The Relations Between q-Deformed Oscillator and Fermionic Oscillator

We study the relevance between the statistical property of q-deformed oscillator and thevalue of q. It is shown that there is the consistency condition |q| = 1. When q is not a realnumber. The q-Fock space is reconstructed so that it is applicable to the case in which q is extended to the complex number. And the general expression of q-deformed oscillator b_q and b_q⁺ in the new g-Fock space is given. It is found that q = ±i correspond to fermionic oscillator(the mix of the ordinary proper and improper one). This conclusion implies that q-deformedoscillator relates naturally and inherently the fermionic oscillator with the bosonic oscillatorby choosing the value of q. Actually, the bosonization structure of the single mode of fermioncan also be obtained in the way. Of course, the solution of the fermionic oscillator given inthis paper is a representation of quantum algebra SU_q(2).     

Wave Relations

The wave equation (free boson) problem is studied from the viewpoint of the relations on the symplectic manifolds associated to the boundary induced by solutions. Unexpectedly, there is still something to say about this simple, well-studied problem. In particular, boundaries which do not allow for a meaningful Hamiltonian evolution are not problematic from the viewpoint of relations. In the two-dimensional Minkowski case, these relations are shown to be Lagrangian. This result is then extended to a wide class of metrics and is conjectured to be true also in higher dimensions for nice enough metrics. A counterexample where the relation is not Lagrangian is provided by the Misner space.     

Isometric Uncertainty Relations

Journal of Statistical Physics - We generalize the link between fluctuation theorems and thermodynamic uncertainty relations by deriving a bound on the variance of fluxes that satisfy an isometric...     

The String Uncertainty Relations Follow from the New Relativity Principle

Stringy corrections to the ordinary Heisenberg uncertainty relations have been known for some time. However, a proper understanding of the underlying new physical principle modifying the ordinary Heisenberg uncertainty relations has not yet emerged. The author has recently proposed a new scale relativity theory as a physical foundation of string and M theories. In this work the stringy uncertainty relations, and corrections thereof, are rigorously derived from this new relativity principle without any ad-hoc assumptions. The precise connection between the Regge trajectory behavior of the string spectrum and the area quantization is also established.     

The Ising Model with Boundaries: Integrability and Functional Relations

In this Letter, we analyse the boundary conditions of the planar Ising model and determine the boundary Boltzmann weights in terms of bulk Boltzmann weights. The commutativity of the transfer matrices and their functional relations are shown.     

Thermodynamic Uncertainty Relations

Bohr and Heisenberg suggested that the thermodynamical quantities of temperature and energy are complementary in the same way as position and momentum in quantum mechanics. Roughly speaking their idea was that a definite temperature can be attributed to a system only if it is submerged in a heat bath, in which case energy fluctuations are unavoidable. On the other hand, a definite energy can be assigned only to systems in thermal isolation, thus excluding the simultaneous determination of its temperature. Rosenfeld extended this analogy with quantum mechanics and obtained a quantitative uncertainty relation in the form U (1/T) k, where k is Boltzmann's constant. The two extreme cases of this relation would then characterize this complementarity between isolation (U definite) and contact with a heat bath (T definite). Other formulations of the thermodynamical uncertainty relations were proposed by Mandelbrot (1956, 1989), Lindhard (1986), and Lavenda (1987, 1991). This work, however, has not led to a consensus in the literature. It is shown here that the uncertainty relation for temperature and energy in the version of Mandelbrot is indeed exactly analogous to modern formulations of the quantum mechanical uncertainty relations. However, his relation holds only for the canonical distribution, describing a system in contact with a heat bath. There is, therefore, no complementarily between this situation and a thermally isolated system.     

