# Dirac Equation & Negative Energy

### #1

Posted 26 May 2010 - 08:19 PM

Part 1: http://openseti.org/...HotsonPart1.pdf

Part 2: http://openseti.org/...HotsonPart2.pdf

this added 06/01/2010

Part 3: http://blog.hasslber.../HotsonIE86.pdf

### #2

Posted 27 May 2010 - 08:41 AM

A publication in 2002 by D.L.Hotson (in two parts) provides a novel interpretation of the negative energy solution of the Dirac Equation. I would like this thread to be a forum to discuss the thoughts of Hotson.

Part 1: http://openseti.org/...HotsonPart1.pdf

Part 2: http://openseti.org/...HotsonPart2.pdf

That was a fun read. Thanks for that Rade.

The problem I see is ironic. D.L. Hotson criticizes the standard model and the big bang theory for using an entire zoo full of parameters (fudge factors), yet seems to adhere to the idea of an undetectable sea of negative energy, a virtual 'medium,' not unlike the aether shown to be inexistent by the Michelson-Morley experiment. In addition to that, Hotson argues for a set of unobservable spin dimensions. But these can be perceived as no less imaginary than any other fudge factor used in the standard model or big bang theory (e.g., dark energy, dark matter). The irony is compounded by Hotson's view that anything we postulate to exist must, in principle, be observable directly (or indirectly, perhaps).

The idea that a potential

*theory of everything*can be derived from Dirac's equations would have stood on less rickety footing had those

*ad hoc*features not been required.

Other details regarding a universal BEC, and consciousness interacting with

*time*, are rather sketchy, shady even. Or maybe I should re-read those parts of the text.

CC

### #3

Posted 27 May 2010 - 04:33 PM

I did a search for the word "undetectable" in the documents. In all cases the word is prefix by the word "virtual". But I do not see that Hotson finds that a sea of negative energy being undetectable to be a fudge factor. There is nothing to fudge, given that the reality of the negative energy sea is a direct result of taking the + and - terms for E (energy) in the Dirac Equation seriously. That is, the negative energy sea exists, the same way an "off pixel" exists in a computer screen---how can we detect the "off pixel" until it is turned "on" ? That is the point of his argument, is it not ? -- that taking the Dirac Equation seriously does not require any fudge factors.The problem I see is ironic. D.L. Hotson criticizes the standard model and the big bang theory for using an entire zoo full of parameters (fudge factors), yet seems to adhere to the idea of an undetectable sea of negative energy, a virtual 'medium,' not unlike the aether shown to be inexistent by the Michelson-Morley experiment.

Here is what he says in the Part 2 document about undetectable negative energy:

*"The sea of negative-energy one-dimensional epos, vibrating in imaginary directions, forms a virtually undetectable background, like “off” pixels in a perfect computer screen. And like a three-way light switch, they “turn on” in three stages, each stage vital to our reality. Epos vibrating in one “real” dimension form the electromagnetic field. Vibrating in two “real” dimensions, they carry angular momentum around at the speed of light: the “photon.” And vibrating in three “real” dimensions, they form matter."*

Concerning the aether, Hotson discusses this in many places. What he claims is that Lorentz died too soon (1928), before Dirac published his equation. Lorentz had a concept of a type of electromagnetic aether that agrees completely with the Dirac Equation and that shows how the results of the Michelson-Morley experiment have been misinterpreted. Here is what Hotson says about this topic, in Part 1 document:

*"Lorentz’s electromagnetic aether (Lorentz, 1904, 1909) answered all of the other objections to a carrier of light, including the results of the Michelson-Morley experiment"*

*"One of the tragedies of science is Lorentz’s death in 1928, just as Dirac’s equation was formulated, as Lorentz surely would have recognized the negative-energy sea as responsible for his electromagnetic aether."*

I have not read the 1904 & 1909 papers by Lorentz about his concept of an electromagnetic aether, so I take it that Hotson has read them and finds they agree with predictions of the Dirac Equation. So, if Lorentz was incorrect about his aether concept--how so, and why does his concept conflict with Dirac Equation--I think these are the interesting questions being raised by Hotson--correct ?

