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
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
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
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.
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.
> For example, ask a lay person about BB and they will tell you it's an explosion, in the classical sense
Give her Weinberg’s book to read and she’ll come to a different
conclusion. Of course those books won’t give you a deep understanding
of cosmology but that doesn’t mean they have to be completely wrong.
Stephen Weinberg’s The First Three Minutes is my favorite example of
pop-sci done right.
All our accepted model of nature assume that in some sense time
exists. But in general relativity it doesn’t make much sense to
speak of a single event. In the same way it doesn’t make much sense
to speak of a single point in the euclidean plane (you can only
localize one point relative to another one).
So some people (like Rovelli) think one should try to formulate
relativity only in terms relations of events. This might be relevant
in a more fundamental theory of spacetime but on a classical level
it’s irrelevant. You can measure the geometry of spacetime in our
solar system w/o disturbing it. The gravitational field of e.g. some
satellite is negligible to the gravitational field of planets and the
sun.
Yeah, science isn’t about being useful and relevant in day to day
life. It’s an important part of human culture and scientist should at
least try to communicate their findings to general public (which is
paying for it!).
I partially agree with the first comment that it isn’t really helpful
to popularize highly speculative – and thus probably false – ideas.
But well established but still “weird” discoveries like GR & QM are
interesting on their own and have huge implications on our
understanding of nature. You don’t need a science degree to be
fascinated by black holes, the big bang and quantum weirdness.
> Sure, but regular quantum mechanics does the same thing. It's
> under-constrained and allows for too many possibilities.
No it doesn’t. There are a few good guiding principles that make it
possible to build models that turn out to be an accurate description
of nature. These guiding principles don’t manifest themselves in a
set of axioms of a mathematical theory nevertheless they are there.
And they are part of the theory, just not set in stone like the
underlying mathematical structure. But that doesn’t matter at all as
long as you are doing science and not math.
Take classical EM, measure the masses of the electron and atomic
nuclei, and write down a suitable Hamiltonians. Viola, you have an
explanation of almost every phenomena that’s relevant in our daily
life and way beyond. Similar principles exist in elementary particle
physics and lead to standard model.