Space Quantization

[Lucious]

In a space with fields [math](\psi,(q) \hat{\psi}(q))[/math] where [math]q[/math] is a generalized coordinate second quantization leads to a description of the creation and annihilation operators [math](a^{-},a^{+})[/math]

 

[math]\psi(q) = \sum_k a^{+}(k) e^{-ikq}[/math]

 

[math]\hat{\psi}(q) = \sum_k a^{-}(k) e^{ikq}[/math]

 

This is the quantization of the length. Written in the discrete form, for [math]k = \frac{2\pi n}{\ell}[/math] and a periodic space interval of 

 

[math]\int^{\frac{1}{2}}_{-\frac{1}{2}} e^{ikq} dq = 2\pi \sum_{k} a^{+}(k) \delta_{nm}[/math]

 

with a value of zero at the centre (simple topological space example). Hitting them with a momentum operator can yield a complexified version of the momentum space in increment of unit length [math]n[/math] - for the case of [math]\delta_{nm}[/math] the operators work on both

 

[math]-i \hbar \frac{\partial}{\partial t} \cdot 2 \pi \sum_k <n|a^{+}|m> \delta_{nm}[/math]

 

Where [math]\delta_{nm} = <\phi_n|\phi_m>[/math] and the increments of [math]n[/math] are acting on the operator [math]a^{+}[/math] which yields the raising operation [math]<n|m + 1> \sqrt{n+1}[/math] ... the same can be done for lowering. 

 

In the Schwinger quantization of charge context, [math]\hbar[/math] is measured in terms of increment [math]<n>[/math] for the square of the charge with using the discrete basis picture, we have the relation [math]<\phi_n|\phi_m>= \delta_{nm}[/math]. This discrete version of the delta function [math]\delta(x−y)[/math] is best seen as the identity matrix [math]\mathcal{I}[/math], where [math]\delta_{nm} =\mathcal{1}[/math] if [math]m=n[/math] , [math]\delta_{nm}[/math] goes to zero if and only if [math]m \ne n[/math]. 

 

If [math]n[/math] acts on the creation operator, we can rewrite it in forms where [math]m=0[/math].

 

In this way, we can our quantization and second quantizated discrete fundamental space and rewrite an equivalence to the raising operator to a smooth periodic function with discplacement always at zero when they converge. 

 

To write this we can give it in a new set of forms, first of all with [math]m = 0[/math] and [math]k[/math] taking on the forms [math]\frac{2 \pi n}{\ell}[/math] would give us the simpler form of

this picture we have adopted one might notice that we are dealing with

 

[math]i \hbar \frac{\partial}{\partial q} \cdot 2 \pi(n \cdot n)\sum_k a^{+}(k) \delta_{nm}[/math]

 

An unit vector dot product with another unit vector gives the value of the identity matrix and they effectively vanish. 

 

[math]-i \hbar \frac{\partial}{\partial q} \cdot 2 \pi \sum_k a^{+}(k)|n> \delta_{nm}[/math]

 

Because the the ket acts on the creation operation space [math]a^{+}(k)|n>[/math] this can also be written as [math]<n|m + 1> \sqrt{n+1}[/math] with [math]m[/math] not acting on [math]m[/math] not acting on the topological space in this case for the creation operator gives us

 

[math]-i \hbar \frac{\partial}{\partial q} \cdot 2 \pi \sum_k a^{+}(k) \delta(q) = \alpha \hbar n[/math]

 

Notice the righthandside is the same form of the Schwinger quantization expression for his quantization of the charge and then used magnetic charge - we know today there are no magnetic monopoles or at least if any exist, inflation made them aloof as they were diluted during the verocious inflational stages of the universe, thanks to the work of Alan Guth. The equation presented before is possible because

 

[math]\sum_k a^{+}(k) \int^{\frac{1}{2}}_{-\frac{1}{2}} e^{-ikq} \delta(q)[/math]

 

and [math]\sum_k a^{-}(k) \int^{\frac{1}{2}}_{-\frac{1}{2}} e^{ikq} \delta(q)[/math]

 

which are equal to the simple expression [math]2 \pi \delta(q)[/math]. 

 

The space topological charge which appears to indicate in previous Weyl quantization equation that there is also a presence of a topological charge-space through a relationship to working with the Weyl quantization equivalence to the generation of mass (aka. gravitational charge) via [math]\hbar c = Gm^2[/math] as mention before from a Motz paper on quantization of the mass [math]\hbar = \frac{Gm^2}{c}[/math] where the numerator plays a synonymous role of the quantization method of Schwinger [math]\frac{e^{2}_{1} - e^{2}_{2}}{c} = \pm \alpha \hbar n[/math] where this time it seen in the more accurate light of taking positive or negative values on the quantization of the angular momentum component. When dealing with the wave functions dictating the operations on the first and second quantized space, it must be kept in mind that they strictly work in the Banach space.