A Unified Field Theory

by Peter Hickman
Email: peter.hickman1@ntlworld.com

From: http://homepage.ntlworld.com/peter.hickman1/

ABSTRACT

 In this paper, the extension of  Riemann geometry to include an asymmetric metric tensor is presented. A new co-variant derivative is derived, and used in the commutator of two co-variant derivatives of a vector. This leads to two equations which describe spin 1 bosons.  The energy-stress tensor arises as a contraction of the curvature tensor, its divergence enables the number  of dimensions of Space-Time to be determined. A weak field approximation gives potential equations for both massless and massive bosons of at least 206.6GeV/c^2.

Affine Connection

The invariant interval between two points on a Riemann manifold is
	1.1
Let be the co-variant derivative, the invariance of ds, requires
	1.2
For the affine connection to be determined by a metric tensor only, two cases arise:
Case I: The metric and affine connection are both symmetric
	1.3
With the conditions given by equation 1.2, and 1.3, the symmetric affine connection are the Christoffel Symbols, see reference [1]
	1.4
Case 2: The metric and affine connection are both asymmetric:
	1.5
With the conditions given by equation 1.2, and 1.5, the asymmetric affine connection is
	1.6
A general affine connection can be formed from equations 1.4 and 1.6
	1.7
where the imaginary part of the connection is asymmetric in m and n
It can be shown that using condition 1.2 with the connection 1.7, that the affine connection is
	1.8
where
	1.9
the asymmetric affine connection is completely asymmetric.

The Field Equations

The commutator of 2 co-variant derivatives of a complex vector using 1.8 for the affine connection gives
	2.1
where
By adding to both sides of equation 2.1 and equating the LHS to zero, and lowering a , two equations arise:
	2.2
	2.3
Contracting equation 2.3 by setting n =b gives
	2.4
In a geodesic frame, this reduces to
and since , it can be shown that is imaginary. Contracting equation 2.2 by letting b =s and n =r gives
	2.5
The vector can be eliminated from 2.4 and 2.5 by using the relations
where are NxN matrices where to give
	2.6
	2.7
For , the euclidean metric, the matrices are found to satisfy the following, (see section 3 for the calculation of N)
where I is the 6x6 unit matrix. A solution for the i=1,2,3 matrices is
where the matrices are given by, see reference [2]
These are the matrices for Spin 1 particles, thus the vector is the wavefunction for spin 1 bosons.

Calculation of N, dimensions of Space-Time

The symmetric connection in equations 2.6 and 2.7 can be eliminated to yield
	3.1
Multiply the expression in parenthesis by hc gives the total energy E,
where from which it follows that
	3.2
where N is the number of dimensions. Contracting equation 3.2 gives G=NS, so equation 3.2 can be written as
	3.3
which can be written as
	3.4
compare with Einsteins field equations of gravitation with a cosmological term, see reference [1]
gives
	3.5
With the aid of the following identities
with f=3 to N, the following expression arises when calculating the divergence of
which vanishes if (1-3/N)=3/N, gives N=6, Space-Time is 6d
The divergence of equation 3.4 becomes

The Weak Field Approximation

From equation 3.6, the equation for the current densities is
	4.1
where
The weak field approximation for the asymmetric metric tensor is
	4.2
where w is scalar field and
To an approximation, equation 4.1 is
	4.3
Using equation 4.2 for the asymmetric metric, applying the gauge condition , and dividing by w , equation 4.3 becomes
	4.4
With w =constant, equation 4.4 reduces to the classical wave-equation. In general equations 4.4 is for interactions mediated by massive bosons. For w constant, equation 4.3 can be written in a more familiar form:
where	4.5
If
	4.6
Maxwells equations of Electro-magnetism are obtained. see reference [3]
For the special case of w =constant, equation 3.2 is found to be, to an approximation:
Compare with Einsteins field equations: see reference [1]
Gives The particular case for f=5,
Is the energy-stress tensor of the Electro-magnetic field. see reference [3]
Equations 4.3 consist of 4 sets of equations, for the index f=3 to 6.
It can be shown that there are 4 'Electric' fields, given by the following equations
	4.7a
	4.7b
	4.7c
There are 4 'Magnetic' fields given by
Where i=1,2,3
The following identification can be made :
are the colour potentials, the electro-magnetic scalar potential and the scalar potentials related to the weak interaction.

Rest Mass Of Bosons

The rest-mass of bosons depends on the scalar field w , which is determined by the following equation
	5.1
A general solution in spherical co-ordinates is
	5.2
where are seperation constants, A is a constant of integration and
Applying the following conditions
with l,m,n integers, gives the particular solution for w
	5.3
A general time-independent solution for the potentials is
	5.4
where are constants of integration.
The integral of the 44-component of the energy-stress tensor for a Boson at rest is to an approximation with 2p r=l is
	5.5
Using Plancks relation for the energy , equation 5.5 gives
	5.6
Equate the gravitational field strength to a and
Substituting for w and let l=m=n, and finally solve for E gives
	5.7
Evaluating E for n=0,1,2 gives
E(0) = 4.87x10^27 eV
E(1) = 3.17x10^19 eV
E(2) = 206.6GeV
Thus, the existence of a massive boson, of 206.6GeV is predicted.

The Cosmological Term

With N=6, equation 3.5 for the cosmological term is
	6.1
Using the Robertson-Walker metric see reference [1], the current value for the cosmological term is
	6.2
where
For k=0, and the present value of the cosmological term is
which agrees well with observations

Modified De Broglie Momentum Equation

It can be shown that
	7.1
	7.2
For , i=1,2,3, then and in the weak field approximation equation 7.2 for the 4th component of momentum reduces to
	7.3
For , , and solving equation 7.3 gives
	7.4
For r/t=c and from equation 7.4
	7.5
Since the LHS of equation 7.5 is positive, then ie greater than the
Planck Length which implies particles have a maximum size of 1.6x10^-35m