Where Infinite Spin Particles are Localizable

Abstract

Particle states transforming in one of the infinite spin representations of the Poincaré group (as classified by E. Wigner) are consistent with fundamental physical principles, but local fields generating them from the vacuum state cannot exist. While it is known that infinite spin states localized in a spacelike cone are dense in the one-particle space, we show here that the subspace of states localized in any double cone is trivial. This implies that the free field theory associated with infinite spin has no observables localized in bounded regions. In an interacting theory, if the vacuum vector is cyclic for a double cone local algebra, then the theory does not contain infinite spin representations. We also prove that if a Doplicher-Haag-Roberts representation (localized in a double cone) of a local net is covariant under a unitary representation of the Poincaré group containing infinite spin, then it has infinite statistics. These results hold under the natural assumption of the Bisognano-Wichmann property, and we give a counter-example (with continuous particle degeneracy) without this property where the conclusions fail. Our results hold true in any spacetime dimension s + 1 where infinite spin representations exist, namely s ≥ 2.

Wigner’s infinite spin representations and inert matter

“…The recent observation that Wigner's infinite spin representation class admits no compact causal localization [4] and that its tightest localized quantum fields "live" on semi-infinite strings (space-like rays) S(x, e) = x + R + e, e 2 = −1 [5] led to several new fundamental problems concerning the foundational properties of quantum fields. Besides the obvious question concerning possible physical manifestations of this new form of matter one could also ask whether a better understanding of the interplay between positivity and causal localization could favor the use of string-localized (sl) fields in a positivity-maintaining formulation of interactions involving finite higher spin particles.…”

“…These considerations were strengthened in [11]. A rigorous proof which excludes the possibility of finding compact localized subalgebras (generated by test-function-smeared pl fields) was finally presented in a seminal paper by Longo et al [4].…”

“…But the recent gain of knowledge from modular localization regarding attempts to unite WS with normal matter under the conceptual roof of AQFT in [4], as well as new insights coming from preliminary perturbative studies of couplings involving string-local fields [17,18], led to a revision of these ideas.…”

“…In [4] it was shown that the attempt to unite normal matter (matter as we know it) together with WS in a nontrivial way 3 under the conceptual roof of algebraic QFT (AQFT) leads to an unexpected loss of the so-called "Reeh-Schlieder property" for compact localized observable algebra. The R-S property states that the set of state vectors obtained by the application of operators from a compact localized subalgebra of local observables to the vacuum is "total" in the vacuum Hilbert space [13].…”

“…This leaves the conclusion that, apart from gravitational backreaction which require the solution of additional renormalization problems of the stress-energy tensor 4 (the substitute for the vacuum-based Wick-product [12]), WS cannot interact with normal matter in a way which complies with the causal localization structure of QFT. In mathematical terminology, WS matter tensor-factorizes with normal matter and the standard properties (including the ReehSchlieder cyclic action of local subalgebras on the vacuum vector [13]) hold only in the tensor factor of normal matter.…”