It is known that vectors change them sing under the influence of parity when $(x,z,y)$ change into $(-x,-z,-y)$:

$P: y_{i} \rightarrow -y_{i}$

where $i=1,2,3$

But what about vectors in Minkowski space? Is it true that

$P: y_{\mu} \rightarrow -y_{\mu}$

where $\mu=0,1,2,3$. If yes how one can check it?


2 Answers 2


There is a charge conjugation operator C, a parity flip P, and a time reversal T. Conventionally, we define P to be only a spatial flip. If you want to flip all four coordinates, you can use the operator PT. People could have defined P to flip all four; it's just a matter of convention.

In an alternate world, they could have defined C and T as in our world, and Q as a flip of all four coordinates. Then to express what we call P, they would have written QT. What we refer to as CPT symmetry in particle physics, they would have called QC.

  • $\begingroup$ Thank you, but may you say what happens with vector if I do PT transform? I saw that y_0 component doesn't change in such case. Is this right? $\endgroup$
    – illuminato
    Jan 22, 2018 at 21:22

Note that a vector (contravariant components) is indicated with up indices, that is $y^{\mu}$.
A parity transformation is the flip in the sign of spatial coordinates. As in Minkowski spacetime a four-vector has a temporal coordinate as well, parity does not simply change the sign of the four vector. You have:
$P: y^{\mu} (+, +, +, +) \rightarrow (+, -, -, -) \neq -y^{\mu}$


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