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…
Best Answer
$PT-, T-, P-$ 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.