Clifford代数中的双曲相位变换群及其在四维相对论时空中的应用

将Clifford代数所定义的双曲复空间R_H和作用在双曲复空间R_H上的双曲相位变换群U₄(H)赋予了明确的物理意义. 双曲复空间R_H同构于四维Minkowski时空，而其上的双曲相位变换群U₄(H)就是四维相对论时空中的洛仑兹(Lorentz)变换群. 进一步，利用U₄(H)群的复合变换性质，自然导出了四维Minkowski时空中Lorentz变换和速度变换的一般表达式. 由此，将狭义相对论中的特殊Lorentz变换作为特例包含其中. 关键词： 双曲复数 双曲相位变换 Minkowski时空 Clifford代数     

Absence of renormalization group pathologies near the critical temperature. Two examples

We consider real-space renormalization group transformations for Ising-type systems which are formally defined by $$\exp \left[ { - H'(\sigma ')} \right] = \sum\limits_\sigma {T(\sigma ,\sigma ')} \exp \left[ { - H(\sigma )} \right]$$ whereT(σ, σ′) is a probability kernel, i.e., ∑_σ′ T(σ,σ′) = 1 for every configuration σ. For each choice of the block spin configuration σ′, let σ′, let μ_σ′ be the measure on spin configurations σ which is formally given by taking the probability of σ to be proportional toT(σ, σ′) exp[?H(σ)]. We give a condition which is sufficient to imply that the renormalized HamiltonianH′ is defined. Roughly speaking, the condition is that the collection of measures μ_σ′ is in the high-temperature phase uniformly in the block spin configuration σ′. The proof of this result uses methods of Olivieri and Picco. We use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples: decimation with spacingb = 2 on the square lattice with β < 1.36β _c and the Kadanoff transformation with parameterp on the trian gular lattice in a subset of the β,p plane that includes values of β greater than β _c .     

Test and optimization of two routine dosemeters for the dose quantity Hp(3)

The individual monitoring service Seibersdorf uses two different passive dosemeter types based on thermo luminescence (TL) detectors for monitoring occupationally exposed persons in Austria. Whole body personal dosemeters for the personal dose equivalent quantities H_p(10) and H_p(0.07) worn on the trunk and dosemeters for the extremities for H_p(0.07) worn on a finger or wrist. Both routine dosemeters were calibrated and tested in terms of the personal dose equivalent H_p(3) assuming that the whole body dosemeter is worn on the chest (without or above a lead apron) and the modified ring/wrist dosemeter using a special strap worn on the forehead near the eyes (head band dosemeter). The test results show that it is possible to measure the dose quantity H_p(3) with these dosemeters that were originally not designed for this dose quantity. Only changes in the dose calculation algorithm and in the choice of the reference radiation quality were necessary to fulfil the requirements given in international standards for passive dosemeters in a wide energy (20 keV–1.3 MeV) and angular range (0°–60°).     

Scattering Processes of Excited Muonic Atoms by Molecular Hydrogen

The coupled-channel approach of an excited exotic atom scattering from a hydrogen molecule in the rigid-rotator approximation is developed. The K-matrix method is used to solve the set of coupled collisional equations in the space-fixed (SF) framework for the systems pμ+H₂, dμ+D₂, pμ+D₂, and dμ+H₂. The elastic and Stark mixing cross-sections for scattering on molecular ortho- and para-states are calculated for the principle quantum number n = 2 and collisional energies E≥0.1 eV.     

The random-field Ising model with asymmetric bimodal probability distribution

The Ising model, in the presence of a random field, is investigated within the mean-field approximation based on Landau expansion. The random field is drawn from the bimodal probability distribution P(h)=pδ(h−h₀)+(1−p)δ(h+h₀), where the probability p assumes any value within the interval [0,1], asymmetric distribution. The prevailing transitions are of second-order but, for some values of p and h₀, first-order phase transitions take place for smaller temperatures and higher h₀, thus confirming the existence of a tricritical point. Also, the possible reentrant phenomena in the phase diagram (T−h₀ plane) occur for appropriate values of p and h₀. Using the variational principle, we determine the equilibrium equation for magnetization and solve it for both transitions and at the tricritical point in order to determine the magnetization profile with respect to h₀.     

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.     

