Free Field Equations Research Articles

Dynamic response of semi-cylindrical depression, cylindrical cavity and type-III crack to SH wave in half-space anisotropic media

In this study, the anti-plane dynamic response of an elastic half-space anisotropic medium containing surface semi-cylindrical depressions and internal cylindrical cavity and type-III crack is solved analytically. The wave function expansion method, the complex function method and the Green's function method can be used to effectively construct the free wave field equation and the scattered wave field equation in the case of SH wave incidence, in which the scattered wave field generated by the crack is constructed by the Green's function method and the "crack cut" method. For the scattered wave field generated by a circular cavity is constructed using the mirror method, where the positional offset coefficient in the mirror method of a half-space anisotropic medium is introduced, which well solves the problem of the asymmetry of the scattered wave field. Finally, the effects of anisotropic strength of the medium, SH wave incidence angle and dimensionless incident wave number on the surface displacement amplitude (|W|), dynamic stress concentration factor (DSCF) at the boundary of the circular cavity and dynamic stress intensity factor (DSIF) at the crack tip are investigated in both frequency and time domains under different working conditions, and conclusions different from those of isotropic medium are obtained, which provide some theoretical references for the earthquake-resistant design of the underground structure with canyon topography and the surface building.

Scattering on self-dual Taub-NUT

We derive exact solutions of massless free field equations and tree-level two-point amplitudes up to spin 2 on self-dual Taub-NUT (SDTN) spacetime, as well as on its single copy, the self-dual dyon (SDD). We use Killing spinors to build analogues of momentum eigenstates, finding that, in the spirit of color-kinematics duality, those for the SDD lift directly to provide states on the SDTN background if one replaces charge with energy. We discover that they are forced to have faster growth at infinity than in flat space due to the topological non-triviality of these backgrounds. The amplitudes for massless scalars and spinning particles in the and helicity configurations vanish for generic kinematics as a consequence of the integrability of the self-dual sector. The amplitudes are non-vanishing and we compute them exactly in the backgrounds, which are treated non-perturbatively. It is explained how spin is easily introduced via a Newman–Janis imaginary shift along the spin-vector leading directly to the well-known exponential factor in the dot product of the spin with the momenta. We also observe a double copy relation between the gluon amplitude on a SDD and graviton amplitude on a SDTN spacetime.

Evolutionary Sequence of Spacetime and Intrinsic Spacetime and Associated Sequence of Geometries in Metric Force Fields II

The geometry of curved ‘three-dimensional’ absolute intrinsic metric space (an absolute intrinsic Riemannian metric space) \(\varnothing\hat{\mathbb{M}}^3\), which is curved (as a curved hyper-surface) toward the absolute time/absolute intrinsic time ‘dimensions’ (along the vertical), and projects a flat three-dimensional absolute proper intrinsic metric space \(\varnothing\hat{\mathbb{E}}^\prime\)ab3 and its outward manifestation namely, the flat absolute proper 3-space \(\hat{\mathbb{E}}^\prime\)ab3, both as flat hyper-surfaces along the horizontal, isolated in the first part of this paper, is subjected to graphical analysis. Two absolute intrinsic metric tensor equations, one of which is of the form of Einstein free space field equations and the other which is a tensorial statement of absolute intrinsic local Euclidean invariance (A\(\varnothing\)LEI) on \(\varnothing\hat{\mathbb{M}}^3\), are derived. Simultaneous (algebraic) solution to the equations yields the absolute intrinsic metric tensor and the absolute intrinsic Ricci tensor of absolute intrinsic Riemann geometry on the curved \(\varnothing\hat{\mathbb{M}}^3\), in terms of a derived absolute intrinsic curvature parameter. A superposition procedure that yields the resultant absolute intrinsic metric tensor and the resultant absolute intrinsic Ricci tensor, when two or a larger number of absolute intrinsic Riemannian metric spaces co-exist (or are superposed) is developed.
 The fact that a curved ‘three-dimensional’ absolute intrinsic metric space \(\varnothing\hat{\mathbb{M}}^3\) is perfectly isotropic and is consequently contracted to a ‘one-dimensional’ isotropic absolute intrinsic metric space, denoted by \(\varnothing\hatρ^3\), which is curved toward the absolute time/absolute intrinsic time ’dimensions’ (\(\hat{c}\)\(\hat{t}\)/\(\varnothing\)\(\hat{c}\)\(\varnothing\)\(\hat{t}\)) along the vertical is derived.

