Spacelike deformations: higher-helicity fields from scalar fields
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Spacelike deformations: higher-helicity fields from scalar fields Vincenzo Morinelli1
· Karl-Henning Rehren2
Received: 25 May 2019 / Revised: 11 March 2020 / Accepted: 28 May 2020 © The Author(s) 2020
Abstract In contrast to Hamiltonian perturbation theory which changes the time evolution, “spacelike deformations” proceed by changing the translations (momentum operators). The free Maxwell theory is only the first member of an infinite family of spacelike deformations of the complex massless Klein–Gordon quantum field into fields of higher helicity. A similar but simpler instance of spacelike deformation allows to increase the mass of scalar fields. Keywords Quantum field theory · Representation theory · Deformation theory · Higher helicity · Operator algebra Mathematics Subject Classification 81R05 · 81R15 · 22E43
1 Introduction The basic idea of Hamiltonian perturbation theory is to start from a time zero algebra (“canonical commutation relations”) equipped with a free time evolution, and perturb the free Hamiltonian such that the observables at later time (t) := ei H t 0 e−i H t (where H is the perturbed Hamiltonian) deviate from the free ones. We present here
Vincenzo Morinelli: Titolare di un Assegno di Ricerca dell’Istituto Nazionale di Alta Matematica (INdAM fellowship).
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Karl-Henning Rehren [email protected] Vincenzo Morinelli [email protected]
1
Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, 1, 00133 Rome, Italy
2
Institut für Theoretische Physik, Universität Göttingen, 37077 Göttingen, Germany
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V. Morinelli, K.-H. Rehren
a “complementary” deformation scheme for free quantum field theories: fixing the algebra along the time axis, we deform the space translations, so as to obtain a different local quantum field theory in Minkowski space. Despite the apparent similarity, there are many differences, though. Hamiltonian Perturbation Theory (PT) is well-known to be obstructed by Haag’s theorem, which implies that the perturbation is possible on the same Hilbert space only locally. Globally, the perturbed vacuum state is not a state in the “free Hilbert space”, so that one is forced to change the representation of the time zero algebra. The need of renormalization of the mass also shows that one is even forced to change the time zero algebra itself. More precisely, interacting quantum fields in general do not even exist as distributions at a fixed time (see, e.g., [14,15]). A recent approach [3], designed to avoid these obstructions, uses instead of a CCR time-zero algebra, an abstract “off-shell” C*-algebra of kinematical fields on spacetime which supports a large class of dynamics (one-parameter groups of time-evolution automorphisms). The invariant states under each dynamics, however, annihilate different ideals of the algebra (“field equations”), such that the corresponding GNS Hilbert spaces cannot be identified for any time-zero subalgebra. In contrast, Wightman quantum fields can always be restricted to the time a
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