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10

Let me first remind you of (or perhaps introduce you to) a couple of aspects of quantum mechanics in general as a model for physical systems. It seems to me that many of your questions can be answered with a better understanding of these general aspects followed by an appeal to how spin systems emerge as a special case. General remarks about quantum states ...


6

The Poincare group has two Casimir Invariants - namely $p^2$ and $W^2$ where $$ W_\mu = \frac{1}{2} \epsilon_{\mu\nu\rho\sigma} J^{\nu\rho} p^\sigma $$ is the Pauli-Lubanski pseudo-vector. Thus, representations of the Lorentz group are labelled by the eigenvalues of both $p^2$ and $W^2$. When $p^2 = -m^2$, we have the property $W^2 = -m^2 {\bf J}^2$. ...


3

To show that $$ \left(\sigma\cdot\mathbf{n}\right)^2=\mathbf n\cdot\mathbf n+i\sigma\cdot\left(\mathbf n\times\mathbf n\right)\tag{1} $$ consider writing the above as \begin{align} \left(\sigma\cdot\mathbf a\right)\left(\sigma\cdot\mathbf b\right)&=\sum_j\sigma_ja_j\sum_k\sigma_kb_k\\ ...


3

Groups are abstract mathematical structures, defined by their topology (in case of continual (Lie) groups) and the multiplication operation. But it is almost impossible to talk about abstract groups. That is why usually elements of groups are mapped onto linear operators acting on some vector space $V$: $$ g \in G \rightarrow \rho(g) \in \text{End}(V), $$ ...


2

As @ACuriousMind pointed out in his comment, the definition of a pure state is one that is a vector of $\mathcal{H}$, in this case $\Bbb{C}^2\otimes\Bbb{C}^2$. If you take for example $\rho=|\varphi^+\rangle\langle\varphi^+|$, the reduced density matrix on the first space is \begin{align}\rho_1&=\operatorname{Tr}_2\rho\\ ...


2

I guess what you are missing is the following: given a representation $\rho(g)$ of $g\in$SU(2) acting on some vector space $V$. We define the representation $\rho_\otimes$ of SU(2) (not of SU(2)$\times$SU(2)) on $V\otimes V$ as $$\rho_\otimes(g) (v_1 \otimes v_2) = \rho(g) v_1 \otimes \rho(g) v_2.$$ So in fact we are defining the tensor product of two ...


2

Fermion wavefunctions are antisymmetric under the interchange of two particles. Spatial inversion flips the spatial coordinate, but does not interchange particles. In other words, let's say we have a two particle wave function, $\psi(x_1, x_2)$ (where $x_1$ is the position of particle 1, and $x_2$ is the position of particle 2). Being odd under parity ...


1

What Qmechanic said in comments is pretty solid, "Lagrangian (2) is not bounded from below because the kinetic term of $A_0$ field has the wrong sign, and hence the theory is not physical in the first place", but I think your Question needs a change of emphasis. Your Lagrangian allows us to construct four equations of motion for four non-interacting fields. ...


1

There are two other interpretation of the Pauli matrices that you might find helpful, although only after you understand JoshPhysics's excellent physical description. The following can be taken more as "funky trivia" (at least I find them interesting) about the Pauli matrices rather than a physical interpretation. 1. As a Basis for $\mathfrak{su}(2)$ The ...


1

Objects in orbit tend to lose their spin on their own axis. However they do not completely lose their rotation and end up rotating with a period that is the same as the orbital period, so that they face always the same side towards the other body. The best know example is the Moon that shows always the same side to Earh. The phenomenon is called tidal ...


1

I seem to have found an answer in the reference On the Variation of Ortho-hydrogen and Para-hydrogen Ratio with Magnetic Field Strength at Low Temperature. At high enough field strength, orthohydrogen rather than parahydrogen is favored.



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