The σ− Cohomology Analysis for Symmetric Higher-Spin Fields

In this paper, we present a complete proof of the so-called First On-Shell Theorem that determines dynamical content of the unfolded equations for free symmetric massless fields of arbitrary integer spin in any dimension and arbitrary integer or half-integer spin in four dimensions. This is achieved by calculation of the respective σ− cohomology both in the tensor language in Minkowski space of any dimension and in terms of spinors in AdS4. In the d-dimensional case Hp(σ−) is computed for any p and interpretation of Hp(σ−) is given both for the original Fronsdal system and for the associated systems of higher form fields.

Quantum field theory of space-like neutrino

We performed a Lorentz covariant quantization of the spin-1/2 fermion field assuming the space-like energy-momentum dispersion relation. We achieved the task in the following steps: (i) determining the unitary realizations of the inhomogenous Lorentz group in the preferred frame scenario by means of the Wigner–Mackey induction procedure and constructing the Fock space; (ii) formulating the theory in a manifestly covariant way by constructing the field amplitudes according to the Weinberg method; (iii) obtaining the final constraints on the amplitudes by postulating a Dirac-like free field equation. Our theory allows to predict all chiral properties of the neutrinos, preserving the Standard Model dynamics. We discussed the form of the fundamental observables, energy and helicity, and show that non-observation of the +tfrac{1}{2} helicity state of the neutrino and the -tfrac{1}{2} helicity state of the antineutrino could be a direct consequence of the “tachyoneity” of neutrinos at the free level. We found that the free field theory of the space-like neutrino is not invariant under the C and P transformations separately but is CP-invariant. We calculated and analyzed the electron energy spectrum in tritium decay within the framework of our theory and found an excellent agreement with the recent measurement of KATRIN. In our formalism the questions of negative/imaginary energies and the causality problem does not appear.

Hidden symmetries for transparent de Sitter space

It is known that odd-dimensional de Sitter space acts as a transparent potential for free fields. Previous studies have explained this phenomena by relating de Sitter free field equations of motion to the time-independent Schrödinger equation with known transparent potentials. In this work we show that de Sitter’s transparency is a consequence of an infinite set of ‘hidden’ symmetries. These symmetries arise from an accidental symmetry for the zero-mode of matter fields, as well as the boost isometry of de Sitter space. For simplicity, we consider the case of massive Klein–Gordon theory. We show that the Noether charges associated with these hidden symmetries distinguish the two linearly-independent solutions of the free field wave equation in the asymptotic past and future of de Sitter. Conservation of these charges requires that the asymptotic behavior of any solution be identical, up to a constant phase, in the future and the past, which is the property of transparency. In the quantized theory, these charges act trivially on particle states belonging to the in/out vacuum Fock space. For particle states constructed from other vacua, the action of the charges generates particles.

Massless Rarita–Schwinger field from a divergenceless antisymmetric tensor-spinor of pure spin-3/2

We construct the Rarita–Schwinger basis vectors, [Formula: see text], spanning the direct product space, [Formula: see text], of a massless four-vector, [Formula: see text], with massless Majorana spinors, [Formula: see text], together with the associated field-strength tensor, [Formula: see text]. The [Formula: see text] space is reducible and contains one massless subspace of a pure spin-[Formula: see text]. We show how to single out the latter in a unique way by acting on [Formula: see text] with an earlier derived momentum independent projector, [Formula: see text], properly constructed from one of the Casimir operators of the algebra [Formula: see text] of the homogeneous Lorentz group. In this way, it becomes possible to describe the irreducible massless [Formula: see text] carrier space by means of the antisymmetric tensor of second rank with Majorana spinor components, defined as [Formula: see text]. The conclusion is that the [Formula: see text] bi-vector spinor field can play the same role with respect to a [Formula: see text] gauge field as the bi-vector, [Formula: see text], associated with the electromagnetic field-strength tensor, [Formula: see text], plays for the Maxwell gauge field, [Formula: see text]. Correspondingly, we find the free electromagnetic field equation, [Formula: see text], is paralleled by the free massless Rarita–Schwinger field equation, [Formula: see text], supplemented by the additional condition, [Formula: see text], a constraint that invokes the Majorana sector.

Periodic System of Atoms in Biquaternionic Representation

Private monochromatic solutions of the free-field equation of electro-gravimagnetic charges and currents are constructed in the differential algebra of biquaternions, which describe elementary particles as standing electro-gravimagnetic waves. The two classes of solutions of this biquaternionic wave equation have been investigated, generated by scalar potentials (pulsars) and vectorial potentials (spinors). Their asymptotic properties are considered, on the base of which they are classified into heavy (boson) and light (lepton) elementary particles. The biquaternion representation of the hydrogen atom is given. The periodic system of elements is produced, which is built on the principle of the musical structure of a simple gamma.

