This overview paper continues original works [1]-[9] started in 2017 on expansion of the theory of Atomic String Functions (ASF) [1]-[19] evolving since the 1970s to the fundamental theories [20]-[43] including General Relativity (GR) [1][8][33], Quantum Mechanics (QM) [9], and now Quantum Field Theories (QFT), Quantum Gravity (QG) and String theories [21]-[31]. ASF are the finite functions possessing the unique feature to exactly compose the polynomials of any order from finite “solitonic atoms” resembling quanta/excitations/strings/loops. Finite ASF functions become the base of the Atomic Series [1]-[5], generalizing widely-used Taylor series, to compose a wide range of analytic functions and solutions of differential equations of mathematical physics. Like in string [28]-[31] and Loop Quantum Gravity (LQG) [26][27], the theory offers a new theorem-based approach to introduce desired finiteness into GR, QFT, and QM theories traditionally operating with infinitely-spread functions based on space points. It offers a new Atomic Quanta model for the fundamental particles from the Standard Model (§6), with inclusion of quantum sizes into the definitions of energy while upholding the QM quantization rules. In application to GR, it leads to the Atomic Spacetime theory [1]-[8] and the new model of a graviton. Using the Atomic Series for both quantum and spacetime fields allows deriving “common blocks” to link GR and Quantum Theories within a framework of AString theory and AString Quantum Gravity, which can help to resolve the quantum gravity problem [26][27] included in the list [41] of unresolved problems in physics. Hope this theory, coming from a mathematical school, would attract the attention of physicists and string theorists to the unique apparatus of atomic functions naturally describing the finite building blocks of nature and fields as shown in this paper.

The long-standing problem of reconciling GR with Quantum Theories [41] is related to the mathematical incompatibility of nonlinear GR with linear QM, or simply the absence of “common mathematical blocks” between the theories [8]. The root cause of it is the use of continuous mathematical analysis based on “point in space” dots and infinitely spread functions [8], sometimes leading to unphysical infinities like GR singularities, QM wavefunction collapse into “a point”, QFT infinities renormalization. As string theorist M. Kaku mentioned [31][43], it is hard to find the commonality between “dots”. String and LQG theories [26]-[31] appeared to replace them with finite strings and loops.

This problem was well understood by A. Einstein, who in his 1933 lecture [25] classified it as “stumbling blocks”of theories operating “…exclusively with continuous functions of space”. Interestingly, in the same lecture he envisaged the solution—“perfectlythinkable” “atomic theory” with “finite regions of space” discussed in [1]-[3] and especially [8]. Development of this “atomic theory” based on atomic functions started in 2018 [1]-[9] is the focus of this paper.

The theory of Atomic Functions (AF) [1]-[19] has been evolving since 1967-1971 when a distinguished mathematician, the author teacher, Academician of NAS of Ukraine V.L. Rvachev¹ (Ukrainian V.L. Rvachov) had proposed a finite pulse function u p ( x ) for which derivatives (also pulses) would conveniently be similar to the original pulse shifted and stretched by the factor of 2:

As noted in the historical survey [14], some elements or similar functions were probably known earlier in 1935, Fabius [44] in 1966, and have been getting reinvented by different scientists even in the XXI century; the similarity of pulses seems quite an obvious idea. After describing u p ( x ) in 1971 as a “one finite function” [10], the full theory of such a new class of functions was mainly developed by V. A. Rvachev [10]-[13] and other collaborators [10]-[19] who extended the theory toward many functions called atomic functions in 1975 [10]-[19] (§4).

The main feature of AF u p ( x ) is the unique ability of 𝑒𝑥𝑎𝑐𝑡 composition of sections of polynomials of any order (§4) from superposition of a limited number N of neighboring “atoms” on a lattice of size a

For example, due to finiteness, only 9 neighboring atoms are required to calculate the value of a parabola x 2 at a given point x

The most significant is that other functions like trigonometric, exponential, and other analytic functions, which by definition [32] are the converging Taylor series of polynomials, can be represented either exactly (like polynomials) or via a converging series of shifts and stretches of AFs. Like from “mathematical atoms”, smooth functions can be composed of the AF superpositions, and because of that, those “atoms” have been called Atomic Functions (AF) in 1975 [10]-[19].

It means that what we perceive as continuous polynomials, lines, shapes, orbits, and trajectories are actually the stretches/interactions of a few finite neighboring atoms, with the hint that nature, which does not know what the functions are, operates in the same simple way—it just “distribute” or “shares” the energy/information in a local network of neighbors (§12).

The unique features of atomic functions attracted attention of many followers from different countries, notably by schools of V.F. Kravchenko [13]-[15], B. Gotovac, H. Gotovac [16][45], and the author [1]-[9], with the number of papers and books observed in [14] has grown to a few hundred.

In 2017, the author noted that AF u p ( x ) (2.1), known since the 1970s, is a composite object/solitonic atom consisting of two kink functions called AStrings [1]-[9], making them more generic and interchangeable (§4.2):

The ability of finite ASF functions to compose smooth polynomials, analytic functions, and solutions of differential equations, including GR, QM and QFT, has led to the novel interpretations of atomic spacetime [1]-[3][5] and uncovering the “common blocks” between General Relativity and Quantum Field Theories within AString Quantum Gravity theory described in §9.

