An Interpretation Of Geometry And Frozen Fields In Which Energy And Matter Is Considered

[Æthelwulf]

Just the last week, I have been twiddling my thumbs, so I put it to paper when I had an idea. I don't honestly believe it has any real physical application on systems, but may have an abstractual application atleast when describing a system to have either an eigenstate of mass or one of energy. To better illustrate, my idea first sprang from these first scetchings I made:

 

A Spinor Wave Function has a [imath]\pm[/imath] in front of it. So the [imath]SO(3)[/imath] understands that there can be [imath]\psi[/imath] and [imath]-\psi[/imath] which will differ by a phase but are actually the same. One similar proposition, a relativistic one, is that matter and energy are the same, so under the same loose concepts we may say that matter and energy also differs by a phase, which would make some kind of physical sense since matter converts to energy and vice versa under a phase transition.

 

Using this one example in polar coordinates, we may state that our system can be described as

 

[imath]\psi_n = \begin{pmatrix} cos \frac{\theta}{2} e^{-i \frac{\phi}{2}} \\ sin \frac{\theta}{2} e^{i \frac{\phi}{2}}\end{pmatrix}[/imath]

 

Moving away from polar coordinates (because it is simply the choice I wish to make), rotating this system [imath]n[/imath] by [imath]2\pi[/imath], we will inexorably find that [imath]n[/imath] does not change (thus the unit vector is a frozen field) but an Eigenstate can change from [imath]+1[/imath] to [imath]-1[/imath] through the rotation of [imath]sin(\theta + 2\pi)[/imath].

 

Perhaps in a system which makes good use of [imath]2 \pi[/imath] as definition could give us some mathematical description of the two eigenstate systems of an inertial mass [imath](\frac{E}{c^2})[/imath] and energy [imath](E)[/imath] (free of an inertial term). So one Eigenstate applies to the presence of energy alone, the other system will imply not only energy but an inertial rest (mass). Consider the well known quantum field postulates:

 

[imath]\int_{\frac{-\ell}{2}}^{\frac{\ell}{2}} e^{ikx} dx = \ell[/imath]

 

if for [imath]k=0[/imath] then it equals [imath]1[/imath] and [imath]k \ne 0[/imath] will equal [imath]0[/imath] so will may deploy the use of the dirac delta function and the sought after expression [imath]2\pi[/imath] arises through the equation

 

[imath]\int_{-\infty}^{\infty} e^{ikx} dx = 2\pi \delta (k)[/imath]

 

A quantum field expression which uses this identity (the time independant case) is

 

[imath]\psi(x) = \sum_k a^{+}_{k} 2\pi \delta(k)[/imath]

 

where [imath]a^{+}[/imath] is identified as the creation operator, which is just a neat mathematical trick to express quantum occupation levels and no more.

 

Between the phase transitions of the [imath]2\pi[/imath] with eigenvalues of [imath]+1[/imath] and [imath]-1[/imath] we should deduct that you can rotate the equation's [imath]\psi(x) = \sum_k a^{+}_{k} 2\pi \delta(k)[/imath] meaning in terms of the eigenstate of mass or energy states but not any appearance of ''n'' itself. So we may describe this as

 

[imath]\psi(x) = \sum_k a^{+}_{k} \begin{pmatrix} -sin \theta \\ -cos \theta \end{pmatrix} \delta(k)[/imath]

 

as a mathematical representation of a system with mass or energy. I think what could be interesting, assuming I have got no concept wrong, is that the phase exists here as an oscillation of the two states of energy and mass. To define a mass state would to have a value nearing or equalling the condition of either the trigonometric identities of [imath]-sin \theta[/imath] or [imath]-cos \theta[/imath].

 

What d'ya think???

 

Cheers!

[Æthelwulf]

I just realized about 20 mins ago that I could fathom another way to view the frozen vector [imath]n[/imath] and the changing eigenvalues as

 

If [imath]n = \phi[/imath] is vector, and is related to another field [imath]\psi[/imath], then identifying that [imath]k = \rho[/imath] then

 

[imath]\phi = \rho e^{i \psi}[/imath]

 

where [imath]\psi[/imath] is the eigen-field of mass or energy such that in a first-approximation we may have

 

[imath]\phi = ke^{i\psi}[/imath]

 

n is frozen as was speculated, so is [imath]\phi[/imath] by our equality. The Langrangian of the field would naturally involve the Covariant derivative

 

[imath]D\phi = \partial \phi + iA \phi[/imath]

 

To take the action of this field you must computate the equation by multiplying [imath]i(\partial \phi + A)ke^{i\phi}[/imath] it with it's conjugate which will yield

 

[imath]k^2(\partial \phi + A)^2[/imath]