# The first Weyl algebra

To cite the weyl package in publications please use Hankin (2022b).

Consider the vector space $${\mathcal A}$$ of linear operators on univariate functions; $${\mathcal A}$$ can be made into an algebra where multiplication (denoted by juxtaposition) is defined as operator composition (Coutinho 1997). That is, given operators $${\mathcal O}_1,{\mathcal O}_2$$ we define their product $${\mathcal O}_1{\mathcal O}_2$$ by

$({\mathcal O}_1{\mathcal O}_2)f={\mathcal O}_1({\mathcal O}_2f).$

where $$f$$ is any univariate function. Here we consider the algebra generated by the set $$\left\lbrace\partial,x\right\rbrace$$ where $$\partial\colon f\longrightarrow f'$$ [that is, $$(\partial f)(x) = f'(x)$$] and $$x\colon f\longrightarrow xf$$ [that is, $$(xf)(x) = xf(x)$$]. This is known as the (first) Weyl algebra. We observe that the Weyl algebra is not commutative: $$\partial xf=(xf)'=f+xf'$$ but $$x\partial f=xf'$$, so $$\partial x=x\partial+1$$. The algebra generated by $$\left\lbrace x,\partial\right\rbrace$$ will include elements such as $$7\partial + 4\partial x\partial^3 x$$, which would map $$f$$ to $$7f' + 4\left(x\left(xf\right)'''\right)'$$. It can be shown that any element of the Weyl algebra can be expressed in the standard form

$\sum_i a_i \partial^{p_i}x^{q_i}$

for real $$a_i$$ and nonnegative integers $$p_i,q_i$$. Converting a general word to standard form is not straightforward but we have

$\partial x^n = x^n\partial + nx^{n-1}$

and

$\partial^n x = x\partial^n + n\partial^{n-1}.$

We can apply these rules recursively to find standard form for products $$(\partial^i x^j)(\partial^l x^m)$$. Alternatively we may follow Wolf (1975) and use the fact that

$(\partial^i x^j)(\partial^lx^m)= \sum_{r=0}^j{j\choose r}{l\choose r}\partial^{i+l-r}x^{j+m-r}.$

These rules can be used to show that $$7\partial + 4\partial x\partial^3 x$$ can be expressed as $$7\partial + 12x\partial^2 + 4x^2\partial d^3$$, which is in standard form.

## The package in use

The package includes functionality to automate the above calculations. In particular, package idiome represents the generating elements $$\partial$$ and $$x$$ of the first Weyl algebra as R objects d and x respectively. These may be manipulated with standard arithemetic operations:

library(weyl)
7*d + 4*x*d^3*x
## A member of the Weyl algebra:
##  x d     val
##  0 1  =    7
##  1 2  =   12
##  2 3  =    4

Above, the result of the input is given in standard form. We see the terms, one per row, with coefficients in the rightmost column (viz $$7,12,4$$). Thus the first row is $$7\partial$$, the second is $$12x\partial^2$$, and the third is $$4x^2\partial^3$$. We may choose to display the result in symbolic form rather than matrix form:

options(polyform=TRUE)
7*d + 4*x*d^3*x
## A member of the Weyl algebra:
## +7*d +12*x*d^2 +4*x^2*d^3

which is arguably a more natural representation. The package allows one to use R semantics. For example, consider $$d_1=\partial x + 2\partial^3$$ and $$d_2=3+7\partial -5x^2\partial^2$$. Observing that $$d_1$$ and $$d_2$$ are in standard form, package idiom to create these operators would be:

(d1 <- d*x + 2*d^3)
## A member of the Weyl algebra:
## 1 +x*d +2*d^3
(d2 <- 3 + 7*d  -5*x^2*d^2)
## A member of the Weyl algebra:
## 3 +7*d -5*x^2*d^2

(object d1 is converted to standard form automatically). Observe that, like the spray package, the order of the terms is not defined. We may apply the usual rules of arithmetic to these objects:

d1*d2
## A member of the Weyl algebra:
## 3 +7*x*d^2 +3*x*d -15*x^2*d^2 -60*x*d^4 -5*x^3*d^3 +7*d +14*d^4 -54*d^3
## -10*x^2*d^5

Standard R semantics operate, and it is possible to work with more complicated expressions:

options(polyform=TRUE)
(d1^2 + d2) * (d2 - 3*d1)
## A member of the Weyl algebra:
## -276*x*d^7 +28*d^7 +5*x^2*d^2 -732*d^6 -636*x*d^4 +28*d^4 -414*d^3
## -63*x^2*d^3 +7*d -20*x^3*d^6 -24*d^9 -70*x*d^2 -20*x^2*d^8 +77*x^3*d^3
## +20*x^4*d^4 -21*x*d +49*d^2 -198*x^2*d^5 +28*x*d^5

### Comparison with mathematica

Mathematica can deal with operators and we may compare the two systems’ results for $$\partial^2x\partial x^2$$:

In[1] := D[D[x*D[x^2*f[x],x],x],x] // Expand

Out[1] := 4 f[x] + 14 x f'[x] + 8 x^2 f''[x] + x^3f'''[x]

x <- weyl(cbind(0,1))
D <- weyl(cbind(1,0))
x^2*D*x*D^2
## A member of the Weyl algebra:
## 4 +x^3*d^3 +8*x^2*d^2 +14*x*d

Above, we see agreement between weyl and Mathematica, although the terms are presented in a different order.

