On the linearization of source free gauge field equations

A scheme for linearizing the source free Yang-Mills field equations is given. An infinite parameter invariance group of the gauge field equations is discovered.     

Yang-Mills equations are formally linearizable

It is proved, in the framework of a recent theory of non-linear group representations elaborated by the authors, that the non-linear Yang-Mills equations are formally linearizable: there exists a formal non-linear operator which intertwins the non-linear Yang-Mills equations with their free (linear) part.     

Symmetry group of generalized vector field equations

The equations describing an electromagnetic field, a Yang-Mills massless field, and a free massive vector field are generalized in a quaternion setting. The generalized equations are invariant under a six-parameter group of transformations, which do not affect the space-time coordinates. In application to the generalized Maxwell equations the indicated group is isomorphic to Zaitsev's group of outer transformations of the electromagnetic field variables.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 45–48, August, 1977.The author is indebted to S. I. Kruglov, Yu. A. Kurochkin, and E. A. Tolkachev for a critical and stimulating discussion of the present results.     

Yang-Mills fields permitted by abelian complete sets of first-order symmetry operators of the Dirac equation

Nonequivalent complete sets of first-order symmetry operators of the Dirac free equation determine the Yang-Mills field, permitting complete variable separation in the Dirac equations with an external Yang-Mills field. Typical representatives of the classes of permissible fields are considered.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 30–34, October, 1989.     

Cylindrical configuration of classical Yang-Mills fields and its symmetrical properties

An explicit construction of classical Yang-Mills equations in euclidean four-space is given on the basis of a single external constant vector ansatz for SU(2) gauge potentials. The presence of a symmetry group leads to an essential simplification of the equations for gauge invariant combinations of structure functions (entering in the field ansatz) and shows the remarkable group properties of the corresponding field structures.     

Symmetries of SU(2) invariant Yang-Mills theories

It is shown that both the field equations and the self-duality equations of SU(2) invariant Yang-Mills theories do not allow any other Lie symmetries than those they have by construction, i.e., the 15 generators of the conformal group and the 3 generators of the gauge group. This is true in Euclidean as well in Minkowskian space-time.     

Extension of the York field decomposition to general gravitationally coupled fields

We extend the York decomposition analysis of the initial value constraints to general gravitationally coupled classical field theories. The decomposition is found to be particularly useful in solving the constraint equations for all theories of current physical interest. These include Einstein gravity or Einstein-Cartan (torsion) gravity coupled to the massive or massless version of the following: general scalar (including Klein-Gordon, Brans-Dicke, and Higgs), Dirac spin 1/2, Maxwell (Proca) and Yang-Mills (any gauge group). We show in detail how the program works for the general Yang-Mills field and for the Einstein-Cartan-Proca field.     

A Universal Regulator?

By using the principle of metrical invariance which requires that all physical laws are independent of the choice of units (alternatively, all physical laws are invariant with respect to scale transformations of space-time coordinates) and Goldstone's theorem, a universal regulator is discovered. The cosmic field is the Yang-Mills field of the local scale transformations. Its physical role is as follows. Cosmon, its quantum, is a massless, spinless, and neutral particle. The cosmic field is created by inertial masses. Therefore it participates in all physical processes and if its presence is taken into account, then the quantum field theory is free from all ultraviolet infinities. From the point of view of Yang-Mills field theory, it is proved that the so-called gravitational masses are identical with inertial masses and the gravitational field is created by inertial masses moving non-inertially. This fact permits to solve satisfactorily the problem of energy-momentum complex of the gravitational field. The system of equations which defines simultaneously the cosmic and gravitational fields is established. A non-Einstein cosmology is outlined.     

Exact solutions of the equations of motions of quarks in a non-Abelian field of a flat color wave

Exact solutions of the equations of motion of quarks in a non-Abelian field of a flat color wave of a spectral configuration are obtained. The gauge field of the wave takes values from the SU(2) group and is an exact solution of the Yang-Mills equations.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 76–78, December, 1986.     

Gravitational instantons

The path-integral approach to quantum field theory assigns special importance to finite action Euclidean solutions of classical field equations. In Yang-Mills gauge theories, the instanton solutions of classical field equations with self-dual field strength have given rise to a new, nonperturbative treatment of the quantum field theory and its vacuum state. Since gravitation is also a species of gauge theory, one might think that similar phenomena would occur in gravity. The authors recently sought and found a new self-dual solution to Euclidean gravity which plays a role parallel to that of the Yang-Mills instanton. Gravitational instantons now promise to yield new insights into the nature of quantum gravity.This essay received the second award from the Gravity Research Foundation for the year 1979-Ed.     

