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A new Approach to Unification of Potential Fields Using GLT Model

There have been many attempts to create an unified theory of the all fundamental laws of physics [1, 2, 3 ..]. The main problem is to put together General Relativity and Quantum Mechanics into one self consistent theory. The most popular candidate for the unified theory is Superstring Theory, in which all particles are just different vibration modes of very small loops of string. It is expected that an unified theory exists at the Planck scale, where all forces of nature are unified and quantum gravity is significant. If an unified theory can be constructed at all, then the first step should be an unification of potential fields. This paper presents one of the possible ways to create it. Starting with the new General Lorentz Transformation model (GLT-model in [4, 5] ) a general covariant energy equation has been derived as a function of two free parameters a and a'. The corresponding covariant energy equation of a gravitational field is obtained from the previous one, after substitution of the identified parameters a and a' from the well-known Einstein equation of a gravitational field [5]. Comparing this new energy equation with a corresponding one of an electromagnetic field, we derived, on the natural way, a scalar and a vector potentials (i.e. a four-potential vector) of a gravitational field. This was very important step of our investigation, because it makes more symmetry between a gravitational and an electromagnetic field. This symmetry makes the possibility to create a new form of tensor of a gravitational field, a Maxwell like equations of a gravitational field, a four-force vector on a particle in a gravitational field, differential equations of a particle motion in a gravitational field, and, what is very important, an application of usual quantisation methods to a gravitational field. Applying the well-known Lorentz - Maxwell condition to the four-potential vector of a gravitational field we confirmed that a gravitational force-carrying particle (graviton) should have a velocity equal to the light velocity in a vacuum. In the case of the Newton limitation (v << c) , this model is reduced to the well-known Newton model of a gravitational field, what we expected that should be. The second aim of our investigation was to create procedures of unification of potential fields, using the results obtained from the applications of GLT model. This paper approach is based on the fact that free parameters a and a' are functions of a state of a potential energy in each space-time point. Generally, a particle can be in a multi-potential field with n-different potentials. Here we have the problem of unification of the all potentials of that potential field. This problem has been solved by an introduction of so called a specific four-potential vector of a certain potential field. These specific potentials have been defined in the form that enables to add the all specific potentials of a multi-potential field. The next step was to connect these specific potentials with free parameters a and a' through the corresponding mathematical relation. Since the all items like field tensors, field equations, energy equations and so on, can be described as functions of free parameters a and a' ( i.e. as functions of an unified specific four-potential vector of multi-potential field) , the trace of unification of different potential fields has been build. In that sense we propose the two new unification procedures. The first one is based on the derivation of a general items of an unified field like a general tensor, a general Maxwell like equations, a general four-force vector on a particle, general differential equations of a particle motion, and a general Hamilton equation of an unified potential field. These items have been derived by employing the parameters a and a' of an unified potential field. As an example, following the proposed unification procedure, an unification of an electromagnetic and a gravitational field has been derived. In order to connect this approach with the quantum mechanics we have to make a quantisation of an unified potential field. It can be done by employing one of the usual quantization procedures for quantisation of a potential fields (let say a canonical quantisation of an electromagnetic field [6]). The second unification procedure is based on the fact that a general line element derived from GLT model [5] is also a function of free parameters a and a', i.e. it is also a function of a state of a potential energy at each space-time point. We have to use a non-diagonal line element because it has got a very important property: a non-singularity in a very strong gravitational field [5]. Starting with a general non-diagonal line element we derived a general energy - momentum equation, a general frequency - wave length relation and a general Klein, Gordon and Fock relativistic equation [7, 8], as functions of free parameters a and a'. After substitution of the parameters a and a' of an unified potential field into the general Klein, Gordon and Fock relativistic equation we obtain a wave function of an unified potential field, with an ability of quantisation on the known way [6]. Since the general non-diagonal line element can be the starting point for derivation of an unified potential field (including gravity), as well as the corresponding wave functions, it seams that a general line element could be the candidate for the basic element of superstring theory. We hope that the proposed approach can help in process of an unification of potential fields.

Unification of Potential Fields; Scalar and Vector Potentials of Gravitational Field

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