Non-relativistic Limit of Dirac Equations in Gravitational Field and Quantum Effects of Gravity

Commun. Theor. Phys. Beijing, China) 45 26) pp. 452 456 c International Academic Publishers Vol. 45, No. 3, March 15, 26 Non-relativistic Limit of Dirac Equations in Gravitational Field and Quantum Effects of Gravity WU Ning Institute of High Energy Physics, P.O. Box 918-1, Beijing 149, China Received June 15, 25) Abstract Based on unified theory of electromagnetic interactions and gravitational interactions, the non-relativistic limit of the equation of motion of a charged Dirac particle in gravitational field is studied. From the Schrödinger equation obtained from this non-relativistic limit, we can see that the classical Newtonian gravitational potential appears as a part of the potential in the Schrödinger equation, which can explain the gravitational phase effects found in COW experiments. And because of this Newtonian gravitational potential, a quantum particle in the earth s gravitational field may form a gravitationally bound quantized state, which has already been detected in experiments. Three different kinds of phase effects related to gravitational interactions are studied in this paper, and these phase effects should be observable in some astrophysical processes. Besides, there exists direct coupling between gravitomagnetic field and quantum spin, and radiation caused by this coupling can be used to directly determine the gravitomagnetic field on the surface of a star. PACS numbers: 4.6.-m, 11.15.-q, 4.8.Cc Key words: quantum gravity, gauge field, quantum effects of gravity 1 Introduction For a long time, experimental studies of gravity are mainly concentrated on classical tests of gravity, which aim to test classical Newton s theory of gravity and Einstein s general relativity. Of course, both classical Newton s theory of gravity and Einstein s general relativity have met with brilliant series of successes in describing phenomena related to classical gravitational interactions. However, just to study gravity at classical level is not enough for us to understand the nature of gravity. From the point of view of quantum field theory, interactions are transmitted by some kinds of fields. After quantization, such fields consist of elementary particles. For example, electromagnetic field consists of photons. Similarly, gravitational field should consist of gravitons. In order to study gravity in quantum mechanics, a quantum theory of gravity which is perturbatively renormalizable is necessary. And in order to test the theory of gravity at quantum level, experimental study on quantum effects of gravity is highly essentially important. Without doubt, studying gravity at quantum level will push our further understanding on the nature of gravity, and should be one of the main streams of the future study on the nature of gravity from the physical point of view. Up to now, there are also some experimental studies on quantum effects of a quantum state in gravitational field, such as gravitational phase effects found in COW experiments [1 3 and gravitationally bound quantized states found recently. [4,5 Because of the tremendous success of the standard model in elementary particle physics, it is now generally believed that the common nature of the fundamental interactions should be gauge. Based on gauge principle, quantum gauge theory of gravity is proposed to study quantum behavior of gravity in the framework of quantum field theory. [6 11 At classical level, quantum gauge theory of gravity gives Einstein s general relativity, [6,7 and can explain classical tests of gravity. [12 At quantum level, it is a perturbatively renormalizable quantum theory of gravity, and can be used to explain quantum behavior of gravitational interactions. In contrast with the traditional gauge gravity, quantum gauge theory of gravity is a physical theory of gravity, not a geometric theory of