The mathematical foundation of decomposition (“atomization”) of physical fields is based on the sequence of Atomic Quantization (Atomization) Theorems summarized in [1]-[3]. It led to the Atomic Series [1]-[3] as universal as Taylor and Fourier series but based on finite functions which can be applied to many physical theories, including GR, QM, QFT (§8-10), Quantum Gravity and String (§11) theories, with a detailed overview in [8].

Atomic Function (AF) (V.L. Rvachev, V.A. Rvachev, [10]) u p ( x ) was introduced in 1967-1971 as a finite compactly supported non-analytic infinitely differentiable function (Figure 1) with the first derivative conveniently expressed via the function itself shifted and stretched by the factor of 2:

(a)

(b)

(c)

Figure 1. (a) Atomic Function pulse with its derivative and integral (AString); (b) Atomic Function pulse (“solitonic atom”) in 3D; (c) Two Atomic Function pulses (“solitonic atoms” or “atomic solitons”).

With exact Fourier series representation [1]-[13],

the values of u p ( x ) can be tabulated in Appendix 2 with scripts [2][4][19].

Higher derivatives u p ( n ) and integrals I m can also be expressed via u p ( x ) [1]-[13][15]

Due to special double symmetry, AF provides a partition of unity [10]-[19] to exactly represent the number 1 by summing up individual overlapping pulses set at regular points ... -2, -1, 0, 1, 2… (Figure 2(a)):

(a)

(b)

(c)

Figure 2. (a) Partition of unity with Atomic Functions; (b) Representation of flat surface via summation of Afs; (c) Curved surface as a superposition of “solitonic atoms”.

Generic AF pulse of width 2 a , height c , and center positions b , d is

Multi-dimensional atomic functions [1]-[14] (Figure 1, Figure 2) can be constructed as either multiplications or radial atomic functions:

As a normalized function, AF possesses statistical properties [10]-[14] of being a weighted density distribution of the random variable ξ composed from independent, equally distributed variables ξ k :

A S t r i n g ( x ) function (Figure 3) was introduced in 2018 [1]-[9] as both the integral (4.3) and “composing branch” of u p ( x ) , noting that due to double-symmetry (4.4) AF u p ( x ) can be composed from two simpler kink functions:

AString is related to the shifted Fabius function [44] but was obtained based on AFs. AString is a solitary kink (Figure 3(a)) which can simply compose a straight line y = x as a translation of the same AString kinks (Figure 3(c)):

The shifted and stretched AString kink function can be generalized as

and their combination can reproduce curved functions and surfaces.

Importantly, the Atomic Function pulse (4.5) can be presented as a sum of two opposite AString kinks (Figure 3(b)), making AStrings and AFs deeply related

AFs and AStrings, or simply Atomics or ASF functions [1]-[3] can compose straight and curved geodesics and fields via superposition of solitonic kinks which inspired the spacetime “atomization”/quantization research [1]-[9] leading to Atomic Spacetime theory (§8) [1]-[3] and extended here to Quantum Fields (§5-7), Quantum Gravity (§10), and string theories where AString becomes the “open string” candidate (§11).

(a)

(b)

(c)

Figure 3. (a) Atomic String Function (AString); (b) Atomic function as a combination of two AStrings; (c) Representation of a straight-line segment by summing of AStrings.

The functions for which derivatives are expressed via the function itself (like exponents ex p ′ ( x ) = exp ( x ) or trigonometric functions si n ′ ( x ) = sin ( π / 2 − x ) become especially useful in mathematics, and ASF functions are not the exception. It turns out that polynomials of any order can be exactly composed from the shifts and stretches of ASF pulses [10]-[18]

Due to finiteness, only a limited number N = 2 n + 1 + 1 of neighboring “atoms” are required to calculate a n -order polynomial value at a given point x (for example, 9 for a parabola). Setting N = ∞ expands the series into x direction for exact representation of continuous polynomials. It means Atomics can also compose—“atomize”—any analytic function (a function representable by converging Taylor power series [32]) with calculable coefficients:

These series called Atomic Series [1][2][7]-[9] become the foundation of Atomic Quantization (Atomization) Theorems described in [2]. They state how Analytic functions, including a wide range of polynomial, trigonometric, exponential, hyperbolic, and other functions, and their sums, derivatives, integrals, reciprocals, multiplications, and superpositions [32] can be “atomized” either exactly (like polynomials) or with predefined precision by subdividing the lattice. Examples are shown in Figure 4 and throughout the paper.

Let’s note that being non-analytic (not representable via converging Taylor series), ASF functions can also compose non-analytic functions; for example, multiplication of y ( x ) (4.13) to the AF u p ( x ) z ( x ) = u p ( x ) y ( x ) would be non-analytic, but representable via AF combinations.

Instead of sums (4.12) and (4.13), we will be using short notation with localized atomic ASF functions A S F k ( x ) and function values y ( k ) at space lattice node k , assuming summation over repeated indices k :

(b)

(b)

(c)

Figure 4. Representing sections of polynomials and analytic functions with AStrings and Atomic Functions: (a) Linear function via 5 AStrings; (b) Parabola via 8 AStrings; (c) Cubic parabola via 8 Atomic Functions.

