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    $\begingroup$ The notation you cite from Tung is, to say the least, not universal. I'm not sure what kind of answer you expect here except that you have to understand the conventions of whatever source you're reading. $\endgroup$ Commented Dec 28, 2025 at 22:38
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    $\begingroup$ I agree with @ACuriousMind that the notation may change from author to author. That being said, Wald also uses a similar notation in his book Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics and discusses it well in the book's appendix (which is pretty much independent from the rest of the book). Maybe it could help you $\endgroup$ Commented Dec 28, 2025 at 23:18
  • $\begingroup$ @ACuriousMind My question is not strictly about understanding the notation in the book is but more about how index notation is used on complex vector spaces. Irrespective of what conventions a source uses, the usual rules of index notation concerning raising and lowring of indices are no longer valid. I want to know how one may extend index notation to cater to these cases $\endgroup$ Commented Dec 29, 2025 at 6:28
  • $\begingroup$ @Níck I took a look and it was actually quite helpful. I'm still hoping to get an answer here but the appendix does resolve some of my queries $\endgroup$ Commented Dec 29, 2025 at 6:29
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    $\begingroup$ My point is more that there is just nothing universal to discuss about "how index notation is used in complex vector spaces" - in contrast to GR/differential geometry there's no de facto standard most texts will be at least close to. Personally I'd usually not even use any particular meaning for upper and lower indices in this case. $\endgroup$ Commented Dec 29, 2025 at 12:33