Newton → Schrödinger equation derivation case; Einstein’s wave 4-vector & Gauss’ Fundamental Theorem of Algebra (FTA)

The main contents of this posting
included in NRF In-depth Policy Report
(as a part of it, published on August 16, 2022),
NRF, MSIT of Korean Government
(Author: Dr. Seongsoo Choi, Chairman of Metacomputing Inc.)
[Update] Officially registered at National Policy Information Office(Council/Service) (2022.08.18.)

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[NRF In-depth Policy Report 2022-1]
The need to understand quantum computing technology trends and reality (theory/simulation) and to recognize the advent of “the era of expanding technological super-gap”, NRF In-depth Policy Report 2022-1, NRF, MSIT (2022)
https://www.linkedin.com/feed/update/urn:li:activity:6965188443685441536
https://www.nrf.re.kr/cms/board/library/view?menu_no=419&nts_no=182514
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[NRF In-depth Policy Report 2022-2]
Trends and considerations of metaverse; “Academic continuation”, usefulness quantification and sustainability innovation, sharing economy supercomputing, NRF In-depth Policy Report 2022-2, NRF, MSIT (2022)
https://www.linkedin.com/feed/update/urn:li:activity:6965174515307819008
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[Relevant Posts]
[Supplementary Post for 2022-1] https://lnkd.in/ggensqU6
[Fields Medalist, Prof. June Huh’s Lecture for General Audience] https://lnkd.in/dcTRkJuV

[Body Text]
Gauss’ Fundamental Theorem of Algebra (FTA) is very beautiful in that the complex number field(we often simply call it “set of complex numbers”) is algebraically closed; wheras the set of natural numbers is non-algebracally closed; we can very easily be aware of the fact here by just additionally considering the elementary operation, minus(-). In this regard, once we start dealing with any kind of algebraic method, it is unable to escape from the “Gaussian World” dominated under FTA.

The fundamental theorem of algebra (FTA), proved by Johann Carl Friedrich Gauss in 18th century (considered the most challenging problem in the century), is as follows.

Ref.: (Wikipedia) Gauss’ Fundamental theorem of algebra(FTA)
└ The fundamental theorem of algebra, also known as d’Alembert’s theorem,^{[Ref. 1]} or the d’Alembert–Gauss theorem,^{[Ref. 2]} states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division. Despite its name, there is no purely algebraic proof of the theorem, since any proof must use some form of the analytic completeness of the real numbers, which is not an algebraic concept.^{[Ref. 3]} Additionally, it is not fundamental for modern algebra; its name was given at a time when algebra was synonymous with theory of equations.

Ref.: (Wikipedia) Gauss-Lucas Theorem: If P is a (nonconstant) polynomial with complex coefficients, all zeros of P′ belong to the convex hull of the set of zeros of P.
└ In complex analysis, a branch of mathematics, the Gauss–Lucas theorem gives a geometric relation between the roots of a polynomial P and the roots of its derivative P′. The set of roots of a real or complex polynomial is a set of points in the complex plane. The theorem states that the roots of P′ all lie within the convex hull of the roots of P, that is the smallest convex polygon containing the roots of P. When P has a single root then this convex hull is a single point and when the roots lie on a line then the convex hull is a segment of this line. The Gauss–Lucas theorem, named after Carl Friedrich Gauss and Félix Lucas, is similar in spirit to Rolle’s theorem.

With FTA in mind, we would feel it comfortable to deal with physics and to accept many theoretic divergence problems requiring renormalization techniques based on analytic continuation in complex analysis (the continuation result is mathematically unique). Now considering a non-local property of a particle in Newtonian physics, it directly gives rise to taking account of eigenvalue problems.

In this posting, we will see that Schroedinger equation could appear in a simple theoretical logic. Based on high school physics, it is necessary to consider a particle to have wave properties (non-local). To specify a physical quantity for a non-local being with respect to a local variable, the wave mechanical expression should be contracted so as to have a specified physical quantity (mainly, we consider the case that the imaginary part: 0). Thus, we can use the following kind of interpretation (ψ: wave function including wave mechanical properties of a physical particle):

As we can see above, obtaining proper wave functions is nothing but solving for eigenvalue equations. Solving for an eigenvalue equation is equivalent to obtaining the zeros of the corresponding polynomial equation (Characteristic eq.). Getting slightly back to a little middle school algebra, we may remember the mathematical fact that the above zeros are actually corresponding to the conditions when the relevant linear system of equations having infinite solutions, related to “Coherent States”.