gravity, in which four kinds of fundamental interactions in nature can be unified in a simple and beautiful way. [13 15 Massive gravitons can also be used to explain possible origin of dark matter and dark energy. [16,17 Gravitational phase effects [1 3 and gravitationally bound quantized states [4,5 can also be explained in a more natural and fundamental way, which will be discussed in this paper. Besides, there exists another interesting quantum phenomenon of gravity, that is, after considering spontaneously symmetry breaking, non-homogenous super currents may shield gravity. [18 Gravitational shielding effects may have already been found in experiments. [19,2 The close analogy between Newton s gravitation law and Coulomb s law of electricity led to the birth of the concept of gravitomagnetism in the nineteenth century. [21 24 Later, in the framework of General Relativity, gravitomagnetism was extensively explored [25 27 and recently some experiments were designed to test gravitomagnetism effects. Some recently reviews on gravitomagnetism can be found in literatures. [28 3 In quantum gauge theory of gravity, gravitoelectric field and gravitomagnetic field are E-mail: wuning@mail.ihep.ac.cn

No. 3 Non-relativistic Limit of Dirac Equations in Gravitational Field and Quantum Effects of Gravity 453 naturally defined as components of field strength of gravitational gauge field. In this paper, non-relativistic limit of Dirac equations in gravitational field is studied, which is the basis in quantum gauge theory of gravity to explain gravitational phase effects found in COW experiments [1 3 and gravitationally bound quantized states found recently. [4,5 Besides, two new kinds of gravitational phase effects, including gravitational Aharonov Bohm effect, are also discussed. The explanation given in this paper is from first principle, not just explained by phenomenologically introducing classical Newtonian gravitational potential into Schrödinger equation. 2 Unified Theory of Electromagnetic Interactions and Gravitational Interactions First, for the sake of integrity, we give a simple introduction to gravitational gauge theory and introduce some notations that are used in this paper. Details on quantum gauge theory of gravity can be found in literatures. [6 11 In gauge theory of gravity, the most fundamental quantity is gravitational gauge field C µ x), which is the gauge potential corresponding to gravitational gauge symmetry. Gauge field C µ x) is a vector in the corresponding Lie algebra, which is called the gravitational Lie algebra. So C µ x) can be expanded as C µ x) = C α µ x) ˆP α, µ, α =, 1, 2, 3), 1) C α µ x) is the component field and ˆP α = i / x α is the generator of the global gravitational gauge group. The gravitational gauge covariant derivative is defined by D µ = µ igc µ x) = G α µ α, 2) g is the gravitational coupling constant and matrix G is given by G = G α µ) = δ α µ gc α µ ) = I gc) α µ. 3) Matrix G is an important quantity in gauge theory of gravity. Its inverse matrix is denoted as G 1, G 1 1 = I gc = G 1µ α ). 4) Using matrix G and G 1, we can define two important quantities, g α = η µν G α µg ν, 5) g α = η µν G 1µ α G 1ν. 6) The field strength of gravitational gauge field is defined by F µν 1 ig [D µ, D ν. 7) Its explicit expression is F µν x) = µ C ν x) ν C µ x) igc µ x)c ν x) + igc ν x)c µ x). 8) F µν is also a vector in gravitational Lie algebra and can be expanded as F µν x) = F α µνx) ˆP α, 9) F α µν = µ C α ν ν C α µ gc µ C α ν + gc ν C α µ. 1) The U1) gauge-invariant field strength of electromagnetic field is given by [13,31 A µν is defined by A µν = A µν + gg 1λ α A λ F α µν, 11) A µν = D µ A ν ) D ν A µ ). 12) The symmetry group of the unified theory of electromagnetic interactions and gravitational interactions is the semi-direct product group of U1) Abelian group and gravitational gauge group. We denote it as GU1) U1) s Gravitational Gauge Group = {gx)}. 