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- 1. Energy Quantization within the Old Quantum Theory Formulation of Gravity in the Schwarzschild case by Amino Adelmo Lavorini aadelmo.lavorini@gmail.com Abstract The physics interpretation of the world is described in terms of fundamental constituents and their interactions. Such description is given in terms of particles with diﬀerent type of charge whose interaction generates all known ﬁelds. Research tell us in Nature exists the hierarchy of fundamental interactions for wich only gravity seems to give no appreciable eﬀects on particle scattering. Simple kinematical arguments will be used to give new perspective to the study of gravitational eﬀects in particle physics whose implications can be more importants. Following the historical development of particle physics, the aim of this paper is to give an expression to the quantization of energy levels for the Schwarzschild ﬁeld. In what follows the old fashion quantum theory prescriptions will be adopted and a very simple expression for the energy levels is given in general and for particular cases. 1 Introduction The strength of interaction between particles is speciﬁed via a dimensionless quantity called coupling constant. In Nature four fundamental interactions are known i.e. Electromagnetic (Em), Weak (W), Strong (S) and Gravitational (G) interaction: their relative strenght is expressed by the following formulas [1] αEm = e2 4πǫ0 c , αW = GF (mc2 )2 ( c)2 , αS(Q2 ) = 1 β0 ln(Q2/Λ2 QCD) , αG = GN m2 4π c , (1) where: GF ( c)3 = 1.1663787(6) × 10−5 GeV −2 , GN = 6.70837(80)( c) × 10−38 GeV c2 −2 , β0 = 11CA − 4TF nf 12π . As illustrated in Ref. [2], GF is known as Fermi coupling constant and GN as gravitational one, β0 is the 1-loop beta-function coeﬃcent, CA = Nc = 3 is the color-factor associated with gluon emission from a gluon and TR = 1/2 is the color factor for a gluon to split to a q¯q pair; ΛQCD = Λ (3) MS = 339(10) MeV is a constant of integration, i.e. the non-perturbative scale of QCD: its value is extracted from the experiments to evaluate the world-average value for αs(M2 Z) in the modiﬁed minimal substraction scheme. A numerical estimation of the coupling constants is given: by using nf = 3, then β0 = 9 4π , and protons as interacting particles one has
- 2. αS(m2 p) αEm 0.686(82) 7.2973525698(24)× 10−3 αW αG 1.026825125(46)×10−5 4.69963(56)×10−39 Table 1: The strength of the Em, W , S and G interaction by using Eq. (1) and Ref. [2] From Tab. 1 the inequalitirs αs(m2 p) > αem > αW >> αG follows, consequently hierarchy of four foundamental forces of Nature tell us how gravity does not play a keyrole in particle physics: as it is well known the Gravitational coupling has a strenght much more weaker compared to the typical ones of the other three; even using LHC energies, 14 T eV in the centre-of-mass system, no man would be able to see gravitational eﬀects in particle interactions. In 1967 Prof. Bryce S. DeWitt published the trilogy of Quantum Theory of Gravity [3, 4, 5], in particular He shown how calculate the diﬀerential cross-section of two identical scalar particles mediated by a single graviton [5] exchange; in the centre-of-mass system and considering extreme relativistic limits was found dσ dΩ ER = 4G2 N E2 cot2 θ 2 + tan2 θ 2 + sin2 θ 4 + 1 2 2 . (2) Eq. (2) shows how hard is to see gravitational eﬀects in a particle collision since dσ dΩ ER ∝ G2 N E2 . However the most important mechanical relativistic applications can be found in the the ﬁeld of nuclear and sub-nuclear physics: in this framework particles collides with typical velocities of order of c, the velocity of light in vacuo. To test the statement above, one can analize the kinematics of Bhabha scattering e+ e− → e+ e− by using the maximum beam energy, for electron, avaible in Ref. [2] as shown in Tab. (2): the kinematics of the process, in the centre-of-mass system with colliding particles moving along the z-axis, gives the following expressions for the energy and (3-)momentum [6]: Ee+ = Ee− = E′ e+ = E′ e− = √ s 2 , |pe+ | = |pe− | = |p′ e+ | = |p′ e− | = s − 4m2 e 2 , (3) the prime is referred to scattered particles. The ratio β = v/c of the outcoming particles, generally expressed by computing ∂E/∂p, with E = p2c2 + m2c4, can be written by means of Eq. (3) as β = 1 − 4m2 e s , √ s = Ee+ + Ee− . (4) VEPP-2000 VEPP-4M BEPC BEPC-II DAΦNE Beam energy [GeV] 1.0 6.0 2.5 1.89÷2.3 0.510 CESR CESR-C LEP ILC CLIC Beam energy [GeV] 6.0 6.0 100÷104.6 250÷500 250÷1500 KEKB PEP-II Super-B SuperKEKB Beam energy [GeV] e− : 8.33 e+ : 3.64 e− : 7÷12 e+ : 2.5÷4 e− : 4.2 e+ : 6.7 e− : 7.0 e+ : 4.0 Table 2: Maximum beam energy from Ref. [2] 2
- 3. By inserting data of Tab. (2) into Eq. (4), results of Tab. (3) below follows: Now, since results of Tab. (3) ∗ VEPP-2000 VEPP-4M BEPC √ s [GeV] 2.0 12.0 5.0 β 0.999999869440039 0.999999996373334 0.999999979110407 BEPC-II DAΦNE CESR √ s [GeV] 3.78÷4.6 1.02 12.0 β 0.999999963450084 0.999999975319479 0.99999949803927 0.999999996373334 CESR-C LEP ILC √ s [GeV] 12.0 200÷209.2 500÷1000 β 0.999999996373334 0.999999999986944 0.999999999988067 0.999999999997911 0.999999999999478 CLIC KEKB PEP-II √ s [GeV] 500÷3000 11.97 9.5÷16.0 β 0.9999999999997911 0.9999999999999942 0.999999996355133 0.999999994213409 0.999999997960001 SuperB SuperKEKB √ s [GeV] 10.9 11.0 β 0.999999995604412 0.999999995683968 Table 3: Velocity values for scattered particles by using Eq. (4), and Tab. 2 and noting that a Gravitational ﬁeld is as weak as the constraint v/c << 1 is satisﬁed [7], One would inquire How can We disregard gravitational eﬀects in the high-energy regime if outcoming particles have β ∼ 1 after a scattering process? In my opinion, the right way to proceed and overcome this diﬃcoulty is to start from ﬁrst principles: we must use Einstein’s equations of gravitational ﬁeld [7, 8] for a point-particle and then to quantize the system † : the quantization of the system in General Relativity start by resolving the gravidynamics equation, the celebrated Wheeler-DeWitt equation [3]: unfortunately there are diﬃcoulties to resolve it, so that how can we proceed now? In this pioneristic work I decided to adopts the prescriptions of the Old Quantum Theory. 