yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010
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Don’t Shake That Solenoid Too Hard: Particle Production From Aharonov-Bohm. Yi-Zen Chu Particle Astrophysics/Cosmology Seminar, ASU Wednesday, 6 October 2010. K. Jones-Smith, H. Mathur , and T. Vachaspati , Phys. Rev. D 81 :043503,2010

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yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 2010

Don’t Shake

That Solenoid

Too Hard:

Particle Production

From Aharonov-Bohm

Yi-Zen Chu

Particle Astrophysics/Cosmology Seminar, ASU

Wednesday, 6 October 2010

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

slide2

Aharonov and Bohm (1959)

  • Quantum Mechanics:
    • Vector potential Aμ is not merely computational crutch but indispensable for quantum dynamics of charged particles.
  • (2010) Quantum Field Theory:
    • Spontaneous pair production of charged particles just by shaking a thin solenoid.
slide3

Setup

z

Shake

the

Solenoid!

B

y

x

slide4

Setup

z

e+

Pop!

e-

e-

B

e+

y

x

slide5

Setup

  • Effective theory of magnetic flux tube: Alford and Wilczek (1989).
  • Bosonic or fermionic quantum electrodynamics (QED)

Bosonic QED

Fermionic QED

slide6

I: Time dependent H

  • The gauge potential Aμ around a moving solenoid is time-dependent.
  • Hamiltonian of QFT is explicitly time-dependent: Hi = ∫d3x AμJμ.
  • Zero particle state (in the Heisenberg picture) at different times not the same vector – i.e. particle creation occurs.
slide7

II: Aharonov-Bohm Interaction

  • However thin the flux tube is, particle production will occur – purely quantum process.
  • Pair production rate contains topological aspect.

B

AB

QM

  • QM: Amp. for paths belonging to different classes will differ by exp[i(integer)eФ]
slide8

II: Aharonov-Bohm Interaction

  • However thin the flux tube is, particle production will occur – purely quantum process.
  • Pair production rate contains topological aspect.

B

AB

QM

  • Expect: Pair production rate to have periodic dependence on AB phase eФ.
slide9

Moving frames

  • Expand scalar operator φ and Dirac operator ψ in terms of the instantaneous eigenstates of the Hamiltonian of the field equations.
    • i.e. Solve the mode functions for the stationary solenoid problem and shift them by ξ, location of moving solenoid.
slide10

Moving frames φ results

  • Rate carries periodic dependence on eФ
  • Non-relativistic
slide11

Moving frames ψ results

  • Rate carries periodic dependence on eФ
  • Non-relativistic
slide12

Relativistic

eФ << 1

φ

φ*

slide13

Small AB phase: eФ << 1

φ

φ*

  • Valid for any flux tube trajectory.
slide15

Small AB phase: eФ << 1

φ

φ*

  • Spins of e+e- anti-correlated along direction determined by their momenta and I+ x I-.
slide17

eФ<<1: Total Power

v0 = 1

v0 = 0.1

v0 = 0.001

  • Similar plot for bosons.
slide19

Cosmic String Loops

  • Motivated by astrophysical observations of anomalous excess of e+e-, some recent particle physics models involve mixing of photon U(1) vector potential with some “dark” sector (spontaneously broken) U(1)’ vector potential.
  • This allows emission of electrically charged particle-antiparticle pairs from cosmic strings via the AB interaction.
  • Consider: Kinky and cuspy loops.
slide20

eФ << 1: Kinky Cosmic String Loop

Kinky loop configuration

  • Like solenoid, there are infinite # of harmonics
  • Both bosonic and fermionic emission show similar linear-in-N behavior.
  • Cut-off determined by finite width of cosmic string loop.
slide21

eФ<<1: Cuspy Cosmic String Loop

Cuspy loop configuration

  • Like solenoid, there are infinite # of harmonics
  • Both bosonic and fermionic emission show similar constant-in-ℓ behavior.
  • Cut-off determined by finite width of cosmic string loop.
slide22

Gravitational “AB” interaction

  • Spacetime metric far from a straight, infinite cosmic string is Minkowski, with a deficit angle determined by string tension.
  • Shaking a cosmic string would generate a time-dependent metric and induce gravitationally induced production of all particle spieces.
    • Scalars, Photons, Fermions
yi zen chu particle astrophysics cosmology seminar asu wednesday 6 october 20101

