# Whittaker 1904 paper on scalar potential functions

http://www.hyiq.org/Library/E.T.Whittaker-1904.pdf [Broken]

"On an Expression of the Electromagnetic Field due to Electrons by Means of Two Scalar Potential Functions" by E. T. Whittaker published in Proceedings of the London Mathematical Society, Vol. 1, 1904)

In the paper Whittaker 1904 seemed to show that all EM fields and waves can be decomposed into two scalar potential functions. It follows that, by assembling two such scalar potential functions in beams, one can produce a "scalar potential interferometer" where the potential beams intersect at a distance. In that interference zone, ordinary transverse EM fields and energy appear.

You may state that it was written prior to special relativity. But note that maxwell equations are naturally lorentz invariant.. so Whittaker paper would still be in full force. Can anyone point out any flaw in the paper that would make the above not possible?

Note Whittaker is one major player and part of early 20th century physics akin to Poincare or Lorentz. Whittaker was, in 1954, selected by the Fellows of the Royal Society to receive the Copley Medal, the highest award granted by the scientific Royal Society of London, "for his distinguished contributions to both pure and applied mathematics and to theoretical physics". Back in 1931 Whittaker had received the Royal Society's Sylvester Medal "for his original contributions to both pure and applied mathematics". What made him write the paper and what is it really all about?

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Nice find! Yeah, it is possible to do this; Whittaker is correct. I think a good birds-eye-view explanation for why this is possible is that the differential form of Maxwell's Equations for electric and magnetic fields are four equations involving two vectors with three components each. This implies that once you solve Maxwell's Equations for a certain charge and current configuration, you are left with exactly two independent degrees of freedom. What Whittaker did was give a specific procedure for finding a certain degree-of-freedom pair, which he calls F and G. (Others are possible, like A = F + G, B = F - G, and so on).

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I look forward to the development of this thread.

Nice find! Yeah, it is possible to do this; Whittaker is correct. I think a good birds-eye-view explanation for why this is possible is that the differential form of Maxwell's Equations for electric and magnetic fields are four equations involving two vectors with three components each. This implies that once you solve Maxwell's Equations for a certain charge and current configuration, you are left with exactly two independent degrees of freedom. What Whittaker did was give a specific procedure for finding a certain degree-of-freedom pair, which he calls F and G. (Others are possible, like A = F + G, B = F - G, and so on).
So how do you implement it in practice... produce a "scalar potential interferometer" where the potential beams intersect at a distance say a mile away, and in that interference zone, make ordinary transverse EM fields (which includes microwaves and x-rays) appear?

Nice find! Yeah, it is possible to do this; Whittaker is correct. I think a good birds-eye-view explanation for why this is possible is that the differential form of Maxwell's Equations for electric and magnetic fields are four equations involving two vectors with three components each. This implies that once you solve Maxwell's Equations for a certain charge and current configuration, you are left with exactly two independent degrees of freedom. What Whittaker did was give a specific procedure for finding a certain degree-of-freedom pair, which he calls F and G. (Others are possible, like A = F + G, B = F - G, and so on).
Mathematically. How are these two independent degrees of freedom related to the electric scalar potential and magnetic vector potential? Or are these two independent degrees of freedom related only to one of them or not related to them at all?

What is the modern equivalent of Whittaker formulation? I think we use 4 components now instead of 3, isn't it.

jtbell
Mentor
So how do you implement it in practice... produce a "scalar potential interferometer" where the potential beams intersect at a distance say a mile away, and in that interference zone, make ordinary transverse EM fields (which includes microwaves and x-rays) appear?
You're the one who claimed it was possible, in post #1. I don't see anything in Whittaker's paper that addresses this scenario. What makes you think it's possible?

You're the one who claimed it was possible, in post #1. I don't see anything in Whittaker's paper that addresses this scenario. What makes you think it's possible?
In Dirac equation and many other equations. What they thought are just mathematical artifacts have real consequences.. for example, the negative energy state is really the positron. Dirac thought it was about a sea of electrons with a hole leftover producing the positive charge. All equations are subject to interpretations. So I thought the scalar potentials mean Electromagnetic field is infolded within them, and this can be recovered by assembling back the scalar potentials. So they can't? Sorry for thinking it can, erronously. But how do you determine or tell whether something in the equation correspond to an element of reality or just artifacts. For example, virtual particles as multivariate variables are just artifacts and don't really exist in nature.. as Arnold Neumaier emphasized.. although others thought they really exist in physical nature. So it appears equations also is subject to interpretations like Dirac thinking the positive charge are holes left by electrons when they are really now understood as positrons?