Composing continuous functions from finite pieces can also be achieved with widely-used polynomial splines, but with the crucial limitations (Streng-Fix theorem [2][3]) that n -order splines can exactly reproduce only n -order polynomials (eq cubic parabola cannot be exactly represented by quadratic splines [7][17]). Infinitely smooth Atomic Splines are more generic and provide the smooth connections between overlapping splines, which is important for precise rather than approximated field theories.

Being solutions of special kinds of nonlinear differential equations with shifted arguments (4.1), (4.8), ASF Functions possess some mathematical properties of topological lattice solitons [4][6][46] called Atomic Solitons, described in detail in [4][6]. AString is a solitonic kink whose particle-like properties exhibit themselves in the composition of lines (4.9), curves (4.13), and kink-antikink “atoms” (4.8) (Figure 3). Being a composite object (4.8) made of two AStrings, AF u p ( x ) is not a true soliton but rather a “solitonic atom”, or “topological dislocation atoms” [1]-[6].

For 50 years, many other Atomic Functions more complex than u p ( x ) [10]-[18] have been discovered. In fact, for any analytic function y ( x ) which is a solution of a linear differential equation L ( y ) = 0 it is possible to construct its own Atomic Function φ ( x ) from the equations with shifted arguments like

Then, the function y ( x ) can be composed from stretches of AF φ ( x )

So, it is possible to build AFs exactly composing sinusoids, exponents, gaussians, and others [10]-[19]. However, the multitude of AFs hides the importance of u p ( x ) as the simplest “mother function” [13]-[15] which, in combination, can exactly compose polynomials, hence the analytic functions which are the converging sums of polynomials [32]. For example, a function expanded via Taylor series as e x = 1 + x 1 ! + x 2 2 ! + x 3 3 ! + ⋯ can be represented by its “own” atomic function e u p ( x ) [13]-[18] but also by a series of AFs

By simply subdividing a lattice of nodes k / 2 , we can define a lattice k / 4 for parabolas x 2 , and then k / 8 for cubic parabolas x 3 , k / 2 − n for x n . It resonates with fractality (divisibility) of nature sizes/lengths [1][19][34], hinting that nature can use the same fractal mechanism to accommodate high variability fields (§12). Choosing a more granular lattice (up to Planck length) allows composing practically any fields from the simplest AF u p ( x ) without the need to invent complex atomic functions. In the future, in Atomic Series (4.14), we would assume that A S F k ( x ) can represent the multitude of Atomic Functions, but they ultimately can be expressed via the simplest and universal AFs u p ( x ) and A S t i r i n g ( x ) .

AF theory [10]-[13] proves that with a norm

u p ( x ) , with minimal wiggles, is “the best” and “simplest” ( a = 1 ) out of finite functions φ ( x ) capable of composing polynomials x n . This property allows deriving u p ( x ) from some important minimization (like least actions or shortest path) physical principles for quantization of fields (§5,8).

Atomic Series and Atomization Theorems (§4.3), described in detail in [2][8], are the theoretical foundation of applying ASF functions in QFT theories. The key idea is simple—because ASF finite functions can exactly compose polynomials of any order, we need to check whether QFT differential equations or Lagrangians L ( A μ ) can be resolved, at least in principle, in multi-dimensional polynomials, or generally, analytic functions which are the sums of polynomials [32]. This can be achieved in a typical power series way by analyzing whether QFT differential operators L ( A μ ) “preserve polynomiality” [1]-[3] and being applied to presumably polynomial operands A μ produce other polynomials matchable to the polynomials for F μ on the right side of a field equation, or schematically

As will be shown below, the majority, if not all, QFT and GR operators and Lagrangians historically made of derivatives and multiplications [2] possess this crucial property of “preservation of polynomiality”—because derivatives, integrals, multiplications, powers, sums, inversions, and superpositions of polynomials would be the polynomials (“polynomials are hard to destroy” [2]). Finally, the resolution, at least in principle, of QFT equations in polynomials would guarantee that the solutions can be atomizable by the Atomic Series via finite ASF functions. This atomization idea can be generalised to analytic functions, which by definition [32] are the converging sum of polynomials. So, the representability of QFT solutions via finite ASF functions is based on the two key features—preservation of polynomiality (and generally analyticity by QFT operators) and exact representation of polynomials by ASF functions. The universality of the Atomic Series is related to the fact that derivatives and integrals of ASF functions are expressed via ASF themselves (4.3), so they can be matched on both sides of the field equations in finite regions of space. Let’s demonstrate the fields' decomposition/atomisation idea for some theories.

Quantum Electrodynamics (QED) [21], as the most successful QFT theory matching experiments with astonishing precision, deals with interactions of electron, photon, muon, and other electromagnetic (EM) fields. Apart from the non-relativistic Schrödinger equation

QED deals with the Klein-Gordon equations for spin-0 particles for the wavefunction

more advanced Dirac Equation for electron/positron spinor ψ fields [21][22]

and relativistic Maxwell Equations for EM field components A μ [21]

It is easy to observe that all these L operators preserve polynomiality and analyticity, and being applied to presumably polynomic 4D field functions would yield other polynomials (because derivatives, integrals, sums, and multiplications of polynomials are the polynomials) which can be matched on both sides to satisfy the field equations according to scheme (5.1). So, the field functions ψ , A μ can be at least in principle resolved in polynomials, which in turn can be exactly represented by Atomic Series via overlapping ASF functions (4.14):

Field analytic functions, which by definition [32] are the converging sum of polynomials, follow the same rules. Physically, it means the QED fields can be decomposed into overlapping interactions of finite quantum-like objects (§6,7), like in string and LQG theories [26]-[31], leading to the AString theory (§11).