2 Bohr-Sommerfeld quantization rule and Schwarzschild solution of the Einstein equations The Old Quantum theory is essentially a general method of calculating quantized quantities, based upon the hypotheses of Bohr and the correspondences principle [9]. The determination of the quantization rules constitues the central problem of the Old Quantum Theory: its solution is the Bohr-Sommerfeld quantization rule pdq = hn (n = 1, 2, · · ·, ∞) , it determines the allowed trajectories of phase space and the corresponding quantized energies. ∗As their consequence we can say that in the limit of high-energies β → 1 †It goes without say to satisfy the statement above one must resolve the scattering problem of a particle in a gravitational ﬁeld for β → 1 3
- 4. Wilson and Sommerfeld have generalized this rule to multiply-periodic system whose motion can be represented by a sequence of functions, namely the conjugate momenta pr, as a function of the coordinate qr only, then the quantization rule are the R relations dqrpr = hnr (r = 1, 2, · · · , R) ; (5) consequently nr deﬁnes the r-th quantized trajectory of the system and the quantized value of the corre- sponding constant of the motion. To use this method, one has to test the periodicity of a given physical system with ﬁnite motion condition is understood fulﬁlled. In particular for systems with several degrees of freedom is necessary a complete separation of variables in the Hamilton-Jacobi’s method [11] then, quantization rules [10] reduces to Eq. (5); In classical mechanics the analogy to quantization rules in the old fashion quantum mechanics is placed of conditionally periodic motion and action variables. 2.1 Motion in a gravitational ﬁeld with central symmetry In this paper the goal is to give the quantizated version of the energy level for the central static gravitational ﬁeld. As it is well known, the solution of such ﬁeld was found by K. Schwarzschild in 1916. He obtains the following expression of the line element ds2 = 1 − rg r c2 dt2 − r2 sin2 θdφ2 + dθ2 − dr2 1 − rg r , rg = 2GN µ c2 . (6) With Eq. (6) one can construct the Lagrangian L = −mc2 1 − rg r 1/2 1 − r2 ( ˙θ2 + sin2 θ ˙φ2 ) c2 1 − rg r − ˙r2 c2 1 − rg r 2 1/2 = −mc2 1 − rg r 1/2 1 − β2 , (7) to calculate the three-momentum components, whose expression in the θ = π 2 plane reads pr = m ˙r 1 − rg r 3/2 1 − β2 , (8) pφ = mr2 ˙φ 1 − rg r 1/2 1 − β2 . (9) The expressions of the angular momentum M and the energy E, i.e. the two conserved quantities of the system, are M = pφ = mr2 ˙φ 1 − rg r 1/2 1 − β2 , E = mc2 1 − rg r 1/2 1 − β2 and, since the line element is ds = 1 − rg r 1/2 cdt 1 − β2, the expressions above reads M = mcr2 dφ ds , (10) E = mc3 1 − rg r dt ds . (11) From Eq. (10), Eq. (11) and Eq. (6) one has the equation of motion for a particle of mass m dr ds 2 = E2 m2c4 − 1 − rg r 1 + M2 m2c2r2 . (12) As it is well known, in a central ﬁeld, the angular momentum can be written as an integral of area. To show this, one can rewrite the expression of the angular momentum, Eq. (10), in the form M = 2mc df ds , (13) 4