I did a search of both documents and found no place where Hotson uses the term "unobservable" as a type of fudge factor. This seems to be a comment related to his use of the term "undetectable" discussed above--but again--saying that the Dirac Equation predicts 6 dimensions that are "imaginary" in superposition with 4 that are "real" does not I think mean the "imaginary" does not exist--correct ? I cannot find that Hotson would not at least view the outcomes of the imaginary spin dimensions to be indirectly observable ?In addition to that, Hotson argues for a set of unobservable spin dimensions. But these can be perceived as no less imaginary than any other fudge factor used in the standard model or big bang theory (e.g., dark energy, dark matter). The irony is compounded by Hotson's view that anything we postulate to exist must, in principle, be observable directly (or indirectly, perhaps).

Thanks CC for your interest.

### #4

Posted 29 May 2010 - 05:38 AM

That is indeed a very informative and thought provoking article.

Hotson's contention that the Dirac equation was never properly interpreted

really does seem to make a lot more sense than todays "standard model".

On to part 2.

Don.

### #5

Posted 30 May 2010 - 12:08 AM

His understanding of Dirac's electron sea/hole theory is very poor- the point of Dirac's theory is that positrons are empty holes in the sea of electrons (like the hole's in a semi-conductor). When an electron meets a positron, it is meeting a lower energy state that is unoccupied, and so it falls in, emitting a photon. This is exactly like creation/annihilation (the formalism is the identical). An identical formalism can give you no different quantitative result, and the sea of electrons cannot be extended to bosons. For bosons, there is no pauli exclusion, so everything would fall into the lowest state- except in this case, there isn't one, the states go all the way down to negative infinity.

Further, his understanding of classical physics is also nonsense- changing Franklin's sign convention on the electron will not change what counts as positive and negative energy, etc. I haven't read the second paper, I can only assume its more of the first.

### #6

Posted 31 May 2010 - 12:03 AM

Thanks for your comments, but I think some are too harsh.....the sea of electrons cannot be extended to bosons. For bosons, there is no pauli exclusion, so everything would fall into the lowest state- except in this case, there isn't one, the states go all the way down to negative infinity.

Your comment above is correct as understood by Dirac when he proposed his "hole theory". But, Hotson has a section in Part 1 document concerning how a different interpretation of the negative energy sea in terms of the Bose-Einstein Condensate (BEC) would allow for the Dirac sea to be extended to bosons. Here are some quotes from Hotson in the Part 1 document:

*"What becomes clear from all this is that the negative energy sea of bosons (epos) called for by the equations must exist in the form of a BEC. According to the equations and everything we know, our reality is surrounded by and immersed in a vast, all pervasive Bose-Einstein Condensate."*

*"Bell’s proof and the experimental facts of electromagnetism and gravitation require a non-local reality. Dirac’s equation, in requiring a universal BEC, provides just that."*

*"Of course, the BEC wasn’t well described in 1934, so it is no mystery why Dirac didn’t see that this is what his equation calls for. Only in the light of more recent findings is it evident that Dirac’s “sea” must be a BEC. For, of course, it fills the crucial needs of Dirac’s sea—it is “full,” so that no positive energy particle can “fall in” unless it first loses all its positive energy, and then only if a balancing antiparticle similarly divests itself. Further, it has no “mass,” hence no inertia or gravitational interaction, so it is virtually undetectable."*

Also, physicists have made attempts to apply the Dirac sea concept directly to show how it does apply to bosons--here is one effort I found:

http://arxiv.org/PS_...8/9808108v1.pdf

It would help if someone that discussed this issue with Dirac could provide some input to this issue. My point is that Dirac did not pass away that long ago, and perhaps someone reading this forum may have discussed with him this issue about bosons and Dirac sea ?

You miss the point he was trying to make. Hotson does not say that Franklin's sign choice to call his electron e- would change energy dynamics. He is only making a statement that, if Franklin had given his electron the e+ sign, that action may have changed human perception and opinion of the Dirac sea. That is, if e+ is the Franklin electron sign then the Dirac sea would have the perception of "positive energy", and a "positive" as a perception of some thing is often more easy for the human mind to accept as a truth statement than a "negative".Further, his understanding of classical physics is also nonsense- changing Franklin's sign convention on the electron will not change what counts as positive and negative energy, etc.

### #7

Posted 31 May 2010 - 07:24 AM

So what I get out of this is the interesting idea of taking seriously the Dirac equation and its implications with respect to negative energy,and certain points about his view on bosons and the Bose-Einstein condensate.

One question, wouldn't this model be one in wich negative and positive energy cancel each other out leaving a vanishing net energy density of the universe=0?

