I hope it's all right to ask a question about some very basic intuition in particle physics. I was looking into these (very nice) notes of Matthew Schwartz, I guess from his course on QFT:


Around the middle of the second page, one finds the lines

'The unitary representations of the Poincare group were classified by Eugene Wigner in 1939. As you might imagine, before that people didn’t really know what the rules were for constructing physical theories, and by trial and error they were coming across all kinds of problems. Wigner showed that irreducible unitary representations are uniquely classified by mass $m$ and spin $J$. $m$ can take any value and spin is a half integer $J = 0, 1/2, 1, 3/2,...$ Moreover, Wigner showed that if $m > 0$ then there are $2J + 1$ basis states in the representation, and if $m = 0$ there are exactly 2, for any $J > 0$. These basis states correspond to linearly independent polarizations of particles with spin $J$.'

Even though I've given a lengthy quotation for clarity of context, my question is about the last line. To avoid confusion, let's first note that the sentence about the number of basis vectors in the representation, taken literally, is wrong. The representation is infinite-dimensional. So I understand the 'representation' in that part of the sentence to mean the representation of the so-called 'little group' out of which one builds up the given representation of the Poincare group. Or, in other words, he is referring to the vectors in the representation corresponding to a fixed character of the translation subgroup. (We will ignore the fact that such a subspace doesn't exist rigorously.)

Now, what I find most interesting is his description of the basis vectors as 'linearly independent polarizations of particles of spin $J$'. I would like to know if this is a completely standard view, whereby the (complex) directions in the representations of the little group are thought of as 'polarizations'. If so, can one give an intuitive account of how they are natural extensions of, say, the polarization of light? Here, I would be grateful for some physical intuition, since the parallel at the level of mathematical formalism is reasonably clear. If there is good reference where this is discussed, I would be equally grateful.


Let's first clarify the physical sense of discrete labels of irreducible representations in massive and massless case belonging to corresponding little groups.

For massive representations, the little group is isomorphic to $SO(3)$. But this is nothing but the subgroup of the Lorentz group in our world associated with the spatial rotations. Its generators $\hat{J}_{i}$ are physically associated with angular momentum operators, the squared operator $\hat{J}_{i}^{2}$ defines the total spin number at rest (since the Lorentz orbit for massless particles are generated by $k_{\mu} = (m, 0, 0, 0)$), while, say, acting of, say, $\hat{J}_{3}$ on the irreducible massive representation $|k,\sigma\rangle$, $\hat{J}_{3}|k, \sigma\rangle = \sigma|k,\sigma\rangle$, defines the projection of the spin on the axis z for the particle at rest.

For massless representations, the little group is isomorphic to $ISO_{+}(2)$, and representations are characterized by the helicity $\lambda$, defined as $$ \tag 1 \hat{W}_{\mu}|p,\lambda\rangle = \lambda \hat{P}_{\mu}|p,\lambda\rangle $$ Here $\hat{W}_{\mu}$ is the Pauli-Lubanski operator, and $\hat{P}_{\mu}$ is translation operator. $\lambda$ is again has sense of the angular momentum, since it is defined das the eigenvalue of a 2-plane rotation generator $\hat{J}_{3}$ (for the standard vector $p_{\mu} = (1,0,0,1)$), $$ \hat{J}_{3}|p,\lambda\rangle = \lambda |p,\lambda\rangle, $$ but now it is rather the projection the total angular momentum on the direction of motion of massless particle, which is told by $(1)$.

Helicity and polarization of light

Finally, let's relate the photon helicity to the physical polarization of light (as example). The relation is not quite obvious since there are many polarizations bases - linear, elliptical, circular, and, to the best of my knowledge, the relation is not so intuitive, since requires to use the equations of motion.

Let's use the equations of motion for the field representing the helicities $\pm 1$ states. It can be shown that such fields are given by antisymmetric tensors $\hat{F}_{\mu\nu}^{\pm}$, obeying standard Maxwell equations of vacuum, $$ \partial^{\mu}\hat{F}^{\pm}_{\mu\nu} = 0, \quad \epsilon^{\mu\nu\alpha\beta}\partial_{\nu}\hat{F}_{\alpha\beta}^{\pm} = 0, $$ and satisfying the constrains $$ \tag 2 \hat{F}^{\pm}_{\mu\nu} = \pm \frac{i}{2}\epsilon_{\mu\nu\alpha\beta}\hat{F}^{\alpha\beta, \pm} \quad \text{for}\quad \lambda = \pm 1 $$ Using representation $\hat{F}_{\mu\nu}^{\pm} = (\hat{\mathbf E}^{\pm}, \hat {\mathbf B}^{\pm})$ in terms of electric and magnetic fields, the constrain $(2)$ can be rewritten in the form $$ \tag 3 \hat{\mathbf E}^{\pm} = \pm i\hat{ \mathbf B}^{\pm} $$ By using the equations of motion for $\hat{F}_{\mu\nu}^{\pm}$ we see that the vector expansion of $\hat{\mathbf E}^{\pm}$ and $\hat{\mathbf B}^{\pm}$ is perpendicular to the direction of the helical EM wave propagation, i.e., it can't be linear.

Next, the constrain $(3)$ together with Maxwell equations gives the precise vector expansion $\mathbf{e}_{\pm} = \frac{1}{\sqrt{2}}(\mathbf{e}_{x} \pm i\mathbf{e}_{y})$ (for travelling along the $z$ axis), which is nothing but the circular polarizations.


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