- 5. where df/ds is the sectorial velocity, since M is a conserved quantity consequently df/ds is a conserved one as well; this is known as constancy of sectorial velocity [11]. Writing df = M 2mc ds , and using the expression of ds from Eq. (12) one has df = M 2mc dr E2 m2c4 − 1 − rg r 1 + M2 m2c2r2 1/2 . (14) The area described by the orbit can be found integrating Eq. (14) f = M 2mc rmax rmin dr dr E2 m2c4 − 1 − rg r 1 + M2 m2c2r2 1/2 ; (15) whose extremes of integration are the roots of the mass-shell relation in polar coordinates E2 = p2 rc2 + M2 c2 r2 + m2 c4 1 − rg r , (16) i.e. by resolving Ar2 − Cr + B = 0 we have rmin = C 2A − √ C2 − 4AB 2A (17) rmax = C 2A + √ C2 − 4AB 2A (18) where A = p2 rc2 + m2 c4 − E2 , B = M2 c2 , C = m2 c4 rg , . The expression of the period T of revolution for a particle moving in such a ﬁeld, can be found by using Eq. (13), Eq. (11) and Eq. (14); by integration it from 0 to T one has T = rmax rmin dr E mc3 1 − rg r dr E2 m2c4 − 1 − rg r 1 + M2 m2c2r2 1/2 . (19) 3 Quantization procedure At this stage one can apply the Bohr-Sommerfeld quantization rule, the Lagrangian allows separeted variables since the classical system shows the same property, then our system is two fold periodic and its period of revolution was calculated. Since we have two constants of motion then there are two quantized quantities, the ﬁrst one follows from Eq. (9), then pφ dφ = M dφ = hℓ → M = ℓ; (20) as usual ℓ, with ℓ ∈ N, is the azimutal quantum number. To calculate the second one we must have the expression for the radial momentum; it follows from Eq. (16) pr = E2 c2 − M2 r2 − m2c2 1 − rg r , (21) then, by equating it to zero one has two conditions r = 0 , (E2 − m2 c4 )r2 + (m2 c4 rg)r − M2 c2 = 0 , thus r± = − m2 c4 rg 2(E2 − m2c4) ± (m2c4rg)2 + 4(E2 − m2c4)M2c2 2(E2 − m2c4) . (22) 5
- 6. The integral to be calculated reads pr dr = r+ r− dr E2 c2 − M2 r2 − m2c2 1 − rg r = hκ ; (23) here κ is the radial quantum number, it is also an integer number. If we deﬁne x as x = 1/r, then Eq. (23) becomes − dx E2 c2 − m2c2 − M2x2 + m2c2rgx x2 , (24) by introducing the notation α = E2 c2 − m2 c2 , β = m2 c2 rg , γ = −M2 , (25) the integral of Eq. (24) can be written as − dx R(x) x2 , R(x) = α + βx + γx2 ; its solution is [12] − dx R(x) x2 = − − R(x) x + β 2 dx x R(x) + γ dx R(x) , with dx x R(x) = − 1 √ α Artanh 2α + βx 2 αR(x) , and dx R(x) = − 1 √ −γ Arsin 2γx + β √ −∆ ∆ = 4αγ − β2 < 0. Since notations of Eq. (25), Eq. (22) reads r± = − β 2α ± √ −∆ 2α , consequently solution of Eq. (23), before restoring r as variable, has the ﬁnal form, prdr = r R(r) + β 2 √ α Artanh 2α + β/r 2 αR(r) + γ √ −γ Arsin 2γ/r + β √ −∆ r+ r− = hκ . (26) Noting that R(r) = pr, see Eq. (21), and lim x→∞ Artanh(x) = −ıπ 2 , the nonzero term in Eq. (26) is the third one only − √ −γ Arsin 2γ/r+ + β √ −∆ − Arsin 2γ/r− + β √ −∆ = hκ . (27) Since β ∝ rg → 0 then 2γ/r± ≃ ∓ √ 4αγ; deﬁning x = 2γ/r+ + β √ −∆ > 0 , y = 2γ/r− + β √ −∆ < 0 with x2 + y2 ∼ 2 > 1 as β → 0 we can use the expression [12] Arsin(x) − Arsin(y) = −π − Arsin x 1 − y2 − y 1 − x2 . 6