The fields’ atomization with the Atomic Series can be validated and demonstrated with some known QED solutions. The core QED Dirac equation (5.2) solution for a position-independent electron (and positron) [21]

includes analytic exponents and trigonometric functions expandable by Taylor series over polynomials [32], exactly representable via ASF series (4.13)

A similar validation can be applied for electron plain-wave solution [21] for spinor ψ , including an analytic exponential function representable by sums of polynomials, hence ASF finite functions on electron field lattice of size s (§6):

Let’s demonstrate how ASF functions can be used in QFT with the known photon QED model for wavefunction components A μ which reduces to a plain-wave solution [21], typically resolvable in analytic functions

One photon excitation wave packet travelling in x direction can be written as a carrier wave frequently oscillating inside an envelope [21] (Figure 5):

with a traditionally used Gaussian envelope

Interestingly, in QED, the envelope function is not enforced but dictated by the initial and decay conditions [21][22], so it is not a QED violation if we choose another envelope that provides additional benefits. Assuming that a generic envelope function is analytical allows composing an envelope from Atomics:

Extracting one atomic pulse from this series would yield the Atomic Photon model based on AF pulse with amplitude ε , width s , and energy/integral E

This APhoton pulse can serve as the simplest elementary atomic excitation/quantumof the QED photon field. More complex envelopes can be constructed by summing Atomic Envelopes (4.13) and (5.13) on some lattice. This A P h o t o n model is better than the traditional Gaussian model (5.12), because from a few Gaussians it is not possible to compose even the simplest fields, like a constant or a line.

(a)

(b)

Figure 5.(a) Atomic Photon envelope wavepacket; (b) Atomic and Gaussian Envelopes.

This Atomic Photon model correlates well with the extensive faint starlight laboratory experiments [21], confirming photon as the simplest (§4.6, 4.7) extremely stable soliton-like (§4.5) electro-magnetic quantum energy excitation/pattern (5.14) moving billions of years over vast distances without disturbances and registered as a single quantum lump of energy (5.14). It can be associated with the moving rearrangement of a few spacetime atoms (§9, 12), or “intricate distortions of spacetime”, as S. Hawking [37] mentioned.

Let’s note that ASF functions can be applied not only to the envelope, but to the whole presumably analytic photon field function (5.11), with a schematic spatial extension sum representation

This APhoton model can be the basis for the experiments to check whether a few photon envelopes are shaped (Figure 5) as u p ( x / s ) envelopes (5.14).

Another core QFT theory of Quantum Chromodynamics (QCD), describing strong nuclear forces and interactions of quark and gluon fields, is based on the gauge formulation with Lagrangian density [21][22]:

The variation of this Lagrangian produces the equations for three Dirac fields interacting with eight gluon fields [21][22]. Despite the complex compactification of differential operators and spinor vectors ψ , one can see that those operators containing derivatives and multiplications preserve the polynomiality and analyticity, and, being applied to polynomial operands, would produce other polynomials for the Lagrangian density [2]. The minimization of it should ultimately produce a polynomial representation for the field ψ , which in turn can be exactly representable by Atomic Series via the finite ASF functions:

So, the Atomic Series is applicable not only for differential equations, but also for the minimization of Lagrangian functionals typically used in QFT.

Scalar Higgs field ϕ , which gives particles a property of mass, is described by gauge QFT theory with Lagrangian density [21]

The operators contain derivatives and multiplications which preserve polynomiality and analyticity, and, being applied to polynomial operands, would produce other polynomials for the Lagrangian density and the solution ϕ , which on a lattice can be exactly atomizable with ASF functions:

For further validation, let’s note that the Higgs theory yields the well-known “Mexican hat” function [21]

which, as a simple polynomial, is exactly representable by Atomic Functions (4.17). As described in §6, the elementary atomic pulse in series (5.19) would represent an atomic Higgs boson (Figure 6), often coined as “the God particle”, popularised in the book [5] (Eremenko, 2020).

(a)

(b)

Figure 6. (a) Sean Carroll’s [33] lecture features the Higgs boson particle in space; (b) Higgs boson representation via atomic soliton pulse up(r).

