A Quantum Weak Energy Inequality¶for Dirac Fields in Curved Spacetime

Quantum fields are well known to violate the weak energy condition of general relativity: the renormalised energy density at any given point is unbounded from below as a function of the quantum state. By contrast, for the scalar and electromagnetic fields it has been shown that weighted averages of the energy density along timelike curves satisfy “quantum weak energy inequalities” (QWEIs) which constitute lower bounds on these quantities. Previously, Dirac QWEIs have been obtained only for massless fields in two-dimensional spacetimes. In this paper we establish QWEIs for the Dirac and Majorana fields of mass m≥ 0 on general four-dimensional globally hyperbolic spacetimes, averaging along arbitrary smooth timelike curves with respect to any of a large class of smooth compactly supported positive weights. Our proof makes essential use of the microlocal characterisation of the class of Hadamard states, for which the energy density may be defined by point-splitting. Received: 21 May 2001 / Accepted: 23 August 2001     

Nuclearity, Local Quasiequivalence and Split Property for Dirac Quantum Fields in Curved Spacetime

We show that a free Dirac quantum field on a globally hyperbolic spacetime has the following structural properties: (a) any two quasifree Hadamard states on the algebra of free Dirac fields are locally quasiequivalent; (b) the split-property holds in the representation of any quasifree Hadamard state; (c) if the underlying spacetime is static, then the nuclearity condition is satisfied, that is, the free energy associated with a finitely extended subsystem (``box') has a linear dependence on the volume of the box and goes like ∝T^s+1 for large temperatures T, where s+1 is the number of dimensions of the spacetime.     

Existence of Local Covariant Time Ordered Products of Quantum Fields in Curved Spacetime

 We establish the existence of local, covariant time ordered products of local Wick polynomials for a free scalar field in curved spacetime. Our time ordered products satisfy all of the hypotheses of our previous uniqueness theorem, so our construction essentially completes the analysis of the existence, uniqueness, and renormalizability of the perturbative expansion for nonlinear quantum field theories in curved spacetime. As a byproduct of our analysis, we derive a scaling expansion of the time ordered products about the total diagonal that expresses them as a sum of products of polynomials in the curvature times Lorentz invariant distributions, plus a remainder term of arbitrarily low scaling degree. Received: 6 December 2001 / Accepted: 10 June 2002 Published online: 21 October 2002     

PCT Theorem for the Operator Product Expansion in Curved Spacetime

We consider the operator product expansion for quantum field theories on general analytic 4-dimensional curved spacetimes within an axiomatic framework. We prove under certain general, model-independent assumptions that such an expansion necessarily has to be invariant under a simultaneous reversal of parity, time, and charge (PCT) in the following sense: The coefficients in the expansion of a product of fields on a curved spacetime with a given choice of time and space orientation are equal (modulo complex conjugation) to the coefficients for the product of the corresponding charge conjugate fields on the spacetime with the opposite time and space orientation. We propose that this result should be viewed as a replacement of the usual PCT theorem in Minkowski spacetime, at least in as far as the algebraic structure of the quantum fields at short distances is concerned.     

The B(3) Field from Curved Spacetime Embedded in the Irreducible Representation of the Einstein Group

The details of this paper build on the previous work of M. Sachs and M. Evans that describe an enhanced form of general relativity which contains an inherently non-abelian field tensor. We focus on a particular field arising from the non-abelian form of electrodynamics. In particular, the form of this field will be demonstrated in a first-order perturbation approach within the context of a simple manifold, and also the leading order contribution to this field due to the presence of matter. The enhanced form of general relativity, as detailed in [1], is that of antisymmetrized general relativity which relies on the irreducible representation of the Einstein translation group. We also discuss the possibility of an inherent energy induced by curvature.     

Stability of Quantum Systems at Three Scales: Passivity, Quantum Weak Energy Inequalities and the Microlocal Spectrum Condition

