Linked Questions
10 questions linked to/from Tensor Operators
10
votes
2
answers
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Why does a Lorentz scalar field transform as $U^{-1}(\Lambda)\phi(x)U(\Lambda) = \phi(\Lambda^{-1}x)$?
This problem is from Srednicki page 19. Why $U^{-1}(\Lambda)\phi(x)U(\Lambda) = \phi(\Lambda^{-1}x)$?
Can anyone derive this?
$\phi$ is a scalar and $\Lambda$ Lorentz transformation.
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11
votes
1
answer
3k
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Lorentz group representations in QFT: what's the vector space?
In QFT, a representation of the Lorentz group is specified as follows:
$$
U^\dagger(\Lambda)\phi(x) U(\Lambda)= R(\Lambda)~\phi(\Lambda^{-1}x)
$$
Where $\Lambda$ is an element of the Lorentz group, $\...
- 2,917
3
votes
2
answers
3k
views
Tensors in Quantum Mechanics and tensors from linear algebra
Consider the following two understandings of tensors:
Given $k$ vector spaces $V_1,\dots,V_k$ one can define the tensor product $V_1\otimes\cdots \otimes V_k$ by means of the universal property: it ...
- 37.4k
6
votes
1
answer
3k
views
Scalar field transformation and generators
When we do a transformation (norm preserving one) for a given quantity, from what I have understood it seems like there is a representation of the group element for each quantity depending how they ...
- 3,134
6
votes
4
answers
7k
views
What type of mathematical object is the "Pauli vector"?
The three pauli matrices $\sigma_x$, $\sigma_y$, $\sigma_z$ are sometimes combined in the "Pauli vector", usually denoted $\boldsymbol{\sigma} = \sigma_{x} \boldsymbol{e_x} + \sigma_y \boldsymbol{e_y} ...
- 1,320
4
votes
1
answer
5k
views
Vector operators in quantum mechanics
Vector operators $\vec{V}$ in quantum mechanics are usually defined as those that commute in a particular way with the spatial Angular Momentum $\vec{L}$: $[L_i,V_j]=i\hbar\varepsilon_{ijk}V_k$. I am ...
- 484
2
votes
3
answers
286
views
What does “transform among themselves” mean?
I'm reading a script on atomic physics, and there's a chapter on irreducible tensors. I can't understand the meaning of "transform among themselves" in this context:
An arbitrary rotation of the ...
- 3,507
0
votes
2
answers
1k
views
What is $\vec \sigma$ in the context of spin?
Leonard susskind's book, quantum mechanics, the theoretical mimimum, statea that we can derive the spin operator
$$\sigma_n=\sigma_x n_x + \sigma_y n_y + \sigma_z n_z$$
Where the $\sigma_i$ are spin ...
- 1,880
1
vote
1
answer
247
views
Commutation of abstract $O(3)$ generators and vectors [closed]
I've been given the following problem, and I'm quite lost with it.
Let $L_1$, $L_2$, and $L_3$ denote the abstract $o(3)$ algebras. You are given that $\vec{A} = (A_1, A_2, A_3)$ and $\vec{B} = (B_1, ...
- 205
2
votes
1
answer
181
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How is a tensor operator defined in terms of commutators?
If $J_i$ represent the angular momentum operators, then a scalar operator $S$ (rank-0 tensor) is defined as an operator which satisfies $$[S,J_i]=0$$ for $i=1,2,3$.
$A_i$ is a vector (rank-1 tensor) ...
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