Fundamental Critical Relations

We consider systems which exhibit typical critical dependence of the specific heat: <artwork name="GPHT31001ei1">) where γ, γ ′ are critical exponents (γ = α for <artwork name="GPHT31001ei2"> for <artwork name="GPHT31001ei3">), as well as, the case when <artwork name="GPHT31001ei4">, uniaxial ferroelectrics; a = 1, liquid He⁴). Starting from the critical behaviour of the specific heat we can exactly find the asymptotic form of the Gibbs (Helmholtz) potential in the vicinity of the critical point for each case separately. We derive in this way many exact critical relations in the limit T → T_C which remain the same for each particular case. They define a new class of universal critical relations independent from the underlying microscopic mechanism and the symmetry of these systems. It, however, means that they are independent from the critical indices, characterizing each particular material. The derived relations are valid for magnetic, ferroelectric and superconducting materials, as well as, for liquid He⁴ and they have very important consequences concerning the mutual relations between critical amplitudes of many thermodynamical quantities near the critical point and therefore can be important and interesting from the experimental and technological point of view.     

普通物理学定律的规律性和内在联系

普通物理学中定律、公式很多，难以理解和记忆．但通过仔细研究，分解出各物理定律的性质、本质，加以区别，弄清它们的共性和个性，掌握其规律性和内在联系，使问题变得清晰和条理化，容易理解，便于记忆．     

Explicit Relations of Physical Potentials Through Generalized Hypervirial and Kramers' Recurrence Relations

Based on a Hamiltonian identity,we study one-dimensional generalized hypervirial theorem,Blanchardlike(non-diagonal case) and Kramers'(diagonal case) recurrence relations for arbitrary x~κ which is independent of the central potential V(x).Some significant results in diagonal case are obtained for special κ in x~κ(κ≥ 2).In particular,we find the orthogonal relation(n_1|n_2) = δ_(n_1n_2)(κ = 0),(n_1|V'(x)\n_2) =(E_(n_1)-E_(n_2))~2〈n_1x|n_2)(κ = 1),E_n =(n/V'(x)x/2|n) +(n|V(x)|n)(κ = 2) and-4E_n(n|x|n) +(n|V'(x)x~2\n〉 +4〈n|V(x)x|n〉 = 0(κ = 3).The latter two formulas can be used directly to calculate the energy levels.We present useful explicit relations for some well known physical potentials without requiring the energy spectra of quantum system.     

Uncertainty Relations for Coherence

Quantum mechanical uncertainty relations are fundamental consequences of the incompatible nature of noncommuting observables. In terms of the coherence measure based on the Wigner-Yanase skew information, we establish several uncertainty relations for coherence with respect to von Neumann measurements, mutually unbiased bases(MUBs), and general symmetric informationally complete positive operator valued measurements(SIC-POVMs),respectively. Since coherence is intimately connected with quantum uncertainties, the obtained uncertainty relations are of intrinsically quantum nature, in contrast to the conventional uncertainty relations expressed in terms of variance,which are of hybrid nature(mixing both classical and quantum uncertainties). From a dual viewpoint, we also derive some uncertainty relations for coherence of quantum states with respect to a fixed measurement. In particular, it is shown that if the density operators representing the quantum states do not commute, then there is no measurement(reference basis) such that the coherence of these states can be simultaneously small.     

πN Partial Wave Relations from Fixed-t Dispersion Relations

Starting from fixed-t dispersion relations we derive a set of relations for the πN partial wave amplitudes, generalizing previous work of OEHME [1], CAPPS and TAKEDA [2], and CHEW , GOLDBERGER , LOW and NAMBU [3]. Our relations contain a single integral kernel, which is agiven in a closed form valid for arbitrary angular momentum. This kernel correlates the imaginary parts of all partial wave amplitudes with the partial wave amplitude under consideration. The partial wave relations give the correct threshold behaviour. The region of convergence is determined in the case of axiomatic field theory and in the case of Mandelstam analyticity. Possible applications are discussed.     

The Green-Kubo Formula and the Onsager Reciprocity Relations in Quantum Statistical Mechanics

We study linear response theory in the general framework of algebraic quantum statistical mechanics and prove the Green-Kubo formula and the Onsager reciprocity relations for heat fluxes generated by temperature differentials. Our derivation is axiomatic and the key assumptions concern ergodic properties of non-equilibrium steady states.     