The triumph of QFT as the main physical theory was the formulation of the Standard Model (SM) of Particle Physics with the Lagrangian including the parts from all QFT theories describing up to 60 fundamental fields [21]-[24]

This compactified formula incorporates the minimizable action Z , the GR Ricci scalar R , electromagnetism F μ v , strong field G μ v , weak field G μ v , Higgs field H , matter fields ψ ¯ i , with an overarching Feynman quantization integral ∫ D   Φ   e i S S M . Let’s analyze what kind of additional commonality can be extracted. First, it describes the distribution of tensor/spinor/scalar field functions f ( x , y , z , t ) coexisting at point x , y , z , t of spacetime. Second, those solutions are based on classical infinite spread functions and operations, which assume the existence of an infinitesimal point in space d x → 0 . Third, the functional-differential operators F μ v , G μ v , W μ v are historically built upon the classical derivatives and multiplication operations (§5.2), for example, i ℏ c ψ ¯ γ μ ∂ μ ψ − m c 2 ψ ¯ ψ for QCD fields (5.16). Those operators preserve polynomiality of arguments due to the powerful feature of polynomials—their multiplications, derivatives, integrals, and superpositions are also polynomials. So, despite their complexity, the QFT gauge field equations (5.21) are, in principle, resolvable in multi-dimensional polynomials or their sums (analytic functions [32]). Composing the infinitely-spread polynomials with ASF functions naturally introduces the quantized smooth finiteness and replaces “a point in space” with “finite region of space”—in line with A. Einstein’s 1933 perspective “atomic theory” [1][2][8][25], (§1.2). In this case, up to 60 quantum fields coexisting in every “point” of space would be replaced by interacting finite pulses of energy, where a field configuration at every “point” would be defined by the interaction pattern of neighboring quanta (§12). It opens the possibility of fields unification (§9-11), including Quantum Gravity (§9), based on a common representation of different fields via the core finite function u p ( x , a , b , c ) (4.5). String theory [26]-[31] also suggests a similar common derivation of all fields from “vibrating strings” (§6.9) based on the universality of Fourier series, but here the core functions are atomic, which leads to the new variant of AString theory (§11).

Formally, introducing finite ASF functions and Atomic Series into QFT Lagrangians follows the standard procedure (§5.1)—first, we evaluate whether the operators preserve polynomiality and are potentially resolvable via polynomials or their sums, and then we engage the Atomic Series, which exactly atomize polynomials of any order. Instead of following this standard way, let’s formulate a more ambitious task—to derive (rather than introduce) ASF functions from the Standard Model equations, so theoretically it could have been done on the onset of QFT and Atomic Functions in the 1980s. A similar approach of deriving Atomics from Einsteinian GR is described in [1]-[3]. This can be achieved with the following theorem.

Theorem(Standard Model Finite Atomic Function Theorem). Assuming the preservation of the polynomiality for SM Lagrangians (5.21), it is possible to find a finite function u p ( x ) ∈ [ − 1 , 1 ] which would represent smooth SM field solutions via sums of shifts and stretches d + c u p ( ( x − b ) / a ) in multiple dimensions. This function should be non-analytic and infinitely differentiable, with the derivatives and integrals expressed via the function itself, with the simplest function u p ′ ( x ) = 2 u p ( 2 x + 1 ) − 2 u p ( 2 x − 1 ) .

Proof. Preservation of polynomiality by SM Lagrangian operators (5.21) implies that multi-dimensional field functions are representable by polynomials P m ( x n ) being injected into these operators would yield the other polynomials, because multiplication of derivatives of polynomials by other polynomials would also be polynomials. So, in some regions of space, the Lagrangian (5.21) can be represented as the complex superposition of many polynomials of any order. Minimization (derivation) of Lagrangians would yield the field equations and their solutions as a series of polynomials. This theorem can be proved if we find some basis spline pulse-like finite function p ( x ) ∈ [ − 1 , 1 ] which, in translation, would exactly compose a polynomial of any order x n = ∑ k = − N k = N c k p ( x − b k a k ) . Evaluating the candidates for p ( x ) , we first have to eliminate the polynomial splines of some order, because they are unable to exactly compose a polynomial of any order ([2], §4.4). The spline function should be a polynomial of “infinite” order, so belonging to class C ( ∞ ) of absolutely smooth functions. Secondly, we have to eliminate smooth trigonometric and other exponent-based analytic functions like Gaussians or sigmoid, because by summing a few pulses, they are unable to exactly reproduce even the simplest polynomials (a line, or a constant). The choice has narrowed to infinitely differentiable finite functions which at least satisfy “partition of unity” c = ∑ k c p ( x − k ) ith derivatives expressed via the function itself p ′ ( x ) = F ( p ( x ) ) , or in the simplest (§4.7) linear form p ′ ( x ) = F ( p ( x ) ) = k p ( a x + b ) − k p ( a x − b ) which, with the symmetry condition p ( x ) = p ( − x ) , normalization p ( 0 ) = 1 and finiteness p ( x ) = 0 , | x | > 1 leads to the Atomic Function p ( x ) = u p ( x ) , u p ′ ( x ) = 2 u p ( 2 x + 1 ) − 2 u p ( 2 x − 1 ) discovered in the 1970s (§4). They can compose a polynomial of any order x n (Theorem §4.3). Due to double-symmetry (4.2), the AF u p ( x ) itself can be represented via the sum of two A S t r i n g kink functions (4.8) u p ( x ) = A S t r i n g ( 2 x + 1 ) − A S t r i n g ( 2 x − 1 ) = A S t r i n g ′ ( x ) which can compose the polynomials of any order too. In summary, due to preservation of polynomiality by all Lagrangian operators, the QFT field solutions can be, at least in principle, resolved in multi-dimensional polynomials, or their sums, which in turn can be exactly representable by a series of shifts and stretches of multiplications of Atomic Functions in many dimensions (4.6), (4.13).

