Sep 03 2008

## The Graviton as a Momentum Operator

The fundamental commutator relation [**x**, **p**] = iћ, between the operators of coordinate and momentum, provides a way to show how a graviton can add linear momentum to an electron.

An energy relation for a synchronous encounter by a graviton with an oncoming electron in an atomic orbital can start with:

[**x**, **p _{g}**]

^{2}= i

^{2}ћ

^{2}= -ћ

^{2},

the added kinetic energy being ћ

^{2}k

_{i}

^{2}/2m

_{i}, and the added momentum -√(ћ

^{2}k

_{i}

^{2}).

The added momentum, as shown by the minus sign, is in the opposite direction of that in which the graviton was traveling at the speed of light in a vacuum before it was absorbed by the electron.

Internal to the electron we can use the {N} representation to form the basis of a set of wavefunctions forming orthonormal vectors: │0 >, │1 >, …, │n >, …, with eigenvalues of N: 0, 1, …, n, … [Messiah, XII.16, pg 436]. The graviton in the process of being absorbed by an electron in a quantum atomic orbital can then be seen as a raising operator, where

**a**

^{†}φ

_{n}= φ

_{n+1}and

**a**

^{†}φ

_{n+1}= φ

_{n+2},

and the release of a graviton a lowering operator, with

**a**φ

_{n}= φ

_{n-1}and

**a**φ

_{n-1}= φ

_{n-2}.

The Hamiltonian for such a system is represented as:

**H**φ

_{n}= ћω

_{0}(

**a**

^{†}

**a**+ ½) φ

_{n}[Liboff, Section 7.2],

with energy eigenvalues

E

_{n}= ћω

_{0}(n + ½) n = (0, 1, 2, …, 68, …n

_{max}).

Here 68 represents the average n value at the face of the earth, and n

_{max}depends on the orbital.

If the mass of the electron diminishes as the gravitational field diminishes, the characteristic wavenumber

*β*of the electron also diminishes. For each graviton internal to the electron

*β*

_{i}

^{2}= m

_{i}ω

_{0}/ћ, and for the mass of the electron at the face of the earth we have Σm

_{i}= m

_{e}= 9.1095 x 10

^{-31}kg.