Graduate Constructive QFT - current status

[Auto-Didact]

Why does the wrong derivation work (I think I've read that it reproduces the right derivation term by term)?

It works heuristically, but not necessarily mathematically.

You just can't use that expansion method directly in the continuum. If you want to prove the relation directly in the continuum there are other methods but they are much more mathematically involved.

Being constructively inclined, I prefer those other methods, i.e. a non-perturbative analysis. I would even go as far to say that mathematically speaking a non-perturbative analysis is necessary in order to prove existence at all since perturbation theory is known to break down for many classes of problems, including many which are asymptotic, not convergent.

The failure of those doing the perturbative expansion is then essentially caused by them not realizing that they are expanding the power series by assuming ad hoc that the independent variable is fixed purely in order to be able to make an empirical comparison i.e. making a mathematically illegitimate assumption which is in a specific sense completely independent of experiment!

There is one side effect of the fact that they are disjoint that shows up when using the free theory to compute the terms. The need to renormalize the terms.

The perturbative series ends up being only asymptotic of course, not convergent. Though that happens in NRQM as well. In lower dimensions for some theories you can use the Borel transform to sum the series and thus existence of the interacting theory can be proved directly from perturbation theory.

In 4D but also for Yang Mills in lower dimensions there are poles in the Borel plane preventing resummation. The poles mean one has to take a contour around them to obtain the interacting theory, but there are infinite such contours introducing an ambiguity of order $\mathcal{O}\left(e^{-\frac{1}{\lambda}}\right)$.

The 'some theories' for which this can be proved require both linearity of the space of solutions as well as linearity of the equations; if one or both of these assumptions fail, then perturbation theory - beyond an initial small semi-accurate range of validity - will quickly fail once the independent variables aren't ad hoc assumed to be fixed anymore. In this sense, perturbation theory is obviously just a more sophisticated version of a heuristic technique such as the small angle approximation.

 

[Auto-Didact]

Another simple argument given in

A. Duncan, The conceptual framework of quantum field
theory, Oxford University Press, Oxford (2012).

uses a finite volume with periodic spatial boundary conditions and works in momentum space. Then the infinite-volume limit is taken at the very end for the transition rates.

"Regularizations" like this or "latticizing" the theory etc. physicists intuitively do in a naive way. It's of course good to know, that one can explain, why this finally works, (more) rigorously.

Using periodic boundary conditions, either before or after a Fourier transform, is an implicit importation of topological phase space analysis, which severely complicates the issue because it introduces new existence and uniqueness issues of the periods whose resolution requires the full arsenal of nondimensionalization, bifurcation theory and index theory.

As Duncan makes explicitly clear in his masterful book, there is good reason to be suspect of the ultimate validity of perturbative analysis, either as non-optimized perturbation theory in which case the regularization based on the Borel transform is generally quite fragile, or optimized perturbation theory which generally isn't useful in the context of field theory.

From applied mathematics, all of these issues are already well known - with physicists often merely introducing novel verbiage which unnecessarily complicate these purely mathematical issues and which tend to be ultimately unjustifiable (e.g. wanting to make a comparison with empirical phenomenology) - which is exactly why non-perturbative analysis was invented in the first place.

Even in physics this is old news; already during the 60s-70s, the recognition that perturbative arguments were unjustifiable lead to a split in the QFT community into field theorists and S-matricists, as Shankar describes during his time under S-matrix purist Geoff Chew at Berkeley. The immediate recognition of the extremely contingent nature of renormalization by constructive mathematicians (and constructively inclined physicists) directly led to the constructive QFT programme, which as we can still see is nowhere near completion.

 

[Jimster41]

Thanks, now it's intuitive to me too.
That's yet another demonstration that Tong's lectures are really great.

