Slide 1

Yi-Zen Chu @ Fermilab Monday, 7 February 2011 Dont Shake That Solenoid Too Hard: Particle Production From Aharonov-Bohm K. Jones-Smith, H. Mathur, and T. Vachaspati, Phys. Rev. D 81:043503,2010 Y.-Z.Chu, H. Mathur, and T. Vachaspati, Phys. Rev. D 82:063515,2010 D.A. Steer, T. Vachaspati, arXiv:1012.1998 [hep-th] Slide 2 Setup B x y z ShaketheSolenoid! Slide 3 B x y ze+e- e+e-Pop! Setup Slide 4 Aharonov and Bohm (AB, 1959) Quantum Mechanics: (AB, 1959) Quantum Mechanics: Vector potential A is not merely computational crutch but indispensable for quantum dynamics of charged particles. Vector potential A is not merely computational crutch but indispensable for quantum dynamics of charged particles. (2010) Quantum Field Theory: (2010) Quantum Field Theory: Spontaneous pair production of charged particles just by shaking a thin solenoid. Spontaneous pair production of charged particles just by shaking a thin solenoid. Slide 5 Aharonov and Bohm (1959) Incomingelectrons Outgoingelectrons Maxwell tensor is zero almost everywhere in thin solenoid limit no classical dynamics. Maxwell tensor is zero almost everywhere in thin solenoid limit no classical dynamics. Non-zero cross section (AB 1959; Alford, Wilczek 1989) Non-zero cross section (AB 1959; Alford, Wilczek 1989) w Purely quantum process. Purely quantum process. Slide 6 Aharonov and Bohm (1959) Incomingelectrons Outgoingelectrons Maxwell tensor is zero almost everywhere in thin solenoid limit no classical dynamics. Maxwell tensor is zero almost everywhere in thin solenoid limit no classical dynamics. Quantum dyanmics: A is non-zero outside solenoid. Quantum dyanmics: A is non-zero outside solenoid. w Purely quantum process. Purely quantum process. Slide 7 Aharonov and Bohm (1959) Incomingelectrons Outgoingelectrons Maxwell tensor is zero almost everywhere in thin solenoid limit no classical dynamics. Maxwell tensor is zero almost everywhere in thin solenoid limit no classical dynamics. Periodic dependence on e topological interaction. Periodic dependence on e topological interaction. w Purely quantum process. Purely quantum process. Slide 8 Aharonov-Bohm Interaction Purely quantum process. Purely quantum process. Topological aspect: Topological aspect: ABQM QM: Amp. for paths that cannot be deformed into each other will differ by exp[i(integer)e] QM: Amp. for paths that cannot be deformed into each other will differ by exp[i(integer)e] Slide 9 Aharonov-Bohm Interaction Purely quantum process. Purely quantum process. Topological aspect: Topological aspect: ABQM QM: Amp. for paths belonging to different classes will differ by exp[i(integer)e] QM: Amp. for paths belonging to different classes will differ by exp[i(integer)e] Slide 10 Aharonov-Bohm Interaction Purely quantum process. Purely quantum process. Topological aspect: Topological aspect: ABQM QM: Amp. for paths belonging to different classes will differ by exp[i(integer)e] QM: Amp. for paths belonging to different classes will differ by exp[i(integer)e] Slide 11 Aharonov-Bohm Interaction Purely quantum process. Purely quantum process. Topological aspect: Topological aspect: ABQM Expect: Pair production rate to have periodic dependence on AB phase e. Expect: Pair production rate to have periodic dependence on AB phase e. Slide 12 B x y ze+e- e+e-Setup Slide 13 Setup Effective theory of magnetic flux tube: Alford and Wilczek (1989). Effective theory of magnetic flux tube: Alford and Wilczek (1989). Bosonic or fermionic quantum electrodynamics (QED) Bosonic or fermionic quantum electrodynamics (QED) Bosonic QED Fermionic QED Slide 14 Why are particles produced? The gauge potential A around a moving solenoid is time-dependent. The gauge potential A around a moving solenoid is time-dependent. Hamiltonian of QFT is explicitly time- dependent: H i = d 3 x A J Hamiltonian of QFT is explicitly time- dependent: H i = d 3 x A J Zero particle state (in the Heisenberg picture) at different times not the same vector i.e. particle creation occurs. Zero particle state (in the Heisenberg picture) at different times not the same vector i.e. particle creation occurs. Slide 15 Moving frames scheme