# Further Weyl algebras

The package supports arbitrary multivariate Weyl algebras. Consider:

options(polyform=FALSE)  # revert to default print method
set.seed(0)
x <- rweyl()
x
## A member of the Weyl algebra:
##   x  y  z dx dy dz     val
##   2  0  1  2  0  1  =    3
##   0  1  2  2  0  1  =    2
##   1  0  2  1  0  1  =    1

Above, object x is a member of the operator algebra generated by $$\left\lbrace\partial_x,\partial_y,\partial_z,x,y,z\right\rbrace$$. Object x might be expressed as $$xz^2\partial_x\partial_z + 3x^2z\partial_x^2\partial_z + 2yz^2\partial_x^2\partial_z^2$$ although as ever the rows are presented in an implementation-dependent order. We may verify associativity of multiplication:

x <- rweyl(n=1,d=2)
y <- rweyl(n=2,d=2)
z <- rweyl(n=3,d=2)
options(polyform=TRUE)
x*(y*z)
## A member of the Weyl algebra:
## +6*x*y^2*dx^2*dy +36*x*y^3*dx*dy^3 +2*x^2*y^4*dx*dy^4
## +4*x^2*y^2*dx*dy^5 +36*x*y^2*dx*dy^2 +x*y^4*dx*dy^3 +4*x*y^3*dx*dy^2
## +x^2*y^4*dx^2*dy^3 +12*x^2*y^2*dx*dy^2 +2*x^2*y^2*dx^2*dy^4
## +2*x*y^2*dx*dy^4 +2*x*y^2*dx*dy +2*x^2*y^2*dx^2*dy +4*x^2*y^3*dx^2*dy^2
## +3*x*y^4*dx^2*dy^3 +12*x^2*y^3*dx*dy^3 +6*x*y^4*dx*dy^4
## +12*x*y^3*dx^2*dy^2
(x*y)*z
## A member of the Weyl algebra:
## +2*x*y^2*dx*dy^4 +2*x^2*y^2*dx^2*dy^4 +3*x*y^4*dx^2*dy^3
## +x^2*y^4*dx^2*dy^3 +12*x^2*y^2*dx*dy^2 +x*y^4*dx*dy^3
## +2*x^2*y^4*dx*dy^4 +4*x^2*y^3*dx^2*dy^2 +2*x^2*y^2*dx^2*dy
## +2*x*y^2*dx*dy +4*x*y^3*dx*dy^2 +12*x^2*y^3*dx*dy^3 +6*x*y^4*dx*dy^4
## +12*x*y^3*dx^2*dy^2 +6*x*y^2*dx^2*dy +36*x*y^3*dx*dy^3
## +36*x*y^2*dx*dy^2 +4*x^2*y^2*dx*dy^5

Comparing the two results above, we see that they apparently differ. But the apparent difference is due to the fact that the terms appear in a different order, a feature that is not algebraically meaningful. We may verify that the expressions are indeed algebraically identical:

x*(y*z) - (x*y)*z
## A member of the Weyl algebra:
## the NULL multinomial of arity 4

The package can deal with arbitrarily high dimensional Weyl algebras. For exmaple:

(x9 <- rweyl(dim=9))
## A member of the Weyl algebra:
## +3*x1*x2^2*x3^2*x4*x6*x7^2*x8^2*x9^2*d2^2*d7^2*d8*d9
## +2*x1^2*x3*x4*x5*x6*x8*x9*d3^2*d4*d6*d7^2*d8*d9^2
## +x3*x4^2*x5*x6*x7*x8^2*x9^2*d1^2*d3*d6^2*d8^2*d9

Above we see a member of the ninth Weyl algebra; see how the column headings no longer use the x y z notation and revert to numeric labels. Symbolic notation is available but can be difficult to read:

options(polyform=TRUE)
x9
## A member of the Weyl algebra:
## +3*x1*x2^2*x3^2*x4*x6*x7^2*x8^2*x9^2*d2^2*d7^2*d8*d9
## +2*x1^2*x3*x4*x5*x6*x8*x9*d3^2*d4*d6*d7^2*d8*d9^2
## +x3*x4^2*x5*x6*x7*x8^2*x9^2*d1^2*d3*d6^2*d8^2*d9
options(polyform=FALSE) # revert to default

