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Missed Opportunities

Freeman Dyson Emmy Nether Nina Byers Improper Power Integrals Improper Integrals Proper Integrals Indefinite Integrals

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#1 LaurieAG



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Posted 26 April 2018 - 03:39 AM

Freeman Dyson delivered a speech in 1972 called "Missed Opportunities" for the Gibbs Lectures. 

The purpose of the Gibbs lectures is officially defined as "to enable the public and the academic community to become aware of the contribution that mathematics is making to present-day thinking and to modern civilization." This puts me in a difficult position. I happen to be a physicist who started life as a mathematician. As a working physicist, I am acutely aware of the fact that the marriage between mathematics and physics, which was so enormously fruitful in past centuries, has recently ended in divorce. I shall examine in detail some examples of missed opportunities, occasions on which mathematicians and physicists lost chances of making discoveries by neglecting to talk to each other.

I'm surprised Freeman Dyson didn't mention Emmy Noether, a German mathematician, and her contribution to the conceptual structures of the mathematics in modern physics in his paper. Nina Byers goes into this in detail in her paper "E. Noether’s Discovery of the Deep Connection Between Symmetries and Conservation Laws" in 1998.

Though the general theory of relativity was completed in 1915, there remained unresolved problems. In particular, the principle of local energy conservation was a vexing issue. In the general theory, energy is not conserved locally as it is in classical field theories - Newtonian gravity, electromagnetism, hydrodynamics, etc.. Energy conservation in the general theory has been perplexing many people for decades. In the early days, Hilbert wrote about this problem as ‘the failure of the energy theorem’. In a correspondence with Klein [3], he asserted that this ‘failure’ is a characteristic feature of the general theory, and that instead of ‘proper energy theorems’ one had ‘improper energy theorems’ in such a theory. This

conjecture was clarified, quantified and proved correct by Emmy Noether.
Before I continue, does anybody else understand what the mathematical differences between Improper IntegralsProper Integrals and also Indefinite Integrals are in Calculus? This is what this thread is about so please give your examples from the link below.
Reference H.J.Keisler "Elementary Calculus an Infinitessimal Approach" PDF link: https://www.math.wis...lc-03-07-17.pdf


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#2 Dubbelosix



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Posted 27 April 2018 - 02:14 PM

Calculus is one of those things if you are not doing it often you tend to forget its complexities.


As for the matter of energy conservation, it appears that energy conservation only holds in a local sense. What the article should say which it appears to have terminology wrong, is that energy is not conserved globally. If an early universe is sufficiently small enough then the dynamics are indeed localized. In this sense we would seek the Raychauduri equation to explain the strong local gravitational features - likewise, the presence of a curvature in the early universe may ensure that particle creation happens in an irreversible way which is actually another example of non-conserved energy processes which tend to be described by thermodynamic properties.


The problem with the idea that general relativity makes it that global energy is not always conserved, is a matter of consequence of the Wheeler de Witt equation in which the right hand side has a vanishing time derivative - the interpretation has been since that time falls out of the theory of general relativity as we come to understand it and the evolution is described by diffeomorphism invariances. However, taking that this interpretation is correct and we should do away with a quantum interpretation of time in the global parameters of our physics, we come along to find additional problems trying to describe the Friedmann equation since it is a general relativistic solution of the field equations ''that appear'' to be explicitly time-dependent (ie. the scale factor is time dependent and the scale factor is radius dependent).


Another way to view this, is that the universe is not fundamentally a classical system and even when it does ''get large'' there is still loads of quantum mechanical processes going on. One such feature we have come to accept is that the universe is getting larger, which means space is expanding - Sean Carrol has noted that the metric in such a way must change according to it's bulk energy. In a similar fashion, I have argued that the appearance of new space and time must mean additional energies, so the problem comes down to, why do we measure vacuum energy something of 10^120 magnitudes too small?


