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From what I understand the spin of elementary particles comes from combining relativity with quantum mechanics so if the speed of light was infinite would elementary particles still have spin?

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The property of spin is independant from relativity and quantum mechanics (well not quite). It is linked with the rotational invariance of space.

As per Noether's theorem, continuous symmetries are associated with invariant quantities. In the case of rotational symmetries, this is angular momentum. For a field that does not transform trivially under the rotation, this can be decomposed into two parts, one part depending on the momentum of the field itself, the other is the spin density. This is true even in classical mechanics, the simplest example being the spin density of the electromagnetic field.

What relativity and quantum mechanics do is that, for relativity, the rotation group changes ($SO^\uparrow(3,1)$ instead of $SO(3)$), and for quantum mechanics, the spin density, which is a continuous quantity, becomes discrete. In the case of the speed of light becoming infinite, the Lorentz group will then just break down to just the 3D rotation group, and we can use the simple spins used in non-relativistic quantum mechanics, such as using the Pauli equation instead of the Dirac equation for spin $\frac{1}{2}$ particles instead of the Dirac equation, with $SU(2)$ spinors instead of $SL(2,\Bbb C)$.

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In the case of infinite speed of light the Poincare group contracts to the semidirect product of Galileo group and the central charge $\hat{M}$, which defines the mass; for simplicity I'll call such group the Galileo group, since "true" Galileo group algebra may be. formally expanded by adding $\hat{M}$ operator.

To find the definition of spin of elementary particle from the position of Galileo group, we need to find the galilean analog of Pauli-Lubanski operator, the square of which defines the spin of the representation of the Poincare group. But we expect that squared spin as Casimir operator for Galileo group exist as well as for Poincare group, since in Poincare case it is defined independently on the frame of representation (i.e., the eigenvalues of $\hat{\vec{P}}$ of spatial translation operator on irreducible representation may be arbitrary, for example, zero).

The Galileo group contains generators of galilean boosts $\hat{C}_{i}$, generators of spatial $\hat{P}_{i}$ and time $\hat{H}$ translations, generators of rotation $\hat{L}_{ij}$ and mass operator $\hat{M} = m\hat{I}$. Again, as in the case of the Poincare group, we may matematically identify the elementary particle with irreducible representation of the Galileo group. To define the classification, we need to find all of Casimir invariants for Galileo group. These invariants are

1) Central charge $\hat{M}$, which is defined through ($C_{i}$ is generator of galilean boosts, $P_{j}$ defines the generator of spatial translation) $$ [C_{i}, P_{j}] = iM\delta_{ij}; $$ it is associated with the mass;

2) $\hat{W} = \hat{M}\hat{H} - \frac{\hat{P}^{2}}{2}$, which is associated with internal energy;

3) Analog of squared Pauli-Lubanski vector, $$ \hat{\vec{W}} = \hat{M}\hat{\vec{L}} + \hat{\vec{P}}\times\hat{\vec{C}}, $$ which defines the square of spin at rest.

I.e., irreps of Galileo group are classified by values of spin, mass and scalar internal energy.

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