Asymptotic Level Spacing of the Laguerre Ensemble: A Coulomb Fluid Approach

We determine the asymptotic level spacing distribution for the Laguerre Ensemble in a single scaled interval, $(0,s)$, containing no levels, E_{\bt}(0,s), via Dyson's Coulomb Fluid approach. For the $\alpha=0$ Unitary-Laguerre Ensemble, we recover the exact spacing distribution found by both Edelman and Forrester, while for $\alpha\neq 0$, the leading terms of $E_{2}(0,s)$, found by Tracy and Widom, are reproduced without the use of the Bessel kernel and the associated Painlev\'e transcendent. In the same approximation, the next leading term, due to a ``finite temperature'' perturbation (\bt\neq 2), is found.Comment: 10pp, LaTe

Periodic orbit theory and spectral rigidity in pseudointegrable systems

We calculate numerically the periodic orbits of pseudointegrable systems of low genus numbers $g$ that arise from rectangular systems with one or two salient corners. From the periodic orbits, we calculate the spectral rigidity $\Delta_3(L)$ using semiclassical quantum mechanics with $L$ reaching up to quite large values. We find that the diagonal approximation is applicable when averaging over a suitable energy interval. Comparing systems of various shapes we find that our results agree well with $\Delta_3$ calculated directly from the eigenvalues by spectral statistics. Therefore, additional terms as e.g. diffraction terms seem to be small in the case of the systems investigated in this work. By reducing the size of the corners, the spectral statistics of our pseudointegrable systems approaches the one of an integrable system, whereas very large differences between integrable and pseudointegrable systems occur, when the salient corners are large. Both types of behavior can be well understood by the properties of the periodic orbits in the system

Anthropic tuning of the weak scale and of m_u/m_d in two-Higgs-doublet models

It is shown that in a model in which up-type and down-type fermions acquire mass from different Higgs doublets, the anthropic tuning of the Higgs mass parameters can explain the fact that the observed masses of the $d$ and $u$ quarks are nearly the same with $d$ slightly heavier. If Yukawa couplings are assumed not to "scan" (vary among domains), this would also help explain why the t quark is much heavier than the b quark. It is also pointed out that the existence of dark matter invalidates some earlier anthropic arguments against the viability of domains where the Standard Model Higgs has positive $\mu^2$, but makes other even stronger arguments possible.Comment: 31 pages, 7 figure

Systematic Renormalization in Hamiltonian Light-Front Field Theory: The Massive Generalization

Hamiltonian light-front field theory can be used to solve for hadron states in QCD. To this end, a method has been developed for systematic renormalization of Hamiltonian light-front field theories, with the hope of applying the method to QCD. It assumed massless particles, so its immediate application to QCD is limited to gluon states or states where quark masses can be neglected. This paper builds on the previous work by including particle masses non-perturbatively, which is necessary for a full treatment of QCD. We show that several subtle new issues are encountered when including masses non-perturbatively. The method with masses is algebraically and conceptually more difficult; however, we focus on how the methods differ. We demonstrate the method using massive phi^3 theory in 5+1 dimensions, which has important similarities to QCD.Comment: 7 pages, 2 figures. Corrected error in Eq. (11), v3: Added extra disclaimer after Eq. (2), and some clarification at end of Sec. 3.3. Final published versio

Eigenvalue correlations on Hyperelliptic Riemann surfaces

In this note we compute the functional derivative of the induced charge density, on a thin conductor, consisting of the union of g+1 disjoint intervals, $J:=\cup_{j=1}^{g+1}(a_j,b_j),$ with respect to an external potential. In the context of random matrix theory this object gives the eigenvalue fluctuations of Hermitian random matrix ensembles where the eigenvalue density is supported on J.Comment: latex 2e, seven pages, one figure. To appear in Journal of Physics

Energy level statistics for models of coupled single-mode Bose--Einstein condensates