The N-Body Problem

in

General Relativity

from

Perturbative (Quantum) Field Theory

Yi-Zen Chu

Particle Astrophysics/Cosmology Seminar, ASU

Wednesday, 6 October 2010

Y.-Z.Chu, Phys. Rev. D 79: 044031, 2009

arXiv: 0812.0012 [gr-qc]

n body problem in gr
n-Body Problem in GR
  • System of n ≥ 2 gravitationally bound compact objects:
    • Planets, neutron stars, black holes, etc.
  • What is their effective gravitational interaction?
n body problem in gr1
n-Body Problem in GR
  • Compact objects ≈ point particles
  • n-body problem: Dynamics for the coordinates of the point particles
  • Assume non-relativistic motion
    • GR corrections to Newtonian gravity: an expansion in (v/c)2

Nomenclature: O[(v/c)2Q] = Q PN

n body problem in gr2
n-Body Problem in GR
  • Note that General Relativity is non-linear.
    • Superposition does not hold
    • 2 body lagrangian is not sufficient to obtain n-body lagrangian

Nomenclature: O[(v/c)2Q] = Q PN

n body problem in gr3
n-Body Problem in GR
  • n-body problem known up to O[(v/c)2]:
    • Einstein-Infeld-Hoffman lagrangian
    • Eqns of motion used regularly to calculate solar system dynamics, etc.
      • Precession of Mercury’s perihelion begins at this order
  • O[(v/c)4] only known partially.
    • Damour, Schafer (1985, 1987)
    • Compute using field theory? (Goldberger, Rothstein, 2004)
motivation i
Motivation I
  • 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?
    • ASTROD, LATOR, GTDM, BEACON, etc.
    • See e.g. Turyshev (2008)
motivation i1
Motivation I
  • n-body Leff 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
    • Metric can be read off its Leff
motivation ii
Motivation II
  • Gravitational wave observatories may need the 2 body Leff beyond O[(v/c)7]:
    • LIGO, VIRGO, etc. can track gravitational waves (GWs) from compact binaries over O[104] orbital cycles.
    • GW detection: Raw data integrated against theoretical templates to search for correlations.
    • Construction of accurate templates requires 2 body dynamics.
    • Currently, 2 body dynamics known up to O[(v/c)7], i.e. 3.5 PN
    • See e.g. Blanchet (2006).
why quantum field theory
Why (Quantum) Field Theory
  • Starting at 3 PN, O[(v/c)6], GR computations of 2 body Leff 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.
  • Renormalization & regularization
  • Computational algorithm – Feynman diagrams with appropriate dimensional analysis.

QFT

Offers:

