Conjecture: Imagine I have a spherical jar of balls. I shake the jar (non-randomly) such that the balls come back to its original configuration in time $t$. If my shaking algorithm takes $1$ unit time and $1$ unit volume to complete $1$ cycle. Then if I scale the volume by $v$, time by $k$ then the number of cycles my system completes is $C = v^{1/3}/k^3$. Assume simple $F = ma$ for equations of motion.
Is this better on computer science stack exchange or PSE?
@ACuriousMind Is symplectic reduction in classical symplectic mechanics related to this "modding out by gauge transformations" business in field theories?
uugh, two ppl on the main site are asking meow for old history. Feynman calculating QED corrections for constant background E field. Can't remember where such tales are...
i hav a charge density $\rho(x,t=0)$, its time derivative $\rho '(x,t=0)$ and the electromagnetic field $F^{\mu \nu} (x, t=0)$. this is insufficient to determine the full time evolution, right?
in this section, it says that we can drop the k label which specifies the state, but it also mentions particles occupying the ground state here. im not sure how these two things relate? k specifies the state in terms of momentum, right?
@Mr.Feynman you have two vertices, and $J$ momenta coming from each (due to the Fourier transform of the $J$ derivatives in the coupling). Each pair of momenta contracts to an $s$, so you get $s^J$
@Mr.Feynman Hm? Remember the Feynman rules - the $\sigma^\mu$ etc. turns into equivalent indices on the propagator for that field, which are the contracted with something suitable with indices on the vertices
and that something here are the momenta from the derivatives in the coupling
We know that in Coulomb's law, the constant $k_e$ is $\frac{1}{4\pi \epsilon_0}$. Where does $4\pi$ come from? Why is it related to the force between two charges?
@Relativisticcucumber i would guess that $k$ and $x$ do refer to "states of definite" momentum and position, respectively. this then allows you to write an arbitrary state (even if your Hamiltonian isn't free) in terms of vacuum $\lvert 0 \rangle$ and creation/annihilation operators
@Relativisticcucumber oh i see who the text is by lol
oh nevermind it does indeed seem like $k$ is used to label the energies and that this discussion is specific to the harmonic oscillator
i feel like philosophers use "Science" like physicists use "Math" lol
i am wondering what the "new technology" is in high energy theory. for instance, it seems like putting field theories into bundle theory language was popular (and continues today) in say the mid-late 1900s (maybe this is inaccurate). what new sort of mathematical structures are being used today to "mathematicize" physical theories?
or rather: What question exactly are you trying to answer here? Bundles etc. are the answer to "what is the proper mathematical formulation of what fields are?", what's the question you expect more structures to appear while answering it?
hm well maybe a question could be like: what sort of mathematical structures are involved in a given supposed theory of quantum gravity (since quantum gravity is what I would guess is a more modern and in development physical theory)
we don't even know the "proper" mathematical formulation of ordinary QFT - the Yang-Mills millenium problem is still, as of today, unsolved
so demanding to know the proper mathematical structure of QG strikes me as one step too far - in all other cases, we have first figured out the correct theory at a physical level of rigor, then formalized it in mathematical terms
and we haven't even quite caught up with "physical" QFT yet
is the Yang-Mills millenium problem just proving the existence of a structure (subject to some carefully defined axioms) that is supposed to be a physically relevant QFT? or is the problem more than that
i mean to more ask: do we have a set of axioms for a "QFT" we think is reasonable, but the hard part right now is proving that a structure even exists satisfying them
yes; the challenge is establishing the mathematically rigorous existence of a realistic QFT - namely non-Abelian Yang-Mills theory in 4d, i.e. essentially the Standard Model - that satisfies any equivalent of the Wightman axioms
are there alternative formulations of classical mechanics that do not use manifold structures? more precisely, assuming we are dealing with a "nice" system: is there a formulation of classical mechanics that does not define a configuration space and then use its tangent or cotangent bundle as the "space of states"?
are our current formulations of algebra and analysis nice languages for creating physical models? Maybe measuring niceness by being able to compute and prove things
we like to pretend there is some unbroken continuity from Newton to us, but it's much more that we can't understand the principia without a lot of effort and context and Newton wouldn't understand any of our math, either
there's something continuous there, sure, but it certainly isn't the way we talk about math
is there a resource which talks about electromagnetism historically? In particular to make sense of the following observations. It sort of seems like the theory of electromagnetism is quite special relative (lol pun not intended) to any other physical theory. (1) it is automatically relativistic, (2) it was the first field theory to be quantized successfully, to my understanding. Or maybe it is not so special really?