13) GU1) gauge-covariant derivative is D µ = µ iea µ igc µ, 14) e is the electromagnetic coupling constant. For electron, e should be used instead of e, for it carries negative electric charge. In the unified theory of electromagnetic interactions and gravitational interactions, after considering Dirac field, the Lagrangian is selected to be [6,7,31 L = ψγ µ D µ + m)ψ 1 4 ηµρ η νσ A µν A ρσ C µνρσ α 1 4 Cµνρσ α F α µνf ρσ, 15) = 1 4 ηµρ η νσ g α + 1 2 ηµρ G 1ν G 1σ α η µρ G 1ν α G 1σ. 16) The action of the system is defined by S = d 4 x det g α L. 17) It can be strictly proved that this action has strict local gravitational gauge symmetry and U1) gauge symmetry. From this action, we can deduce the Euler Lagrangian equation of motion of Dirac field, which is [γ µ µ iea µ gc α µ α ) + mψ =. 18) This equation of motion describes the electromagnetic interactions and gravitational interactions of a Dirac particle. In this paper, we will discuss its non-relativistic limit, which is the basis of our understanding on many quantum effects of gravity that are detected by experiments. 3 Non-relativistic Limit of Dirac Equation We start our discussion from Eq. 18), which can be written as another form, Define [γ µ D µ iea µ ) + mψ =. 19) α i = γ γ i, 2)

454 WU Ning Vol. 45 then, we can change Eq. 19) into [D iea ) + α i D i iea i ) γ mψ =. 21) Multiply both sides of Eq. 21) with we will get [D iea ) α i D i iea i ) + γ m, 22) {D iea ) 2 [ α D ie A) 2 + m 2 Define [7,32 + D iea )[ α D ie A) [ α D ie A) D iea )}ψ =. 23) E i = F i = E α i ˆP α, 24) E ei = A i, 25) B i = 1 2 ε ijkf jk = B α i ˆP α, 26) B ei = 1 2 ε ijka jk. 27) Ei α is the gravitoelectric field and Bi α is the gravitomagnetic field, while E ei is the electromagnetic electric field and B ei is the electromagnetic magnetic field. The α = components of gravitoelectromagnetic fields Ei and Bi correspond to those discussed in literatures. [25 3 Using the following relation α A) α B) = A B) + 2i Σ A B), 28) Σ is the spin matrix of Dirac field, we can get [ α D ie A) 2 = D ie A) 2 + 2i Σ [ D ie A) D ie A) = D ie A) 2 2g Σ B + 2e Σ B e. 29) On the other hand, it is easy to prove that D iea )[ α D ie A) [ α D ie A)D iea ) = ig α E + ie α E e. 3) Then, Dirac equation 23) is changed into [D iea ) 2 D ie A) 2 + 2g Σ B 2e Σ B e Define ig α E + ie α E e + m 2 ψ =. 31) ψ = ) ϕ, 32) χ ϕ is the wave function for the Dirac particle and χ is for its antiparticle. Both of them are two-component spinors. Using ) σ α =, 33) σ Σ = 1 ) σ, 34) 2 σ we can obtain the following equation for the field ϕ, [D iea ) 2 D ie A) 2 + g σ B e σ B e + ig σ E ie σ E e + m 2 ϕ =. 35) Equation 35) is a strict relativistic equation. Now, let us discuss its non-relativistic limit. Suppose that the moving speed of the Dirac particle is slow, then, in nonrelativistic limit, we can define ϕ x, t) = Ψ x, t) e imt, 36) Ψ x, t) satisfies the Schrödinger equation i Ψ x, t) = EΨ x, t) 37) t with E the kinematical energy of the Dirac particle, which is much smaller than its mass m. Using the following equation, we can change Eq. 35) into i Ψ [ 1 t = 2m D iea ) 2 ψ = {[ 2im 2emA m 2 1 2gC )Ψ} e imt, 38) D ie A) 2 ψ = {[ D ie A + img C 2 Ψ} e imt, 39) i D e A + mg C ) 2 + mgc ea g 2 σ B e 2m σ B e ig 2 σ E ie 2m σ E e Ψ, 4) C = C i ). 41) It can be seen that the classical Newtonian gravitational potential naturally enters the Schrödinger equation. Besides, there is direct coupling between spin and gravitomagnetic field, no matter whether the Dirac particle carries electric charge or not. 4 Quantum Effects of Gravity In most cases, the gravitational field is very weak, for example, on the earth, gc = GM r 7 1 1. 42) In these cases, the leading order approximation gives D Ψ = d dt Ψ, D iψ = x i Ψ. 43) For neutron, it carries no electric charge, therefore, equation 4) is changed into i d dt Ψ = [ 1 2m i + mg C ) 2 + mgc g 2 σ B ig 2 σ E Ψ. 44) In the above equation, the Planck constant is clearly written out. Now, starting from this Schrödinger equation, we will discuss related quantum effects of gravity. From Eq. 44),

