In experimentally confirmed quantum field theory, the vacuum can only be polarized above a threshold electric field strength on the order 10^20 v/m. This is basically the IR cutoff of the field, the lower limit at which electromagnetism involves vacuum pair production and polarization phenomena. The pairs produced are polarized within a range of 1 fm from an electron, shielding part of the charge as seen at long distances. If there was not a threshold of 10^20 v/m, and Maxwell's theory was correct, then the vacuum would be capable of polarization (and charge shielding) so long as any uncancelled charge remained. If Maxwell's theory was correct, all charges would be cancelled out within a fraction of a millimetre by vacuum polarization. It isn't correct: there are no displacement currents in the vacuum due to vacuum charge polarization except above the IR cutoff or in an exceptionally intense static or oscillating electromagnetic field. In ordinary radio waves and transmission lines, Maxwell's displacement current is a travesty of the facts. The equation still holds when IR cutoff effects are ignored, but the mechanism for the equation below the IR cutoff is electromagnetic radiation exchange between fields!

Maxwell’s Displacement and Einstein’s Trace
(http://www.mathpages.com/home/kmath103/kmath103.htm):

'There’s a remarkable parallel between Maxwell’s development of the field equations of electromagnetism and Einstein’s development of the field equations of general relativity. Recall that when Maxwell began his work there had already been established a set of relations describing the forces of electricity and magnetism. ... Maxwell then added another term to Ampere’s law, which he called the displacement current. The magnitude of this term was far too small to have been perceptible in the experiments performed by Ampere and Faraday, so its inclusion by Maxwell was motivated purely by theoretical considerations.'

This is a falsehood. Every time you turn on the switch, electricity proceeds towards the light bulb at light speed. The circuit is open until completed by the arrival of electricity at the bulb, because the resistance of the circuit is determined by the bulb. Hence Ohm's laws and Kirchoff's laws don't apply to any transient real system like turning on the light, they are only steady-state approximations.

The major fallacy in the claim that electricity is a pushing of electrons down a wire is simply shown by the case of a brief logic step, 1 foot long, which takes 1 ns to pass you. What is pushing the last electron in that 1 foot length pulse, and how is that last electron able to push all the others without slowing down due to resistance? A second fallacy in electricity as electron 'pushing theory' (McEwan's explanation that electrons flow like a row of ball bearings lined up in contact being nudged at one end) is his model's complete inability to explain why the logic step speed is that of the insulator around and between the two conductors.

IN ALL REAL SITUATIONS, 'DISPLACEMENT CURRENT' (WHICH IS DUE TO RADIATION, NOT THE MECHANISM MAXWELL THOUGHT) IS VITAL. It allows these transients to flow before the pulse of electric current has completed the circuit. For example, the diagram below (which is not entirely accurate) purports to show how a capacitor charges up. (Maxwell didn't know how fast electricity goes, so he ignored the spread of electricity along the capacitor plates, which reflects back off the far end and adds to further incoming energy.)




This illustration (due to Ivor Catt, Wireless World December 1978) is false because Catt draws the steps as being vertical increments. This is because he falsely assumes zero rise-time at the front of the long pulse energy current flowing into the capacitor (which is an physics error that goes right back to Heaviside). I've already corrected it:

D = permittivity*E (electric field strength)

dD/dt = permittivity*dE/dt

I distinguish between D and dD/dt in the illustration above.

Ivor's "Catt Question" diagram states dD/dt = 0 at all places except the vertical step front where dD/dt = infinity!

Catt claims a capacitor is a transmission line, where dD/dt would represent charging. But how can dD/dt represent real charging, when dD/dt in Catt's diagram is always 0 or infinity?

To get around this problem of the discontinuity, Ivor needs to admit that he has never seen a discontinuity in your life, and his oscilloscope traces of logic steps are not square waves. Hence, he was just duping himself and trying to dupe others with his "question" based on the falsehood of a discontinuity which can never actually occur in the real world.

These people pretend to be interested in science, but instead of welcoming the facts and revising their arguments, instead they just obfuscate and ignore the facts. (See comments section.)

