Accidental Isomorphisms: Spinning Spacetime with Special Matrices

Presenter Type

UNO Graduate Student (Masters)

Major/Field of Study

Mathematics

Other

Mathematics

Advisor Information

Faculty Sponsor

Location

MBSC Ballroom Poster # 208 - G (Masters)

Presentation Type

Poster

Start Date

24-3-2023 2:30 PM

End Date

24-3-2023 3:45 PM

Abstract

For rotations, 0º=360º, but not for "spins." The topology behind this can be illustrated with a simple belt/plate trick (Dirac). While rotation matrices preserve x²+y² (Pythagoras), other classical groups are designed to preserve more general forms (Weyl). In low dimensions, special circumstances allow us to represent spin groups, and their action on spinors, nicely using classical groups. Best known are SU(2)=Spin(3) in quantum mechanics and SL(2,C)=Spin(3,1) in special relativity, visualized variously using the Bloch / Riemann / "celestial" sphere along with the Hopf bundle or Mobius transformations. My future thesis aims to survey the whole story in one place, generalizing to both multiple space and time dimensions (p+q≤6), employing new number systems like quaternions and split complex numbers.

Additional Information (Optional)

I am also listed as an author for the talk

"A Student-led Initiative of Providing Accessible Higher-Level Math Problems"

with Jordan Sahs and I can't present both at the same time.

Scheduling

9:15-10:30 a.m., 10:45 a.m.-Noon, 1-2:15 p.m., 2:30 -3:45 p.m.

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COinS
 
Mar 24th, 2:30 PM Mar 24th, 3:45 PM

Accidental Isomorphisms: Spinning Spacetime with Special Matrices

MBSC Ballroom Poster # 208 - G (Masters)

For rotations, 0º=360º, but not for "spins." The topology behind this can be illustrated with a simple belt/plate trick (Dirac). While rotation matrices preserve x²+y² (Pythagoras), other classical groups are designed to preserve more general forms (Weyl). In low dimensions, special circumstances allow us to represent spin groups, and their action on spinors, nicely using classical groups. Best known are SU(2)=Spin(3) in quantum mechanics and SL(2,C)=Spin(3,1) in special relativity, visualized variously using the Bloch / Riemann / "celestial" sphere along with the Hopf bundle or Mobius transformations. My future thesis aims to survey the whole story in one place, generalizing to both multiple space and time dimensions (p+q≤6), employing new number systems like quaternions and split complex numbers.