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Some "clarification"

To begin with, I'm not even talking about relativity so, in the following, rotations always act on the Euclidean space or only the space subpart of the Minkowski space.

Second, here parity means we multiply all the space components by a minus sign.

The question

Mathematically, it would be quite easy for me to understand that parity is an improper rotation in the odd dimension, but is a proper rotation in the even dimension. We just need to consider the definition of being proper, $\det=1$.

But, how to comprehend this conclusion physically? And is there any physical consequence from this difference? For example, is there any physical phenomenology which is totally different for 2D and 3D and which is due to this difference?

Qmechanic
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Qi Tianluo
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1 Answers1

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This has to do with that in $n$ spatial dimensions, the parity/point-reflection is represented by the matrix $-{\bf 1}_{n\times n}$. We can chip away 2 diagonal minus signs in the $i$th and $j$th spatial direction by making a $180^{\circ}$ rotation in the $(i,j)$ plane. In this way we can get rid of an even number of minus signs. Therefore in even dimensions $n$, we can rotate $-{\bf 1}_{n\times n}$ to $+{\bf 1}_{n\times n}$, but in odd dimensions $n$, there remains 1 minus sign that signifies an additional mirror-reflection.

For higher-dimensional rotations, see also e.g. this & this related Phys.SE posts.

Qmechanic
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    Thank you very much for your answer, but to my understanding, this explanation is still mathematical, but not physical. So how to understand this from the point of view of physicist? – Qi Tianluo May 21 '22 at 09:11