What actually happens in the sloping part of the real logic step is that electrons are accelerated in non-zero time, and in so doing radiate energy like a radio transmitter antenna. Because the current variation in each conductor is an exact inversion of that in the other, the fields from the radio waves each transmits is capable of exactly cancelling the fields from the signal from the opposite conductor. Seen from a large distance, therefore, there is no radio transmission of energy whatsoever. But at short distances, between the conductors there is exchange of radio wave energy between conductors in the rising portion of the step. This exchange powers the logic step mechanism. Behind the rise part of the step, there dD/dt = 0 so the mechanism for the logic pulse and electric current drift there is the magnetic field effects of each conductor upon the other. The electron drift in each conductor is there due to the magnetic field from electron drift in the other conductor. Using these two mechanisms (radio emission producing the effects attributed to "displacement current" dD/dt, and magnetic field from current in the other conductor inducing current where dD/dt = 0), electricity flows. That's why the speed of a logic pulse is the speed of light for the insulator between and around the conductors.

When electric current (or a long flat-topped logic pulse), enters a capacitor plate, Maxwell thought it continued straight on to the other plate, without any change of direction, via aethereal displacement current. Hence in Maxwell's displacement current of the vacuum, i = e.dE/dt = dD/dt, the direction of both i and dD/dt is perpendicular to the plane of the plates, so it is from one plate to the other.

But since Maxwell got the direction wrong (the current spreads along the plates), and since x-rays/radioactivity showed wires to be like an aether (nuclear atoms with enormous spacesbetween electrons, not by any means a 'solid metal), the 'displacement current' is actually a displacement of real charge in the conductor itself. Hence in the equation i = dD/dt, the direction of i and dD/dt is parallel to the conductor, and i is real current not aethereal current.

The 90 degree direction change of current is vital, see http://www.wbabin.net/physics/cook.htm:

'This "capacitor is a transmission line" conclusion directly contradicts Maxwell, Article 610:

"One of the chief peculiarities of this treatise is the doctrine which asserts, that the true electric current, I, that on whichthe electromagnetic phenomena depend, is not the same thing as i, thecurrent of conduction, but...

I = i + dD/dt (Equation of True Currents)."

'This quotation pins down the gross falsehood in today's physical science, based on (Maxwell's) electromagnetic theory; the correct equation due to Catt, Davidson, and Walton is: 'I = i = dD/dt

'In this equation, there is an "=" sign whereas in Maxwell's equation there is a "+" sign. This says it all. In other words, Maxwell treats wire electricity (i) as being different to the current flow in the vacuum dielectric of a charging or discharging capacitor (dD/dt), whereas Catt. Davidson, and Walton have proven that there is no distinction for pulses ofelectromagnetic energy in wires. Hence, Maxwell is mathematically wrong.' {However, Ivor Catt refuses to scientifically comment on this clarification of his writings, or on the diagram above.}

The problem that people may have is the direction of electric field E. The potential involts is varying from 0 to v over a distance x along the plate. Once a current has been induced on the other plate, there is a charge there and so there is then a field gradient between the plates (with E pointing from one plate towards the other, the E vector being perpendicular to the plane of the capactor plates or transmission line wires). The E field I'm talking about is that parallel to the plates, because the current must turn 90 degrees and must spread along the plate after entering it (not proceed as Maxwell thought straight in the direction of one plate to the other).

The mathematician claims: 'Maxwell decided that this [Ampere's law] equation was incomplete, and the right hand side needed to be augmented by an additional term which he called the displacement current. ... His explanation evolved over the years, as his ideas about suitable mechanisms changed, but in essence he argued that the current density j at a given location does not actually represent the total current flow at that location (even though that is essentially its definition). According to Maxwell, a dielectric medium can be considered to consist of couples of positive and negative charges, and an electric field E pulls these charges in opposite directions, stretching the links between them until they achieve some kind of equilibrium. If the strength of the field is increased, the charges are pulled further apart, so during periods when the electric field is changing there is movement of the electric charge elements of the dielectric medium. This movement of charge is what Maxwell calls the displacement current, proportional to dE/dt, which he adds to Ampere’s original formula ...

