THE INSTITUTE FOR SCIENTIFIC COMPUTING AND APPLIED MATHEMATICS

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1 Spectral Theory of Differential Operators and Energy Levels of Subatomic Particles Tian Ma and Shouhong Wang January 6, 2014 Preprint No.: 1402 THE INSTITUTE FOR SCIENTIFIC COMPUTING AND APPLIED MATHEMATICS INDIANA UNIVERSITY

2 SPECTRAL THEORY OF DIFFERENTIAL OPERATORS AND ENERGY LEVELS OF SUBATOMIC PARTICLES TIAN MA AND SHOUHONG WANG Abstract. Motivated by the Bohr atomic model, in this article we establish a mathematical theory to study energy levels, corresponding to bounds states, for subatomic particles. We show that the energy levels of each subatomic particle are finite and discrete, and corresponds to negative eigenvalues of the related eigenvalue problem. Consequently there are both upper and lower bounds of the energy levels for all sub-atomic particles. In particular, the energy level theory implies that the frequencies of mediators such as photons and gluons are also discrete and finite. Both the total number N of energy levels and the average energy level gradient for two adjacent energy levels) are rigorously estimated in terms of some physical parameters. These estimates show that the energy level gradient is extremely small, consistent with the fact that it is hard to notice the discrete behavior of the frequency of subatomic particles. Contents 1. Introduction 2 Part 1. Mathematics 4 2. Spectrum of Elliptic Operators Physical bacground Mathematical preliminaries Conditions for existence of negative eigenvalues Estimates for the number of negative eigenvalues 7 3. Spectrum of Weyl operators Wave equations for massless particles Spectral theory of Weyl operators Estimates on number of negative eigenvalues 14 Part 2. Physics Spectral Equations of Subatomic Particles Constituents of subatomic particles Bound potentials of subatomic particles Wave equations of bound states Spectral equations of bound states 19 Date: January 6, Key words and phrases. spectrum of differential operators, energy levels, Dirac operator, Weyl operator, subatomic particles. The wor was supported in part by the Office of Naval Research, by the US National Science Foundation, and by the Chinese National Science Foundation. 1

3 2 MA AND WANG 5. Energy Levels of Subatomic Particles Charged leptons and quars Baryons Mediators Number of energy levels Physical conclusions 26 References Introduction As we now, all matters in the universe are made of atoms, and atoms are made up of sub-atomic particles. There are six classes of sub-atomic particles: charged leptons, quars, baryons, mesons, internal bosons, and mediators. This article is motivated 1) by the atomic energy level theory, 2) the weaton model of elementary particles, and 3) unified field model for interactions of Nature. The classical atomic energy level theory demonstrates that there are finite number of energy levels for an atom given by E n = E 0 + λ n, n = 1,, N, where λ n are the negative eigenvalues of the Schrödinger operator, representing the bound energies of the atom, holding the orbital electrons, due to the electromagnetism. The weaton model of elementary particles and the unified field theory are developed recently by the authors [6, 7]. The field model is based on two recently postulated principles: the principle of interaction dynamics PID) and the principle of representation invariance PRI). The main objectives of this article are 1) to introduce the energy levels for all subatomic particles, 2) to develop a mathematical theory to study energy levels, and 3) to derive physical implications and predictions of the theory. Hereafter we explore the ey ingredients and the main results in this article. First, we develop a mathematical theory for estimating the number of negative eigenvalues for a class of differential operators, and consequently the theory is applied to study the energy levels for subatomic particles. Second, the constituents of subatomic particles are spin- 1 2 fermions, which are bound together by either wea or strong interactions. Hence the starting point of the study is the layered wea and strong potentials derived recently by the authors [6], which play the similar role as the Coulomb potential for the electromagnetic force which bounds the orbital electrons moving around the nucleons. Third, dynamic equations of massless particles are the Weyl equations, and the dynamic equations for massive particles are the Dirac equations. The bound energies of all subatomic particles are the negative eigenvalues of the corresponding Dirac and Weyl operators, and the bound states are the corresponding eigenfunctions. The Weyl equations were introduced by H. Weyl in 1929 to describe massless spin- 1 2 free particles [11], which is now considered as the basic dynamic equations of neutrino [3, 4, 8]; see also [2]. For a bound state of a massless particle under either wea or strong interaction potentials, we derive then a new set of dynamic equations 3.11) based on quantum mechanics principles, which are still called the Weyl equations; see Section 3 for the detailed derivation.

4 3 Fourth, with bound state equations for both massless and massive particles, we derive the corresponding spectral equations for the bound states. With the mathematical theory developed in this article on the related eigenvalue problems, we show that the energy levels of each subatomic particle are finite and discrete: 0 < E 1 < < E N <, and each energy level E n corresponds to a negative eigenvalue λ n of the related eigenvalue problem. Physically, λ n represents the bound energy of the particle, and are related to the energy level E n with the following relation: 1.1) E n = E 0 + λ n, λ n < 0 for 1 n N. Here E 0 is the intrinsic potential energy of the constituents of a subatomic particle such as the weatons. Fifth, one important consequence of the above derived energy level theory is that there are both upper and lower bounds of the energy levels for all sub-atomic particles, and the largest and smallest energy levels are given by 1.2) 0 < E min = E 0 + λ 1 < E max = E 0 + λ N <. In particular, it follows from the energy level theory that the frequencies of mediators such as photons and gluons are also discrete and finite, and are given by ω n = E n / n = 1,, N). In the Planc classical quantum assumption that the energy is discrete for a fixed frequency, and the frequency is continuous. Our results are different in two aspects. One is that the energy levels have an upper bound. Two is that the frequencies are also discrete and finite. Hence the derived results in this article will have significant implications for field quantizations. Sixth, based on the mathematical theory developed in this article, we have rigorously derived estimates on the total number N of energy levels for subatomic particles, which cam be summarized as follows: 4 ρ 2 0A w m w c g 2 ) 3/2 w for charged leptons and quars, λ 1 ρ w c N = 4 ρ 2 1A s m q c g 2 ) 3/2 s for baryons, λ 1 ρ q c Aw ρ γ g 2 ) 3 w for photons. ρ w c β 1 Here g w and g s are the wea and strong charges, m w is the mass of the constituent weaton, m q is the mass of the constituent quar, ρ w and ρ q are the radii of the constituent weaton and quar respectively, ρ 0, ρ 1, ρ γ are the radii of the wea attraction, the strong attraction and the photon respectively, and A w, A s are nondimensional constants. Seventh, by estimating the average energy level gradient for two adjacent energy levels), we are able to show that E ev, which is too small to be practically measurable. This is consistent with the fact that it is hard to notice the discrete behavior of the frequency of subatomic particles.

