PhyslibAlpha.Mathematics.LadderSystem.Vacuum

Given a ladder system $L$ over a field $K$ and a vector space $V$ with creation operators $a^\dagger_i$ for $i \in \{1, \dots, d\}$, the function `word` takes a list of indices $v = [i_1, i_2, \dots, i_n]$ and a vector $x \in V$, and returns the vector obtained by the sequential application of the corresponding creation operators to $x$: $$a^\dagger_{i_1} a^\dagger_{i_2} \dots a^\dagger_{i_n} x$$

Let $L$ be a ladder system over a field $K$ and a vector space $V$ with creation operators $a^\dagger_j$ for $j \in \{1, \dots, d\}$. For any index $i \in \{1, \dots, d\}$, any list of indices $v = [i_1, \dots, i_n]$, and any vector $x \in V$, the creation-operator word associated with the list obtained by prepending $i$ to $v$ applied to $x$ is equal to the creation operator $a^\dagger_i$ acting on the word associated with $v$ applied to $x$: $$\text{word}(i :: v, x) = a^\dagger_i (\text{word}(v, x))$$

Let $L$ be a ladder system over a field $K$ and a vector space $V$ with $d$ degrees of freedom, where $a^\dagger_i$ denotes the creation operator for index $i \in \{1, \dots, d\}$. For any vector $x \in V$ and any two lists of indices $v = [i_1, \dots, i_n]$ and $v' = [j_1, \dots, j_n]$ that are permutations of each other, the resulting creation-operator words applied to $x$ are equal: $$a^\dagger_{i_1} a^\dagger_{i_2} \dots a^\dagger_{i_n} x = a^\dagger_{j_1} a^\dagger_{j_2} \dots a^\dagger_{j_n} x$$

Let $L$ be a ladder system over a field $K$ and a vector space $V$ with $d$ degrees of freedom. Let $a_i$ and $a_i^\dagger$ denote the annihilation and creation operators respectively for index $i \in \{1, \dots, d\}$. Suppose $x \in V$ is a vector such that $a_i x = 0$. For any sequence of indices $v = [i_1, i_2, \dots, i_n]$, the action of the annihilation operator $a_i$ on the creation-operator word $a^\dagger_{i_1} a^\dagger_{i_2} \dots a^\dagger_{i_n} x$ is given by: $$a_i (a^\dagger_{i_1} a^\dagger_{i_2} \dots a^\dagger_{i_n} x) = \text{count}(i, v) \cdot (a^\dagger_{j_1} a^\dagger_{j_2} \dots a^\dagger_{j_{n-1}} x)$$ where $\text{count}(i, v)$ is the number of times the index $i$ appears in the sequence $v$, and the word on the right-hand side is obtained by removing one occurrence of the index $i$ from the original sequence.

Let $L$ be a ladder system on a $K$-vector space $V$ with $d$ modes. Suppose $x \in V$ is a vacuum state, such that $a_k x = 0$ for all $k \in \{0, \dots, d-1\}$. For any list of indices $v = [v_1, \dots, v_n]$ and indices $i, j \in \{0, \dots, d-1\}$, let the creation-operator word be defined as $a^\dagger_v x = a^\dagger_{v_1} a^\dagger_{v_2} \dots a^\dagger_{v_n} x$. Then the action of the operator $E_{ij} = a^\dagger_i a_j$ on this word is given by $$E_{ij} (a^\dagger_v x) = \text{count}(j, v) \cdot a^\dagger_{i :: \text{erase}(j, v)} x$$ where $\text{count}(j, v)$ is the number of times the index $j$ appears in the list $v$, and $i :: \text{erase}(j, v)$ is the list formed by removing one occurrence of $j$ from $v$ and prepending $i$.

Let $L$ be a ladder system on a $K$-vector space $V$ with $d$ modes. Let $x \in V$ be a vacuum state such that $a_k x = 0$ for all $k \in \{0, \dots, d-1\}$. For any mode $i \in \{0, \dots, d-1\}$ and any list of indices $v = [v_1, v_2, \dots, v_n]$, let the creation-operator word be $a^\dagger_v x = a^\dagger_{v_1} a^\dagger_{v_2} \dots a^\dagger_{v_n} x$. Then the action of the number operator $N_i$ on this word is given by $$N_i (a^\dagger_v x) = \text{count}(i, v) \cdot a^\dagger_v x$$ where $\text{count}(i, v)$ denotes the number of occurrences of the index $i$ in the list $v$. This implies that creation-operator words are eigenvectors of the number operator $N_i$ with eigenvalue equal to the count of the corresponding mode in the word.

Given a number of colors $d \in \mathbb{N}$ and a count function $\alpha : \{0, \dots, d-1\} \to \mathbb{N}$, this function constructs a list (or "word") of indices by repeating each color $c$ exactly $\alpha(c)$ times. The resulting list is ordered such that all occurrences of color $c$ appear before those of color $c+1$, resulting in a word of the form $[\underbrace{0, \dots, 0}_{\alpha(0)}, \underbrace{1, \dots, 1}_{\alpha(1)}, \dots, \underbrace{d-1, \dots, d-1}_{\alpha(d-1)}]$.