No. 3 Non-relativistic Limit of Dirac Equations in Gravitational Field and Quantum Effects of Gravity 455 we can see that when a neutron is moving in gravitational field, the term mgc will contribute a phase factor δφ 1 = mg r C r ) d r. 45) Suppose that there is an interference neutron beam split into two beams at point A and recombined at point B. One beam goes along the path C 1 and another goes along the path C 2. When two beams recombined at point B, the phase difference between two beams will be δφ 1 = mg ) C r ) d r C 1 C 2 = mg C r ) d r, 46) C C is the closed path formed by path C 1 and C 2. Define Φ = C r ) d r = B d σ, 47) C therefore, the phase difference can be written as δφ 1 = mg c Φ. 48) This is the gravitational Aharonov Bohm effect. The classical Newtonian gravitational potential mgc will contribute another phase factor, δφ 2 = mg C dt = mg C dx v, 49) v is the velocity of the neutron. For the closed path C discussed above, the classical Newtonian gravitational potential mgc will cause a phase difference, δφ 2 = mg ) C dx. 5) C 1 C 2 v δφ 2 is just the gravitational phase detected in COW experiments. [1 3 The spin coupling term g/2) σ B in the Schrödinger equation 44) will contribute the third phase factor δφ 3 = g σ 2 B dt = g dx σ B 2 v. 51) For the closed path C, this term will cause a phase differernce δφ 3 = g 2 C 1 C 2 ) σ B dx. 52) v This phase originates from the coupling between spin and gravitomagnetic field. When the moving speed of neutron is very slow and the gravitational field is weak, the magnitude of last two terms on the right-hand side of the Schrodinger equation is much smaller than that of the classical Newtonian gravitational potential mgc. So, the above Schrödinger equation can be further simplified to i d dt Ψ = [ 2 2m + mgc Ψ. 53) This is just the Schrödinger equation that is widely used. For a static problem, this Schrödinger equation becomes 2m + mgc Ψ = E Ψ. 54) In a local region in the earth s surface, the Newtonian gravitational potential is a linear function of the altitude z, that is mgc = V + mg a z, 55) g a is the gravitational acceleration and V is the gravitational potential at the origin. For experiments on the earth, V is the gravitational potential at earth s surface. Then, the Schrödinger equation 54) will be changed into 2m + mg az Ψ = EΨ, 56) E = E V. Its eigenvalue equation is [33 2m + mg az Ψ n = E n Ψ n, 57) the energy eigenvalue E n is 2 mg 2 ) 1/3 [ a 3π E n = n 1 2/3. 58) 2 2 4) The gravitationally bound quantized states found recently [4,5 are just the eigenstates Ψ n of the above Schrödinger equation. When there is strong gravitomagnetic field but no electromagnetic magnetic field, there is coupling between spin and gravitomagnetic field, which has the following coupling energy g 2 σ B. 59) When spin transforms from down to up in gravity, it will radiate the following energy E = g B. 6) Detecting such kind of radiation can directly measure the gravitomagnetic field on the surface of the star. [34 5 Summary and Discussions In this paper, the Schrödinger equation for a Dirac particle in gravitational field is obtained from the nonrelativistic limit of Dirac equations. In this equation, the traditional Newtonian potential appears as the potential of Schrödinger equation, which can lead to the formation of a bound state when a particle is in gravitational field. Besides, there is a direct coupling term between spin and gravitomagnetic field. Possible quantum effects of a particle in gravitational field, including three different kinds of phase effects, are studied. In fact, there are two kinds of quantum effects of gravity, one is the effect of the quantized gravitational field, another is the quantum effect of a particle in gravitational field. In this paper, the second kind of quantum effects of gravity is studied. These effects should be observable in some astrophysical processes. [34,35 Gravitational wave and gravitational shielding effects [18 belong to the effects of quantized gravitational field. From the study of the elementary particle physics, we know that the effective coupling constant of a fundamental