5 4 MA AND WANG Eighth, it is now classical that the electromagnetic, wea and strong interactions are described by gauge fields. Consequently, they all enjoy a common property that moving charges generate magnetism. Namely, in the same spirit as the electric charge e producing magnetism, the wea and strong charges g w, g s can also produce similar effects, which we also call magnetism. For a massless particle, the term g σ curl A in the Weyl equations reflects the magnetic force generated by the wea or strong interaction, where σ = σ 1, σ 2, σ 3 ) is the Pauli matrix operator, and A is the special component of the combined wea or strong potential A µ = A 0, A) using PRI first introduced in [6]. For a massive particle, magnetic effect is reflected by the term µ curl A in the Dirac equations, where µ = g σ/2m represents magnetic moment, generated by either wea or strong interaction. Ninth, we are able to establish a relationship between the spectrum and the bound state of subatomic particles in terms of related physical parameters. For example, for an electron, the eigenvalue λ e and and the certain quantities of the corresponding bound state ψ e enjoy the following relation: λ e = 1 2 m wv 2 + 2g2 w r e 2A wg 2 w ρ w κ e, where 1 2 m wv 2 is the inetic energy of each constituent weaton in the electron, r e is the radius of the electron, and κ e is a new parameter for the electron; see 5.10). This article is divided into two parts, consisting of Sections 2 and 3. The first part deals with the mathematical theory for eigenvalue problems originated from the Dirac and Weyl equations for bound states of subatomic particles. Section 2 studies spectrum of general elliptic differential operators, and Section 3 addresses the spectrum of the Weyl operators. Part 2 consists of Sections 4 and 5, and studies the energy levels and spectrum of the Dirac and Weyl equations applied to subatomic particles. Section 4 focuses on the spectral problems, and Section 5 studies energy levels of subatomic particles. Part 1. Mathematics 2. Spectrum of Elliptic Operators 2.1. Physical bacground. Based on the Bohr atomic model, an atom consists of a proton and its orbital electron, bounded by electromagnetic energy. Due to the quantum effect, the orbital electron is in proper discrete energy levels: 2.1) 0 < E 1 < < E N, which can be expressed as 2.2) E n = E 0 + λ n λ n < 0), where λ n 1 n N) are the negative eigenvalues of a symmetric elliptic operator. Here E 0 stands for the intrinsic energy, and λ n stands for the bound energy of the

6 5 atom, holding the orbital electrons, due to the electromagnetism. Hence there are only N energy levels E n for the atom, which are certainly discrete. To see this, let Z be the atomic number of an atom. Then the potential energy for electrons is given by V r) = Ze2 r. With this potential, the wave function ψ of an orbital electron satisfies the Schrödinger equation 2.3) i ψ t ψ + Ze2 2m 0 r ψ = 0. Let ψ tae the form ψ = e iλt/ ϕx), where λ is the bound energy. Putting ψ into 2.3) leads to 2 2 ϕ Ze2 2m 0 r ϕ = λϕ. Since the orbital electrons are bound in the interior of the atom, the following condition holds true: ϕ = 0 for x > r 0, where r 0 is the radius of an atom. Thus, if ignoring the electromagnetic interactions between orbital electrons, then the bound energy of an electron is a negative eigenvalue of the following elliptic boundary problem 2.4) 2 2 ϕ Ze2 2m 0 r ϕ = λϕ for x B r 0, ϕ = 0 for x B r0, where B r0 is a ball with the atom radius r 0. According to the spectral theory for elliptic operators, the number of negative eigenvalues of 2.4) is finite. Hence, it is natural that the energy levels in 2.2) are finite and discrete. In fact, we shall demonstrate in this article that all subatomic particles possess discrete and finite energy levels as electrons. The study is based on a unified field model for interactions of Nature [5, 6] and an elementary particle theory, called the weaton model in [7, 6], both developed recently by the authors. Based on these two models, two or three of weatons and quars are bound together to form a subatomic particle under the wea and strong interactions. Thus, naturally these particles have a sequence of energy levels, which are described by the negative eigenvalues of some linear differential operators Mathematical preliminaries. Consider the eigenvalue problem of linear elliptic operators given by 2.5) D 2 ψ + Aψ = λψ for x, ψ = 0 for x, where R n is a bounded domain, ψ = ψ 1,, ψ m ) T : C m is the unwon wave function, 2.6) D = + i B, B = B1,, B n ), and A, B 1 n) are m m Hermitian matrix-valued functions.

7 6 MA AND WANG Let λ 0 be an eigenvalue of 2.5). The corresponding eigenspace: E λ0 = {ψ L 2, C m ) ψ satisfy 2.5) with λ = λ 0 } is finite dimensional, and its dimension σ = dim E λ0 is called the multiplicity of λ 0. Physically, σ is also called degeneracy. Usually, we count the multiplicity σ of λ 0 as σ eigenvalues: λ 1 = = λ σ = λ. The following spectral theorem is classical. Theorem 2.1. Assume that the functions in the matrices A and B j 1 j n) are square integrable. Then the following assertions hold true: 1) All eigenvalues of 2.5) are real with finite multiplicities, and form an infinite consequence as follows: < λ 1 λ 2 λ n, λ n as n, where λ n is counting the multiplicity. 2) The eigenfunctions ψ n corresponding to λ n are orthogonal to each other: ψ nψ l dx = 0 n l. In particular, {ψ n } form an orthogonal basis of L 2, C m ). 3) There are only finite number of negative eigenvalues in {λ n }, 2.7) < λ 1 λ N < 0, and the number N of negative eigenvalues depends on A, B j 1 j n) and the domain. Remar 2.1. For the energy levels of particles, we are mainly interested in the negative eigenvalues of 2.5) and the estimates for the number N in 2.7) Conditions for existence of negative eigenvalues. The following theorem provides a necessary and sufficient condition for the existence of negative eigenvalues of 2.5), and a criterion to estimate the number of negative eigenvalues. Theorem 2.2. For the eigenvalue problem 2.5), the following assertions hold true: 1) Equations 2.5) have negative eigenvalues if and only if there is a function ψ H0 1, C m ), such that 2.8) [Dψ) Dψ) + ψ Aψ]dx < 0, where D is as in 2.6). 2) If there are K linear independent functions ψ 1,, ψ K H 1 0, C m ), such that 2.9) ψ satisfies 2.8) for any ψ E K = span {ψ 1,, ψ K }, then the number N of negative eigenvalues is larger than K, i.e., N K.