Adiabatic approximation Mode expansion: Solve the mode functions for the stationary solenoid problem and shift them by , location of moving solenoid. Mode expansion: Solve the mode functions for the stationary solenoid problem and shift them by , location of moving solenoid. Slide 16 Moving frames scheme Compute 0 particle to 2 particle transition amplitude Compute 0 particle to 2 particle transition amplitude Slide 17 Moving frames results Rate carries periodic dependence on e Rate carries periodic dependence on e Non-relativistic Non-relativistic Slide 18 Moving frames results Rate carries periodic dependence on e Rate carries periodic dependence on e Non-relativistic Non-relativistic Slide 19 Relativistic e e >m o Similar plot for bosons. v 0 = 0.001 v 0 = 0.1 v 0 = 1 o = e Slide 25 e >m, v 0 ~1, k z =0, k xy =k xy Slide 26 Gravitational Aharonov-Bohm x y z g = Spacetime curvature is zero outside string. Spacetime curvature is zero outside string. = 8G = 8G Non-trivial QM phase due to deficit angle. Non-trivial QM phase due to deficit angle. Slide 27 Shake a cosmic string: gravitationally induced production of all particle spieces. Shake a cosmic string: gravitationally induced production of all particle spieces. Scalars Scalars Photons Photons Fermions Fermions x y z Gravitational Aharonov-Bohm Slide 28 Shake a cosmic string: gravitationally induced production of all particle spieces. Shake a cosmic string: gravitationally induced production of all particle spieces. Scalars Scalars Photons Photons Fermions Fermions Gravitational Aharonov-Bohm Quantum scattering of cosmic background photons and neutrinos. Quantum scattering of cosmic background photons and neutrinos. Slide 29 The N-Body Problem in General Relativity from Perturbative (Quantum) Field Theory Y.-Z.Chu, Phys. Rev. D 79: 044031, 2009 arXiv: 0812.0012 [gr-qc] Yi-Zen Chu @ Fermilab Monday, 7 February 2011 Slide 30 System of n 2 gravitationally bound compact objects: System of n 2 gravitationally bound compact objects: Planets, neutron stars, black holes, etc. Planets, neutron stars, black holes, etc. What is their effective gravitational interaction? What is their effective gravitational interaction? Slide 31 Compact objects point particles Compact objects point particles n-body problem: Dynamics for the coordinates of the point particles n-body problem: Dynamics for the coordinates of the point particles Assume non-relativistic motion Assume non-relativistic motion GR corrections to Newtonian gravity: an expansion in (v/c) 2 GR corrections to Newtonian gravity: an expansion in (v/c) 2 Nomenclature: O[(v/c) 2Q ] = Q PN Slide 32 Note that General Relativity is non-linear. Note that General Relativity is non-linear. Superposition does not hold Superposition does not hold 2 body lagrangian is not sufficient to obtain n-body lagrangian 2 body lagrangian is not sufficient to obtain n-body lagrangian Nomenclature: O[(v/c) 2Q ] = Q PN Slide 33 n-body problem known up to O[(v/c) 2 ]: n-body problem known up to O[(v/c) 2 ]: Einstein-Infeld-Hoffman lagrangian Einstein-Infeld-Hoffman lagrangian Eqns of motion used regularly to calculate solar system dynamics, etc. Eqns of motion used regularly to calculate solar system dynamics, etc. Precession of Mercurys perihelion begins at this order Precession of Mercurys perihelion begins at this order O[(v/c) 4 ] only known partially. O[(v/c) 4 ] only known partially. Damour, Schafer (1985, 1987) Damour, Schafer (1985, 1987) Compute using field theory? (Goldberger, Rothstein, 2004) Compute using field theory? (Goldberger, Rothstein, 2004) Slide 34 Solar system probes of GR beginning to go beyond O[(v/c) 2 ]: Solar system probes of GR beginning to go beyond O[(v/c) 2 ]: New lunar laser ranging observatory APOLLO; Mars and/or Mercury laser ranging missions? New lunar laser ranging observatory APOLLO; Mars and/or Mercury laser ranging missions? ASTROD, LATOR, GTDM, BEACON, etc. ASTROD, LATOR, GTDM, BEACON, etc. See e.g. Turyshev (2008) See e.g. Turyshev (2008) Slide 35 n-body L eff gives not only dynamics but also geometry. n-body L eff gives not only dynamics but also geometry. Add a test