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My experience with index notation is strictly in the context of special relativity where the real vector space $V$ is equipped with a symmetric, non-degenerate bilinear form $g:V\times V\to \mathbb{R}$. However, in reading a book on Group theory in physics, I have come across the following strange conventions (see, for example, pg 293 of Group Theory in Physics by Wu-ki Tung)

$x_i^\dagger=\bar{x^i}$

$D^{\dagger}\ ^{i}\ _j =\bar{D^j}_i $

which make me question how the rules of the formalism need to be modified.

In particular, the whole concept of raising and lowering of indices relies on the existence of linear (musical) isomorphisms between $V$ and $V^*$. For a complex vector space equipped with a non-degenerate Hermitian form, these are no longer linear and the bilinear form no longer symmetric. How do the usual rules like $$g(V,W)=V^\alpha W_\alpha $$ $$T^i\ _j=g^{ik} T_{kj}$$ need to be changed?

I would be grateful if someone can explains this.

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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of what conventions a source uses, the usual rules of index notation concerning raising and lowring of indices are no longer valid.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

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When a vector space is over the field of complex numbers the anti-linearity of the first slot of the inner product $\langle {\bf x}| {\bf y}\rangle$ (defined by the non-degenerate hermitian form) means we can no longer make a simple identification of $V$ with $V^*$. Instead there is an anti-linear corresponence between the two spaces. The vector ${\bf x} \in V$ is mapped to $\langle {\bf x} |{\phantom x }\rangle $ which, since it returns a number when a vector is inserted into its vacant slot, is an element of $V^*$. This mapping is anti-linear because
$$ \lambda{\bf x}+\mu {\bf y} \mapsto \langle \lambda{\bf x}+\mu {\bf y}|{\phantom x}\rangle = \lambda^* \langle {\bf x}|{\phantom x}\rangle +\mu^*\langle {\bf y}|{\phantom x}\rangle . $$

This is best described by using Dirac notation where $V$ is the space of Dirac's "ket" vectors $|{\psi}\rangle $ and $V^*$ the space of "bra" vectors $\langle{\psi}|$. To each vector $|{\psi}\rangle $ we use the $(\ldots)^\dagger$ map to assign it a dual vector $$ |{\psi}\rangle \mapsto |{\psi}\rangle ^\dagger\,\,\equiv\,\, \langle {\psi}|$$ having the same labels. Dirac denoted the number resulting from the pairing of the covector $\langle{\psi}|$ with the vector $|{\chi}\rangle $ by the "bra-c-ket" symbol $\langle{\psi}|\chi\rangle$.

Using Dirac notations you can do all the "musical" maps without having to raise and lower indices.

You can find a fuller account in appendix A 3.3 of our book "Mathematics for Physics". A nonpaywalled draft verion is available at https://people.physics.illinois.edu/stone/bookmaster.pdf

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  • $\begingroup$ Thank you for your answer. I actually have a good idea of how how to work with complex vectorspaces using braket notation. My question is more about how index notation is used in such cases. In QFT in particular, I believe index notation is quite common even though the nondegenerate form (the innerproduct) is not bilinear $\endgroup$ Commented Dec 29, 2025 at 19:28
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    $\begingroup$ I agree that the other responses have addressed this better than I. There just isn't any real standard here. The example you give is as good as any. $\endgroup$ Commented Dec 29, 2025 at 21:53

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