8 7 Proof. Assertion 1) follows directly from the following classical formula for the first eigenvalue λ 1 of 2.5): 1 [ 2.10) λ 1 = min Dψ) Dψ) + ψ Aψ ] dx. ψ H0 1,Cm ) ψ L 2 We now prove Assertion 2) by contradiction. Assume that it is not true, then K > N. By Theorem 2.1, the K functions ψ j in 2.9) can be expended as N 2.11) ψ j = α ji e i + β jl ϕ l for 1 j K, i=1 l=1 where e i 1 i N) and ϕ l are eigenfunctions corresponding to negative and nonnegative eigenvalues. Since K > N, there exists a K-th order matrix P such that ) ) P α =, where α 11 α 1N α =.. with α ij as in 2.11). α K1 α KN Thus, under the transformation P, ) ψ 2.13) ψ = P E K, ψ = ψ 0 1,, ψ N ) T, where E K is as in 2.9). However, by 2.11) and 2.12), the first term ψ 1 in 2.13) can be expressed in the form 2.14) ψ1 = θ l ϕ l E K. l=1 Note that ϕ l are the eigenfunctions corresponding to the nonnegative eigenvalues of 2.5). Hence we have 2.15) [D ψ 1 ) D ψ 1 ) + ψ 1 A ψ 1 ]dx = ψ 1 D2 ψ1 + A ψ 1 )dx = θ l 2 λ l > 0. Here λ l 0 are the nonnegative eigenvalues of 2.5). Hence we derive, from 2.14) and 2.15), a contradiction with the assumption in Assertion 2). Thus, the proof of the theorem is complete Estimates for the number of negative eigenvalues. For simplicity, it is physically sufficient for us to consider the eigenvalue problem of the Laplace operators, given by 2.16) where B r R n is a ball with radius r. l=1 2 ψ + V x)ψ = λψ for x B r, ψ = 0 for x B r,

9 8 MA AND WANG In physics, V represents a potential function and taes negative value in a bound state, ensuring by Theorem 2.2 that 2.16) possesses negative eigenvalues. Here, for the potential function V x), we assume that 2.17) V ρx) ρ α V 0 x) α > 2), where V 0 x) is defined in the unit ball B 1, and 2.18) = {x B 1 V 0 x) < 0}. Let θ > 0 be defined by 2.19) θ = inf ψ L 2,C m ) 1 ψ L 2 V x) ψ 2 dx. The main result in this section is the following theorem, which provides a relation between N, θ and r, where N is the number of negative eigenvalues of 2.16). Let λ 1 be the first eigenvalue of the equation e = λe for x, 2.20) e = 0 for x, where B 1 is as defined by 2.18). Theorem 2.3. Under the assumptions of 2.17) and 2.18), the number N of the negative eigenvalues of 2.16) satisfies the following approximative relation ) θr 2+α n/2 2.21) N, λ 1 provided that θr 2+α /λ 1 1 is sufficiently large, where r and θ are as in 2.16) and 2.19), and λ 1 is the first eigenvalue of 2.20). Remar 2.2. The estimates 2.21) is also valid for Problem 2.16) in a shell region 2.22) B = {x R n r 0 < x < r 1 }, if r 0 r 1. The spectral equations for subatomic particles are defined in such shellregions as 2.22). To prove Theorem 2.3, we need to introduce a lemma, which is due to H. Weyl [10]; see also [1, 9]. Lemma 2.1 H. Weyl). Let λ N be the N-th eigenvalue of the m-th order elliptic operator 2.23) then λ N has the asymtptotical relation 1) m m e = λe for x R n, D e = 0 for 0 m 1, 2.24) λ N λ 1 N 2m/n, where λ 1 is the first eigenvalue of 2.23). Proof of Theorem 2.3. The ball B r can be written as B r = {y = rx x B 1 }. Note that / y = r 1 / x, 2.16) can be equivalently expressed as 2.25) ϕ + r 2 V rx)ϕ = βϕ for x B 1, ϕ = 0 for x B 1,

10 9 and the eigenvalue λ of 2.16) is λ = 1 β, where β is the eigenvalue of 2.25). r2 Hence the number of negative eigenvalues of 2.16) is the same as that of 2.25), and we only need to prove 2.21) for 2.25). By 2.17), the equation 2.25) is approximatively in the form 2.26) ϕ + r 2+α V 0 x)ϕ = βϕ for x B 1, ϕ = 0 for x B 1. Based on Assertion 2) in Theorem 2.2, we need to find N linear independent functions ϕ n H 1 0 B 1 ) 1 n N) satisfying 2.27) B 1 [ ϕ 2 + r 2+α V 0 x)ϕ 2] dx < 0, for any ϕ span {ϕ 1,, ϕ N } with ϕ L 2 = 1. To this end, we tae the eigenvalues {λ n } and eigenfunctions {e n } of 2.20) such that and 0 < λ 1 λ N < λ N+1, 2.28) λ N < θr 2+α λ N+1. For the eigenfunctions e n, we mae the extension ϕ n = { en for x, 0 for x B 1 /. It is nown that ϕ n is wealy differentiable, and ϕ n H 1 0 ). These functions ϕ n 1 n N) are what we need. Let ϕ = N α n ϕ n, ϕ L 2 = 1. n=1 By Assertion 2) in Theorem 2.1, ϕ n 1 n N) are orthonormal: Therefore we have ϕ i ϕ j dx = B ) ϕ L 2 = e i e j dx = δ ij. N αn 2 = 1. n=1

11 10 MA AND WANG Thus the integral in 2.27) is [ ϕ 2 + r 2+α V 0 x)ϕ 2 ]dx B 1 N ) N ) = α n e n α n e n dx + r 2+α = n=1 N αnλ 2 n + r 2+α n=1 n=1 V 0 x)ϕ 2 dx N αnλ 2 n θr 2+α by 2.19) n=1 < 0 by 2.28) and 2.29). V 0 x)ϕ 2 dx It follows from Theorem 2.2 that there are at least N negative eigenvalues for 2.26). When θr 2+α 1 is sufficiently large, the relation 2.28) implies that 2.30) λ N θr 2+α. The result 2.21) follows from 2.24) and 2.30). The proof is complete. Remar 2.3. In Part 2, we shall see that for particles with mass m, the parameters in 2.21) are α = 0, r = 1, θ = 4mρ 2 1g 2 / 2 ρ, where g = g w or g s is the wea or strong interaction charge, ρ is the particle radius, and ρ 1 is the wea or strong attracting radius. Hence the number of energy levels of massive particles is given by N = [ 4 ρ 2 1 λ 1 A ρ mc g 2 c ] 3/2, where λ 1 is the first eigenvalue of in the unit ball B 1, and A is a constant. 3. Spectrum of Weyl operators 3.1. Wave equations for massless particles. First, we recall a basic postulate of quantum mechanics. Basic Postulate of Quantum Mechanics. For a quantum system with observable Hermit operators L 1,, L m, if the physical quantities l corresponding to L 1 m) satisfy a relation: 3.1) Hl 1,, l m ) = 0, then the following equation induced from 3.1) 3.2) HL 1,, L m )ψ = 0 describes the system provided the operator HL 1,, L m ) in 3.2) is Hermitian. The classical quantum mechanical equations, such as the Schrödinger equation, the Klein-Gordon equation and the Dirac equations, are based on this Basic postulate.

