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Register a new userQuantum systems in a cut Fock space
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Standard quantum mechanics is viewed as a limit of a cut system with artificially restricted dimension of a Hilbert space. Exact spectrum of cut momentum and coordinate operators is derived and the limiting transition to the infinite dimensional Hilbert space is studied in detail. The difference between systems with discrete and continuous energy spectra is emphasized. In particular a new scaling law, characteristic for nonlocalized, states is found. Some applications for supersymmetric quantum mechanics are briefly outlined.
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Maciej Trzetrzelewski (M. Smoluchowski Institute of Physics
, n Jagellonian University
, Krakow
2004
A recently introduced numerical approach to quantum systems is analyzed. The basis of a Fock space is restricted and represented in an algebraic program. Convergence with increasing size of basis is proved and the difference between discrete and continuous spectrum is stressed. In particular a new scaling low for nonlocalized states is obtained. Exact solutions for several cases as well as general properties of the method are given.
Motivated by the computation of loop space quantum mechanics as indicated in [7], here we seek a better understanding of the tubular geometry of loop space ${cal L}{cal M}$ corresponding to a Riemannian manifold ${cal M}$ around the submanifold of vanishing loops. Our approach is to first compute the tubular metric of $({cal M}^{2N+1})_{C}$ around the diagonal submanifold, where $({cal M}^N)_{C}$ is the Cartesian product of $N$ copies of ${cal M}$ with a cyclic ordering. This gives an infinite sequence of tubular metrics such that the one relevant to ${cal L}{cal M}$ can be obtained by taking the limit $Nto infty$. Such metrics are computed by adopting an indirect method where the general tubular expansion theorem of [12] is crucially used. We discuss how the complete reparametrization isometry of loop space arises in the large-$N$ limit and verify that the corresponding Killing equation is satisfied to all orders in tubular expansion. These tubular metrics can alternatively be interpreted as some natural Riemannian metrics on certain bundles of tangent spaces of ${cal M}$ which, for ${cal M} times {cal M}$, is the tangent bundle $T{cal M}$.
The self-consistent Hartree-Fock-Bogoliubov problem in large boxes can be solved accurately in the coordinate space with the recently developed solvers HFB-AX (2D) and MADNESS-HFB (3D). This is essential for the description of superfluid Fermi systems with complicated topologies and significant spatial extend, such as fissioning nuclei, weakly-bound nuclei, nuclear matter in the neutron star rust, and ultracold Fermi atoms in elongated traps. The HFB-AX solver based on B-spline techniques uses a hybrid MPI and OpenMP programming model for parallel computation for distributed parallel computation, within a node multi-threaded LAPACK and BLAS libraries are used to further enable parallel calculations of large eigensystems. The MADNESS-HFB solver uses a novel multi-resolution analysis based adaptive pseudo-spectral techniques to enable fully parallel 3D calculations of very large systems. In this work we present benchmark results for HFB-AX and MADNESS-HFB on ultracold trapped fermions.
Prompted by recent results on Susy-U(N)-invariant quantum mechanics in the large N limit by Veneziano and Wosiek, we have examined the planar spectrum in the full Hilbert space of U(N)-invariant states built on the Fock vacuum by applying any U(N)-invariant combinations of creation-operators. We present results about 1) the supersymmetric model in the bosonic sector, 2) the standard quartic Hamiltonian. This latter is useful to check our techniques against the exact result of Brezin et al. The SuSy case is where Fock space methods prove to be the most efficient: it turns out that the problem is separable and the exact planar spectrum can be expressed in terms of the single-trace spectrum. In the case of the anharmonic oscillator, on the other hand, the Fock space analysis is quite cumbersome due to the presence of large off-diagonal O(N) terms coupling subspaces with different number of traces; these terms should be absorbed before taking the planar limit and recovering the known planar spectrum. We give analytical and numerical evidence that good qualitative information on the spectrum can be obtained this way.
Within the framework of the covariant formulation of light-front dynamics, we develop a general non-perturbative renormalization scheme based on the Fock decomposition of the state vector and its truncation. The counterterms and bare parameters needed to renormalize the theory depend on the Fock sectors. We present a general strategy in order to calculate these quantities, as well as state vectors of physical systems, in a truncated Fock space. The explicit dependence of our formalism on the orientation of the light front plane is essential in order to analyze the structure of the counterterms. We apply our formalism to the two-body (one fermion and one boson) truncation in the Yukawa model and in QED, and to the three-body truncation in a scalar model. In QED, we recover analytically, without any perturbative expansion, the renormalization of the electric charge, according to the requirements of the Ward identity.
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