Free Field Equations Research Articles

Canonical connections on Riemannian symmetric spaces and solutions to the Einstein–Yang–Mills equations

It is shown that for any principal bundle over a Riemannian symmetric space G/G0 which admits G as automorphism group, the canonical G-invariant connection satisfies the source free gauge field equations. Extending this to product manifolds V×G/G0 and assuming the metric and gauge fields decompose in a natural way, this result is still valid and the Einstein equations with gauge fields as source may also be satisfied. For G/G0, this is so automatically, but with a cosmological term present. For dimV=1 or 2, solutions are found, yielding metrics of the Robertson–Walker and Reissner–Nordstrom type.

Magnetic-monopole interactions

We study the long-range interactions between an arbitrary number of finite-energy monopoles in non-Abelian gauge theories with spontaneous symmetry breakdown. The Higgs fields may belong to either the adjoint representation or to an arbitrary representation of the gauge group $G$ with residual symmetry group U(1). We use the conservation properties of the stress-energy tensor to calculate the instantaneous force on monopoles starting from given initial field configurations. We show that when the Higgs fields belong to the adjoint representation of $G$, the stress-energy tensor vanishes everywhere in the Prasad-Sommerfield limit leading to a "no-interaction" result for a system of monopoles (or antimonopoles, but not both). When the monopoles are widely separated, one may picture each of them as consisting of a "core" outside which (the exterior region) the Yang-Mills gauge potentials obey free-field Yang-Mills equations. We find exact solutions in the exterior region (exterior solutions) and show how they determine the desired long-range interactions through the stress-energy tensor. One is led to a very simple physical interpretation of the interactions as consisting of Coulomb-type attractive or repulsive forces due to magnetic charges and Newtonian or Yukawa-type attractive forces due to Higgs fields. We show how these forces differ when we have massive and massless Higgs fields. In the massless case, the Coulomb and Newtonian forces do not have the same strength in general. From this result and also the mass spectrum, we find that the conjectured symmetry between gauge particles and finite-energy monopoles is limited to the case when the Higgs fields belong to the adjoint representation.

Representations of the duals of gauge field tensors

Classical electromagnetic fields admit a perfectly symmetrical description in terms of the electromagnetic field tensors and the duals thereof, this phenomenon being a simple consequence of Maxwell’s equations for free fields. Since the Yang–Mills equations for free gauge fields possess a somewhat analogous structure, the possibility of a similar symmetrical description of gauge fields is investigated. It is shown that source-free Yang–Mills equations do, in fact, imply the existence of a new gauge in terms of which the duals of the field tensors admit a representation; however, the latter does not in general possess a structure which is identical with that of the field tensors in terms of the original gauge.

Conformal relativity

The free-field equations for the metric bosons, that is the linearized field equations for the conformal metric, are solved and the solutions classified in this paper. The metric bosons comprise symmetric,h αβ, and skew-symmetric,f αβ, 5-tensors, where the latter is a 5-curl:f αβ=∂α f β−∂β f α. As for the fermion case (part III) the solutions fall into two general types: massless, or massive resonance families with discrete bare mass spectra. Solutions which fit the photon, the graviton, the ρ, ω etc. spin-1 mesons, the π, η etc. spin-0 mesons and the spin-2 f-meson are found, among others.

Some global properties of massless free fields

Elementary group-theoretical considerations show that global solutions to the massless free field equations are functions on the bundle of twistor dyads, rather than the bundle of conformal spin frames. Only in certain degenerate cases may they be thought of as ordinary spinor fields. This is the origin of the “Grgin discontinuity”.

Free electromagnetic fields on a compact Lie group manifold

Maxwell’s equations for free fields are studied on the underlying C∞ manifold of a compact Lie group. The formulation is in terms of exterior differential forms as given by Wheeler. It is found that on a compact connected Lie group there are no free electromagnetic fields. The results obtained are essentially a physical interpretation of the well-known theorem that the second Betti number of a compact semisimple Lie group is zero.