Remarks on the Poisson equations of motion

In respect of Poisson's equations of motion, we give solutions of the form F_o(H)exp{iU(p, q)}, which have a physical significance and are of importance for the calculation of phase operators in quantum mechanics.     

Electrical conduction properties of rare earth orthophosphates under reducing conditions

Electrical conduction properties of Sr-doped PrPO₄, NdPO₄ and SmPO₄ with the monazite structure were investigated at 500–925 °C under H₂/H₂O reducing conditions. From H/D isotope effects and p(H₂)-dependencies of the conductivities, it was found that the materials dominantly conducted protons under the wet reducing conditions although the materials slightly showed n-type electronic conduction at higher temperatures. Based on p(H₂O) and p(H₂)-dependencies of the conductivities, defect structures of the materials under the reducing conditions were discussed. As a result, it was considered that major positive defects were oxygen deficits at higher temperatures and protons at lower temperatures.     

Polar Phase of Superfluid <Superscript>3</Superscript>He: Dirac Lines in the Parameter and Momentum Spaces

The time reversal symmetric polar phase of the spin-triplet superfluid ³He has two types of Dirac nodal lines. In addition to the Dirac loop in the spectrum of the fermionic Bogoliubov quasiparticles in the momentum space (p _x , p _y , p _z ), the spectrum of bosons (magnons) has Dirac loop in the 3D space of parameters—the components of magnetic field (H _x ,H _y ,H _z ). The bosonic Dirac system lives on the border between the type-I and type-II.     

Specific heat studies of randomly diluted magnets evidence for fractal properties of the infinite percolation cluster

We present specific heat data on the system Fe_pZn_1-pL₆(ClO₄)₂, which is an experimental realization of the sc, $\text{S =}\text{1}\text{2}$, diluted Ising antiferromagnet. Here L = C₅H₅NO. The results are compared with earlier data on the diluted sc, $\text{S =}\text{1}\text{2}$, XY antiferromagnet Co_pZn_1-pL₆(Cl)₄)₂ and the diluted bcc, $\text{S =}\text{1}\text{2}$, Ising antiferromagnet Co_pZn_1-pCs₃Cl₅. The specific heat behaviour of the diluted Ising systems is found to be completely different from that of the diluted XY magnet, for which very strong short-range order effects have been observed. An explanation is given in terms of the fractal properties of the infinite percolation cluster in the temperature range above T_c(p). It is shown that there exists a strong analogy between the specific heat behaviour of a diluted magnet and that of a system of weakly coupled magnetic chains. In this picture a decrease of p corresponds with a decrease of the coupling between the chains. The differences in behaviour observed between the Ising systems on the one hand, and the (isotropic) XY or Heisenberg systems on the other, can then be explained by the fact that the intrachain correlation length depends exponentially on temperature for the Ising chain, whereas it depends much less strongly on temperature (∝ 1/T) for the XY and Heisenberg chains.     

Hyperfine magnetic interactions of 57Fe nuclei in NaFeAs arsenide

The results of the Mössbauer effect studies of layered NaFeAs arsenide in a wide temperature range are presented. The measurements at T > T _N demonstrate that the main part (～90%) of iron atoms are in the low-spin state Fe²⁺. The other atoms can be attributed to the impurity NaFe₂As₂ phase or to the extended defects in NaFeAs. The structural phase transition (at T _S ≈ 55 K) does not produce any effect on hyperfine parameters (δ, Δ) of iron atoms. At T < T _N, the spectra exhibit the existence of a certain distribution of the hyperfine magnetic field (H _Fe) at ⁵⁷Fe nuclei, indicating the inhomogeneity of the magnetic environment around iron cations. The analysis of the temperature behavior of the distribution function p(H _Fe) allows us to determine the temperature of the magnetic phase transition (T _N = 46 ± 2 K). It has been found that the magnetic ordering in the iron sublattice has a two-dimensional type. The analysis of the H _Fe(T) dependence in the framework of the Bean-Rodbell model reveals a first-order magnetic phase transition accompanied by a drastic change in the electron contributions to the main component (V _ZZ ) and the asymmetry parameter (η) of the tensor describing the electric field gradient at ⁵⁷Fe nuclei.     

