Constructive QFT - current status

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It's interesting that Tong listed quaternions as part of Hamilton's unhappiness. In his days they were a great success for him as a respected mathematician. It was considered even a kind of rebellion when some people started to use vectors, among the first Heaviside, Gibbs, and Helmholtz ;-).
It still always feels a bit paradoxical to recognize that quaternion algebra was invented before vector calculus and to fully realize how new linear algebra really is. Hamilton was really way ahead of his time.

I have a feeling that I'm in a small minority of physicists who have a strong aesthetic preference for quaternion algebra over vector calculus for non-classical physics, seeing how neatly it dovetails with the spinor calculus and the theory of differential forms, while the limitations of vector calculus are by now painfully obvious.
 
  • #52
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I'm in a small minority of physicists who have a strong aesthetic preference for quaternion algebra over vector calculus for non-classical physics, seeing how neatly it dovetails with the spinor calculus and the theory of differential forms, while the limitations of vector calculus are by now painfully obvious.
???

The even more painful limitations of quaternions would become obvious if you would discuss classical electrodynamics in terms of quaternions only.
 
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  • #53
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Notice how I said 'for non-classical physics'. Quaternion algebra is an area of study in pure mathematics which still has plenty of new gifts to give to mathematics and so form a breeding ground for new physical theories, as opposed to the by now sterile contribution of vector calculus.

I'd even make the case that the theory of vector calculus is in a sense the study of the properties of classical physics and its direct application outside of physics as a general form of applied mathematics (e.g. in economics) is an unwarranted and ultimately unjustifiable extension.
 
  • #54
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Notice how I said 'for non-classical physics'. Quaternion algebra is an area of study in pure mathematics which still has plenty of new gifts to give to mathematics and so form a breeding ground for new physical theories, as opposed to the by now sterile contribution of vector calculus.

I'd even make the case that the theory of vector calculus is in a sense the study of the properties of classical physics and its direct application outside of physics as a general form of applied mathematics (e.g. in economics) is an unwarranted and ultimately unjustifiable extension.
Nobody is doing quantum physics with quaternions.

The spinor calculus is far prefermenu://applications/Administration/lxterminal.desktop
able to quaternions, but only for fermions. Doing spin 1 quantum fields is awkward with spinors, let alone with quaternions.
 
  • #55
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Nobody is doing quantum physics with quaternions.
There are research programmes in mathematical physics, theoretical physics, applied mathematics, pure mathematics and interdisciplinary efforts of all of the above which all try to extend or go beyond QT and orthodox mathematical theory using the insights gained from quaternion algebra; modern group theory itself is indirectly such a product developed in the 20th century.

The role that quaternion algebra plays in (non-commutative) local class field theory is evidenced by it being an arithmetic instance of the local Langlands correspondence; if for example a novel physical theory based upon this isn't an example of world class mathematical physics, then what is? (rhetorical question)
The spinor calculus is far preferable to quaternions, but only for fermions. Doing spin 1 quantum fields is awkward with spinors, let alone with quaternions.
This just means that (most) researchers so far haven't been creative enough to actually discover the pattern which removes the awkwardness. This actually implies that they do not know enough existing mathematics not part of their curriculum which doesn't seem to have any relation to their field, but in actuality is very intimately connected yet still merely unrecognized by contemporary practitioners of being so. Recognizing these connections would illuminate the pattern and remove all the awkwardness.

You seem in this argument to have a strong focus on the direct applicability of mathematics in practice - i.e. applied (theoretical) physics - up to the actual exclusion of untread unconventional mathematical pathways for the discovery of actually novel physical theories. Adopting such an applied strategy directly flies in the face of the main goal of theoretical physics, which is of course instead going beyond the current theory by directly acknowledging its limitations. To quote Sheldon Glashow: In fact, QFT is just wrong! Quantum mechanics is all-encompassing. A correct theory must include quantum gravity and QFT is not up to the task.
 
  • #56
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This just means that (most) researchers so far haven't been creative enough to actually discover [...]
You seem in this argument to have a strong focus on the direct applicability of mathematics in practice
I have a strong focus on what has been discovered, rather than on speculations what one might (or more likely might not) discover.

I don't see any evidence why quaternions should ever be more relevant in physics than they are now - namely for doing certain computations in SO(3) and SU(2), and for classifying certain families of simple Lie groups.
if for example a novel physical theory based upon this isn't an example of world class mathematical physics, then what is?
It is pure speculation that such a theory would be useful.

World class mathematical physics would be the rigorous construction of QED or QCD. This needs tools quite different from what you speculate about.
 
