Narrow Results

It is argued that the severe consequences of Haag’s inconsistency theorem for relativistic quantum field theories can be successfully evaded in the direct-action approach. Some recent favorable comments of John Wheeler, often mistakenly presumed to have abandoned his own (and Feynman’s) direct-action theory, together with the remarkable immunity of direct-action quantum electrodynamics to Haag’s theorem, suggest that it may well be a good time to rehabilitate direct action theories. It is also noted that, as extra dividends, direct-action QED is immune to the self-energy problem of standard gauge field QED, and can also provide a solution to the problem of gauge arbitrariness.

We report on a study of baryon-baryon scattering by applying time-ordered perturbation theory to the covariant chiral perturbation theory. Through working out the rules of time-ordered diagrams, we built up a systematic framework to investigate the chiral potentials in the SU(3) sector. At leading order, we found that the baryon-baryon potentials are perturbatively renormalizable, and the corresponding integral equations have unique solutions in all partial waves. We also discussed the issue of additional finite subtractions required, e.g., in the ³P₀ partial waves to improve the ultraviolet convergence of loop integrals for the nucleon-nucleon and lambda-nucleon scatterings.

We consider the renormalization of the one-loop effective action for the Yukawa interaction. We compute the beta functions in the generalized DeWitt-Schwinger subtraction scheme. For the quantized scalar field we obtain that all the beta functions exhibit decoupling for heavy fields as stated by the Appelquist-Carazzone theorem including also the gravitational couplings. For the quantized Dirac field, decoupling appears for almost all of them. We obtain the atypical result that the mass parameter of the background scalar field, does not decouple.

I discuss the difficulties of lattice power-counting for staggered fermions. The recently formulated staggered fermion power-counting theorem is summarized.

RPA and its quasiparticle generalization (QRPA) have been widely used to study electromagnetic transitions and beta decays in medium and heavy nuclei, being the pn-QRPA charge exchange mode extensively employed in the description of single and double beta decays in vibrational nuclei. However develops a collapse, i.e. it presents imaginary eigenvalues for strengths beyond a critical value of the force. Extensions called renormalized QRPA (RQRPA) do not develop any collapse going beyond the simplest quasiboson approximation, however they present several drawbacks which will be analyzed.

It is shown that classical gravity coupled with quantum fields can be renormalized with a finite number of independent couplings, plus field redefinitions, without introducing higher-derivative kinetic terms in the gravitational sector, but adding vertices that couple the matter stress-tensor with the Ricci tensor. The theory predicts the violation of causality at high energies.

The renormalization of singular potentials leads to a new power counting in nuclear chiral effective field theory.

We analyze the renormalization of the nucleon–nucleon interaction at low energies in coordinate space for both one and two pion exchange chiral potentials. The singularity structure of the long range potential and the requirement of orthogonality respectively determines, once renormalizability is imposed, the minimum and maximum number of counterterms allowed in the effective description of the nucleon–nucleon interaction in a non-perturbative context.

We recall some models for noncommutative space-time and discuss quantum field theories on these deformed spaces. We describe the IR/UV mixing disease, which implies nonrenormalizability for a large class of models. We review the power-counting analysis of field theories on the Moyal plane in momentum space and our recent renormalization proof of noncommutative φ⁴-theory based on renormalization group techniques for dynamical matrix models. Some further developments of dynamical matrix models are mentioned.

After a short introduction into the formulation of noncommutative field theory and the discussion of the IR/UV mixing, I review the main ideas and techniques of our proof with Raimar Wulkenhaar that the duality-covariant four-dimensional noncommutative scalar model is renormalizable to all orders. Next I discuss the calculation of the one-loop contribution to the beta function and emphasize the taming of the Landau ghost.

I continue with the formulation of fermion models as well as gauge models, where less results are worked out.

For fields defined over a deformed Minkowski space-time the property of Wedge-locality replacing locality is mentioned.

We review our proof that in a scaling limit, the time evolution of a quantum particle in a static random environment leads to a diffusion equation. In particular, we discuss the role of Feynman graph expansions and of renormalization.

Renormalization played a central role in reviving classical dynamics as an exciting area of research during the second half of the twentieth century, and continues being an invaluable theoretical tool in the twenty-first century. Its defining characteristic is dynamical self-similarity, which allows one reliably to probe asymptotically small distances and asymptotically long orbits in phase space. We consider two applications of renormalization in nonlinear dynamics. The first is the historic discovery by Feigenbaum of the universal period-doubling route to chaos. The second is a more recent treatment of local and global scaling behavior of a periodically kicked harmonic oscillator. Some of these models exhibit the phenomenon of pseudo-chaos, where complex fractal structure in phase space is generated without the exponential divergence of nearby orbits which characterizes true chaos.