Conformally invariant equations of the quantum field theory

The equations invariant under the transformation of the conformal algebra are obtained using the Casimir operators. The connections among the components of the field are explicitly derived in the case of indecomposable representations of the conformal algebra which give rise to e.g. the Maxwell equations with currents. The free field equations are also incorporated in the conformally covariant scheme.

Solution of the zero rest-mass equations for free fields in the einstein static universe

Solutions to the source-free spins zero rest-mass equations in the Einstein static universe are obtained by means of the Newman-Penrose spin coefficient formalism.

Induced Representations and Free Field Equations

Abstract We give a simple and genuine method for the derivation of free field equations. Within our approach, the field equations represent necessary and sufficient conditions for the field representation to be a unitary and irreducible representation of a covering group of the full Poincare group. The method is applied to particles with spin less than or equal to one.

Equal-Time Commutation Relations for Heisenberg Fields in the Lee Model

>~ 4 > has been successfully applied to problems in both high energy physics5> and the area of many-body problems, particularly in superconductivity 6> and ferromagnetism. 7> In this paper we apply this method to the solvable Dirac­ Lee model which, unlike the pair model considered in Ref. 3), has mass and coupling constant renormalizations. The equal-time commutation relations among Heisenberg fields are derived and not assumed. Further we show that the wave­ function renormalization constant is determined from microcausality. This result sheds light on the question why in Ref. 3) the existence of the~ bound state was found to be closely connected with microcausality. The self-consistent method exploits the duality between basic fields and observed particles concretely exhibited in the presence of interactions. The basic Heisenberg fields obey nonlinear field equations which reflect the laws of nature, while the physical particles are the ones that appear in observations. These physical fields obey free field equations and define the physical Fock space. The field equations for the Heisenberg fields are to be regarded as operator equations in this physical Fock space. The Heisenberg field is related to the physical field by means of a mapping known as the dynamical map.4> The set ?f physical fields is taken to form an irreducible operator ring. Therefore, any operator of the Fock space can be written as a sum of normal products of physical flelds. The ability of this method to predict new particles rests on this postulate. The asymptotic condition may tell us that the physical fields are not complete. The completion of this set of fields then requires the existence of other particles at the level of observation and should be incorporated. 8>·8> The success of this method therefore rests on the proper choice of the set of physical fields.

Superfield Equations

AbstractCovariant free field equations are systematically derived for irreducible and bilinear superfields from representation theory of the Wess‐Zumino algebra. Interactions are introduced by covariance arguments alone or by the principle of local gauge invariance. Finally we construct propagators for massive superfields.

Covariance properties of fields under inversion and finite conformal transformations

We study the covariance properties of fields which according to Mack and Salam's terminology transform as conformal tensors of class Ib and II. A theorem is proved which states the conditions in order that a vector basis of class Ib fields can be turned into a basis of class Ia by subtracting derivatives of the same basic fields. A consequence of this result is a simple rule which allows us to write down all the possible conformal covariant free-field equations. Finally, the conditions for the existence of a unitary inversion operator, as well as the transformation properties of fields under it, have been explicitly determined.

The quantization of Gel'fand-Yaglom equations for free fields of arbitrary spin

The author considers the Gel'fand-Yaglom equations for free fields of arbitrary spin, in the particular case when the field transforms according to a direct sum of inequivalent irreducible finite representations of the proper Lorentz group. Under the assumption that the theory carries neither physical states of zero charge or energy density and that the mass-spin states are non-degenerate, he obtains the precise forms of the minimal and characteristic polynomials of the s blocks of the L0 matrix, which are then used to obtain new necessary and sufficient conditions that the theory be quantizable. The representation according to which the field transforms can be depicted graphically in a simple way and he takes advantage of this to use some simple ideas of graph theory to obtain results. The conclusion is that it will probably be necessary to allow repeated irreducible representations of the proper Lorentz group for theories of spin greater than eight to be quantizable.

