-one correspondence in such a case.

Do you have any evidence to support that?

If that is true, using the plane-wane decomposition of a spherical wave to explain moving point-source Doppler effect is questionable.

It is also unnecessary. I derived it above without any such decomposition.

In fact, I don't think that a plane wave spectrum helps in the Doppler analysis at all. What is the net Doppler shift for an infinite sum of Doppler shifts?

 

[keji8341]

Do you have any evidence to support that?

I guess the Fourier image integral and the original function are actually not one-to-one correspondence in such a case,
------because the original spherical wave has a singularity, and the corresponding Fourier integral is supposed to have poles in the complex plane. The Fourier integral is not certain before a contour is given. This is my guess.

 

[keji8341]

...It is also unnecessary. I derived it above without any such decomposition.

In fact, I don't think that a plane wave spectrum helps in the Doppler analysis at all. What is the net Doppler shift for an infinite sum of Doppler shifts?

Some mainstrain scientist said to me, “The plane wave decomposition is mathematically universal” , then the Einstein's plane-wave Doppler formula should be applicable to any cases. I am referring to that scientist.

 

[PhilDSP]

2. "Can you somehow replace the frequency term with a partial derivative (involving dr/dt with a constant c)?"
I think that's a different problem. My question: Can "expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave" be represented as a sum of uniform plane waves? I guess the Fourier image integral and the original function are actually not one-to-one correspondence in such a case. Like the 4d invariant Green's function, the outgoing wave, incoming wave, and outgoing+incoming are all corresponding to the same Fourier integral, just taking different contours. If that is true, using the plane-wane decomposition of a spherical wave to explain moving point-source Doppler effect is questionable.

The problem with obtaining a FT for the function as you've written it (as I see it) is that the FT results in a series of frequency components. Since you've already given the frequency component(s) you've short circuited the transformation process. In this context the Fourier Transform translates movement across space (in a period of time) into frequency (and phase).

 

[keji8341]

The problem with obtaining a FT for the function as you've written it (as I see it) is that the FT results in a series of frequency components. Since you've already given the frequency component(s) you've short circuited the transformation process. In this context the Fourier Transform translates movement across space (in a period of time) into frequency (and phase).

Sorry, I don't understand what you said. My questions is: Can "expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave" be represented as a sum of uniform plane waves?

Your conclusion is "can" or "cannot"? or not sure?

 

[PhilDSP]

There should exist a decomposition of plane waves for the function, but it probably won't be simple enough to provide any insight into the situation. The frequency you state will likely only map into one particular ray, not all possible rays of all possible plane waves. Since the function isn't written in a form that I can readily visualize I'd hesitate to offer any simple analysis.

 

[keji8341]

There should exist a decomposition of plane waves for the function, but it probably won't be simple enough to provide any insight into the situation. The frequency you state will likely only map into one particular ray, not all possible rays of all possible plane waves. Since the function isn't written in a form that I can readily visualize I'd hesitate to offer any simple analysis.

If it is simple, I won't post it here.
Never mind.
Thanks a lot.

 

[Dale]

Some mainstrain scientist said to me, “The plane wave decomposition is mathematically universal” , then the Einstein's plane-wave Doppler formula should be applicable to any cases. I am referring to that scientist.

I suggest you take it up with them. I don't see the need or the use.

 

[Dale]

keji8341, earlier I had mentioned:

In fact, I don't think that a plane wave spectrum helps in the Doppler analysis at all. What is the net Doppler shift for an infinite sum of Doppler shifts?

I would like to elaborate a bit.

So, if you separate a spherical wave into an infinite sum of plane waves you will have plane waves in all different directions. Each direction will have a different Doppler shift, some might have a factor of 2, others a factor of 1.1, others a factor of 0.7. How do you combine all of those to get the Doppler shift at that point? Do you do a weighted sum? Do you add them in quadrature? Do you multiply them? What?

I just see no benefit to this approach. It is neither necessary nor helpful.

 

[keji8341]

keji8341, earlier I had mentioned:I would like to elaborate a bit.

So, if you separate a spherical wave into an infinite sum of plane waves you will have plane waves in all different directions. Each direction will have a different Doppler shift, some might have a factor of 2, others a factor of 1.1, others a factor of 0.7. How do you combine all of those to get the Doppler shift at that point? Do you do a weighted sum? Do you add them in quadrature? Do you multiply them? What?

I just see no benefit to this approach. It is neither necessary nor helpful.

I have the same question as yours.

 

[keji8341]

There should exist a decomposition of plane waves for the function, but it probably won't be simple enough to provide any insight into the situation. ...

My questions is: Can "expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave" be represented as a sum of uniform plane waves? Note: r=0 is a singularity.

This is actually the potential function produced by an ideal radiation electric dipole. [J. D. Jackson, Classical Electrodynamics, 3rd edition, (John Wiley & Sons, NJ, 1999), Chapter 9, p. 410, Eq. (9.16).]

(i) A spherical-wave decomposition of a plane wave is presented in textbook; for example, J. D. Jackson, Classical Electrodynamics, 3rd edition, (John Wiley & Sons, NJ, 1999), p. 471, Eq. (10.44) in Chapter 10.

