- #1
tannin
- 3
- 0
Suppose we have a vector (contravariant) and we want to build an invariant.
a) we may take the direct product of the vector with some covariant vector (1-form obtained through metric tensor) and contract. The result is scalar.
b) we may take it's product with an axial vector (built with Levi-Civita symbol from antisymmetric tensor). The result is pseudoscalar.
I wonder if our vector may be acted on with some object giving result, that is neither scalar nor pseudoscalar, but having the following property:
with inversion of coordinates (in 3D) it acquires factor of e^{\imath \phi} ?
Does it imply necessarily that the metric should be complex?
a) we may take the direct product of the vector with some covariant vector (1-form obtained through metric tensor) and contract. The result is scalar.
b) we may take it's product with an axial vector (built with Levi-Civita symbol from antisymmetric tensor). The result is pseudoscalar.
I wonder if our vector may be acted on with some object giving result, that is neither scalar nor pseudoscalar, but having the following property:
with inversion of coordinates (in 3D) it acquires factor of e^{\imath \phi} ?
Does it imply necessarily that the metric should be complex?