'However, Maxwell’s rationalization of this extra term is questionable in at least two respects. First, it’s reasonable to ask why the displacement current is not already included as part of the total current density j at the given point. By definition, j is supposed to represent the flow of electric charge at a given location and time. Since Maxwell conceives of the displacement current as literally a flow of electric charge, one could argue that it should already be included in j, especially since the experimental results did not indicate the need for any additional term. Second, after introducing the concept of displacement current in dielectric media (where the existence of coupled electric charges is somewhat plausible), Maxwell goes on to apply the extra term to the vacuum, where the presence of coupled electric charges (being pulled apart and held in equilibrium by a stationary electric field) is questionable. He certainly could not point to any evidence of such disembodied charges existing in the vacuum. It’s true that some aspects of modern quantum field theory can be expressed in terms of pairs of oppositely charged virtual particles in the vacuum, flashing in and out of existence within the limits of the uncertainty principle, but surely virtual particles were not what Maxwell had in mind when he conceived of his tangible mechanistic models of the luminiferous ether. Without the uncertainty relations such particles would violate conservation of charge ...'

Charge conservation implies div.j = -d(charge density)/dt, which in plain English states that if you diverge an electric current, spreading the charge outward, then the charge density falls with time. It isn't a mathematical law, just plain common sense to anybody with any.

The mathematician then states 'Coulomb's law is div.E = charge density...' which is not directly Coulomb's law but is Gauss' law (Coulomb's is the inverse square force law for charges, whereas Gauss' is the mathematical law of electric field). Anyway, he uses Gauss's law to show that div.(dE/dt) = d(charge density)/dt, and then comments that this should be added to charge conservation law to give div.(j + dE/dt) = 0. But anybody can see where this is wrong.

Charge conservation div.j = -d(charge density)/dt combines with div.(dE/dt) = d(charge density)/dt not by addition but by equality, showing div.j = -div.(dE/dt), or j = -dE/dt. This is the whole point. The mathematical nonsense can't understand that j (displacement current) is not necessarily real, and since it is dimensionally equivalent to dE/dt (multiplied by the permittivity of free space of course, which the mathematican invariably ignores as too down to earth or trivial), as a physicist you have to suspect that what is going on is a field mechanism not an aethereal displacement current. I'll come back to this later. The mathematician goes on:

'Thus the combination of charge conservation and Coulomb’s law implies that the divergence of [div.B = j] vanishes, whereas the divergence of equation [div.B = j + dE/dt] does not vanish. This immediately shows that equation [div.B = j + dE/dt] must be correct, i.e., we must add dE/dt to Ampere’s law, purely for mathematical consistency, because the left hand sides of equations [div.B = j + dE/dt] and [div.B = j] are the curl of the magnetic field, and it’s easy to show that the divergence of the curl of any vector field is identically zero.'

The mathematician then writes out the equations to show that a divergence of a curl is zero, but if you switch to physical thinking you can do it without equations as such. A wire carrying a current is encircled by B field lines which have curl because they are circles, closed loops. The B field line at every point is an equal distance from the wire, hence it doesn't diverge outward. Thus, the divergence of the curling magnetic field is physically equal to zero. But of course this is too arcane for the great mathematicians, who want to make things crazy:

'One finds in the literature three basic justifications for introducing the “displacement current” term to Ampere’s law. First, it is sometimes claimed that it can be justified simply on the grounds of symmetry, i.e., since Faraday’s law indicates that a changing magnetic field is associated with an electric field, we would expect by symmetry that a changing electric field should be associated with a magnetic field. However, the glaring asymmetry due to the absence of magnetic monopoles tends to undermine the cogency of this argument. The second justification, found especially in historical treatments, is Maxwell’s heuristic rationale based on the idea of a dielectric medium consisting of charge couples that are pulled apart by an electric field. Lastly, the most common justification is consistency with Coulomb’s law and charge conservation, noting that the divergence of the curl of B must vanish. Thus we begin with Ampere’s hypothesis that the curl of B equals j, but then we note that the divergence of j does not vanish, whereas the vector j + dE/dt does have vanishing divergence (due to Coulomb’s law and the conservation of charge), so we add this term to complete the field equations of electromagnetism in a mathematically and physically self-consistent way.