Instantons and confinement

The Euclidean vacuum functional of Yang-Mills theory is saturated by infinitely large two-dimensional sheets of calculable universal thickness. The gauge fields inside the sheets are characterised by a distribution of instantons held together by a smooth solution of the field equations. The sheets cannot grow in the transverse directions. Within each sheet the orientation in group space of the average field is correlated to the spatial orientation of the area, while different sheets are totally uncorrelated. We exhibit a class of fluctuations above this background which confine quarks. No assumption about the infra-red behaviour of the Gell-Mann-Low function is used; all quantities are calculable and no free parameters exist beyond the group structure constants.     

A connection between the Einstein and Yang-Mills equations

It is our purpose here to show an unusual relationship between the Einstein equations and the Yang-Mills equations. We give a correspondence between solutions of the self-dual Einstein vacuum equations and the self-dual Yang-Mills equations with a special choice of gauge group. The extension of the argument to the full Yang-Mills equations yields Einstein's unifield equations. We try to incorporate the full Einstein vacuum equations, but the approach is incomplete. We first consider Yang-Mills theory for an arbitrary Lie-algebra with the condition that the connection 1-form and curvature are constant on Minkowski space. This leads to a set of algebraic equations on the connection components. We then specialize the Lie-algebra to be the (infinite dimensional) Lie-algebra of a group of diffeomorphisms of some manifold. The algebraic equations then become differential equations for four vector fields on the manifold on which the diffeomorphisms act. In the self-dual case, if we choose the connection components from the Lie-algebra of the volume preserving 4-dimensional diffeomorphism group, the resulting equations are the same as those obtained by Ashtekar, Jacobsen and Smolin, in their remarkable simplification of the self-dual Einstein vacuum equations. (An alternative derivation of the same equations begins with the self-dual Yang-Mills connection now depending only on the time, then choosing the Lie algebra as that of the volume preserving 3-dimensional diffeomorphisms.) When the reduced full Yang-Mills equations are used in the same context, we get Einstein's equations for his unified theory based on absolute parallelism. To incorporate the full Einsteinvacuum equations we use as the Lie group the semi-direct product of the diffeomorphism group of a 4-dimensional manifold with the group of frame rotations of anSO(1, 3) bundle over the 4-manifold. This last approach, however, yields equations more general than the vacuum equations.Andrew Mellon Postdoctoral fellow and Fulbright ScholarSupported in part by NSF grant no. PHY 80023     

Unified gravitational and Yang-Mills fields

We unify the gravitational and Yang-Mills fields by extending the diffeomorphisms in (N=4+n)-dimensional space-time to a larger group, called the conservation group. This is the largest group of coordinate transformations under which conservation laws are covariant statements. We present two theories that are invariant under the conservation group. Both theories have field equations that imply the validity of Einstein's equations for general relativity with the stress-energy tensor of a non-Abelian Yang-Mills field (with massive quanta) and associated currents. Both provide a geometrical foundation for string theory and admit solutions that describe the direct product of a compactn-dimensional space and flat four-dimensional space-time. One of the theories requires that the cosmological constant shall vanish. The conservation group symmetry is so large that there is reason to believe the theories are finite or renormalizable.     

Approximate radiative solutions of the Yang-Mills field equations

In this paper we look for the asymptotic radiative solutions of the Yang-Mills field equations. Considering the potential of the Yang-Mills field as a connection in a principal fibre bundle gives us a fully covariant formalism similar to the formalism of the General Relativity. Then we apply directly the results obtained by Mme Choquet-Bruhat for the gravitational field by means of the W.K.B. method. After deriving the equations for the asymptotic waves and interpreting the zero-order conditions as the initial conditions, we consider some known trivial solutions of the Yang-Mills field equations as the background field and construct the asymptotic waves explicitly. All the solutions considered turn out to be of the electromagnetic type, with some extra restrictions of the algebraic type.     

规范场与主纤维丛上的联络

本文较详细地给出物理上的规范场与数学上的主纤维丛上的联络论之间的对应关系,从而以联络论的观点统一地处理杨振宁所说的规范场的“微分方法”与“积分”方法,并指出存在有更广泛的规范场的可能性。此外,讨论了引力场如何作为规范场及比较了一些已知的引力场方程。     

The scattering of certain Yang-Mills fields

The Yang-Mills fields considered by us in an earlier paper are asymptotically non-interacting. Also any free field is an incoming field for some Yang-Mills field.     

Renormalization group equations and the continuum limit of some renormalizable quantum field theories

The renormalization group equations for asymptotically and non-asymptotically free theories are discussed by exploiting the general properties of their integral curves. Some constraints are derived on the existence of a consistent continuum limit for pure Yang-Mills theories in an infinite quantization volume. An analogy between the far infrared behaviour of non-abelian gauge theories and the deep ultraviolet one of asymptotically free theories is discussed.