Currents and the energy-momentum tensor in classical field theory: a fresh look at an old problem

We give a comprehensive review of various methods to define currents and the energy-momentum tensor in classical field theory, with emphasis on a geometric point of view. The necessity of “improving” the expressions provided by the canonical Noether procedure is addressed and given an adequate geometric framework. The main new ingredient is the explicit formulation of a principle of “ultralocality” with respect to the symmetry generators, which is shown to fix the ambiguity inherent in the procedure of improvement and guide it towards a unique answer: when combined with the appropriate splitting of the fields into sectors, it leads to the well-known expressions for the current as the variational derivative of the matter field Lagrangian with respect to the gauge field and for the energy-momentum tensor as the variational derivative of the matter field Lagrangian with respect to the metric tensor. In the second case, the procedure is shown to work even when the matter field Lagrangian depends explicitly on the curvature, thus establishing the correct relation between scale invariance, in the form of local Weyl invariance “on shell”, and tracelessness of the energy-momentum tensor, required for a consistent definition of the concept of a conformal field theory.     

Nonlinear massive spin-2 field generated by higher derivative gravity

We present a systematic exposition of the Lagrangian field theory for the massive spin-2 field generated in higher-derivative gravity upon reduction to a second-order theory by means of the appropriate Legendre transformation. It has been noticed by various authors that this nonlinear field overcomes the well-known inconsistency of the theory for a linear massive spin-2 field interacting with Einstein’s gravity. Starting from a Lagrangian quadratically depending on the Ricci tensor of the metric, we explore the two possible second-order pictures usually called “(Helmholtz-)Jordan frame” and “Einstein frame.” In spite of their mathematical equivalence, the two frames have different structural properties: in Einstein frame, the spin-2 field is minimally coupled to gravity, while in the other frame it is necessarily coupled to the curvature, without a separate kinetic term. We prove that the theory admits a unique and linearly stable ground state solution, and that the equations of motion are consistent, showing that these results can be obtained independently in either frame (each frame therefore provides a self-contained theory). The full equations of motion and the (variational) energy-momentum tensor for the spin-2 field in Einstein frame are given, and a simple but non-trivial exact solution to these equations is found. The comparison of the energy-momentum tensors for the spin-2 field in the two frames suggests that the Einstein frame is physically more acceptable. We point out that the energy-momentum tensor generated by the Lagrangian of the linearized theory is unrelated to the corresponding tensor of the full theory. It is then argued that the ghost-like nature of the nonlinear spin-2 field, found long ago in the linear approximation, may not be so harmful to classical stability issues, as has been expected.     

Some remarks on the energy-momentum tensor in general relativity

In general relativity, the energy-momentum tensor of a classical tensor field can be constructed by varying the action of the field with respect to the background metric. This paper suggests an alternative interpretation of the construction which also makes sense for spinor fields, and which gives some insight into the locality of energy-momentum operators in generally covariant quantum field theory.     

Invariance under conformal coordinate and point transformations

Coordinate and point transformations are studied in the context of conformal symmetry. When invariance requirements on arbitrary rank tensors are involved in both contexts, the similitudes and differences in transformation laws and invariance conditions are analysed in connection with those on tensor densities of weight W. Physically interesting tensors like the metric tensor, the electromagnetic field and the energy-momentum tensor are specifically examined. Some remarks on scalar fields and densities are added.     

Why dilatation generators do not generate dilatations

We show that the Ward identities associated with broken scale invariance contain anomalies in renormalized perturbation theory. In low orders, these anomalies can be absorbed into a redefinition of the scale dimensions of the fields in the theory, but in higher orders this is not possible. Also, these anomalies cannot be removed by studying the Green's functions for objects other than canonical fields, e.g., currents. These results are established to first nontrivial order in perturbation theory by explicit Feynman calculations (which give us information at all momentum transfers), and in higher orders by the method of Callan and Symanzik (which gives information only at zero momentum transfer). The two approaches are consistent within their common domain of validity. Two appendices contain self-contained treatments of the formal canonical theory of scale and conformal transformations and of the derivation of Ward identities. In another appendix, we derive the Callan-Symanzik equations for Green's functions of currents, and show that no redefinition of scale dimension is necessary for these objects, although the other anomalies remain.     

