# Parity inversion and time reversal transformation in Lorentz group

I understand the Lorentz group have 4 connected components, one of them being the proper orthochronous group $$SO(1,3)^{+}$$, where its matrix elements have determinant +1 and the component at column $$0$$ and row $$0$$ is $$\geq1$$ (i.e. $$\Lambda_0^0\geq1$$). The other three components will be given by:

{$$T(SO(1,3)^{+})$$ , $$P(SO(1,3)^{+})$$, $$T(P(SO(1,3)^{+}))$$} where $$T$$ is a linear transformation that changes the sign of $$\Lambda_0^0$$ of a Lorentz transformation $$\Lambda$$, and $$P$$ is a linear transformation that changes the sign of the determinant of a lorentz transformation. If we consider:

$$\Lambda_{P}=\begin{pmatrix}1&&&\\&-1&&\\&&-1&\\&&&-1\end{pmatrix}.$$

$$\Lambda_{T}=\begin{pmatrix}-1&&&\\&1&&\\&&1&\\&&&1\end{pmatrix}.$$ I thought $$T$$ and $$P$$ would be defined by:

$$T(A) = \Lambda_TA$$ for any $$A$$ in $$SO(1,3)^{+}$$

$$P(A) = \Lambda_PA$$ for any $$A$$ in $$SO(1,3)^{+}$$

I was trying to verify this was true but most authors seem to exclude stating them more explicitly. I found in Wikipedia that these transformations actually behave as an adjoint action $$Ad_{\Lambda_P}$$, you can check here. It seems they are defined by:

$$T(A) = \Lambda_T^{-1}A\Lambda_T$$, $$P(A) = \Lambda_P^{-1}A\Lambda_P$$ , for any $$A$$ in $$SO(1,3)^{+}$$.

Is this true? If so, what is the intuition behind them having to be similarity transformations? why can't they just be a single matrix multiplication (from the left) with the given $$\Lambda_T$$ and $$\Lambda_P$$ as I thought?

I was looking for a more reputable source that stated P and T more explicitly, if you can give one I would appreciate it.

• Who is A ? An element of the full Lorentz group, or a geometrical object subject to a full Lorentz group transformation? @Qmechanic addresses the second possibility. Dec 5, 2018 at 13:09

## 1 Answer

It seems that OP is essentially asking

How do a physical quantity $A$ transforms under the Lorentz group $G=O(3,1)$, in particular wrt. the discrete generators $P$ and $T$?

Brief answer: That depends on the physical quantity $A$, and which $G$-representation it transforms under. To mention one simple example: A pseudoscalar is a quantity that changes sign under $P$, while a true scalar does not.