Reflection Relations and Fermionic Basis

There are two approaches to computing the one-point functions for sine-Gordon model in infinite volume. One is based on the use of the reflection relations, this is a bootstrap type procedure. Another is based on using the fermionic basis which originated in the study of lattice model. We show that the two procedures are deeply interrelated.     

Exts and the AGT Relations

We prove the connection between the Nekrasov partition function of \({\mathcal{N}=2}\) super-symmetric U(2) gauge theory with adjoint matter and conformal blocks for the Virasoro algebra, as predicted by the Alday–Gaiotto–Tachikawa relations. Mathematically, this is achieved by relating the Carlsson–Okounkov Ext vector bundle on the moduli space of rank 2 sheaves with Liouville vertex operators. Our approach is geometric in nature, and uses a new method for intersection-theoretic computations of the Ext operator.     

Integral Relations for Multichannel Reactions

Multichannel scattering reactions can be described through two integral relations based on the Kohn variational principle. When used in combination with the hyperspherical adiabatic expansion method the pattern of convergence of the corresponding ${{\mathcal K}}$ -matrix is similar to the one found when applied to obtain bound states.     

Monogamy Relations of Measurement-Induced Disturbance

The standard monogamy imposes severe limitations to sharing quantum correlations in multipartite quantum systems, which is a star topology and is established by Coffman, Kundu and Wootters. In this work, we discuss some monogamy relations beyond it, and focus on the measurement-induced disturbance (MID) which quantifies the multipartite quantum correlation. We prove exactly that MID obeys the property of discarding quantum systems never increases in an arbitrary quantum state. Moreover, we define a new kind of sharper monogamy relation which shows that the sum of all bipartite MID can not exceed the amount of total MID. This restriction is similarly called a mesh monogamy. We numerically study how MID is distributed in a 4-qubit mixed state, and which relation exists between the mesh monogamy of MID and the level of obeying the standard monogamy.     

Relations between site percolation thresholds

By decomposing certain lattices into two sublattices, and examining at percolation threshold the structure of their infinite clusters, an approximate relation between p _c ⁰ , of the original lattice and p _c ¹ , of the sublattice is established: p _c ⁰ (p _c ¹ )^1/2. It is conjectured that an inequality always holds: p _c ⁰ (p _c ¹ )^1/2, and heuristic arguments are given to substantiate it. By similar considerations good estimates forp _c of certain correlated percolation problems are also obtained.     

Fluctuation Relations for Diffusion Processes

The paper presents a unified approach to different fluctuation relations for classical nonequilibrium dynamics described by diffusion processes. Such relations compare the statistics of fluctuations of the entropy production or work in the original process to the similar statistics in the time-reversed process. The origin of a variety of fluctuation relations is traced to the use of different time reversals. It is also shown how the application of the presented approach to the tangent process describing the joint evolution of infinitesimally close trajectories of the original process leads to a multiplicative extension of the fluctuation relations. Member of C.N.R.S.     

Complementarity Relations for Multi-Qubit Systems

No Heading We derive two complementarity relations that constrain the individual and bipartite properties that may simultaneously exist in a multi-qubit system. The first expression, valid for an arbitrary pure state of n qubits, demonstrates that the degree to which single particle properties are possessed by an individual member of the system is limited by the bipartite entanglement that exists between that qubit and the remainder of the system. This result implies that the phenomenon of entanglement sharing is one specific consequence of complementarity. The second expression, which holds for an arbitrary state of two qubits, pure or mixed, quantifies a tradeoff between the amounts of entanglement, separable uncertainty, and single particle properties that are encoded in the quantum state. The separable uncertainty is a natural measure of our ignorance about the properties possessed by individual subsystems, and may be used to completely characterize the relationship between entanglement and mixedness in two-qubit systems. The two-qubit complementarity relation yields a useful geometric picture in which the root mean square values of local subsystem properties act like coordinates in the space of density matrices, and suggests possible insights into the problem of interpreting quantum mechanics.