dynamics action
Dynamics: Action
  • GR: Einstein-Hilbert
  • n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line
dynamics action1
Dynamics: Action
  • GR: Einstein-Hilbert
  • n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line
  • –M∫ds describes structureless point particle
dynamics action2
Dynamics: Action
  • GR: Einstein-Hilbert
  • n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line
  • Non-minimal terms encode information on the non-trivial structure of individual objects.
dynamics action3
Dynamics: Action
  • GR: Einstein-Hilbert
  • n point particles: any scalar functional of geometric tensors, d-velocities, etc.integrated along world line
  • Coefficients {cx} have to be tuned to match physical observables from full description of objects.
  • E.g. n non-rotating black holes.
dynamics action4
Dynamics: Action
  • GR: Einstein-Hilbert
  • n point particles: any scalar functional of geometric tensors, d-velocities, etc.integrated along world line
  • Point particle approximation gives us computational control.
  • Infinite series of actions truncated based on desired accuracy of theoretical prediction.
dynamics action5
Dynamics: Action
  • GR: Einstein-Hilbert
  • n point particles: any scalar functional of geometric tensors, d-velocities, etc. integrated along world line
  • For non-rotating compact objects, up to O[(v/c)8], only minimal terms -Ma∫dsa needed
perturbation theory
Perturbation Theory
  • Expand GR and point particle action in powers of graviton fields hμν …
perturbation theory1
Perturbation Theory
  • Expand GR and point particle action in powers of graviton fields hμν …
  • ∞ terms just from Einstein-Hilbert and -Ma∫dsa.
dimensional analysis
Dimensional Analysis
  • … but some dimensional analysis before computation makes perturbation theory much more systematic
  • The scales in the n-body problem
    • r – typical separation between n bodies.
    • v – typical speed of point particles
    • r/v – typical time scale of n-body system
dimensional analysis1
Dimensional Analysis
  • Lowest order effective action
  • Schematically, conservative part of effective action is a series:
  • Virial theorem
dimensional analysis2
Dimensional Analysis
  • Look at Re[Graviton propagator], non-relativistic limit:
dimensional analysis3
Dimensional Analysis
  • Look at Re[Graviton propagator], non-relativistic limit:
dimensional analysis4
Dimensional Analysis
  • n-graviton piece of -Ma∫dsa with χ powers of velocities scales as
  • n-graviton piece of Einstein-Hilbert action with ψ time derivatives scales as
  • With n(w) world line terms -Ma∫dsa,
  • With n(v) Einstein-Hilbert action terms,
  • With N total gravitons,
  • Every Feynman diagram scales as
dimensional analysis5
Dimensional Analysis
  • n-graviton piece of -Ma∫dsa with χ powers of velocities scales as
  • n-graviton piece of Einstein-Hilbert action with ψ time derivatives scales as
  • With n(w) world line terms -Ma∫dsa,
  • With n(v) Einstein-Hilbert action terms,
  • With N total gravitons,
  • Every Feynman diagram scales as
  • Know exactly which terms in action & diagrams are necessary.

=1

for classical problem

Q PN

superposition
Superposition
  • Every Feynman diagram scales as
  • 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
    • 1 PN, O[(v/c)2]: 3 body problem
    • 2 PN, O[(v/c)4]: 4 body problem
    • 3 PN, O[(v/c)6]: 5 body problem
o v c 2 1 pn n 3 body problem
O[(v/c)2]: 1 PNn = 3 Body Problem

Einstein-Infeld-Hoffman

d-spacetime dimensions

2 body

diagrams

3 body

diagrams

o v c 2 1 pn n 3 body problem1
O[(v/c)2]: 1 PNn = 3 Body Problem

Relativistic corrections

Einstein-Infeld-Hoffman

d-spacetime dimensions

2 body

diagrams

3 body

diagrams

o v c 2 1 pn n 3 body problem2
O[(v/c)2]: 1 PNn = 3 Body Problem

Gravitational

1/r2 potentials

Einstein-Infeld-Hoffman

d-spacetime dimensions

2 body

diagrams

3 body

diagrams

o v c 2 1 pn n 3 body problem3
O[(v/c)2]: 1 PNn = 3 Body Problem
  • Non-linearities of GR
  • Three body force

Einstein-Infeld-Hoffman

d-spacetime dimensions

2 body

diagrams

3 body

diagrams

o v c 2 1 pn n 3 body problem4
O[(v/c)2]: 1 PNn = 3 Body Problem
  • Time derivative of Runge-Lenz vector gives perihelion precession of planetary orbits.

Einstein-Infeld-Hoffman

d-spacetime dimensions

2 body

diagrams

3 body

diagrams

o v c 4 2 pn n 4 body problem 2 body diagrams
O[(v/c)4]: 2 PNn = 4 Body Problem2 body diagrams

No graviton

vertices

Graviton

vertices

o v c 4 2 pn n 4 body problem 3 body diagrams
O[(v/c)4]: 2 PNn = 4 Body Problem3 body diagrams

No graviton

vertices

Graviton

vertices

o v c 4 2 pn 4 body diagrams
O[(v/c)4]: 2 PN4 Body Diagrams

Graviton

vertices

No graviton

vertices

o v c 4 2 pn n 4 body problem1
O[(v/c)4]: 2 PNn = 4 Body Problem

Work in progress (with Roman Buniy)

the road ahead
The road ahead
  • N-body problem for rotating masses, multi-pole moments, gravitational radiation, etc.
  • Higher PN computation:
    • Different field variables: ADM, Kol-Smolkin-Kaluza-Klein.
    • Different gravitational lagrangian: Bern-Grant.
    • Are there recursion relations for off-shell gravitational amplitudes?
    • Software development.
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