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5 votes

Why is half-integer spin not observed classically?

The other answers are great, and it's very much true that you can totally write down a classical field theory with half-integer-spin fields. That notion still feels kind of odd, though, because all ...
Rokas Veitas's user avatar
3 votes
Accepted

Given a representation $(n, m)$ of the Lorentz group, is the little group representation just the tensor product $n \otimes m$?

Consider the restricted Lorentz group $SO^+(3,1;\mathbb{R})$ and its complexification $$SO(3,1;\mathbb{C}).\tag{1}$$ Picking the COM frame the massive little group becomes the 3D rotation group $SO(3,\...
Qmechanic's user avatar
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3 votes
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Generators of rotations: $[J_i, J_j] = \epsilon_{ijk} J_k$ and $(J_i)_{jk} = -\epsilon_{ijk}$. Is this a coincidence?

This is not a coincidence. The three matrices you have chosen as generators for your rotations live in the adjoint representation of $\mathfrak{so}(3)$. This explicit matrix elements of the adjoint ...
CStarAlgebra's user avatar
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3 votes

Rotation of spherical harmonics

You are probably replicating common errors of basis conversion from Cartesian to the spherical basis and back (!), but it's harder to descend into the error instead of doing it right. You know from ...
Cosmas Zachos's user avatar
2 votes
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The meaning of a representation in one-dimensional quantum mechanics

In abstract terms the meaning of representation in these cases is the ordinary meaning of representation in the sense of representation theory. Concretely, you are representing the Heisenberg algebra ...
ACuriousMind's user avatar
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2 votes
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Orthogonal singlet states?

Is there an issue in my reasoning or in my understanding of spin states? In both. You are abusing Dirac notation, meant to defocus from side issues and omit other labels (and bases of tensor calculus ...
Cosmas Zachos's user avatar
2 votes
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How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$?

Physically, you can solve the Heisenberg equations of motion. It is equivalent to all the general arguments of the adjoint action, but at least it cuts to the chase and gives your formula explicitly. ...
LPZ's user avatar
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2 votes
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$Ad\circ\exp=\exp\circ ad$ and $e^{i(\theta/2)\hat{n}\cdot\sigma}\sigma e^{-i(\theta/2)\hat{n}\cdot\sigma}=e^{\theta\hat{n}\cdot J}\sigma$

The original question is really almost there. The answer is that we can almost prove $(1)$ directly from $(2)$. All we need in addition is a few facts about the pauli matrices $\sigma$ and the ...
Jagerber48's user avatar
  • 14.3k
2 votes

Why is half-integer spin not observed classically?

Why is half-integer spin not observed classically? Why do you single out half-integer spin? Are you implying that integer spin can be observed classically? If neither integer spin nor half-integer ...
MadMax's user avatar
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2 votes

Given a representation $(n, m)$ of the Lorentz group, is the little group representation just the tensor product $n \otimes m$?

I am not going to read Weinberg's book with you to your satisfaction, nor should I. Indeed, the little group of a massive particle (go to its rest frame) is the rotation group SO(3), sharing a Lie ...
Cosmas Zachos's user avatar
1 vote

Why is half-integer spin not observed classically?

Ferromagnets are a classical manifestation of electron spin. MRI is a classical manifestation of nuclear, mostly proton, spin. Both are spin half. What you are looking for is a -1 factor upon 360$^\...
my2cts's user avatar
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1 vote

How to prove $e^{+i(\theta/2)(\hat{n}\cdot \sigma)}\sigma e^{-i(\theta/2)(\hat{n}\cdot \sigma)} = e^{\theta \hat{n}\cdot J}\sigma$?

You appear to have two unrelated gaps, the final one being the rotation formula for the vector (triplet) representation, the celebrated Rodrigues rotation formula. $\vec \sigma$ rotates like a vector. ...
Cosmas Zachos's user avatar
1 vote
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Is the Dirac adjoint in the representation dual to Dirac spinor?

In order for $\overline{\psi} \psi$ to be a Lorentz scalar, we should have $$\lambda^{\dagger}\gamma^0\lambda = \gamma^0$$ or equivalently (assuming $(\gamma^0)^2 = 1$) $$\lambda^{-1} = \gamma^0\...
MadMax's user avatar
  • 4,452

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