Missed Opportunities

[LaurieAG]

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

 

https://pdfs.semanticscholar.org/682a/6a48f857e9a19ef0448c00f14ae8089d908f.pdf

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.

 

https://arxiv.org/abs/physics/9807044v2

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.

 

https://en.wikipedia.org/wiki/Emmy_Noether

 

Before I continue, does anybody else understand what the mathematical differences between Improper Integrals, Proper 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.wisc.edu/~keisler/keislercalc-03-07-17.pdf

 

[LaurieAG]

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. 

 

https://en.wikipedia.org/wiki/Arthur_Compton

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.

 

https://en.wikipedia.org/wiki/Compton_wavelength

 

https://en.wikipedia.org/wiki/Compton_wavelength#Reduced_Compton_wavelength

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. 

 

https://en.wikipedia.org/wiki/Compton_wavelength#Relationship_between_the_reduced_and_non-reduced_Compton_wavelength

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.

 

 

 

[LaurieAG]

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: 

 

https://arxiv.org/pdf/1601.00921.pdf

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 2π radians for the 360 degrees within a circle. While 1 Radian for 360 degrees might be convenient for things like dimensional analysis, where 2π 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 2π 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. 

[LaurieAG]

Here is my take on the mathematical differences between Improper Integrals, Proper 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.wisc.edu/~keisler/keislercalc-03-07-17.pdf