Orbital restriction on the functional form of the Lagrangian

Peter Woit

Masao Matsumoto

From the viewpoint of the SU(2) coherent states (CS) and their path integrals (PI) labeled by a full set of Euler angles (ϕ, θ, ψ) which we developed in the previous paper, we study the relations between gauge symmetries of Lagrangians and allowed quantum states; We investigate permissible types of fiducial vectors (FV) in the full quantum dynamics in terms of SU(2) CS for typical Lagrangians. We propose a general framework for a Lagrangian having a certain gauge symmetry with respect to one of the Euler angles ψ. We find that for the case FV are so restricted that they belong to the eigenstates of Ŝ3or to the orbits of them under the action of the SU(2); And the strength of a fictitious monopole, which appears in the Lagrangian, is a multiple of ½. In this case Dirac strings are permitted. Our formulations and results deepen those of the preceding work by Stone that has piloted us; We illustrate the relation between the two methods. The reasoning here does not work for a Lagrangian without the gauge symmetry. This suggests a new possibility about monopole charge quantization. Besides analogies to field theory and entanglements in quantum information (QI) are briefly mentioned.

Alexander Komech

Janusz Grabowski, Marek Kuś, Giuseppe Marmo +1 more

Abstract We develop a geometric approach to quantum mechanics based on the concept of the Tulczyjew triple. Our approach is genuinely infinite-dimensional, i.e. we do not restrict considerations to finite-dimensional Hilbert spaces, contrary to many other works on the geometry of quantum mechanics, and include a Lagrangian formalism in which self-adjoint (Schrödinger) operators are obtained as Lagrangian submanifolds associated with the Lagrangian. As a byproduct we also obtain results concerning coadjoint orbits of the unitary group in infinite dimensions, embedding of pure states in the unitary group, and self-adjoint extensions of symmetric relations.

Katsuhisa Ohta

R. Chatterjee, J.M. Dixon

The origin of spin–orbit coupling has been investigated in the Schrödinger picture with the constraint of Galilean invariance. A Lagrangian has been constructed with this group symmetry which contains a term that has been interpreted as the damping of the translational motion of the electron in a spinor field. From the Lagrangian a Hamiltonian [Formula: see text] has been formed by classical methods. A unitary transformation has been found which transforms [Formula: see text] into a Hamiltonian [Formula: see text] which contains not only the Zeeman terms, with the correct magnetic moment, but also the spin–orbit coupling.