1
$\begingroup$

A tensor $T$ is an object which is invariant under all coordinate transformations: $$ T\mapsto T = e^{i0}T. $$ A pseudotensor $P$ is an object which changes its sign under the inversion of a coordinate axis: $$ P\mapsto P = e^{i\pi}P. $$ As shown above, these are two special cases of the more general mapping $$ G \mapsto e^{i\theta}G $$ for some generalised object $G$ under the inversion of a coordinate axis. Do any objects exist which have $\theta\neq 0, \pi$? If so, where do they arise and what are they called?

$\endgroup$
0
$\begingroup$

A tensor, unless it is scalar, is in general not invariant under coodinate transformations. It is co-variant.

Also, what you are asking for above is that, I presume, a real-valued tensor, would undergo coordinate transform, also real-valued, and would then become complex-valued. This is not possible. What is possible however is to work in sufficiently exotic basis where tensors no-longer have real-valued components.

For example, if you switch from XYZ-basis to circular basis, i.e. +-0, then rotation by angle $\theta$ around z-axis will act on a tensor $V^{\alpha_1\dots\alpha_n}_{\beta_1\dots\beta_n}$ as:

$V^{\alpha_1\dots\alpha_n}_{\beta_1\dots\beta_n} \overset{R_z\left(\theta\right)}{\to} \exp\left(i\theta\cdot\left(\alpha_1+\alpha_2+\dots\alpha_n-\beta_1-\beta_2-\dots\beta_n\right)\right)V^{\alpha_1\dots\alpha_n}_{\beta_1\dots\beta_n}$

Where $\alpha_k=\pm 1,0$ etc. But this is coordinate-specific.

Similar, but not so-simple expressions, can be worked out for reflections/inversions, but since I doubt this is what you are after, I will not develop it further for now.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.