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

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 $$\langle a_1, \ldots, a_n \mid X_i(u) \mid a_1', \ldots, a_n' \rangle$$?

• 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 Mar 1, 2019 at 20:04
• @zeldredge, thank you very much. What are the $a_i$'s? Mar 1, 2019 at 22:42
• 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. Mar 2, 2019 at 19:36

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 $$|a_1,...,a_n⟩$$.