Spinor description of the curvatures of D = 5 gauge fields

Spinor description of the curvatures of D = 5 Yang–Mills, Rarita–Schwinger and gravitational fields is considered. Restrictions imposed on the curvature spinors by the dynamical equations and Bianchi identities are studied. In the absence of sources symmetric curvature spinors with 2s indices obey first-order equations that in the linearized limit reduce to Dirac-type equations for massless free fields. These equations allow for a higher-spin generalization similarly to 4d case. Their solution in the form of the integral over Lorentz-harmonic variables parametrizing coset manifold SO(1, 4)/(SO(1,1) × ISO(3)) isomorphic to the three-sphere is considered.

Bulk Local Operators, Conformal Descendants, and Radial Quantization

We establish a construction of the bulk local operators in AdS by considering CFT at finite energy scale. Without assuming any prior knowledge about the bulk, the solution to the bulk free field equation automatically appears in the field theory arguments. In the radial quantization formalism, we find a properly regularized version of our initial construction. Possible generalizations beyond pure AdS are also discussed.

Spinor description of D = 5 massless low-spin gauge fields

Spinor description for the curvatures of D = 5 Yang–Mills, Rarita–Schwinger and gravitational fields is elaborated. Restrictions imposed on the curvature spinors by the dynamical equations and Bianchi identities are analyzed. In the absence of sources symmetric curvature spinors with indices obey first-order equations that in the linearized limit reduce to Dirac-type equations for massless free fields. These equations allow for a higher-spin generalization similarly to case. Their solution in the form of the integral over Lorentz-harmonic variables parametrizing coset manifold isomorphic to the three-sphere is considered. Superparticle model that contains such Lorentz harmonics as dynamical variables, as well as harmonics parametrizing the two-sphere is proposed. The states in its spectrum are given by the functions on S3 that upon integrating over the Lorentz harmonics reproduce on-shell symmetric curvature spinors for various supermultiplets of D = 5 space–time supersymmetry.

Bulk locality and boundary creating operators

We formulate a minimum requirement for CFT operators to be localized in the dual AdS. In any spacetime dimensions, we show that a general solution to the requirement is a linear superposition of operators creating spherical boundaries in CFT, with the dilatation by the imaginary unit from their centers. This generalizes the recent proposal by Miyaji et al. for bulk local operators in the three dimensional AdS. We show that Ishibashi states for the global conformal symmetry in any dimensions and with the imaginary di-latation obey free field equations in AdS and that incorporating bulk interactions require their superpositions. We also comment on the recent proposals by Kabat et al., and by H. Verlinde.

Higher spins in hyperspace

We consider the Sp(2n) invariant formulation of higher spin fields on flat and curved backgrounds of constant curvature.In this formulation an infinite number of higher spin fields are packed into single scalar and spinor master fields (hyperfields) propagating on extended spaces, to be called hyperspaces, parametrized by tensorial coordinates.We show that the free field equations on flat and AdS-like hyperspaces are related to each other by a generalized conformal transformation of the scalar and spinor master fields. We compute the four--point functions on a flat hyperspace for both scalar and spinor master fields, thus extending the two-- and three--point function results of arXiv:hep-th/0312244. Then using the generalized conformal transformation we derive two--, three-- and four--point functions on AdS--like hyperspace from the corresponding correlators on the flat hyperspace.

Newton's Laws of Motion Based Substantial Aether Theory for Electro-Magnetic Wave

AbstractEven though electro-magnetic wave can be calculated from Maxwell's equations, the cause of electro-magnetic waves has not been fully understood. This paper proposes a Newton's laws of motion based aether theory to derive identical results as those from Maxwell's equations for free field. The authors suggest that every aether particle has a mass and occupies a volume in space. Every aether particle has translational movement and particle spin movement. The translational movement is similar to the gas particle moving in the air and it does not produce an electro-magnetic wave. The particle spin movement generates shear and a spin wave that will be shown to have the same results as Maxwell's equations. Detailed derivation of electro-magnetic wave solutions from the proposed aether theory and Maxwell's equations is presented in this paper to show the validation of this model.

