What are some examples of mixed tensors in physics?

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    Mixed Physics Tensors
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Discussion Overview

The discussion centers on examples of mixed tensors in physics, specifically seeking instances that do not fall into purely covariant or contravariant categories. The scope includes theoretical and applied aspects of tensor analysis in various physical contexts.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant requests examples of mixed tensors, emphasizing the need for specificity beyond covariant or contravariant tensors.
  • Another participant cites the mixed tensor \(\delta^\mu_\nu\) as an example.
  • A participant mentions a mixed tensor of physical nature, though specifics are not provided.
  • Another example given is \(\kappa^\mu_\nu=\kappa\,\delta^\mu_\nu\), where \(\kappa\) is identified as the gravitational constant.
  • Riemann Curvature \(R_{abc}{}^d\) is presented as a mixed tensor, with a note that its trace leads to the Ricci Tensor relevant in Einstein's Field Equation, alongside a mention of Torsion \(T^a{}_{bc}\).
  • A participant introduces the representation of spin 3/2 particles using \(\psi_{\mu}^{\alpha}\), indicating that the spinorial index \(\alpha\) complicates the classification.
  • It is proposed that every operator in quantum mechanics can be viewed as a mixed tensor, with an example of a density matrix operator for spin 1/2 and the representation of a rigid rotation as a tensor \(R^i_j\).

Areas of Agreement / Disagreement

Participants present multiple examples of mixed tensors, but there is no consensus on a definitive list or classification. The discussion remains open with various viewpoints and examples provided.

Contextual Notes

Some examples may depend on specific definitions or contexts within physics, and the classification of certain entities as mixed tensors may be subject to interpretation.

quasar987
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Can anyone give me examples of mixed tensors that appear in physics? I'm looking for mixed specifically here: purely covariant or contravariant ones won't do.
 
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\delta^\mu_\nu
 
tensors of a physical nature...
 
\kappa^\mu_\nu=\kappa\,\delta^\mu_\nu

\kappa - gravitational constant
 
Riemann Curvature R_{abc}{}^d (whose trace gives the Ricci Tensor appearing in Einstein's Field Equation) and (maybe) Torsion T^a{}_{bc}.
 
Perhaps not entirely "tensors", but spin 3/2 particles represented by

<br /> \psi_{\mu}^{\alpha}<br />
where alpha is a spinorial index.
 
Every operator in quantum mechanics is a mixed tensor. In a basis it is represented as a tensor A^i_j. For instance, if we want to stay in finite dimensions, a density matrix operator for spin 1/2.

For space, every 3-dimensional active rigid rotation is an operator. It can be considered as a tensor R^i_j.
 

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