Quantum Gravity with the Klein-Gordon equation

[Linker]

Hi,

 

I am developing a new quantum gravity theory. The concept of the theory sounds simple, but in reality, this theory is not so simple. Here, the main ideas of the theory:

 

- No extra dimensions, no spin foam

 

- The Klein-Gordon-Lagrangian looks similar to the Lagrangian of General Relativity (GR), because the maximum derivative is twice and both Lagrangians containing inner product of metric tensor with another tensor expression

 

- To fit KG-Lagrangian on GR-Lagrangian, the scalar field in KG-Lagrangian is not only a simple complex number; it is a more complicated algebraic object similar to matrices

 

- I'm using Feynman's path integral quantization with the following treatment of algebraic scalar fields: Instead of integration over all possible values of the field, integrations/summations are performed over all possible algebraic

 

structurizings and the values, that are invariant on algebraic structure modification

 

- All geometric quantities like volume element or metric can be expressed completely in dependence on the algebraic scalar field

 

- For the classical limit, the KG-Lagrangian in Feynman path integral can be transformed into the GR-Lagrangian

 

 

 

What are you think about this theory?

Edited October 27, 2012 by Linker

[chandragupta]

Very difficult to understand your idea.please make it simple.

[ajb]

- The Klein-Gordon-Lagrangian looks similar to the Lagrangian of General Relativity (GR), because the maximum derivative is twice and both Lagrangians containing inner product of metric tensor with another tensor expression

 

You want there to be at most second order in derivatives so that in the quantum theory you avoid ghosts. I don't think this links the KG equation with the Field Equations of GR.

 

- To fit KG-Lagrangian on GR-Lagrangian, the scalar field in KG-Lagrangian is not only a simple complex number; it is a more complicated algebraic object similar to matrices

 

Can you tell us more about these objects?

[Linker]

My theory is the following:

 

The metric tensor [latex]g_{\mu \nu}[/latex] can be expressed in dependence on a 4x4-matrix field [latex]\omega[/latex] with some algebraic classifications (my matrices are classified by the degree of commutativity).

So, the metric tensor can be expressed as [latex]g_{\mu \nu} =e_\mu \omega e_\nu + O(\omega \omega)[/latex] with characteristic unit vecors [latex]e_\mu[/latex].

 

The degree of commutativity is the following: May be [latex]A,B[/latex] two matrices. Then, the entries of that matrices determine the commutator result [latex]C:=AB-BA[/latex]. Different commutator results are possible: The matrices can commute ([latex]C=0[/latex]) or anticommute ([latex]C= 2AB[/latex] or [latex]C=-2AB[/latex]), etc. The matrix norm of the commutator is a possible variable for the degree of commutativity. Furthermore, the four eigenvalues of the matrix are gravitation field variables. Because every matrix can be written as [latex]\omega = Tdiag(\lambda_1,...,\lambda_4)T^{-1}[/latex], other field variables are algebraic variables depending on the structure of the diagonalization matrix [latex]T[/latex].

 

The action functional of my theory has the form:

 

[latex]S=\int d^4x \sqrt{-g(\omega)}(g^{\mu \nu}(\omega)tr(\partial_\mu \omega^\dagger \partial_\nu \omega) + L_{source}(\omega))[/latex]

 

But that's not all. The path integral function has the form:

 

[latex]K = \int d \lambda_1... \int d \lambda_4 \oint d [a] e^{iS} [/latex]

 

Hence, the integration over all algebraic possibilities [latex][a][/latex] makes the theory so complicated. In General Relativity, the gravitational field couples on the 4x4 = 16 functions of the energy-momentum-tensor. In this quantum gravity theory the coupling is similar to GR (remind that the noncommutative algebra of matrices caused by its non-diagonal elements), but at first the complicated algebra integration must be performed. With the approximation

 

[latex]\oint d [a] e^{iS} \approx \sum_{[a]}e^{iS} > (\prod_{[a]} e^{iS})^{\frac{1}{N_{[a]}}} = exp(\frac{i}{N_{[a]}}\sum_{[a]} S) [/latex]