'It’s interesting how similar this is to the process by which Einstein arrived at the final field equations of general relativity. The simplest hypothesis involving only the metric coefficients and their first and second derivatives, is that the Ricci tensor R_uv equals the stress energy tensor T_uv, but then we notice that the divergence of T_uv does not vanish as it should in order to satisfy local conservation of mass-energy. However, the tensor T_uv - (1/2)g_uvT does have vanishing divergence (due to Bianchi’s identity), so we include the “trace” term -(1/2)g_uvT to give the complete and mathematically consistent field equations of general relativity

R_uv = T_uv - (1/2)g_uvT

which can also be written in the equivalent form

R_uv - (1/2)g_uv = T_uv

'Just as the inclusion of the “displacement current” in Ampere’s formula was the key to a Maxwell’s self-consistent field theory of electrodynamics, so the inclusion of the “trace stress-energy” in the expression for the Ricci tensor was the key to Einstein’s self-consistent field theory of gravitation. In both cases, the extra term was added in order to give a divergenceless field.

'Incidentally, to the three common justifications for the displacement current discussed above, we might add a fourth, namely, the fact that the inclusion of the term dE/dt in Ampere’s equation leads to transverse electromagnetic waves propagating in a vacuum at the speed of light. Of course, this is ordinarily presented (triumphantly) as a consequence of the added term, rather than as a justification or motivation for it. However, someone as mathematically astute as Maxwell could hardly have failed to notice that the standard wave equation would result from the known system of electromagnetic equations if only Ampere’s law contained a term of the form dE/dt. Indeed Faraday (Maxwell’s primary source and inspiration) had speculated that the electromagnetic ether and the luminiferous ether might well turn out to be the same thing, suggesting that light actually is a propagating electromagnetic disturbance. Also, Weber had shown that a speed on the order of the speed of light is given by a simple combination of electromagnetic constants, and many other people (including Riemann) had pursued the same idea. The objective of explaining the wave properties of light was certainly “in the air” at that time. Is it conceivable that Maxwell actually reverse-engineered the displacement current precisely so that the equations of electromagnetism would support transverse waves at the speed of light in a vacuum? If so, he would have been consistent with a long tradition [of pretenders]...'

The mathematician then quotes Einstein's comment on displacement current in a letter to Michele Besso in 1918:

'No genuinely useful and profound theory has ever really been found purely speculatively. The closest case would be Maxwell’s hypothesis for displacement current. But there it involved accounting for the fact of the propagation of light (& open circuits).'

I'm going to repeat the end of my last post on this blog here:

‘What they now care about, as physicists, is (a) mastery of the mathematical formalism, i.e., of the instrument, and (b) its applications; and they care for nothing else.’ – Karl R. Popper, Conjectures and Refutations, R.K.P., 1969, p100.

Displacement current, i = Permittivity x Voltage divided into (Time x Distance) or, i = eV/(tx)

This equation is simplified to a constant gradient slope in the electric field (voltage/distance), so we get away from differential equations.

It vital that in order for there to be displacement current, voltage increment V occurs over: (a) time increment t, and (b) distance increment x. Because electricity goes at speed c, the time increment t is given by: t =x/c. Hence displacement current i = eV/(tx) = eVc/x². Alternatively, written in terms of rise time t, we get displacement current i = eV/(tx) = eV/(ct²). This is a very useful formula. The displacement current for a uniform rise in voltage V over time t is equal to i = eV/(ct²).

The voltage variation with time and distance in the ramp at the front of electric pulse delivered by a transmission line causes the electrons in that ramp part to be accelerated. As a result they transmit energy perpendicular to the direction of their acceleration, which also occurs in radio transmission from an aerial (radio emission occurs in proportion not to thecurrent in the aerial, but to the rate of change of the current, hence to the acceleration of the electrons). Therefore, there is electromagnetic energy emission from the accelerating electrons in one conductor to those in the other. This constitutes the mechanism for the effects normally attributed to "displacement current". All the objections people can think up against this new mechanism are discredited here.

So, to be scientific, we must examine what the quantitative role of traditional displacement current is over this radiation mechanism. Maxwell-type displacement current in the vacuum (motion of aether particles) is trivial compared to radiation. I'm pointing out that it is not the most important mechanism. This is not an all or nothing situation. Displacement current in the vacuum must exist, but the evidence I have is that it is trivial compared to the radiation mechanism which delivers the energy. You have to accept two mechanisms for displacement current: radiation and Maxwell's charge polarisation.

