All Questions
Tagged with group-theory operators
77
questions
2
votes
1
answer
98
views
$Ad\circ\exp=\exp\circ ad$ and $e^{i(\theta/2)\hat{n}\cdot\sigma}\sigma e^{-i(\theta/2)\hat{n}\cdot\sigma}=e^{\theta\hat{n}\cdot J}\sigma$
This question is inspired by my recent question How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$? with answer https://...
- 14.4k
4
votes
1
answer
128
views
Is the factorization method of Hamiltonian related to the theory of Lie groups?
I was learning about algebraic methods to solve the H atom, when I came across the factorization method. It is mentioned in various textbooks, notes and papers, like the one from Infeld and Hull.
I am ...
- 105
3
votes
0
answers
48
views
How does the Hamiltonian act on the multiplicity space of irreps?
My question in the title stems from not completely understanding the last three lines of this answer. I list specific questions at the end of this post.
Setup. Consider a quantum system described over ...
- 171
2
votes
1
answer
56
views
Helicity operator in spinor-helicity variables
How do I prove that the helicity operator is
$$
H = \frac{1}{2} (\tilde{\lambda}_\dot{\alpha} \frac{\partial}{\partial \tilde{\lambda}_\dot{\alpha}} - \lambda_\alpha \frac{\partial}{\partial \lambda_\...
- 191
1
vote
1
answer
72
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Generator of two-qubit quantum gate
I would like to know how to derive the explicit form of the GENERATOR of a general two-qubit gate (also here), e.g., controlled-rotation Y.
From the definition: $$\exp(-i\theta G) \ ,$$
I see it is: $$...
0
votes
2
answers
65
views
Can the generators of a Lie group furnish its adjoint representation?
For generators of the Lie group under an arbitrary representation: $[T^a,T^b]=if^{abc}T^c$
$[T_A^c]^{ab}=-if^{cab}$ is the generator of the adjoint representation.
Is $\ \ e^{i\theta^d T^d}T^ae^{-i\...
- 619
1
vote
0
answers
52
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What kind of combinations of field components are equal under $SO(9)$ symmetry?
My question is a bit long and chaotic since I haven't learnt group theory systematically.
I am looking at the Banks-Fischler-Shenker-Susskind (BFSS) matrix model. It consists of 9 bosonic matrices $...
- 368
1
vote
0
answers
52
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Doubt about the diagonalization of a state
I'm studing the unitary irreducible representations of the deSitter group in two dimension.The generators of its double cover, the group $SL(2,R)$, are the following
$$
H_2 = \frac{\sigma_2}{2} \\
F_1 ...
- 191
2
votes
2
answers
171
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Irreps of symmetric group: Are projection operators always elements of the group algebra?
In the past, I studied the irreps of the symmetric group using chapter 5 and Appendix III of Wu-Ki Tung's book "Group Theory in Physics." I always thought that the reason that the theory ...
user7154
0
votes
0
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34
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Are elements of different invariant subspaces of a self-adjoint set orthogonal?
I know that self-adjoint operators have orthogonal eigenspaces, but how does that generalize to the orthogonality of invariant subsapces?
I am reading Fonda's Symmetry Principles in Quantum Physics ...
- 1,095
1
vote
1
answer
51
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Why we can set Lorentz generators to be antisymmeric?
I am following Weinberg 2.4 and I understand why $\omega_{\mu \nu}$ is antisymmetric, but he says after expanding $$U(1 + \omega, \epsilon) = 1 + \frac{1}{2}i\omega_{\rho \sigma}J^{\rho \sigma} - i\...
0
votes
0
answers
66
views
What is the action of generators on operators?
I have been following "Lie Algebras in Particle Physics: From Isospin to Unified Theories". In the textbook, the chapter on tensor operators defines them as operators that transform under ...
- 193
0
votes
1
answer
98
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Dimensionality of unitary representation of Lorentz group
I am reading the textbook "Lectures on Quantum Field Theory", second edition by Ashok Das. On page 137, he talks about the non-unitarity of the finite-dimensional representation of the ...
- 513
2
votes
2
answers
153
views
Power Series Expansion of Unitary Operators in Weinberg
For a Lie group $T(\theta)$ depending on a finite set of real parameters $\theta^a$,Weinberg (in his QFT book - Equations 2.2.17, 2.2.18, 2.2.19, p54) expands the unitary representations of $T$ in the ...
- 143
2
votes
2
answers
75
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Very simple translational invariant Hamiltonian and adaptation for physically sensible hamiltonian
If we consider the Hamiltonian of a system as given by
$$H_0 = \hat{p}.\tag{1}$$
Then the unitary evolution operator will be given by
$$U(t) = \exp\left({-\frac{iH_0t}{\hbar}}\right)= \exp\left({-\...
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