Twistor space

In mathematics, twistor space is the complex vector space of solutions of the twistor equation ∇ A ′ ( A Ω B ) = 0 {displaystyle abla _{A'}^{(A}Omega ^{B)}=0} . It was described in the 1960s by Roger Penrose and Malcolm MacCallum. According to Andrew Hodges, twistor space is useful for conceptualizing the way photons travel through space, using four complex numbers. He also posits that twistor space may aid in understanding the asymmetry of the weak nuclear force. In mathematics, twistor space is the complex vector space of solutions of the twistor equation ∇ A ′ ( A Ω B ) = 0 {displaystyle abla _{A'}^{(A}Omega ^{B)}=0} . It was described in the 1960s by Roger Penrose and Malcolm MacCallum. According to Andrew Hodges, twistor space is useful for conceptualizing the way photons travel through space, using four complex numbers. He also posits that twistor space may aid in understanding the asymmetry of the weak nuclear force. For Minkowski space, denoted M {displaystyle mathbb {M} } , the solutions to the twistor equation are of the form where ω A {displaystyle omega ^{A}} and π A ′ {displaystyle pi _{A'}} are two constant Weyl spinors and x A A ′ = σ μ A A ′ x μ {displaystyle x^{AA'}=sigma _{mu }^{AA'}x^{mu }} is a point in Minkowski space. This twistor space is a four-dimensional complex vector space, whose points are denoted by Z α = ( ω A , π A ′ ) {displaystyle Z^{alpha }=(omega ^{A},pi _{A'})} , and with a hermitian form