What Constitutes a Natural Completion of a Physical Theory?

[Auto-Didact]

This thread is about the process of theory completion in the practice of theoretical physics, and therefore mostly directed towards practicing theoretical and mathematical physicists, but others can chime in as well.

a) What exactly constitutes a natural completion of a physical theory?
b) Is the process of completing a theory more aligned with the 'job description' of a mathematical physicist or of a theoretical physicist?
c) Are there any specific criteria for what it means to complete a theory?
d) What is meant by a completion as opposed to a natural completion?
e) Is completion merely equivalent to extension? Or is completion more akin to some kind of particular unique extension of a theory?

You don't necessarily need to answer all the questions, but for the sake of clarity please specify which question(s) you are answering.

It seems quite obvious that theory completion is used in a much wider context than in purely the effective field theory sense a la UV completion, obviously because theories have been completed far before effective field theory (EFT) was invented, eg. Einstein already spoke about completing quantum theory decades before anyone had even heard of EFT. Opposed to theory completion, theory extension on the other hand seems to be pretty straightforward: any modification of a theory's original equations, axioms or unspecified numerical parameters.

With regard to natural completion, I should also clarify by 'natural' I am not necessarily talking about 'naturalness' as is used to refer to dimensionless ratios being close to 1. That is a specific use of the term natural which doesn't seem at all to really capture all the forms in which the term can be and is used in the context of physics, which is why I see no good reason for an a priori insistence upon that particular usage of the term.

By analogy, the term 'uncertainty' does not necessarily refer to (an effect of) stochasticity, stochasticity instead merely being a specific form of uncertainty, pretty similar to how kinetic energy is a form of energy; perhaps something similar applies to UV completion and completion? Is there some existing hierarchical listing of all forms of theory completion?

 

[LURCH]

I am not a practicing theoretical physicist, but I do have a small talent for communication (enough to really enjoy phrases like “practicing theoretical physicist”), and this thread seems to be largely concerned with how certain terms are used. From the OP, would I be correct in concluding that you have heard the term “natural completion “ used in ways that seem vague, inconsistent, or even somewhat contradictory? If so, an example of two such statements might serve to get this conversation rolling, and rolling in the right direction.

In what contexts have you heard this phrase, what did it appear that the speakers meant when they said it, and what, if anything, seemed incongruous, to your ear?

 

[Auto-Didact]

Not simply two, but too many cases to count or name, at least hundreds to thousands, a full listing defies compilation. Back when I was in university, I remember some physics professors using the phrase when referring to the 'reverse' of progressing from a more general theory to some limiting case through a Taylor expansion of the appropriate quantities; especially the grad students in theoretical and mathematical physics were keen on using this particular phraseology.

Freeman Dyson, in his seminal paper 'Missed Opportunities' (1972), refers to the same idea in the following manner:

After Newton’s laws of gravitational dynamics had been promulgated in 1687, the mathematicians of the eighteenth century seized hold of these laws and generalized them into the powerful mathematical theory of analytical mechanics. Through the work of Euler, Lagrange and Hamilton, the equations of Newton were analyzed and understood in depth. Out of this deep exploration of Newtonian physics, new branches of pure mathematics ultimately emerged. Lagrange distilled from the extremal properties of dynamical integrals the general principles of the calculus of variations. Fifty years later the work of Euler on geodesic motions led Gauss to the creation of differential geometry. Another fifty years later, the generalization of the Hamilton-Jacobi formulation of dynamics led Lie to the invention of Lie groups. And finally, the last gift of Newtonian physics to pure mathematics was the work of Poincaré on the qualitative behavior of orbits which led to the birth of modern topology. But the mathematicians of the nineteenth century failed miserably to grasp the equally great opportunity offered to them in 1865 by Maxwell. If they had taken Maxwell’s equations to heart as Euler took Newton’s, they would have discovered, among other things, Einstein’s theory of special relativity, the theory of topological groups and their linear representations, and probably large pieces of the theory of hyperbolic differential equations and functional analysis A great part of twentieth century physics and mathematics could have been created in the nineteenth century, simply by exploring to the end the mathematical concepts to which Maxwell’s equations naturally lead.

Here is another particularly striking example:
https://arxiv.org/abs/1711.08643
This is one of the only handful of papers I found by a quick search on the arxiv which explicitly repeatedly use the phrase.

The exact meaning of what this phrase entails and how to do this never seems to be explained, just referred to by some analogy. This implies that theory completion is possibly only a belief or intuition about some kind of nontrivial extension of a theory, possibly even unique with regard to some particular mathematical framework. From an optimistic point of view this can seemingly even be objective if multiple practitioners can independently come to the same or similar conclusion, while a pessimist would argue that this a completely subjective matter with no formal or rigorous basis whatsoever.

