Dirac equation path integral: Interpreting the Grassmann variables

Summary A functional integral for a particle obeying the Dirac equation is presented. In earlier work (reviewed here) we showed that 1) such a particle could be described as a massless particle randomly flipping direction and helicity at a complex ratei/m and 2) its between-flips propagation could be written as a sum over paths for a Grassmann variable valued stochastic process. We here extend the earlier work by providing a geometrical interpretation of the Grassmann variables as forms onSU(2). With this interpretation we clarify the supersymmetric correspondence relating products of Grassmann variables to spatial coordinates. To speed up publication, the authors have agreed not to receive proofs which have been supervised by the Scientific Committee.     

Grassmann variables on quantum spaces

Attention is focused on antisymmetrized versions of quantum spaces that are of particular importance in physics, i.e. two-dimensional quantum plane, q-deformed Euclidean space in three or four dimensions as well as q-deformed Minkowski space. For each case standard techniques for dealing with q-deformed Grassmann variables are developed. Formulae for multiplying supernumbers are given. The actions of symmetry generators and fermionic derivatives upon antisymmetrized quantum spaces are calculated. The complete Hopf structure for all types of quantum space generators is written down. From the formulae for the coproduct a realization of the L-matrices in terms of symmetry generators can be read off. The L-matrices together with the action of symmetry generators determine how quantum spaces of different type have to be fused together. Arrival of the final proofs: 6 December 2005     

Fermionic path integration and Grassmann Brownian motion

A Grassmann probability theory, with anticommuting random variables and stochastic processes, is developed using an extension of Berezin integration to infinite dimensional spaces. A Kolmogorov-type consistency condition allows integration on spaces of paths in anticommuting space. One particular stochastic process, Grassmann Brownian motion, is described and the associated measure used to give a path-integral formula for the kernel of the evolution operator in fermionic quantum mechanics. The Fourier mode expansion of Grassmann Brownian motion is derived.Research supported by the Science and Engineering Research Council of Great Britain under advanced research fellowship number B/AF/687     

Phase transition in the spanning-hyperforest model on complete hypergraphs

By using our novel Grassmann formulation we study the phase transition of the spanning-hyperforest model of the k-uniform complete hypergraph for any k2. The case k=2 reduces to the spanning-forest model on the complete graph. Different k are studied at once by using a microcanonical ensemble in which the number of hypertrees is fixed. The low-temperature phase is characterized by the appearance of a giant hyperforest. The phase transition occurs when the number of hyperforests is a fraction (k−1)/k of the total number of vertices. The behaviour at criticality is also studied by means of the coalescence of two saddle points. As the Grassmann formulation exhibits a global supersymmetry we show that the phase transition is second order and is associated to supersymmetry breaking and we explore the pure thermodynamical phase at low temperature by introducing an explicit breaking field.     

Growth of resistance with density of scatterers in one dimensional wave propagation

We study wave propagation in a one-dimensional disordered array of scattering potentials. We consider three different ensembles of scatterer configurations: anN-ensemble with a fixed numberN of scatterers, anL-ensemble with a varying number of scatterers distributed over a fixed lengthL, and anNL-ensemble where bothN andL are fixed. The latter ensemble allows a detailed study of the mean resistance and its variance for a fixed lengthL as the number of scatterersN increases. We find that the Landauer result, which predicts an exponential increase of the mean resistance withN, is valid only in the low-density regime. At high density the mean resistance grows exponentially with N and the concept of optical potential applies. In the crossover regime we find an interesting resonance.     

Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces

The Airy distribution function describes the probability distribution of the area under a Brownian excursion over a unit interval. Surprisingly, this function has appeared in a number of seemingly unrelated problems, mostly in computer science and graph theory. In this paper, we show that this distribution function also appears in a rather well studied physical system, namely the fluctuating interfaces. We present an exact solution for the distribution P(h_m,L) of the maximal height h_m (measured with respect to the average spatial height) in the steady state of a fluctuating interface in a one dimensional system of size L with both periodic and free boundary conditions. For the periodic case, we show that P(h_m,L)=L^−1/2f(h_m L^−1/2) for all L>0 where the function f(x) is the Airy distribution function. This result is valid for both the Edwards–Wilkinson (EW) and the Kardar–Parisi–Zhang interfaces. For the free boundary case, the same scaling holds P(h_m,L)=L^−1/2F(h_m L^−1/2), but the scaling function F(x) is different from that of the periodic case. We compute this scaling function explicitly for the EW interface and call it the F-Airy distribution function. Numerical simulations are in excellent agreement with our analytical results. Our results provide a rather rare exactly solvable case for the distribution of extremum of a set of strongly correlated random variables. Some of these results were announced in a recent Letter [S.N. Majumdar and A. Comtet, Phys. Rev. Lett. 92: 225501 (2004)].     