Quantum weak energy inequalities have recently been extensively discussed as a condition on the dynamical stability of quantum field states, particularly on curved spacetimes. We formulate the notion of a quantum weak energy inequality for general dynamical systems on static background spacetimes and establish a connection between quantum weak energy inequalities and thermodynamics. Namely, for such a dynamical system, we show that the existence of a class of states satisfying a quantum weak inequality implies that passive states (e.g., mixtures of ground- and thermal equilibrium states) exist for the time-evolution of the system and, therefore, that the second law of thermodynamics holds. As a model system, we consider the free scalar quantum field on a static spacetime. Although the Weyl algebra does not satisfy our general assumptions, our abstract results do apply to a related algebra which we construct, following a general method which we carefully describe, in Hilbert-space representations induced by quasifree Hadamard states. We discuss the problem of reconstructing states on the Weyl algebra from states on the new algebra and give conditions under which this may be accomplished. Previous results for linear quantum fields show that, on one hand, quantum weak energy inequalities follow from the Hadamard condition (or microlocal spectrum condition) imposed on the states, and on the other hand, that the existence of passive states implies that there is a class of states fulfilling the microlocal spectrum condition. Thus, the results of this paper indicate that these three conditions of dynamical stability are essentially equivalent. This observation is significant because the three conditions become effective at different length scales: The microlocal spectrum condition constrains the short-distance behaviour of quantum states (microscopic stability), quantum weak energy inequalities impose conditions at finite distance (mesoscopic stability), and the existence of passive states is a statement on the global thermodynamic stability of the system (macroscopic stability).Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany. verch@mis.mpg.de     

Moving Branes with Background Massless and Tachyon Fields in the Compact Spacetime

In this article, we shall obtain the boundary state associated with a moving Dp-brane in the presence of the Kalb–Ramond field B _μν , an internal U(1) gauge field A _α and a tachyon field, in the compact spacetime. According to this state, properties of the brane and a closed string, with mixed boundary conditions emitted from it, will be obtained. Using this boundary state, we calculate the interaction amplitude of two moving Dp ₁ and Dp ₂-branes with above background fields in a partially compact spacetime. They are parallel or perpendicular to each other. Properties of the interaction amplitude will be analyzed, and contribution of the massless states to the interaction will be extracted.     

Casimir Effect of Massive Scalar Field with Hybrid Boundary Condition in （1＋1）-Dimensional Spacetime

The Casimir energy of massive scalar field with hybrid （Diriehlet-Neumann） boundary condition is calculated. In order to regularize the model, the typical methods named as mode summation method and Green＇s function method are used respectively. It is found that the regularized zero-point energy density depends on the scalar field＇s mass. When the field is massless, the result is consistent with previous literatures.     

Solenoid Configurations and Gravitational Free Energy of the AdS–Melvin Spacetime

In this paper we explore a solenoid configuration involving a magnetic universe solution embedded in an empty Anti-de Sitter (AdS) spacetime. This requires a non-trivial surface current at the interface between the two spacetimes, which can be provided by a charged scalar field. When the interface is taken to the AdS boundary, we recover the full AdS–Melvin spacetime. The stability of the AdS–Melvin solution is also studied by computing the gravitational free energy from the Euclidean action.     

The Spinning and Curving of Spacetime: The Electromagnetic and Gravitational Fields in the Evans Field Theory

The unification of the gravitational and electromagnetic fields achieved geometrically in the generally covariant unified field theory of Evans implies that electromagnetism is the spinning of spacetime and gravitation is the curving of spacetime. The homogeneous unified field equation of Evans is a balance of spacetime spin and curvature and governs the influence of electromagnetism on gravitation using the first Bianchi identity of differential geometry. The second Bianchi identity of differential geometry is shown to lead to the conservation law of the Evans unified field, and also to a generalization of the Einstein field equation for the unified field. Rigorous mathematical proofs are given in appendices of the four equations of differential geometry which are the cornerstones of the Evans unified field theory: the first and second Maurer-Cartan structure relations and the first and second Bianchi identities. As an example of the theory, the origin of wavenumber and frequency is traced to elements of the torsion tensor of spinning spacetime.     

Equivalence of the Two Results for the Free Energy of the Chiral Potts Model

The free energy of the chiral Potts model has been obtained in two ways. The first used only the star-triangle relation, symmetries, and invariances, and led to a system of equations that implicitly define the free energy, and from which the critical behavior can be obtained The second used the functional relations derived by Bazhanov and Stroganov, solving them to obtain the free energy explicitly as a double integral. Here we obtain, for the first time, a direct verification that the two results are identical at all temperatures.     

The Time-space-dependent Power Spectrum for the Description of Nonstationary and Inhomogeneous Electromagnetic Fields

The nonstationary analogue to the stationary Wiener Khintchine theorem is considered and applied to the multi-dimensional case of temporally nonstationary and spatially inhomogeneous electromagnetic fields. The time-space-dependent power spectrum of these fields is introduced, and its connection with important measurable quantities is discussed. The dispersions relations (the mutual dependence of the Fourier variables ω, k belonging to the time and space coordinates t, ??) are included into the consideration. The variable Fourier amplitudes are compared with the conventionally used slowly varying field amplitudes. The transition to the stationary limiting case is carried out.     