On Matter and Metric in Affine Theory of Gravity

The affine theory was conceived as a geometric model, wherein the connection field is the primary structure of the space-time. According to the program lying on the basis of this theory, metric and some sort of matter are somehow to be deduced from the connection field. In the present paper, we point out classical ways to a realization of this program. It is shown that, even in that case where the introduction of the metric seems to exclude the coupling of gravity to matter, the situation is not so hopeless as one may assume. In particular, for a symmetric Einstein tensor, it is answered the old question as to a self-consistent introduction of a metric and a metrical energy-momentum tensor controversially debated by Einstein, Eddington, and Weyl.     

Arbitrary tensor concomitants of a bivector and a metric in a space-time manifold

In the general theory of relativity the energy-momentum tensor due to an electromagnetic field is taken to be a symmetric rank-two tensor concomitant of a bivector and the metric tensor. As a step in the discussion of the possible uniqueness of this tensor we display a method for finding all tensor concomitants of a bivector and a metric in a space-time manifold.     

Renormalization in the theory of a quantized scalar field interacting with a robertson-walker spacetime

We demonstrate the possibility of removing the divergences in the energy-momentum tensor by identifying divergent terms with renormalizations of the coupling constants in the gravitational field equation, modified to include a cosmological term and terms quadratic in the curvature. The model studied is that of a classical Robertson-Walker metric and a quantized minimally coupled neutral scalar field. The theory is constructed first with an ultraviolet cutoff as a phenomenological ansatz. The cutoff is then removed in an attempt to obtain a more fundamental theory, whereupon the question arises of the covariance and uniqueness of the resulting renormalized energy-momentum tensor. In the case of a massless field in a spatially flat universe, an apparent infrared divergence is discussed from the point of view of operational determination of the renormalized coupling constants. In the other cases, although the divergences are successfully accounted for by renormalization, we are left with finite leading terms which do not appear to be identifiable with geometrical tensors; the significance of this result is under investigation. If these anomalous terms are dropped, the renormalized energy-momentum tensor agrees with that defined by adiabatic regularization, provided that the limit of slow time variation taken in that method is generalized to a limit of “spacetime flatness.”     

New developments in the theory of gravity interacting with quantized fields

Renormalization in the theory of a quantized scalar field interacting with the classical Einstein gravitational field is discussed. The scalar field obeys the generalization of the Klein-Gordon equation which is conformally invariant in the limit of vanishing mass. A generalized Kasner metric corresponding to an anisotropic expansion of the universe is considered. Results obtained in collaboration with S.A. Fulling and B.L. Hu are described, which show explicitly how the infinities appearing in the expectation value of the energy-momentum tensor can be absorbed through renormalization of the cosmological constant and the coefficients of a quadratic tensor appearing in a slightly generalized form of the Einstein equation. There is also a finite renormalization of the gravitational constant.     

Conformal invariance in quantum gravity

Trace anomalies in a conformal invariant theory do not arise when its conformal invariance in four dimensions is extended to an arbitrary number n of space-time dimensions: the theory can be made finite in any order of perturbation theory by conformal invariant counterterms in n dimensions. Such an extension of conformal invariance is possible provided one works in the framework of spontaneously broken conformal invariance. This is shown explicitly by working out several examples at the one-loop level and by examining the Ward identities which lead to a general proof.We speculate upon possible consequences of these results on the nature of gravitation and other fundamental interactions.     

