1,367 research outputs found

### The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory

The Jacobian conjecture is an old unsolved problem in mathematics, which has
been unsuccessfully attacked from many different angles. We add here another
point of view pertaining to the so called formal inverse approach, that of
perturbative quantum field theory.Comment: 22 pages, 13 diagram

### Bosonic Monocluster Expansion

We compute connected Green's functions of a Bosonic field theory with cutoffs
by means of a ``minimal'' expansion which in a single move, interpolating a
generalized propagator, performs the usual tasks of the cluster and Mayer
expansion. In this way it allows a direct construction of the infinite volume
or thermodynamic limit and it brings constructive Bosonic expansions closer to
constructive Fermionic expansions and to perturbation theory.Comment: 30 pages, 1 figur

### RTT relations, a modified braid equation and noncommutative planes

With the known group relations for the elements $(a,b,c,d)$ of a quantum
matrix $T$ as input a general solution of the $RTT$ relations is sought without
imposing the Yang - Baxter constraint for $R$ or the braid equation for
$\hat{R} = PR$. For three biparametric deformatios, $GL_{(p,q)}(2),
GL_{(g,h)}(2)$ and $GL_{(q,h)}(1/1)$, the standard,the nonstandard and the
hybrid one respectively, $R$ or $\hat{R}$ is found to depend, apart from the
two parameters defining the deformation in question, on an extra free parameter
$K$,such that only for two values of $K$, given explicitly for each case, one
has the braid equation. Arbitray $K$ corresponds to a class (conserving the
group relations independent of $K$) of the MQYBE or modified quantum YB
equations studied by Gerstenhaber, Giaquinto and Schak. Various properties of
the triparametric $\hat{R}(K;p,q)$, $\hat{R}(K;g,h)$ and $\hat{R}(K;q,h)$ are
studied. In the larger space of the modified braid equation (MBE) even
$\hat{R}(K;p,q)$ can satisfy $\hat{R}^2 = 1$ outside braid equation (BE)
subspace. A generalized, $K$- dependent, Hecke condition is satisfied by each
3-parameter $\hat{R}$. The role of $K$ in noncommutative geometries of the
$(K;p,q)$,$(K;g,h)$ and $(K;q,h)$ deformed planes is studied. K is found to
introduce a "soft symmetry breaking", preserving most interesting properties
and leading to new interesting ones. Further aspects to be explored are
indicated.Comment: Latex, 17 pages, minor change

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