hansen

To be a bit pedantic: the identification of tensors with multilinear forms requires finite dimensions (or reflexive topological spaces).

Physicists and philosophers of science give those terms different meanings. E.g. what a philosopher of science would call a theory is called model by physicists. What physicists call a theory is more a broad framework in which more concrete models can be build.

From the view of a philosopher of science string theory isn’t even a hypothesis as a hypothesis needs to be testable

Spacetime is extremely stiff or equivalently gravity extremely weak. Only these extreme events result in detectable gravitational waves.

I doubt there are many users out there who care if someones analytics stuff breaks.

The standard (but rather heavy and difficult) is

Abraham, Marsden: Foundations of Mechanics

More gentle is

Vladimir Arnold: Mathematical Methods of Classical Mechanics

I don’t think the typical “change of variables” definition is bad. You take the derivative of L along the fiber of the tangent bundle. If the derivative is non-singular it defines an isomorphism in each point of the tangent space with the cotangent space. And that’s the important thing, going from the tangent bundle to the cotangent bundle. Now we can use all the beauty of symplectic geometry

Mathematicians messed up here… A manifold with boundary is not a manifold. But a manifold is a manifold with boundary (the empty set).

Gravitational waves are disturbances in a background spacetime, i.e. you linearize the Einstein equations around a given solution. The analogy is less bad as one might initially think

I always thought it's more about caching

The problems in the classical theory are easily understood. The charge and current densities of point particles are not smooth functions but distributions (think of the Dirac δ-“function”). If they act as the sources of the EM field the EM field itself becomes singular. Now if you try to solve the full Maxwell equations including the backreaction of matter & radiation fields you would have to multiply distributions which is ill defined.

There are similar problems in the quantum theory but the divergences are less severe and can be dealt with in a systematic way. Most physicist believe they will totally disappear in some more fundamental underlying theory. From a mathematicians point of view there is the hope that at least some QFTs are finite and the divergences are just an artifact of the construction & pertubation theory.

> Given the trajectories of charges it will tell you what the electromagnetic field will be, and given the electromagnetic field it will tell you how charges will move. Unfortunately, these two parts of the theory seem to be incompatible, and the theory will not tell you how fields + charges will evolve in time.

The Problem is not coupling charged matter fields to the EM field. You get a well defined set of coupled and (now) non-linear PDEs. The problems arise when you try to model point charges. Then the theory is plagued by infinities that originate in the infinite charge and current densities.

The infinities in QED are actually far less problematic than those in the classic theory. They just seem to be more problematic because you cant (approximately) ignore the backreaction of EM and matter fields.

And all those Emacs jokes don't work anymore

If you want Cᵏ to be a normed space you do

|f| = ∑ᵢ₌₀ᵏ sup|f⁽ⁱ⁾|.

On open domains you can also use the topology that forces uniform convergence on all compact subsets. But this will only give you a metric space, no Banach space (but you’ll include unbouded functions). This is needed for studying Brownian motions with an unbounded time domain.

Regarding the other question: If you take Cᵏ⁻¹ as a the co-domain differentiation will be continuous. IIRC to get unbounded linear maps defined on the whole Banach space you need the axiom of choice, you won’t be able to write one down.

The problem with differential operators is that they are usually only defined on a dense subset of the domain, and there they are not bounded. E.g. in quantum mechanics the space of states is L² but all the interesting observables are differential operators. You can weasel out of this situation by defining them on a subset of “physical states” (e.g. smooth wave functions of rapid decay). But they aren’t continuous anymore (the spectrum is unbounded). But on Hilbert spaces everything mostly works out fine. Physicist usually ignore those technical problems and still don’t make mistakes.

> there is enough experimental evidence that is anomalous to the theories

Huh? I have more the impression that Λ-CDM & SM are working just too damn good. I’m not aware of any observation that hints that we need something completely different than GR & local QFT.

But “settled” and “well established” doesn’t mean that humans will never find a more fundamental theory. Classical mechanics is settled, well established and correct as it was in Newtons times. Now we just better understand its limits and when to use different models.

GR, QM & QFT are correct and won’t go away. At some point in the future we will have a better understanding of their limits too. But that won’t make them wrong.

> The fact that we don't know and that our current suite of theories is incomplete and hence could very well be wrong in both minor and major ways should be driving us onward into understanding the universe around us. It really is not a problem if we don't understand. It should mean that we strive to learn more.

I’m not objecting exploring new speculative models. I’m worried about presenting highly speculative ideas as facts or probable solutions to the public. A lot of the pop-sci articles you read these days are very misleading. And IMHO Rovelli, Green, Susskind, Smolin and the like aren’t doing anyone a favor with their pop-sci books.