12 11 It is nown that the basic equations describing a free and massless particles are the Weyl equations: ψ 3.3) = c σ )ψ, t where ψ = ψ 1, ψ 2 ) T is a two-component Weyl spinor, = 1, 2, 3 ) is the gradient operator, σ = σ 1, σ 2, σ 3 ) is the Pauli matrix operator, and the Pauli matrices σ 1 3) are given by σ 1 = ), σ 2 = ) ) 0 i 1 0, σ i 0 3 =. 0 1 The Weyl equations 3.3) also enjoy the Basic Postulate. In fact, the energy and momentum operators E and P for the Weyl spinor are given by 3.4) E = i, P = i σ ). t By the de Broglie relation we get that E = ω, P = /λ, c = ωλ, 3.5) E = cp, and 3.3) follows from 3.4) and 3.5). For a massless particle system in a bound state by either wea or strong interaction, 3.4) is replaced by 3.6) E = i V, P = i σ D), t where V = ga 0 is the potential, g is the wea or strong charge, A µ = A 0, A 1, A 2, A 3 ) represents the combined gauge potential for either wea or strong interaction derived using PRI [6], and the differential operator D = D 1, D 2, D 3 ) is defined by D j = j + i g c A j. By the weaton model developed in [6, 7], the mediators such as photons and gluons consist of two massless weatons, which are bound in a small ball B r by the wea and strong interactions. Hence the Weyl spinor ψ of each weaton is restricted in a small ball, i.e. which implies the boundary condition ψ = 0 x B r. 3.7) ψ Br = 0. However, in mathematics the boundary problem for the Weyl equations generated by 3.5) and 3.6) given by 3.8) i ψ t = i c σ D)ψ + V x)ψ ψ Br = 0 is in general not well-posed. The main reason is that equations 3.8) are first-order differential equations and the Dirichlet boundary condition is over-determined.

13 12 MA AND WANG Hence, for the massless particle system with the boundary condition 3.7), we have to consider the relation 3.9) P E = cp 2, which is of first order in t. It is nown that the operator P E is Hermitian if It implies P E should be in the form P E = EP. 3.10) P E = P E P V + V P ), E 0 = i t. Thus the boundary problem for a massless system is generated by 3.9) with 3.10), and is given in the following general form ψ σ D) 3.11) t = c σ D) 2 ψ ig [ σ 2 D)A 0 + A 0 σ D) ] ψ, ψ = 0, where R n is a bounded domain, and D is as in 3.6) Spectral theory of Weyl operators. The equations 3.11) describe bound states of a massless particle system. The solutions ψ of the bound state equations 3.11) tae in the following form ) ϕ 1 ψ = e iλt/ ϕ, ϕ = Then equations 3.11) are reduced to the following eigenvalue problem c σ 3.12) D ) 2 ig ϕ + 2 { σ D), A 0 }ϕ = iλ σ D)ϕ, ϕ = 0, where { σ D), A 0 } is the anti-commutator defined by ϕ 2 { σ D), A 0 } = σ D)A 0 + A 0 σ D). Definition 3.1. A real number λ and a two-component wave function ϕ H0 1, C 2 ) are called the eigenvalue and eigenfunction of 3.12), if λ, ϕ) satisfies 3.12) and 3.13) ϕ [i σ D)ϕ]dx > 0. Remar 3.1. The physical significance of 3.13) is that the inetic energy E = cp is positive, i.e. E > 0. Remar 3.2. The equation 3.12) is essentially an eigenvalue problem of the first order differential operator: i c σ D) + ga 0, which is called the Weyl operator. In addition, the operator σ D) 2 is elliptic and can be rewritten as 3.14) σ D) 2 = D 2 g c σ curl A. The ellipticity of 3.14) ensures the existence of eigenvalues of 3.12)..

14 13 Note also that A = A 1, A 2, A 3 ) represents the magnetic-component of the wea or strong interaction. Hence, in 3.13), the term g σ curl A stands for magnetic energy of the wea or the strong interaction. Theorem 3.1. For 3.12), the following assertions hold true: 1) The eigenvalues of 3.12) are real, with finite multiplicities, and satisfy < λ 1 λ n, λ n for n. 2) The eigenfunctions are orthogonal in the sense that 3.15) [i σ D)ψ ] l dx = 0 n l ψ n 3) The number of negative eigenvalues is finite < λ 1 λ N < 0. 4) Equations 3.12) have negative eigenvalues if and only if there exists a function ϕ H0 1, C 2 ) satisfying 3.13) such that 3.16) [ c σ D)ϕ 2 + ig2 { ϕ σ D), } ] A 0 ϕ dx < 0. Proof. It is clear that the operator L = i σ D) : H 1 0, C 2 ) L 2, C 2 ), is a Hermitian operator. Consider a functional F : H0 1, C 2 ) R defined by [ F ψ) = c Lψ 2 + ig ] 2 ψ { σ D), A 0 }ψ dx. By 3.14), the operator L 2 = σ D) 2 is elliptic. Hence F is has the following lower bound on S: { } S = ψ H0 1, C 2 ) ψ Lψdx = 1. Namely, min F ψ) >. ψ S Based on the Lagrange multiplier theorem of constraint minimization, the first eigenvalue λ 1 and the first eigenfunction ψ 1 S satisfy 3.17) λ 1 = F ψ 1 ) = min ψ S F ψ). In addition, if λ 1 λ 2 λ m are the first m eigenvalues with eigenfunctions ϕ, 1 m, then we have 3.18) λ m+1 = F ψ m+1 ) = min F ψ), ψ S,ψ Hm where H m = span{ψ 1,, ψ m }, and Hm is the orthogonal complement of H m in the sense of 3.15). Assertions 1)-4) follow from 3.17) and 3.18), and the proof is complete.

15 14 MA AND WANG 3.3. Estimates on number of negative eigenvalues. If the interaction potential A µ = { K, 0, 0, 0} and K > 0 is a constant, then the equation 3.12) becomes 3.19) ϕ = iλ + K) σ )ϕ in R n, ϕ = 0 on. It is clear that the number N of negative eigenvalues of 3.19) satisfies that 3.20) β N < K β N+1, and β is the -th eigenvalue of the equation 3.21) ϕ = iβ σ )ϕ, ϕ = 0. For 3.21) we have the Weyl asymptotic relation as 3.22) β N β 1 N 1/n. Hence from 3.20) and 3.21) we obtain an estimate on the number N of of negative eigenvalues: ) n K 3.23) N, where β 1 is the first eigenvalue of 3.21). Remar 3.3. For the mediators such as the photons and gluons, the number N in 3.23) is A ρ 1 g 2 ) 3 w N =, β 1 ρ c where ρ 1, ρ are as in Remar 2.3, β 1 is the first eigenvalue of 3.21) on the unit ball = B 1, and A is a constant. Part 2. Physics 4. Spectral Equations of Subatomic Particles 4.1. Constituents of subatomic particles. We first recall the weaton model introduced in [6]. The starting point of the model is the puzzling decay and reaction behavior of subatomic particles. For example, the electron radiations and the electron-positron annihilation into photons or quar-antiquar pair clearly shows that there must be interior structure of electrons, and the constituents of an electron contribute to the maing of photon or the quar in the hadrons formed in the process. In fact, all sub-atomic decays and reactions show clearly that there must be interior structure of charged leptons, quars and mediators. A careful examination of these subatomic decays and reactions leads us to propose a weaton model in [6], which is briefly recapitulated as follows. The weaton model is basic for the energy level theory. Weatons. There are six elementary particles, which we call weatons, and their anti-particles: 4.1) β 1 w, w 1, w 2, ν e, ν µ, ν τ, w, w 1, w 2, ν e, ν µ, ν τ,