# Derivations

A derivation $$D$$ of an algebra $${\mathcal A}$$ is a linear operator that satisfies $$D(d_1d_2)=d_1D(d_2) + D(d_1)d_2$$, for every $$d_1,d_2\in{\mathcal A}$$. If a derivation is of the form $$D(d) = [d,f] = df-fd$$ for some fixed $$f\in{\mathcal A}$$, we say that $$D$$ is an inner derivation:

$D(d_1d_2) = d_1d_2f-fd_1d_2 = d_1d_2f-d_1fd_2 + d_1fd_2-fd_1d_2 = d_1(d_2f-fd_2) + (d_1f-fd_1)d_2 = d_1D(d_2) + D(d_1)d_2$

Dirac showed that all derivations are inner derivations for some $$f\in{\mathcal A}$$. The package supports derivations:

f <- rweyl()
D <- as.der(f)  # D(x) = xf-fx

Then

d1 <- rweyl()
d2 <- rweyl()
D(d1*d2) == d1*D(d2) + D(d1)*d2
## [1] TRUE

# Low-level considerations and generalizations

In the package, the product is customisable. In general, product a*b [where a and b are weyl objects] is dispatched to the following sequence of functions:

• weyl_prod_multivariate_nrow_allcolumns()
• weyl_prod_multivariate_onerow_allcolumns()
• weyl_prod_multivariate_onerow_singlecolumn()
• weyl_prod_univariate_onerow()
• weyl_prod_helper3() [default]

In the above, “univariate” means “generated by $$\left\lbrace x,\partial_x\right\rbrace$$” [so the corresponding spray object has two columns]; and “multivariate” means that the algebra is generated by more than one variable, typically something like $$\left\lbrace x,y,z,\partial_x,\partial_y,\partial_z\right\rbrace$$.

The penultimate function weyl_prod_univariate_onerow() is sensitive to option prodfunc which specifies the recurrence relation used. This defaults to weyl_prod_helper3():

weyl_prod_helper3
## function (a, b, c, d)
## {
##     f <- function(r) {
##         factorial(r) * choose(b, r) * choose(c, r)
##     }
##     ind <- numeric(0)
##     val <- numeric(0)
##     for (r in 0:b) {
##         ind <- rbind(ind, c(a + c - r, b + d - r))
##         val <- c(val, f(r))
##     }
##     spray(ind, val, addrepeats = TRUE)
## }
## <bytecode: 0x55581bf31b80>
## <environment: namespace:weyl>

Function weyl_prod_helper3() follows Wolf. This gives the univariate concatenation product $$(\partial^a x^b)(\partial^c x^d)$$ in terms of standard generators:

$\partial^a x^b \partial^c x^d=\sum_{r=0}^b r!{b\choose r}{c\choose r} \partial^{a+c-r}x^{b+d-r}$

The package also includes lower-level function weyl_prod_helper1() implementing $$\partial^a x^b \partial^c x^d=\partial^ax^{b-1}\partial^cx^{d+1} + c\partial^ax^{b-1}\partial^{c-1}x^d$$ (together with suitable bottoming-out). I expected function weyl_prod_helper3() to be much faster than weyl_prod_helper1() but there doesn’t seem to be much difference between the two.

# Generalized commutator relations

We can exploit this package customisability by considering, instead of $$\left\lbrace x,\partial\right\rbrace$$, the algebra generated by $$\left\lbrace e,\partial\right\rbrace$$, where $$e$$ maps $$f$$ to $$e^xf$$: if $$f$$ maps $$x$$ to $$f(x)$$, then $$ef$$ maps $$x$$ to $$e^xf(x)$$. We see that $$\partial e-e\partial=e$$. With this, we can prove that $$\partial^ne=e(1+\partial)^n$$ and $$e^n\partial=e^n\partial+ne^n$$ and, thus

$(e^a\partial^b)(e^c\partial^d) =e^{a+1}(1+\partial)^be^{c-1}\partial^d =e^{a}\partial^{b-1}e^{c}\partial^{d+1}+ce^{a}\partial^{b-1}e^{c}\partial^d$

We may implement this set in package idiom as follows:

weyl_e_prod <- function(a,b,c,d){
if(c==0){return(spray(cbind(a,b+d)))}
if(b==0){return(spray(cbind(a+c,d)))}
return(
Recall(a,b-1,c,d+1) +
c*Recall(a,b-1,c,d)  # cf: c*Recall(a,b-1,c-1,d)) for regular Weyl algebra
)
}