Let us take into consideration some middle school math just a little:

To help understanding, if the system were corresponing to a sound wave mechanical system, then the case as above would provide kind of “Good Sound”. In practice, the same kind of mathematical structure appears when dealing with natural frequencies.

Notice that without FTA, it is even impossible to safely define such equations and solutions (No algebraic “exception handling”).

Now, let us see an example in which a quantum physical equation appears based on Einstein’s theories & Gauss’ FTA from the energy conservation relation in (Korean) middle school & high school physics textbooks.

According to the definition of the Einstein’s 4-wavevector, Electromagnetic wave function (without its amplitude coefficient) having the Einstein’s photon energy below is defined as follows:

Rest mass = 0:

Note that the above relation still holds for a photon-like case of the rest mass ≈ 0.

The above energy and the corresponding wave function are the (slightly different) expectation value defined above (eigenvalue) and the corresponding eigenfunction.

Gauss’ FTA actually guarantees the existence of the corresponding linear operator with the following pair (E, ψ) as well.

————

＊Note 1: The use of (linear) wave mechanics is premised (as is in Maxwell’s equations).

＊Note 2: Contradictory results would appear if we regarded the above product operator as a nonlinear function.

※ The following proposition holds.
Every operator is a function (However, the reverse does not hold).
A function could be an operator or not.

＊Note 3: The linear operator above exists for any pair of (E, Ψ) as above where E is an expectation value corresponding to the physical state Ψ.

※ In general, we deal with the case when the imaginary part of E is 0 (for that, we call it observable) along with Ψ, a wave mechanically coherent state, which is mathematically expressed as an eigenfunction with its corresponding eigenvalue (we are already awared of it above with some middle school algebra). For any pair of (E, Ψ) above with the imaginary part of E being 0, the above linear operator is mathematically self-adjoint or (if the target physical system is expressed as a finite dimensional linear algebraic problem) hermitian.

* Supplementary note for a question (introduced in the LinkedIn post’s comment for the relevant explanation):

————

Therefore, we have the corresponding energy operator as follows:

As in high school physics book, the energy conservation law in Newtonian physics is expressed as

Considering a photon-like case of ψ for the rest mass ≈ 0 (Imposing ψ → ψ for the rest mass = 0),

Since we deal with a scalar potential, the following holds:

Likewise, FTA guarantees to have the momentum operator as above (Rest mass ≈ 0 (Photon-like)):

Finally, we have

and the following complex function‘s nontrivial zero is the Schroedinger eq.:

Here, it is worth noting the fact that the Schroedinger eq. could appear without “imposing (at the very beginning)” the following commutation relation

(The necessity of the “imposing” actually means Schroedinger eq. is nothing but just a sort of experimental eq. As seen here, the above noncommutative relation could appear “as a result” without the axiom-like “imposing”.).

In this regard, we may as well make the meaning of uncertanty principle clearer for better understanding quantum physics.

Let us define “L=0 case” and “L=1 case” as follows:

Here, “L=0 case” is kind of “Plato’s Idea” covering “L=1 case” as well. We may be able to understand of what it is meant by the above “L=0, 1 cases” – thinking that the following green horse-like any kinds of horses could infinitely be created in the world of ideas.

Notice that “L=1 case” includes “a sort of hidden variables” related to operator’s time-series properties.

What “L=0 case” include are just (purely mathematical, not related to “actual time variables“) non-commutative relations.

Such non-commutative relations also exist in Newtonian physics if we don’t neglect any kind of effect related to observation(by measurment – its mathematical correspondance is operation) so that we make classical error terms 0 (which actually can be said “still existing“). For reference, light or sonic measurements also touch the state of classical particles (very very weakly).

For fun, we might as well challenge managing nonlinearities inherent in general relativity to derive quantum mechanical equations. Suppose that we deal with the following kinds of analogy (※ In fact, the above quantum physical eq. independently deal with time and space variables repectively. See the difference in between the definition of time in Relativity.):

When developing the above kind of analogy-based logic, the problem of the nonlinear relationship between the coordinate components is encountered, but through the linear approximation of the nonlinearity and the special relativitistic/classical approximation of the general relativity, we could see that equations of quantum physics also appear as we could have seen their appearance above. Fortunately, linearized types of general relativistic equations are also showing very accurate results when utilizing them computationally. Managing non-linearity is known a very tough task (especially for analytic theoretical studies).

