2,005 research outputs found
Commutators and squares in free groups
Let F_2 be the free group generated by x and y. In this article, we prove
that the commutator of x^m and y^n is a product of two squares if and only if
mn is even. We also show using topological methods that there are infinitely
many obstructions for an element in F_2 to be a product of two squares.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-27.abs.htm
Quasi-Linear Cellular Automata
Simulating a cellular automaton (CA) for t time-steps into the future
requires t^2 serial computation steps or t parallel ones. However, certain CAs
based on an Abelian group, such as addition mod 2, are termed ``linear''
because they obey a principle of superposition. This allows them to be
predicted efficiently, in serial time O(t) or O(log t) in parallel.
In this paper, we generalize this by looking at CAs with a variety of
algebraic structures, including quasigroups, non-Abelian groups, Steiner
systems, and others. We show that in many cases, an efficient algorithm exists
even though these CAs are not linear in the previous sense; we term them
``quasilinear.'' We find examples which can be predicted in serial time
proportional to t, t log t, t log^2 t, and t^a for a < 2, and parallel time log
t, log t log log t and log^2 t.
We also discuss what algebraic properties are required or implied by the
existence of scaling relations and principles of superposition, and exhibit
several novel ``vector-valued'' CAs.Comment: 41 pages with figures, To appear in Physica
N=4 SYM to Two Loops: Compact Expressions for the Non-Compact Symmetry Algebra of the su(1,1|2) Sector
We begin a study of higher-loop corrections to the dilatation generator of
N=4 SYM in non-compact sectors. In these sectors, the dilatation generator
contains infinitely many interactions, and therefore one expects very
complicated higher-loop corrections. Remarkably, we find a short and simple
expression for the two-loop dilatation generator. Our solution for the
non-compact su(1,1|2) sector consists of nested commutators of four O(g)
generators and one simple auxiliary generator. Moreover, the solution does not
require the planar limit; we conjecture that it is valid for any gauge group.
To obtain the two-loop dilatation generator, we find the complete O(g^3)
symmetry algebra for this sector, which is also given by concise expressions.
We check our solution using published results of direct field theory
calculations. By applying the expression for the two-loop dilatation generator
to compute selected anomalous dimensions and the bosonic sl(2) sector internal
S-matrix, we confirm recent conjectures of the higher-loop Bethe ansatz of
hep-th/0412188.Comment: 28 pages, v2: additional checks against direct field theory
calculations, references added, minor corrections, v3: additional minor
correction
Second Quantization of the Wilson Loop
Treating the QCD Wilson loop as amplitude for the propagation of the first
quantized particle we develop the second quantization of the same propagation.
The operator of the particle position (the endpoint of the
"open string") is introduced as a limit of the large Hermitean matrix. We
then derive the set of equations for the expectation values of the vertex
operators \VEV{ V(k_1)\dots V(k_n)} . The remarkable property of these
equations is that they can be expanded at small momenta (less than the QCD mass
scale), and solved for expansion coefficients. This provides the relations for
multiple commutators of position operator, which can be used to construct this
operator. We employ the noncommutative probability theory and find the
expansion of the operator in terms of products of creation
operators . In general, there are some free parameters left
in this expansion. In two dimensions we fix parameters uniquely from the
symplectic invariance. The Fock space of our theory is much smaller than that
of perturbative QCD, where the creation and annihilation operators were
labelled by continuous momenta. In our case this is a space generated by creation operators. The corresponding states are given by all sentences made
of the four letter words. We discuss the implication of this construction for
the mass spectra of mesons and glueballs.Comment: 41 pages, latex, 3 figures and 3 Mathematica files uuencode
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