Then, for example, to calculate $$\partial^2e=e(1+2\partial+\partial^2)$$:

options(prodfunc = weyl_e_prod)
options(weylvars = "e")  # changes print method
d <- weyl(spray(cbind(0,1)))
e <- weyl(spray(cbind(1,0)))
d*d*e
## A member of the Weyl algebra:
##  e d     val
##  1 0  =    1
##  1 1  =    2
##  1 2  =    1
d^2*e
## A member of the Weyl algebra:
##  e d     val
##  1 0  =    1
##  1 1  =    2
##  1 2  =    1

By way of verification:

d^5*e == e*(1+d)^5
## [1] TRUE

which verifies that indeed $$\partial^5e=e(1+\partial)^5$$. Another verification would be to cross-check with Mathematica, here working with $$\partial e\partial^2e$$:

In[1] := D[Exp[x]*D[D[Exp[x]*f[x],x],x],x]

Out[1] := 2E^2x f[x] + 5E^2x f'[x] + 4E^2xf''[x] + E^2x f'''[x]

options(polyform = TRUE)
d*e*d^2*e
## A member of the Weyl algebra:
## +2*e^2 +5*e^2*d +4*e^2*d^2 +e^2*d^3

We can manipulate more complicated expressions too. Suppose we want to evaluate $$(1+e^2\partial)(1-5e^3\partial^3)$$:

o1 <- weyl(spray(cbind(2,1)))
o2 <- weyl(spray(cbind(3,3)))
options(polyform = FALSE)
(1+o1)*(1-5*o2)
## A member of the Weyl algebra:
##  e d     val
##  5 3  =  -15
##  3 3  =   -5
##  5 4  =   -5
##  2 1  =    1
##  0 0  =    1

And of course we can display the result in symbolic form:

options(polyform = TRUE)
(1+o1)*(1-5*o2)
## A member of the Weyl algebra:
## 1 -15*e^5*d^3 -5*e^3*d^3 -5*e^5*d^4 +e^2*d

# Note on use of disordR package

The coefficients of a weyl object, and the rows of its spray matrix, are stored in an implementation-specific order. Thus, extraction and replacement use the disordR package (Hankin 2022a). A short example follows in the context of the weyl package; a much more extensive and detailed discussion is given in the disordR vignette and in Hankin (2022a). First we create a moderately complicated weyl object:

options(weylvars = NULL)  # revert to default names
(W <- weyl(spray(matrix(c(0,1,1,1,1,2,1,0),2,4),2:3))^2)
## A member of the Weyl algebra:
##   x  y dx dy     val
##   2  2  4  0  =    9
##   2  2  3  0  =   18
##   0  2  2  2  =    4
##   2  2  2  0  =    9
##   1  2  3  1  =   12
##   1  2  2  0  =    6
##   1  2  3  0  =    6
##   1  2  2  1  =    6
##   0  2  2  1  =    4

The coefficients of W may be extracted:

coeffs(W)
## A disord object with hash ef3b76da15a19ac6dd3ba83e2ec6b436a0f975f6 and elements
## [1]  9 18  4  9 12  6  6  6  4
## (in some order)

The object returned is a disord object. There is no way to extract (e.g.) the first coefficient, for the order of the matrix rows is not defined. If we try we will get an error:

coeffs(W)[1]
## Error in .local(x, i, j = j, ..., drop): if using a regular index to extract, must extract each element once and once only (or none of them)

However, it is perfectly well defined to ask “give all coefficients greater than 6”:

o <- coeffs(W)
o[o>6]
## A disord object with hash ebf6f5624c14b9166f702b460f5f202351ed5026 and elements
## [1]  9 18  9 12
## (in some order)

Extraction works as expected. Using recent improvements in the disordR package, we take all coefficients less than 7 and add 100 to them:

coeffs(W)[coeffs(W)<7] <- coeffs(W)[coeffs(W)<7] + 100
W
## A member of the Weyl algebra:
##   x  y dx dy     val
##   0  2  2  1  =  104
##   1  2  2  1  =  106
##   1  2  3  0  =  106
##   1  2  2  0  =  106
##   1  2  3  1  =   12
##   2  2  2  0  =    9
##   0  2  2  2  =  104
##   2  2  3  0  =   18
##   2  2  4  0  =    9

## References

Coutinho, S. C. 1997. “The Many Avatars of a Simple Algebra.” The American Mathematical Monthly 104 (7): 593–604. https://doi.org/https://doi.org/10.1080/00029890.1997.11990687.

Hankin, Robin K. S. 2022a. “Disordered Vectors in R: Introducing the disordR Package.” https://arxiv.org/abs/2210.03856; arXiv. https://doi.org/10.48550/ARXIV.2210.03856.

———. 2022b. “Quantum Algebra in R: The Weyl Package.” arXiv. https://doi.org/10.48550/ARXIV.2212.09230.

Wolf, K. B. 1975. The Heisenberg-Weyl Ring in Quantum Mechanics. Vol. III. Group Theory and Its Applications. Academic Press.