To cite the `weyl`

package in publications please use
Hankin 2022. The `weyl`

package provides R-centric
functionality for working with Weyl algebras of arbitrary dimension. A
detailed vignette is provided in the package.

The Weyl algebra is a noncommutative algebra which is used in quantum
mechanics and the theory of differential equations (Coutinho 1997). The
`weyl`

package offers a consistent and documented suite of
R-centric software. It is based on the `spray`

package for
sparse arrays for computational efficiency.

The Weyl algebra is arguably the simplest noncommutative algebra and is useful in quantum mechanics. It is isomorphic to the quotient ring of the free algebra on two elements

One usually writes the Weyl algebra in terms of operators

The Weyl algebra is also known as the symplectic Clifford algebra.

You can install the released version of the weyl package from CRAN with:

```
# install.packages("weyl") # uncomment this to install the package
library("weyl")
set.seed(0)
```

`weyl`

package in
useThe basic creation function is `weyl()`

, which takes a
`spray`

object and returns a member of the Weyl algebra.

```
<- spray(rbind(c(1,0,0,1,1,0),c(0,1,1,3,2,0)) ,1:2)
S
S#> val
#> 0 1 1 3 2 0 = 2
#> 1 0 0 1 1 0 = 1
```

Above, object `S`

is a standard `spray`

object
but to work with Weyl algebra we need to coerce it to a
`weyl`

object with `weyl()`

:

```
<- weyl(S)
W
W#> A member of the Weyl algebra:
#> x y z dx dy dz val
#> 0 1 1 3 2 0 = 2
#> 1 0 0 1 1 0 = 1
```

Above, object `W`

is a member of the third Weyl algebra:
that is, the algebra generated by

We might ask what

```
<- W*W
Wsquared
Wsquared#> A member of the Weyl algebra:
#> x y z dx dy dz val
#> 0 2 2 6 4 0 = 4
#> 0 1 2 6 3 0 = 8
#> 0 1 1 3 3 0 = 6
#> 1 1 1 4 3 0 = 4
#> 2 0 0 2 2 0 = 1
#> 1 0 1 4 2 0 = 2
#> 1 0 0 1 2 0 = 1
```

This is a more complicated operator. However, we might wish to display it in symbolic form:

```
options(polyform=TRUE)
Wsquared#> A member of the Weyl algebra:
#> +4*y^2*z^2*dx^6*dy^4 +8*y*z^2*dx^6*dy^3 +6*y*z*dx^3*dy^3
#> +4*x*y*z*dx^4*dy^3 +x^2*dx^2*dy^2 +2*x*z*dx^4*dy^2 +x*dx*dy^2
```

S. C. Coutinho 1997.

*The many avatars of a simple algebra*. The American Mathematical Monthly, 104(7):593-604. DOI https://doi.org/10.1080/00029890.1997.11990687.Hankin 2022.

*Quantum algebra in R: the weyl package*. Arxiv, DOI https://doi.org/10.48550/ARXIV.2212.09230.

For more detail, see the package vignette

`vignette("weyl")`