Random Matrix Theory and L-Functions at s= 1/2

Recent results of Katz and Sarnak [8, 9] suggest that the low-lying zeros of families of L-functions display the statistics of the eigenvalues of one of the compact groups of matrices U(N), O(N) or USp(2N). We here explore the link between the value distributions of the L-functions within these families at the central point s= 1/2 and those of the characteristic polynomials Z(U,θ) of matrices U with respect to averages over SO(2N) and USp(2N) at the corresponding point θ= 0, using techniques previously developed for U(N) in [10]. For any matrix size N we find exact expressions for the moments of Z(U,0) for each ensemble, and hence calculate the asymptotic (large N) value distributions for Z(U,0) and log Z(U,0). The asymptotic results for the integer moments agree precisely with the few corresponding values known for L-functions. The value distributions suggest consequences for the non-vanishing of L-functions at the central point. Received: 1 February 2000 / Accepted: 24 March 2000     

Characteristic polynomials of random Hermitian matrices and Duistermaat–Heckman localisation on non-compact Kähler manifolds

We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N×N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard “supersymmetry” approach, we integrate out Grassmann variables at the early stage and circumvent the use of the Hubbard–Stratonovich transformation in the “bosonic” sector. The method, suggested recently by J.V. Fyodorov [Nucl. Phys. B 621 [PM] (2002) 643], is shown to be capable of calculation when reinforced with a generalisation of the Itzykson–Zuber integral to a non-compact integration manifold. We arrive to such a generalisation by discussing the Duistermaat–Heckman localisation principle for integrals over non-compact homogeneous Kähler manifolds. In the limit of large-N the asymptotic expression for the correlation function reproduces the result outlined earlier by A.V. Andreev and B.D. Simons [Phys. Rev. Lett. 75 (1995) 2304].     

On the Reflection Coefficient in a Superbarrier Passage of a Quantum Particle

The problem on the reflection coefficient is considered for a quantum particle passing over a potential barrier. A rigorous treatment of this problem is not available in the literature. We have developed a consecutive method of finding the pre-exponential multiplier in solving the problem on the probability of the passage in a quasiclassical case, including a correct choice of the singular point. Its novelty in comparison to the earlier used methods is that it involves some rules for the most expedient analytic continuation of the wave function to the complex region. Our method does not use the conventional subdivision of the incident wave function into two ones: penetrating and reflected. When considering the action integral L = pdx = L ₁ + iL ₂, we obtain a bundle of trajectories with L ₂ = const: one extreme member of this bundle is the real axis and the other extreme member is a curve which is indefinitely close to one of the singular points. This singular point plays the leaging role in finding the asymptote of the reflection coefficient R having a physical meaning. Five examples that explain the theory are considered.     

Asymmetric Diffusion and the Energy Gap Above¶the 111 Ground State of the Quantum XXZ Model

We consider the anisotropic three dimensional XXZ Heisenberg ferromagnet in a cylinder with axis along the 111 direction and boundary conditions that induce ground states describing an interface orthogonal to the cylinder axis. Let L be the linear size of the basis of the cylinder. Because of the breaking of the continuous symmetry around the axis, the Goldstone theorem implies that the spectral gap above such ground states must tend to zero as L→∞. In [3] it was proved that, by perturbing in a sub-cylinder with basis of linear size R≪L the interface ground state, it is possible to construct excited states whose energy gap shrinks as R ^-2. Here we prove that, uniformly in the height of the cylinder and in the location of the interface, the energy gap above the interface ground state is bounded from above and below by const.L ^-2. We prove the result by first mapping the problem into an asymmetric simple exclusion process on ℤ³ and then by adapting to the latter the recursive analysis to estimate from below the spectral gap of the associated Markov generator developed in [7]. Along the way we improve some bounds on the equivalence of ensembles already discussed in [3] and we establish an upper bound on the density of states close to the bottom of the spectrum. Received: 9 August 2001 / Accepted: 29 October 2001     

Quantum Grassmann manifolds

Orbits of the quantum dressing transformation forSU _q (N) acting on its solvable dual are introduced. The case is considered when the corresponding classical orbits coincide with Grassmann manifolds. Quantization of the Poisson bracket on a Zariski open subset of the Grassmann manifold yields a *-algebra generated by the quantum coordinate functions. The commutation relations are written in a compact form with the help of theR-matrix. Finite-dimensional irreducible representations ofU _h are derived from the *-algebra structure.     

