DSpace at VNU: Method for the partial wave scattering problem for the quantum field theory

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DSpace at VNU: Method for the partial wave scattering problem for the quantum field theory

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Method for the partial wave scattering problem for the quantum field theory Nguyen Dinh Thinh Hanoi University of Science, VNU; Faculty of Physics Major: Theoretical Physics - Mathematical Physics Code: 60 44 01 Supervisors: Prof.PhD Nguyen Xuan Han Date of Presenting Thesis: 2011 Abstract Nghiên cứu phương pháp giải phương trình Schrodinger ho ̣c lươ ̣ng tử: phương pháp khai triể n theo sóng riêng phầ n ; phương pháp hàm Green; phương pháp chuẩ n cổ điể n ; mối liên hệ biên độ tán xạ theo sóng riêng phần và biên độ tán xạ eikonal Trình bày sơ đồ mối liên hệ phương pháp bài tốn tán xạ Phân tích các hiê ̣u ứng hấ p dẫn và điê ̣n từ bài toán tán xa ̣ ở lươ ̣ng Plangck tán xa ̣ toàn phầ n toàn phầ n hấ p dẫn; cực điể m tán xạ; tán xạ hấp dẫn có kể thêm tương tác điện từ Keywords Sóng; Vật lý lý thuyết; Tán xạ; Trường lượng tử; Vật lý toán Content In recent years there have been important advances in our understanding of scattering at the Planck scale energy in quantum field theory / 1-10 / To study this process in the theory of quantum gravity will provide a scientific basis to be aware of physical phenomena such as the birth of the singularity and the formation of black holes, the loss of information as well as the improvement variable's string theory of gravity The results obtained are confirmed Planck scattering amplitude of high-energy particles in the size (where s is the energy of the hat, is the Planck mass, - is the gravitational constant) and t-squared pulse of transmission is small, within the limits of the form eikonal representation representation Glauber (leading term) phase depends on energy Additional terms of (non-leading terms) in the scattering problem has been many domestic and foreign scientific research interest over 20 years, including the Department of Theoretical Physics National University of Hanoi Initial results of the -1- Department of Physics theory is to find the first-order terms added to the terms of the amplitude eikonal scattering amplitude in the theory of quantum gravity, using both methods are different methods of analysis distributed gossip content and the standard equation / 8-9 / Finding other methods for this problem is still topical issues - It offers three methods of solving the Schrodinger equation to find the scattering amplitude in which the partial wave method A comparison of three methods to help us have different directions for the scattering problem in quantum mechanics - Method of partial waves used in quantum mechanics is generalized, then it is used to study the scattering problem in the Planck energy theory of quantum gravity I The Schrodinger equation method 1.Phuong method developed by the partial wave Schrodinger equation: Total wave function describing the motion of the r ù r r é h2 êúy (r ) = E y (r ) Ñ + U ( r ) ê 2m ú ë û particle toparticle scattering and at large distances (r> a) for interest equal to the sum of scattering and wave to the scattered wave: r e ikr ikz Y(r ) = e + f ( q, j ) r Return to R equations obtained radial equation of the form: ö m d ổ ỗỗr dR ữ ữữ 2R+ lR = r dr ỗố dr ữ ứ r scattering amplitude in partial waves ¥ f ( q) = å gl Pl (cos q) = l= -2- 2ik ¥ å l= (2l + 1)(e 2i dl - 1)Pl (cos q) Green function method Schrodinger equation: r r ur éÑ + k ùy (r ) = U (r )y (r ) ú ëê û , Equation can be rewritten as integral equation: r r y (r ) = f (r ) + r ur r r d r ' G ( r , r ') U ( r ') y ( r ') ò , Under the boundary conditions, the wave function must includetwo components: component waves to the plane wave traveling inthe positive z axis and the rest is scattered spherical wave Sorewritten as: ur r r i k r y (r ) = A0e 4p r r ik r - r ' r r e ò d r ' r r U (r )y (r ') r- r' Scattering amplitude in partial waves f (0) ( q) = k i ò ¥ b ' db ' J 0(kb ' q) éêe i c (b ') - 1ù ú ë û The standard method of classical Also derived from the Schrodinger equation (and test of the equation of the form: y = e iS(x)/ h So the Schrodinger equation we obtain: é h2 ¶ ù iS/ h êúe + U ( x ) = Ee iS/ h ê 2m ¶ x ú ë û -3- h S ''+ S '2 = E - U 2m i 2m In the classical limit, and instead we have: S '(x )2 h 2k = - U (x ) 2m 2m Integral expression S = h z òU 2m U ( b2 + z '2 )dz '+ const h k2 - - L Derived from the wave function of the form: y = e ikze 3/ (2p ) - z im òU( h k b2 + z '2 )dz ' - L ( The scattering amplitude is written: f (k ', k ) = é im ´ exp êê- êë h k 2m 4p h r r rr - ik ' x ' 2 ikx ' ò d x ' e v( b + z )e ù ò U ( b + z ' )dz 'úúú - L û z' 2 The amplitude of scattering is calculated according to standardclassical f (0) ( q) = k i ò ¥ bdbJ (kbq) éêe i c (b) - 1ù ú ë û Contact between the scattering amplitude in partial wave and the eikonal scattering amplitude f(q) = As calculated above, the ¥ (2l + 1)Pl (cos q) éëe2idl - 1ùû å 2ik l= scattering amplitude obtained by of partial waves of the -4- the method form: f(k, q) = ¥ dl(2l + 1)Pl (cos q) éëe2idl (k) - 1ù ò û 2ik With the problem of high energy scattering, is considered to belarge we can replace the summation by the integral l l When the angle is small, we have: ỉq 2k ỉq 2l + ỉq (2l + 1) sin ỗỗ ữ = (2l + 1) sin ỗỗ ữ = 2k sin ỗỗ ữ ữ ữ ữ ữ ÷ ÷ è2 ø 2k è2 ø è2 ø 2k II The gravitational effects Gravitational scattering completely Starting from the general covariant equation Klein-Gordon formassless -g D m( - g g mnDmY) = , Dm = ¶ m - ieAm particles - such as nuclear "test" in the gravitational field and electromagnetic field: , that - gg mn Am(x ) g = det gmn (x ) = , as the electromagnetic field First we consider the gravitational scattering completely, that is,consider the scattering of neutral particles So we set in Where the classical Schwarzschild beer slow motion of the particle (the particle mass M is considered to be small beercompared to) obtained by the experiment of Einstein equations,the form: ỉ 2GM ÷ ÷ ds = - ỗỗ1 dt + ữ ữ ỗố r ứ - ổ 2GM ữ ỗỗ1 ữ dr + r (d q2 + sin qd f ) ữ ữ ỗố r ứ energy value is the center of mass is very strange: Gs = i él(l + 1) - N (N + 1)ù ú û (2N + 1) êë -5- The formula above allows to draw out some of the majorrejection, and has been used in the eikonal limit by theasymptotic expansion of the function variables associated with increased Gamma inverse exponent of l The result we get é (Gs ) + O ỗổ1 ửữữ 1ự ỳ+ dl ằ - Gs ờlog l ỗ 3ữ ữ ỗốl ứ 2l ỳ ë û 2l Culmination of the scattering amplitude First, we will re-expression eikonal scattering amplitude asobtained in the first chapter: f (s, t ) = i s 2p ò ¥ d 2be ikb éêe ë 2i dl - 1ù ú û And with attention Mandelstam variables As such, we will rewrite the complementary expression of the scatteringamplitude: 1- iGs ö i s - iGs iGs G(1 - iGs ) ổ ỗỗ ữ ữ f (s, t ) = s p ữ ữ 2p G(iGs ) ỗố- t ø (0) 2Gs = t iGs ö G(1 - iGs ) ổ ỗỗ- t ữ ữ ữ ữ G(1 + iGs ) ỗố s ứ 1 ö2 Gs - iGs iGs G(2 - iGs ) æ çç ÷ ÷ f (1) (s, t ) = s p ữ p G(21 + iGs ) ỗố- t ÷ ø - iGs iGs 2Gs G(2 - iGs ) ổ ỗỗ- t ữ ữ = ữ ứ - t G(21 + iGs ) ỗố s ữ Scatter more attractive since the electromagnetic interaction The first one considers the scattering of neutral particles inexternal test metric ReissnerNordstom by the static charge.Klein Gordon equation for the fast moving particles can also beobtained by replacing the derivative in spacetime covariantderivative compatible with metric Reissner-6- Nordstom: , æ 2GM GQ ữ ds = - ỗỗỗ1 + ữ dt + ữ ữ ỗố r r ứ - ổ 2GM GQ ỗỗ1 ữ + ÷ dr + r 2d W2 ÷ çèç ÷ r r ø i (Gs - QQ ') ö 2(Gs - QQ ') G(1 - i(Gs - QQ ')) ổ ỗỗ- t ữ ữ f (s, t ) = ữ G(1 + i(Gs - QQ ')) ỗố s ÷ ø -t (1) - Method of partial waves used in quantum mechanics isgeneralized, then it is used to study the scattering problem in thePlanck energy theory of quantum gravity - Have shown that for neutral particles, the peak of