Eikonal Blog

2010.02.13

Twistors

Filed under: mathematics, physics — Tags: — sandokan65 @ 01:09

Twistor variables:

  • direct \{{\cal Z}^\alpha|\alpha=1,2\},
  • conjugated: \{\bar{{\cal Z}}_\beta|\beta=1,2\}.

Quantization: [{\cal Z}^\alpha, \bar{{\cal Z}}_\beta] = \hbar \delta^\alpha_\beta.

Canonical representation: \bar{{\cal Z}}_\alpha \rightarrow -\hbar \frac{\partial}{\partial {\cal Z}^\alpha}.

Helicity: s :\equiv \frac12 \left\{{\cal Z}^\alpha,\bar{{\cal Z}}_\alpha\right\} = -\frac12\hbar \left(2+ {\cal Y} \right),
where {\cal Y} :\equiv {\cal Z}^\alpha\frac{\partial}{\partial {\cal Z}^\alpha} is the Euler’s homogeneity operator.

Source: Roger Penrose “The Road to Reality”

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