it would be interesting to find something that talked about how the empirical observations which led to the development of electromagnetism are really observations that could only be explained if they took into account relativity/quantum mechanical effects or something like that
I don't know a specific resource but really the core of what distinguishes EM is that it is the only part of the Standard Model that is better approximated in terms of waves and fields in the classical regime than particles due to photons being the only massless unconfined objects
and then if you look at the EM waves you inevitably will notice that their speed in vacuum is fixed by the form of Maxwell's equations, whether you wrote them in a manifestly relativistic formalism or not
I am having a hard time seeing how the 2-form defined in (13) is a 2-form. In local coordinates it looks like it takes in a tangent vector from $TM$ and a tangent vector from $T(T^*M)$
but presumably the (will be called symplectic) 2-form should take in two tangent vectors from the tangent space of phase space $T(T^*M)$ (being loose with global v. local notation here)
@SillyGoose no, it's just a 2-form on $T^\ast M$. The coordinates of $T^\ast M$ are the $(q^i,p_i)$, so $\mathrm{d}q^i \wedge \mathrm{d}p_i$ is a perfectly ordinary 2-form on it
because the partials themselves are the "basis" so actually i don't even understand what it would mean to write the basis itself as if it were a vector in a basis as (9) seems to do
i wonder why the notes i was working off of wrote it like that
so can i think of $D\pi$ like a linear map from linear algebra? so its action is specified by its action on the basis vectors. so us specifying its action as you wrote is all the data we need to define the map
also it seems like this wedge product is additional structure than the cotangent bundle itself
is that true?
or i more mean to ask: where does the structure allowing us to now talk about differential $k$-forms come from if we only start with a cotangent bundle of some manifold
ohhh i think i keep conflating the properties of a tensor with the properties of the determinant as a particular type of tensor... and the determinant is antisymmetric or "alternating"
okay i see
okay so this idea of a tensor is inherited from the vector space structure of the tangent spaces or cotangent spaces
(or that is how i am thinking of it, perhaps there are other ways)
then of course we can restrict to particular types of tensors, the antisymmetric ones, and we give them the name differential forms
sure, a tensor field (often just called "a tensor" when the context is clear) is just a tensor in the sense of linear algebra on the (co)tangent spaces of a manifold at every point
okay so at each point $x \in M$ we have a tangent space $T_xM$. from this tangent space we construct a sequence of tensor products of this space $\{ \otimes_{i=1}^k T_xM\}$. We can construct the dual sequence as $\{\otimes_{i=1}^k T^*_kM \}$. These constructions are just fine since this is usual linear algebra. This is the data we were after to talk about differential forms
and so then we define $\Omega^k(M)$ to be the space of (global?) sections of $\otimes^k_{i=1} T^*M$ such that at every point the tensor (living in the tensor product of cotangent spaces) is antisymmetric.
the trick is that the terms that spoil proper tensorial behaviour of the derivatives of arbitrary tensors (the ones the Christoffels of a connection cancel!) are symmetric
so antisymmetrization kills them, and the exterior derivative is well-defined without a connection
it tells you whether there are closed forms that are not exact :P
for instance, whether there are divergence-free vector fields that are not the curl of a vector potential
e.g. there are none on $\mathbb{R}^n$, but in the case of the idealized solenoid that removes the origin from the space under consideration, you have a magnetic field that's not the curl of a single vector potential
@ACuriousMind hm that seems like very course grain data. in a physical situation would the mere existence of a non-exact closed form really matter? since if you are working with particular potentials, it seems you would have to check anyways whether the potential you are working with is non-exact and closed.
in some contexts, e.g. dimensional reduction, you will be interested in harmonic forms $\Delta \omega = 0$, i.e. zero modes of the Laplacian. There is exactly one harmonic form in each cohomology class
but like with much math it's more that it's an organizing principle rather than the only way to arrive at some fact
for instance for the solenoid and its magnetic field: you can very explicitly show that there is no vector potential on all of $\mathbb{R}^3 \setminus 0$ for it without using much machinery at all
but then it remains somewhat mysterious why it can sometimes happen that such magnetic fields exist and sometimes not
the Poincaré lemma and deRham cohomology are the answer - this happens if and only if your space has certain non-trivial cohomologies