Оператор для спиноров

Оператор группы Лоренца

После построения отдельных полуоператоров по осям, как для быстрот так и для углов поворотов, построим полуоператоры преобразования Лоренца для бустов в произвольном направлении для спиноров.

Для этого подставим все три параметра быстроты

trigreduce(smakeop(psi[x]/2,psi[y]/2,psi[z]/2,
  0,0,0));

Для упрощения автоматическими средствами системы Maxima добавим предварительные определения для упрощения:

tellsimpafter(
  psi[x]^2+psi[y]^2+psi[z]^2,
  psi^2);
tellsimpafter(
  phi[x]^2+phi[y]^2+phi[z]^2,
  phi^2);
tellsimpafter(
  phi[z]^2/4+phi[y]^2/4+phi[x]^2/4,
  phi^2/4);
tellsimpafter(
  atan2(0,-psi^2/4),
  %pi);

Учитывая четность гиперболических функций и введя направляющие косинусы для вектора быстроты $$ \begin{array}{c} \sqrt{\psi_x^2+\psi_y^2+\psi_z^2}=\psi \\ \psi_x/|\psi|=\cos(\chi_x) \\ \psi_y/|\psi|=\cos(\chi_y) \\ \psi_z/|\psi|=\cos(\chi_z) \\ \end{array} $$ Получаем полуоператор $$ \left( \begin{array}{cc} \mathrm{ch}(\psi/2)-\mathrm{sh}(\psi/2)\cos(\chi_x) & -\mathrm{sh}(\psi/2)\cos(\chi_z)+i\mathrm{sh}(\psi/2)\cos(\chi_y)\\ -\mathrm{sh}(\psi/2)\cos(\chi_z)-i\mathrm{sh}(\psi/2)\cos(\chi_y) & \mathrm{ch}(\psi/2)+\mathrm{sh}(\psi/2)\cos(\chi_x) \end{array} \right) $$ Для получения полуоператора поворота с тремя параметрами подставим их:

trigreduce(smakeop(0,0,0,
  phi[x]/2,phi[y]/2,phi[z]/2));

Учтем четность тригонометрических функций и сделав замену на направляющие косинусы для вектора угола поворота: $$ \begin{array}{c} \sqrt{\varphi_x^2+\varphi_y^2+\varphi_z^2}=\varphi \\ \varphi_x/|\varphi|=\cos(\chi_x) \\ \varphi_y/|\varphi|=\cos(\chi_y) \\ \varphi_z/|\varphi|=\cos(\chi_z) \\ \end{array} $$ Получаем полуоператор поворота с тремя параметрами: $$ \left( \begin{array}{cc} \cos(\varphi/2)+i\sin(\varphi/2)\cos(\chi_x) & \sin(\varphi/2)\cos(\chi_y)+i\sin(\varphi/2)\cos(\chi_z) \\ -\sin(\varphi/2)\cos(\chi_y)+i\sin(\varphi/2)\cos(\chi_z) & \cos(\varphi/2)-i\sin(\varphi/2)\cos(\chi_x) \end{array} \right) $$ Для получения оператора преобраования Лоренца для спиноров при 6-ти параметрах понадобится не только подставить 6 параметров, но и предварительно задать правила упрощения:

tellsimpafter(atan2((phi[z]*psi[z])/2+
  (phi[y]*psi[y])/2+(phi[x]*psi[x])/2,
  -psi[z]^2/4+phi[z]^2/4-psi[y]^2/4+
  phi[y]^2/4-psi[x]^2/4+phi[x]^2/4),Theta);
tellsimpafter(-psi[z]^2+phi[z]^2-psi[y]^2+
  phi[y]^2-psi[x]^2+phi[x]^2,phi^2-psi^2);
tellsimpafter(psi[z]^4+2*phi[z]^2*psi[z]^2+
  2*psi[y]^2*psi[z]^2-
  2*phi[y]^2*psi[z]^2+2*psi[x]^2*psi[z]^2-
  2*phi[x]^2*psi[z]^2+
  8*phi[y]*psi[y]*phi[z]*psi[z]+
  8*phi[x]*psi[x]*phi[z]*psi[z]+
  phi[z]^4-2*psi[y]^2*phi[z]^2+
  2*phi[y]^2*phi[z]^2-2*psi[x]^2*phi[z]^2+
  2*phi[x]^2*phi[z]^2+psi[y]^4+
  2*phi[y]^2*psi[y]^2+2*psi[x]^2*psi[y]^2-
  2*phi[x]^2*psi[y]^2+
  8*phi[x]*psi[x]*phi[y]*psi[y]+phi[y]^4-
  2*psi[x]^2*phi[y]^2+
  2*phi[x]^2*phi[y]^2+psi[x]^4+
  2*phi[x]^2*psi[x]^2+phi[x]^4,
  (psi^2-phi^2)^2+4*(psi[x]*phi[x]+
  psi[y]*phi[y]+psi[z]*phi[z])^2);
tellsimpafter((-psi[z]^2/4+phi[z]^2/4-
  psi[y]^2/4+phi[y]^2/4-
  psi[x]^2/4+phi[x]^2/4)^2+
  ((phi[z]*psi[z])/2+(phi[y]*psi[y])/2+
  (phi[x]*psi[x])/2)^2,alpha);

