On the Kolmogorov--Wiener--Masani spectrum of a multi-mode weakly stationary quantum process

We introduce the notion of a $k$-mode weakly stationary quantum process $\varrho$ based on the canonical Schr\"odinger pairs of position and momentum observables in copies of $L^2(\mathbb{R}^k)$, indexed by an additive abelian group $D$ of countable cardinality. Such observables admit an autocovariance map $\widetilde{K}$ from $D$ into the space of real $2k \times 2k$ matrices. The map $\widetilde{K}$ on the discrete group $D$ admits a spectral representation as the Fourier transform of a $2k \times 2k$ complex Hermitain matrix-valued totally finite measure $\Phi$ on the compact character group $\widehat{D}$, called the Kolmogorov-Wiener-Masani (KWM) spectrum of the process $\varrho$. Necessary and sufficient conditions on a $2k \times 2k$ complex Hermitian matrix-valued measure $\Phi$ on $\widehat{D}$ to be the KWM spectrum of a process $\varrho$ are obtained. This enables the construction of examples. Our theorem reveals the dramatic influence of the uncertainty relations among the position and momentum observables on the KWM spectrum of the process $\varrho$. In particular, KWM spectrum cannot admit a gap of positive Haar measure in $\widehat{D}$. The relationship between the number of photons in a particular mode at any site of the process and its KWM spectrum needs further investigation.Comment: 17 pages, added Theorem 4.2 and some remarks. Comments welcome. Keywords: Weakly stationary quantum process, Kolmogorov-Wiener-Masani spectrum, autocovariance map, spectral representation, uncertainty relation

Fock spaces corresponding to positive definite linear transformations

Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner product space $V$. The reproducing kernel for the Fock space of square integrable holomorphic functions on $V$ relative to the Gaussian measure $d\mu_A(z)=\frac {\sqrt {\det A}} {\pi^n}e^{-{\rm Re}} dz$ is described in terms of the holomorphic--antiholomorphic decomposition of the linear operator $A$. Moreover, if $A$ commutes with a conjugation on $V$, then a restriction mapping to the real vectors in $V$ is polarized to obtain a Segal--Bargmann transform, which we also study in the Gaussian-measure setting

Did Prepayments Sustain the Subprime Market?

This paper demonstrates that the reason for widespread default of mortgages in the subprime market was a sudden reversal in the house price appreciation of the early 2000's. Using loan-level data on subprime mortgages, we observe that the majority of subprime loans were hybrid adjustable rate mortgages, designed to impose substantial financial burden on reset to the fully indexed rate. In a regime of rising house prices, a financially distressed borrower could avoid default by prepaying the loan and our results indicate that subprime mortgages originated between 1998 and 2005 had extremely high prepayment rates. Most important, prepayment rates on subprime mortgages were extremely high (i) not just for ARMs but FRMs as well, (ii) even before the reset dates on hybrid-ARMs and (iii) despite prepayment penalties on the contract. However, a sudden reversal in house price appreciation increased default in this market because it made this prepayment exit option cost-prohibitive. In short, prepayments sustained the subprime boom and the extremely high default rates on 2006-2007 vintages were largely due to the inability of these mortgages to prepay (an option that was available for mortgages of earlier vintages).mortgages;subprime;refinance;prepayment;crisis

Muon (g-2) from the bulk neutrino field in a warped extra dimensional model

In the Randall-Sundrum model, a bulk neutrino field in the 5-dimensional space-time can give rise to tiny Dirac masses to neutrinos. In such a scenario, we have computed the contribution of the bulk neutrino field to the anomalous magnetic moment $(g-2)_\mu$ of muon. We have computed this contribution in the 't Hooft-Feynman gauge and have found that the contribution has the right sign to fit the current discrepancy between the experiment and the standard model value of $(g-2)_\mu$. We have also studied possible constraints on the model parameters by including contributions to $(g-2)_\mu$ from other sources such as bulk gravitons.Comment: 18 pages, 3 figures, 2 tables, minor changes, this version has been published in Physical Review

SU(2) gauge theory of gravity with topological invariants

The most general gravity Lagrangian in four dimensions contains three topological densities, namely Nieh-Yan, Pontryagin and Euler, in addition to the Hilbert-Palatini term. We set up a Hamiltonian formulation based on this Lagrangian. The resulting canonical theory depends on three parameters which are coefficients of these terms and is shown to admit a real SU(2) gauge theoretic interpretation with a set of seven first-class constraints. Thus, in addition to the Newton's constant, the theory of gravity contains three (topological) coupling constants, which might have non-trivial imports in the quantum theory.Comment: Based on a talk at Loops-11, Madrid, Spain; To appear in Journal of Physics: Conference Serie

Tuning the conductance of Dirac fermions on the surface of a topological insulator

We study the transport properties of the Dirac fermions with Fermi velocity $v_F$ on the surface of a topological insulator across a ferromagnetic strip providing an exchange field ${\mathcal J}$ over a region of width $d$. We show that the conductance of such a junction changes from oscillatory to a monotonically decreasing function of $d$ beyond a critical ${\mathcal J}$. This leads to the possible realization of a magnetic switch using these junctions. We also study the conductance of these Dirac fermions across a potential barrier of width $d$ and potential $V_0$ in the presence of such a ferromagnetic strip and show that beyond a critical ${\mathcal J}$, the criteria of conductance maxima changes from $\chi= e V_0 d/\hbar v_F = n \pi$ to $\chi= (n+1/2)\pi$ for integer $n$. We point out that these novel phenomena have no analogs in graphene and suggest experiments which can probe them.Comment: v1 4 pages 5 fig