  • #57
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I have a strong focus on what has been discovered, rather than on speculations what one might (or more likely might not) discover.
Exactly. It is like someone outside in the dark who has lost their keys and is searching for them, but then deliberately chooses to only search in the small area illuminated by a street light purely because it is the only place they are able to see anything at all. New mathematics isn't discovered by those cautious researchers who dare not tread into the dark - or worse, are so afraid that they actively prevent other researchers from doing so as well.

Instead, a level of fearlessness and a certain amount of boldness is required to venture out directly into the dark and commit illegal mathematical procedures from the point of view of contemporary mathematics. This is of course historically exactly what mathematical outlaws like Euler and Gauss did (e.g. with complex numbers and non-Euclidean geometry) and so ended up inventing the core theory underlying most of modern mathematics. It is impossible to say a priori if such an endeavor will be successful.

Almost any form of naively employed likelihood analysis in the case of the success of such situations is premature due to the unstated presumptions made about the choices that such researchers make in their discovery strategies, purely in order to carry out a successful likelihood analysis; in fact, the process of unforeseen mathematical discovery is a Pareto-distributed phenomenon and thereby already a priori defies almost all commonly employed conventional statistical tools to treat likeliness.
I don't see any evidence why quaternions should ever be more relevant in physics than they are now - namely for doing certain computations in SO(3) and SU(2), and for classifying certain families of simple Lie groups.
The use comes from them being natural mathematical objects - e.g. such as complex numbers, polynomials and polyhedra are natural mathematical objects. Looking at the history and philosophy of mathematics and physics, it has been shown time and time again that natural objects are prime candidates used to marry different seperate fields in mathematics; the most well-known example of course is Descartes naturally unifying geometry and algebra and so inventing analytic geometry, eventually leading to analysis.

Not recognizing this vitally important aspect of natural objects in the relationship between mathematics and physics is actually a failure of academia and education. Suffice to say, unification based on natural objects was for much of history the goal of the main school of pure mathematics in the pre-Bourbakian era, which sadly died in the divorce between mathematics and physics; I am merely someone who in the modern era still follows that school of thought.
It is pure speculation that such a theory would be useful.
It is equally pure speculation that it wouldn't be useful. Moreover, 'usefulness' is a very contigent concept which means many different things to different people: with theories this applies as well, just because some mathematical theory isn't directly useful in some practitioners opinion doesn't mean it isn't useful at all.

By that logic the theory of differential forms isn't useful either because vector calculus does largely the same thing and has the added benefit of being far more widely known. Funny enough I have actually met physicists - usually working in applied physics or experimental physics - who have made exactly such claims.

In any case, finding the correct use for some form of mathematics is a matter of creativity and trial and error, not a matter of conforming to strict rationality or routine logic as the more mundane applications of mathematics as are.
World class mathematical physics would be the rigorous construction of QED or QCD. This needs tools quite different from what you speculate about.
It is funny that you would say so as what I am speculating about is actually part of a practically abandoned research direction in constructive QFT; the reason for abandoning this direction wasn't lack of results but instead lack of funding due to explosively dominant competing programs (read: string theory). The researchers moved onto applied mathematics, made their mark there and the original connection of their work with physics was forgotten. This just shows how much of a fad-ridden endeavor contemporary theoretical physics has become.
 
  • #58
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There are research programmes in mathematical physics, theoretical physics, applied mathematics, pure mathematics and interdisciplinary efforts of all of the above which all try to extend or go beyond QT and orthodox mathematical theory using the insights gained from quaternion algebra; modern group theory itself is indirectly such a product developed in the 20th century.

The role that quaternion algebra plays in (non-commutative) local class field theory is evidenced by it being an arithmetic instance of the local Langlands correspondence; if for example a novel physical theory based upon this isn't an example of world class mathematical physics, then what is?
Do you have any references to actual papers that use quaternions in the ways you describe? If so, please provide them. If not, your claims are personal speculation and are off topic here, and if you continue to make them without references, you will receive a warning.
 
  • #59
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Instead, a level of fearlessness and a certain amount of boldness is required to venture out directly into the dark
This is futile without knowing where to go; chances that one finds new physics this way are virtually zero. Discovering something new is quite different from searching for something lost - since we do not even know what to search for!
This is of course historically exactly what mathematical outlaws like Euler and Gauss did (e.g. with complex numbers and non-Euclidean geometry)
On the contrary, they searched for more general structure apparent in what they knew already - the standard way of expanding knowledge. And they discovered math, not physics. The applications to physics came much later.
finding the correct use for some form of mathematics is a matter of creativity and trial and error
No, it is forced upon one by the structure of the problems in physics to be solved.
what I am speculating about is actually part of a practically abandoned research direction in constructive QFT
Constructive QFT has had so far no use at all for quaternions. You could as well speculate that p-adic numbers are useful for constructive QFT.
 
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Thread closed for moderation.
 
  • #61
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The thread has run its course and will remain closed.
 

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