 

([latex]N_{[a]} [/latex] is the effective number of algebraic possibilities) there can be approximated the GR Lagrangian. By applying the Gaussian law in the Lagrangian with vanishing boundary terms

[latex]S=\int d^4x \sqrt{-g(\omega)}g^{\mu \nu}(\omega)tr(\partial_\mu \omega^\dagger \partial_\nu \omega) = \int dS_\mu (...)^\mu - \int d^4x \sqrt{-g(\omega)}g^{\mu \nu}(\omega)tr(\omega^\dagger \partial_\mu \partial_\nu \omega) - \int d^4x \sqrt{-g(\omega)}\partial_\mu (g^{\mu \nu})(\omega)tr(\omega^\dagger \partial_\nu \omega) [/latex]

the action principle contains the elements of the linear part of the ricci tensor in the second term and the elements of the nonlinearities in the third term. Note that not the KG equation, but the sum over [latex][a][/latex] creates the structure of the ricci tensor.

[ajb]

The metric tensor [latex]g_{\mu \nu}[/latex] can be expressed in dependence on a 4x4-matrix field [latex]\omega[/latex] with some algebraic classifications (my matrices are classified by the degree of commutativity).

So, the metric tensor can be expressed as [latex]g_{\mu \nu} =e_\mu \omega e_\nu + O(\omega \omega)[/latex] with characteristic unit vecors [latex]e_\mu[/latex].

 

Can you give us some specific examples of your matrix field? For Minkowski space-time, is this just diag(-1,1,1,1)? What about the Schwarzschild metric?

 

This will help clear up your ideas for us.

[Linker]

Now I have a better idea. The spacetime is characterized by algebraic expressions. Assuming a polynomial function

[latex]f(x)= (x-a_1)(x-a_2)...(x-a_n)[/latex]

where the zeroes are given by

[latex]a_k = r_k e^{\frac{2 \pi i}{k}}[/latex].

With increasing k the zeroes fill a closed curve in the complex plane. And this closed curve can be extended to the 4-dimensional spacetime.

For the metric tensor, the following relation holds:

[latex]g_{\mu \nu} = \frac{\partial \vec r}{\partial x^\mu}\frac{\partial \vec r}{\partial x^\nu}=|\frac{\partial \vec r}{\partial x^\mu}||\frac{\partial \vec r}{\partial x^\nu}|cos(\frac{\partial \vec r}{\partial x^\mu},\frac{\partial \vec r}{\partial x^\nu})[/latex].

So, there can be defined a vector potential

[latex]g_\mu:=|\frac{\partial \vec r}{\partial x^\mu}|[/latex]

and the cosine between two vectors can be quantized by the zeroes of polynomials. Because the spacetime geometry is not determined by energy-momentum-tensor as in GR (all shapes of spacetime are allowed but the shapes have quantum-mechanical probabilities),

the probability for any angle between two vectors is given by the probability that the vectors lying on the zeroes that have this angle difference. Algebraic theories like the Galois theory can be used to define quantizations of angles.

 

For the quantized vector field [latex]g_\mu[/latex] (the field of characteristic lengths) there can be used the relationship for vector calculus

[latex]\frac{\partial \vec r}{\partial x^\mu}=g_\mu cos(\frac{\partial \vec r}{\partial x^\mu},\vec e_\alpha) \vec e^\alpha[/latex]

The quantized metric tensor is invariant under the rotation of the zeroes in the polynomial function, i.e. the transformation [latex]\frac{\partial \vec r}{\partial x^\mu} \mapsto T \frac{\partial \vec r}{\partial x^\mu}[/latex] with the orthogonal transformation matrix T. All other quantities like Christoffel's symbols, Riemann's tensor, etc. can be derived from the characteristic lengths and the angles between them. The action functional must be invariant under the rotation of the zeroes.

 

For the path integral quantization, there must be integrated over

- the vector potential [latex]g_\mu[/latex]

- all possible algebraic realizations of polynomials

- this algebraic realizations including the angles lying between two characteristic lengths

[Singer]

Not easy to understand.