- 7. Since 2γ r± = − 4αγ β ∓ √ −∆ , then Eq. (27) reads 1 √ −∆ β − 4αγ β − √ −∆ 1 − 1 −∆ β − 4αγ β + √ −∆ 2 + − β − 4αγ β + √ −∆ 1 − 1 −∆ β − 4αγ β − √ −∆ 2 = − sin hκ √ −γ ; (28) identity sin(α + β) = sin(α) cos(β) + cos(α) sin(β) was used to keep in account the π-term. At this stage we can also simplify the expression above: writing √ −∆ = β2 − 4αγ = β2 + A2 (γ = −M2 < 0) then by using √ 1 + x ≃ 1 + x/2 with x = β2 /A2 we have β − 4αγ β ± √ −∆ ≃ β + A2 β ± A + β2 2A . For its square and keeping only β2 terms, we have two cases for plus and minus sign respectively; these two have the same solution, in fact since β + A2 β ± β2 + A2 2 ≃ β + A ±1 + β A 2 , and deﬁning x = β/A, we have β + A 1 + β A 2 ∼ β + A 1 − β A 2 = A2 for the plus sign and, β + A −1 + β A 2 = β − A 1 − β A 2 ∼ β − A 1 + β A 2 = A2 for the minus one; in the ﬁnal result, 1/(1 ± x) ≃ 1 ∓ x was used for the ± case respectively. Then 1 − 1 −∆ β − 4αγ β ± √ −∆ 2 ≃ 1 − A2 A2 1 + β2 A2 ≃ β2 A2 = β A , since 1 β2 + A2 ≃ 1 A 1 − β2 2A2 we can write Eq. (28) as β A 1 − β4 4A4 = sin hκ √ −γ . (29) Since Eq. (25) and Eq. (20), √ −γ = ℓ then A2 = 4M2 E2 c2 − m2 c2 = 4 2 ℓ(ℓ + 1) E2 c2 − m2 c2 . It follows A = 2M E2 c2 − m2 c2 1/2 = 2 ℓ E2 c2 − m2 c2 1/2 7
- 8. and A4 = 16M4 E2 c2 − m2 c2 2 = 16 4 ℓ2 (ℓ + 1)2 E2 c2 − m2 c2 2 from ˆl2 |l = 2 ℓ(ℓ + 1)|l then (ˆl2 )2 |l = 4 ℓ2 (ℓ + 1)2 |l . Eq. (29) then reads m2 c2 rg 2 ℓ E2 c2 − m2c2 1/2 1 − (m2 c2 rg)4 64 4ℓ2(ℓ + 1)2 E2 c2 − m2c2 2 = sin 2πκ ℓ . (30) 3.1 Special cases As it is well known sin(α) = sin(α+2πk), (k ∈ Z) then sin 2πκ ℓ = sin 2πn ℓ where n = κ+ℓ is the principal quantum number; n/ℓ = 1 + κ/ℓ, then n/ℓ ≥ 1. If n/ℓ = ξ ∈ N then sin(2πξ) = 0, consequently from Eq. (30) we have 1 − (m2 c2 rg)4 64 4ℓ2(ℓ + 1)2 E2 c2 − m2c2 2 = 0 then En,ℓ = ±mc2 1 + ξm2 µ2 4n(ℓ + 1)m4 P (31) where µ is the mass of the central body and mP is the Planck mass. For n, ℓ → ∞ we have lim n,ℓ→∞ En,ℓ = ±mc2 (Energy of continuous spectrum). For κ = 0 with n/ℓ = 1 for n, ℓ → 0 we have lim n,ℓ→0 En,ℓ = ±mc2 1 + m2 µ2 4m4 P (Energy of ground state). If n/ℓ = α/2, an odd half-integer, with α ∈ N > 2 we have sin(πα) = 0, then En,ℓ = ±mc2 1 + αm2 µ2 8n(ℓ + 1)m4 P . (32) Disregarding the r4 g-term in Eq. (30) we have En,ℓ = ±mc2 1 + m2 µ2 2ℓ(ℓ + 1)m4 P sin2 2πn ℓ , (n/ℓ ∈ Q, ℓ = 1, 2). (33) To draw the spectrum of Schwarzschild ﬁeld we get a particle whose Schwarzschild radius is equal to the Planck length, i.e. a particle with m = mP /2. For the sake of simplicity we also set µ = m; by using Eq. (31), Eq. (32) and Eq. (33) we have n ℓ En,l/mc2 n ℓ En,l/mc2 0 0 1.0625 4 3 1.0035 1 1 1.0078 5 3 1.0035 2 1 1.0078 5 4 1.0016 3 1 1.0078 6 3 1.0013 3 2 1.0026 6 4 1.0008 4 2 1.0026 6 5 1.0012 Table 4: Spectrum of Schwarzschild ﬁeld with m = µ = mP from Eq. (31), Eq. (32) and Eq. (33). 8
