Improper and non-orthochronous group

  • Context: Graduate 
  • Thread starter Thread starter PRANAV UPADHYAY
  • Start date Start date
  • Tags Tags
    Group
Join the discussion
Ask a follow-up here, or get your own question answered by working scientists, mathematicians and engineers — people, not an autocomplete.
Real named experts · corrections over time · the nuance an AI answer skips
1 reply · 2K views
PRANAV UPADHYAY
Messages
1
Reaction score
0
will it be correct to say
00)2 – (Λ11)2 – (Λ22)2 – (Λ33)2 = 1

if Λ - group is neither proper nor orthochronous
 
Physics news on Phys.org
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).
 
  • Like
Likes   Reactions: PRANAV UPADHYAY and Orodruin