16 where ν e, ν µ, ν τ are the three generation neutrinos, and w, w 1, w 2 are three new particles, called w-weatons. These are massless, spin- 1 2 particles with one unit of wea charge g w. Both w and w are the only weatons carrying strong charge g s. Six classes of subatomic particles. 1) Charged Leptons: 2) Quars: 3) Baryons: 4) Mesons: 5) Internal Bosons: 6) Mediators: e ±, µ ±, τ ±. u, d, s, c, b, t, ū, d, s, c, b, t. p, n, Λ, σ ±, Σ 0, ++, ±, 0, Ξ ±, Ξ 0, etc. π ±, π 0, K ±, K 0, η, ρ ±, ρ 0, K,±, K 0, ω, etc. W ±, Z 0, H ±, H 0 H ±,0 the Higgs particles). γ, g, ν, φ γ, φ g the dual bosons of γ and g). Constituents of subatomic particles. 1) The weaton constituents of charged leptons and quars are given by 4.2) e = ν e w 1 w 2, µ = ν µ w 1 w 2, τ = ν τ w 1 w 2, u = w w 1 w 1, c = w w 2 w 2, t = w w 2 w 2, d = w w 1 w 2, s = w w 1 w 2, b = w w 1 w 2, where c, t and d, s, b are distinguished by their spin arrangements as follows: u = w w 1 w 1,,, ), t = w w 2 w 2, ), s = w w 1 w 2, ), c = w w 2 w 2, ), d = w w 1 w 2, ), b = w w 1 w 2, ). 2) Hadrons include baryons and mesons. Baryons consist of three quars, and mesons consist of a quar and an antiquar: 4.3) Baryons = qqq, Mesons = q q. 4.4) 4.5) 3) Internal bosons consist of and the dual particles W + = w 1 w 2, ), W = w 1 w 2, ), Z 0 = cos θ w w 2 w 2 + sin θ w w 1 w 1, ), H + = w 1 w 2, ), H = w 1 w 2, ), H 0 = cos θ w w 2 w 2 + sin θ w w 1 w 1, ). 15

17 16 MA AND WANG 4.6) where θ w = is the Weinberg angle. 4) Mediators: γ = cos θ w w 1 w 1 sin θ w w 2 w 2, ), φ γ = cos θ w w 1 w 1 sin θ w w 2 w 2, ), ν = l α l ν l ν l ), g = w w, ), φ g = w w, ), αl 2 = Bound potentials of subatomic particles. The main bound energy holding the weatons and quars to form subatomic particles is due to the wea and strong interactions. According to the unified field theory developed in [6], the sources of the wea and strong forces are the strong and wea charges: g w g s the wea charge, the strong charge. The gauge theories for strong and wea interaction involve 3 wea interaction gauge potentials W a µ a = 1, 2, 3), and 8 strong interaction gauge potentials S µ = 1,, 8). The PRI applied to both gauge theories leads to the following 4-dimensional potentials: 4.7) W µ = α a W a µ = W 0, W 1, W 2, W 3 ), S µ = β S µ = S 0, S 1, S 2, S 3 ), where W a µ 1 a 3) and S µ 1 8) are the SU2) and SU3) gauge potentials, α a 1 a 3) and β 1 8) are the SU2) and SU3) tensors. The acting forces are given by 4.8) and the magnetic forces are 4.9) Wea force = g w W 0, Strong force = g s S 0, Wea magnetism = g w curl W, Strong magnetism = g s curl S, where W 0, W ) and S 0, S) are as in 4.7). By 4.8) and 4.9), the wea and strong potentials 4.7) generate the bound states 4.2)-4.6) of subatomic particles, with corresponding interactions given as follows: Charged Leptons and Quars: wea interaction, 4.10) Hadrons: Internal Bosons: Gluons g and dual gluons Φ g : Other Mediators: l strong interaction, wea interaction, wea and strong interactions, wea interaction. Wea interaction potentials. The layered wea interaction potentials are [ 1 W 0 = g w ρ)e r/r0 r A w 1 + 2r ) ] e r/r0, ρ r ) ) 3 ρw g w ρ) = g w, ρ

18 17 where ρ w and ρ are the radii of the constituent weaton and the particle, A w is a constant depending on the types of particles, and r 0 is the radius of wea interaction: r cm. Based on 4.11), the wea potential generated by two particles with N 1, N 2 wea charges and radii ρ 1, ρ 2 is given by [ ) Φ w = N 1 N 2 g w ρ 1 )g w ρ 2 )e r/r0 r Āw 1 + 2r ) ] e r/r0, ρ1 ρ 2 r 0 where Āw is a constant depending on the two particles, and g w ρ ) = 1, 2) are as in 4.11). According to the standard model, the coupling constant G w of the wea interaction and the Fermi constant G f have the following relation G 2 w = 8 mw c ) 2 Gf, 4.13) 2 G f = 10 5 mp c ) 2 c/, where m w and m p are the masses of the W ± particles and the proton. gauge theory, we have 4.14) g w ρ n ) = G w. It follows from 4.13) and 4.14) that 1 gwρ 2 n ) = G 2 w = 8 ) 2 mw 10 5 c. 2 m p Then, by 4.11), we deduce the wea charge g w as ) ) gw 2 ρn = 0.63 c, where ρ n is the nucleon radius. ρ w By the Strong interaction potentials. The layered strong interaction potentials are [ 1 S 0 = g s ρ) r A s 1 + r ) ] e r/r, ρ R 4.16) ) 3 ρw g s ρ) = g s, ρ where A s is a constant depending on the particle type, and R is the attracting radius of strong interactions given by { cm for w and quars, R = cm for hadrons. 1 In our previous wor [6], we called gwρ n) and g sρ n) as the basic units for wea and strong charges. However, with the weaton theory of elementary particles [7, 6], it is more natural to g wρ w) and g sρ w) as the basic units of wea and strong charges.

