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Consider a relativistic quantum theory in d-dimensional flat spacetime. Neglecting possible internal symmetries, a particle is defined as a system whose Hilbert space furnishes the support of an irreducible unitary representation of the universal covering group of the Poincaré group $\mathcal P$. We know that there is a 1-1 correspondence between such irreps and the irreps of the possible little groups: if the Casimir $P^2$ takes a positive, negative or zero value on the irrep we're studying, the little group is, respectively, $Spin(d-2, 1)$, $Spin(d-1)$ or $ISpin(d-2)$. Thus we can equivalently define a particle by specifying its mass squared (which fixes the little group), and the specific irrep of the little group.

Now, for a massive particle in $d=4$, the irreps of $Spin(d-1)=Spin(3)=SU(2)$ are classifiable by a single number $j\in \mathbb{Z}/2$, which we refer to as spin. The name is no coincidence since in $d=4$, the second Casimir of $\mathcal P$, in the rest frame of the particle, is proportional to the square of the generator of rotations in the Lorentz algebra, with eigenvalues $j(j+1)$ for $j\in \mathbb{Z}/2$.

My question is: is there a way to work out the same reasoning in generic spacetime dimensions?

I'd expect that, considering for example the massive case, since the irreps of $Spin(d-1)$ are classified by more than a single Casimir, we would be forced to use a set of numbers to unambiguously fix the irrep we're referring to. In this case it is no longer clear to me what is meant by a "spin j" particle. Would it make sense to consider how such an irrep branches when viewed as an representation of a $SU(2)$ subgroup of $Spin(d-1)$ and then declare (somewhat arbitrarily) the spin of the particle to be the highest of those appearing in this decomposition?

Jack Euler
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    The massive little group is $Spin(d-1)$ instead of $SU(d-2)$ (of course these agree for $d=4$). Related/possible duplicate: https://physics.stackexchange.com/q/221881/84967 – AccidentalFourierTransform Nov 30 '23 at 01:45
  • You can still have massive spin $s$ UIRs in $d$ dimensions, but these are special cases when the UIRs of the little group are classified by single row rectangular Young tableaux of length $s$. For the more general spin representations, where multiple spin labels are needed, you can read e.g. https://arxiv.org/abs/hep-th/0611263. At a field theoretic level the latter are realised by tensors of a mixed symmetry type, whereas the former are typically realised on symmetric traceless rank-$s$ tensors. – NormalsNotFar Nov 30 '23 at 10:37
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    Another possible duplicate: https://physics.stackexchange.com/q/381752/50583 – ACuriousMind Nov 30 '23 at 16:09

1 Answers1

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The spin of a particle is an intrinsic quantum property that is associated with its intrinsic angular momentum. In the context of quantum field theory and d-dimensional Minkowski spacetime, the spin of a particle is described by representations of the Lorentz group, which is the symmetry group of special relativity.

The specific definition of a particle's spin depends on the type of particle and the associated Lorentz group representation. Here's an overview:

  1. Spin of elementary particles:

    • Fermions (particles with semi-integer spin): These include quarks and leptons. In (d)-dimensional Minkowski spacetime, fermions are described by spinor representations.
    • Bosons (particles with integer spin): These include photons, gluons and W/Z bosons. Bosons are described by tensor representations of the Lorentz group.

  2. Spin of composite particles:

    • For composite particles such as mesons or baryons, the total spin can be determined by the combination of the spins of their constituents and the orbital angular momentum.

In Minkowski spacetime, the Lorentz group is associated with Lorentz transformations, which include spatial rotations and boost transformations (changes in relative velocity). Lorentz group representations describe how quantum fields transform under these transformations.

For example, in the case of fermions, spinor representations can be characterized by the spin quantum number (s), where (s) is a semi-integer number ((s = 1/2, 3/ 2, \ldots)). In the case of bosons, tensor representations can have integer spin quantum numbers ((s = 0, 1, 2, \ldots)).

It is important to note that the full description of a particle's spin involves more advanced concepts of quantum field theory and Lie algebra, which go beyond an introductory description.

José Torres
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