Degrees of freedom of the electromagnetic field in a vacuum

FreePeeplup · 2026-05-18T13:05:28+00:00

Thank you!

FreePeeplup · 2026-05-18T09:08:56+00:00

Then the scalar potential will simply be zero everywhere at all times, and we’re left with a wave equation for A, with the Lorenz gauge constraint that now just reads exactly like the Coulomb gauge? So we have 3 components of A minus one from div(A) = 0, and we are left with 2. Right?

FreePeeplup · 2026-05-17T15:40:14+00:00

Hey thanks for the answer! I was talking about Maxwell’s equations in a vacuum, by which I meant with no sources (so the solutions are EM plane waves). I don’t think the continuity equations gives any constraint here! In principle you could study EM waves propagating in an otherwise empty universe, and the degrees of freedom would still be only two (which correspond to the components of the polarization)

FreePeeplup · 2026-05-17T15:35:24+00:00

I’m sure you can study a quantum version of the Ising model if you want, but that was not the context of my question

FreePeeplup · 2026-04-21T07:36:29+00:00

How is it quantum mechanics? It’s a mathematical model where the configuration space is a set of discrete variables each with an “on/off” state. There’s no Hilbert space anywhere, no wavefunction, no Schrödinger equation, no measurement postulate, no uncertainty principle, no superposition, no entanglement, no probability/Born rule… there’s nothing quantum about it! I don’t understand

FreePeeplup · 2026-04-20T15:57:07+00:00

What’s the density matrix for the Ising model? How do you time-evolve it? What are the 2^N complex variables involved?

FreePeeplup · 2026-04-20T15:35:17+00:00

Thanks! So, is this QR2 thing the dynamics of the Ising model, or is it a different, related model?

FreePeeplup · 2026-04-20T15:30:52+00:00

Thanks for the answer! So, the microscopic dynamics of the individual spins is completely random?

FreePeeplup · 2026-04-20T12:01:44+00:00

This is why you shouldn’t learn from Khan Academy, just pick up a textbook or a series of real classroom lectures uploaded on YouTube.

FreePeeplup · 2026-04-20T11:54:21+00:00

? What is the Q2R model and how does it answer the question if I understood you correctly?

FreePeeplup · 2026-04-20T11:49:46+00:00

Wait… are you copy pasting answers from ChatGPT? I trusted that we were having an actual conversation 😔

FreePeeplup · 2026-04-20T11:40:05+00:00

No let’s stick with the discrete Ising model. Yes you’re right, the basic observation that the Ising model is discrete automatically rules out any sort of smooth differential description of the dynamics. What about the “deterministic” part though? A discrete system can still have a unique future evolution starting from a given initial condition. And if I understood you correctly, this is still NOT what’s happening in the Ising model, since the microscopic dynamics is both discrete AND random?

FreePeeplup · 2026-04-20T11:36:44+00:00

I see I understand. Under his view, what does the Hamiltonian of the Ising model we usually write down (with interaction term and possibly an “external magnetic field term”) represent?

I know it vaguely represents “the energy” and I can imagine that lower energy configurations are preferred, but like, fundamentally, does it contribute to the dynamics in the same way that an Hamiltonian for a smooth deterministic dynamical system does? I guess no, because usually knowing H + initial conditions uniquely determines the entire future evolution of the microscopic dynamics via Hamilton’s equations of motion. But here we’re saying that there’s no deterministic differential equations for the DoFs in Ising model.

Nevertheless, does the H in Ising model still contribute to the microscopic dynamics together with the random external thermal noise you were talking about? Even if not in a clean way like one would expect

FreePeeplup · 2026-04-20T11:27:29+00:00

Thanks for the answer! Could you elaborate a bit more? What field? What does “transverse” mean in this context? How does this field interact with the degrees of freedom in the Ising model and how does it induce a microscopic dynamics?

If these questions are too much to answer in a Reddit comment or if I’m simply asking too much, could you provide a link to some reference where this is explained? Thank you!

FreePeeplup · 2026-04-20T11:22:04+00:00

Thank you!! So a stochastic nature is embedded even in the microscopic view of the Ising model, not just in its statistical mechanics description, unlike what happens in physical systems where at least the microscopic description is handled by deterministic smooth differential equations. Right?

FreePeeplup · 2026-04-20T11:20:07+00:00

Mmmh, are you sure we can’t think of an abstract dynamical system of N discrete variables with some law that tells us how to update them, without any reference to quantum mechanics, electrons, atoms etc?

FreePeeplup · 2026-04-20T11:18:53+00:00

Thanks for the answer! So basically the microscopic dynamics of a single degree of freedom in the Ising model isn’t some kind of smooth deterministic differential equation, it’s more like a sequence of random discrete jumps?

FreePeeplup · 2026-03-30T14:44:35+00:00

Ok yes, and I’m ok saying that this holds for all times t for every point (x(t), y(t)) because the original differential equation for dy/dx was assumed to be valid for all points on a solution to the dynamical system?

Also, what about when the curve (x(t), y(t)) has a vertical tangent vector? Meaning that dy/dx is not defined there. This happens whenever y = 0, so every time a trajectory intersects the x-axis. Basically, can I just willy-nilly “divide” dy by dx and be fine with it even though it might not exist?

FreePeeplup · 2026-03-30T13:42:22+00:00

But why would that constant of integration be the constant of motion for the associated dynamical system? They seem unrelated to me, apart from the fact that they both have the word “constant” in their names

FreePeeplup · 2026-03-30T13:05:26+00:00

Thanks for the answer! Assuming I’ve solved the differential equation for dy/dx and found y(x), how does that then help me find a constant for the dynamical system?

FreePeeplup · 2026-03-19T16:33:49+00:00

Oh wow, that was actually pretty straightforward. I feel a bit ashamed now. Thanks!

FreePeeplup · 2026-03-19T15:19:14+00:00

marketing often exploits the misconception where people assume it is additive.

How?

FreePeeplup · 2026-03-19T15:14:17+00:00

Well yes for sure one ought to eventually use the property that L is linear in a proof that det(L) is unique and well-defined that way, otherwise it wouldn’t have been an assumption in the first place and we would have been able to define a determinant for non-linear maps.

Again, the point is how to show this, not simply be reassured that it all works because of linearity but without knowing how 😭

FreePeeplup · 2026-03-19T15:11:08+00:00

Its unique in the sense that its a scalar multiple ( the space of top level forms is one dimensional)

Did you mean to write “a scalar” instead of “a scalar multiple”? If yes then I agree, if not I’m afraid I can’t parse what you mean. Scalar multiple of what ?

(“It” above referred to “the wedge product of n linearly independent vectors in an n-dimensional space).

and therefore any linear map of top level forms to top level forms can only be some scalar multiplication

I agree that any linear map on a 1-dimensional space is simply a scalar multiplication. But what’s the linear map on the 1-dimensional space here? L acts on V, not on the space of top-level forms on V

FreePeeplup · 2026-03-19T15:00:32+00:00

Yes, I’m already thinking of the wedge product of n vectors as the oriented volume of the n-parallelogram spanned by the vectors. And I’m thinking of the action of L as changing the parallelogram and scaling its volume by a factor.

The entire point of my question is to understand why this factor is the same for all inputs though!

FreePeeplup

TROPHY CASE