> for every action (force) in nature there is an equal and opposite reaction
So if you have two charged bodies A and B and a field F then the change in momentum of any one body is equal and opposite to the change in momentum of the rest of the system.
My view is that yes it's just a statement of conservation of momentum, but I reserve the right to be wrong if there's some subtle difference that I can't remember.
For example I would say the first and second laws are the same - the first is just a special case of the second. But I get corrected by people explaining that there is a subtle difference.
@satan29 We've plenty of questions about that already, see e.g. physics.stackexchange.com/q/114466/50583, physics.stackexchange.com/q/136835/50583. You're entirely correct that the third law fails in electromagnetism - the third law is a statement about conservation of momentum of particles/bodies with mass, but in EM, there's more than just massive bodies that can carry momentum.
There seems to be quite a lot of questions that demand for textbook answers. These are technically not homework questions - i.e., they are not asking for solutions to problems - but grounded in not having read the relevant chapter of a textbook. They are not necessarily trivial either - i.e., the...
If I take derivative like $$\frac{\partial}{\partial t}\langle a|e^{\hat A t}|b\rangle=\hat A\langle a|e^{\hat At}|b\rangle$$ for what reason can the operator $\hat A$ be brought outside of the inner product?
I am assuming a product rule essentially applies here, $\partial_t (abc)=a'bc+ab'c+abc'$ and that $|a\rangle$ and $|b\rangle$ are not time dependent.
In fact I could be even more precise about the problem, that $\langle \vec v, \hat A \vec w\rangle\neq \hat A \langle \vec v,\vec w\rangle$
Well the context is basically showing that the propagator satisfies something resembling the Schrodinger equation, $K(q_a,q_b;T)=\langle q_b|e^{-\frac{i}{\hbar}HT}|q_a\rangle$, $i\hbar \partial_T K(q_b,q_a;T)=HK(q_a,q_b;T)$
What you've just said is basically my objection, but this is a fairly standard thing to show and seems odd because it just doesn't seem to add up
Maybe the problem is trying to use operators like derivatives in the same place as braket notation, I think I've seen posts on PSE about this before, specifically mentioning the Schrodinger equation
@Charlie Well, that's technically also wrong, the propagator is a Green's function and there should be a $\delta(q_b - q_a)$ somewhere, see Wiki for the correct equation
note the subscript $H_x$ they're using - this is not some equation of operators, it is a specific differential equation in position space for the function/distribution $K(q_a, q_b;t_a, t_b)$.
I mean, all of this is complete "physics math" still since we're using the "states" $\lvert x\rangle$ to begin with, don't expect this to make rigorous sense
This post is about this question that was closed (for good reason):
Is there such a concept as "opposite colors"?
I think the question is very interesting when one gets rid of all the ... (not sure how to describe it in a polite way).
Please tell me if the fix is good enough so as to reopen.
This is a bit of a wild swing because I have no idea how to answer it myself, but do you lose any information during Wick rotation? As in is there any downside whatsoever to Wick rotating, performing the integral and then "un"-Wick rotating?
the real underpinning of Wick rotation is the Osterwalder-Schrader reconstruction theorem, which says that given a Euclidean field theory obeying a list of axioms, the analytic continuation of the Euclidean fields defines a (relativistic, Wightman or Haag-Kastler) QFT and the analytic continuations of the n-point functions of the Euclidean theory are the correct n-point functions for this corresponding Minkowskian theory
that is, the fact that Wick rotation works is highly specific to the structure of quantum field theories - it's not something you do to a single integral, it's more an operation on the entire theory
Ah, it's usually introduced pretty ad-hoc to look at the 1-loop divergences in scalar field theory, I did wonder why it wasn't just applied to everything else
sure, the theorems don't "really" apply to most real QFTs anyway because we don't have a construction of Euclidean fields obeying the OS axioms in dimension >3 last time I looked
This post is about this question that was closed (for good reason):
Is there such a concept as "opposite colors"?
I think the question is very interesting when one gets rid of all the ... (not sure how to describe it in a polite way).
Please tell me if the fix is good enough so as to reopen.
Does Wick rotation work for quantum gravity? The Euclidean Einstein-Hilbert action isn't bounded from below.
