# What type of fields are continuous spin representations?

Continuous spin representations (infinite dimensional representations of the Lorentz group) are pretty rarely discussed, and usually not in that much mathematical details. And usually it is done in a very "physicist" way. The field operators for a CSR is given as $\hat \varphi_{\theta_n}(x)$, a field with a family of continuous index (apparently transforming as a 4-component spinor), that transforms as

$$U(\Lambda)\hat \varphi_{\theta_n}(x) U^{-1}(\Lambda)\rightarrow \hat \varphi_{\Gamma_{mn}(\Lambda)\theta_n}(\Lambda x)$$

With the matrices $\Gamma$ a rep of the Lorentz group, written as

$$\Gamma_{mn}(\Lambda) = (e^{-i\alpha J_3}e^{-i\beta J_2}e^{-i\gamma K_3})_{mn}$$

$J$ and $K$ the usual spin transformation matrix.

What object would the field itself correspond to, though? What kind of fiber would correspond to an infinite (continuous) dimensional manifold transforming as such under the Lorentz group, in the same way that a scalar field corresponds to the line bundle, a vector field to the tangent bundle, etc? It's not any tensor bundle, obviously.

• semi-infinite spacelike strings : See arxiv.org/pdf/1601.02477.pdf – jjcale May 12 '16 at 16:03
• Lots of detail and references about continuous spin representations can be found at physicsoverflow.org/24751 – Arnold Neumaier May 31 '16 at 12:50