The effects of quantum fluctuations in fields confined by background configurations may be simply and transparently computed using the Green's function approach pioneered by Schwinger. Not only can total energies and surface forces be computed in this way, but local energy densities, and in general, all components of the vacuum expectation value of the energy-momentum tensor may be calculated. For simple geometries this approach may be carried out exactly, which yields insight into what happens in less tractable situations. In this talk I will concentrate on the example of a scalar field in a circular cylindrical delta-function background. This situation is quite similar to that of a spherical delta-function background. The local energy density in these cases diverges as the surface of the background is approached, but these divergences are integrable. The total energy is finite in strong coupling, but in weak coupling a divergence occurs in third order. This universal feature is shown to reflect a divergence in the energy associated with the surface, the integrated local energy density within the shell itself, which surface energy should be removable by a process of renormalization.

A fractal method for the renormalization theory of an infinite dimensional Clifford algebra is presented and the renormalized Dirac operator is given. A fractal set K which is defined by four self similar mappings between a rectangle is introduced, which is called a self similar fractal set of Peano type. We show that the Hilbert space L²(K, dµ^D) with respect to the Hausdorff measure µ^D is regarded as the renormalization space of an infinite dimensional Clifford algebra. The concept of degrees is introduced for basis of the Hilbert space and is orthonormalized by different degrees. The obtained space is denoted by . The following results are obtained:

(1) An orthonormal basis of is constructed (Theorem I).

(2) Infinite dimensional Clifford algebra Cl_Ω(∞, C) is defined by a sequence of inclusions Ω = {ω_2p-1}, ω_2p-1 : Cl(2p-1, C) ↦ Cl(2p+l,C) and its representation is constructed on (Theorem II).

(3) The derivation structures are introduced on and the Euclidean spaces of arbitrary dimensions can be realized as subspaces of which preserve the derivation structures (Theorem III).

(4) The Dirac operator is defined for ClΩ(∞, C) and its renormalization is given. The embedding of the Dirac operator of ClΩ(∞, C) into is given (Theorem IV).

We summarize recent results for the Gribov-Zwanziger Lagrangian which includes the effect of restricting the path integral to the first Gribov region. These include the two loop and one loop MOM gap equations for the Gribov mass.

We discuss how to extract information about the cosmological constant from the Wheeler-DeWitt equation, considered as an eigenvalue of a Sturm-Liouville problem. A generalization to a f(R) theory is taken under examination. The equation is approximated to one loop with the help of a variational approach with Gaussian trial wave functionals. We use a zeta function regularization to handle with divergences. A renormalization procedure is introduced to remove the infinities together with a renormalization group equation.

In this article we briefly discuss the adiabatic renormalization program for spin 1/2 fields in expanding universes. We introduce the method and provide explicit expressions for the renormalized vacuum expectation value of the stress-energy tensor. Then, we discuss its application to some cosmological scenario of physical interest. We end up sketching out the proof that adiabatic and DeWitt-Schwinger point-splitting schemes provide the same renormalized expectation values of the stress-energy tensor for Dirac fields.

We consider a massive scalar field living on the recently found exact quantum space-time corresponding to vacuum spherically symmetric loop quantum gravity. The discreteness of the quantum space time naturally regularizes the scalar field, eliminating divergences. However, the resulting finite theory depends on the details of the micro physics. We argue that such dependence can be eliminated through a finite renormalization and discuss its nature. This is an example of how quantum field theories on quantum space times deal with the issues of divergences in quantum field theories.

The goal of this work is to characterize the fine structures of fractals that are in the frontier between order and chaos or whose dynamics are chaotic.

We present an one-to-one correspondence between (i) Pinto's golden tilings; (ii) smooth conjugacy classes of golden diffeomorphism of the circle that are fixed points of renormalization; (iii) smooth conjugacy classes of Anosov difeomorphisms, with an invariant measure absolutely continuous with respect to the Lebesgue measure, that are topologically conjugated to the Anosov automorphism G(x, y) = (x + y, x); (iv) solenoid functions. As we will explain, the solenoid functions give a parametrization of the infinite dimensional space consisting of the mathematical objects described in the above equivalences.