The Blackbody Radiation Spectrum Follows from Zero-Point Radiation and the Structure of Relativistic Spacetime in Classical Physics

The analysis of this article is entirely within classical physics. Any attempt to describe nature within classical physics requires the presence of Lorentz-invariant classical electromagnetic zero-point radiation so as to account for the Casimir forces between parallel conducting plates at low temperatures. Furthermore, conformal symmetry carries solutions of Maxwell’s equations into solutions. In an inertial frame, conformal symmetry leaves zero-point radiation invariant and does not connect it to non-zero-temperature; time-dilating conformal transformations carry the Lorentz-invariant zero-point radiation spectrum into zero-point radiation and carry the thermal radiation spectrum at non-zero temperature into thermal radiation at a different non-zero temperature. However, in a non-inertial frame, a time-dilating conformal transformation carries classical zero-point radiation into thermal radiation at a finite non-zero-temperature. By taking the no-acceleration limit, one can obtain the Planck radiation spectrum for blackbody radiation in an inertial frame from the thermal radiation spectrum in an accelerating frame. Here this connection between zero-point radiation and thermal radiation is illustrated for a scalar radiation field in a Rindler frame undergoing relativistic uniform proper acceleration through flat spacetime in two spacetime dimensions. The analysis indicates that the Planck radiation spectrum for thermal radiation follows from zero-point radiation and the structure of relativistic spacetime in classical physics.     

SiC在辐射场中的物理特性和光谱研究

本文采用密度泛函理论的B3LYP方法,在6-311G++(d,p)基组水平上优化了不同外电场(0-0.04a.u.)下碳化硅分子的基态稳定构型,在此基础上利用同样的方法计算了碳化硅分子的分子结构、偶极矩、总能量、能隙以及红外光谱、拉曼光谱、紫外-可见吸收光谱强度.结果表明,在外电场的作用下,分子结构变化明显,与电场呈现强烈的依赖关系.碳化硅分子键长一直在增大,电偶极矩先减小后增大,分子总能量先增大后减小,能隙E_G先减小后增大再减小再增大,红外光谱吸收峰出现红移现象,拉曼光谱出现蓝移现象.随着外电场的加强,分子紫外可见吸收光谱振子强度出现先增大后减小的反复变化,其波峰也出现蓝移现象.     

GaN在辐射场中的物理特性和光谱研究

采用密度泛函理论的B3LYP方法, 在6-311G++(d,p)基组水平上优化了不同外电场(0～0.025 a.u.)下氮化镓分子的基态稳定构型, 在此基础上利用同样的方法计算了氮化镓分子的分子结构、偶极矩、总能量、能隙以及红外光谱, 拉曼光谱, 紫外-可见吸收光谱强度。结果表明, 分子的结构的变化与电场大小呈现强烈的依赖关系。随着正向外加电场的增加, GaN分子键长不断减小, 电偶极距不断减小, 分子总能量不断增大, 分子能隙不断减小, 红外光谱吸收峰出现蓝移现象, 拉曼光谱出现蓝移现象。随着外电场的加强, 分子紫外-可见吸收光谱振子强度出现先减小后增大再减小的反复变化, 其波峰则出现红移现象。     

Generalised Split Octonions and Their Transformation in SO(7) Symmetry

Generators of \(\operatorname{SO}(8)\) group have been described by using direct product of the Gamma matrices and the Pauli Sigma matrices. We have obtained these generators in terms of generalized split octonion also. These generators have been used to describe the rotational transformation of vectors in \(\operatorname{SO}(7)\) symmetry group.     

Energy Distribution for a Power Model of the Plane Symmetric Spacetime in f(R) Gravity

The study of the energy localization in f(R) theories of gravity has attracted much interest in recent years. In this paper, the vacuum solutions of the modified field equations for a power model of plane symmetric metric are studied in metric f(R) gravity with the assumption of constant Ricci scalar. Next, we determine the energy-momentum complexes in f(R) theories of gravity for this spacetime for some important models. We also show that these models satisfy the stability and constant curvature conditions.     

Energy Distribution for a Power Model of the Plane Symmetric Spacetime in f (R) Gravity

The study of the energy localization in f(R)theories of gravity has attracted much interest in recent years.In this paper,the vacuum solutions of the modified field equations for a power model of plane symmetric metric are studied in metric f(R)gravity with the assumption of constant Ricci scalar.Next,we determine the energy-momentum complexes in f(R)theories of gravity for this spacetime for some important models.We also show that these models satisfy the stability and constant curvature conditions.