The random field Ising model with an asymmetric trimodal probability distribution

The Ising model in the presence of a random field is investigated within the mean field approximation based on Landau expansion. The random field is drawn from the trimodal probability distribution P(h_i)=pδ(h_i−h₀)+qδ(h_i+h₀)+rδ(h_i), where the probabilities p,q,r take on values within the interval [0,1] consistent with the constraint p+q+r=1 (asymmetric distribution), h_i is the random field variable and h₀ the respective strength. This probability distribution is an extension of the bimodal one allowing for the existence in the lattice of non magnetic particles or vacant sites. The current random field Ising system displays second order phase transitions, which, for some values of p,q and h₀, are followed by first order phase transitions, thus confirming the existence of a tricritical point and in some cases two tricritical points. Also, reentrance can be seen for appropriate ranges of the aforementioned variables. Using the variational principle, we determine the equilibrium equation for magnetization, solve it for both transitions and at the tricritical point in order to determine the magnetization profile with respect to h₀.     

Calculation of transverse and longitudinal space charge effects within the framework of the fully six-dimensional formalism

In the following report we describe a method for calculating the envelope of a particle bunch in linear coupled storage rings and transport systems in the presence of transverse and longitudinal space charge forces using the (canonical) variablesx, p _x ,z, p _z , σ=s?v ₀·t,p _σ=ΔE/E ₀ of the fully six-dimensional formalism. This work is an extension of earlier calculations on transverse space charge forces [1] to include the synchrotron oscillations. The extension is achieved by defining a 6-dimensional ellipsoid in thex?p _x ?z?p _z ?σ?p _σ space. The motion of this ellipsoid under the influence of the external fields and the instantaneous space charge forces can be described by six generating orbit vectors which can be combined into a 6-dimensional matrixB(s). This “bunch-shape matrix”,B(s), contains complete information about the configuration of the bunch. The solution of the equations of motion is carried through in the thin lens approximation. The formalism can also encompass acceleration by cavity fields.     

The initial stage in the nonlinear motion of a domain wall in garnet films

A study of the domain-wall motion in single-crystal garnet films of the YBiFeGa system with a perpendicular magnetic anisotropy, activated by a constant in-plane bias field H _p parallel to the wall plane and a pulsed drive field H _g of an amplitude corresponding to the nonlinear region of the domain-wall velocity vs. the H _g relation is reported. The earlier data suggesting the existence of an initial phase of motion, where the wall is accelerated to a high instantaneous velocity, have been confirmed. The wall behavior in the initial phase has been shown to be affected by the field H _p and the drive-field pulse rise time. A possible mechanism of the wall structure transformation after the application of the H _g pulse is considered. It has been established that the dependence of the wall velocity on H _p in the saturation region disagrees with theory.     

Low temperature phase diagrams for quantum perturbations of classical spin systems

We consider a quantum spin system with Hamiltonian $$H = H^{(0)} + \lambda V,$$ whereH ⁽⁰⁾ is diagonal in a basis ∣s〉=? _x ∣s _x 〉 which may be labeled by the configurationss={s_x} of a suitable classical spin system on ? ^d , $$H^{(0)} |s\rangle = H^{(0)} (s)|s\rangle .$$ We assume thatH ⁽⁰⁾(s) is a finite range Hamiltonian with finitely many ground states and a suitable Peierls condition for excitation, whileV is a finite range or exponentially decaying quantum perturbation. Mapping thed dimensional quantum system onto aclassical contour system on ad+1 dimensional lattice, we use standard Pirogov-Sinai theory to show that the low temperature phase diagram of the quantum spin system is a small perturbation of the zero temperature phase diagram of the classical HamiltonianH ⁽⁰⁾, provided λ is sufficiently small. Our method can be applied to bosonic systems without substantial change. The extension to fermionic systems will be discussed in a subsequent paper.     