### #8

Posted 31 May 2010 - 08:24 AM

The Dirac sea was the first idea for interpreting the solutions of the equation, a quite brilliant idea too which mathematically works perfectly. It is not at all disproved by the M&M experiment, in fact it came later, in the context of SR and the nexus with the aether is quite arbitrary; however it bears an obvious ontological snag and a bit less obviously leads to a conundrum that I once pointed out somewhere on these boards.

The Dirac sea is not at all necessary, in RQFT it is replaced by PCT, Spin and Statistics, and All That.

### #9

Posted 31 May 2010 - 10:14 AM

CC

### #10

Posted 31 May 2010 - 11:38 AM

Your comment above is correct as understood by Dirac when he proposed his "hole theory". But, Hotson has a section in Part 1 document concerning how a different interpretation of the negative energy sea in terms of the Bose-Einstein Condensate (BEC) would allow for the Dirac sea to be extended to bosons.

No, he doesn't. The fundamental difficulty is this- electrons stack up (i.e. pauli exclusion principal prevents two electrons from being in the same state). So if we fill all the negative states, no other electron can enter enter a negative energy state.

Bosons do not have such an exclusion principal. Even if there is a boson in every negative state, a boson can lower its energy and join them. Further, bosons can bose condense, which means that they all start to fall into the lowest energy state possible. Unfortunately, in the "dirac sea' there is no lowest state, so the bosons continue dropping energy forever, releasing an infinite amount of energy. Hotson does not discuss this problem, and has no resolution to it.

My point is that Dirac did not pass away that long ago, and perhaps someone reading this forum may have discussed with him this issue about bosons and Dirac sea ?

Dirac moved away from the electron/hole model when it became clear that other interpretations allowed more progress (such as QED).

You miss the point he was trying to make. Hotson does not say that Franklin's sign choice to call his electron e- would change energy dynamics. He is only making a statement that, if Franklin had given his electron the e+ sign, that action may have changed human perception and opinion of the Dirac sea. That is, if e+ is the Franklin electron sign then the Dirac sea would have the perception of "positive energy", and a "positive" as a perception of some thing is often more easy for the human mind to accept as a truth statement than a "negative".

And my point is that flipping the signs on the charges will NOT flip the signs of the dirac sea, it wll not flip the signs of binding energies, etc. Binding energies are relative energies- they are negative if you choose the convention that two unbound particles have 0 energy.

Obviously bound particles have less energy then free particles, so the energy goes negative. HOWEVER, this is not absolute energy (the energy measured by relativity, and hence the energy in Dirac's equation). Now, two particles far apart have an energy 2mc^2, and two bound particles have slightly less, but that energy is still positive.

This has nothing to do with the sign of the charge. The problem of interpretation is with a negative absolute energy.

### #11

Posted 31 May 2010 - 11:08 PM

Also, while I agree with you that Dirac hole theory initially was not applied to bosons---such is not the case at this time. There are published attempts to identify the ground state for bosons within a Dirac sea hole theory, so they do not need to drop forever releasing infinite energy. So, while Hotson has not discussed this--others have. I provided one paper in my last post--here is another:

68-81

===

Here are some personal comments I have and I welcome any comments, pro or con.

I posted this thread because I think Hotson does a great service to raise awareness of the reality of both positive and negative energy present in quantum entities.

Hotson makes a valid point that it is the current thinking by some (many ?) physicists that a Dirac sea with negative energy does not exist. Clearly this is false. Heisenberg and others were incorrect to hand wave away the negative energy solution of the Dirac Equation and Einstein Equation E = + - mc^2. How many physics textbooks discuss this ? How many students realize that both a (+) and (-) solution to the Einstein Equation exists and that one solution refers to matter and the other to antimatter ? Antimatter is an important topic today, much more so than at time of Heisenberg. Even string theory predicts that a portion of any string mass must be from a minimum amount of vibration as allowed by quantum uncertainty--the zero-point energy....but....the contribution of the zero-point energy to the mass of the string is a negative energy ! Negative energy is predicted by all known ways to explain quantum entities.

I find the issue of negative energy as it relates to antimatter to be especially important now that experiment has documented many more types of antimatter than the positive electron (e + positron). We now have the anti-proton, anti-helium 3, anti-hydrogen 3, anti-deuterium--all have been created at various labs. The first three are fermions, and anti-deuterium is a boson.

