The Klein-Gorden Equation and the Mass-Sqaured Factor -a question on natural units

[ManyNames]

Homework Statement

A Consistency question regarding the dimensions of the KG Equation.

Homework Equations

$$M^2=\frac{\hbar c}{G}$$ for a quantization of gravitational charge on matter

The Attempt at a Solution

I know a few things however concerning the KG equation. It is directly related to a relativistic equation which is directly a consequence or rather an absolute solution to the Dirac Equation.

The Klein-Gorden Equation if my memory serves me correctly can be given under natural units... and looking quickly through some notes, this is given as:

$$-\partial^{2}_{t} \psi + \nabla^2 \psi=M^2 \psi$$

Now since $M^2$ has a natural unit dimensionality, the equation if derived independantly was:

$$M^2=\frac{\hbar c}{G}$$

So since the natural units of dimensionality cannot be disputed with the Klein-Gorden Equation under the same consistency, it seems that the mass-squared signature would not alter the equation, but the equation would have a new quantized interpretation.

Now, since the KG Equation is used for a spinless quantum field, which should not i would have suspected alter the issue i raised, because it turns out that spin really isn't a physical spin at all, but rather the element for particles with some angular momentum. Nevertheless, since the equation i modified, (which i will simply name as the 'Modified Klein-Gorden Equation,') the KG Equation itself does describe the quantum amplitude for finding a point particle in various places in spacetime, which is of course, perfectly consistent for any quantized locality.

The modified equation i gave, was, as i take you through the derivations of a quantized charge:

$$GM^2=\hbar c$$

thus

$$M^2=\frac{\hbar c}{G}$$

Therego, my question enhanced is, surely this must be true, keeping to usual rules yes?

$$\mathcal{L} = \frac{1}{2} \left(\partial_{\mu} \phi \right)^2 - \frac{1}{2} \frac{\hbar c}{G} \phi^2$$

So here i have replaced the $$M^2$$ signature with the quantization of $$\frac{\hbar c}{G}$$ whilst pertinently remaining with the use of natural units. Thus, the modified equation would describe a qantization of the gravitational charge on a mass in some field $$\phi$$.

It seems reasonable no? Or am i missing something pertient?

 

[ManyNames]

Homework Statement

A Consistency question regarding the dimensions of the KG Equation.

Homework Equations

$$M^2=\frac{\hbar c}{G}$$ for a quantization of gravitational charge on matter

The Attempt at a Solution

I know a few things however concerning the KG equation. It is directly related to a relativistic equation which is directly a consequence or rather an absolute solution to the Dirac Equation.

The Klein-Gorden Equation if my memory serves me correctly can be given under natural units... and looking quickly through some notes, this is given as:

$$-\partial^{2}_{t} \psi + \nabla^2 \psi=M^2 \psi$$

Now since $M^2$ has a natural unit dimensionality, the equation if derived independantly was:

$$M^2=\frac{\hbar c}{G}$$

So since the natural units of dimensionality cannot be disputed with the Klein-Gorden Equation under the same consistency, it seems that the mass-squared signature would not alter the equation, but the equation would have a new quantized interpretation.

Now, since the KG Equation is used for a spinless quantum field, which should not i would have suspected alter the issue i raised, because it turns out that spin really isn't a physical spin at all, but rather the element for particles with some angular momentum. Nevertheless, since the equation i modified, (which i will simply name as the 'Modified Klein-Gorden Equation,') the KG Equation itself does describe the quantum amplitude for finding a point particle in various places in spacetime, which is of course, perfectly consistent for any quantized locality.

The modified equation i gave, was, as i take you through the derivations of a quantized charge:

$$GM^2=\hbar c$$

thus

$$M^2=\frac{\hbar c}{G}$$

Therego, my question enhanced is, surely this must be true, keeping to usual rules yes?

$$\mathcal{L} = \frac{1}{2} \left(\partial_{\mu} \phi \right)^2 - \frac{1}{2} \frac{\hbar c}{G} \phi^2$$

So here i have replaced the $$M^2$$ signature with the quantization of $$\frac{\hbar c}{G}$$ whilst pertinently remaining with the use of natural units. Thus, the modified equation would describe a qantization of the gravitational charge on a mass in some field $$\phi$$.

It seems reasonable no? Or am i missing something pertient?

Can no one answer this for me... it's very imortant, and i would appreciate it.

Thanks.