Boltzmann'sH theorem and the Loschmidt and the Zermelo paradoxes

The Umkehreinwand of Loschmidt and the Wiederkehreinwand of Zermelo have been reexamined. The former paradox depends on the augument that for a dynamical system, upon the reversal of the velocities of all the molecules, theH function retraces its sequence of values so thatdH/dt will change its sign. The latter paradox depends on the argument that theH function returns infinitely close to its value after a Poincare' quasi-period and therefore cannot be decreasing all the time. While the main contention of the two paradoxes is correct, that theH theorem is inconsistent with classical dynamical laws, the arguments there can be considerably simplified and the “paradoxes” answered more directly. If the distribution functionf(q _K ,p _K ,t) is governed by an equation which is time-reversal invariant (such as the Liouville equation for a closed dynamical system), then it can be shown immediately thatdH/dt=0,H=cons. In this case, both paradoxes disappear, but together with them, thedH/dt<0 part of theH theorem also has disappeared, i.e., there is no second law of thermodynamics. Iff(q _K ,p _K ,t) is governed by an equation which is not time-reversal invariant (such as the Boltzmann equation, or the Master Equation for Markovian processes), then (1) there is no argument forf andH(t) to retrace their sequence of values upon the reversal of all the velocities of the system, (2) there is no quasiperiod in whichf andH(t) return to their earlier values. In this case, both paradoxes disappear also, but then one must go beyond classical dynamics in order to maintain theH theorem.     

Supersymmetry and the multi-instanton measure

We propose explicit formulae for the integration measure on the moduli space of charge-n ADHM multi-instantons in N = 1 and N = 2 supersymmetric gauge theories. The form of this measure is fixed by its (super) symmetries as well as the physical requirement of clustering in the limit of large spacetime separation between instantons. We test our proposals against known expressions for n ⩽ 2. Knowledge of the measure for all n allows us to revisit, and strengthen, earlier N = 2 results, chiefly: (1) For any number of flavors N_F, we provide a closed formula for F_n, the n-instanton contribution to the Seiberg-Witten propotential, as a finite-dimensional collective coordinate integral. This amounts to a solution, in quadratures, of the Seiberg-Witten models, without appeal to electric-magnetic duality. (2) In the conformal case N_F = 4, this means reducing to quadratures the previously unknown finite renormalization that relates the microscopic and effective coupling constants, τ_micro and τ_eff. (3) Similar expressions are given for the 4-derivative/8-fermion term in the gradient expansion of N = 2 supersymmetric QCD.     

Supermembranes and the signature of spacetime

We consider extended objects with s space and t time world-volume dimensions moving in a spacetime with S⩾ s space and T⩾t time dimensions. The requirements of spacetime supersymmetry and world-volume fermionic gauge invariance severely restrict the possible values of S and T. If we furthermore insist that the transverse group SO(S − s, T−t) be compact to avoid ghosts, then t=T. The results may be interpreted as a set of superconformal field theories with s+t⩽6 and N⩽8 whose superconformal groups are in one-to-one correspondence with those in Nahm's classification. Although the choice t = T = 1 is not uniquely singled out, it does seem to play a preferred role.     

Factorization and the couplings of the vacuum trajectory

The decoupling theorems associated with an isolated factorizable pomeron pole of unit intercept are re-examined. It is found that the coupling of three such poles, Γ(t, t, 0), need not vanish, precisely at the point t = 0. This is demonstrated by summing only over states in the appropriate unitarity sum, and sum rule, which are consistent with the M², s/M² → ∞ limit. The triple-Regge region then makes a constant contribution to σ_total, insteadsb of the ln lns result obtained if the isolated pole is assumed to couple also to states such that s/M² = constant. The physical implications regarding factorization and the pole-cut relationship are discussed. The relationship between higher order optical theorems (Mueller discontinuities) and particular terms in the unitarity sum for the two → two absorptive part A₂₂ is exploited. Consistent contributions to the triple-Regge region contribute constant vertex corrections to pure pole behaviour in A₂₂. There is no cut contribution and the magnitude of the vertex corrections reflects the relative amount of diffractive production. The analysis is extended to multiple fireball production where pure multipole structures emerge. The series naturally terminates if the diffractive component is sufficiently small. The implications for the behaviour of the total cross section at machine energies are discussed.     

