@bolbteppa :I don't agree. I find the bundle picture of instantons using transition functions on the four sphere clearer than the physical interpretation using the "pure gauge at infinity"
The physical picture after you start thinking about it seriously often falls apart because some details are missing
Does anybody know a physical argument behind equivalence of dimensional regularization and momentum cutoff regularizations? I talked to a person who says to have faintly remembering about it in some talk by Howard Georgi where he talks of "shells" of mass being peeled off and idk...
@bolbteppa do you mean for covariant vectors or defining connections in general? I'm more comfortable with the differential geometric definition of connections on vector bundles, but having said that I repeat that it's not about necessity
For me it's just that I don't like doing math intuitively. I like doing things rigorously, which doesn't mean that I need the most advanced proof available, just a self-consistent rigorous one. Then again, I think the difference is the same as speaking informal English vs higher register English
You wouldn't speak informally to the PoTUS in a formal meeting, would you? :P
I have read about a dozen pieces of text on gauge transformations, diffeomorphisms and co-ordinate transformations... it's all gone over my head. does anyone know if there's a gentle introduction somewhere that relates the different usages in different contexts (EM, GR, cosmology)?
Carroll says "you have to cheat somewhat by equating components of tensors in two different coordinate systems" to treat gauge transformations as coordinate transformations and not diffeos. wald says the active/ passive viewpoints (diffeos v coordinate transformations) are really equivalent.
@qwerty i guess the outcome of a diffeomorphism is indistinguishable from that of a co ordinate change. if u take a co ordinate change, and interpret the co ordinates as the "literal points" instead of labels of points, u end up interpreting the co ordinate change as an active diffeomorphism
but there may be subtle differences.. maybe they can be ignored
@RyderRude i don't understand the intuition behind gradient, divergence and curl. I am thinking about these since 4 years. But still no progress. I have learned many threads and books but no clue i found.
@RyderRude oops sorry new to this chat thing. yes i suppose that is the point that Wald makes in his textbook. yet e.g. bertschinger claims things like the volume element in GR is invariant under diffeos, but not coordinate transformations; so they cant be treated the same.
@123 consider a fluid, draw a closed surface as you like. the divergence tells you how much stuff is entering or leaving the surface
I want to understand this step by step. Like we have vector field (e.g. gravitational field, electric field) then if we partial differentiate this field at some point which gives slope/gradient of this field keeping other variables constant.
@qwerty This topic is a horrible can of worms and really people do not seem to agree with each other. Personally I'm on Wald's and the mathematicians' side that you really only need to talk about diffeomorphisms and the active/passive distinction is a trap you don't need to fall into.
@ACuriousMind is there any intro that you might recommend in particular? the more i read the more confused i get; i feel like the fact that people often start talking about gauge transformations without defining what they mean (and it seems there are a few related meanings? at least when people say "gauge invariant") only adds to the confusion.
yes, that's true; I don't think there is a single source that does it perfectly
I think the best work on gauge theory that's consistent from start to finish is Quantization of Gauge Systems by Henneaux and Bunster, but it's almost entirely in the Hamiltonian formalism and you need to do a bit of work to connect it to what other people are doing
alas, nothing will solve the problem that people use words inconsistently - as you say, people often try to talk about "gauge transformations" only handwavingly defining them as some kind of "spacetime-dependent transformations" when you really need a more solid definition to be able to work with it
why do you think that people don't seem to agree with each other? is it inconsistent definitions? i somehow dont think the definitions are inconsistent (at least when talking about specifically cosmology)
also thanks for the book rec. i'll give it a look.
hmmm i'm inclined to agree in general!! but in this case both carroll and bertschinger seemed to be precise in what they were saying with respect to diffeos/coordinates, or at least as much as wald; even though i'm still confused.
@qwerty Well, the first problem here is: Could you write down what exactly the definition of "diffeomorphism" and "coordinate transformation" is that each of these sources are using? Like, a proper, rigorous definition, not just a handwaving sentence that sounds like a definition?
at this point I feel like I'm linking this once per week but I explain this in a bit more detail (in particular footnote 2 and the expansion of $F_\omega$) in this answer of mine
i see... Carroll mentioned in appendix b of his textbook actually, but how gauge invariance is related(ish; i dont know if he agrees - he says it is "garbled") to general covariance in appendix b of his textbook...
