Improper and non-orthochronous group

[PRANAV UPADHYAY]

will it be correct to say
(Λ⁰₀)² – (Λ¹₁)² – (Λ²₂)² – (Λ³₃)² = 1

if Λ - group is neither proper nor orthochronous

 

[vanhees71]

That's never correct. What you mean is
$$\eta_{\mu \nu} {\Lambda^{\mu}}_{\rho} {\Lambda^{\nu}}_{\sigma}=\eta_{\rho \sigma},$$
which defines the full Lorentz group $\mathrm{O}(1,3)$.

The special Lorentz group has the additional constraint that $\mathrm{det} ({\Lambda^{\mu}}_{\nu})=+1$. The corresponding subgroup is called $\mathrm{SO}(1,3)$.

Then there is the orthochronous Lorentz group $\mathrm{O}(1,3)^{\uparrow}$, for which ${\Lambda^{0}}_0 \geq 1$.

Finally there's the subgroup which is the part that is continuously connected with the idendity, and that's the proper orthochronous Lorentz group. It consists of all orthochronous special Lorentz transformations, forming the group $\mathrm{SO}(1,3)^{\uparrow}$, and it's the group that is describes a fundamental symmetry of nature (together with the space-time translations it's the symmetry group of Minkowski space that is realized as a symmetry group of nature, the proper orthochronous Poincare group).