@lucabtz oh i wish i could find a good reference for the theoretical framework of these spinor BECs.
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
@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
> 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.
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
@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.
@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.
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+$F_{\mu \nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$