@qwerty in standard model, gauge fields arise by making global internal symmetries into local and writing the covariant derivative, and then writing the free term of the gauge field using its curvature
so this is a unified view of gauge symmetries other than GR
a common element of guage fields is that theyre connections
@ACuriousMind yeah.. i was just saying that this trick is how gauge fields of the known forces happen to arise, but we dont have a good explanation for why this trick should work
If the Maxwell tensor in EM is the analogue of the Riemann curvature tensor in GR, is $A_\mu$ the Levi-Civita connection, or some other choice of connection? And if not, is there some reason why we don't use the L-C connection in EM (like it doesn't exist, or we just don't introduce a metric in E&M)?
Another question I was thinking about this morning, when we mention the connection coefficients $\Gamma$ in GR we have to be careful about the non-tensorial transformation rules and emphasising that two of the components label the Lie algebra element (matrix) components, but if the Riemann curvature tensor is also a Lie algebra-valued 2-form it also has two indices which denote the indices on a Lie algebra element, but it doesn't have the same transformation property and we treat it as a
completely ordinary tensor, what gives?
It's part of the reason I'd never connected the Riemann CT and the connection 2-form, because I've always been taught it's just a regular tensor
recall that also in generic gauge theory, while the $A$ transforms non-linearly under gauge transformations: $A\mapsto g^{-1}Ag + g^{-1}\mathrm{d}g$, the curvature transforms linearly: $F\mapsto g^{-1}Fg$
i feel the matter field is also technically a gauge field
because it has just as many redundancies. it also transforms under the gauge transform
so if we define "guage field" to mean "redundant field", we hav to include these matter fields
but one crucial distinction is that this redundancy is not there in the free theory of the matter field, while it's still there in the free theory of the EM field
but technically, the redundancy is still there in the free matter field theory too. it's just that the curvature of the connection is zero
but when we write the free field solution of the matter field in qft, we have already chosen the trivial gauge. so no more degrees of freedom need to be removed
@SillyGoose it is far more likely that a theoretical physicist cranked out a few computations and then gets bothered by some regularities and then sets out to discover that they are a single result. Mathematical physicists are much less likely to actually be involved in a computation of that form.
homology considers maps from standard r-simplices to the manifold and makes it a group, and homotopy considers maps from n-cubes to the manifold with boundary identified, and makes it a group
what is the generalisation of these ideas
why consider maps from those two spaces
is there a general idea here or do these maps just happen to be useful
There are all kinds of notations, the most common notation is certainly $\frac{\partial^2}{\partial x\partial z}$ but the first two examples you've given probably make the most sense to the average person who knows what a partial derivative is
I've never seen your last example, since it doesn't suppress anything beyond $\frac{\partial^2}{\partial x\partial z}$, it's just more keystrokes to write basically the same thing
well this was a waste of time because in the problem statement they forgot the parentheses in this expression $\nabla\times\nabla\times\vec{A} = -\nabla^2\vec{A}+\nabla(\nabla\cdot\vec{A})$
should be $\nabla\times(\nabla\times\vec{A})=-\nabla^2\vec{A}+\nabla(\nabla\cdot\vec{A})$
@Charlie this derivation of the Riemann tensor shows how to see the Riemann tensor as a 2-form with two additional indices naturally, and combining with my post on connections yesterday, you can see that if $\mathbf{A} = A^{\rho} \mathbf{e}_{\rho}$ lives in the same space as your $x^{\mu}$, then it has all space-time indices,
however if you take your vector (the $\mathbf{A}$ now denoted) $ \mathbf{V}$ to live in some other space say with $SU(N)$ indices, you'd get Yang-Mills curvature with the connection (now denoted $\mathbf{A}$ as usual for YM connections) as I explained yesterday
@Mr.Feynman I found a complicated way to do it and I haven't written it up nicely yet
Can someone help me understand this statement: A 1st order phase transition is therefore a transition where the first derivative (dG/dT) is discontinuous, which is equivalent to a latent heat at the phase transition.
If the system has a 2nd order phase transition, does that mean that the latent heat is equal to the 2nd derivative w.r.t Temperature of Gibs free energy ?