thescattering amplitude in partial wave method lies on the imaginaryaxis energymomentum The culmination of this was distributed atlocations other than where they appear in the eikonalapproximation - For particles with electric charge, the effects of electromagnetic and gravitational fields remain separate when using the eikonal approximation, and obtained the additionalterms of the momentum transfer The effects of electromagneticand gravitational disturbances would be together as additionalprimary research at higher levels References I Vietnameses Nguyễn Ngọc Giao (1999), Lý thuyết trường hấp dẫn, Đại học Quốc gia TPHCM Nguyễn Ngọc Giao (1999), Hạt bản, Trường ĐHKH Tự Nhiên, Hà nội Nguyễn Xuân Hãn (1998), Cơ học lượng tử, ĐHQG Hà Nội, Hà nội -7- Nguyễn Xuân Hãn (1998), Cơ sở lý thuyết trường lượng tử, ĐHQG Hà Nội, Hà nội II English t Hoof, (1988) “On the Factorization of Universal Poles in a Theory of Gravitating Point Particles”, Nucl Phys B304, pp 867-876 D.Amati, M.Ciafaloni and G.Veneziano, (1988) “Classical and Quantum Gravity Effects from Planckian Energy Superstring”, Int J Mod Phys A3, pp1615-1561 H Verlinde and E Verlinde, (1992)” Scattering at Planckian Energies”, Nucl.Phys.B371, pp 246-252 D.Kabat and M Ortiz, (1992) “Eikonal Gravity and Planckian Scattering”, Nucl.Phys.B388, pp.570-592 Nguyen Suan Han and Eap Ponna; (1997) “ Straight-Line Path Approximation for the Studying Planckian-Scattering in Quantum Gravity”, Nuo Cim A, N110A pp 459-473 Nguyen Suan Han, (2000) “Straight-Line Path Approximation for the High-Energy Elastic and Inelastic Scattering in Quantum Gravity” Euro Phys J C, vol.16, N3 p.547-553 Proceedings of the 4th International Workshop on Graviton and Astrophysics heid in Beijing, from October 1015, 1999 at the Beijing Normal University, China, Ed Liao Liu, et al World Scientific Singapore (2000)pp.319-333 S Das and P Majumdar, (1998) “Aspects of Planckian Scattering Beyon the Eikonal ” Journal Pramana, India, 51, pp 413-418 Nguyen Suan Han and Nguyen Nhu Xuan (2002), “Planckian Scattering Beyon the Eikonal Approximation in the Functional Approach” E-print arxiv: gr-qc/0203054, 15 mar 2002, 15p; European physical journal c, Vol 24, pp.643-651 Nguyen Suan Han and Nguyen Nhu Xuan, (2008)“ Planckian Scattering Beyon the Eikonal Approximation in the Quasi-Potential Approach” E-print -8- arxiv: 0804.3432 v2 [quant-ph] To be published in european physical journal c (2008) 10 Rosenfelder r (2008), “Path Integrals for Potential Scattering”,E-print arxiv: 0806.3217v2[nucl-th] 11 Charles Poole Herbert Goldstien and John Safko Classical Mechanics Addison Wesley 12 Robert A Leacock and Michael J Padgett “Hamilton-Jacobi Theory and the Quantum Action Variable” Physical Review Letters, 50(1):3–6, 1983 13 Robert A Leacock and Michael J Padgett “Hamilton-Jacobi/action-angle quantum mechanics” Physical Review D, 28(10):2491–2502, 1983 Marco Roncadelli and L.S Schulman “Quantum Hamilton-Jacobi 14 Theory” Physical Review Letters, 99(17), 2007 15 t Hoof, (1988) “On the Factorization of Universal Poles in a Theory of Gravitating Point Particles”, Nucl Phys B304, pp 867-876 A.K Kapoor R.S Bhalla and P.K Panigrahi “Quantum Hamilton-Jacobi formalism and the bound state spectra” arXiv, quant-ph/9512018v2, 1996 16 J.J Sakurai Modern Quantum Mechanics Pearson Education, 2007 -9- ... then it is used to study the scattering problem in the Planck energy theory of quantum gravity I The Schrodinger equation method 1.Phuong method developed by the partial wave Schrodinger equation:... in which the partial wave method A comparison of three methods to help us have different directions for the scattering problem in quantum mechanics - Method of partial waves used in quantum mechanics... -t (1) - Method of partial waves used in quantum mechanics isgeneralized, then it is used to study the scattering problem in thePlanck energy theory of quantum gravity - Have shown that for neutral

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