И,конечно, зададим все 6 параметров:

smakeop(psi[x]/2,psi[y]/2,psi[z]/2,
  phi[x]/2,phi[y]/2,phi[z]/2);

Получаем оператор преобразования с 6-ю параметрами при подстановке промежуточных переменных, образуемых из параметров: $$ \frac{-\psi_x^2-\psi_y^2-\psi_z^2+ \varphi_x^2+\varphi_y^2+\varphi_z^2}{4}+ \frac{\varphi_x\psi_x+\varphi_y\psi_y+\varphi_z\psi_z}{4}=\alpha $$ $$ \mathrm{arctg}(2\frac{\varphi_x\psi_x+\varphi_y\psi_y+\varphi_z\psi_z} {-\psi_x^2-\psi_y^2-\psi_z^2+ \varphi_x^2+\varphi_y^2+\varphi_z^2})=\Theta $$ $$ \begin{array}{c} A_{11}=i(((\sin(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})+\\ \cos(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\varphi_x)/(2\alpha^{1/4})\\ -((\cos(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})-\\ \sin(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\psi_x)/(2\alpha^{1/4})\\ -\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4}))\\ -((\sin(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})+\\ \cos(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\psi_x)/(2\alpha^{1/4})\\ +((\cos(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})-\\ \sin(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\varphi_x)/(2\alpha^{1/4})\\ +\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}) \end{array} $$ $$ \begin{array}{c} A_{12}=i(((\sin(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})+\\ \cos(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\varphi_z)/(2\alpha^{1/4})\\ -((\cos(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})-\\ \sin(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\psi_z)/(2\alpha^{1/4})\\ +((\sin(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})+\\ \cos(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\psi_y)/(2\alpha^{1/4})\\ +((\cos(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})-\\ \sin(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\varphi_y)/(2\alpha^{1/4}))\\ -(((\sin(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})+\\ \cos(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\psi_z)/(2\alpha^{1/4}))\\ +((\cos(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})-\\ \sin(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\varphi_z)/(2\alpha^{1/4}))\\ +((\sin(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})+\\ \cos(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\varphi_y)/(2\alpha^{1/4}))\\ -((\cos(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})-\\ \sin(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\psi_y)/(2\alpha^{1/4})) \end{array} $$ $$ \begin{array}{c} A_{21}=i(((\sin(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})+\\ \cos(\Theta)\sin(\cos(\Theta)\alpha^{1/4})\mathrm{ch}(\sin(\Theta)\alpha^{1/4} ))\varphi_z)/(2\alpha^{1/4})\\ -((\cos(\Theta)\cos(\cos(\Theta)\alpha^{1/4})\mathrm{sh}(\sin(\Theta)\alpha^{1/4 })-\\ \sin(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\psi_z)/(2\alpha^{1/4})\\ -((\sin(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})+\\ \cos(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\psi_y)/(2\alpha^{1/4})\\ +((\cos(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})-\\ \sin(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\varphi_y)/(2\alpha^{1/4}))\\ -((\sin(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})+\\ \cos(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\psi_z)/(2\alpha^{1/4})\\ +((\cos(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})-\\ \sin(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\varphi_z)/(2\alpha^{1/4})\\ -((\sin(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})+\\ \cos(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\varphi_y)/(2\alpha^{1/4})\\ -((\cos(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})-\\ \sin(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\psi_y)/(2\alpha^{1/4}) \end{array} $$ $$ \begin{array}{c} A_{22}=i(-((\sin(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})+\\ \cos(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\varphi_x)/(2\alpha^{1/4})\\ -((\cos(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})-\\ \sin(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\psi_x)/(2\alpha^{1/4})\\ -\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4}))\\ +((\sin(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})+\\ \cos(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\psi_x)/(2\alpha^{1/4})\\ +((\cos(\Theta)\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{sh}(\sin(\Theta)\alpha^{1/4})-\\ \sin(\Theta)\sin(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}))\varphi_x)/(2\alpha^{1/4})\\ +\cos(\cos(\Theta)\alpha^{1/4}) \mathrm{ch}(\sin(\Theta)\alpha^{1/4}) \end{array} $$

Оператор группы Лоренца