Invariant scalar products and quantization of general poincaré-invariant wave equations

The question of second-quantizing the recently derived most general types of proper Poincare-invariant Schrodinger wave equations (employing locally covariant wave functions without redundant components and describing arbitrary-spin particles) is considered. As a prelude to carrying out this program the associated relativistically invariant scalar products (with respect to which the Poincare group is unitary) are determined. Then it is shown that from amongst the infinity of Poincare-invariantc-number wave equations just two survive the test of second quantization consistent with the microcausality criterion—one being suitable for half-integer spins alone and the other for integer spins only. Our analysis establishes that just the two constraints of proper Poincare invariance and quantizability lead to the emergence ofseparate T, C, P invariance (in the case of free-field equations) as well as the correct spin-statistics connection.

Kerr geometry as complexified Schwarzschild geometry

We present a simple derivation of the Schwarzschild and Kerr geometries by simplifying the Einstein free space field equations for the algebraically special form of metric studied by Kerr. This results in a system of two partial differential equations, the Laplace and eikonal equations, for a complex generating function. The metric tensor is a simple explicit functional of this generating function. The simplest solution generates the Schwarzschild geometry, while a displacement of the origin by ia in this solution generates the Kerr geometry.

On Conformally Covariant Spinor Field Equations

Conformally covariant equations for free spinor fields are determined uniquely by carrying out a descent to Minkowski space from the most general first-order rotationally covariant spinor equations in a six-dimensional flat space. It is found that the introduction of the concept of the "conformally invariant mass" is not possible for spinor fields even if the fields are defined not only on the null hyperquadric but over the entire manifold of coordinates in six-dimensional space.

Unitary Representations of the Homogeneous Lorentz Group in an O(1,1)⊗O(2) Basis and Some Applications to Relativistic Equations

Unitary irreducible representations of the homogeneous Lorentz group O(3, 1) belonging to the principal series are reduced with respect to the subgroup O(1,1) ⊗ O(2). As an application we determine the mixed basis matrix elements between O(3) and O(1,1) ⊗ O(2) bases and derive recurrence relations for them. This set of functions is then used to obtain invariant expansions of solutions of the Dirac and Proca free field equations. These expansions are shown to have the correct nonrelativistic limit.

Gauge fields and gravitational field equations

It is shown that the gravitational field equations in free space have a similar form to the free Yang-Mills field equations, where the group SL (2, C) replaces the group SU(2). The Ricci rotation coefficients take the role of the Yang-Mills like potentials, whereas the Riemann tensor takes the role of the gauge fields.

On the Harish-Chandra condition for first-order relativistically-invariant free field equations

It has hitherto been accepted that the degree of the Harish-Chandra condition applying to single-mass equations of arbitrary spin is determined by the maximum spin appearing in the representation ofSL(2,C) which acts on the field. The present paper demonstrates a fallacy in the published arguments which lead to the above conclusion, and gives the correct conclusion which can be deduced from the hypotheses. A counter-example of an irreducible, single-mass, spin 3/2 equation which does not satisfy the accepted theory is provided in an appendix.

General relativity from gauge invariance

AbstractThe free field equations for particles with spin are invariant under a group SL(2, c) whose transformations correspond to changes of representation of the twocomponent spinor algebra. The generalization of the equations which extends this invariance to a guage invariance in the Yang–Mills sense necessitates the introduction of auxiliary fields (which are also necessary to maintain Lorentz covariance). These fields can be interpreted as the potentials of a spin-2 field, just as the auxiliary fields for the charge gauge group are the potentials of a spin-l field (electromagnetism); this spin-2 field is then self-interacting. The Bargmann–Wigner formulation of the linear spin-2 field, when modified by the proposed self-interaction, yields a non-linear theory of a spin-2 field which is shown to be identical with Einstein's gravitational theory. With this interpretation the auxiliary fields take on an extra role of Yang–Mills field for the general coordinate transformation group – that is, they are the components of the affine connexion.