For any natural number $d$, a count function $\alpha : \{0, \dots, d-1\} \to \mathbb{N}$, and an index $i \in \{0, \dots, d-1\}$, the number of occurrences (count) of the index $i$ in the canonical word $\text{countWord}(d, \alpha)$ is equal to $\alpha(i)$.

Let $v$ be a list of indices (representing colors or modes) from the set $\{0, 1, \dots, d-1\}$. For each index $c$, let $\text{count}(v, c)$ denote the number of times $c$ appears in the list $v$. Then the sum of these counts over all possible indices $c$ is equal to the total length of the list $v$: $$\sum_{c=0}^{d-1} \text{count}(v, c) = \text{length}(v)$$

Given a ladder system $L$ over a field $K$ and a vector space $V$ with $d$ modes, let $\Omega \in V$ be a vector (typically the vacuum state). The $n$-th excitation-number sector, denoted as `vacuumSpan L Ω n`, is the $K$-linear submodule of $V$ spanned by the set of all vectors obtained by applying a word of $n$ creation operators to $\Omega$. That is, it is the span of all vectors of the form $$a^\dagger_{i_1} a^\dagger_{i_2} \dots a^\dagger_{i_n} \Omega$$ where each index $i_j \in \{0, 1, \dots, d-1\}$.

Let $l$ be a list of elements from the finite set $\{0, 1, \dots, d-1\}$ with length $n$. There exists a function $w : \{0, 1, \dots, n-1\} \to \{0, 1, \dots, d-1\}$ such that the list $l$ is equal to the list generated by evaluating $w$ on $0, 1, \dots, n-1$ (denoted as $[w(0), w(1), \dots, w(n-1)]$).

Let $L$ be a ladder system over a field $K$ and a vector space $V$ with $d$ modes. Let $\Omega \in V$ be a vector (typically the vacuum state). For any list of indices $l = [i_1, i_2, \dots, i_k]$ where each $i_j \in \{0, 1, \dots, d-1\}$, if the length of the list is $n$ (i.e., $k = n$), then the creation-operator word $$a^\dagger_{i_1} a^\dagger_{i_2} \dots a^\dagger_{i_n} \Omega$$ is an element of the $n$-th excitation-number sector $\text{vacuumSpan}(L, \Omega, n)$.

Let $L$ be a ladder system on a $K$-vector space $V$ with $d$ modes, and let $W \subseteq V$ be a $K$-submodule. If $W$ is invariant under the operators $E_{ij} = a^\dagger_i a_j$ for all $i, j \in \{0, \dots, d-1\}$, then $W$ is also invariant under the action of any matrix $x \in \text{M}_d(K)$ via the linear map $\rho: \text{M}_d(K) \to \text{End}_K(V)$ that sends matrix units to $E_{ij}$. Specifically, for any $x \in \text{M}_d(K)$ and $w \in W$, we have $\rho(x)w \in W$.

Let $V$ be a $K$-vector space equipped with a ladder system $L$ of $d$ modes. Let $\Omega \in V$ be a vacuum state, meaning $\Omega \neq 0$ and $a_k \Omega = 0$ for all $k \in \{0, \dots, d-1\}$. Let $\text{vacuumSpan}(L, \Omega, n)$ be the $n$-th excitation-number sector, which is the $K$-linear subspace of $V$ spanned by all vectors obtained by applying a word of $n$ creation operators to the vacuum: $$\text{span}_K \{ a^\dagger_{i_1} a^\dagger_{i_2} \dots a^\dagger_{i_n} \Omega \mid i_1, \dots, i_n \in \{0, \dots, d-1\} \}.$$ For any natural number $n$ and any indices $i, j \in \{0, \dots, d-1\}$, the operator $E_{ij} = a^\dagger_i a_j$ preserves the $n$-th excitation-number sector; that is, for every vector $v \in \text{vacuumSpan}(L, \Omega, n)$, it holds that $E_{ij} v \in \text{vacuumSpan}(L, \Omega, n)$.

Let $L$ be a ladder system on a $K$-vector space $V$ with $d$ modes. Given a vacuum state $\Omega \in V$ (satisfying $a_i \Omega = 0$ for all $i$) and a natural number $n$, the $n$-th excitation-number sector $\text{vacuumSpan}(L, \Omega, n)$ is defined as the $K$-linear span of all vectors of the form $a^\dagger_{i_1} a^\dagger_{i_2} \dots a^\dagger_{i_n} \Omega$ where $i_j \in \{0, \dots, d-1\}$. This definition bundles that submodule as a Lie submodule of $V$ under the representation of the general linear Lie algebra $\mathfrak{gl}_d(K)$, where the action of a matrix $X \in \mathfrak{gl}_d(K)$ is determined by mapping the elementary matrix $E_{ij}$ to the operator $a^\dagger_i a_j$.

Let $V$ be a $K$-vector space equipped with a ladder system $L$ of $d$ modes. Given a vacuum state $\Omega \in V$ (satisfying $a_i \Omega = 0$ for all $i$) and a natural number $n$, the underlying $K$-linear submodule of the Lie submodule $\text{vacuumSpanLieSubmodule}(L, P, n)$ is equal to the $n$-th excitation-number sector $\text{vacuumSpan}(L, \Omega, n)$, which is the span of all vectors of the form $a^\dagger_{i_1} a^\dagger_{i_2} \dots a^\dagger_{i_n} \Omega$.