The number of independent traces and supertraces on symplectic reflection algebras

It is shown that A:= H₁, _η (G), the sympectic reflection algebra over ?, has T_G independent traces, where T_G is the number of conjugacy classes of elements without eigenvalue 1 belonging to the finite group G ? Sp(2N) ? End(?^2N) generated by the system of symplectic reflections.

Simultaneously, we show that the algebra A, considered as a superalgebra with a natural parity, has S_G independent supertraces, where S_G is the number of conjugacy classes of elements without eigenvalue -1 belonging to G.

We consider also A as a Lie algebra A^L and as a Lie superalgebra A^S.

It is shown that if A is a simple associative algebra, then the supercommutant [A^S, A^S] is a simple Lie superalgebra having at least S_G independent supersymmetric invariant non-degenerate bilinear forms, and the quotient [A^L, A^L]/([A^L, A^L] ∩ ?) is a simple Lie algebra having at least T_G independent symmetric invariant non-degenerate bilinear forms.     

Electroweak currents as a consequence of the gauge symmetry of lepton flavors and of the seesaw spectrum of neutrino masses

Consequences of the hypothesis that the Sp(pn/2) symplectic group is a broken gauge group of (n) lepton flavors are considered. An invariant Majorana mass is impossible in Sp(pn/2). A dynamical spontaneous breaking of Sp(pn/2) is admissible only if the number of flavors is n = 6and if, simultaneously, parity (R,L symmetry) violation occurs. The action of the seesaw mechanism generates here three light and three heavy Dirac neutrinos. The disregard of heavy particles in an R,L-symmetric system of weak and electromagnetic interactions (R + L vector currents) leads to a theory featuring parity nonconservation and axial anomalies. Only a weak left-handed (L) and a total (R + L) electromagnetic current do not have anomalies and remain independent of high-mass physics. These are precisely Standard Model currents.     

Optical model of plant tissue

A P _L model of a planar multiply scattering multilayer plant tissue is developed based on the expansion of radiation intensity in spherical harmonics. The dependences of differential backscattering and fluorescence coefficients on the chlorophyll concentration are numerically studied in the first-order P _L approximation. It is shown that the P _L approximation yields the results that are close to the numerical Monte Carlo solution (the deviations do not exceed 5.3%). The contribution of fluorescence to the backscattering intensity is calculated to reach 16% at high chlorophyll concentrations.     

L X-ray energy shifts and intensity ratios in tantalum with C and N ions — multiple vacancies in M, N and O shells

The energy shifts and intensity ratios of different L X-ray components in tantalum element due to 10 MeV carbon and 12 MeV nitrogen ions are estimated. From the observed energy shifts, the possible number of simultaneous vacancies in M shell are estimated. A comparison of L _α /L _β2.15, L _β1/L _γ1 and L _γ2.3/L _γ4.4 with the ratios due to Scofield theoretical transition rates indicate that the number of multiple vacancies in N shell are higher than the vacancies in M and O shell. Employing Larkin’s statistical scaling procedure, the number of possible multiple vacancies in N and O shells are estimated quantitatively.     

A rigorous theory of finite-size scaling at first-order phase transitions

A large class of classical lattice models describing the coexistence of a finite number of stable states at low temperatures is considered. The dependence of the finite-volume magnetizationM _per(h, L) in cubes of sizeL ^d under periodic boundary conditions on the external fieldh is analyzed. For the case where two phases coexist at the infinite-volume transition pointh _t , we prove that, independent of the details of the model, the finite-volume magnetization per lattice site behaves likeM _per(h _t )+M tanh[ML ^d (h–h_t)] withM _per(h) denoting the infinite-volume magnetization and M=1/2[M _per(h _t +0)–M _per(h _t –0)]. Introducing the finite-size transition pointh _m (L) as the point where the finite-volume susceptibility attains the maximum, we show that, in the case of asymmetric field-driven transitions, its shift ish _t –h _m (L)=O(L ^–2d ), in contrast to claims in the literature. Starting from the obvious observation that the number of stable phases has a local maximum at the transition point, we propose a new way of determining the pointh _t from finite-size data with a shift that is exponentially small inL. Finally, the finite-size effects are discussed also in the case where more than two phases coexist.On leave from: Institut für Theoretische Physik, FU-Berlin, D-1000 Berlin 33, Federal Republic of Germany.     

Functional integral via functional equation

Discretization ofp-adic Grassmann-valued -model leads to a hierarchical model with the Hamtilonian given by a nontrivial functional integral over the Grassmann variables. Using renormalization group arguments, we reduce the calculation of this integral to a functional equation. The problem of the convergence of the perturbation expansion of this integral, realized as a small-divisors problem, is investigated.