Well, what I have come to learn is that the vacuum energy hypothesis has been treated [incorrectly] - since the energy we measure in the observable horizon corresponds to on-shell particles. Zero point energy consists of what is called ''off-shell'' particles and these ''virtual particles'' do not obey the usual rules of energy and momentum. In fact, it is considered by many that off-shell particles corresponds to ''unobservable dynamics'' happening at scales which are far to small in both area and in time to be measured which renders them different to the usual matter and energy we encounter on a daily basis. In fact, these fluctuations are an example of many kinds of relationships of physics and relativity that are deeply related to each other. These ground state fields exist even when you remove all the visible matter and energy in a vacuum! This is what remains when you think there is nothing left! What is the residual energy, but other than the thing predicted by quantum mechanics itself, that space is not truly empty and never is truly empty since there is always a residual motion associated to the ground state of the field. Relativity calls it ''relative motion,'' that is, all things are in motion with everything else. In quantum mecanics, it is called the uncertainty principle, that is, there is no such thing as a true rest system and the thing we call rest is but an approximation - there is no rest system because as you will know, locations in space are subject to a complimentary law related to the momentum of the system. The same quantum motion is the motion attributed to the ground state. And as we learn from classical mechanics, that motion will generate a temperature - you could argue that open space shows us what this general temperature is as it is spread over space - since the vacuum is never at absolute zero, then the universe does posses a temperature, even in the so-called 'voids of space'' where hardly any visible matter or energy can be found.


I'd argue though, this is not a failure of relativity as is claimed in the article, but instead, a failure of our understanding of our physics. The Wheeler de Witt equation falls out of physics from a path quantization on the general field equations. The problem with doing this is that gravity is not even a real field from the prospect of relativity and so direct quantization will no doubt lead to some essential problems, some of those include:


1) UV Divergence, ie. Singularities

2) The Problem of Time

3) Conflicting theories


Certainly, it has been said for a very long time now that relativity and quantum mechanics are ''conflicting theories.'' This doesn't appear to be the case at all, only that our ''interpretations'' of the physics is conflicting, even right down to the discrete concept found in quantum mechanics compared with the continuous vacuum in relativity ~ recently it has been argued that something can be both discrete and continuous! So yeah, sorry for posting such a long post, I just love the sujbect of cosmic energy.

#3 LaurieAG



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Posted 29 April 2018 - 11:52 PM

Thanks Dubbelosix, you have summarised some of the issues that have arisen due to divergences between the early proponents of QM and the more modern interpretations that seem to disavow wave particle duality. I will go into the actual calculus later and try to keep this thread relatively mathematics free for the present. 

While Emmy Noether provided the conceptual symmetries of relativity and proved Hilbert's conjecture correct Arthur Compton provided the final piece of the relativity puzzle by experimentally and theoretically identifying the wave and particle natures of electromagnetic particles between 1922-23. 

Arthur Holly Compton (September 10, 1892 – March 15, 1962) was an American physicist who won the Nobel Prize in Physics in 1927 for his 1923 discovery of the Compton effect, which demonstrated the particle nature of electromagnetic radiation. It was a sensational discovery at the time: the wave nature of light had been well-demonstrated, but the idea that light had both wave and particle properties was not easily accepted.



When the Compton wavelength is divided by 2π, one obtains the "reduced" Compton wavelength ƛ (barred lambda), i.e. the Compton wavelength for 1 radian instead of 2π radians:

The Compton wavelength λ (conversion of mass into energy, or to the wavelengths of photons interacting with mass) and the reduced Compton wavelength ƛ (barred lambda, a natural representation for mass on the quantum scale) are both named after him as a result and the relationship between the two form the underlying difference between classical/modern physics and later variants such as QM. 

The reduced Compton wavelength is a natural representation for mass on the quantum scale. Equations that pertain to inertial mass like Klein-Gordon and Schrödinger's, use the reduced Compton wavelength. The non-reduced Compton wavelength is a natural representation for mass that has been converted into energy. Equations that pertain to the conversion of mass into energy, or to the wavelengths of photons interacting with mass, use the non-reduced Compton wavelength.

A particle of mass m has a rest energy of E = mc2. The non-reduced Compton wavelength for this particle is the wavelength of a photon of the same energy. For photons of frequency f, energy is given by

                     E = h/f = hc/λ = mc2.

which yields the non-reduced or standard Compton wavelength formula if solved for λ.



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#4 LaurieAG



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Posted 02 May 2018 - 02:59 AM

While Emmy Noether and Arthur Compton wrapped up many of the loose ends with regards to relativity things seemed to diverge a little after them and, although mankind was yet to create atomic weapons or get to the moon and back safely, applied advances progressed at a relatively smooth rate until Freeman Dyson wrote his paper about 'Missed Opportunities' in 1972. 
If we jump forward to the late great Stephen Hawking et al's 2016 paper "Soft Hair on Black Holes" we can get several insights into where and how the divergence occurred. Firstly, on page 2 footnote 1, the authors note: 

This is related to the lack of a canonical definition of angular momentum in general relativity.

