# BosonNormalOrder

BosonNormalOrder is a Mathematica code I wrote to normal-order bosonic operators, e.g., like the following:

$$aa^{\dagger k}a^la^\dagger=a^{\dagger(k+1)}a^{l+1}+(k+l+1)a^{\dagger k}a^l+kla^{\dagger(k-1)}a^{l-1}\,.$$

The implementation relies on what I call a "weighted sequence" (wseq), which is of the type

{w, {k, l, m, n}}


where w is the weight and k, l, m, n, etc. (the list can be of any size) are the exponents of a product of annihilation and creation operators, always starting with an annihilation one on the right (normal order), so, e.g.,

{1, {1, 1}}


corresponds to $\ud{a}a$ while

{2, {1, 1, 0}}


is $2a\ud{a}$. One can make more complex sequences in this way, e.g.,

{-1, {1, 2, 3, 1}}


is $-\ud{a}a^2\ud{a}^3a$. Various such weighted sequences in a list are understood as a sum of them, e.g.

{{-1, {1, 2, 3, 1}}, {2, {1, 1, 0}}}


is $-\ud{a}a^2\ud{a}^3a+2a\ud{a}$. Etc. What the module does is, given an expression in any order, it returns it normally-ordered version, e.g.:

BosonNormalOrder[{{1, {1, 1, 0}}}]

{{1, {0}}, {1, {1, 1}}}


For simplicity, if there is only one correlator only, instead of passing the full wseq, one can use:

BosonNormalOrderCorrelator[{1, 1, 0}]

{{1, {0}}, {1, {1, 1}}}


This is $a^2\ud{a}^2=2+4\ud{a}a+\ud{a}^2a^2$:

BosonNormalOrderCorrelator[{2, 2, 0}]

{{2, {0}}, {4, {1, 1}}, {1, {2, 2}}}


More details on the code and notebook can be foudn in this blog post.

## Versions

1. older versions of the hack were stored in undocumented notebooks.
2. v°0.11: 21 May (2022).
3. v°0.2: Revised 31 May (2022).
4. v°1.0 (current): Bug fixed and Wikilaussy page (here) created 13 June (2023).