Editing Lorentz transformation (section)

=== Spinors ===
Equation {{EquationNote|(T1)}} hold unmodified for any representation of the Lorentz group, including the [[bispinor]] representation. In {{EquationNote|(T2)}} one simply replaces all occurrences of {{math|Λ}} by the bispinor representation {{math|Π(Λ)}},

{{Equation box 1
 |indent =:
 |equation =
<math>\begin{align}
u \otimes v \rightarrow \Pi(\Lambda) u \otimes \Pi(\Lambda) v &=
  {\Pi(\Lambda)^\alpha}_\beta u^\beta \otimes {\Pi(\Lambda)^\rho}_\sigma v^\sigma\\ &=
  {\Pi(\Lambda)^\alpha}_\beta {\Pi(\Lambda)^\rho}_\sigma u^\beta \otimes v^\sigma\\ &\equiv
  {\Pi(\Lambda)^\alpha}_\beta {\Pi(\Lambda)^\rho}_\sigma w^{\beta\sigma}
\end{align}</math> {{spaces|13}} {{EquationRef|(T4)}}
 |cellpadding= 6
 |border
 |border colour = #0073CF
 |bgcolor=#F9FFF7
}}

The above equation could, for instance, be the transformation of a state in [[Fock space]] describing two free electrons.

==== Transformation of general fields ====
A general ''noninteracting'' multi-particle state (Fock space state) in [[quantum field theory]] transforms according to the rule<ref>{{harvnb|Weinberg|2002|loc=Chapter 3}}</ref>
{{NumBlk||<math display="block">\begin{align}
         &U(\Lambda, a) \Psi_{p_1\sigma_1 n_1; p_2\sigma_2 n_2; \cdots} \\
    = {} &e^{-ia_\mu \left[(\Lambda p_1)^\mu + (\Lambda p_2)^\mu + \cdots\right]}
            \sqrt{\frac{(\Lambda p_1)^0(\Lambda p_2)^0\cdots}{p_1^0 p_2^0 \cdots}}
            \left( \sum_{\sigma_1'\sigma_2' \cdots} D_{\sigma_1'\sigma_1}^{(j_1)}\left[W(\Lambda, p_1)\right] D_{\sigma_2'\sigma_2}^{(j_2)}\left[W(\Lambda, p_2)\right] \cdots \right)
            \Psi_{\Lambda p_1 \sigma_1' n_1; \Lambda p_2 \sigma_2' n_2; \cdots},
  \end{align}</math>
 | {{EquationRef|1}}
}}
where {{math|''W''(Λ, ''p'')}} is the [[Wigner's classification|Wigner's little group]]<ref>{{Cite web |title=INSPIRE |url=https://inspirehep.net/literature/26312 |access-date=2024-09-04 |website=inspirehep.net}}</ref> and {{math|''D''{{sup|(''j'')}}}} is the {{nowrap|{{math|(2''j'' + 1)}}-dimensional}} representation of {{math|SO(3)}}.