Why would GR and its derivatives not have a canonical definition of angular momentum? Is there some issue with the work Arthur Compton did that means there is some measure of doubt with regards to his Nobel prize or is there some other reason that has arisen since then? 
The answer to this seemingly only post 1923 conundrum lies in a basic examination of SI Units (Systeme Internationale in French, the International System of Units for everybody else apart from the US, Liberia and Burma) particularly the Radian. The SI derived unit for the Radian is a measure of angles that has  radians for the 360 degrees within a circle. While 1 Radian for 360 degrees might be convenient for things like dimensional analysis, where  often cancels out, it is not exactly good for technical measurements because it is not the SI derived Radian definition used by the International Scientific community.
Secondly, at the bottom of page 13 of the "Soft Hair on Black Holes" paper we read:

However, no particle can be localised in a region smaller than either its Compton wavelength [math]\frac {\hbar}{M}[/math] ...

While it may also be a convention among theoretical physicists and cosmologists to only consider that they are working with the reduced Compton wavelength or 1 Radian per 360 degrees with a unitary c, and call it what they like, it is not surprising that agreement cannot be reached over a canonical definition of angular momentum in GR. The main problem with using 1 Radian instead of 2π Radians is that you effectively need to multiply your standard Compton wavelength by  to keep the relationship between the reduced and standard forms of the Compton wavelength consistent in GR or any other variant including QM.
This exact problem rears its vexing and ugly head in what is called dark matter under the ΛCDM (Lambda cold dark matter) model, the substance whose constituents can not be identified in our present physical system but, ask almost any theoretical physicist or cosmologist, it is supposedly at the core of our ever expanding universe. If they went back to basics they might find that if they multiply the Ordinary matter component of ΛCDM by 2π they will get the Total universal matter component of ΛCDM, within their own error bars.
But it is all too easy for those of blind faith to cry 'numerology' in unison and perpetuate their lucrative quest for something that doesn't exist! Searching for this type of perpetual philosophers stone should not be funded by any public purse. If the theoretical sciences of physics and cosmology wish to miss further opportunities they can continue to retain solidarity with their peers in Liberia and Burma or they could go back to the basics and review everything they have produced since 1923, clean things up, take out the trash and add the required caveats, while rebuilding their cloisters on solid foundations based on proper scrutiny. 
If they did so they might just finish in time for the centenary of Arthur Compton's discovery of the Compton effect, which demonstrated the particle nature of electromagnetic radiation. 

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#5 LaurieAG



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Posted 09 February 2019 - 12:48 AM

Here is my take on the mathematical differences between Improper IntegralsProper Integrals and also Indefinite Integrals in Calculus.


At a conceptual structural level improper integrals in physics can be piecewise continuous integrals, with limits from +infinity to -infinity, that converge. Refer H.J. Keisler, p367, Definition to p369, examples 7, 8, and 9. If they are continuous and don't converge then they are indefinite integrals which are entirely different. Refer H.J. Keisler, p370, example 10, diagram 6.7.10 "It is tempting to argue that the positive area to the right of the origin and the negative area to the left exactly cancel each other out so that the improper integral is zero. But this leads to a paradox... So we do not give the integral ... the value 0, instead leave it undefined."

That doesn't mean that indefinite integrals don't play a part in our physics as an indefinite integral that cycles between +infinity and -infinity at its limits, as a sub function of a higher level function, is a valid proper use of indefinite integrals as definite integrals by change of variables. Refer H.J. Keisler, p224-5, Definition and example 8, diagram 4.4.6 second equation with u and substitute infinite limits. "We do not know how to find the indefinite integrals in this example. Nevertheless the answer is 0 because on changing variables both limits of integration become the same."

A valid proper integral of any form is not equivalent to a valid improper integral because that is the underlying conceptual difference between classical and modern physics as discussed by Hilbert and Klein above.

Reference H.J.Keisler "Elementary Calculus an Infinitessimal Approach" PDF link: https://www.math.wis...lc-03-07-17.pdf