I find that in the same way the Dirac Sea relates to the interaction of the e- and e+ to help explain spin and magnetic moment, the Dirac Equation could be extended to interactions between proton and anti-proton (two fermions), helium 3 and anti-helium 3 (two fermions), hydrogen 3 and anti-hydrogen 3 (two fermions), deuterium and anti-deuterium (two bosons). Consider also how the concept of a Dirac Sea of negative energy could be applied to possible interactions of mass asymmetrical fermions and anti-bosons (or anti-fermions and matter bosons) now that we know they exist, such as the interaction of helium 3 (fermion)+ anti-deuterium (boson) ! What does current Standard Model say about this possible interaction ?--clearly it cannot be a complete annihilation because the two particles are not mirror opposite. How would this reaction relate to Dirac equation if the interaction was between unified quark bags, and not at the level of individual quarks ?

### #12

Posted 01 June 2010 - 01:31 AM

His factual statements about solutions to the dirac equation are false to start (I contend Hotson does not understand the Dirac equation), which makes his conjectures most likely false.

[quote]There are published attempts to identify the ground state for bosons within a Dirac sea hole theory, so they do not need to drop forever releasing infinite energy. So, while Hotson has not discussed this--others have. I provided one paper in my last post--here is another:

68-81 [/quote]

Have you read these papers? They make it clear their correspondence to a hole theory is entirely formal, the vacuum require putting -1 particle in each negative energy state. Surely you don't think adding a negative particle number is anything more than a formal operation?

[quote]Hotson makes a valid point that it is the current thinking by some (many ?) physicists that a Dirac sea with negative energy does not exist. Clearly this is false. Heisenberg and others were incorrect to hand wave away the negative energy solution of the Dirac Equation and Einstein Equation E = + - mc^2. How many physics textbooks discuss this ?[/quote]

Every textbook that discusses the Dirac equation has a discussion of the negative energies, and a discussion of a quantization method that avoids the need for negative energies and is easily capable of extending to bosons. The hole theory does not extend easily, and so is used as an intuitive starting point, but eventually dropped. Also, extensions that avoid the sea of negative particles can easily be shown to be entirely equivalent to those that don't- i.e. removing the negative sea doesn't change any numberss but makes the the theory much easier to interpret.

[qupte]How many students realize that both a (+) and (-) solution to the Einstein Equation exists and that one solution refers to matter and the other to antimatter ? [/quote]

Any reasonable student of the subject, much attention is given to this idea in the context of the Dirac equation.

[quote]the Dirac Equation could be extended to interactions between proton and anti-proton (two fermions), helium 3 and anti-helium 3 (two fermions), hydrogen 3 and anti-hydrogen 3 (two fermions), deuterium and anti-deuterium (two bosons). [/quote]

This extension is tricky- the dirac equation requires a g factor of exactly 2. Nucleons are composite entities and so have g factors quite different than 2. You can extend the equation, but it becomes much more approximate. Also, bosons will never follow a dirac equation (which has spin built in), but instead a Klein-Gordon equation.

### #13

Posted 01 June 2010 - 04:57 AM

What's even cooler is that the topic can be found in many books that treat particle physics much more correctly than that author's paper. I just chose the one that has the most relevant matter in its very title. I'm not expecting everybody around here to rush out and buy it, 'specially as it ain't the most introductive one.It's kind of cool, that for a meager price of $31.50 ($3.50 or 10% off) you can save yourself the trouble of having to write- and others the trouble of having to sift through, two big PDF files.

That author is trying to dig a hole in water.

As has been said, it should be regarded as mathematical rather than physical. Quantum Field Theory is the way to go; the energy symmetry of the equations translates into time reversal symmetry and there is no boson-antiboson distinction.Also, while I agree with you that Dirac hole theory initially was not applied to bosons---such is not the case at this time. There are published attempts to identify the ground state for bosons within a Dirac sea hole theory, so they do not need to drop forever releasing infinite energy. So, while Hotson has not discussed this--others have.

Sometimes you also say things that surprise me. Magnetic moment doesn't come into the Dirac equation, so how does it require the exact g-factor value? In my studies, the Dirac equation was good for all free fermions and Klein-Gordon for all free bosons. They are used as the free terms in the complete Lagrangian for any process where electric charge, and hance also magnetic moment, come in with the EM coupling term.This extension is tricky- the dirac equation requires a g factor of exactly 2. Nucleons are composite entities and so have g factors quite different than 2. You can extend the equation, but it becomes much more approximate.