particle, M->0: it moves along geodesic in the spacetime metric generated by the rest of the n masses Add a test particle, M->0: it moves along geodesic in the spacetime metric generated by the rest of the n masses Metric can be read off its L eff Metric can be read off its L eff Slide 36 Gravitational wave observatories may need the 2 body L eff beyond O[(v/c) 7 ]: Gravitational wave observatories may need the 2 body L eff beyond O[(v/c) 7 ]: LIGO, VIRGO, etc. can track gravitational waves (GWs) from compact binaries over O[10 4 ] orbital cycles. LIGO, VIRGO, etc. can track gravitational waves (GWs) from compact binaries over O[10 4 ] orbital cycles. GWs: Need theoretical templates to integrate against raw data to search for correlations. GWs: Need theoretical templates to integrate against raw data to search for correlations. Construction of accurate templates requires 2 body dynamics. Construction of accurate templates requires 2 body dynamics. Currently, 2 body dynamics known up to O[(v/c) 7 ], i.e. 3.5 PN Currently, 2 body dynamics known up to O[(v/c) 7 ], i.e. 3.5 PN See e.g. Blanchet (2006). See e.g. Blanchet (2006). Slide 37 Starting at 3 PN, O[(v/c) 6 ], GR computations of 2 body L eff start to give divergences due to the point particle approximation that were eventually handled by dimensional regularization. Starting at 3 PN, O[(v/c) 6 ], GR computations of 2 body L eff start to give divergences due to the point particle approximation that were eventually handled by dimensional regularization. Perturbation theory beyond O[(v/c) 7 ] requires systematic, efficient methods. Perturbation theory beyond O[(v/c) 7 ] requires systematic, efficient methods. Renormalization & regularization Renormalization & regularization Computational algorithm Feynman diagrams with appropriate dimensional analysis. Computational algorithm Feynman diagrams with appropriate dimensional analysis. QFTOffers: Slide 38 GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line Slide 39 GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors, d-velocities, etc.integrated along world line n point particles: any scalar functional of geometric tensors, d-velocities, etc.integrated along world line Point particle approximation gives us computational control. Point particle approximation gives us computational control. Infinite series of actions truncated based on desired accuracy of theoretical prediction. Infinite series of actions truncated based on desired accuracy of theoretical prediction. Slide 40 Mds describes structureless point particle Mds describes structureless point particle GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line Slide 41 Non-minimal terms encode information on the non-trivial structure of individual objects. Non-minimal terms encode information on the non-trivial structure of individual objects. GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line Slide 42 GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors, d-velocities, etc.integrated along world line n point particles: any scalar functional of geometric tensors, d-velocities, etc.integrated along world line Coefficients {c x } have to be tuned to match physical observables from full description of objects. Coefficients {c x } have to be tuned to match physical observables from full description of objects. E.g. n non-rotating black holes. E.g. n non-rotating black holes. Slide 43 For non-rotating compact objects, up to O[(v/c) 8 ], only minimal terms -M a ds a needed For non-rotating compact objects, up to O[(v/c) 8 ], only minimal terms -M a ds a needed GR: Einstein-Hilbert GR: Einstein-Hilbert n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line Slide 44 Expand GR and point particle action in powers of graviton fields h Expand GR and point particle action in powers of graviton fields h Slide 45 terms just from Einstein-Hilbert and -M a ds a. terms just from Einstein-Hilbert and -M a ds a. Slide 46 but some dimensional analysis before computation makes perturbation theory much more systematic but some dimensional analysis before computation makes perturbation theory much more systematic The scales in the n-body