Duality diagrams and the decays of the ψ and ψ′

We apply duality diagrams to the ψ and ψ′ decays. We propose that the ψ → 3π decay is described by a Virasoro formula, and show that the low mass ππ suppression in ψ′ → ψ2π is a consequence of the threshold factor contained in the Pomeron-2π coupling. We stress the importance of the ψ and ψ′ decays for the study of heavy mesonic resonances.     

Beta-delayed fission and neutron emission: Consequences for the astrophysicalr-process and the age of the galaxy

Beta-delayed neutron emission and beta-delayed fission probabilities (P _n andP _{β d j} ) were calculated for neutron-rich nuclei between the beta stability line and the neutron-drip line in the range 10≦Z≦100 and 75≦Z≦100, respectively. These results were obtained by applying recentβ-strength function calculations, fission barrier height predictions, and a neutron optical potential from infinite nuclear matter calculations. An area of ～100% fission probability is predicted around Z=94,N=168 extending beyond the well-known island of spontaneous fission in that region. Astrophysicalr-process calculations were performed including the calculatedP _n and Pβ d f values. This puts the method to determine the age of the Galaxy by the actinide chronometers on a reliable basis. An excellent overall agreement with the observedr-abundance distribution is obtained. The predictedr-process production ratios for the Chronometrie pairs²³²Th/²³⁸U,²³⁵U/²³⁸U and²⁴⁴Pu/²³⁸U result in an age of the Galaxy oft _G =(20.8± ₄ ² )×10⁹ a, which is by almost a factor two larger than earlier predictions by this method, but in accordance with recent astronomical observations from globular clusters. The predicted island of 100%β-delayed fission acts as a sink to ther-process with the consequence that no superheavy elements are produced in nature.     

Spectral bundles and the DRY-Conjecture

Supersymmetric heterotic string models, built from a Calabi-Yau threefold X endowed with a stable vector bundle V, usually start from a phenomenologically motivated choice of a bundle V_v in the visible sector, the spectral cover construction on an elliptically fibered X being a prominent example. The ensuing anomaly mismatch between c₂(V_v) and c₂(X), or rather the corresponding differential forms, is often ‘solved’, on the cohomological level, by including a fivebrane. This leads to the question whether the difference can be alternatively realized by a further stable bundle. The ‘DRY’-conjecture of Douglas, Reinbacher and Yau in math.AG/0604597 gives a sufficient condition on cohomology classes on X to be realized as the Chern classes of a stable sheaf. In 1010.1644 [hep-th], we showed that infinitely many classes on X exist for which the conjecture is true. In this note, we give the sufficient condition for the mentioned fivebrane classes to be realized by a further stable bundle in the hidden sector. Using a result obtained in 1011.6246 [hep-th], we show that corresponding bundles exist, thereby confirming this version of the DRY-Conjecture.     

Supersymmetry and the division algebras

We show how spinors in space-times of dimension D = t + s, where t is the time dimension, are associated for s - t = 1, 2, 4, 8 (and if t = 0, 1, 2) with the number systems (division algebras), $\text{|R, C, H, O}$. For t = 1 and s - t = 1, 2, 4 this association is “realized” by the sequence of Lorentz groups S1(2,$\text{|R}$), S1(2;$\text{|C}$), S1(2;$\text{|H}$) for D = 3, 4, 6 respectively. We discuss how octonions may be related to D = 10. For D = 6 we give details of S1(2; $\text{|H}$) spinors and construct supersymmetric models with them. These results explain various “empirical” observations in the literature relating quaternions and supersymmetry.     

On the trace and the relativistic scalar products involving Dirac matricesγ i and the 4 × 4 unit matrix

In this paper we have dealt with the relativistic scalar products likeγ _i uγ _i,Tr(γ _i u) Tr(γ _i v) and Tr(γ _i γ _j v). Here any string likeu andv involve Dirac matrices in the manneru=Π _n ^r=1 a _r where the element \(a_r = a_{r_i } \gamma _i + ia_{r_5 }\) i.e.,a _r in general involves a term \(ia_{r_5 }\) which is multiplied by a 4 × 4 unit matrix. We have further evaluated Tr (γ _b u) Tr (γ _b v) where the suffix ‘b’, unlike the dummy suffixesi andj, does not imply any summation and can assume any specific value from 1 to 5. Some reduction formulae for the evaluation of Tr (γ₅ u) and Tr(u) have been obtained in this paper.     