I'm interested in history of electrodynamics from Maxwell to present and came across Whittaker. I want to understand how the Whittaker approach as classical electrodynamics compares to the modern approach. But Maxwell Equations didn't actually have much modifications after more than 150 years in spite of SR and GR and QM, isn't it. In fact Einstein based SR on Maxwell Equation in his light thought experiment. That is why the Maxwell Equations seem to be pretty solid and can be reference for other theories like SR.

So how do you implement it in practice... produce a "scalar potential interferometer" where the potential beams intersect at a distance say a mile away, and in that interference zone, make ordinary transverse EM fields (which includes microwaves and x-rays) appear?
It doesn't work that way. The electric and magnetic fields exist at all points in space, just as they always did. Whittaker is saying that you can write the six total components in terms of derivatives of only two scalar fields. It is the same as saying that the electrostatic field can be derived from a single scalar field, the familiar electric potential, or voltage. The voltage is a different kind of field from the electric field; they are linked only in that you can derive one from the other mathematically.

Mathematically. How are these two independent degrees of freedom related to the electric scalar potential and magnetic vector potential? Or are these two independent degrees of freedom related only to one of them or not related to them at all?
Well, Whittaker shows us how fairly clearly. I haven't read the paper closely enough to get the exact mechanism, but he's basically using Maxwell's (four) equations to eliminate four of the six degrees of freedom in the E and B fields, leaving two independent degrees of freedom F and G.

In Dirac equation and many other equations. What they thought are just mathematical artifacts have real consequences.
This is a different situation. The Dirac equation was the expression of genuinely new physics -- no one had thought about a field that obeyed the Dirac equation before. Whereas here, Whittaker is "only" rearranging Maxwell's equations in a novel way. Physics doesn't care about our labels, only our unique physical predictions of the outcomes of experiments. And Whittaker's paper amounts to a complicated relabeling scheme for the exact same physics as described by Maxwell.

So I thought the scalar potentials mean Electromagnetic field is infolded within them, and this can be recovered by assembling back the scalar potentials.
Take a look at the last page of the paper. This contains the formulas used to derive the E and B fields (what he calls the d and h fields) from F and G.

you cant reduce the magnetic field to the gradient of a scalar field

It doesn't work that way. The electric and magnetic fields exist at all points in space, just as they always did. Whittaker is saying that you can write the six total components in terms of derivatives of only two scalar fields. It is the same as saying that the electrostatic field can be derived from a single scalar field, the familiar electric potential, or voltage. The voltage is a different kind of field from the electric field; they are linked only in that you can derive one from the other mathematically.

Well, Whittaker shows us how fairly clearly. I haven't read the paper closely enough to get the exact mechanism, but he's basically using Maxwell's (four) equations to eliminate four of the six degrees of freedom in the E and B fields, leaving two independent degrees of freedom F and G.

This is a different situation. The Dirac equation was the expression of genuinely new physics -- no one had thought about a field that obeyed the Dirac equation before. Whereas here, Whittaker is "only" rearranging Maxwell's equations in a novel way. Physics doesn't care about our labels, only our unique physical predictions of the outcomes of experiments. And Whittaker's paper amounts to a complicated relabeling scheme for the exact same physics as described by Maxwell.
What can you say about the Quaternions that Maxwell originally wanted to use. Later Heavyside and Gibbs removed the scalar component of the Quaternions which could hav emake the Whittaker approach more obvious. And present day electrodynamics use the more restrictive Heavyside and Gibbs (and Hertz) formulation. Note Maxwell's original EM theory was written in quaternions, which are an extension to the complex number theory and an independent system of mathematics. In short, since the quaternion is a hypernumber, Maxwell's theory was a hyperspatial theory -- not just the limited three-dimensional subset that was extracted and expressed by Heaviside and Gibbs in terms of an abbreviated, incomplete vector mathematics

Take a look at the last page of the paper. This contains the formulas used to derive the E and B fields (what he calls the d and h fields) from F and G.
In a paper one year before it, Whittaker described how to do the opposite... infold E and B fields into F and G.

http://www.hyiq.org/Library/E.T.Whittaker-1903.pdf [Broken]

Wonder if you have seen the paper. It is partner to his 1904 paper. The two papers are some of the most interesting I encountered in my research into the complete history of electrodynamics.