Anisotropic phase field equations of arbitrary order

We derive a set of higher order phase field equations using a microscopic interaction Hamiltonian with detailed anisotropy in the interactions of the form $a_{0}+\delta\sum_{n=1}^{N}{a_{n}\cos( 2n\theta) + b_{n}\sin( 2n\theta) }$ where $\theta$ is the angle with respect to a fixed axis, and $\delta$ is a parameter. The Hamiltonian is expanded using complex Fourier series, and leads to a free energy and phase field equation with arbitrarily high order derivatives in the spatial variable. Formal asymptotic analysis is performed on these phase field equation in terms of the interface thickness in order to obtain the interfacial conditions. One can capture $2N$-fold anisotropy by retaining at least $2N^{th}$ degree phase field equation. We derive, in the limit of small $\delta,$ the classical result $( T-T_{E} ) [s]_{E}=-\kappa {\sigma( \theta ) + \sigma^{''}( \theta) }$ where $T-T_{E}$ is the difference between the temperature at the interface and the equilibrium temperature between phases, $[s]_{E}$ is the entropy difference between phases, $\sigma$ is the surface tension and $\kappa$ is the curvature. If there is only one mode in the anisotropy [i.e., the sum contains only one term: $A_{n}\cos( 2n\theta) $] then this identity is exact (valid for any magnitude of $\delta$) if the surface tension is interpreted as the sharp interface limit of excess free energy obtained by the solution of the $2N^{th}$ degree differential equation. The techniques rely on rewriting the sums of derivatives using complex variables and combinatorial identities, and performing formal asymptotic analyses for differential equations of arbitrary order.

PARTICLE SPECTRUM OF NON-ABELIAN TENSOR GAUGE FIELDS

We review the non-Abelian tensor gauge field theory and analyze its free field equations for lower rank gauge fields when the interaction coupling constant tends to zero. The free field equations are written in terms of the first-order derivatives of extended field strength tensors similar to the electrodynamics and non-Abelian gauge theories. We determine the particle content of the free field equations and count the propagating modes which they describe. In four-dimensional spacetime the rank-2 gauge field describes propagating modes of helicity two and zero. We show that the rank-3 gauge field describes propagating modes of helicity-three and a doublet of helicity-one gauge bosons. Only four-dimensional spacetime is physically acceptable, because in five- and higher-dimensional spacetime the equation has solutions with negative norm states. We discuss the structure of the particle spectrum for higher rank gauge fields.

Solution of free field equations in non-Abelian tensor gauge field theory

Recently proposed Lagrangian for non-Abelian tensor gauge fields contains quadratic kinetic terms, as well as cubic and quartic terms describing non-linear interaction of tensor gauge fields with dimensionless coupling constant g. We analyze the free field equations for the lower rank non-Abelian tensor gauge fields. These equations are written in terms of the first order derivatives of extended field strength tensors, similarly to the electrodynamics and Yang–Mills theory. In four-dimensional space–time the rank-2 gauge field describes propagating modes of helicity two and zero. We show that the rank-3 gauge field describes propagating modes of helicity-three and a doublet of helicity-one gauge bosons. We present the free field equation for the general rank-(s+1) tensor gauge field and its higher helicity solution.

INTERPRETING THE M2-BRANE ACTION

The world-volume theory of multiple M2-branes proposed recently has a free scalar field. For large vev of this scalar field the world-volume action reduces to that of multiple D2-branes with Yang–Mills coupling proportional to the vev. We suggest that the correct interpretation of this scalar field is as the radial position of the M2-brane center of mass in a cylindrical polar coordinate system. Regarding the azimuthal angle as compact we can regard this as a set of coincident D2-branes in type IIA string theory with varying dilaton and metric. We find that the effective world-volume theory on the D2-branes has Yang–Mills coupling proportional to the radial coordinate; furthermore the radial coordinate satisfies free field equations of motion. This agrees with the corresponding results derived from the M2-brane world-volume theory.

Initial energy density and gluon distribution from the glasma in heavy-ion collisions

We estimate the energy density and the gluon distribution associated with the classical fields describing the early-time dynamics of heavy-ion collisions. In the McLerran-Venugopalan model, we first decompose the energy density into the momentum components exactly, with the use of the Wilson line correlators. Then we evolve the energy density with the free-field equation, which is justified by the dominance of the ultraviolet modes near the collision point. We also discuss the improvement that occurs with the inclusion of nonlinear terms into the time evolution. Our numerical results at RHIC energy are fairly consistent with the empirical values.

New counterterms induced by trans-Planckian physics in semiclassical gravity

We consider free and self-interacting quantum scalar fields satisfying modified dispersion relations in the framework of Einstein-Aether theory. Using adiabatic regularization, we study the renormalization of the equation for the mean value of the field in the self-interacting case, and the renormalization of the semiclassical Einstein-Aether equations for free fields. In both cases we consider Bianchi type I background spacetimes. Contrary to what happens for free fields in flat Robertson-Walker spacetimes, the self-interaction and/or the anisotropy produce nonpurely geometric terms in the adiabatic expansion, i.e. terms that involve both the metric g{sub {mu}}{sub {nu}} and the aether field u{sub {mu}}. We argue that, in a general spacetime, the renormalization of the theory would involve new counterterms constructed with g{sub {mu}}{sub {nu}} and u{sub {mu}}, generating a fine-tuning problem for the Einstein-Aether theory.