Use of tensor densities / pseudo tensors in physics

[burakumin]

Hello

I've sometime read physics texts that mention tensor densities (or pseudo-tensors). I find they are quite an ugly notion and I'm not sure to understand their necessity in physics. I have realized that tensor densities with an integer weight can be expressed differently with standard tensors and that the use of pseudotensors in this context is only a sort of "simplification of notation". But i'd really like to know if tensor densities with non-integer weights are really used in physics and in which context.

Thank you

 

[netheril96]

Two most used tensor densities: $$\det g_{\mu\nu}$$ and Levi-Civita symbol.

 

[burakumin]

Thank you netheril96 but I'm asking about tensor densities with non integer weights. To me things like $\det(g)$ and $\epsilon_{i_1i_2\ldots{}i_n}$ can be thought as real tensors. I'm looking for entities really used in physics that could not be seen this way.

Should this question be asked in the relativity forum as tensor calculus is frequent in this field ?

 

[netheril96]

Thank you netheril96 but I'm asking about tensor densities with non integer weights. To me things like $\det(g)$ and $\epsilon_{i_1i_2\ldots{}i_n}$ can be thought as real tensors. I'm looking for entities really used in physics that could not be seen this way.

Should this question be asked in the relativity forum as tensor calculus is frequent in this field ?

My bad. I personally have never encountered any tensor density of non-integer weight.

 

[robphy]

Wavefunctions in quantum mechanics are "densities of weight 1/2". (Google the quoted phrase.)

 

[netheril96]

Wavefunctions in quantum mechanics are "densities of weight 1/2". (Google the quoted phrase.)

Googled, and nothing interesting showed up.

And why are wavefunctions tensor densities?

 

[burakumin]

Wavefunctions in quantum mechanics are "densities of weight 1/2". (Google the quoted phrase.)

I'm not sure to understand why tensor terminology would be relevant in the case of Wavefunctions.

I personally have never encountered any tensor density of non-integer weight.

Ok. The article in wikipedia says "The transformations for even and odd tensor densities have the benefit of being well defined even when W is not an integer" but does not show any example of such objects. So I was wondering about their usefullness. The books I've skimed that mentioned tensor densities were a bit old so I guess this is not concept widely used anymore.

 

[robphy]

consult
Geroch's "Geometrical Quantum Mechanics",
III Quantum Mechanics
14. Densities. Integrals.
15. States

 

[burakumin]

consult
Geroch's "Geometrical Quantum Mechanics",
III Quantum Mechanics
14. Densities. Integrals.
15. States

Ok. The definition given for densities of weight W (W-homogenous applications from the maximum exterior power to a tensor space) is interesting and much cleaner than anything I've read before. Still I'm wondering if it is possible for a given (complex ?) vector space to define its "tensorial square root" so that densities of weights 1/2 could be defined as tensors on that space...

 

[burakumin]

W-homogenous applications from the maximum exterior power to a tensor space ...

I've suddenly realized something : using the definition of the article, tensor densities are ill-defined for numbers $\alpha < 0$ as $\alpha^s$ has no meaning for $s \not \in \mathbb{Z}$. And considering complex numbers does not solve the problem given that there is no unique possible solution ...