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19 votes
Accepted

Is intrinsic spin a quantum or/and a relativistic phenomenon?

SPIN ORIGIN Spin is a purely relativistic property. It comes in fact from the representation theory of the Lorentz group (the relativistic symmetries group). In classical mechanics, you have ...
LolloBoldo's user avatar
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13 votes

Is intrinsic spin a quantum or/and a relativistic phenomenon?

There is nothing quantum about spin. When the great mathematician Élie Cartan first introduced the concept of spinor in 1913, it's a purely classical concept of geometry. Quantum mechanics has not ...
MadMax's user avatar
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7 votes

Is intrinsic spin a quantum or/and a relativistic phenomenon?

The classical electromagnetic field has spin $S=1$. In my paper A theory of electromagnetism with uniquely defined potential and covariant conserved spin I proved this and defined the electromagnetic ...
my2cts's user avatar
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5 votes

Have all the symmetries of the standard model of particle physics been found?

My answer is also ‘No’ but by direct construction: over the past few years, various research groups have understood new symmetries of the Standard Model. Over the past decade, field theorists have ...
SethK's user avatar
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4 votes
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From any element of $\mathrm{SO}(8)$, can we always find one corresponding $\mathrm{SU}(3)$ element?

It seems the answer is negative: given one arbitrary $\mathrm{SO}(8)$ element, we cannot always find one corresponding $\mathrm{SU}(3)$ element in the sense of the transformation of the coefficients ...
narip's user avatar
  • 307
3 votes

Can you ever obtain a pure rotation from composing Lorentz transformations?

Ever? Always, of course, for infinitesimal boosts. Review your Wigner rotations but consider the evidently superior 2$\times$2 matrix representation of the spinor map, which protects you from the busy ...
Cosmas Zachos's user avatar
3 votes

Is intrinsic spin a quantum or/and a relativistic phenomenon?

There is nothing that distinguishes the spin from the magnetic dipole Spin was introduced into physics when it was realized that, in addition to its interaction with electric fields, the electron also ...
HolgerFiedler's user avatar
3 votes

Is intrinsic spin a quantum or/and a relativistic phenomenon?

I do not know how to discuss this for general spin, but I will try to give a concrete answer for spin 1/2. In non-relativistic quantum mechanics spin is usually discussed as some independent ...
Maik H.'s user avatar
  • 88
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
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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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
2 votes
Accepted

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
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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

Does all symmetry breaking have corresponding unitary group?

In general you can have broken symmetry groups different from $U(n)$. For example the quantum Ising model has a discrete symmetry (enacted by $P=\prod_j \sigma^z_j$) that breaks spontaneously in the ...
lcv's user avatar
  • 2,474
1 vote

Does all symmetry breaking have corresponding unitary group?

As in this SE post, you can have the Lagrangian with $SO(N)$ symmetry $${\mathcal{L}} = \frac{1}{2}(\partial_\mu \Phi)^T (\partial^\mu \Phi) - \left(\frac{1}{2}\mu^2 \Phi^T \Phi + \frac{1}{4}\lambda (...
Gabriel Ybarra Marcaida's user avatar

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