Belinfante Energy-momentum Tensors Research Articles

Spin hydrodynamics and symmetric energy-momentum tensors – A current induced by the spin vorticity –

We discuss a puzzle in relativistic spin hydrodynamics; in the previous formulation the spin source from the antisymmetric part of the canonical energy-momentum tensor (EMT) is crucial. The Belinfante improved EMT is pseudo-gauge transformed from the canonical EMT and is usually a physically sensible choice especially when gauge fields are coupled as in magnetohydrodynamics, but the Belinfante EMT has no antisymmetric part. We find that pseudo-transformed entropy currents are physically inequivalent in nonequilibrium situations. We also identify a current induced by the spin vorticity read from the Belinfante symmetric EMT.

Double-logarithmic nonlinear electrodynamics

A new model of nonlinear electrodynamics is introduced and investigated. The theory carries one dimensionful parameter β as in Born-Infeld electrodynamics. It is shown that the dual symmetry and dilatation (scale) symmetry are broken in the proposed model. The electric field of a point-like charge is derived for this model, showing that it is non-singular at the origin. Using this electric field, the static electric energy of a point-like charge is calculated. In the presence of an external magnetic field, the theory shows a phenomenon known as vacuum birefringence. The refraction index of two polarizations, parallel and perpendicular to the external magnetic induction field, are calculated. The canonical and symmetrical Belinfante energy-momentum tensors are obtained. Using the causality and unitarity principles, the regions where the theory becomes causal and unitary are found.

Some QCD/gravity intersections

Gravitational form factors are the matrix elements of the Belinfante energy momentum tensor (EMT) which naturally incorporate the hadron structure and the equivalence principle. The relocalization property allowing to transform EMT to the Belinfante form provides the “kinematical” counterpart of the famous [Formula: see text] problem. The equivalence principle may be approximately valid for quarks and gluons separately in non-perturbative (NP)QCD, and this conjecture is supported by the experimental and lattice data. The extra-dimensional gravity leading to holographic AdS/QCD is supporting the relation of quark transverse momentum to the Regge slope, discovered by V.N. Gribov.

Modified Nonlinear Model of Arcsin-Electrodynamics

A new modified model of nonlinear arcsin-electrodynamics with two parameters is proposed and analyzed. We obtain the corrections to the Coulomb law. The effect of vacuum birefringence takes place when the external constant magnetic field is present. We calculate indices of refraction for two perpendicular polarizations of electromagnetic waves and estimate bounds on the parameter γ from the BMV and PVLAS experiments. It is shown that the electric field of a point-like charge is finite at the origin. We calculate the finite static electric energy of point-like particles and demonstrate that the electron mass can have the pure electromagnetic nature. The symmetrical Belinfante energy-momentum tensor and dilatation current are found. We show that the dilatation symmetry and dual symmetry are broken in the model suggested. We have investigated the gauge covariant quantization of the nonlinear electrodynamics fields as well as the gauge fixing approach based on Dirac's brackets.

On generalized logarithmic electrodynamics

The generalized logarithmic electrodynamics with two parameters \(\beta \) and \(\gamma \) is considered. The indexes of refraction of light in the external magnetic field are calculated. In the case \(\beta =\gamma \) we come to results obtained by Gaete and Helayël-Neto (Eur Phys J C 74:2816, 2014). The bound on the values of \(\beta \), \(\gamma \) was obtained from the Biréfringence Magnétique du Vide (BMV) experiment. The symmetrical Belinfante energy-momentum tensor and dilatation current are found.

Nonlinear electrodynamics with birefringence

A new model of nonlinear electrodynamics with three parameters is suggested. The phenomena of vacuum birefringence takes place when there is the external constant magnetic field. We calculate the indices of refraction for two polarizations of electromagnetic waves, parallel and perpendicular to the magnetic induction field. From the Bir\'{e}fringence Magn\'{e}tique du Vide (BMV) experiment one of the coefficients, $\gamma\approx 10^{-20}$ T$^{-2}$, was estimated. The canonical, symmetrical Belinfante energy-momentum tensors and dilatation current were obtained. The dilatation symmetry and the dual symmetry are broken in the model considered.

