The fifteen local generators

Displayed formulas in this passage

• Oai282-B@1#eq:CK · line 773

  $${\partial }_{\mu }{X}_{\nu }+{\partial }_{\nu }{X}_{\mu }=2{\sigma }_X{\eta }_{\mu \nu },{\sigma }_X=\frac{1}{4}{\partial }_{\rho }{X}^{\rho }.$$

• Oai282-B@1#eq:CKaux · line 799

  $$\square X^{\mu }=-2{\partial }^{\mu }{\sigma }_X,\square {\sigma }_X=0.$$

• Oai282-B@1#eq:waveintertwine · line 813

  $$\square d_X=(d_X+2{\sigma }_X)\square .$$

• unlabeled · line 781

  $$\begin{array}{lll}\text{translation:}& X^{\mu }=a^{\mu },& {\sigma }_X=0,\\ \text{Lorentz:}& X^{\mu }={\omega }^{\mu }{}_{\nu }{x}^{\nu },{\omega }_{\mu \nu }=-{\omega }_{\nu \mu },& {\sigma }_X=0,\\ \text{dilation:}& X^{\mu }=\lambda x^{\mu },& {\sigma }_X=\lambda ,\\ \text{special conformal:}& X^{\mu }=2(b\cdot x)x^{\mu }-b^{\mu }{x}^2,& {\sigma }_X=2b\cdot x.\end{array}$$

• unlabeled · line 807

  $$d_X=X^{\mu }{\partial }_{\mu }+{\sigma }_X,v_X\varphi =d_X\varphi .$$

• unlabeled · line 820

  $${j_X}^{\mu }=t^{\mu \nu }{X}_{\nu }$$

• unlabeled · line 824

  $${\partial }_{\mu }{j_X}^{\mu }=({\partial }_{\mu }{t}^{\mu \nu })X_{\nu }+t^{\mu \nu }{\partial }_{\mu }{X}_{\nu }={\sigma }_X{t}^{\mu }{}_{\mu }=0.$$