Dilaton effective lagrangian in gluodynamics

The lagrangian of gluodynamics does not contain dimensional parameters. As a consequence, there emerges an infinite set of Ward identities connecting n-point functions induced by the operator σ = θ_μμ, where θ_μv stands for the regularized energy-momentum tensor. We construct an effective (tree) lagrangian which includes one scalar meson field and saturates automatically all the Ward identities.     

Two theorems on energy-momentum conservation

The mathematical meaning of the law of conservation of energy-momentum is examined. A distinction is made between the intrinsic properties of the metric tensor (i.e., those properties that are independent of the coordinate system), and the nonintrinsic properties of this tensor (i.e., those properties that depend upon the coordinate system). The covariance of the energy-momentum law is used to demonstrate that if one is given (a) any analytic contravariant energy-momentum tensor density in a given coordinate systemx and (b) an analytic specification of the intrinsic properties of the metric tensor, no matter what these properties may be, one can always choose the nonintrinsic properties of the metric tensor in such manner as to satisfy the law of conservation of energy-momentum in the coordinate systemx and thereby in every coordinate system. This result is proved only in the case where the contravariant components of the energy-momentum tensor density are given. Neither the covariant, nor the mixed energy-momentum tensor densities are considered. Other theorems similar to that described above are also derived. Many of the results obtained are nontrivial even when space-time is flat.     

Renormalization of an abelian gauge theory in stochastic quantization

The renormalization of an abelian gauge field coupled to a complex scalar field is disccused in the stochastic quantization method. The supper space formulation of the stochastic quantization method is used to derived the Ward Takahashi identities assocoated with supersymmetry. These Ward Takahashi identities together with previously derived Ward Takahshi identities associated with gauge invariance are shown to be sufficient to fix all the renormalization constant in temrs of scaling of the fields and of the parameters appearing in the stochastic theory.     

From massive gravity to modified general relativity

Massive gravity which has been constructed from a cohomological formulation of gauge invariance by means of the descent equations is here investigated in the classical limit. Gauge invariance requires a vector-graviton field v coupled to the massive tensor field h _μν . In the limit of vanishing graviton mass the v-field does not decouple. On the classical level this leads to a modification of general relativity. The contribution of the v-field to the energy-momentum tensor can be interpreted as dark matter density and pressure. We solve the modified field equations in the simplest spherically symmetric geometry.     

Hawking Radiation from NUT Kerr Newman Kusuya Black Hole via Effective Action and Covariant Anomalies

Using the theory of the anomalous (chiral) effective action and covariant anomalies method, the Hawking radiation from NUT-Kerr-Newman-Kusuya black hole is researched. In this paper, the electric charge parameter and magnetic monopole parameter are rewritten as equivalent parameter. In addition, we simplify the metric as 1+1 dimensional effective metric. Finally, with the method of anomalous effective action and covariant anomalies respectively, we calculate the chiral covariant current and covariant energy-momentum tensor.     

Ward Identities and Chiral Anomaly in the Luttinger Liquid

Systems of interacting non-relativistic fermions in d =1, as well as spin chains or interacting two dimensional Ising models, verify an hidden approximate Gauge invariance which can be used to derive suitable Ward identities. Despite the presence of corrections and anomalies, such Ward identities can be implemented in a Renormalization Group approach and used to exploit nontrivial cancellations which allow to control the flow of the running coupling constants; in particular this is achieved combining Ward identities, Dyson equations and suitable correction identities for the extra terms appearing in the Ward identities, due to the presence of cutoffs breaking the local gauge symmetry. The correlations can be computed and show a Luttinger liquid behavior characterized by non-universal critical indices, so that the general Luttinger liquid construction for one dimensional systems is completed without any use of exact solutions. The ultraviolet cutoff can be removed and a Quantum Field Theory corresponding to the Thirring model is also constructed.     