- 9. 2 /mcnl E -1 -0.5 0 0.5 1 〉|0,0 〉|1,1 〉|2,1 〉|3,1 〉|3,2 〉|4,2 〉|4,3 〉|5,3 〉|5,4 〉|6,3 〉|6,4 〉|6,5 Dirac Sea Positive Continuos Spectrum Negative Continuos Spectrum 〉|0,0 〉|1,1 〉|2,1 〉|3,1 〉|3,2 〉|4,2 〉|4,3 〉|5,3 〉|5,4 〉|6,3 〉|6,4 〉|6,5 Figure 1: Spectrum of Schwarzschild ﬁeld from Tab. (4): Entire. 2 /mcnlE 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 〉|0,0 〉|1,1 〉|2,1 〉|3,1 〉|3,2 〉|4,2 〉|4,3 〉|5,3 〉|5,4 〉|6,3 〉|6,4 〉|6,5 Positive Continuos Spectrum Figure 2: Spectrum of Schwarzschild ﬁeld from Tab. (4): Positive. 9
- 10. 2 /mcnlE -1.07 -1.06 -1.05 -1.04 -1.03 -1.02 -1.01 -1 -0.99 〉|0,0 〉|1,1 〉|2,1 〉|3,1 〉|3,2 〉|4,2 〉|4,3 〉|5,3 〉|5,4 〉|6,3 〉|6,4 〉|6,5 Negative Continuos Spectrum Figure 3: Spectrum of Schwarzschild ﬁeld from Tab. (4): Negative. 4 Conclusions and Outlooks It is shown with simple kinematical arguments how the concept of weakness of gravity in particle physics interactions must be revised. The naive old fashion quantum theory can be used to give an expression to the quantization of energy levels for the Schwarzschild ﬁeld. The framework used in this paper can be used to quantize the most general expression of the line element ds for the central ﬁeld with special focus on solutions of Einstein equations for point-like particle will be my next goal. I would thank Prof. H. Koppelaar and V. Ny´am´adi for their precious suggestions and warm encourage- ment. References [1] D. H. Perkins, Introduction to High Energy Physics, 4th edition, Cambridge University Press, Cambridge (2000), ISBN 0-521-62196-8. [2] J. Beringer et al. (Particle Data Group), Phys. Rev. D 86, 010001 (2012). [3] B. S. DeWitt, Physical Review 162 5 (1967): Quantum Theory of Gravity. I. The Canonical Theory. [4] B. S. DeWitt, Physical Review 162 5 (1967): Quantum Theory of Gravity. II. The Manifestly Covariant Theory. [5] B. S. DeWitt, Physical Review 162 5 (1967): Quantum Theory of Gravity. III. Applications of the Covariant Theory. [6] V. Barone, Relativity Principles and Applications (in Italian), Ed. Bollati Boringhieri, Bollati Bor- inghieri editore (2004), ISBN 88-339-5757-8. 10
- 11. [7] L. D. Landau, E. M. Lifˇsits, Course of Theoretical Physics vol. II, 3th edition (in Italian), Editori Riuniti, Edizioni Mir (2004), ISBN 88-359-5599-8. [8] A. J. Kox, M. J. Klein, R. Schulmann, The Collected Papers of Albert Einstein, Vol. 6, Doc. 30: The Foundation of the General Theory of Relativity, Princeton University Press, Princeton (1997). [9] A. Messiah, Quantum Mechanics vol. I, Ist edition, Noth-Holland Publishing Company, Amsterdam (1961). [10] L. D. Landau, E. M. Lifˇsits, Course of Theoretical Physics vol. III, IInd edition (in Italian), Editori Riuniti, Edizioni Mir (1994), ISBN 88-359-3474-5. [11] L. D. Landau, E. M. Lifˇsits, Course of Theoretical Physics vol. I, Ist edition (in Italian), Editori Riuniti, Edizioni Mir (1982). [12] I. S. Gradshtetn, I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press (1980). 11