1,278,087 research outputs found

### Is Hilbert space discrete?

We show that discretization of spacetime naturally suggests discretization of
Hilbert space itself. Specifically, in a universe with a minimal length (for
example, due to quantum gravity), no experiment can exclude the possibility
that Hilbert space is discrete. We give some simple examples involving qubits
and the Schrodinger wavefunction, and discuss implications for quantum
information and quantum gravity.Comment: 4 pages, revtex, 1 figur

### Strings with Discrete Target Space

We investigate the field theory of strings having as a target space an
arbitrary discrete one-dimensional manifold. The existence of the continuum
limit is guaranteed if the target space is a Dynkin diagram of a simply laced
Lie algebra or its affine extension. In this case the theory can be mapped onto
the theory of strings embedded in the infinite discrete line $\Z$ which is the
target space of the SOS model. On the regular lattice this mapping is known as
Coulomb gas picture. ... Once the classical background is known, the amplitudes
involving propagation of strings can be evaluated by perturbative expansion
around the saddle point of the functional integral. For example, the partition
function of the noninteracting closed string (toroidal world sheet) is the
contribution of the gaussian fluctuations of the string field. The vertices in
the corresponding Feynman diagram technique are constructed as the loop
amplitudes in a random matrix model with suitably chosen potential.Comment: 65 pages (Sept. 91

### Discrete space-time geometry and skeleton conception of particle dynamics

It is shown that properties of a discrete space-time geometry distinguish
from properties of the Riemannian space-time geometry. The discrete geometry is
a physical geometry, which is described completely by the world function. The
discrete geometry is nonaxiomatizable and multivariant. The equivalence
relation is intransitive in the discrete geometry. The particles are described
by world chains (broken lines with finite length of links), because in the
discrete space-time geometry there are no infinitesimal lengths. Motion of
particles is stochastic, and statistical description of them leads to the
Schr\"{o}dinger equation, if the elementary length of the discrete geometry
depends on the quantum constant in a proper way.Comment: 22 pages, 0 figure

### On discrete analytic functions: Products, Rational Functions, and some Associated Reproducing Kernel Hilbert Spaces

We introduce a family of discrete analytic functions, called expandable
discrete analytic functions, which includes discrete analytic polynomials, and
define two products in this family. The first one is defined in a way similar
to the Cauchy-Kovalevskaya product of hyperholomorphic functions, and allows us
to define rational discrete analytic functions. To define the second product we
need a new space of entire functions which is contractively included in the
Fock space. We study in this space some counterparts of Schur analysis

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