1.8.3 The Quantum Flowing Point

We now con­sid­er the ex­am­ple of a mas­sive object that is bound to the geom­e­try of space­time and rel­a­tiv­i­ty in the guise of the elec­tron. The elec­tron is a sub­atom­ic par­ti­cle from the lepton par­ti­cle family and car­ries a neg­a­tive el­e­men­tary charge. Lep­tons are as­sumed in physics to be fun­da­men­tal par­ti­cles, having no known sub­struc­ture or other com­po­nen­try. It exists as a point in space and time, im­ply­ing math­e­mat­i­cal­ly that it is zero-di­men­sion­al. This make us ponder how is it so im­pact­ful on di­men­sion­al forms? The answer lies in the math­e­mat­i­cal struc­ture un­der­ly­ing its form, which con­fers the prop­er­ties of charge, spin, and po­si­tion. A more recent de­vel­op­ment in physics has been that the elec­tron has a sub­struc­ture, found through the spin-charge sep­a­ra­tion process. At tem­per­a­tures near ab­solute zero, the elec­tron may split into three quasi-par­ti­cles: spin­ons, holons, and or­bitons¹. The elec­tron quasi-par­ti­cle is an ex­am­ple of the fun­da­men­tal role math­e­mat­ics plays in the uni­verse, where the spinon car­ries the elec­tron's holon in its charge, and the or­biton its or­bital lo­ca­tion, which are all zero-di­men­sion­al math­e­mat­i­cal prop­er­ties.

Waves are de­scribed in both spa­tial and tem­po­ral re­la­tion­ships. In tem­po­ral space, we con­sid­er a wave's fre­quen­cy as the number of waves oc­cur­ring per unit of time and is re­lat­ed to the period ($T$) be­tween each wave in time as

An­gu­lar fre­quen­cy $\omega$ is a scalar value taken from the an­gu­lar ve­loc­i­ty, which mea­sures, in ra­di­ans, the amount of ro­ta­tion per unit time

noting its re­la­tion­ship to the wave's fre­quen­cy. An anal­o­gous spa­tial fre­quen­cy is de­scribed by con­sid­er­ing the number of cycles or ra­di­ans per unit of dis­tance and is re­ferred to as the wave number $k$, which has the fol­low­ing equiv­a­lence

for $\lambda$, the wave­length or dis­tance in space the wave covers for each com­plete cycle.

For mas­sive par­ti­cles, where $h,p,m,\ \text{and}\ v$ define Planck's con­stant, mo­men­tum, mass, and the par­ti­cle's ve­loc­i­ty, the de Broglie wave­length is de­fined as

which was first pro­posed by Louis de Broglie² in 1924 to de­scribe the wave nature of matter. From this re­la­tion­ship, we can sub­sti­tute the value for $\lambda$ into the wavenum­ber ex­pres­sion to find

where $\hslash$ is the re­duced Plank's con­stant

Schrödinger's equa­tion is a linear par­tial dif­fer­en­tial func­tion de­scrib­ing a quan­tum-me­chan­i­cal system's wave func­tion evo­lu­tion over time. It is named after Erwin Schrödinger³, who first pos­tu­lat­ed the equa­tion in 1925 and then pub­lished it in 1926. The time-de­pen­dent form of the equa­tion is

where $\psi(X,\tau)$ is the wave­func­tion $\in \mathbb{C}$, which equates to a com­plex number for each point $X$ at time $\tau$, $\hslash$ is the re­duced Plank con­stant, $m$ is the par­ti­cle's mass, and $V(X,\tau)$ is the po­ten­tial energy im­part­ed to the par­ti­cle over time and po­si­tion.

We can ex­press a wave­form gen­er­a­tor matrix, $\psi(X,\tau)$, as

which has the form of our $G\left( X_{1} \right)$ group gen­er­a­tor, and set about how to con­sid­er the evo­lu­tion of this wave­func­tion over time in the sim­pli­fied case for no po­ten­tial field, $V(X,\tau).$Re­call­ing how we pre­vi­ous­ly de­fined dif­fer­en­tial op­er­a­tors for ma­tri­ces in the book, the matrix time-de­riv­a­tive op­er­a­tor spe­cif­ic to the $X_{1}$-type al­ge­bra can be de­fined as

for which the $\widehat{\partial\tau}$ op­er­a­tor's hat is con­sis­tent with the op­er­a­tor in quan­tum me­chan­ics. This dif­fer­en­tial op­er­a­tor is con­sis­tent with how the $X_{1}$ Lie al­ge­bra evolves into the in­fi­nite series of terms in the cal­cu­la­tion process of the matrix ex­po­nen­tial, lead­ing to the $G\left( X_{1} \right)$ group gen­er­a­tor.