### #14

Posted 01 June 2010 - 09:25 AM

Thanks for all of your useful comments. I understand that the Dirac negative energy hole concept can be "quantized away" using other quantum models. But, if we are to take the position that quantum field theory is the way to go, then, is it not true that we must take seriously those that publish new ways to bring the Dirac Sea negative energy view into quantum field theory--such as this 2007 publication:Quantum Field Theory is the way to go; the energy symmetry of the equations translates into time reversal symmetry and there is no boson-antiboson distinction.

http://arxiv.org/abs...um field theory

Here, the authors bring in the pilot-wave model of quantum theory of David Bohm and relate it to Dirac Equation (negative energy) and quantum field theory. It is only for fermions, but at the end of the paper they discuss how their approach could be extended to bosons.

This is a topic Hotson discusses--taking the Dirac equation to mean (in reality) that negative energy is present in all quantum entities and that this negative energy allows Bohm's interpretation of quantum theory concerning pilot-waves (with "information" moving faster than speed of light) to be taken seriously. I does appear that when the Dirac equation is taken to mean that negative energy is a useful concept, then the Bohm pilot wave interpretation of the Bell experiment is seen in a new light (not that it is made any more valid, only that the explanation is clarified). Does anyone else see this connection ?

For me, rather then mud the waters, I find that this "interpretation" of quantum theory (presence of negative energy via E = + - mc^2, combined with negative energy holes of Dirac Sea, combined with "information" from General Relativity {mass & energy + spacetime} linked via pilot-waves moving > c) lets us ask questions about interactions between different types of matter and antimatter fermions and bosons in new ways (see my previous post).

==

I appreciate the interest by all that have posted to this thread. It does appear that Hotson receives a much more negative review than positive up to this point in the discussion. Are there any that see any value to what Hotson claims ?-- and please give specifics.

==

Edit: Well, this was fast--it appears one answer to my question is available--plus a response by Hotson--his Part III. So, I will also add Hotson Part III to the OP:

Hotson Part III (with introduction):

http://blog.hasslber.../HotsonIE86.pdf

==

Another review of Hotson that has some positive comments:

http://www.dpedtech.com/HotsonRev.pdf

### #15

Posted 01 June 2010 - 02:58 PM

Sometimes you also say things that surprise me. Magnetic moment doesn't come into the Dirac equation, so how does it require the exact g-factor value? In my studies, the Dirac equation was good for all free fermions and Klein-Gordon for all free bosons. They are used as the free terms in the complete Lagrangian for any process where electric charge, and hance also magnetic moment, come in with the EM coupling term.

Generally, the Dirac equation refers to

[math] i\gamma^\mu (\partial_\mu -i q A_\mu) \psi -m\psi = 0 [/math]

With a free particle being the special case where the potential is 0. This is the analog of the Schroedinger equation, and why you often hear that the Dirac equation is exactly solvable in the case of hydrogen. Anyway, this is the way to couple a dirac particle to an EM field. This structure predicts that for any spin 1/2 particle g is exactly 2.

Hence, extending the dirac equation to a composite charged object like a proton (with g not equal to 2) is at best an approximation.

### #16

Posted 03 June 2010 - 07:03 AM

Goodness! That explains a lot! I hadn't read those .pdf docs as far as references to the Bohm pilot wave, which runs into its own difficulies anyway.Here, the authors bring in the pilot-wave model of quantum theory of David Bohm and relate it to Dirac Equation (negative energy) and quantum field theory.

Well, if you mean the Dirac equation that way, for a fixed EM field and no other interaction terms, then I see what you meant. I do however recall it as being, more conventionally, meant as in my avatar with no "special status" for that interaction term; what you write strikes me a specific alteration for a specific purpose. I'll try to remember and have a look in my battered old copy of Itzykson-Zuber when I get back home.Generally, the Dirac equation refers to

[math] i\gamma^\mu (\partial_\mu -i q A_\mu) \psi -m\psi = 0 [/math]

With a free particle being the special case where the potential is 0.

After posting, I had thought a bit further and supposed you might have meant simply that it's easier to treat a process involving hadrons by considering them as single fermions but this requires some tweaking for the g factor. This certainly makes more sense than replacing every hadron with its bunch of constituents and, yes, I'd perfectly agree with it.

### #17

Posted 15 June 2017 - 06:31 AM

It has been shown by the LIGO that the gravitational effects of a black hole merger millions or billions of light years away can decrease the earth's distance from the sun by the width of a hydrogen atom.

By that notion, the closest thing to negative energy is the evaporation of a black hole, which would increase earth's distance from the sun by the width of a hydrogen atom. This is the very mechanism for which we get the Casimir Effect.

**Edited by Super Polymath, 15 June 2017 - 06:32 AM.**