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I am from the math department and reading Belavin & Gebner's On the Algebraic Approach to Solvable Lattice Models.

I am trying to understand the left-hand side of Equation (2.2) on page 4.

What is meaning of the notation $\langle a_1, \ldots, a_n \mid X_i(u) \mid a_1', \ldots, a_n' \rangle$?

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    $\begingroup$ The $\left|\dots\right\rangle$ is a vector in Hilbert space (a ket), the $X_i(u)$ is some operator on that Hilbert space, and $\left\langle \dots \right|$ is a vector in the dual space. Overall, it's an inner product. See: en.wikipedia.org/wiki/Bra%E2%80%93ket_notation $\endgroup$
    – zeldredge
    Commented Mar 1, 2019 at 20:04
  • $\begingroup$ @zeldredge, thank you very much. What are the $a_i$'s? $\endgroup$ Commented Mar 1, 2019 at 22:42
  • $\begingroup$ This is very common notation used in quantum mechanics. Every undergraduate text should have at least a chapter on this, if it doesn't use it throughout the whole of the text. $\endgroup$
    – Kyle Kanos
    Commented Mar 2, 2019 at 19:36

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Physically, the $|a_1',...,a_n'⟩$ term represents a state of your system. Mathematically, it is a vector of the Hilbert space.

Each $a_i$ represents the occupation number of the $i$-th site of your lattice, ie. the number of particles on that site. Here, this notation would imply that your lattice has $n$ different sites, and that a state is fully characterized by the number of particles in each lattice site.

If you know second quantization, this state can also be written as $(b_1^\dagger)^{a_1'}...(b_n^\dagger)^{a_n'}|vac⟩$, where $|vac⟩$ is a state with no particles in the system at all, and each of the $b_i^\dagger$ operators adds one particle to the $i$-th site.

Then, $X_i(u)$ is an operator acting on your state, and $⟨a_1, ..., a_n|$ is also a state of the system but in the dual space.

One way to understand the full expression is the following. If your lattice starts from the state $|a_1',...,a_n'⟩$ and you apply the operator $X_i(u)$ on your system, you will end up with the projection of this new state of your system on the state $|a_1,...,a_n⟩$.

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