Given the pos­si­bil­i­ty for non-com­mu­ta­tive al­ge­bras and the dif­fer­en­tial op­er­a­tor being de­fined as a right-sided mul­ti­pli­er, we ex­press the time de­riv­a­tive of the wave­func­tion in the cor­rect form for our treat­ment as $\psi(X,)\widehat{\partial\tau}$. Pro­ceed­ing, we cal­cu­late the time-de­riv­a­tive of our matrix wave­func­tion, $\psi(X,\tau)$, as fol­lows

The matrix space-de­riv­a­tive op­er­a­tor spe­cif­ic to the $X_{1}$-type al­ge­bra is sim­i­lar­ly

which then takes the fol­low­ing form for its second de­riv­a­tive

Using this op­er­a­tor, we can now cal­cu­late the second space-de­riv­a­tive of our matrix wave­func­tion $\psi(X,\tau)$, as fol­lows

For our quan­tum system evolv­ing in time and not acted on by a po­ten­tial, Schrödinger's equa­tion is

which after the math­e­mat­i­cal action of the time and space dif­fer­en­tial op­er­a­tors on the wave­func­tion $\psi(X,\tau)\widehat{\partial\tau},\ \psi(X,\tau)\widehat{\partial X^{2}}$, the equa­tion be­comes

By dis­trib­ut­ing $i$ in the matrix on the left side and by fac­tor­ing out the $- k^{2}$ on the right side, the Schrödinger's equa­tion for our system be­comes

for which the ma­tri­ces on both sides are equiv­a­lent and equal to our $\psi(X,\tau)$ wave­func­tion, which we sim­pli­fy to rep­re­sent it as

and as an equal­i­ty, it holds if

To show that $\hslash\theta$ and $\frac{\hslash^{2}k^{2}}{2m}$ are equal, we must con­sid­er matter waves in more depth. The value $\theta$ from $\hslash\theta$ rep­re­sents the an­gu­lar fre­quen­cy we de­fined pre­vi­ous­ly as $\omega$, and $\theta$ will equal

where $f$ is the fre­quen­cy. The Plank-Ein­stein re­la­tion de­fines energy $E$ as the prod­uct of the Plank's con­stant and the fre­quen­cy

and given that

energy $E$ can be ex­pressed in terms of the an­gu­lar fre­quen­cy $\theta$, by

The wave number $k$ is equal to the mo­men­tum $p$ di­vid­ed by the re­duced Plank's con­stant

and mo­men­tum is de­fined

where $m$ is the par­ti­cle's rest mass and $v$, its ve­loc­i­ty. Mo­men­tum is de­fined as

where $E$ is the energy rep­re­sen­ta­tion in the ki­net­ic form, which equals

This equiv­a­lence is de­rived start­ing with the ex­pres­sion for mo­men­tum

for which we mul­ti­ply both sides by $\frac{v}{2}$

then both sides by $m$

yield­ing the stated equiv­a­lence

The wave number $k$ can now be ex­pressed as

and after in­sert­ing this back into the ini­tial ex­pres­sion

we find it equals $E$. As we have shown, $\hslash\theta = E$, and there­fore the ex­pres­sion $\left( \hslash\theta - \frac{\hslash^{2}k^{2}}{(2m)} \right)$ equals zero,

in­di­cat­ing the time-de­pen­dent Schrödinger's equa­tion

holds true for the gen­er­a­tor wave­func­tion

Next, we take an ap­proach that might be more fa­mil­iar to quan­tum me­chan­ics pre­sen­ta­tions by having the gen­er­a­tor wave­func­tion act on a co­or­di­nate vector

and con­sid­er the system's evo­lu­tion over time

We use the same dif­fer­en­tial op­er­a­tors to act on the gen­er­a­tor, first for the time-de­riv­a­tive on the left-hand side

and then the spa­tial de­riv­a­tive on the right-hand side

Equat­ing the left and right-hand sides of our time-de­pen­dent Schrödinger's equa­tion

shows equal plane waves in the vector form, along with the same equal­i­ty that we just proved

In quan­tum me­chan­ics pre­sen­ta­tions, the vector-based time-de­pen­dent Schrödinger's equa­tion will take the form

where $\overrightarrow{x}$ in­di­cates a three-di­men­sion­al space co­or­di­nate vector, and $t$ is the time com­po­nent, which is fre­quent­ly ex­pressed as

Above, the term $H$ de­notes the Hamil­ton­ian, a quan­tum me­chan­i­cal op­er­a­tor cor­re­spond­ing to the total energy of the system and is clas­si­cal­ly the op­er­a­tor that de­fines the system's total energy, in­clud­ing the ki­net­ic and po­ten­tial en­er­gies. In the op­er­a­tor form of the Schrödinger equa­tion the Hamil­ton­ian is ex­pressed as

where $\widehat{V}$ rep­re­sents the po­ten­tial energy op­er­a­tor

and $\widehat{T}$, the ki­net­ic energy op­er­a­tor, rep­re­sents

for which the dot, $`` \bullet ",$in­di­cates the dot prod­uct, and the Lapla­cian $\nabla^{2}$ is the dot prod­uct of del op­er­a­tors $\nabla \bullet \nabla$, which in Carte­sian co­or­di­nates is