@ACuriousMind i started skimming through "Quantization of gauge systems", and it's quite helpful, thank you again! although now im wondering why some of the introductory content isnt covered in classical mechanics texts like goldstein...
I think it's because in the particle mechanics context you can usually get rid of first-class constraints by fully gauge-fixing them and then get rid of all the second-class constraints by passing to a reduced phase space; if that works you can just start with the reduced description without having to develop the theory of constraints at all
The point of the first chapter is to show you there's around 50 pages of motivation and background all based on insane stuff like this formula for the basic models they set up from first principles in the second chapter on
@RyderRude What do you mean by your first statement on homology. To my understanding homology considers a functor which maps from one category into the category of chain complexes, then you talk about n-cycles that are not n-boundaries
And the space of n-cycles that are not n-boundaries is the nth homology class and through some theorems and etc tell you something about the space you functored into a chain complex
"the Laplacian" acts on functions, not vectors, the $\nabla^2 A$ is the "vector Laplacian" where you just apply the Laplacian to every component of the vector
just because the symbol $\nabla^2$ is the same it doesn't mean it's exactly the same operator
"the Laplacian" applied to a scalar field returns a scalar field
but you can also consider the "vector Laplacian" applied to a vector field, which returns a vector field where each component is the scalar Laplacian applied to the component of the original vector field
@ACuriousMind well stuff like density operator formalism of QM isn't new either but still not in textbooks on UG or grad quantu
sans a single section or two
@Obliv you can do whatever you would like. the $\langle - , - \rangle$ notation sucks in my opinion because this symbol is reserved for inner products in my head
i think in physics it is common to write $\vec{A} = (A_1, A_2, ..., A_n)$ and denote the $i^{th}$ component by $A_i$
if you are talking about a vector whose components correspond to $x, y, z$ directions, just define that $1 \mapsto x$, $2 \mapsto y$, and $3 \mapsto z$
if your coordinates are not Cartesian, then the Laplacian (either scalar or vector) is not $\partial_\mu \partial^\mu$, but includes terms incurred from non-zero Christoffels
ok I know we already discussed $\int dx (\dots)$ vs $\int (\dots)dx$ notation but what about $\hat{i}\left[blah\right]$ vs $\left[blah\right]\hat{i}$ notation
I kinda like the first one simply because I think it's more readable
heuristically: since you're doing electromagnetism, which is a relativistic theory, we will have four coordinates: one coordinate for time $x^0$ and three coordinates for space $x^i$ where $i = 1, 2, 3$. We combine the $0$ index and the $1,2,3$ indices into a single free index denoted by a greek index. This is for instance $\mu$ as ACM writes. Then, if I have an expression $A^\mu A_\mu$ in which the indices are repeated (and also on different floors), then this expression is short hand for
$\sum_{\mu = 0}^3 A^\mu A_\mu$
where we are writing $\vec{A} = (A^0, A^1, A^2, A^3)$ in terms of its components as $A^\mu$
@Obliv so this is not the $\nu$th partial of the $\nu$th component, it is the second partial derivative w.r.t. $x_\nu$
well the idea is not complicated. you start with a vector space $V$. it has elements $v \in V$. consider the set of linear maps $\omega: V \to \mathbb{R}$. we given this map a name: a linear (because it is...linear) functional (because it is a function of vectors).
the set of all such $\omega$ turns out to naturally have the structure of a vector space itself, and we call it the dual space $V^*$ to $V$
I wonder if $\mathbb{R}^*$ meaning the group of the nonzero reals under mult. and $\mathbb{R}^{**}$ are duals. yeah probably not since $\mathbb{R}$ isn't a group by itself under mult.
it needs to be linear and take in vectors as input and spits out real numbers (more generally elements of the field $V$ is defined over)
this is relevant because $A^\mu(x^\mu)$ at each point $x^\mu$ lives in a vector space $V$ while $A_\mu(x^\mu)$ at each point $x^\mu$ lives in the space dual to $V$, i.e. $V^*$
so then it actually makes sense what $A^\mu A_\mu$ is.
this is at least how i think about it but other people are more familiar with this sort of stuff :P.
