O. Teoman Turgut

Point interactions in two- and three-dimensional Riemannian manifolds

We present a non-perturbative renormalization of the bound state problem of n bosons interacting with finitely many Dirac-delta interactions on two- and three-dimensional Riemannian manifolds using the heat kernel. We formulate the problem in terms of a new operator called the principal or characteristic operator ?(E). In order to investigate the problem in more detail, we then restrict the problem to one particle sector. The lower bound of the ground state energy is found for a general class of manifolds, e.g. for compact and Cartan?Hadamard manifolds. The estimate of the bound state energies in the tunneling regime is calculated by perturbation theory. Non-degeneracy and uniqueness of the ground state is proven by the Perron?Frobenius theorem. Moreover, the pointwise bounds on the wave function is given and all these results are consistent with the one given in standard quantum mechanics. Renormalization procedure does not lead to any radical change in these cases. Finally, renormalization group equations are derived and the ? function is exactly calculated. This work is a natural continuation of our previous work based on a novel approach to the renormalization of point interactions, developed by Rajeev.

Nonrelativistic Lee model in three dimensional Riemannian manifolds

In this work, we construct the nonrelativistic Lee model on some class of three dimensional Riemannian manifolds by following a novel approach introduced by S. G. Rajeev (e-print hep-th∕9902025). This approach together with the help of heat kernel allows us to perform the renormalization nonperturbatively and explicitly. For completeness, we show that the ground state energy is bounded from below for different classes of manifolds, using the upper bound estimates on the heat kernel. Finally, we apply a kind of mean field approximation to the model for compact and noncompact manifolds separately and discover that the ground state energy grows linearly with the number of bosons n.

Finitely many Dirac-delta interactions on Riemannian manifolds

This work is intended as an attempt to study the nonperturbative renormalization of bound state problem of finitely many Dirac-delta interactions on Riemannian manifolds, S2, H2, and H3. We formulate the problem in terms of a finite dimensional matrix, called the characteristic matrix Φ. The bound state energies can be found from the characteristic equation Φ(−ν2)A=0. The characteristic matrix can be found after a regularization and renormalization by using a sharp cut-off in the eigenvalue spectrum of the Laplacian, as it is done in the flat space, or using the heat kernel method. These two approaches are equivalent in the case of compact manifolds. The heat kernel method has a general advantage to find lower bounds on the spectrum even for compact manifolds as shown in the case of S2. The heat kernels for H2 and H3 are known explicitly, thus we can calculate the characteristic matrix Φ. Using the result, we give lower bound estimates of the discrete spectrum.

Non-relativistic Lee model in two-dimensional Riemannian manifolds

This work is a continuation of our previous work [F. Erman and O. T. Turgut, J. Math. Phys. 48, 122103 (2007)10.1063/1.2813026], where we constructed the non-relativistic Lee model in three-dimensional Riemannian manifolds. Here we renormalize the two-dimensional version by using the same methods and the results are shortly given since the calculations are basically the same as in the three-dimensional model. We also show that the ground state energy is bounded from below due to the upper bound of the heat kernel for compact and Cartan-Hadamard manifolds. In contrast to the construction of the model and the proof of the lower bound of the ground state energy, the mean field approximation to the two-dimensional model is not similar to the one in three dimensions and it requires a deeper analysis, which is the main result of this paper.

Large N limit of SO(N) scalar gauge theory

In this paper we study the large Nc limit of SO(Nc) gauge theory coupled to a real scalar field following ideas of Rajeev [Int. J. Mod. Phys. A 9, 5583 (1994)]. We will see that the phase space of this resulting classical theory is Sp1(H)/U(H+) which is the analog of the Siegel disk in infinite dimensions. The linearized equations of motion give us a version of the well-known ’t Hooft equation of two dimensional quantum chromodynamics.

New minimal hypersurfaces in R(k+1)(2k+1) and S2k2+3k

We find a class of minimal hypersurfaces H(k) as the zero level set of Pfaffians, resp. determinants of real 2k+2 dimensional antisymmetric matrices. While H(1) and H(2) are congruent to a 6-dimensional quadratic cone resp. Hsiangs cubic su(4) invariant in R15, H(k>2) (special harmonic so(2k+2)-invariant cones of degree>3) seem to be new.

Singular interactions supported by embedded curves

In this work, singular interactions supported by embedded curves on Riemannian manifolds are discussed from a more direct and physical perspective, via the heat kernel approach. We show that the renormalized problem is well defined, the ground state is finite and the corresponding wavefunction is positive. The renormalization group invariance of the model is also discussed.

Large N limit of SO(N) gauge theory of fermions and Bosons

In this paper we study the large Nc limit of SO(Nc) gauge theory coupled to a Majorana field and a real scalar field in 1+1 dimensions extending ideas of Rajeev [Int. J. Mod. Phys. A 9, 5583 (1994)]. We show that the phase space of the resulting classical theory of bilinears, which are the mesonic operators of this theory, is OSp1(H|H)/U(H+|H+), where H|H refers to the underlying complex graded space of combined one-particle states of fermions and bosons and H+|H+ corresponds to the positive frequency subspace. In the begining to simplify our presentation we discuss in detail the case with Majorana fermions only [the purely bosonic case is treated in Toprak and Turgut, J. Math. Phys. 43, 1340 (2002)]. In the Majorana fermion case the phase space is given by O1(H)/U(H+), where H refers to the complex one-particle states and H+ to its positive frequency subspace. The meson spectrum in the linear approximation again obeys a variant of the ’t Hooft equation. The linear approximation to the boson/fermion coupl...

Boundary effects on Bose-Einstein condensation in ultra-static space-times

The boundary effects on the Bose-Einstein condensation with a nonvanishing chemical potential on an ultra-static space-time are studied. High temperature regime, which is the relevant regime for the relativistic gas, is studied through the heat kernel expansion for both Dirichlet and Neumann boundary conditions. The high temperature expansion in the presence of a chemical potential is generated via the Mellin transform method as applied to the harmonic sums representing the free energy and the depletion coefficient. The effects of boundary conditions on the relation between the depletion coefficient and the temperature are analyzed. Both charged and neutral bosons are considered.

A many-body problem with point interactions on two-dimensional manifolds

A non-perturbative renormalization of a many-body problem, where nonrelativistic bosons living on a two-dimensional Riemannian manifold interact with each other via the two-body Dirac delta potential, is given by the help of the heat kernel defined on the manifold. After this renormalization procedure, the resolvent becomes a well-defined operator expressed in terms of an operator (called principal operator) which includes all the information about the spectrum. Then, the ground state energy is found in the mean-field approximation and we prove that it grows exponentially with the number of bosons. The renormalization group equation (or Callan‐Symanzik equation) for the principal operator of the model is derived and the β function is exactly calculated for the general case, which includes all particle numbers.