this question is motivated by something @Relativisticcucumber said. I am wondering if every single two level system can be interpreted as a spin-1/2 system.

In particular, let the two levels be $\lvert a \rangle, \lvert b \rangle$. Then, we can define $\sigma_z = \lvert a \rangle \langle a \lvert - \lvert b \rangle \langle b \lvert$ and analogous define $\sigma_x$ and $\sigma_y$

supposing that the algebra of operators is induced by experimental choices (choices of what things to measure), then it seems we can perhaps always construct some choices of operators corresponding to the above definitions to treat our system like a spin-1/2 system

hehe i have finally finished my chern simons notes

h o n k 👍

H O N K ~

@SillyGoose you can; however, $\sigma_x$ and $\sigma_y$ might be meaningless in such a system.

@Sanjana hi. i meant ur recent post about symmetries. and {} is the Poisson bracket in my comment

@Sanjana one can make H time dependent to make it commute with boost (e.g. H=mX-Pt)$ but it wudnt b very interesting becuz the eigenfunctions become time dependent

@SillyGoose after including the identity matrix, Pauli matrices form a basis for 2x2 Hermitian matrices. so u can write the Hamiltonian of any 2x2 quantum system using them

see this physics.stackexchange.com/questions/205524/…

but the coefficients must b real to span Hermitian matrices

I just made a polar plot of the distance of the centre of the Sun from the SSB (Solar System barycentre). astronomy.stackexchange.com/a/44903/16685

> In this plot, the radius gives the Sun-SSB distance (in millions of km), the angle (anticlockwise, from the +X axis) gives the date. The date wraps around with a period of 65295.5 days, so each degree corresponds to ~2 years. Each cycle is plotted in a different colour of the rainbow, starting at red.

that's a pretty butterfly~

I was tempted to do more cycles, but it gets a bit messy. The Sun's speed relative to the SSB is actually fairly small. From astronomy.stackexchange.com/a/28036/16685

oh, that is unexpectedly human-relateable

about a bike ride's speed?

@naturallyInconsistent Indeed! It's positively glacial compared to usual solar system orbit speeds.

@RyderRude ACM said that this might not be possible even for time dependent Hamiltonians because $[H,J]=-\partial_t J$ is non-zero, for boost invariant systems.

Somehow, his words manipulated me to stop thinking about systems not invariant under boosts :p

@RyderRude Yes. But the formulation I am looking for can be more general. See SillyGoose's comments on the post and discussions in this chat.

@Sanjana oh

makes sense. Valter Moretti's definition of time dependent symmetries isnt satisfied either if we set H=J

@SillyGoose What does "treat like a spin-1/2 system" mean? All you're really doing here is observing that the space of 2-by-2 matrices is spanned by the Pauli matrices and the identity, nothing to do with spin as such.

in this song, Beatles deviate from the popular song structure by cutting things short

but kind of hypocritical of Lennon to write this stuff when he was abusing his family around that time

the @SillyGoose mobile

what do u think about the philosophy where all humans are the same organism

and I got stuck with commutators again :P

is the following result correct: $$ [S_i,S_{1j}S_{2j}] = [S_{1i} + S_{2i}, S_{1j}S_{2j}] = \cdots = i\hbar( \epsilon_{ijk} S_{1k} \otimes S_{2j} ) + i\hbar( S_{1j} \otimes \epsilon_{ijk}S_{2k} )$$

Indices 1,2 refer to the subspaces of two particles $\mathcal{E}_1, \mathcal{E}_2$ and $i,j,k$ are the ordinary cartesian components $x,y,z$

Wait, I have a hunch it should be zero

forgot to point out that $\mathbf{S} = \mathbf{S}_1 + \mathbf{S}_2$ altough it is highly inferable :P

This comes from trying to prove $[\mathbf{J},\mathbf{S}_1 \cdot \mathbf{S}_2] = 0$ via the fundamental commutation relations

and not by noting that that dot product is a scalar quantity which obv commutes with $\mathbf{J}$

Why does deriving Maxwell equation from $\mathcal{L}=-\frac{1}{4}F_{\mu \nu}(\partial^ \mu A^\nu-\partial^ \nu A^\mu)$ where $A$ and $F$ are independent variables, does not work? E.g. I get the correct Maxwell equation by varying the action w.r.t. $A$ but variation w.r.t. $F$ gives $\partial_\mu A_\nu=\partial_\nu A_\mu$.

A similar exercise from a different Lagrangian density $\mathcal{L}=\frac{1}{4} F_{\mu \nu} F^{\mu \nu} -\frac{1}{2}(\partial^\mu A^\nu-\partial^\nu A^\mu)$ seems to work all fine. It gives Maxwell's equation+$F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$