A model of nonlinear electrodynamics

A new model of nonlinear electrodynamics with two parameters is investigated. We also consider a model with one dimensional parameter. It was shown that the electric field of a point-like charge is not singular at the origin and there is the finiteness of the static electric energy of point-like charged particle. We obtain the canonical and symmetrical Belinfante energy–momentum tensors and dilatation currents. It is demonstrated that the dilatation symmetry and dual symmetry are broken in the models suggested. We have calculated the static electric energy of point-like particles.

Localized energetics of linear gravity: Theoretical development

We recently developed a local description of the energy, momentum and angular momentum carried by the linearized gravitational field, wherein the gravitational energy-momentum tensor displays positive energy density and causal energy flux, and the gravitational spin tensor describes purely spatial spin [1,2]. We now investigate the role these tensors play in a broader theoretical context, demonstrating for the first time that (a) they do indeed constitute Noether currents associated with the symmetry of the linearized gravitational field under translation and rotation and (b) they are themselves a source of gravity, analogous to the energy momentum and spin of matter. To prove (a) we construct a Lagrangian for linearized gravity (a covariantized Fierz-Pauli Lagrangian for a massless spin-2 field) and show that our tensors can be obtained from this Lagrangian using a standard variational technique for calculating Noether currents. This approach generates formulae that uniquely generalize our gravitational energy-momentum tensor and spin tensor beyond harmonic gauge: we show that no other generalization can be obtained from a covariantized Fierz-Pauli Lagrangian without introducing second derivatives in the energy-momentum tensor. We then construct the Belinfante energy-momentum tensor associated with our framework (combining spin and energy momentum into a single object) and as our first demonstration of (b) we establish that this Belinfante tensor appears as the second-order contribution to a perturbative expansion of the Einstein field equations, generating the gravitational field in a manner equivalent to the (Belinfante) energy-momentum tensor of matter. By considering a perturbative expansion of the Einstein-Cartan field equations, we then demonstrate that (b) can be realized without forming the Belinfante tensor: our energy-momentum tensor and spin tensor appear as the quadratic terms in separate field equations, generating gravity as distinct entities. Finally, we examine the role of field redefinitions within these perturbative expansions; in contrast to our tensors, the Landau-Lifshitz tensor is found to require a nonlocal field redefinition in order to be cast as a source of the gravitational field. In an appendix, we also give a brief treatment of the global quantities that our framework defines and verify their equivalence (within the quadratic approximation) to the energy momentum and angular momentum of Arnowitt, Deser, and Misner.

Maxwell–Chern–Simons Topologically Massive Gauge Fields in the First-Order Formalism

We find the canonical and Belinfante energy-momentum tensors and their nonzero traces. We note that the dilatation symmetry is broken and the divergence of the dilatation current is proportional to the topological mass of the gauge field. It was demonstrated that the gauge field possesses the `scale dimensionality' d=1/2. Maxwell - Chern - Simons topologically massive gauge field theory in 2+1 dimensions is formulated in the first-order formalism. It is shown that 6x6-matrices of the relativistic wave equation obey the Duffin - Kemmer - Petiau algebra. The Hermitianizing matrix of the relativistic wave equation is given. The projection operators extracting solutions of field equations for states with definite energy-momentum and spin are obtained. The 5x5-matrix Schrodinger form of the equation is derived after the exclusion of non-dynamical components, and the quantum-mechanical Hamiltonian is obtained. Projection operators extracting physical states in the Schrodinger picture are found.

FIELD THEORY OF MASSIVE AND MASSLESS VECTOR PARTICLES IN THE DUFFIN–KEMMER–PETIAU FORMALISM

Field theory of massive and massless vector particles is considered in the first-order formalism. The Hamiltonian form of equations is obtained after the exclusion of nondynamical components. We obtain the canonical and symmetrical Belinfante energy–momentum tensors and their nonzero traces. We note that the dilatation symmetry is broken in the massive case but in the massless case the modified dilatation current is conserved. The canonical quantization is performed and the propagator of the massive fields is found in the Duffin–Kemmer–Petiau formalism.

