554 research outputs found
Renormalization : A number theoretical model
We analyse the Dirichlet convolution ring of arithmetic number theoretic
functions. It turns out to fail to be a Hopf algebra on the diagonal, due to
the lack of complete multiplicativity of the product and coproduct. A related
Hopf algebra can be established, which however overcounts the diagonal. We
argue that the mechanism of renormalization in quantum field theory is modelled
after the same principle. Singularities hence arise as a (now continuously
indexed) overcounting on the diagonals. Renormalization is given by the map
from the auxiliary Hopf algebra to the weaker multiplicative structure, called
Hopf gebra, rescaling the diagonals.Comment: 15 pages, extended version of talks delivered at SLC55 Bertinoro,Sep
2005, and the Bob Delbourgo QFT Fest in Hobart, Dec 200
Hopf algebras and characters of classical groups
Schur functions provide an integral basis of the ring of symmetric functions.
It is shown that this ring has a natural Hopf algebra structure by identifying
the appropriate product, coproduct, unit, counit and antipode, and their
properties. Characters of covariant tensor irreducible representations of the
classical groups GL(n), O(n) and Sp(n) are then expressed in terms of Schur
functions, and the Hopf algebra is exploited in the determination of
group-subgroup branching rules and the decomposition of tensor products. The
analysis is carried out in terms of n-independent universal characters. The
corresponding rings, CharGL, CharO and CharSp, of universal characters each
have their own natural Hopf algebra structure. The appropriate product,
coproduct, unit, counit and antipode are identified in each case.Comment: 9 pages. Uses jpconf.cls and jpconf11.clo. Presented by RCK at
SSPCM'07, Myczkowce, Poland, Sept 200
Products, coproducts and singular value decomposition
Products and coproducts may be recognized as morphisms in a monoidal tensor
category of vector spaces. To gain invariant data of these morphisms, we can
use singular value decomposition which attaches singular values, ie generalized
eigenvalues, to these maps. We show, for the case of Grassmann and Clifford
products, that twist maps significantly alter these data reducing degeneracies.
Since non group like coproducts give rise to non classical behavior of the
algebra of functions, ie make them noncommutative, we hope to be able to learn
more about such geometries. Remarkably the coproduct for positive singular
values of eigenvectors in yields directly corresponding eigenvectors in
A\otimes A.Comment: 17 pages, three eps-figure
A Hopf laboratory for symmetric functions
An analysis of symmetric function theory is given from the perspective of the
underlying Hopf and bi-algebraic structures. These are presented explicitly in
terms of standard symmetric function operations. Particular attention is
focussed on Laplace pairing, Sweedler cohomology for 1- and 2-cochains, and
twisted products (Rota cliffordizations) induced by branching operators in the
symmetric function context. The latter are shown to include the algebras of
symmetric functions of orthogonal and symplectic type. A commentary on related
issues in the combinatorial approach to quantum field theory is given.Comment: 29 pages, LaTeX, uses amsmat
On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form
Clifford algebras are naturally associated with quadratic forms. These
algebras are Z_2-graded by construction. However, only a Z_n-gradation induced
by a choice of a basis, or even better, by a Chevalley vector space isomorphism
Cl(V) \bigwedge V and an ordering, guarantees a multi-vector decomposition
into scalars, vectors, tensors, and so on, mandatory in physics. We show that
the Chevalley isomorphism theorem cannot be generalized to algebras if the
Z_n-grading or other structures are added, e.g., a linear form. We work with
pairs consisting of a Clifford algebra and a linear form or a Z_n-grading which
we now call 'Clifford algebras of multi-vectors' or 'quantum Clifford
algebras'. It turns out, that in this sense, all multi-vector Clifford algebras
of the same quadratic but different bilinear forms are non-isomorphic. The
usefulness of such algebras in quantum field theory and superconductivity was
shown elsewhere. Allowing for arbitrary bilinear forms however spoils their
diagonalizability which has a considerable effect on the tensor decomposition
of the Clifford algebras governed by the periodicity theorems, including the
Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cl_{p,q} which
can be decomposed in the symmetric case into a tensor product Cl_{p-1,q-1}
\otimes Cl_{1,1}. The general case used in quantum field theory lacks this
feature. Theories with non-symmetric bilinear forms are however needed in the
analysis of multi-particle states in interacting theories. A connection to
q-deformed structures through nontrivial vacuum states in quantum theories is
outlined.Comment: 25 pages, 1 figure, LaTeX, {Paper presented at the 5th International
Conference on Clifford Algebras and their Applications in Mathematical
Physics, Ixtapa, Mexico, June 27 - July 4, 199
Z_2-gradings of Clifford algebras and multivector structures
Let Cl(V,g) be the real Clifford algebra associated to the real vector space
V, endowed with a nondegenerate metric g. In this paper, we study the class of
Z_2-gradings of Cl(V,g) which are somehow compatible with the multivector
structure of the Grassmann algebra over V. A complete characterization for such
Z_2-gradings is obtained by classifying all the even subalgebras coming from
them. An expression relating such subalgebras to the usual even part of Cl(V,g)
is also obtained. Finally, we employ this framework to define spinor spaces,
and to parametrize all the possible signature changes on Cl(V,g) by
Z_2-gradings of this algebra.Comment: 10 pages, LaTeX; v2 accepted for publication in J. Phys.
