on integrability of spinning particle motion in higher-dimensional rotating black hole spacetimes
DESCRIPTION
On integrability of spinning particle motion in higher-dimensional rotating black hole spacetimes. David Kubiz ňák (Perimeter Institute). Relativity and Gravitation 100 Years after Einstein in Prague Prague, Czech Republic June 25 – June 29, 2012. Plan of the talk. - PowerPoint PPT PresentationTRANSCRIPT
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On integrability of spinning particle motion in higher-dimensional rotating
black hole spacetimesDavid Kubizňák
(Perimeter Institute)
Relativity and Gravitation100 Years after Einstein in PraguePrague, Czech Republic June 25 – June 29, 2012
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Plan of the talk
I. Spinning particle in curved rotating BH background
II. Semiclassical theory of spinning particleI. Hamiltonian formulationII. Non-generic superinvariants: “SUSY in the sky”III. On integrability in all dimensions
III. Conclusions
Based on: • DK, M. Cariglia, Phys. Rev. Lett. 108, 051104 (2012); arXiv:1110.0495.• M. Cariglia, P. Krtous, DK, in preparation.
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I) Spinning particle in curved rotating BH
background
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a) Quantum description: Dirac equation
• Separable!
• “Enough integrals of motion 2 symmetry operators”
obey decoupled 2nd-order ODEs
complete set of mutually commuting operators
See Marco’s talk!
Spinning particle in curved rotating BH background
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b) Classical GR description: Papapetrou’s Eq.
Chaotic motion!
gauge fixing (not unique)
(even in Schwarzchild due to spin-orb. int.)
Spinning particle in curved rotating BH background
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c) SUSY semi-classical spinning particle
“Classical Hamiltonian system”
Spinning particle in curved rotating BH background
Integrable?
“bosonic”
“fermionic”
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Spinning particle in curved rotating BH background
Quantum
Separable!
complete set of comm.ops
Classical
Chaotic!
SUSY: spinning
Integrable?!
Klein-Gordon Eq. Separable!
Geodesic Eq. Carter: Completely
integrable!
No spin (nontriv)
WKB
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II) Semiclassical theory of spinning particle
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A little more about spinning particle
Hamiltonian formulation:
•
• Poisson bracket
• SUSY
• Physical (gauge) conditions
covariant
canonical
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Nongeneric superinvariants: SUSY in the skyGibbons, Rietdijk, van Holten, Nucl. Phys. B404 (1993) 42; hep-th/9303112.
Automatically an integral of motion
Linear in momenta superinvariants
Killing-Yano 2-form
•
•
•
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SUSY in the sky: Kerr geometry
Set of commuting operators:
“bosonic” “fermionic”(no classical analogue)
termsBosonic set of commuting operators :
• SUSY in the sky• can take a limit and recover Carter’s result
Problem: “integrates” only bosonic equations. What about fermionic?
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SUSY in “astral spheres”? Kerr-NUT-AdS geometryLinear superinvariants
Although there is a whole tower of these (Valeri’s talk), they do not commute!
However, in all D dimensions one can construct D bosonic integrals of mutually commuting integrals of motion
making the bosonic part of the motion integrable.
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Conclusions1) We have shown the existence of D mutually commuting bosonic
integrals of spinning motion in Kerr-NUT-AdS black hole spacetimes in all dimensions D. This generalizes the previous result on complete integrability of geodesic motion. Non-spinning limit can be easily taken.
2) Integrability of “fermionic sector” remains unclear at the moment.
3) There are interesting connections to “quantum” and “classical” descriptions:
• Grassmann algebra s Clifford algebra
• operator ordering
(satisfies Lorentz algebra)
(Integrals OK to linear order)
a) Dirac limit:
b) Papapetrou’s limit: