Time reversal and parity symmetry I was previously under the misapprehension that time $T$ and parity $P$ symmetries in conjunction ($PT$) were a reflection in $(3+1)$-dimensional space-time, where
$$P: \vec x \to -\vec x$$
$$T: t \to -t$$
but, in fact, because it has determinant $(-1)^4=1$, $PT$ is merely a rotation (by $\pi$). Is this correct? Shouldn't the symmetry we seek be a reflection in space time, to generalize parity so that it includes time?
In other words, if $P$ is a reflection in $3$ spatial dimensions, why isn't $PT$ defined to be a reflection in $(3+1)$-dimensional space-time? 
My thought is that $P$ is an intuitive symmetry in spatial dimensions. In the spirit of relativity, why don't we generalize $P$ from a reflection in space to a reflection in space-time (called, say, $PT$)? Why are $P$ and $T$ reflections considered separately? It's problematic to me because their combination ($PT$)  is not a reflection. I suppose I know that $P$ and $T$ are separate because only $T$ must be anti-unitary etc...
 A: $PT-, T-, P-$ transformations refer to subgroup of discrete transformations of the Lorentz group. They transform connected components of the Lorentz group between each other ($PT$ transformation transforms $L^{\uparrow}_{+}$ representation to $L^{\downarrow}_{+}$). In general, they can't be represented as the special case of rotation, which refer to subgroup of continuous transformations. You can't get some other connected component from orthochronous group by acting of any continuous transtormation's matrix on your representation. So, by nature, it can't be rotation.
A: 
why isn't PT defined to be a reflection in (3+1)-dimensional space-time?

It is a matter of defining what is a reflection. In the following paper: About Chirality in Minkowski Spacetime, Symmetry, 2019, 11[10], 1320 (https://doi.org/10.3390/sym11101320), P, T, and PT, are all mirrors.
A: Because the fourth dimension of spacetime,PT with transformations of left handed into right handed and vice-versa,then the transformations if Given by 2pi.with the violation of Pt occur the junction of space and time into spacetime continuos.then cpt is conserved,the violation of Pt generate the charge and appear a displacement currents for the propagations of electromagnetic Waves.