Proof obtained. The importance of this theorem is that it allows not only introducing but deducing finite ASF functions from QFT, noticing the crucial property of QFT operators to preserve polynomials and the ability of ASF Functions to exactly compose them. Basically, it tells that presumably smooth analytic QFT fields can be represented/atomized via sums of ASF pulses in many dimensions (§6-9). Different fields governed by their own equations would have different distributions (§6), but, like in a string theory [29]-[31], the underlying common mathematical entity would be the Atomic Functions leading to AString theory (§11). As we see, the Standard Model (5.21) can incorporate another “layer of commonality” based on universal finitization/atomization of point-like QFT field functions with finite ASF functions resembling strings/loops/excitations described hereafter.

While representing the fields via some universal series like Fourier or Atomic Series (4.14) has some theoretical significance, the other opportunity is important as well—to examine the fundamental elementary processes/laws/principles based on analysis of the individual series components. For example, the universal trigonometric Fourier series gave rise to string theories where elementary “building blocks” of fields are some vibrating strings [28]-[31]. Atomic Series offers its own unique features—smooth quantized finiteness and the fields’ composition from interacting elementary “building blocks”. It is quite natural to assume that finite ASF functions would better describe finite strings, loops, excitations, and quanta. QFT [21]-[24] conveniently postulates that fields are made of ripples, and fundamental “particles”/quanta are the elementary excitations of quantum fields. Because AF u p ( x ) is a finite pulse (Figures 1-4), it is tempting to use it to describe the “elementary ripple”, or an Atomic Quantum of a field, as described hereafter.

The previous chapters provide the theoretical foundation on how the Atomic Series can represent/atomize any QFT field, with the Atomic Photon (§5.3) and Higgs (§5.6) models being the examples. The most important question is what an atomic pulse actually describes in the QFT context.

Generic 1D AF pulse (4.5) (Figure 1,Figure 2) has 4 parameters—pulse centre location x = b , y = d = 0 , amplitude c , and width 2 a :

A few neighbouring atoms overlap and produce a continuous field

for example, a parabolic or any order polynomial (4.12) field

Individual pulses in series (6.2), (6.3) have different shifts and amplitudes, but the same lattice width a . It allows associating the pulse width a with some fundamental field “size” parameter s , a = s , upholding the main QFT [21]-[24] and string theories assumptions [29]-[31] where fields differ by different scales and frequencies of vibrating strings. For example, the size s of a spacetime field can be associated with the Planck length [21] s = l p = 1.61 × 10 − 35   m while the QFT electron field (§5.3) can be associated with either radius of an electron or better with Compton quantum wavelength [21] s = λ c = ℏ m c ≈ 2.426 × 10 − 12   m . In summary, in the AQFT model, the difference between fields is encapsulated in the lattice width/size a = s of the elementary atomic pulse c u p ( x − b a ) .

Next, let’s elaborate on the atomic pulse c u p ( x − b a ) amplitude c . It has units of a field function (the function u p ( x − b a ) is unitless) the theory is applied for, with a few useful options.

In application to QM, the meaning and units of a pulse amplitude c is the probabilistic wavefunction ψ amplitude, so the atomic pulse and QM wavefunction field on some lattice of nodes can be conceptualized as

In this way, the ASF functions were introduced into Atomic Quantum Mechanics [9].

In more advanced QFT, the QM wavefunction is replaced by a complex field operator ϕ encapsulating spinor fields for fermions, vector fields for bosons, or scalar fields for Higgs [21]. In a nutshell, ϕ is just a space-defined complex operator function which can be decomposed into finite atomic functions, so the quantum fields on some lattice of nodes can be conceptualized as

For example, in application to Dirac QED electron fields, the field operator ϕ would have the meaning of a Dirac spinor ψ (5.4) decomposable into finite atomic pulses.

Physically, all fundamental fields are the carriers of energy. In classical physics, energy is stored in the field configuration, like electromagnetic energy density. In QFT, the fields become quantized, meaning a field can only have discrete excitations of energy associated with “particles” [21]. To apply the Atomic Functions theory to QFT fields, it is useful to associate the atomic pulses with the pulses of energy, so the amplitude c of atomic pulses c u p ( x − b s ) can be related to some energy parameter ε , c = ε . It can be done in the following way [8]. First, we need to understand how AFs would appear in many dimensions. Let's assume that an n-dimensional field is described by an analytic function f i e l d ( x i ) , i = 1 , ⋯ , n . By definition, it is representable by n -dimensional polynomials P m n which are the multiplication of some 1D polynomials representable by 1D AFs. For example, the polynomial component x 2 y 5 z can be exactly atomized by the sums of multiplication of AFs in the relevant dimension:

Due to finiteness, those multiplications would be non-zero only inside the finite regions of space

Selecting identical pulses in every dimension allows introducing n -dimensional AF (4.6) and its integral

Finally, let’s make a reasonable QFT assumption that the energy E of a pulse is related to the integral of an excitation pulse, which, due to (4.6) in the 1D case, would be E = ε s or E = ε in n -dimensional theory:

Interestingly, the atomic pulse c u p ( x − b s ) amplitude c has a convenient meaning and units of energy density ε (Joules per unit of volume) frequently used in many classical and QFT theories, and, importantly, in Lagrangians. It means the atomic quantum fields representations (6.2) would have a meaning of energy density distributions ε ( x 1 , ⋯ , x n ) on some lattice, or schematically

From the Atomic Series sum (6.8), it is possible to extract an elementary pulse identical in all directions, which would describe an elementary atomic excitation/ripple of a field, which in QFT theory is typically called quantum [21][22]. Because it is described by an Atomic Function, it is reasonable to name it as Atomic Quantum or simply A Q u a n t which in n and 1 dimensions can be written as

This pulse has the shape of a multi-dimensional Atomic Function (Figure 1, Figure 3, Figure 6), units of energy density, and two key parameters—constant field-dependent size/width s and energy density ε as a pulse amplitude. This A Q u a n t pulse can be defined in n dimensions (simple 1D, QFT 4D spacetime, or up to 11 in some string theories). The spatial integral (6.7) of this finite pulse has the units and meaning of energy. In summary, the description of an Atomic Quantum as an elementary excitation of a quantum field can be encapsulated in the following formula

This key formula unites the energy E , size/width s , and energy density ε of an Atomic Function u p n pulse which describes an elementary field excitation/quantum with n dimensions. Fields differ from each other by the presumably constant size s of quanta (similar to different string frequencies in string theory [29]-[31]). Variable energy density ε , as the amplitude of a quantum pulse, is proportional to the energy level of a field in a given pulse region.

Atomic Quanta (6.10) are not just the mathematical abstractions but can be conveniently matched to the experimental values based on known sizes (like Compton quantum lengths [21]) and energy levels, for example, for a few Atomic Quanta particles in Table 1.

Table 1.Energy densities and sizes of atomic quanta for some fields.

In the historical context, the formula (6.10) unites the 55-year-old theory of Atomic Functions with the 80-year history of Quantum Field Theory as the major theory of physics, enriching both theories. It not only introduces the new Atomic Quantum and fields atomization concepts into QFT theory, but also leads to AString Quantum Gravity and string theories described in §9-11.

Let’s highlight the novelty of Atomic Function representations (6.1)-(6.10) in comparison with traditional QFT theories based on point-in-space and infinitely spread functions, which, according to A. Einstein [25] may create the “stumbling blocks” towards unification (§1.2), notably renormalization of infinities [21]. Atomic QFT theory not only offers the easy and universal way of quantization of fields based on Atomic Series over finite ASF functions envisaged by A. Einstein in his “atomic theory” [25] but also introduces a useful field fundamental size s parameter absent in QFT to distinguish different fields and match it to quantum, string or loop sizes from string and LQG theories [26]-[31] (§9-11).

Atomic Quantum (6.10) is the elementary quantum of a field based on one AF pulse (Figure 1,Figure 3,Figure 6). In reality, fields are variable (Figure 7) and can be composed from the superposition/sum of finite quanta of different intensities located at some lattice k nodes, or schematically in the 1D case

(a)

(b)

(c)

(d)

Figure 7. Different field configurations composed of atomic functions.

Due to AF partition of unity (4.4), equally spaced quanta of identical intensity ε k = ε produce a constant field (Figure 2) with energy (6.10):

This simulates the constant (eq., Higgs, gluon, or electron) energy field measurable in experiments, for example, on the Large Hadron Collider (see Table 1). But generally, the fields are varied and composed of atomic quanta of different intensities (6.11). Examples of different variable fields are shown in Figure 7.

Each field has its own size parameter s , like a Compton or other length [21], typically measured in experiments, for example, s = 2.43 × 10 − 12   m for an electron field (see Table 1), or Planck length s = 1.6 × 10 − 35   m for the spacetime field first described with ASF Functions [1]-[8] in 2018. So, generic A Q u a n t definition (6.10), (6.11) can be introduced for every field type, for example

Figure 8 schematically shows different energy levels for different quanta, and different distributions of fields described by Atomic Functions [1]-[9]. The benefits of introducing separate Atomic Quanta are that the engineers in each field can work with their own quanta instead of generic representations for all fields.

(a)

(b)

(c)

Figure 8.(a) Different fields have quanta of different energy levels; (b) Constant field as superpositions of a few atomic pulses; (c) Composing a growing field/brane from AString functions.

The energetic definition of Atomic Quanta (§6.4) is especially useful for establishing relationships between energies and other matter characteristics. Following Einstein’s energy-mass relationship E = m c 2 , it is possible to express a field atomic quantum (6.10) in terms of mass:

This formula not only unites Einsteinian Special Relativity and Atomic Functions theories but also correlates with Einstein’s 1933 perspective “atomic theory” [1]-[3][25]. It also emphasizes mass as an interaction rather than a matter collection phenomenon.

Instead of associating Atomic Quantum energy (6.10) with mass (6.14), it is possible to choose another universal Planck–Einstein energy-frequency relation [21] E = ℏ ν . In this case, Atomic energy distributions (6.8) would have the meaning of the fields of vibrating strings

with elementary atomic pulse/quantum

or, in terms of frequencies,

Let’s note that we are not postulating that the coefficient ℏ is a constant for every field. It just relates energy to a frequency, and we know that for a photon field, it is the Planck constant [21].