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you cant reduce the magnetic field to the gradient of a scalar field
Yes, but that's not what Whittaker did. He writes the E and B fields (not just the B field) in terms of a more complicated differential operator (not the gradient) of two different scalar functions. To see how this is possible, read the paper!

does this only work in 3 dimensions?

does this only work in 3 dimensions?
Whittaker starts with the standard formulation of EM in terms of Φ - the electric potential, and A, the vector potential. He defines 3 scalar fields which by taking some space and time derivatives are related to Φ and A. He eliminates one of the scalar fields by choosing an appropriate gauge. A similar analysis in the case of more than 3 spatial dimension would seem to require more scalar fields (since one has the same number of equations of constraint but an extra degree of freedom for each additional spatial dimension). I expect for n spatial dimensions, n-1 scalar fields would be required.

One major reason that this way of describing EM is not used very often is probably that it is not lorentz covariant (in fact it doesn't even treat all of the spatial dimensions the same). Also, I imagine it would be difficult to incorporate these scalar fields into a lagrangian (much less a covariant one). All things considered, this two scalar field version of EM seems less fundamental than the usual 1 scalar field, 1 vector field version.

What can you say about the Quaternions that Maxwell originally wanted to use.... [and other questions]
I've never studied quaternions, actually, and it looks like I'm not as knowledgeable about the history of E&M as you. But my understanding is that all these different formulations lead to exactly the same physical predictions, although as IsometricPion notes, some formulations are more aesthetically pleasing and convenient than others.

I've never studied quaternions, actually, and it looks like I'm not as knowledgeable about the history of E&M as you. But my understanding is that all these different formulations lead to exactly the same physical predictions, although as IsometricPion notes, some formulations are more aesthetically pleasing and convenient than others.
My knowledge of history of E&M is only by words. I don't understand all the maths fully. That is why I'm asking in this thread. Prior to the 1904 paper, Whittaker has written the first paper http://www.hyiq.org/Library/E.T.Whittaker-1903.pdf [Broken]

If you have time, pls. look into it as it completed what Whittaker was talking about. The way I understood it. Whittaker seemed to be showing that a standing scalar potential wave can be decomposed into a special set of bidirectional EM waves that convolute or add into a standing scalar potential wave. But you said the electric and magnetic fields exist at all points in space, just as they always did. But what Whittaker seemed to be implying or demonstrating was some kind of hidden bidirectional EM wave structure in the scalar potential of vacuum, and showing how to produce a standing scalar EM potential wave? Isn't it like interpreting the math of Dirac Equation as having negative energy state or antimatter? What is the math really saying (if you won't be contrained by early 20th century limited physics)? You have described only the 1904 paper. Not yet the 1903 paper which introduced the idea.

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My knowledge of history of E&M is only by words. I don't understand all the maths fully.
Sorry to be blunt, but that is like reading about the history of classical music without knowing how to read sheet music or play an instrument. You might read about different composers and their music, but you couldn't get a real understanding of what they were doing. I recommend you pick up a book like Introduction to Electrodynamics by Griffiths and do some problems. There is no better way to build precise understanding of a physics topic than by working through specific examples. You can't just read papers, skip the math, and think that you know what is happening.

Whittaker has written the first paper...
I think this paper is showing something different from the 1904 one. In this one, he's showing that an electrostatic potential which is constant in time can be written as a sum of potentials which individually vary with time. It's sort of like decomposing a function into a Fourier series, as he notes. The physical implications of this are actually kind of interesting -- I think what he's doing is anticipating the modern quantum-field-theory motivated idea that potentials between charged particles arise from the exchange of virtual photons.