If the above explanations are interesting, you can read our NRF In-depth Report (published by R&D Policy Team, NRF, MSIT of the Korean Government) of which topics are related to quantum computing & metaverse; a great deal of effort has been put into all the explanations in the reports in order to provide deep differentiated insights as easy as possible (even high school students could understand) through detailed theoretical logic developments and practical simulations.

By better understanding complex numbers and prime numbers, it is possible to have deeper insights.

For example, the Langlands Program in Mathematics (kind of study on “Grand Unified Field Theory” in math) is considered a little.

Considering

to bottom-up construct the complex number field (if you don’t know abstract algebra, just think of it as “set of complex numbers”) with the following:

Automorphic forms (continuous function) ⇔ Eigenvalues (discrete value)
(※ Automorphic form: Simply thinking, f(az) = b f(z) like form)

⇒ Prime number structure inherent in the linear polynomial K.

⇒ Microscopic structure of complex number field

Very interesting thing is that the important proofs in the Langlands program exactly show very important structures in physics.

https://en.wikipedia.org/wiki/Hitchin_system

https://en.wikipedia.org/wiki/Geometric_Langlands_correspondence

https://en.wikipedia.org/wiki/S-duality

Now, let us introduce some interesting stuffs included in our above NRF report:

Note that all of the above n are prime numbers. Isn’t that interesting?

Actually (somewhat roughly speaking), the above sort of mathematical structure is used to prove the Fermat’s Last Theorem.

Furthermore, that is also related to the Grand Riemann Hypothesis.

For reference in relation to the above, the following contents are included in the above report.

“Every L-function, motivic or automorphic, is equal to a standard L-function”
– R. P. Langlands, L-Functions and Automorphic Representations, Talk at Helsinki ICM (1978)

As for the above Étale Comparison Theorem, you may want to study Étale Cohomology & Alexander Grothendieck‘s Algebraic Geometry.

Grotendik linked number theory with algebraic geometry by proving that, given an algebraic manifold defined on any algebraic number field, a Galois expression can also be given.

In the end, as the automorphic form defined above the Galois expression achieves Langlands Correspondence, it becomes possible to look at number from a geometrical and analytic perspective, and to be able to gain insight from the opposite point of view as well. It is very interesting that the automorphic form has a complex analytic definition and its equivalent algebraic geometric definition. If we think of it a little more, we can see that the “discrete being”, algebraic number, has a geometric relationship with the complex analytically defined automorphic form that is regarded as a “continuous being”.

In relation to this, the fact that can be immediately recalled from a physical point of view is that objects that appear to be continuous “somewhat macroscopically” are composed of discrete-unit (quantum) beings from “a microscopic physics (of, so to speak, atomic scale)” point of view, and their states and energy levels are also quantized. In practice, automorphic forms including modular forms are importantly treated in modern physics including string theory.

Mathematics is REALLY ACTUALLY VERY VERY CLOSELY RELATED to physics.

Owing to Gauss’ work as above, we can have a sort of counterexample that implies the fact that we should really be careful when regarding the Schroedinger equation just as a sort of “experimental equation” that never appears from theory. Probably, most people would admit it is not an overstatement to say, “mathematics is the best friend of physics“; as she is in Einstein‘s General Relativity; it is essential for us to be awared of the notion of Riemannian geometry, which is non-euclidean, when trying to manage Einstein‘s theory of relativity.

In the past when Gauss was alive, to prove FTA was thought of as a sort of the 18th century’s version of the “Riemann Hypothesis”, the greatest mathematical challenge at that time. I believe you may want to hear his voice, and so let us listen to what he said in the past.

I protest against the use of infinite magnitude as something completed, which in mathematics is never permissible. Infinity is merely a facon de parler(manner of speaking), the real meaning being a limit which certain ratios approach indefinitely near, while others are permitted to increase without restriction.

God does arithmetic. You have no idea, how much poetry there is in the calculation of a table of logarithms!

Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.

Johann Carl Friedrich Gauss (German: Gauß; 30 April 1777 – 23 February 1855)

It is very grateful that you have read such a long posting here in this webpage.
So, thanks for your kind interests, and hope you to have some interests & insights here!

With best wishes,

Dr. Seongsoo Choi
Chairman of Metacomputing Inc.