Noether symmetries, energy–momentum tensors, and conformal invariance in classical field theory

In the framework of classical field theory, we first review the Noether theory of symmetries, with simple rederivations of its essential results, with special emphasis given to the Noether identities for gauge theories. With this baggage on board, we next discuss in detail, for Poincaré invariant theories in flat spacetime, the differences between the Belinfante energy–momentum tensor and a family of Hilbert energy–momentum tensors. All these tensors coincide on shell but they split their duties in the following sense: Belinfante's tensor is the one to use in order to obtain the generators of Poincaré symmetries and it is a basic ingredient of the generators of other eventual spacetime symmetries which may happen to exist. Instead, Hilbert tensors are the means to test whether a theory contains other spacetime symmetries beyond Poincaré. We discuss at length the case of scale and conformal symmetry, of which we give some examples. We show, for Poincaré invariant Lagrangians, that the realization of scale invariance selects a unique Hilbert tensor which allows for an easy test as to whether conformal invariance is also realized. Finally we make some basic remarks on metric generally covariant theories and classical field theory in a fixed curved background.

KALB–RAMOND FIELDS IN THE PETIAU–DUFFIN–KEMMER FORMALISM AND SCALE INVARIANCE

Kalb–Ramond equations for massive and massless particles are considered in the framework of the Petiau–Duffin–Kemmer formalism. We obtain 10×10 matrices of the relativistic wave equation of the first-order and solutions in the form of density matrix. The canonical and Belinfante energy–momentum tensors are found. We investigate the scale invariance and obtain the conserved dilatation current. It was demonstrated that the conformal symmetry is broken even for massless fields.

On generalized Born–Infeld electrodynamics

The generalized Born–Infeld electrodynamics with two parameters is investigated. In this model the propagation of a linearly polarized laser beam in the external transverse magnetic field is considered. It was shown that there is the effect of vacuum birefringence, and we evaluate induced ellipticity. The upper bounds on the combination of parameters introduced from the experimental data of BRST and PVLAS Collaborations were obtained. When two parameters are equal to each other, we arrive at Born–Infeld electrodynamics and the effect of vacuum birefringence vanishes. We find the canonical and symmetrical Belinfante energy–momentum tensors. The trace of the energy–momentum tensor is not zero and the dilatation symmetry is broken. The four-divergence of the dilatation current is equal to the trace of the Belinfante energy–momentum tensor and is proportional to the parameter (with the dimension of the field strength) of the model. The dual symmetry is also broken in the model considered.

Note on Dirac–Kähler massless fields

We obtain the canonical and symmetrical Belinfante energy-momentum tensors of Dirac--K\"{a}hler's fields. It is shown that the traces of the energy-momentum tensors are not equal to zero. We find the canonical and Belinfante dilatation currents which are not conserved, but a new conserved dilatation current is obtained. It is pointed out that the conformal symmetry is broken. The canonical quantization is performed and the propagator of the massless fields in the first-order formalism is found.

On first-order generalized Maxwell equations

The generalized Maxwell equations including an additional scalar field are considered in the first-order formalism. The gauge invariance of the Lagrangian and the equations is broken resulting in the appearance of a scalar field. We find the canonical and symmetrical Belinfante energy-momentum tensors. It is shown that the traces of the energy-momentum tensors are not equal to zero and the dilatation symmetry is broken in the theory considered. The matrix Hamiltonian form of equations is obtained after the exclusion of the nondynamical components. The canonical quantization is performed and the propagator of the fields is found in the first-order formalism.PACS Nos.: 03.65.Pm, 03.70.+k, 04.20.Fy

On the energy–momentum tensor

We clarify the relation among canonical, metric and Belinfante's energy-momentum tensors for general tensor field theories. For any tensor field T, we define a new tensor field $\til {\bm T}$, in terms of which the Belinfante tensor is readily computed. We show that the latter is the one that arises naturally from Noether Theorem for an arbitrary spacetime and it coincides on-shell with the metric one.

Can QCD Be a Perturbation Theory of Hadrons?

We have found that the S-matrix for atoms and hadrons depends on a gauge as the elementary particles are off mass-shell in the bound states. The S-matrix for bound states one should to construct by the projection of the Belinfante energy-momentum tensor on the Gauss equation solution for the time component with the time-axis chosen as the eigenvector of the bound state total momentum operator. We have shown that this QCD Hamiltonian determined in infrared region by the rising potential ansatz, besides the parton model in the specific gauge, contains also the nonrelativistic potential model for heavy quarkonia, the chiral Lagrangians for light quarkonia with their spectrum, the glueball physics, and the small effective coupling constant in the whole region of transversal momenta.