To avoid possible confusion, I am not so much concerned with what I think the correct definition of this phrase is or should be, but 1) whether or not they (geometers, algebraists, analysts, group theorists, relativists, particle physicists, string theorists, field theorists, physicists, mathematicians, etc) actually agree upon what they are saying or just think that they agree because they happen to be using the same phraseology and 2) whether there actually is any canonical basis for this agreement codified in the practice of theoretical physics, mathematical physics or more generally in pure mathematics. To summarize, the issue isn't just merely that there are multiple paths, seemingly an infinite number of them, leading to multiple places, some possibly leading to Rome, but indeed, whether Rome even exists.

 

[Nugatory]

Mentor's note: a number of generally unhelpful post have been removed from this thread. Please, please everyone... report instead of replying.

 

[bhobba]

IMHO there is no answer - its philosophy or personal opinion depending on how you look at it. Dirac and Heisenberg discussed the issue of open/closed theories:
http://philsci-archive.pitt.edu/1614/1/Open_or_Closed-preprint.pdf

I personally side with Dirac and think no theory is ever complete - we simply make progress - but its just me - I could be wrong - as I said I do not think there is a definite answer.

Thanks
Bill

 

[Mister T]

Perhaps the only theories that are complete are the ones that are obsolete. As long as a theory is being used it's open to at least the possibility of being added to.

 

[Auto-Didact]

Since the thread isn't getting a lot of responses, I'll pose my own view of what is meant, which I tried to avoid in order not to bias the discussion. In any case:

By a (natural) completion is meant something analogous to the generalization of the correspondence principle in reverse, i.e. not referring to the older theory as a limiting case of the newer theory, but to the newer theory as an unlimiting of the older. The problem is exactly which kind of unlimiting or generalization is correct, since there seems to be an infinite number of not necessarily exclusionary ways of generalizing matters.

In my opinion, intuitively this process is strongly reminiscent of the difference between integration and differentiation, differentiation being straightforward and leading to a unique function, while (indefinite) integration leads to a infinite family of slightly different functions. This is why psychologically carrying out differentiation feels like just simply following some mechanical set of rules, while carrying out integration is more of an art.

The simple kind of generalizations to make in a theory, tend to be the obvious ways in which the mathematical criteria can be changed. These changes don't need any real creativity and after they have been carried out do not lead to qualitatively radically different kind of theories, while the correct generalization seems to have exactly the opposite characteristics, effectively changing the entire mathematical structure underlying the original theory.

One type of generalization I'm particularly fond of is non-linearization, i.e. the process of transforming a set of linear partial differential equations into a set of nonlinear partial differential equations, i.e. the reverse strategy that is used in trying to solve a set of nonlinear differential equations by linearization. The problem here again is that there doesn't seem to be some general and effectively completely algorithmic procedure for always carrying this process out succesfully for any arbitrary nonlinear PDE, which is exactly why solving nonlinear PDE's can be so devilishly difficult.

Reducing nonlinearieties in a physical theory through linearization actually often leads to a qualitatively different kind of mathematical picture, one that may produce numerically close answers, but that is often blatantly inconsistent with the dynamical properties of the original equation. This is reflected in the fact that the linearization is necessarily some ideal limit of the original nonlinear equation describing the phenomenon in question.

IMHO there is no answer - its philosophy or personal opinion depending on how you look at it. Dirac and Heisenberg discussed the issue of open/closed theories:
http://philsci-archive.pitt.edu/1614/1/Open_or_Closed-preprint.pdf

I personally side with Dirac and think no theory is ever complete - we simply make progress - but its just me - I could be wrong - as I said I do not think there is a definite answer.

Thanks
Bill

I think their disagreement is more one of contextual perspective than of actually incompatible principles, i.e. I think, viewed from the correct mathematical perspective, that the radical change Heisenberg is talking about is exactly a natural, small but nontrivial change to an existing theory that Dirac is proposing. Indeed, I think the non-linear perspective is precisely the correct mathematical perspective and that the above equivalence is the description of a second order phase transition from nonlinear dynamics.

Moreover, it can't be just a matter of philosophy. The question is central to the actual goal of theoretical physics, namely finding some mathematical model which can effectively capture the description of all phenomena in nature.

Perhaps the only theories that are complete are the ones that are obsolete. As long as a theory is being used it's open to at least the possibility of being added to.

I doubt that severely, I'm perhaps more drawn to the view that theory completeness implies formal inconsistency in the Gödelian sense. Then again, because a precise definition of theory completion seems to be lacking I'm not so sure all theorems within some theory can even be effectively enumerated by some procedure, nor whether physical theory completeness actually even implies completeness in the mathematical logic sense.

Moreover, theory completion does seem to also imply theory unification, but the two don't necessarily seem to be equivalent procedures. I think it would be instructive to (attempt to) list all types of generalization schemes used generally in the practice of theoretical physics.