Effective potential methods and the Higgs mass spectrum in the MSSM

We generalize the analytical expressions for the two-loop leading-log neutral Higgs boson masses and mixing angles to the case of general left- and right-handed soft supersymmetry breaking stop and sbottom masses and left-right mixing mass parameters (m_Q, m_U, m_D, A_t, A_b). This generalization is essential for the computation of Higgs masses and couplings in the presence of light stops. At high scales we use the minimal supersymmetric standard model effective potential, while at low scales we consider the two-Higgs doublet model (renormalization group improved) effective potential, with general matching conditions at the thresholds where the squarks decouple. We define physical (pole) masses for the top quark, by including QCD self-energies, and for the neutral Higgs bosons, by including the leading one-loop electroweak self-energies where the top/stop and bottom/sbottom sectors propagate. For m_Q = m_U = m_D and moderate left-right mixing mass parameters, for which the mass expansion in terms of renormalizable Higgs quartic couplings is reliable, we find excellent agreement with previously obtained results.     

Remarks concerning the connection between properties of the 4-point-function and the Wilson-Zimmermann expansion

This paper concerns an investigation of the Wilson-Zimmermann (or “short distance”) expansion forA(x)A(y) withx →y whereA(x) is a real scalar field fulfilling Wightman's axioms. If one assumes that such an expansion exists, where the terms of the expansion are operators relatively local toA(x), then the singularities arising in the 4-point-function forx ₃ →x ₄ must control the singularities of then-point functions (n=4, 5, 6, ...) arising forx _j →x _x+1,j=1,2,...,n?1. A similar consequence can be drawn if the terms of the expansion are assumed to exist only as bilinear-forms (Section 2). For certain classes of fields one can show that this condition necessary for the short distance expansion is indeed fulfilled (Section 3). The result of the last section is that the above mentioned condition is also sufficient for the Wilson-Zimmermann expansion, interpreted as an expansion into bilinear forms, and also as an operator expansion in a somewhat modified sense.     

On the relationship between the Fisher-Frieden-Soffer arrow of time,and the behaviour of the Boltzmann and Kullback entropies

We provide a new derivation of the H-theorem satisfied by Fisher's information measure (I) for the diffusion equation and show that the ensuing monotonically decreasing behaviour of I yields, in turn, an H-theorem involving the second time derivative of the standard Boltzmann-Gibbs-Shannon entropy S. An upper bound for S(t) is obtained from this latter relation. Additionally, it is demonstrated that, under certain conditions, the behaviour of I may be related to that of the relative information measure of Kullback. This latter result is generalized to the case of the Fokker-Planck equation.     

Perturbation theory and the Rayleigh quotient

The characteristic frequencies ω of the vibrations of an elastic solid subject to boundary conditions of either zero displacement or zero traction are given by the Rayleigh quotient expressed in terms of the corresponding exact eigenfunctions. In problems that can be analytically expanded in a small parameter ε, it is shown that when an approximate eigenfunction is known with an error O(ε^N), the Rayleigh quotient gives the frequency with an error O(ε^2N), a gain of N orders. This result generalizes a well-known theorem for N=1. A non-trivial example is presented for N=4, whereby knowledge of the 3rd-order eigenfunction (error being 4th order) gives the eigenvalue with an error that is 8th order; the 6th-order term thus determined provides an unambiguous derivation of the shear coefficient in Timoshenko beam theory.     

Density isomerism and the Primakoff effect

We show that the decay width for π⁰→2γ in nuclear matter could be used as a signal for phase transitions in nuclear matter. The decay width of the π⁰ is experimentally measured using pion photoproduction off heavy nuclei by observing the Primakoff peak in the differential cross sections. We present calculations for the differential cross section with corrections to the γ?γ?π⁰ vertex arising from the nuclear medium within the nuclear radius when the medium is in the abnormal matter phase.     

Experimental determination of the electron-avalanche and the electron-ion recombination coefficient

The electron-ion recombination coefficient γ and the avalanche coefficient δ = (α ? a) · v_d, where α and a are the ionizat ion and attachment coefficients respectively and v_d the drift velocity of the electrons, have been experimentally determined in a self-sustained CO₂-laser system (1:1:3 mixture) as a function of the E/N value. For low voltages we found the expected decrease of the recombination coefficient for increasing E/N values. However, it appears that for larger voltage the recombination coefficient increases sharply for increasing E/N values. The measurements of δ show a much smaller value than expected from theoretical calculations. This must be explained by a lower value of the electron-energy distribution function for higher energies, which may be consistent with our measured high recombination probability for electrons having high energy.     