Again, and I cannot stress this enough, neither paper contains new physics. He is only reformulating Maxwell's Equations in various interesting ways, and working through some specific examples. What you have to understand is that physics can have multiple interpretations and still make objective statements about the world. For example, many different unit systems are possible. The speed of light can be expressed as 186000 miles per second, or 300000000 meters per second. But that does not make the speed of light a matter of opinion! Once we agree on a unit system, all observers should agree on a certain measurement of the speed of light. Another example is coordinate systems -- what directions we choose as x, y, and z are completely arbitrary, but once we agree on a coordinate system, the motion of a particle should look the same to everyone.

In field theory, the same kinds of things happen. You can look at either potentials as "fundamental" or vector fields as "fundamental," and you can set up conversions between potentials and fields in various ways. You can also write a static field as a sum of time-varying fields, if you want, or you can not bother to do this and just consider the static field the basic object. But all of these are choices that can be made by the people doing the calculations; they do not affect the physics.

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Sorry to be blunt, but that is like reading about the history of classical music without knowing how to read sheet music or play an instrument. You might read about different composers and their music, but you couldn't get a real understanding of what they were doing. I recommend you pick up a book like Introduction to Electrodynamics by Griffiths and do some problems. There is no better way to build precise understanding of a physics topic than by working through specific examples. You can't just read papers, skip the math, and think that you know what is happening.
Yes. I know. And I'm reading up on Feynman Lectures of Physics now.

I think this paper is showing something different from the 1904 one. In this one, he's showing that an electrostatic potential which is constant in time can be written as a sum of potentials which individually vary with time. It's sort of like decomposing a function into a Fourier series, as he notes. The physical implications of this are actually kind of interesting -- I think what he's doing is anticipating the modern quantum-field-theory motivated idea that potentials between charged particles arise from the exchange of virtual photons.
Earlier in the thread you mentioned the 1904 paper "contains the formulas used to derive the E and B fields (what he calls the d and h fields) from F and G.". But in the 1903 paper we are talking about now. Was he not trying to give formulas to infold the E and B fields into F and G (opposite of the 1904 paper)?. You interpreted the paper about electrostatic potential which is constant in time (meaning scalar) can be written as a sum of potentials which individually vary with time (the same entity as EM wave). Won't the same math be also describing infolding E and B fields into F ang G (or opposite to that of the 1904 paper)? Or won't the math allow it at all and why? Or. Is the math in the 1903 paper not the same as the 1904 paper? Thanks a lot.

Again, and I cannot stress this enough, neither paper contains new physics. He is only reformulating Maxwell's Equations in various interesting ways, and working through some specific examples. What you have to understand is that physics can have multiple interpretations and still make objective statements about the world. For example, many different unit systems are possible. The speed of light can be expressed as 186000 miles per second, or 300000000 meters per second. But that does not make the speed of light a matter of opinion! Once we agree on a unit system, all observers should agree on a certain measurement of the speed of light. Another example is coordinate systems -- what directions we choose as x, y, and z are completely arbitrary, but once we agree on a coordinate system, the motion of a particle should look the same to everyone.

In field theory, the same kinds of things happen. You can look at either potentials as "fundamental" or vector fields as "fundamental," and you can set up conversions between potentials and fields in various ways. You can also write a static field as a sum of time-varying fields, if you want, or you can not bother to do this and just consider the static field the basic object. But all of these are choices that can be made by the people doing the calculations; they do not affect the physics.
Don't forget that in quantum mechanics. The same year it came into being, there were Matrix Mechanics, Heisenberg Mechanics, and Wave Mechanics. The math may be the same but by arranging it differently or in other ways (like the Whittaker papers), what is hidden may seem manifest or is more obvious (like the Minkowski illumination of Einstein 1905 SR).

Also remember in Relativity, there is the kinematics and dynamics interpretation. In the former it's all geometry and math abstraction. In the latter, it's dynamical or involving actual physical entities. The Whittaker papers if interpretated in dynamical relativity would suggest the existence of physical scalar waves of pure potential (trapped) EM energy, where this potential energy contained hidden, deterministically structured, internal bidirectional EM wave patterns — yet no external E- and B-field force resultants appeared in the overall vector summation envelope.