Gravitational field of a radiating rotating body

A nonstationary solution of the Einstein field equations, corresponding to the field of a radiating rotating body, is presented. The solution is algebraically special of Petrov type II with a twisting, shear-free, null congruence identical to that of the Kerr metric. The new metric bears the same relation to the Kerr metric as does Vaidya's metric to the Schwarzschild metric, in the sense that in both cases the radiating solution is generated from the nonradiating one by replacing the mass parameter by an arbitrary function of a retarded time coordinate. The energy-momentum tensor in the present case, however, has two terms, a Vaidya type radiative one and an additional nonradiative residual term. Due to the presence of the nonradiative term in this case, however, the energy-momentum tensor becomes Vaidya-like asymptotically only, thus allowing for a geometrical optics interpretation. Asymptotically, part of the radiation field is purely electromagnetic with a Maxwell tensor which admits only one principal null direction corresponding to the undirectional flow of radiation.     

Scalar torsion and a new symmetry of general relativity

We reformulate the general theory of relativity in the language of Riemann–Cartan geometry. We start from the assumption that the space-time can be described as a non-Riemannian manifold, which, in addition to the metric field, is endowed with torsion. In this new framework, the gravitational field is represented not only by the metric, but also by the torsion, which is completely determined by a geometric scalar field. We show that in this formulation general relativity has a new kind of invariance, whose invariance group consists of a set of conformal and gauge transformations, called Cartan transformations. These involve both the metric tensor and the torsion vector field, and are similar to the well known Weyl gauge transformations. By making use of the concept of Cartan gauges, we show that, under Cartan transformations, the new formalism leads to different pictures of the same gravitational phenomena. We illustrate this fact by looking at the one of the classical tests of general relativity theory, namely the gravitational spectral shift. Finally, we extend the concept of space-time symmetry to Riemann–Cartan space-times with scalar torsion and obtain the conservation laws for auto-parallel motions in a static spherically symmetric vacuum space-time in a Cartan gauge, whose orbits are identical to Schwarzschild orbits in general relativity.     

Can the Local Energy-Momentum Conservation Laws be Derived Solely from Field Equations?

The vanishing of the divergence of the matter stress-energy tensor for General Relativity is a particular case of a general identity, which follows from the covariance of the matter Lagrangian in much the same way as (generalized) Bianchi identities follow from the covariance of the purely gravitational Lagrangian. This identity, holding for any covariant theory of gravitating matter, relates the divergence of the stress tensor with a combination of the field equations and their derivatives. One could thus wonder if, according to a recent suggestion [1], the energy-momentum tensor for gravitating fields can be computed through a suitable rearrangement of the matter field equations, without relying on the variational definition. We show that this can be done only in particular cases, while in general it leads to ambiguities and possibly to wrong results. Moreover, in nontrivial cases the computations turn out to be more difficult than the standard variational technique.     

Palatini formulation of <Emphasis Type="Italic">f</Emphasis>(<Emphasis Type="Italic">R</Emphasis>, <Emphasis Type="Italic">T</Emphasis>) gravity theory,and its cosmological implications

We consider the Palatini formulation of f(R, T) gravity theory, in which a non-minimal coupling between the Ricci scalar and the trace of the energy-momentum tensor is introduced, by considering the metric and the affine connection as independent field variables. The field equations and the equations of motion for massive test particles are derived, and we show that the independent connection can be expressed as the Levi-Civita connection of an auxiliary, energy-momentum trace dependent metric, related to the physical metric by a conformal transformation. Similar to the metric case, the field equations impose the non-conservation of the energy-momentum tensor. We obtain the explicit form of the equations of motion for massive test particles in the case of a perfect fluid, and the expression of the extra force, which is identical to the one obtained in the metric case. The thermodynamic interpretation of the theory is also briefly discussed. We investigate in detail the cosmological implications of the theory, and we obtain the generalized Friedmann equations of the f(R, T) gravity in the Palatini formulation. Cosmological models with Lagrangians of the type \(f=R-\alpha ^2/R+g(T)\) and \(f=R+\alpha ^2R^2+g(T)\) are investigated. These models lead to evolution equations whose solutions describe accelerating Universes at late times.