7,871 research outputs found
Towards topological quantum computer
One of the principal obstacles on the way to quantum computers is the lack of
distinguished basis in the space of unitary evolutions and thus the lack of the
commonly accepted set of basic operations (universal gates). A natural choice,
however, is at hand: it is provided by the quantum R-matrices, the entangling
deformations of non-entangling (classical) permutations, distinguished from the
points of view of group theory, integrable systems and modern theory of
non-perturbative calculations in quantum field and string theory. Observables
in this case are (square modules of) the knot polynomials, and their pronounced
integrality properties could provide a key to error correction. We suggest to
use R-matrices acting in the space of irreducible representations, which are
unitary for the real-valued couplings in Chern-Simons theory, to build a
topological version of quantum computing.Comment: 14 page
Knot invariants from Virasoro related representation and pretzel knots
We remind the method to calculate colored Jones polynomials for the plat
representations of knot diagrams from the knowledge of modular transformation
(monodromies) of Virasoro conformal blocks with insertions of degenerate
fields. As an illustration we use a rich family of pretzel knots, lying on a
surface of arbitrary genus g, which was recently analyzed by the evolution
method. Further generalizations can be to generic Virasoro modular
transformations, provided by integral kernels, which can lead to the Hikami
invariants.Comment: 29 page
On skew tau-functions in higher spin theory
Recent studies of higher spin theory in three dimensions concentrate on
Wilson loops in Chern-Simons theory, which in the classical limit reduce to
peculiar corner matrix elements between the highest and lowest weight states in
a given representation of SL(N). Despite these "skew" tau-functions can seem
very different from conventional ones, which are the matrix elements between
the two highest weight states, they also satisfy the Toda recursion between
different fundamental representations. Moreover, in the most popular examples
they possess simple representations in terms of matrix models and Schur
functions. We provide a brief introduction to this new interesting field,
which, after quantization, can serve as an additional bridge between knot and
integrability theories.Comment: 36 page
S-Duality and Modular Transformation as a non-perturbative deformation of the ordinary pq-duality
A recent claim that the S-duality between 4d SUSY gauge theories, which is
AGT related to the modular transformations of 2d conformal blocks, is no more
than an ordinary Fourier transform at the perturbative level, is further traced
down to the commutation relation [P,Q]=-i\hbar between the check-operator
monodromies of the exponential resolvent operator in the underlying
Dotsenko-Fateev matrix models and beta-ensembles. To this end, we treat the
conformal blocks as eigenfunctions of the monodromy check operators, what is
especially simple in the case of one-point toric block. The kernel of the
modular transformation is then defined as the intertwiner of the two
monodromies, and can be obtained straightforwardly, even when the eigenfunction
interpretation of the blocks themselves is technically tedious. In this way, we
provide an elementary derivation of the old expression for the modular kernel
for the one-point toric conformal block.Comment: 15 page
Octonic Electrodynamics
In this paper we present eight-component values "octons", generating
associative noncommutative algebra. It is shown that the electromagnetic field
in a vacuum can be described by a generalized octonic equation, which leads
both to the wave equations for potentials and fields and to the system of
Maxwell's equations. The octonic algebra allows one to perform compact combined
calculations simultaneously with scalars, vectors, pseudoscalars and
pseudovectors. Examples of such calculations are demonstrated by deriving the
relations for energy, momentum and Lorentz invariants of the electromagnetic
field. The generalized octonic equation for electromagnetic field in a matter
is formulated.Comment: 12 pages, 1 figur
Sedeonic relativistic quantum mechanics
We represent sixteen-component values "sedeons", generating associative
noncommutative space-time algebra. We demonstrate a generalization of
relativistic quantum mechanics using sedeonic wave functions and sedeonic
space-time operators. It is shown that the sedeonic second-order equation for
the sedeonic wave function, obtained from the Einstein relation for energy and
momentum, describes particles with spin 1/2. We show that for the special types
of wave functions the sedeonic second-order equation can be reduced to the set
of sedeonic first-order equations analogous to the Dirac equation. At the same
time it is shown that these sedeonic equations differ in space-time properties
and describe several types of massive and corresponding massless particles. In
particular we proposed four different equations, which could describe four
types of neutrinos.Comment: 22 pages, 3 table
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