Here in this paper (and several others), the notation [d] is used. For instance here is Lemma 9 of the paper:

Let $T_d\in R[x]$ be the d-th Chebyshev polynomial of the first kind. Let $Φ\in R_d$ be such that $φ_1=(1−d)π/2$, and for all $i\in[d] \backslash \{1\}$ let $φ_i:=π/2$. Using this $Φ$ in equation (14) we get that $P = T_d$.

I assume $[d]$ means $\{0,...,d\}$, but I am not sure. Also, is the way they use the notation here common in math or quantum physics? To my knowledge, $[\cdot]$ usually denotes equivalence classes or the floor function.

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    $\begingroup$ I think it means $\{1,\ldots ,d\}$. $\endgroup$
    – Mark S
    Aug 26, 2022 at 11:42

1 Answer 1


In the paper the notation $[x]$ is being used in more than one way. As mentioned in the comments for counting dummy variables $[k]$ is a way for expressing the set $\{1,\dots,k\}$ etc. However there is for instance the notation $\mathbb{C}[x]$, that denotes a set of complex polynomials with respect to the family $[x]$ ranging from zero to $k$. The notation $\mathbb{R}[x]$ denotes the polynomial but with real coefficients. See the beginning of section 2.

You are correct in general we have that $[A]$ can be the equivalence class of all elements that are equivalent to $A$ with respect to an equivalence relation $\sim$ and a set $\mathcal{A}$. Its not the case here. Moreover it is also common to use this notation for the floor function, and once more is not the case. Note that the same notation is even being used in another context for generic matrix elements in the text, so it is to be understood by the reader each case to be considered.

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    $\begingroup$ @MarkS edited; thanks $\endgroup$
    – R.W
    Aug 27, 2022 at 5:49
  • $\begingroup$ Thank you very much for your helpful answer. Regarding the notation of the authors with [k]={1,…,k}, this is totally non-standard use, right? Or have you seen this notation used before in this way? If only the authors have used this notation in this new way, in my view they should have defined the new notation in their paper. $\endgroup$
    – Jadzia
    Aug 27, 2022 at 10:23
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    $\begingroup$ This notation is very common yes. Both notation $[a] \equiv \{1,\dots,a\}$ and $\underline{a} \equiv \{1,\dots,a\}$ are common; I have used in papers and seen used in other papers as well. $\endgroup$
    – R.W
    Aug 27, 2022 at 11:32
  • $\begingroup$ That's good to know, thanks. Maybe this usage should be added to Wikipedia: en.wikipedia.org/wiki/… $\endgroup$
    – Jadzia
    Aug 27, 2022 at 12:24

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