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### Using Lie theory to understand $U=e^{i\mathcal{H}t}$ [duplicate]

Can we use the exponential map (lie theory) to understand how the Hamiltonian $\mathcal{H}$ gives rise to the unitary, and therefore compliments an essential property of the unitary operator (ie to ...
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### Norm of Classical (Poissonian) Hamiltonian Operator

In the Poissonian formulation of classical mechanics, one finds that the time evolution of the phase space vector $\eta = (q_1,q_2\cdots q_n ; p_1, p_2\cdots p_n)^T$ can be put in terms of the ...
374 views

### Does Operator Product Expansion form an algebra?

The operator product algebra in CFT is defined as $$\mathcal{O}_i(z,\bar{z})\mathcal{O}_j(\omega,\bar{\omega}) = \sum_{k} C^k_{ij}(z-\omega,\bar{z}-\bar{\omega})\mathcal{O}_k(\omega,\bar{\omega}).$$ ...
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### Commutator relationships and the exponential

I am currently trying to prove that the two following commutator relationships are equivalent (for an operator $\hat{A}(s)$ that depends on a continuous parameter $s$), so if one holds the other one ...
177 views

### Nonabelian global symmetries, $SO(N)$ charges in terms of creation and annihilation operators

Consider an $SO(N)$ symmetric theory of $N$ real scalar fields,$$\mathcal{L} = {1\over2} \partial_\mu \Phi^a \partial^\mu \Phi^a - {1\over2} m^2 \Phi^a \Phi^a - {1\over4} \lambda (\Phi^a \Phi^a)^2.$$...
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### Calculate mean number of particles of time evolution coherent state [closed]

I seem to be missing some identities. I know you need to calculate P_n = |<n|alpha_t>|^2 and mean number of particles is the infinite sum of nP_n. However I ...
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### Relationship between those two “exponentials”

Let $G$ be a Lie group and $L(G)$ it's Lie algebra. We know that every left-invariant vector field $X$ in $G$ is complete, and so one can consider the integral curve defined for all $t\in \mathbb{R}$ ...
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### Uniqueness of expression of a Lie group element

Just take the SU(2) group as an example. The three generators are $J_z$, $J_+$, and $J_-$. For an element $g$, sometimes we want to express it as $$g = e^{i a J_+} e^{i b J_z} e^{i c J_-} .$$ ...
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### Is there a connection between Lie Groups and observable quantities in physics?

Good evening everybody. I have some questions about the relation between Lie groups and observables in physics. Indeed, taking the example of spin formalism of Quantum mechanics I know that Pauli's ...
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### Eigenvalues of Angular Momentum in Quantum Mechanics

The eigenvalue equation of the $L^2$ operator is given by $$L^2f_l^m = \hbar ^2l(l+1)f_l^m$$ Side: So a determinate state for some observable $Q$ is a state where every measurement of $Q$ returns ...
324 views

### Why does the raising and lowering operator not affect total angular momentum?

My notes define: $$L_{\pm} = L_{x} \pm i L_{y}$$ and states: $$[L_{z},L_{\pm}] = \pm \hbar L_{\pm}$$ I'm fine with this as it's easy to show the result with some ugly algebra. It then says: ...
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### Commutator summation notation

I have the relation $e^L M e^{-L}=\sum_{n=0}^\infty \frac 1{n!} [L,M]_{(n)}$ where $L$ and $M$ are operators. What does the subscript $n$ after the commutator bracket denote?
In Holstein-Primakoff and Dyson-Maleev representation, spin operators are represented by bosonic operators. Roughly speaking, a state with $S^z=S-m$ corresponds to a state containing $m$ bosons. In ...
How does one determine the members of an simultaneously commuting set (of operators)? For example, I have read that for orbital angular momentum, the set is {$H,L^2,L_z$}. How does one know that these ...