(ii) The converse: A plane-wave decomposition of spherical waves is given by MacPhie and Ke-Li Wu, “A Plane Wave Expansion of Spherical Wave Functions for Modal Analysis of Guided Wave Structures and Scatterers”, IEEE Trans. Antennas and Propagation 51, 2801 (2003). Note: The spherical waves are analytical at r=0.
------------
I thought it over carefully.
Physically speaking, the answer is “no”.
Mathematically speaking, the answer is “yes”.
Why? Here are my explanations.

There are two kinds of Fourier transforms.

1 “Sum of real plane waves”. 1D-example: f(x)=Inte{F(k)*exp(-ikx)*dk}, where F(k)*exp(-ikx)*dk is real plane wave component, with k is real. The integration is carried out from –infinity<k<+infinity, and the integral is convergent.

2 “Math correspondence”. 1D-example: f(x)=Inte{F(k)*exp(-ikx)*dk}, where F(k)*exp(-ikx)*dk is NOT a real plane wave, because k is set to be complex to make the integral converge. The integration is carried out in a complex plane by designating a contour for poles. Such Fourier transform is usually used to solve differential equations.

Therefore, "expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave" CANNOT represented as a sum of REAL uniform plane waves, because its Fourier integration must be carried out in a complex plane by designation a contour for poles, and its Fourier transform has no physical meaning, just a kind of math correspondence.

 

[Dale]

Therefore, "expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave" CANNOT represented as a sum of REAL uniform plane waves, because its Fourier integration must be carried out in a complex plane by designation a contour for poles, and its Fourier transform has no physical meaning, just a kind of math correspondence.

The Fourier transform is indeed physical, including applications from MRI to synthetic aperture radar to quantum mechanics. Perhaps you should reconsider your assumptions, namely that complex numbers are non physical.

 

[keji8341]

The Fourier transform is indeed physical, including applications from MRI to synthetic aperture radar to quantum mechanics. Perhaps you should reconsider your assumptions, namely that complex numbers are non physical.

1. In the plane-wave factor expi(wt-k.x), if w and k is complex, then the plane wave decays or grows exponentally with time and position. In free space, such a plane wave is not consistent with energy conservation law.

2.When w is complex, the Planck constant should be proportional to the conjugate complex of w, if E=hbar*w is real..., ha, a lot of new results...

3. If w is complex, Einstein's Doppler formula is still applicable?

Of course, for many theories, it is not required for every intermediate math operation to have physical meaning, especially in quantum mechanics. Assigning a physical explanation is just for being easy to remenber sometimes.

 

[Dale]

You are thinking of the Laplace transform, not the Fourier transform. The Laplace transform also has physical applications, but not in this context.

 

[keji8341]

You are thinking of the Laplace transform, not the Fourier transform. The Laplace transform also has physical applications, but not in this context.

No, I mean the Fourier transform.

 

[Dale]

w and k are real in the Fourier transform. They are only complex in the Laplace transform. If that is your objection to the "physical-ness" of the Fourier transform then it is simply not applicable.

 

[keji8341]

w and k are real in the Fourier transform. They are only complex in the Laplace transform. If that is your objection to the "physical-ness" of the Fourier transform then it is simply not applicable.

Not really.
Do you remember that the Lorentz invariant Green function in the relativistic electrodynamics is obtained by Fourier-transform approach? The Fourier integration is carried out in the complex plane by designating a contour for poles. Please check with the well-known textbook by J. D. Jackson, Classical Electrodynamics, 3rd edition, (John Wiley & Sons, NJ, 1999), Chapter 12, p. 612, Eq. (12.129).

 

[Dale]

Kindly stop double-posting. It is really irritating to have to deal with the same nonsense twice.

 

[PhilDSP]

My questions is: Can "expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave" be represented as a sum of uniform plane waves? Note: r=0 is a singularity.
------------
I thought it over carefully.
Physically speaking, the answer is “no”.
Mathematically speaking, the answer is “yes”.
Why? Here are my explanations.
...
Therefore, "expi[wt-(w/c)*r]/r with r=sqrt(x**2+y**2+z**2), the frequency w is given, monochromatic wave" CANNOT represented as a sum of REAL uniform plane waves, because its Fourier integration must be carried out in a complex plane by designation a contour for poles, and its Fourier transform has no physical meaning, just a kind of math correspondence.

If by "physically speaking" you mean natural, obvious and easily measured then you seem to be approaching a better understanding of the situation. But as Dalespam is arguing, there is nothing at all "unreal" about the mathematics despite the ironic and easy-to-misconstrue use of the terms "real" and "imaginary" when talking about complex numbers (which are fundamental in a Fourier Transform). If you determine mathematically that a plane wave component will have a certain frequency in a certain direction for a particular point some distance from the radiation source, you can no doubt devise a test to measure that component positively. But that wouldn't give you a very satisfying or insightful picture of the entire situation.

 

[Dale]

Hi PhilDSP, note that keji8341 is duplicating all of his recent posts here also in a different thread: https://www.physicsforums.com/showthread.php?t=527847