@antimony yes, this requires a kind of long-term planning and social coordination (a single person can't really usefully do agriculture like this) that doesn't follow from the raw ability to understand mechanisms like water pressure or that seeds grow into plants
and it took us a while!
it's wild to realize that "modern humans" have been around for something like 200000 years but we really only started "civilization" like 10000 years ago
but "wave vector" is a term that means the vector describing a scalar-valued wave in 3 dimension, for a QM wavefunction it's essentially just the momentum vector
I think this is basically about Gaussian elimination. Consider a subspace $V$ of $\mathbb C^{2\nu+1}$ with $\dim(V) \leq \nu$ (eg. any isotropic subspace). Then $V$ can be defined by a system $(\Sigma)$ of $m \geq \nu + 1$ independent linear equations:
$$(\Sigma) : \forall 1 \leq i \leq m, \quad ...
@DIRAC1930 thoughts on the answer there in justifying where $\eta_0 = 0$ comes from?
Say, there is a space of $5{m^3}$ in a room. (Is space synonymous with volume?) Now, we drop a ball of $2m^3$ in the room. How much space is left in the room? I would argue $3m^3$. When the solid ball is dropped in the room, it occupies $2m^3$ of space, and that $2m^3$ of space ceases to exist.
N...
@antimony Although photons have no mass they still have energy. However, photons don't occupy space in the same way that matter does. In particular, matter is made of fermions, which obey the Pauli exclusion principle, which affects how much matter you can pack into a given space.
But photons are bosons, so they don't obey Pauli exclusion, and you can "pack" as many photons as you like into a given volume. In fact, the more the merrier.
Eventually, you'd create a Kugelblitz: a black hole made of light. However, we don't know of any way to create such an insanely high concentration of light. It's hard enough making black holes out of matter, which is a lot denser than your average light source. ;)
@bolbteppa This is weird communication: Where's the robot? If I understand what's happening correctly, this is just a clump of gallium they're controlling with magnetic fields applied from the outside
Can the continuum hypothesis be settled in physics? In a lecture mathematician Woodin considers the possibility:
Develops the mathematical physics of a mathematical understanding of
the physical universe. If it starts to need large Cardinals remember
large carnal axioms with finite-istic conseq...
Carnap was a "late" positivist in that the problem of induction had been such a hurdle for establishing a process of verification that he switched to the idea of "degrees of confirmation" instead of the ability to establish absolute truth, arguably a step in the direction of Popper's falsificationism (which is why Slereah brought him up when I mentioned Popper last time)
this is another "QFT and renormalization" moment :P Just because you first learned about the relation between linear operator theory and differential equations via physics/QM doesn't mean it's physics that relates the two
mathematicians teach functional analysis courses in which differential equations are studied via linear operator theory without ever mentioning physics
purely mathematically, there is a relation between linear operators and (linear) differential equation
so of course when you use a linear differential equation in your physical theory, you can equivalently think about it from the linear operator viewpoint
of course if you notice that two apparently different mathematical frameworks seem to produce the same statements you might start looking into whether or not these frameworks are mathematically related - you don't need the physical observation that the statements they produce actually describe reality for this
quantum mechanics could have been entirely false and yet the observation that matrix mechanics is equivalent to wave mechanics as a physical theory - even one that is false -would still hold
this doesn't have anything to do with observation or reality
of course if you notice that two apparently different mathematical frameworks seem to produce the same statements you might start looking into whether or not these frameworks are mathematically related - you don't need the physical observation that the statements they produce actually describe reality for this
whether or not your $M_i$ are observations you actually make or just observations the $D_i$ predict but you haven't observed is completely irrelevant - your confidence that $D_1$ and $D_2$ are equivalent should not depend on whether or not they correctly describe reality
@ACuriousMind okay consider an example like qft where you don't know how many Terms u have to sum to calculate something like the magentic dipole moment of an electron
There are always infinitely many mathematical structures that can describe a given physical theory, and if you pick the wrong one, you're gonna have a nasty surprise when you find out
so? you're just saying there might be a perturbative and a non-perturbative version of a theory
again, the equivalence between the perturbative and non-perturbative version of the theory does not depend on whether or not these theories actually describe reality
I feel you're not engaging with my objection at all
@Slereah If both sides of this conversation agreed that mathematics does not inherently relate to reality or observation at all there would be no conversation :P
we abort at the order where either computation becomes to annoying for us to bother or where the higher order terms would only contribute below what our specific measurement setup can resolve
I have two theories and when I ask them "give me predictions for the observed value of X up to 10 decimal places" they both spit out numbers (with uncertainties since they probably had to use experimental input to do that)
if the two predictions do not match within their uncertainties, then the two theories are obviously not equivalent. If they do then they might be
all the experiment did so far in this setup is influence the "up to 10 decimal places" bit - you were talking about the order in perturbation theory at which we abort, after all
@MoreAnonymous I am aware of the asymptotic nature of the perturbation series, I explicitly mentioned it above. What does it have to do with our discussion?
@MoreAnonymous no, Slereah is right, you keep trying to make weird technical points about QFT when my arguments against your ideas are extremely general points about the relation between mathematics and reality.
hello, i was wondering why, is magnetism is a relativistic effect, it is a big deal that there is no static magnetic charge? i get the idea that the equations for electric and magnetic field resemble each other in some ways, but is this the only reason?
if you believe in the gospel of the principle of least action it's actually not at all mysterious that there are no magnetic monopoles: One half of Maxwell's equations are equations of motion (and hence you can add an electric current just by adding a term to the EM action) but the other are off-shell identities just following from the definition of the EM field as the derivative of the 4-potential
in order to add magnetic monopoles you need to completely change this theory so that the "no magnetic monopoles" part of Maxwell's equations is no longer an off-shell identity
or you need to claim that the monopoles are somehow "holes in space" from the viewpoint of EM theory, which works if you really do think they're a kind of topological defect but not if you want them to be "normal particles" like electrically charged ones are
that is interesting. yeah i was thinking no mag monopoles seems to be a necessary consequence of how the entities in the equations are defined, but i wasnt sure if i was missing smth
of course this isn't really a "reason" monopoles don't exist: After all, we developed EM theory according to the observation that there aren't any monopoles
but it is interesting that it is not easy to fit them into the more theroetical formulations of EM
yes, the topological version of the monopole forces quantization of electric charge via the condition that the AB effect due to the Dirac string must be unobservable
one way to have electric and magnetic charges is to propose that there are two copies of EM and an equation of motion that equates the field strength tensor of one copy with the dual field strength of the other
Can the continuum hypothesis be settled in physics? In a lecture mathematician Woodin considers the possibility:
Develops the mathematical physics of a mathematical understanding of
the physical universe. If it starts to need large Cardinals remember
large carnal axioms with finite-istic conseq...
