Poincaré Invariant Quantum Theory

Wayne Nicholas Polyzou, Ch. Elster, W. Glöckle +5 more

Russell K. McCormmach

Akira Matsumura

We consider how the reduced dynamics of an open quantum system coupled to an environment is realized in the Poincar\'e symmetry. The reduced dynamics is described by a dynamical map, which is a quantum channel (a completely positive and trace-preserving linear map) given by tracing out the environment from the total unitary evolution without initial correlations. We investigate the dynamical map invariant under the Poincar\'e transformations and discuss how the invariance constrains the form of the map. Based on the unitary representation theory of the Poincar\'e group, we develop a systematic way to construct the dynamical map with the Poincar\'e invariance. Using this method, we derive such a dynamical map for a spinless massive particle, and the conservation of the Poincar\'e generators is discussed. We then find the map with the Poincar\'e invariance and the four-momentum conservation. Further, we show that the conservation of the angular momentum and the boost operator makes the map of a spinless massive particle unitary.

Wayne Nicholas Polyzou

It is shown that the assumptions of Poincar\'e invariance and manifest covariance in quantum mechanics requires an interaction-dependent scalar product. Properties of the kernel of the inner product based on these assumptions are derived. Additional assumptions further restrict the form of this kernel. The relationship of this result to local quantum field theory and covariant constraint dynamics is discussed. A formal representation of a kernel satisfying all of these properties is given in terms of two-body constraints.

Eugen Fick, Günter Sauermann

Vladimir A Bordovitsyn, I. M. Ternov