Recent content by sophiatev

Sorry, the PD code maps to a unique *link* diagram, so the Hopf link is a valid diagram (also, a ring is the unknot so it's still a knot, right? The Hopf link is just two linked unknots)

The PD code [(2, 3, 1, 4), (4, 1, 3, 2)] seems to map to a non-unique knot diagram. I can describe the following two Hopf links with different orientations with this same PD code. As I understand it, while a link diagram does not have a unique PD code, a given PD code should map to just one knot...

This is very interesting. I often wondered about what makes the "future" and "past" lightcones distinguishable - I guess this is one answer to that question. Thank you very much for all your help in general in this thread (as well as @vanhees71 and @Ibix), I feel I have a deeper understanding of...

I think I see what you guys mean. The idea is that because "experimentally" we know that we must advance in time, we want the notion of "time-progression", which I guess in this case relates to "timelike", to be distinct from other types of progression (namely spatial). So we can accomplish this...

Sorry, I should have been more specific. I'm using Carroll's definition from An Introduction to General Relativity. I'm not sure what a good way to do this is so I will just post images of the relevant pages (pg. 73-74) Above is his discussion of what "canonical form" means. Later he goes on...

Correct me if I'm wrong, but from my understanding "Lorentzian signature" simply means that when we put the metric into its canonical form, there is exactly one minus in the signature (and all the rest of the eigenvalues are positive). Is it simply a matter of convention that we always take the...

In Minkowski space, with line element $$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$$ (and ##c = 1##) we take spacelike trajectories to have ##ds^2 > 0##, null trajectories to have ##ds^2 = 0##, and timelike trajectories to have ##ds^2 < 0##. This makes sense given our definition of the line element...

Was that last equation meant to be ##G = x p_y - y p_x##? But otherwise, I think that makes sense, thank you. Sounds like I was interpreting the ##\delta q^j## and ##\delta p_j## as the differential changes in the respective variables when taking a derivative (i.e. ##\delta G = \sum \partial G /...

In Classical Mechanics by Kibble and Berkshire, in chapter 12.4 which focuses on symmetries and conservation laws (starting on page 291 here), the authors introduce the concept of a generator function G, where the transformation generated by G is given by (equation 12.29 on page 292 in the text)...

So it seems the typical way to approach this problem is to consider the sphere when it has charge q and radius r. With uniform charge density ##\rho##, this becomes ##q = 4/3 \pi r^3 \rho## and so ##dq = 4 \pi r^2 dr \rho##. Using our expression for the potential outside of the sphere, we find...

Turns out I had some reading to do to be able to understand your reasoning, but now that I'm caught up, that made sense, thank you :smile:. One thing I wanted to clarify - when you impose the canonical commutation relations on ##\hat{\phi}## and ##\hat{\Pi}##, you do so "axiomatically", correct...

In Quantum Field Theory and the Standard Model by Schwartz, he defines the Hamiltonian for the free electromagnetic field as (page 20, here's a link to the book). This follows (in my understanding) from the fact that the amplitude of the field at a given point in space oscillates as a simple...

In Henley and Garcia's Subatomic Physics, they introduce phase space in chapter 10 by considering all the possible locations a particle can occupy in a plot of ##p_x \ vs. x##, ##p_x## being the momentum of the particle in the x direction. They next consider an area pL on this plot, and state...

Ah, okay, I see. This counterexample was quite helpful, thank you!

In Griffith's Introduction to Electrodynamics, chapter 12, he discusses how electromagnetic fields transform when we move from one inertial reference frame to another. On page 553, he claims He then considers how the electric field inside a conductor made up of two parallel rectangular plates...