The time-dependent Schroedinger equation and the Dirac equation (which is a relativistic time-dependent Schroedinger equation) are both statements of Maxwell's displacement current equation, when you grasp the radiation mechanism for the field that I'm explaining.

The rate of change of the field's wavefunction determines the energy delivery in the Schroedinger/Dirac time-dependent equations. The physics of displacement current is not current = e.dE/dt where e is permittivity, but net energy transfer rate (which is caused by or causes an electric field in a real conductor) is proportional to the rate at which the field varies.

On the topic of the advancement of understanding in general relativity, see D R Lunsford's comment at http://www.math.columbia.edu/~woit/wordpress/?p=319#comment-7160:

D R Lunsford Says:

January 3rd, 2006 at 11:14 pm

Lee Smolin said:"The issue is not about knowing how to write the Einstein’s equations down or find simple solutions. It is about the interpretation of observables on the space of solutions."

This is absolutely not the issue! and it is flat wrong for you to annouce it as such.

I think you people are not very bright - you completely miss the point about GR, which is how you end up abusing it so much.

The issue is the measurement problem without assuming an apparatus = a background. This does honor to both GR and QM. You will instantly understand if you think for 10 minutes that the entire ethos of the measurement problem is antithetical to the idea of background independence. Any attempt to go farther than this is doomed. You must either change one, or the other. Your crowd ignores the actual physical import of GR because it is easier to hide one’s canoe in the metaphyical tributaries of the of “interpretation”.

“Wave function of the universe” - case closed!The only people who take both GR and QM seriously are Finkelstein, Dirac, Einstein, Pauli, and Schroedinger. The very people who get ingored now.

-drl

http://www.math.columbia.edu/~woit/wordpress/?p=307#comment-6424:

Chris Oakley Says: December 8th, 2005 at 8:23 am

"Why do referees do such a poor job? They block the good and interesting, and let through confused stuff like this."

This may have something to do with the fact that LS has a prestigious job at a prestigious university.

The same comment I made about Weinberg also applies here: if one is in a position of influence then one is duty bound to try to encourage people to do one’s subject rather than to try to scare them off. Telling people that String theory is nonsense but still the only idea in particle physics worthy of study is not going to make anyone want to sign up.

Of course, there are alternatives to String theory, but the String-theory-dominated research establishment does its utmost to make sure that these particular suckers never get an even break. In the long run, though, if leaders like Susskind and Weinberg convince taxpayers that theoretical particle physics is not worth funding, then who are the real suckers?

Luboš Motl, http://motls.blogspot.com/2005/12/shut-up-and-calculate.html:

Monday, December 12, 2005

Shut up and calculate

I would not promote overly technical lecture notes, especially not about things covered in many books. But the interpretation of quantum mechanics in general and decoherence in particular - a subject that belongs both to physics as well as advanced philosophy - is usually not given a sufficient amount of space in the textbooks, and some people may be interested in Lecture23.pdf.

http://www.math.columbia.edu/~woit/wordpress/?p=316#comment-6808:

anon Says: December 25th, 2005 at 12:09 pm

Lunsford, a turkey breeding (or string theory) community can’t afford a diversity that would make the most popular theory (strings/turkeys) look silly. I suggest you give up on trying to reform dictators; they just try to suppress all criticism and shoot the messengers. Instead of telling them what they don’t like to hear and having them doing the shooting, you need to adopt more sturdy methods and go on a turkey cull. So I suggest you try building on alternatives to strings until one succeeds in doing more than string theory, then shoot, stuff and slowly roast the turkey.

Merry Christmas

What does Dr Motl define as a deep idea? A piece of string?

Extract from Michael Faraday's letter to James Clerk Maxwell, dated 13 November 1857:

November 13th, 1857.

My dear Sir,

There is one thing that I would be glad to ask you. When a mathematician engaged in investigating physical actions and results has arrived at his own conclusions, may they not be expressed in common language, as fully, clearly and definitely as in mathematical formulae? If so, would it not be a great boon to such as we to express them so - translating them out of their hieroglyphics that we might work upon them by experiment? ...

If this be possible would it not be a good thing if mathematicians, writing on these subjects, were to give us their results in this popular useful working state as in that which is their own and proper to them?

Ever, my dear Sir, most truly yours.