I’m enjoying the lecture in the link but it’s already confusing when he says “little things affect big things. Big things don’t affect little things”. Well, big things certainly affect big things don’t they? Else classical mechanics was useless. Obviously false. But there are no big things not made of little things. Granted. So big things affect big things but only through little things. So wouldn’t it have been better to say that the map between all things big and little is little. Or something like that. The real effect being that you have to allow for big things to affect little things as the big things interact with each other. Co-evolution of phenotypes via the genome is a pretty important example of how it happens at least in the evolution of one regime of the physical world. What precludes the possibility it is more common. To me it seems potentially relevant to the subsequent question of what causes such things as discontinuous phase changes in continuously evolving systems - like some back reaction, some big thing shaking the almost boiling pot.

Moreover, given how confusing that was I am distracted when he then goes off into the (somewhat familiar now) partition function statistics based description of the Ising model and its free energy. I am distracted by that because it already assumes a notion of entropy maximization, the natural Hamiltonian as it were, classical a-priori probability - which, as sensible as it is, seems to me a puzzle that shouldn’t be given as axiomatic if the question is how to understand what makes-up microscopic space-time. I get that we started with an entropy gradient in our universe but there seems to be this weird way that always gets invoked as natural and then taken as an input to models. I get this is super practical and not wrong but doesn’t it potentially confuse the question of how that entropy gradient is managed microscopically, how it relates to the evolution of real things in “proper time” and therefore to the puzzle of differential microscopic GR (twin age difference).

 

[Jimster41]

The lecture is definitely more interesting now getting into universality classes.

 

[Demystifier]

“little things affect big things. Big things don’t affect little things”.

Today I killed a mosquito, so the second statement is clearly wrong.

 

[Jimster41]

Today I killed a mosquito, so the second statement is clearly wrong.

okay... just to be clear. I was quoting the lecturer. P.3
And I agree, as I said in the next sentence, I don’t know why he said that. It was distracting and seems wrong.

Also, the QM ensemble associated with you, (there is no other you) killed that poor mosquito’s ensemble. Or perhaps you two are now a tiny tiny bit entangled. ☹

 

[Demystifier]

I don’t know why he said that.

I think Tong said that to discredit astrology. The full quote should reveal it:
"This is something you already know: particle physics underlies nuclear and atomic
physics; atomic physics underlies condensed matter and chemistry; and so on up the
chain. It’s certainly true that it can be difficult to make the leap from one level to the
next, and new creative ideas are needed at each step, but this doesn’t change the fact
that there is an ordering. Big things don’t affect little things. This is the reason there
are no astrology departments in universities."

 

[Jimster41]

I'm not defending Astrology for heaven sake. But I don't care for it because it doesn't propose any explanatory mechanism - it just asserts the importance of certain patterns. But his approach to discrediting it back-fired almost entirely for me. Or are you equating departments of Economics and Literature, Business, International Relations and Strategic Warfare with Astrology because they don't start with Quantum Mechanics?

Again I wish he had said something more like "little things are the mechanism all interactions including interactions between big things depend on".

Reductionism is clearly very powerful. Clever human beings have discovered (at great pain) that dividing things into Beables is a big key. But so is integration and so are practical models of systems of little things - like classical mechanics. Even stronger, my understanding is that there are properties of big things that aren't in any reducible way properties of the little things they are made of i.e. they are strongly emergent. The dynamics of those may of course feedback down to the little things - like that poor poor innocent little mosquito.

 

[Jimster41]

Here's a question: Why do they call something "projective construction". I'm looking at this book on "Quantum Field Theory of Many Body Systems" and in Chapter 9 he begins a "Projective construction of spin liguid states". I assume he means "rigorously hypothetical" or constructed using a practical but un-proven application of established theory. IOW spin liquids have not been shown to exist?

 

[vanhees71]

Which book on QFT many-body systems? There are zillions of books about this topic, which is not surprising since QFT is the most elegant description of many-body systems...

 

[Demystifier]

Again I wish he had said something more like "little things are the mechanism all interactions including interactions between big things depend on".