456 WU Ning Vol. 45 interactions is not a constant, it is a function of the energy scale E, that is g = ge). For classical gravitational interactions, gravity is transmitted by virtual graviton. In this case, the effective coupling constant is the infrared limit of ge), that is g). For quantum gravitational interactions, when the energy scale is much larger than zero, the effective coupling constant should be much larger than g), and the quantum effects of the quantized gravitational field can be much larger than traditionally expected. Generally speaking, it is not correct to use Newtonian gravitational coupling constant to estimate the order of the magnitude of the quantum gravitational interactions. For the present situation, it is important to calculate the - function of gravitational interactions based on quantum gauge theory of gravity. References [1 R. Colella, A.W. Overhauser, and S.A. Werner, Phys. Rev. Lett. 34 1975) 1472. [2 S.A. Werner, R. Colella, and A.W. Overhauser, Phys. Rev. Lett. 35 1975) 153. [3 A.W. Overhauser and R. Colella, Phys. Rev. Lett. 33 1974) 1237. [4 V.V. Nesvizhevsky, H.G. Borner, et al., Phys. Rev. D 67 23) 122. [5 A. Westphal, Diploma Thesis, Institute of Physics, University of Heidelberg, July 21). [6 Ning WU, Commun. Theor. Phys. Beijing, China) 42 24) 543. [7 Ning WU, gr-qc/3941. [8 Ning WU, hep-th/19145. [9 Ning WU, Commun. Theor. Phys. Beijing, China) 38 22) 151. [1 Ning WU, hep-th/11262. [11 Ning WU, Quantum Gauge Theory of Gravity, Talk Given at Meeting of the Devision of Particles and Fields of American Physical Society at the College of William and Mary DPF22), May 24 28, 22), Williamsburg, Virgia, USA; hepth/27254; Transparancy can be obtained from: http://dpf22.velopers.net/talks pdf/33talk.pdf. [12 Ning WU and Da-Hua ZHANG, gr-qc/589. [13 Ning WU, Commun. Theor. Phys. Beijing, China) 38 22) 322. [14 Ning WU, Commun. Theor. Phys. Beijing, China) 38 22) 455. [15 Ning WU, Commun. Theor. Phys. Beijing, China) 39 23) 561. [16 Ning WU, Commun. Theor. Phys. Beijing, China) 36 21) 169. [17 Ning WU, Commun. Theor. Phys. Beijing, China) 39 23) 671. [18 Ning WU, Commun. Theor. Phys. Beijing, China) 41 24) 567; hep-th/37225. [19 E. Podkletnov and R. Nieminen, Physica C 23 1992) 271. [2 E. Podkletnov, Weak gravitational shielding properties of composite bulk YBa 2Cu 3O 7 x superconductor below 7 K under electro-magnetic field, report MSU-chem 95, cond-mat/97174). [21 J.C. Maxwell, Phil. Trans. 155 1865) 492. [22 G. Holzmuller, Z. Math. Phys. 15 187) 69. [23 F. Tisserand, Compte Rendu hebdomadaire des sceances de L Academie des sciences 75 1872) 76; 11 189) 313. [24 O. Heaviside, Electromagnetic Theory, The Electrician Printing and Publishing Co., London 1894) Vol. I, Appendix B; O. Heaviside, The Electrician 31 1893) 281, 359. [25 A. Einstein, Phys. Z. 14 1913) 1261. [26 H. Thirring, Phys. Z. 19 1918) 24; 19 1918) 33; 22 1921) 29. [27 J. Lense and H. Thirring, Phys. Z. 19 1918) 156; B. Mashhoon, F.W. Hehl, and D.S. Thesis, Gen. Rel. Grav. 16 1984) 711. [28 M.L. Ruggiero and A. Tartaglia, gr-qc/2765. [29 B. Mashhoon, gr-qc/3113. [3 G. Schafer, gr-qc/47116. [31 Ning WU, Commun. Theor. Phys. Beijing, China) 41 24) 381. [32 Ning WU, Commun. Theor. Phys. Beijing, China) 44 25) 883; gr-qc/5339. [33 Jin-Yan Zeng, Quantum Mechanics, Science Press, Beijing 21) in Chinese). [34 Ning WU and Da-Hua ZHANG, Commun. Theor. Phys. Beijing, China) has been accepted); gr-qc/575. [35 Ning WU, Searching for quantum effects of gravity through astrophysical observations in preparation).