Quantization of 10D Massless Superparticle by Lado Razmadze
Delve into the fascinating world of quantizing gauge theories and relativistically invariant theories in curved spacetime through the study of 10D massless superparticles. Explore the algebraic approach using the Faddeev-Jackiw method, covariant quantization principles, and the use of Darboux's theorem. Understand the intricacies of canonical variables, Poisson brackets, and canonical commutator relations, all leading to the quantization of the 10D massless superparticle.
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Presentation Transcript
Quantization of 10d massless Quantization of 10d massless superparticle superparticle Lado Razmadze Supervisor: Prof. George Jorjadze Free University of Tbilisi 8 th Georgian German School and Workshop in Basic Science Tbilisi 23.08.2018
Motivation Strings and Superstrings in AdS space Quantizing relativistically invariant theories in curved spacetime involve solving nonlinear Klein-Gordon equations Algebraic approach using Faddeev-Jackiw method is favorable. LR - Quantization of 10d massless superparticle
Quantizing gauge theories LR - Quantization of 10d massless superparticle
Quantizing gauge theories Covariant quantization first quantizes the gauge theory and then reduces the space of states by the condition ? corresponds to classical constraints. Hamiltonian reduction is done via gauge fixing or by choosing a complete set of gauge invariant variables. ? ?? = 0 where ? LR - Quantization of 10d massless superparticle
Quantizing gauge theories Using Darboux s theorem we can always change to canonical variables ?????= ?? ?,? ????? Q and P are canonical coordinates of our reduced phase space LR - Quantization of 10d massless superparticle
Quantizing gauge theories We can then rewrite our conserved quantities in terms of ?? and ?? Define canonical Poisson brackets as ??,??= 1 Use this to define canonical commutator relations After this we define Inner product so that the operators defined by the conserved quantities are self-adjoint LR - Quantization of 10d massless superparticle
Brink-Schwartz action Consider superspace with coordinates ??,?? where ?? M10 and ?? are fermionic coordinates given by 32-component Majorana Majorana- -Weyl spinors Weyl spinors. ?? are Grassmann Grassmann numbers LR - Quantization of 10d massless superparticle
Brink-Schwartz action 2 ?? ????? ? 2? ? = ?? ?<0 is associated with worldline metric ? = 0,1, ,9 ?0is defined as ?0= ? ?? matrices ? = 1, ,9 form Euclidean 9d Clifford Algebra Clifford Algebra ????+ ????= 2???? LR - Quantization of 10d massless superparticle
Super-Algebra Our action is invariant under the transformations of the form ?? ??+ ??, ?? ??+? ? ? ? ? +? 2??????? +1 2?????, ? ? +1 2 ?? ??+ ?????, 2?0???? ??? 1 4??,??, ?,? = 1,2, ,9 LR - Quantization of 10d massless superparticle
Super-Algebra Introducing the momentum variables ?? ????? ? ? ??= Corresponding charges are ??= ?? ??? ??= ?????? ???? ???= ???? ???? ????? ?0?= ??? ???0 1 ??? 2????? LR - Quantization of 10d massless superparticle
Super-Algebra Introducing first order formalism ?? ????? ? +? 2?2 ? = ?? ?? ? plays a role of Lagrange multiplier and its variation ?2= 0 and defines energy of the particle ? = ?0 LR - Quantization of 10d massless superparticle
Super-Algebra ?? ??, ??= ??? ? = 1,2, ,8 ?? are orthonormal solutions of ?? ??? ?= ??? ????? ? Factor of ? is added for convenience LR - Quantization of 10d massless superparticle
Generator construction We take the presymplectic form ? = ????? ??????? We choose ??,?0?,??,?0,??as our variables ?0?+ ???0+1 ??? 2????? ??= ? ?0? ? ? ? = ??? 2?????? LR - Quantization of 10d massless superparticle
Generator construction Using the solutions of ?? ? = ?????+? 2????? ? ??=?0? ? ???? ??? ??????, ??= ?? 2?? ?? ? ?????? We neglected d ???? LR - Quantization of 10d massless superparticle
Generator construction We use first 8x8 block of ??matrices ?= 0, 9 ??? 1 ? < 8, ??? = ??? 8 orthonormal vectors can be chosen as 1 ?? = ? ?? ????+ ????? 2? ? + ?9 ? = 1,2, ,8 , ? = 1,2, ,16 LR - Quantization of 10d massless superparticle
Generator construction ?? Using the defined vectors we write out ?? ??=?????? ???? 4? ? + ?9 ?? ?? Now we rewrite rotation generator using invariant variables ???= ???? ????+? 1 2??????? ???? ???? ????? ?? ???? ??????+ ???? 2 ???? LR - Quantization of 10d massless superparticle
Thank you! Thanks to VW-Stiftung for the opportunity of continuing my studies in Bonn I intend to continue working on superstings in AdS LR - Quantization of 10d massless superparticle
