On L2L^2 -functions with bounded spectrum

We consider the class $PW(\mathbb R^n)$ of functions in $L^2(\mathbb R^n)$, whose Fourier transform has bounded support. We obtain a description of continuous maps $\varphi : \mathbb R^m\rightarrow\mathbb R^n$ such that $f\circ\varphi\in PW(\mathbb R^m)$ for every function $f\in PW(\mathbb R^n)$. Only injective affine maps $\varphi$ have this property

Conductance Phases in Aharonov-Bohm Ring Quantum Dots

The regimes of growing phases (for electron numbers N~0-8) that pass into regions of self-returning phases (for N>8), found recently in quantum dot conductances by the Weizmann group are accounted for by an elementary Green function formalism, appropriate to an equi-spaced ladder structure (with at least three rungs) of electronic levels in the quantum dot. The key features of the theory are physically a dissipation rate that increases linearly with the level number (and tentatively linked to coupling to longitudinal optical phonons) and a set of Fano-like meta-stable levels, which disturb the unitarity, and mathematically the change over of the position of the complex transmission amplitude-zeros from the upper-half in the complex gap-voltage plane to the lower half of that plane. The two regimes are identified with (respectively) the Blaschke-term and the Kramers-Kronig integral term in the theory of complex variables.Comment: 20 pages, 4 figure

Photon wave mechanics and position eigenvectors

One and two photon wave functions are derived by projecting the quantum state vector onto simultaneous eigenvectors of the number operator and a recently constructed photon position operator [Phys. Rev A 59, 954 (1999)] that couples spin and orbital angular momentum. While only the Landau-Peierls wave function defines a positive definite photon density, a similarity transformation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell's equations preserves eigenvalues and expectation values. We show that this real space description of photons is compatible with all of the usual rules of quantum mechanics and provides a framework for understanding the relationships amongst different forms of the photon wave function in the literature. It also gives a quantum picture of the optical angular momentum of beams that applies to both one photon and coherent states. According to the rules of qunatum mechanics, this wave function gives the probability to count a photon at any position in space.Comment: 14 pages, to be published in Phys. Rev.

On the inconsistency of the Bohm-Gadella theory with quantum mechanics

The Bohm-Gadella theory, sometimes referred to as the Time Asymmetric Quantum Theory of Scattering and Decay, is based on the Hardy axiom. The Hardy axiom asserts that the solutions of the Lippmann-Schwinger equation are functionals over spaces of Hardy functions. The preparation-registration arrow of time provides the physical justification for the Hardy axiom. In this paper, it is shown that the Hardy axiom is incorrect, because the solutions of the Lippmann-Schwinger equation do not act on spaces of Hardy functions. It is also shown that the derivation of the preparation-registration arrow of time is flawed. Thus, Hardy functions neither appear when we solve the Lippmann-Schwinger equation nor they should appear. It is also shown that the Bohm-Gadella theory does not rest on the same physical principles as quantum mechanics, and that it does not solve any problem that quantum mechanics cannot solve. The Bohm-Gadella theory must therefore be abandoned.Comment: 16 page

Out of Equilibrium Thermal Field Theories - Finite Time after Switching on the Interaction - Wigner Transforms of Projected Functions

We study out of equilibrium thermal field theories with switching on the interaction occurring at finite time using the Wigner transforms (in relative space-time) of two-point functions. For two-point functions we define the concept of projected function: it is zero if any of times refers to the time before switching on the interaction, otherwise it depends only on the relative coordinates. This definition includes bare propagators, one-loop self-energies, etc. For the infinite-average-time limit of the Wigner transforms of projected functions we define the analyticity assumptions: (1) The function of energy is analytic above (below) the real axis. (2) The function goes to zero as the absolute value of energy approaches infinity in the upper (lower) semiplane. Without use of the gradient expansion, we obtain the convolution product of projected functions. We sum the Schwinger-Dyson series in closed form. In the calculation of the Keldysh component (both, resummed and single self-energy insertion approximation) contributions appear which are not the Wigner transforms of projected functions, signaling the limitations of the method. In the Feynman diagrams there is no explicit energy conservation at vertices, there is an overall energy-smearing factor taking care of the uncertainty relations. The relation between the theories with the Keldysh time path and with the finite time path enables one to rederive the results, such as the cancellation of pinching, collinear, and infrared singularities, hard thermal loop resummation, etc.Comment: 23 pages + 1 figure, Latex, corrected version, improved presentation, version accepted for publication in Phys. Rev.

Self-adjoint Lyapunov variables, temporal ordering and irreversible representations of Schroedinger evolution

In non relativistic quantum mechanics time enters as a parameter in the Schroedinger equation. However, there are various situations where the need arises to view time as a dynamical variable. In this paper we consider the dynamical role of time through the construction of a Lyapunov variable - i.e., a self-adjoint quantum observable whose expectation value varies monotonically as time increases. It is shown, in a constructive way, that a certain class of models admit a Lyapunov variable and that the existence of a Lyapunov variable implies the existence of a transformation mapping the original quantum mechanical problem to an equivalent irreversible representation. In addition, it is proved that in the irreversible representation there exists a natural time ordering observable splitting the Hilbert space at each t>0 into past and future subspaces.Comment: Accepted for publication in JMP. Supercedes arXiv:0710.3604. Discussion expanded to include the case of Hamiltonians with an infinitely degenerate spectru

The Relationship Between Daytime Sedentary Behavior and Sleep Health in Desk-based Workers

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On the nonlinearity interpretation of q- and f-deformation and some applications

q-oscillators are associated to the simplest non-commutative example of Hopf algebra and may be considered to be the basic building blocks for the symmetry algebras of completely integrable theories. They may also be interpreted as a special type of spectral nonlinearity, which may be generalized to a wider class of f-oscillator algebras. In the framework of this nonlinear interpretation, we discuss the structure of the stochastic process associated to q-deformation, the role of the q-oscillator as a spectrum-generating algebra for fast growing point spectrum, the deformation of fermion operators in solid-state models and the charge-dependent mass of excitations in f-deformed relativistic quantum fields.Comment: 11 pages Late

Constructions of regular algebras Lpw(G)L_p^w(G)

Criterion of (Shilov) regularity for weighted algebras $L_1^w(G)$ on a locally compact abelian group $G$ is known by works of Beurling (1949) and Domar (1956). In the present paper this criterion is extended to translation invariant weighted algebras $L_p^w(G)$ with $p>1$. Regular algebras $L_p^w(G)$ are constructed on any sigma-compact abelian group $G$. It was proved earlier by the author that sigma-compactness is necessary (in the abelian case) for the existence of weighted algebras $L_p^w(G)$ with $p>1$.Comment: Submitted to Mat. Sborni