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
The change of variable definition, as actually presented in the textbooks everyone teaches from, is horrible. That's the topic of the blog post. Yes, that definition can be made clear after introducing a bunch of machinery of symplectic geometry, but I'm doubtful this is good pedagogy and I'm confident that, due to time constraints, it could never be taught to most physicists.
> Mathematicians messed up here… A manifold with boundary is not a manifold. But a manifold is a manifold with boundary (the empty set).
It depends on which mathematicians! Plenty of differential geometers allow manifolds to have boundary, and say "closed manifold" (https://en.wikipedia.org/wiki/Closed_manifold) to emphasise when they are dealing with a (compact) manifold without boundary (or, as you point out, really a manifold whose boundary is empty).
I thought a manifold with a boundary would still be a manifold, but its boundary has to satisfy a dimensionality condition. For example, the 2D disk is a 2-manifold with a 1-dimensional boundary. Strictly speaking, this is a _topological manifold with a boundary_, though.
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
> 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.
what you expounding upon is [1] of the indicators that we dont completely understand physics at the "point charge" scale, and very possible there is no such thing as a point charge, rather there is a centroid of field intensty/probability I.E. a wave function.
point charges are likely an overly simplified view, and artefactual convienience of extrapolation.
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.
I don't think that kind of memory usage is necessarily common - I get sessions that grow for whatever reason but typically I can have several emacs instances running at once using less than 100MB each - the ones I treat like notepads are often less than 10MB.
Hmm, why would one run several emacs instances? Emacs supports the client-server model, which I find very convenient and fast. Basically having `EDITOR="emacsclient -n"` allows you to use the one emacs everywhere.
I run them with different color themes and fonts (randomly chosen from a list on startup) so I can instantly see which window is being used for which project (I juggle several at work - I also take advantage of Windows 10's virtual desktops so get them spread out all over the place). If one of them goes haywire I don't feel bad about killing it since the others remain unaffected, so I never really considered using just a single server instance. Since I don't run out of memory and have CPU cycles to spare most of the time, it works well enough for my current needs.
I also tend to run the windows with partial transparency so I can see bits and pieces of other sessions/browsers/etc behind them - something I was slightly hesitant to start doing but that has made using the computer feel much more "lightweight" (for lack of a better term). I've found it surprisingly useful to have emacs show a second buffer slightly faded directly through the first one.
I too, for a time, had setup my system so that windows were partially transparent. It looked cool, and helped to find windows when they were hidden behind other ones. Then, one day, I turned the partial transparency off, and found it to be very relaxing on my sight and brain. I didn't feel any strain when I started, but I guess it accumulated.
For context, I use a tiling window manager with an average of 4 to 7 windows per workspace-monitor combo. Back then, I also set up a moving background (a screensaver always on in the root window). In fact, I think that was the primary motivator. I had gone years without a wallpaper because having a tiling window manager made it meaningless and I was missing having some eye candy. Having partial transparency would let me see the moving background through the windows. Anyway, I digress and I'm back to no transparency, no background, in any case.
