Poincaré invariant representation of the spin in quantum theory

Rudolf Haag

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

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.

Joseph E. Johnson

Covariant four-vector position operators ${X}^{\ensuremath{\mu}}$ are proposed, which form a natural operator generalization of the four-position in relativistic classical mechanics. These ${X}^{\ensuremath{\mu}}$ are defined by specifying commutation relations of the ${X}^{\ensuremath{\mu}}$ with the Poincar\'e generators ${P}^{\ensuremath{\mu}}$ and ${M}^{\ensuremath{\mu}\ensuremath{\nu}}$, and thereby extending the Poincar\'e algebra to a larger algebra whose representations are subsequently found. The ${X}^{\ensuremath{\mu}}$ are shown to be acceptable relativistic position operators within a proper-time dynamical framework. A single Hamiltonian is used for all spins, with a covariant proper-time description. The dynamics is capable of describing the time evolution of states which are not mass eigenstates. An automatic Foldy-Wouthuysen-type diagonalization is achieved for all spin representations, with spin and orbital angular momentum being separately conserved. The connection with the standard theory is made via the specific field equations. In making this connection to the standard theory of half-integral spins, the origin of Zitterbewegung and the nonseparate conservation of spin and angular momentum are clarified. The connection to Maxwell's equations provides an interesting statement of those equations and of gauge invariance. The unphysical representations of negative and imaginary mass and continuous spin are not present in this formalism. Other features are discussed.

Fabián H. Gaioli, Edgardo T. Garcia Alvarez

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.