We consider a representation of the path groupoid over the group of diffeomorphisms of the base space of a Hilbert bundle (equipped with a unitary parallel transport). The Schroedinger representation in (Weyl form) appears as a flat and measure-invariant particular case of this representation. We prove that our representation induces a representation of Mensky's path group and that generators of the representation of one-parameter groups of the path group are given by the covariant derivative plus a volume correction. We prove that, as in the case of locally compact groups, the imprimitivity systems of this representation is irreducible if and only if the diffeomorphisms group acts transitively on the manifold and the inducing representation is irreducible. In our case the inducing subgroup is the loop group and the inducing representation is the holonomy group of the parallel transport.
We consider a representation of the path groupoid over the group of diffeomorphisms of the base space of a Hilbert bundle (equipped with a unitary parallel transport). The Schroedinger representation in (Weyl form) appears as a flat and measure-invariant particular case of this representation. We prove that our representation induces a representation of Mensky's path group and that generators of the representation of one-parameter groups of the path group are given by the covariant derivative plus a volume correction. We prove that, as in the case of locally compact groups, the imprimitivity systems of this representation is irreducible if and only if the diffeomorphisms group acts transitively on the manifold and the inducing representation is irreducible. In our case the inducing subgroup is the loop group and the inducing representation is the holonomy group of the parallel transport.
The building blocks of Hudson-Parthasarathy quantum stochastic calculus start with Weyl operators on a symmetric Fock space. To realize a relativistically covariant version of the calculus we construct representations of Poincare group in terms of Weyl operators on suitably constructed, Bosonic or Fermionic based on the mass and spin of the fundamental particle, Fock spaces. We proceed by describing the orbits of homogeneous Lorentz group on R4 and build fiber bundle representations of Poincaré group induced from the stabilizer subgroups (little groups) and build the Boson Fock space of the Hilbert space formed from the sections of the bundle. Our Weyl operators are constructed on symmetric Fock space of this space and the corresponding annihilation, creation, and conservation operators are synthesized in the usual fashion in relativistic theories for space-like, time-like, and light-like fields. We achieve this by constructing transitive systems of imprimitivity (second-quantized SI), which are dynamical systems with trajectories dense in the configuration space, by induced representations. We provide the details of the field operators for the case of massive Bosons as the rest are similar in construction and indicate the ways to construct adapted processes paving way for building covariant quantum stochastic calculus.
We study the Hilbert bundle description of stochastic quantum mechanics in curved spacetime developed by Prugovecki, which gives a powerful new framework for exploring the quantum mechanical propagation of states in curved spacetime. We concentrate on the quantum transport law in the bundle, specifically on the information which can be obtained from the flat space limit. We give a detailed proof that quantum transport coincides with parallel transport in the bundle in this limit, confirming statements of Prugovecki. Furthermore, we show that the quantum-geometric propagator in curved spacetime proposed by Prugovecki, yielding a Feynman path integral-like formula involving integrations over intermediate phase-space variables, is Poincaré gauge-covariant (i.e. is gauge-invariant except for transformations at the endpoints of the path) provided the integration measure is interpreted as a `contact point measure' in the soldered stochastic phase-space bundle raised over curved spacetime.
A systematic study is made of path integral representations for a particle constrained to move on a manifold diffeomorphic to S D by applying the irreducible representations of the Dirac algebra on it. Especially, we derive two types of path integral representation for this system, one of which is of a new form with a simple and compact expression, and the other is a rigorous version of the Faddeev–Senjanovic formula. It is also shown that the parameter α [0, 1) specifying the irreducible representation for D = 1 has a close connection with the Aharonov–Bohm gauge potential produced by the magnetic flux .
We consider an ``integral'' extension of the classical notion of affine connection providing a correspondence between paths in the manifold and diffeomorphisms of the manifold. These path-diffeomorphisms are a generalization of parallel translations along paths via the connection. In this way, one can translate nonanalytic functions and distributions rather than tangent vectors. We describe the integral holonomy and the integral curvature. On the symplectic or quantum level this construction makes up the symplectic or quantum paths, as well quantum connection, quantum curvature and quantum holonomy. The construction of path-diffeomorphisms, being applied to trajectories of a given dynamical system produces a transformation of the system to a new one which has an a priori chosen equilibrium point. In the symplectic (quantum) case, this dynamic localization provides a coherent-type representation of the quantum flow.
A number of representations of the group of volume preserving diffeomorphisms of M=S3 in the finite-volume case and M=R3 in the infinite-volume case that describe a Bose gas in the presence of a vortex filament are examined. The core of the vortex is taken to lie along a curve γ. Let Dif f γμ(M) be the group of volume preserving diffeomorphisms that map γ onto itself, Mγ the manifold M with γ removed, and Map(Mγ,T) the space of smooth maps of Mγ into the complex numbers of modulus unity. It is shown, using the formalism of Klauder [J. Math. Phys. 11, 233 (1970)], that an infinite number of inequivalent representations of Dif f γμ(M) exist in the infinite-volume limit using cyclic vectors, which are coherent states based on elements in Map(Mγ,T). It is found, however, that these states have questionable physical significance since they appear to violate the continuity equation for quantum probability flow. This leads to the postulation that Dif f γμ(M) acts as gauge group. The representations of Dif f γμ(M) are then used to construct induced representations of the full group volume preserving diffeomorphisms of M, which are realized in the Hilbert space over the space of unparametrized loops in M in a natural way. These latter representations are found to be essentially the ones under current study in quantum hydrodynamics [G. A. Goldin, R. Menikoff, and D. H. Sharp, Phys. Rev. Lett. 58, 2162 (1987)].
The conventional quantization of the harmonic oscillator in terms of operators Q and P can be implemented with the help of irreducible unitary representations of the Heisenberg-Weyl group which acts transitively and effectively on the simply connected classical phase space S q,p ≊ 2 . In the description of the harmonic oscillator in terms of angle and action variables ϕ and I the associated phase space S ϕ,I corresponds to the multiply connected punctured plane 2 – {0}, on which the 3-dimensional symplectic group Sp(2, ) acts transitively, leaving the origin invariant. As this group contains the compact subgroup U (1) it has infinitely many covering groups. In the here relevant irreducible unitary representations (positive discrete series) the self-adjoint generator K o of U (l) represents the classical action variable I . It has the possible spectra n + k, n = 0,1,...; k > 0, where k depends on the covering group. This implies different possible spectra for the action variable Hamiltonian ℏωK 0 of the harmonic oscillator. On the other hand, expressing the operators Q and P (non-linearly) in terms of the three generators K 0 etc. of Sp (2, ) leads to the usual framework. Possible physical (experimental) implications and generalizations to higher dimensions are discussed briefly.