problem The scales in the n-body problem r typical separation between n bodies. r typical separation between n bodies. v typical speed of point particles v typical speed of point particles r/v typical time scale of n-body system r/v typical time scale of n-body system Slide 47 Lowest order effective action Lowest order effective action Schematically, conservative part of effective action is a series: Schematically, conservative part of effective action is a series: Virial theorem Virial theorem Slide 48 Look at Re[Graviton propagator], non- relativistic limit: Look at Re[Graviton propagator], non- relativistic limit: Slide 49 Slide 50 n-graviton piece of -M a ds a with powers of velocities scales as n-graviton piece of -M a ds a with powers of velocities scales as n-graviton piece of Einstein-Hilbert action with time derivatives scales as n-graviton piece of Einstein-Hilbert action with time derivatives scales as With n (w) world line terms -M a ds a, With n (w) world line terms -M a ds a, With n (v) Einstein-Hilbert action terms, With n (v) Einstein-Hilbert action terms, With N total gravitons, With N total gravitons, Every Feynman diagram scales as Every Feynman diagram scales as Slide 51 n-graviton piece of -M a ds a with powers of velocities scales as n-graviton piece of -M a ds a with powers of velocities scales as n-graviton piece of Einstein-Hilbert action with time derivatives scales as n-graviton piece of Einstein-Hilbert action with time derivatives scales as =1 for classical problem Q PN With n (w) world line terms -M a ds a, With n (w) world line terms -M a ds a, With n (v) Einstein-Hilbert action terms, With n (v) Einstein-Hilbert action terms, With N total gravitons, With N total gravitons, Every Feynman diagram scales as Every Feynman diagram scales as Know exactly which terms in action & diagrams are necessary. Know exactly which terms in action & diagrams are necessary. Slide 52 Limited form of superposition holds Limited form of superposition holds At Q PN, i.e. O[(v/c) 2Q ], max number of distinct point particles in a given diagram is Q+2 At Q PN, i.e. O[(v/c) 2Q ], max number of distinct point particles in a given diagram is Q+2 1 PN, O[(v/c) 2 ]: 3 body problem 1 PN, O[(v/c) 2 ]: 3 body problem 2 PN, O[(v/c) 4 ]: 4 body problem 2 PN, O[(v/c) 4 ]: 4 body problem 3 PN, O[(v/c) 6 ]: 5 body problem 3 PN, O[(v/c) 6 ]: 5 body problem Every Feynman diagram scales as Every Feynman diagram scales as Slide 53 Slide 54 2 body diagrams 3 body diagrams Einstein-Infeld-Hoffman d-spacetime dimensions Slide 55 2 body diagrams 3 body diagrams Einstein-Infeld-Hoffman d-spacetime dimensions Relativistic corrections Slide 56 2 body diagrams 3 body diagrams Einstein-Infeld-Hoffman d-spacetime dimensions Gravitational 1/r 2 potentials Slide 57 2 body diagrams 3 body diagrams Einstein-Infeld-Hoffman d-spacetime dimensions Non-linearities of GR Non-linearities of GR Three body force Three body force Slide 58 2 body diagrams 3 body diagrams Einstein-Infeld-Hoffman d-spacetime dimensions Time derivative of Runge-Lenz vector gives perihelion precession of planetary orbits. Time derivative of Runge-Lenz vector gives perihelion precession of planetary orbits. Slide 59 No graviton vertices Gravitonvertices Slide 60 vertices Gravitonvertices Relativistic corrections Slide 61 No graviton vertices Gravitonvertices Gravitational 1/r 3 potentials Slide 62 No graviton vertices Gravitonvertices Non-linearities in GR Slide 63 No graviton vertices Gravitonvertices Slide 64 vertices Gravitonvertices Slide 65 Slide 66 Slide 67 Slide 68 Slide 69 Slide 70 Slide 71 Slide 72 Slide 73 Slide 74 Slide 75 Slide 76 Physics issues: Physics issues: Rotation, multipoles, tails, etc. Rotation, multipoles, tails, etc. Gravitational radiation Gravitational radiation Theoretical / Technical issues: Theoretical / Technical issues: Different field variables: ADM, Kol- Smolkin-Kaluza-Klein. Different field variables: ADM, Kol- Smolkin-Kaluza-Klein. Different grav. lagrangian: Bern- Dennen-Huang-Kiermaier, Bern-Grant Different grav. lagrangian: Bern- Dennen-Huang-Kiermaier, Bern-Grant Software development. Software development.