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

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

Grobner Bases for Finite-temperature Quantum Computing and their Complexity

Following the recent approach of using order domains to construct Grobner bases from general projective varieties, we examine the parity and time-reversal arguments relating de Witt and Lyman's assertion that all path weights associated with homotopy in dimensions d <= 2 form a faithful representation of the fundamental group of a quantum system. We then show how the most general polynomial ring obtained for a fermionic quantum system does not, in fact, admit a faithful representation, and so give a general prescription for calcluating Grobner bases for finite temperature many-body quantum system and show that their complexity class is BQP

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

Interacting Frobenius Algebras are Hopf

Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise (BSZ 2014) by including non-trivial dynamics of the underlying object---the so-called phase group---and investigate the effects of finite dimensionality of the underlying model. We recover the system of Bonchi et al as a subtheory in the prime power dimensional case, but the more general theory does not arise from a distributive law.Comment: 32 pages; submitte

Free androgen index and leptin are the most prominent endocrine predictors of ovarian response during clomiphene citrate induction of ovulation in normogonadotropic oligoamenorrheic infertility

We have previously demonstrated that obese hyperandrogenic amenorrheic women are less likely to ovulate after clomiphene citrate (CC) medication. The present study was designed to identify whether additional endocrine screening characteristics, all potentially involved in ovarian dysfunction in 182 normogonadotropic oligoamenorrheic infertile women, are associated with ovarian response, which may improve overall prediction of CC-resistant anovulation. Standardized endocrine screening took place before initiation of CC medication (50 mg/day; increasing doses up to 150 mg/day if required) from cycle days 3-7. Screening included serum assays for fasting insulin and glucose, insulin-like growth factor I (IGF-I), IGF-binding protein-1 (IGFBP-1), IGFBP-3, free IGF-I, inhibin B, leptin, and vascular endothelial growth factor. Forty-two women (22% of the total group) did not ovulate at the end of follow-up (a total number of 325 cycles were analyzed). Fasting serum insulin, insulin/glucose ratio, IGFBP-1, and leptin were all significantly different in univariate analyses (P ≤ 0.02), comparing CC responders vs. nonresponders. Forward stepwise multivariate analyses in combination with factors reported earlier for prediction of patients remaining anovulatory after CC revealed a prediction model including 1) free androgen index (FAI = testosterone/sex hormone-binding globulin ratio), 2) cycle history (oligomenorrhea or amenorrhea), 3) leptin level, and 4) mean ovarian volume. These data suggest that decreased insulin sensitivity, hyperandrogenemia, and obesity, all associated with polycystic ovary syndrome, are prominent factors involved in ovarian dysfunction, preventing these ovaries from responding to stimulation by raised endogenous FSH levels due to CC medication. By using leptin instead of body mass index or waist to hip ratio, the previous model for prediction of patients remaining anovulatory aider CC medication could be slightly improved (area under the curve from 0.82-0.85). This may indicate that leptin is more directly involved in ovarian dysfunction in these patients. The capability of insulin and IGFBP-1 to predict patients who remain anovulatory after CC disappears when FAT enters into the model due to a significant correlation between FAT and these endocrine parameters. This suggests that markers for insulin sensitivity (e.g. IGFBP-1 and insulin) are associated with abnormal ovarian function through its correlation with androgens, whereas leptin is directly involved in ovarian dysfunction

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

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. $\Gamma\leq 40$ 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.