19 18 MA AND WANG By the Yuawa theory, we deduce that 4.17) g 2 s = 2 38 e e) ρn ρ w ) 6 g 2, where e is the base of the natural logarithm, g is the classical coupled constant of strong interactions between nucleons, and experimentally g c. The strong interaction potential between two particles with N 1, N 2 charges g s and radii ρ 1, ρ 2 is given by 4.18) Φ s = N 1 N 2 g s ρ 1 )g s ρ 2 ) [ 1 r Ās 1 + r ) ] e r/r, ρ1 ρ 2 R where g s ρ ) = 1, 2) are as in 4.16), and Ās is a constant depending on the types of two particle involved. The wea and strong forces F w and F s between two particles are given by F w = Φ w, F s = Φ s Wave equations of bound states. We infer from 4.2)-4.6) that a subatomic particle consists of two or three fermions with wave functions: 4.19) Ψ 1,, Ψ N for N = 2, 3. Because the bound energy of each fermion is approximately the superposition of the remaining N 1 particles. Thus the bound potential for each fermion taes the form { N 1)Wµ for wea interaction, 4.20) A µ = N 1)S µ for strong interaction, where W µ, S µ are as in 4.7). Dynamic equations of bound states for massless particles. The wave equations for the massless case are the modified Weyl equations given by 3.11). Dynamic equations of bound states for massive particles. constituent fermions be given as follows m 1 0 m = m N Let the masses of the Then the N wave functions 4.19) satisfy the following Dirac equation 4.21) iγ µ D µ mc ) Ψ = 0, where Ψ = Ψ 1,, Ψ N ) T, and D µ = µ + i g c A µ with A µ be given in 4.20). We now that each wave function Ψ in 4.19) is a four-component Dirac spinor: Ψ = Ψ 1, Ψ 2, Ψ 3, Ψ 4) T for 1 N.

20 19 Hence equations 4.21) tae the following equivalent form i ) ) ) t ga Ψ 0 c 2 1 Ψ m = ic σ D) 3, 4.22) Ψ 2 i ) ) t ga Ψ 0 + c 2 3 m Ψ 4 where σ = σ 1, σ 2, σ 3 ) is the Pauli matrix operator. Ψ 4 ) Ψ = ic σ D) 1, 4.4. Spectral equations of bound states. We now derive spectral equations for both massive and massless bound states from 4.22) and 3.11). A. Massive bound states. Consider the case where m 0. Let the solutions of 4.22) be written in the form as Ψ = e iλ+m c 2 )t/ ψ. Then equations 4.22) become ) ) ψ 1 ψ λ ga 0 ) = ic σ D) ), 4.24) ψ 2 ψ 4 Ψ 2 ) ) ψ λ ga 0 + 2m c 2 3 ψ ) = ic σ D) 1, for 1 N. The equations 4.24) can be rewritten as ) ψ ) = i 1 + λ ga ) ) 1 0 ψ 2m c 2m c 2 σ D) 1. ψ 4 In physics, λ is the energy, and λ ga 0 is the inetic energy ψ 4 λ ga 0 = 1 2 m v 2. For the case 2 where v 2 /c 2 0, 4.25) can be approximately expressed as ) ψ 3 = i σ 2m c D ) ) ψ1. ψ 4 Inserting this equation into 4.23), we deduce that ) ) ψ ) λ ga 0 ) = 2 ψ σ 2m D) 2 1. ψ 2 Now, we need to give the expression of σ D) 2. To this end, note that the Pauli matrices satisfy σ 2 = 1, σ σ j = σ j σ = iε jl σ l. Hence we obtain 3 ) ) σ D) 2 = σ D = D 2 + i σ D D). =1 2 Note that the general case can be studied as well involving variable coefficient elliptic operators, and the same results hold true. ψ 2 ψ 2 ψ 2 ψ 2

21 20 MA AND WANG With D = + i g c A, we derive that D D = i g c Note that as an operator we have Hence we get Thus, 4.27) can be written as [ A + A ]. A = curl A A. D D = i g c curl A. 4.28) σ D) 2 = D 2 g c σ curl A. This is the expression given in 3.14). By 4.28), the spectral equations 4.26) are in the form ] 4.29) [ ) ) ) 2 ψ D 2 1 ψ + ga 0 + µ curla 2m 1 ψ 1 = λ, where D = D 1, D 2, D 3 ), A 0, A 1, A 2, A 3 ) is as in 4.20), and ψ ) µ = g 2m σ, D j = j + i g c A j for 1 j 3. The equations 4.29) are supplemented with the Dirichlet boundary conditions: 4.31) ψ 1, ψ 2 ) = 0 where R 3. B. Massless bound states. For massless systems, the spectral equations are given by 3.12), which, by 4.28), can be rewritten as [ cd 2 + g σ curl ] ) ψ1 A + ig { σ ψ2 2 D), } ) ψ1 A 0 ψ2 4.32) = iλ σ D ) ) ψ1 for 1 N, ψ 1, ψ 2 ) = 0, where {A, B} = AB + BA is the anti-commutator. ψ 2 C. Magnetism of wea and strong interactions. In the equations 4.29)-4.30) and 4.32) for bound states, we see that there are terms 4.33) 4.34) µ curl A for massive particle systems, g σ curl A ψ 2 ψ 2 for massless particle systems, where µ = g 2m σ represents magnetic moment, and the term in 4.34) is magnetic force generated by either wea or strong interaction. In other words, in the same spirit as the electric charge e producing magnetism, the wea and strong charges g w, g s can also produce similar effects, which we also call magnetism. Indeed, all three interactions electromagnetic, wea and strong interactions enjoy a common property that moving charges generate magnetism, due mainly to the fact that they are all gauge fields with gauge groups U1), SU2) and SU3), respectively.

22 21 5. Energy Levels of Subatomic Particles 5.1. Charged leptons and quars. In 4.2), charged leptons and quars are made up of three weatons, with masses caused by the deceleration of the constituent weatons. Let the masses of the constituent weatons be m 1, m 2, m 3, and the wave functions of these weatons be given by ) ψ ψ 1 = for = 1, 2, 3. ψ 2 Here ψ 1 and ψ 2 represent the left-hand and right-hand states. The bound states are due to the wea interaction, and the potential in 4.20) taes the form A µ = 2W µ = 2W 0, W ). By 4.29)-4.31), the spectral equations for charged leptons and quars are 2 j + i 2g ) 2 w 2m c W j ψ + 2g w W 0 + µ w curl W )ψ 5.1) = λψ in ρ w < x < ρ, ψ Bρw = 0, ψ Bρ = 0, for 1 3, where ρ w is the weaton radius, ρ is the attracting radius of wea interaction, W 0 is given by 4.11), and µ w = g w 2m σ is the wea magnetic moment. With 5.1), we are in position now to derive a few results on energy levels for charged leptons and quars. A. Bound states and energy levels. We now that the negative eigenvalues and eigenfunctions of 5.1) correspond to the bound energies and bound states. Let 5.2) < λ 1 λ N < 0 be all negative eigenvalues of 5.1) with eigenfunctions ψ 1,, ψ N for ψ n = ψ 1 n, ψ 2 n, ψ 3 n) T. Each bound state ψ n satisfies 5.3) ψn 2 dx = 1 for 1 3, 1 n N. B ρ Then we deduce from 5.1) and 5.3) that λ n = + i g w 5.4) W 2m B ρ c )ψn 2 dx + 2 ψn µ w curla)ψ ndx + 2 g w W 0 ψ n 2 dx. B ρ B ρ In the right-hand side of 5.4), the first term stands for the inetic energy, the second term stands for the wea magnetic energy, and the third term stands for the wea potential energy. By 4.11), the potential energy in 5.4) is negative. Hence the bound energy can be written as 5.5) λ n = inetics energy+ magnetic energy+ potential energy.