Correlation analysis of the shifts of the electronic spectra of anthracene, phenanthrene, and fluorene as functions of the temperature in the gas phase and of the refractive index in n-paraffin solvents

The temperature-induced shifts of the maxima of the longest-wavelength bands of the S ₁ ← S ₀ and S ₂ ← S ₀ transitions in phenanthrene and the S ₁ ← S ₀ transition in fluorene in the gas phase are measured. A correlation analysis of these data and similar data on the S ₁ ← S ₀ and S ₁ → S ₀ transitions in anthracene is fulfilled. The positions of the maxima of analogous bands of the S ₁ ← S ₀ transition in anthracene, S ₁ ← S ₀ and S ₂ ← S ₀ transitions in phenanthrene, and S ₁ ←S ₀ transition in fluorene dissolved in liquid n-paraffins are also measured and analyzed as functions of (n ² ? 1)/(n ² + 2), where n is the refractive index of the solvent. The results confirm the conclusion of the Mirumyantz-Demchuk theory about the linear temperature dependence of the wave number of a band maximum but contradict the prediction that, upon extrapolation to T = 0 K, the positions of the maxima correspond to the electronic transitions in the molecule unperturbed by out-of-plane vibrations of the aromatic core.     

Euler elasticae in the plane and the Whitney-Graustein theorem

In this paper, we study normal forms of plane curves and knots. We investigate the Euler functional E (the integral of the square of the curvature along the given curve) for closed plane curves, and introduce a closely related functional A, defined for polygonal curves in the plane ?² and its modified version A _R , defined for polygonal knots in Euclidean space ?³. For closed plane curves, we find the critical points of E and, among them, distinguish the minima of E, which give us the normal forms of plane curves. The minimization of the functional A for plane curves, implemented in a computer animation, gives a very visual approximation of the process of gradient descent along the Euler functional E and, thereby, illustrates the homotopy in the proof of the classical Whitney-Graustein theorem. In ?³, the minimization of A _R (implemented in a 3D animation) shows how classical knots (or more precisely thin knotted solid tori, which model resilient closed wire curves in space) are isotoped to normal forms.     

Non-Analyticity and the van der Waals Limit

We study the analyticity properties of the free energy f _γ (m) of the Kac model at points of first order phase transition, in the van der Waals limit γ↘0. We show that there exists an inverse temperature β ₀ and γ ₀>0 such that for all β≥β ₀ and for all γ∈(0,γ ₀), f _γ (m) has no analytic continuation along the path m↘m ^* (m ^* denotes spontaneous magnetization). The proof consists in studying high order derivatives of the pressure p _γ (h), which is related to the free energy f _γ (m) by a Legendre transform.     

The quantum harmonic oscillator on the sphere and the hyperbolic plane

A nonlinear model of the quantum harmonic oscillator on two-dimensional space of constant curvature is exactly solved. This model depends on a parameter λ that is related with the curvature of the space. First, the relation with other approaches is discussed and then the classical system is quantized by analyzing the symmetries of the metric (Killing vectors), obtaining a λ-dependent invariant measure dμ_λ and expressing the Hamiltonian as a function of the Noether momenta. In the second part, the quantum superintegrability of the Hamiltonian and the multiple separability of the Schrödinger equation is studied. Two λ-dependent Sturm-Liouville problems, related with two different λ-deformations of the Hermite equation, are obtained. This leads to the study of two λ-dependent families of orthogonal polynomials both related with the Hermite polynomials. Finally the wave functions Ψ_m,n and the energies E_m,n of the bound states are exactly obtained in both the sphere S² and the hyperbolic plane H².     

On the renormalization of the quantum field theory of point-like monopoles and charges

The quantum field theory of point-like monopoles and charges is first formulated on a euclidean lattice for a convenient regularization. The regularization preserves the peculiar features of the theory, namely those related to the invariance and to the quantization condition. The partition function is expressed as a path integral over the particle's closed paths and the action is constructed in terms of arbitrary surfaces having those paths as boundaries. The possible divergences of the continuum limit are discussed, in particular the vacuum polarization ones. It is found that, although both the electric charge Q and the magnetic charge G are renormalized as Q = Z_QQ_R and G = Z_GG_R, the quantization condition is preserved by the renormalization i.e. Z_QZ_G = 1 so that QG = Q_RG_R = 2πn. Due to the dual symmetry of the theory, then, for Q = G we get Z_Q = Z_G = 1.