For the ki­net­ic energy op­er­a­tor, the mo­men­tum op­er­a­tor, $\widehat{p}$, is equal to

and the del op­er­a­tor $\nabla$ is math­e­mat­i­cal­ly a vector de­riv­a­tive. So, the dot prod­uct $\widehat{p} \bullet \widehat{p}$ be ex­pressed in re­la­tion to the ki­net­ic energy op­er­a­tor as

We men­tion the Hamil­ton­ian now, given the es­sen­tial role it plays in study­ing me­chan­i­cal sys­tems in physics. In brief, the Schrödinger equa­tion is not what fails the litmus test of spe­cial rel­a­tiv­i­ty, but in­stead, it is the Hamil­ton­ian that does. The equa­tion pro­vides very pre­cise pre­dic­tions for a non-rel­a­tivis­tic system, but a multi-par­ti­cle frame­work is re­quired to en­com­pass the fact that energy can be con­vert­ed to par­ti­cles and par­ti­cles can con­vert to energy.

The single-par­ti­cle ap­proach to quan­tum me­chan­ics must be cast aside to com­bine Lorentz in­vari­ance, the ob­ser­va­tion­al sym­me­try of spe­cial rel­a­tiv­i­ty, and quan­tum theory. Quan­tum field theory (QFT) is cur­rent­ly the pro­posed so­lu­tion, which presents a dra­mat­ic de­par­ture from quan­tum me­chan­ics' par­ti­cle per­spec­tive. Its frame­work treats par­ti­cles as ex­cit­ed states of an un­der­ly­ing quan­tum field and treats the field as more fun­da­men­tal than the par­ti­cle.

The per­spec­tive that the field is the fun­da­men­tal aspect of quan­tum theory is co­he­sive and trans­lates math­e­mat­i­cal­ly into our dis­course, to view the gen­er­a­tor as fun­da­men­tal and the co­or­di­nates as ex­cit­ed par­ti­cles caused by the gen­er­a­tor field.

We present the wave­func­tion in this sec­tion $\psi(X,\tau)$ as the Lie group gen­er­a­tor

which is the fun­da­men­tal cause to how the par­ti­cle state evolves. The par­ti­cle, as an ex­ci­ta­tion state, is simply the co­or­di­nate vector with qual­i­ties im­part­ed to it caused by the gen­er­a­tor, rep­re­sent­ed in our ex­am­ple system as the vector

We recall that the gen­er­a­tor rep­re­sents an ex­po­nen­ti­at­ed Lie al­ge­bra present on the sur­face of the in­fi­nite sphere of tan­gent spaces, the cause of which is non-local, mean­ing it is out­side of space­time. The dy­nam­ics of geo­met­ric primes inside the sin­gu­lar­i­ty give rise to the in­fi­nite sphere of tan­gent spaces, and the sin­gu­lar­i­ty that we have worked to de­scribe in these hun­dreds of pages is the sine qua non of the quan­tum field. The Uni­fied Field is the in­fi­nite canvas of these sin­gu­lar math­e­mat­i­cal forms, which give rise to space, time, rel­a­tiv­i­ty, forces, and ul­ti­mate­ly, the phys­i­cal vessel we co­or­di­nate. Con­scious­ness is not a co­or­di­nate vector driven by a gen­er­a­tor; it is the apex fun­da­men­tal form com­posed of op­ti­mized geo­met­ric num­bers. It is the gen­er­a­tor of the field of gen­er­a­tors, and its ex­is­tence is not pred­i­cat­ed upon space­time and its phys­i­cal im­pli­ca­tions. Its seat is non-local but drives the local phe­nom­e­nol­o­gy. In up­com­ing works, es­pe­cial­ly those con­cern­ing meta­physics, we will delve deeper into con­scious­ness, and it will be re­vealed/ex­plained that its roots lie in un­con­scious­ness, an un­ac­tu­al­ized po­ten­tial state of ex­is­tence.

1. *Merali, Z. "Not-quite-so el­e­men­tary, my dear elec­tron.", Nature (2012). https://doi.org/10.1038/nature.2012.10471*
2. L. de Broglie, "Recherch­es sur la théorie des quanta", Thesis (Paris), 1924; L. de Broglie, Ann. Phys. (Paris) 3, 22 (1925).
3. E. Schrödinger, "An Un­du­la­to­ry Theory of the Me­chan­ics of Atoms and Mol­e­cules", Phys. Rev. 28, 1049 – Pub­lished 1 De­cem­ber 1926