A philosopher would say it more elegantly: Big things supervene on little things.
https://en.wikipedia.org/wiki/Supervenience

 

[Demystifier]

Which book on QFT many-body systems? There are zillions of books about this topic, which is not surprising since QFT is the most elegant description of many-body systems...

I think only one book has exactly this title, the one by Wen:
https://www.amazon.com/dp/019922725X/?tag=pfamazon01-20

 

[Jimster41]

Which book on QFT many-body systems? There are zillions of books about this topic, which is not surprising since QFT is the most elegant description of many-body systems...

Yeah, the one by Wen

 

[WWGD]

Yeah, the one by Wen

Wasnt it the one by Wen, Watt, Wei and Weir? I think Hu wrote it. Hu? Yes, he let the dogs out.

 

[king vitamin]

Here's a question: Why do they call something "projective construction". I'm looking at this book on "Quantum Field Theory of Many Body Systems" and in Chapter 9 he begins a "Projective construction of spin liguid states". I assume he means "rigorously hypothetical" or constructed using a practical but un-proven application of established theory. IOW spin liquids have not been shown to exist?

In that context, "projective construction" is just jargon used to mean a slave-boson or slave-fermion approach (some people also call these "parton constructions"). It doesn't really have much to do with QFT, the idea is just that you write spins in a form like $\vec{S} = b^{\dagger} \vec{\sigma} b$, and then actual symmetries acting on the spin will be realized projectively on the bosonic operators $b,b^{\dagger}$ (which additionally will have some gauge invariance if you consider a full many-body spin model).

There are certainly models that exist which rigorously realize spin liquids (whether these describe real materials is another matter).

 

[andresB]

...E.g., Dirac's δ distribution (however already defined and used by Sommerfeld around 1910) ...

THat's interesting. I would like to read more about Sommerfeld's δ

 

[Demystifier]

Another funny quote of Tong, from lectures on Classical Dynamics:
"Unfortunately, the later years of Hamilton’s life were not happy ones. The woman he loved married another and he spent much time depressed, mired in drink, bad poetry and quaternions."

And another one:
"Jacobi identity: {f, {g, h}} + {g, {h, f }} + {h, {f, g}} = 0. To prove this you need a large piece of paper and a hot cup of coffee."

 

[vanhees71]

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 ;-).

 

[Demystifier]

It's interesting that Tong listed quaternions as part of Hamilton's unhappiness.

I think it was a joke. But speaking of quaternions, here is another quote of Tong (this time from his lectures on Electromagnetism) interesting from a historical point of view:
"You might think that the world changed when Maxwell published his work. In act, no one cared. The equations were too hard for physicists, the physics too hard for mathematicians. Things improved marginally in 1873 when Maxwell reduced his equations to just four, albeit written in quaternion notation. The modern version of Maxwell equations, written in vector calculus notation, is due to Oliver Heaviside in 1881."

 

[vanhees71]

It's well known that Maxwell's Treatise was an enigma to many physicists at the time, because the "math load" was significantly higher than usual at this time, and the methods also not well known. In Sommerfeld's great lectures (vol. III on electromagnetism) he gives the following anecdote:

1 The great student of electrolysis, Wilhelm Hittorf, who had heard much of the
new theory of electricity, in advanced years attempted to study the Treatise, but
was unable to find his way through the unfamiliar mass of equations and concepts.
He was thus led into a state of deep depression. His colleagues in Munster persuaded
him to take a vacation trip to the Harz Mountains. However when just before
his departure they checked his luggage tbey found in it-the two volumes of the
Treatise on Electricity and Magnetism by James Clerk Maxwell. (As told by A.
Heidweiller .)

 

[Auto-Didact]

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.

 

[A. Neumaier]

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.

 

[Auto-Didact]

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.

 

[A. Neumaier]

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.

 

[Auto-Didact]

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.

 

[A. Neumaier]

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.

 

[Auto-Didact]

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 separate 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.

 

[PeterDonis]

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.

 

[A. Neumaier]

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.

 

[PeterDonis]

Thread closed for moderation.