23 22 MA AND WANG B. Masses. Each particle corresponds to an energy level E n, and its mass M n and λ n satisfy the following relation 3 5.6) M n = m i + 3g2 w ρ w c 2 + λ n c 2, i=1 where g 2 w/ρ w represents the intrinsic energy of the constituent weaton. C. Parameters of electrons. We recall the wea interaction potential 4.11) for the weatons, which is written as [ 1 5.7) W 0 = g w r A w 1 + 2r ) ] e r/r0 e r/r0. ρ w r 0 where A w is the constant for weatons. Assume that the masses of three constituent weatons are the same, and we ignore the magnetism, i.e. let W = 0. Then 5.1) is reduced to 5.8) 2 ψ + 2g w W 0 ψ = λψ 2m w for ρ w < x < ρ, ψ = 0 for x = ρ w, ρ. We shall apply 5.7) and 5.8) to derive some basic parameters and their relations for bound states of electrons. Let λ e and ψ e be a spectrum and the corresponding bound state of an electron, which satisfy 5.8). It follows from 5.7) and 5.8) that 5.9) λ e = 1 2 m wv 2 + 2g2 w r e 2A wg 2 w ρ w κ e, where 1 2 m wv 2 is the inetic energy of each constituent weaton in the electron, r e is the radius of the naed electron, and κ e is a new parameter for the electron: 1 r e = ψ B ρ r e r/r0 e 2 dr, 5.10) κ e = 1 + 2r ) e 2r/r0 ψ e 2 dx, B ρ r 0 mw v ) 2 = ψ e 2 dx. B ρ These three parameters are related with the energy levels of an electron, i.e. with the λ n and ψ n. The most important case is the lowest energy level state. We shall use the spherical coordinate to discuss the first eigenvalue λ 1 of 5.8). Let the first eigenfunction ψ e be in the form ψ e = ϕ 0 r)y θ, ϕ). Then ϕ 0 and Y satisfy 5.11) and 5.12) 2 1 d d 2m w r 2 dr r2 dr )ϕ 0 + 2g s W 0 ϕ 0 + β r 2 ϕ 0 = λ e ϕ 0, ϕ 0 ρ w ) = ϕ 0 ρ) = 0, [ 1 sin θ sin θ ) + 1 θ θ sin 2 θ 2 ] ϕ 2 Y = β Y,

24 23 where β = + 1), = 0, 1,. Because λ e is the minimal eigenvalue, it implies that β = β 0 = 0 in 5.11). The eigenfunction Y 0 of 5.12) is given by Thus ψ e is as follows Y 0 = 1 4π. 5.13) ψ e = 1 4π ϕ 0 r), and λ e and ϕ 0 are the first eigenvalue and eigenfunction of the following equation 2 1 d 5.14) 2m w r 2 r 2 d ) ϕ 0 + 2g w W 0 ϕ 0 = λ 1 ϕ 0, dr dr ϕ 0 ρ w ) = ϕ 0 ρ) = 0. In this case, the parameters in 5.10) are simplified as 5.15) r e = K e = mw ν ρ 0 ρ 0 rϕ 2 0r)dr, r r ) 2 = ρ 0 r 0 r 2 dϕ0 dr ) e 2r/r0 ϕ 2 0r)dr, ) 2 dr Baryons. By 4.3), baryons consist of three quars 5.16) Baryon = qqq, and by 4.2), quars consist of three w-weatons q = w ww and q = w w w. Each quar possesses one strong charge g s and three wea charges 3g w. It loos as if the bound energy of baryons is provided by both wea and strong interactions. However, since the wea interactions is short-ranged, i.e. and the radii of baryons are range of wea force cm, r > cm. Hence the main force to hold three quars together is the strong interaction. Let m 1, m 2, m 3 be the masses of the three constituent quars in 5.16), and ψ = ψ1, ψ2 ) T 1 3) be the wave functions. Then the spectral equations 4.29)- 4.31) for baryons are in the form 5.17) 2 2m + i 2g s c S ) 2 ψ + 2g s S 0 ψ + 2 µ s curl Sψ = λψ for 1 3, ρ 0 < x < ρ 1, ψ = 0 for x = ρ 0, ρ 1,

25 24 MA AND WANG where ρ 0 is the quar radius, ρ 1 is the strong attracting radius, S µ = S 0, S) as in 4.7) is a 4-dimensional strong potential, and µ s = g w σ/2m is the strong magnetic moment. The wave functions ψ are normalized: ψ dx = 1 with = {x R 3 ρ 0 < x < ρ 1 }. By 4.16), the strong interaction potential for quars is taen in the form ) 3 [ ρw ) S 0 = g s ρ 0 r A ) ] s 1 + rr0 e r/r0, ρ 0 where r 0 = cm. Since the proton has a long lifetime, we are interested in its physical parameters. As in the case of electrons, we also have the following formula for proton: λ p = 1 [ 2g 2 m qv s 2A sg 2 ] ) 6 s ρw κ p, r p ρ 0 ρ ) r p = κ p = mq v ρ1 0 ρ1 0 rϕ 2 pr)dr, r r r 0 ) e r/r0 ϕ 2 pr)dr, ) 2 = ρ1 0 ) 2 r 2 dϕp dr, dr where m q is the quar mass, r p is the proton radius, A s as in 5.18), and λ p, ϕ p are the first eigenvalue and eigenfunctions of the following equation 2 1 d 5.20) 2m q r 2 r 2 d ) ϕ p + 2g s S 0 ϕ p = λ p ϕ p, dr dr ϕ p = 0 for r = ρ 0, ρ Mediators. By 4.6), the constituents of mediators consist of two weatons. In this section, we only consider gluons and photon. Gluons. The weaton constituents of gluons are given by 5.21) g = w w. Based on the weaton model, w contains a wea charge g w and a strong charge g s. By 4.15) and 4.17), g w and g s have the same order. Therefore, the interactions for gluons 5.21) are both wea and strong forces, i.e. the 4-dimensional potential A µ = A 0, A) is given by 5.22) ga 0 = g w W 0 + g s S 0, g A = g w W + gs S. In 5.21), we only need to consider the bound states for a single weaton, i.e. N = 1 in 4.32). Then the spectral equations are written as [ cd 2 ψ + g w σ curlw + g s σ curls ] ψ + i 2 g w { σ D), } W 0 ψ 5.23) + i 2 g s{ σ D), S 0 }ψ = iλ σ D)ψ for ρ w < x < ρ g, ψ = 0 for x = ρ w, ρ g,