We study the distribution of energy level spacings in two models describing coupled single-mode Bose-Einstein condensates. Both models have a fixed number of degrees of freedom, which is small compared to the number of interaction parameters, and is independent of the dimensionality of the Hilbert space. We find that the distribution follows a universal Poisson form independent of the choice of coupling parameters, which is indicative of the integrability of both models. These results complement those for integrable lattice models where the number of degrees of freedom increases with increasing dimensionality of the Hilbert space. Finally, we also show that for one model the inclusion of an additional interaction which breaks the integrability leads to a non-Poisson distribution.Comment: 5 pages, 4 figures, revte

Distribution of the Riemann zeros represented by the Fermi gas

The multiparticle density matrices for degenerate, ideal Fermi gas system in any dimension are calculated. The results are expressed as a determinant form, in which a correlation kernel plays a vital role. Interestingly, the correlation structure of one-dimensional Fermi gas system is essentially equivalent to that observed for the eigenvalue distribution of random unitary matrices, and thus to that conjectured for the distribution of the non-trivial zeros of the Riemann zeta function. Implications of the present findings are discussed briefly.Comment: 7 page

Low-Temperature Properties of Two-Dimensional Ideal Ferromagnets

The manifestation of the spin-wave interaction in the low-temperature series of the partition function has been investigated extensively over more than seven decades in the case of the three-dimensional ferromagnet. Surprisingly, the same problem regarding ferromagnets in two spatial dimensions, to the best of our knowledge, has never been addressed in a systematic way so far. In the present paper the low-temperature properties of two-dimensional ideal ferromagnets are analyzed within the model-independent method of effective Lagrangians. The low-temperature expansion of the partition function is evaluated up to two-loop order and the general structure of this series is discussed, including the effect of a weak external magnetic field. Our results apply to two-dimensional ideal ferromagnets which exhibit a spontaneously broken spin rotation symmetry O(3) $\to$ O(2) and are defined on a square, honeycomb, triangular or Kagom\'e lattice. Remarkably, the spin-wave interaction only sets in at three-loop order. In particular, there is no interaction term of order $T^3$ in the low-temperature series for the free energy density. This is the analog of the statement that, in the case of three-dimensional ferromagnets, there is no interaction term of order $T^4$ in the free energy density. We also provide a careful discussion of the implications of the Mermin-Wagner theorem in the present context and thereby put our low-temperature expansions on safe grounds.Comment: 24 pages, 3 figure

Quantum Heisenberg Chain with Long-Range Ferromagnetic Interactions at Low Temperature

A modified spin-wave theory is applied to the one-dimensional quantum Heisenberg model with long-range ferromagnetic interactions. Low-temperature properties of this model are investigated. The susceptibility and the specific heat are calculated; the relation between their behaviors and strength of the long-range interactions is obtained. This model includes both the Haldane-Shastry model and the nearest-neighbor Heisenberg model; the corresponding results in this paper are in agreement with the solutions of both the models. It is shown that there exists an ordering transition in the region where the model has longer-range interactions than the HS model. The critical temperature is estimated.Comment: 17 pages(LaTeX REVTeX), 1 figure appended (PostScript), Technical Report of ISSP A-274

Stability and Instability of Relativistic Electrons in Classical Electro magnetic Fields

The stability of matter composed of electrons and static nuclei is investigated for a relativistic dynamics for the electrons given by a suitably projected Dirac operator and with Coulomb interactions. In addition there is an arbitrary classical magnetic field of finite energy. Despite the previously known facts that ordinary nonrelativistic matter with magnetic fields, or relativistic matter without magnetic fields is already unstable when the fine structure constant, is too large it is noteworthy that the combination of the two is still stable provided the projection onto the positive energy states of the Dirac operator, which defines the electron, is chosen properly. A good choice is to include the magnetic field in the definition. A bad choice, which always leads to instability, is the usual one in which the positive energy states are defined by the free Dirac operator. Both assertions are proved here.Comment: LaTeX fil