Proof of the Atiyah-Singer index theorem by canonical quantum mechanics

Zhou, Zixian1,2; Duan, Xiuqing3; Sun, Kai-Jia3,4

2018-08-03

10.1088/1751-8121/aac546

Journal of Physics A-Mathematical and Theoretical   影响因子和分区

1751-8113

1751-8121

We show that the Atiyah-Singer index theorem of Dirac operator can be directly proved in the canonical formulation of quantum mechanics, without using the path-integral technique. This proof takes advantage of an algebraic isomorphism between Clifford algebra and exterior algebra in small tau (high temperature) limit, together with simple properties of quantum harmonic oscillator. Compared to the proofs given by full symbol calculus and heat kernel, we try to prove this theorem in a more quantum mechanical way.

Atiyah-Singer index theorem topology supersymmetry chiral magnetic effects

SCI

英语

第一

National Natural Science Foundation of China[11574100]

Physics

Physics, Multidisciplinary ; Physics, Mathematical

WOS:000436117200001

IOP PUBLISHING LTD

Web of Science

1.Southern Univ Sci & Technol, Dept Phys, Shenzhen 518055, Peoples R China
2.Southern Univ Sci & Technol, Shenzhen Inst Quantum Sci & Engn, Shenzhen 518055, Peoples R China
3.Shanghai Jiao Tong Univ, Sch Phys & Astron, Shanghai 200240, Peoples R China
4.Shanghai Jiao Tong Univ, Shanghai Key Lab Particle Phys & Cosmol, Shanghai 200240, Peoples R China

Zhou, Zixian,Duan, Xiuqing,Sun, Kai-Jia. Proof of the Atiyah-Singer index theorem by canonical quantum mechanics[J]. Journal of Physics A-Mathematical and Theoretical,2018,51(31).

Zhou, Zixian,Duan, Xiuqing,&Sun, Kai-Jia.(2018).Proof of the Atiyah-Singer index theorem by canonical quantum mechanics.Journal of Physics A-Mathematical and Theoretical,51(31).

Zhou, Zixian,et al."Proof of the Atiyah-Singer index theorem by canonical quantum mechanics".Journal of Physics A-Mathematical and Theoretical 51.31(2018).