The mass-shell and gauge-fixing conditions for the free relativistic massive quantum particle

Working in the context of the Weyl group, which describes off-mass-shell relativistic particles, we impose “gauge-fixing” constraints involvingR ⁰,R ⁺, andD as matrix element conditions to be satisfied by the on-mass-shell states of a massive particle. We evaluate the matrix elements inp-space using five sets of co-ordinates: (p ²,p), (p ²,p ⁺,p _T ), (p ²,p ^?,p _T ), (p ²,π), and (p ²,π ⁺,π _T ) where \(\pi ^\mu \equiv p^\mu /(p^2 )^{\tfrac{1}{2}} \) . We find that, only in the case ofR ⁰ with (p ²,p) coordinates,R ⁺ with (p ²,p ⁺,p _T ) coordinates, andD with (p ², π) or (p ²,π ⁺,π _T ) coordinates, can the condition be satisfied by arbitrary on-mass-shell states. In all other cases, the condition can be satisfied only by states belonging to a subset of subspaces of the on-mass-shell Hilbert space, i.e it forces a violation of the superposition principle. These results constitute thep-space quantum version of Shanmugadhasan's theorem for constrained classical systems which states that there exists, at least locally in phase space, a canonical transformation to a set of variables in which the second-class constraints become canonical pairs equal to zero with the other canonical coordinates independent of the second-class constraints.     

The Variational Principle for a Class of Asymptotically Abelian C*-Algebras

Let (A,α) be a C^*-dynamical system. We introduce the notion of pressure P _α(H) of the automorphism α at a self-adjoint operator H∈A. Then we consider the class of AF-systems satisfying the following condition: there exists a dense α-invariant *-subalgebra ? of A such that for all pairs a,b∈? the C^*-algebra they generate is finite dimensional, and there is p=p(a,b)∈ℕ such that [α ^j (a),b]= 0 for |j|≥p. For systems in this class we prove the variational principle, i.e. show that P _α(H) is the supremum of the quantities h _φ(α) −φ(H), where h _φ(α) is the Connes–Narnhofer–Thirring dynamical entropy of α with respect to the α-invariant state φ. If H∈A, and P _α(H) is finite, we show that any state on which the supremum is attained is a KMS-state with respect to a one-parameter automorphism group naturally associated with H. In particular, Voiculescu's topological entropy is equal to the supremum of h _φ(α), and any state of finite maximal entropy is a trace. Received: 19 April 2000 / Accepted: 14 June 2000     

A calorimetric study of the dimerized phase of C60 fullerene

The temperature dependence of the heat capacity at a constant pressure C _p ⁰ = f(T) for the dimerized phase of the C₆₀ fullerene in the temperature range 300–575 K and the thermodynamic characteristics for depolymerization of this phase under normal pressure are investigated using precision differential scanning calorimetry. It is established that thermal depolymerization is a kinetically hindered process. The final products of thermal depolymerization are identified as a partially crystalline monomer face-centered cubic phase of C₆₀ with a degree of crystallinity α = 67 mol %. The results obtained in this study and our previous experimental data on the low-temperature heat capacity are used in the calculations of standard thermodynamic functions for the (C₆₀)₂ crystalline dimer, namely, the heat capacity C _p ⁰ (T), the enthalpy H ⁰(T) ? H ⁰(0), the entropy S ⁰(T), and the Gibbs function G ⁰(T) ? H ⁰(0) in the temperature range from T → 0 to 394 K.     

Effect of tensile stress on the variation of magnetic field of low-alloy steel

The surface magnetic field intensity H_p(y) of low-carbon steel plate specimen was measured after tensile test and unloading; variation of H_p(y) was studied. It was found that H_p(y) signals first decreased with increase in stress, and then reversed to the initial field when the stress was greater than 160 MPa. Under yield stress, H_p(y) reached its maximum, and then decreased slightly with further increase in stress. The initial magnetic signals have great impact on the variation of magnetic field.     

Liouville dynamics for optimal stochastic phase-space representations of quantum mechanics

It is shown that the space P(Γ_s) of Γ_s-distribution functions ?(q, p) (Husimi transforms) can be described without reference to any conventional representation of the density operator ?. A Liouville-type differential equation governing the free time-evolution of ?_t(q, p) is derived and solved explicitly; the time dependence of this solution supports the thesis that ?(q, p) is a bona fide probability density observable with optimally accurate apparatus for the simultaneous measurement of position and momentum. Liouville-type equations are derived also for the case when local interactions described by analytic potentials are present. Probability currents corresponding to ?(q, p) are defined and it is shown that they obey a continuity equation at space-time points. Reduced Γ_s-distribution functions are defined and it is shown that they obey a BBGKY hierarchy of equations. A Brownian-motion experimental test of the underlying theory of measurement is suggested.