A question on the Lorentz group and proper, orthochronous transformations Why only the subgroup of orthochronous and proper Lorentz transformations (i.e. those simultaneously satisfying $\Lambda_{\;0}^{0}> 0$ and $\text{det}\,\Lambda =1$, respectively) are considered to be physically realisable transformations?
Is it simply because by Lorentz symmetry, all Lorentz transformations must be orthochronous, since if they were not, then one could distinguish between two different inertial frames (time would be running in the opposite sense in one frame relative to another)? 
Assuming the above statement is true, then, since all orthochronous Lorentz transformations are continuously connected to the identity, it follows that the physically realizable Lorentz transformations must also be proper, since improper, orthochronous Lorentz transformations cannot be continuously connected to the identity. 
 A: The full Lorentz group features multiple branches, between which no continuous transformations can interpolate. They are related through discrete transformations though. The restriction to proper orthochronous Lorentz transformations ensures that the transformations we deal with are a continuous branch of a Lie group, for which the representation theory is easier.
Note that the transformations used to restrict the Lorentz group are not always realised in the system. Time reversal and parity are violated in the Standard Model, for example. But if one keeps these subtleties in mind, the restriction is w.l.o.g. (with which I mean that the objects in our theory have to be (not necessarily trivial) representations under T and P).
A: I wouldn't say that orthochronous proper Lorentz transformations are the only physically realizable. The rest of the components of the Lorentz group can be reached via parity or/and time reversal operating on the orthochronus proper Lorentz subgroup, and we know that these symmeties exist in our universe.
A different thing is that because of this relation between the different components of the Lorentz group, we can limit the study of the Lorentz group to the Lorentz orthochronous proper subgroup.