26 25 where ρ w, ρ g are the radii of weatons and gluons, and D = + ig c A, where g, A are given in 5.22). Photons. The photon consists of a pair of weaton and anti-weaton: γ = w w. Because the weatons w and w only contain a wea charge, the interaction in γ is the wea force, i.e. A µ = W µ = W 0, W ). In this case, the spectral equations for a photon are in the form 5.24) c + ig w c W ) 2 ψ + g w σ curl W ψ + i 2 g w{[ σ + ig w c W )], W 0 }ψ = iλ[ σ + ig w c W )]ψ for ψ w < x < ρ γ, ψ = 0 on x = ρ w, ρ γ, where ρ γ is the photon radius Number of energy levels. Based on the spectral theorems, Theorems 2.1 and 3.1, number of energy levels of a subatomic particle is finite. Hence all subatomic particles are in finite and discrete energy states. By 2.21) and 3.23), we can derive estimates for the numbers of energy levels for various subatomic particles. Leptons and quars. To estimate the numbers of energy levels, we always ignore the effect of magnetism, i.e., we set W = 0 in 5.1). In this case the spectral equations 5.1) are reduced to 5.25) 2 ψ + 2g w W 0 ψ = λψ 2m w for ρ w < x < ρ 0, ψ = 0 on x = ρ w, ρ 0, where m w is the mass of the constituent weatons. Formulas in 4.11) for weatons can be approximatively written as 5.26) W 0 = g wa w ρ w. Tae a scaling transformation 5.27) x = ρ 0 x 1, where ρ 0, as in 5.25), is the attracting radius of the wea interaction. Note that the weaton radius ρ w is much smaller than ρ 0, 5.28) ρ w ρ 0, Hence, by 5.26)-5.27), the problem 5.25) can be approximately expressed in the form 5.29) ψ + 4m wa w ρ g ρ wψ 2 = λψ w for x < 1, ψ = 0 on x = 1.

27 26 MA AND WANG It is clear that the parameters in 2.21) are given by θ = 4m wa w ρ ρ w g 2 w, r = 1, α = 0. Thus the number N of the energy levels for charged leptons and quars is approximately given by [ 4 ρ ) N = 0A w m w c g 2 ] 3/2 w, λ 1 ρ w c where λ 1 is the first eigenvalue of in the unit ball B 1 R 3. Baryons. For baryons, similar to 5.29) we can transform the spectral equations 5.17) into the following form 5.31) ψ + 4m qa s ρ g ρ sψ 2 = λψ q for x < 1, ψ = 0 on x = 1, where ρ q is the quar radius, ρ 1 is the attracting radius of strong interaction, m q is the quar mass, and A s is the constant as in 5.18). Then, by 5.31), we derive an estimate of N: [ 4 ρ ) N = 1A s m q c g 2 ] 3/2 s, λ 1 ρ q c where λ 1 is the same as in 5.30). Photons. For simplicity, here we only consider the photon. Then the spectral equations 5.24) are reduced to ϕ = i λ + A ) wρ γ gw 2 σ 5.33) cρ )ϕ for x < 1, w ϕ = 0 on x = 1, where ρ γ is the photon radius. Then, by 5.33), we see that the parameter K in 3.23) is K = A wρ γ ρ w g 2 w c. Thus we derive from 3.23) an estimate for the number of photon energy levels: [ Aw ρ γ g 2 ] 3 w 5.34) N =, ρ w c where β 1 is the first eigenvalue of 3.21) in B 1 R 3. β Physical conclusions. To compute the numbers N in 5.30), 5.32) and 5.34), we need to recall the values of gw 2 and gs. 2 By 4.15) we have g 2 ) 6 w 5.35) c = 0.63 ρn, and in 4.17) taing g 2 = 5 c, then we have ρ w 5.36) g 2 s c = 0.32 ρn ρ w ) 6,

28 27 where ρ n is the radius of nucleons. Based on the energy level theory for subatomic particles established above, we deduce the following physical conclusions. First, the energy levels for each subatomic particle are discrete, and the number of energy levels is finite: 0 < E 1 < < E N <, and each energy level E n corresponds to an eigenvalue λ n of the related eigenvalue problem as 5.23) or 5.24). Physically, λ n represents the bound energy of the particle, and is related to the energy level E n with the following relation: 5.37) E n = E 0 + λ n for 1 n N. Here E 0 is the intrinsic potential energy of the constituent weatons, which is given by g 2 w/ρ w. Second, by 5.37), we deduce immediately the following upper and lower bounds of the energy levels of sub-atomic particles: 5.38) E 0 + λ 1 E n < E 0 for 1 n N. It is clear that the largest and smallest energy levels are given by 5.39) E max = E 0 + λ N, E min = E 0 + λ 1. The total energy level difference is E max E min = λ N λ 1, and the average energy level gradient for two adjacent energy levels) is approximately given by E = E max E min λ 1 N N. In particular, for photons, by λ 1 K c/ρ γ, the energy gradient can be estimated by E c ρ γ K 2 1 ρ γ c ) ev, g 2 w c if we tae the following approximate values of the related parameters: ρ γ = ρ n cm, ρ w One important consequence of finite and discrete energy levels for sub-atomic particles is that the frequency is also discrete and finite, and are given by ω n = E n / n = 1,, N). This result shall play an important role in field quantizations. References [1] S. Agmon, Lectures on elliptic boundary value problems, Van Nostrand, [2] W. Greiner, Relativistic quantum mechanics, 2000). [3] L. Landau, On the conservation laws for wea interactions, Nuclear Physics, ), pp [4] T. D. Lee and C. N. Yang, Parity nonconservation and a two-component theory of the neutrino, Phys. Rev., ), p [5] T. Ma and S. Wang, Unified field equations coupling four forces and principle of interaction dynamics, arxiv: , 2012).

29 28 MA AND WANG [6], Unified field theory and principle of representation invariance, arxiv: ; version 1 appeared in Applied Mathematics and Optimization, DOI: /s , 33pp., 2012). [7], Weaton model of elementary particles and decay mechanisms, Indiana University Institute for Scientific Computing and Applied Mathematics Preprint Series, #1304: iscam/preprint/1304.pdf, May 30, 2013). [8] A. Salam, On parity conservation and neutrino mass, Nuovo Cimento, ), pp [9] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, vol. 68 of Applied Mathematical Sciences, Springer-Verlag, New Yor, second ed., [10] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., ), pp [11] H. Weyl, Electron and gravitation, Z. Phys., ), pp TM) Department of Mathematics, Sichuan University, Chengdu, P. R. China SW) Department of Mathematics, Indiana University, Bloomington, IN address: showang@indiana.edu, fluid