Quantum field theory and Hopf algebra cohomology
We exhibit a Hopf superalgebra structure of the algebra of field operators of
quantum field theory (QFT) with the normal product. Based on this we construct
the operator product and the time-ordered product as a twist deformation in the
sense of Drinfeld. Our approach yields formulas for (perturbative) products and
expectation values that allow for a significant enhancement in computational
efficiency as compared to traditional methods. Employing Hopf algebra
cohomology sheds new light on the structure of QFT and allows the extension to
interacting (not necessarily perturbative) QFT. We give a reconstruction
theorem for time-ordered products in the spirit of Streater and Wightman and
recover the distinction between free and interacting theory from a property of
the underlying cocycle. We also demonstrate how non-trivial vacua are described
in our approach solving a problem in quantum chemistry.Comment: 39 pages, no figures, LaTeX + AMS macros; title changed, minor
corrections, references update
Off shell behaviour of the in medium nucleon-nucleon cross section
The properties of nucleon-nucleon scattering inside dense nuclear matter are
investigated. We use the relativistic Brueckner-Hartree-Fock model to determine
on-shell and half off-shell in-medium transition amplitudes and cross sections.
At finite densities the on-shell cross sections are generally suppressed. This
reduction is, however, less pronounced than found in previous works. In the
case that the outgoing momenta are allowed to be off energy shell the
amplitudes show a strong variation with momentum. This description allows to
determine in-medium cross sections beyond the quasi-particle approximation
accounting thereby for the finite width which nucleons acquire in the dense
nuclear medium. For reasonable choices of the in-medium nuclear spectral width,
i.e. MeV, the resulting total cross sections are, however,
reduced by not more than about 25% compared to the on-shell values. Off-shell
effect are generally more pronounced at large nuclear matter densities.Comment: 31 pages Revtex, 12 figures, typos corrected, to appear in Phys. Rev.
The structure of Green functions in quantum field theory with a general state
In quantum field theory, the Green function is usually calculated as the
expectation value of the time-ordered product of fields over the vacuum. In
some cases, especially in degenerate systems, expectation values over general
states are required. The corresponding Green functions are essentially more
complex than in the vacuum, because they cannot be written in terms of standard
Feynman diagrams. Here, a method is proposed to determine the structure of
these Green functions and to derive nonperturbative equations for them. The
main idea is to transform the cumulants describing correlations into
interaction terms.Comment: 13 pages, 6 figure
HtrA1 Mediated Intracellular Effects on Tubulin Using a Polarized RPE Disease Model
Age-related macular degeneration (AMD) is the leading cause of irreversible vision loss. The protein HtrA1 is enriched in retinal pigment epithelial (RPE) cells isolated from AMD patients and in drusen deposits. However, it is poorly understood how increased levels of HtrA1 affect the physiological function of the RPE at the intracellular level. Here, we developed hfRPE (human fetal retinal pigment epithelial) cell culture model where cells fully differentiated into a polarized functional monolayer. In this model, we fine-tuned the cellular levels of HtrA1 by targeted overexpression. Our data show that HtrA1 enzymatic activity leads to intracellular degradation of tubulin with a corresponding reduction in the number of microtubules, and consequently to an altered mechanical cell phenotype. HtrA1 overexpression further leads to impaired apical processes and decreased phagocytosis, an essential function for photoreceptor survival. These cellular alterations correlate with the AMD phenotype and thus highlight HtrA1 as an intracellular target for therapeutic interventions towards AMD treatment
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