These formulae link the Atomic Functions theory with string theory [28]-[31], assuming that atomic field distributions can be expressed in terms of frequencies of vibrating strings. For example, the parabolic field around point x (4.12) can be expressed as the superposition of 9 ( N = 4 ) AF pulses/notes/keys

with amplitudes ( k 2 64 − 1 36 ) which can be related to some string frequencies. The prospects of AString theory are described in §11.

Chronal fields were hypothesized in the 1980s by A. Veinik [39], who suggested that every fundamental field, including time and space, should have its own quanta—chronants and mertriants—the presence of which gives matter the properties of the “time run rate”, “size”, and “order of location”. Later development of string theories [28]-[31] correlates with these hypotheses because vibrating strings define the frequencies of timing processes and locations of strings with size. Chronal intensity τ is also measured in frequencies [39], and replacing ν with τ in atomic quanta (6.16) and string fields (6.17) definitions allows deriving the atomic chronal fields with a quantum of energy

or, in terms of chronal intensity,

Here, ℏ is not strictly speaking a Planck constant but a proportionality coefficient to match the chronal intensity (of strings) to energy E . Interestingly, this line of research may lead to the theories where energies E = ε s n = ℏ τ are not fundamental but emergent from chronal and metric fields, in correlation with some popularised theories where energy may be related to “time”.

Atomic Quantum (6.10) has a smooth pulse/excitation/ripple shape described by the Atomic Function (Figure 1,Figure 6,Figure 8), generally in n dimensions. The key question is why can’t we use the other pulse looking functions, for example, Gaussian G ( x ) = a e − b ( x − c ) 2 , or some known spline. The answer is that from translations of a few Gaussians, it is not possible to compose even the simplest polynomial fields (like a line or parabola), while shifts and stretches of a few AF pulses compose a polynomial of any order, hence analytic field functions.

Another important question is the atomic function shape invariance in 3D space—the 3D pulses ε u p ( x / s ) u p ( y / s ) u p ( z / s ) may look different if we rotate the coordinate system x , y , z , or see the pulses from a different angle (Figure 6, Figure 8). Here, it is important to note that atomic pulses are actually the “atomized” versions of analytic functions appearing in solutions of differential QFT field equations (§5,6), typically invariant against coordinate rotations and Lorentzian transformations [21]. Final analysis of the solutions requires the selection of a coordinate system and writing the functions, “atomization” of which would lead to an elementary ε u p ( x / s ) u p ( y / s ) u p ( z / s ) ripple/atom/quantum. It does not mean that there is a “rigid” object like a 3D Lego block that physically exists somewhere in space. Nature does not know what is x , y , z , and the AF Quantum pulse (6.10) expresses some relationship/distribution/feature of a mathematical solution, in our case, the composition of any polynomial in any coordinate system from finite pieces.

Energy of a quantum excitation is an integral (6.10) of an elementary AF excitation/quantum, matching the QFT convention. This exactly calculable integral (4.5), (4.6) is a product of two parameters—amplitude ε and size/width/volume s n factor; in 3D, it is V = s 3 . The bigger and more intense the excitation, the more energy/mass it has. Association of energy with mass (§6.8), based on Einsteinian E = m c 2 , means that Atomic Quantum carries some mass m = ε s n / c 2 . Noting that V = s n is a volume and is a “matter” density, we can write

It reminds the Einsteinian E = m c 2 , but expressed in terms of an energy density ε of an Atomic Quantum pulse.

Each fundamental field (photon, gluon, electron, Higgs) has its own (minimal) atomic quantum size s . Despite the s constancy, we need to be careful about “fixing” it. According to Einstein’s Special Relativity and Lorentzian transformations, the sizes of a moving object can shrink. It means the quantum pulse ε u p ( x s ) width s is a “dynamic” uniformly shrinking constant, the same for all quanta. In QFT, the particles are not the “rigid” matter objects with fixed size, but the “excitations” of fields, and the Atomic Quantum model (6.10) describes exactly that.

The key intensity parameter ε of an Atomic Quantum (6.10) ε u p ( x s ) have a meaning of energy density ε , which should be treated as the amplitude of a pulse that can fall into positive or negative territory [8].

Let’s elaborate on the important relationship E = ε s n between Atomic Quantum energy E and quantum intensity ε , remembering that size s n depends on the field type. For example, the Higgs field has energy levels of 125GeV, while electromagnetic fields are much weaker. It does not mean that quanta intensities ε obey the same proportions, because Atomic Quanta of different fields have different s n sizes which are the true field-dependent constants. Using the string theory [29]-[31] analogy, the energy of strings should depend not only on frequencies but also on the sizes s n of strings. So, energy, as a universal concept for many theories, becomes a product of two parameters, creating multiple opportunities discussed in §9-11.

Another parameter of the Atomic Quantum model (6.10) is the number of dimensions n appearing in the definition of a size/volume s n and n -dimensional Atomic Function u p n ( x n s ) . The ASF theory does not impose the number of dimensions, and is applicable for the familiar 4D spacetime or for 11 dimensions in some string theories.

Atomic Functions can also describe loops presumed to be the building blocks of spacetime in Loop Quantum Gravity (LQG) theory [26]-[28]. Radial AF (4.6) (Figure 1,Figure 6)