Pattern matching with two variables - derivatives on 4-vectors

Question

I am interested in pattern matching with two variables and not just one, and more specifically implementing derivatives on 4-vectors. In the case of interest the 4-vectors have two indices; one is in Euclidean spacetime and the other is just such that it counts sites.

So, for the raising and lowering of the spacetime indices we have: $$ \begin{equation} \begin{aligned} x^{\mu} &= \delta^{\mu \nu} x_{\nu} \, , \\ x_{\mu} &= \delta_{\mu \nu} x^{\nu} \end{aligned} \end{equation} $$ where in the above $\delta$ denotes the Kronecker delta. Greek indices are Euclidean spacetime indices, and latin indices correspond to site-counting.

In practice, the derivatives in which I am interested obey the following rules $$ \begin{equation} \begin{aligned} \partial^{\mu}_k x^{\nu}_1 &= \delta^{\mu \nu} \delta_{k ~ 1} \, , \\ \partial^{\mu}_1 x^{\nu}_1 &= \delta^{\mu \nu} \delta_{1 ~ 1} = \delta^{\mu \nu} \, , \\ \partial^{\mu}_2 x^{\nu}_1 &= \delta^{\mu \nu} \delta_{2 ~ 1} = 0 \, , \\ \partial_{k,\mu} x^2_1 &= \partial_{k,\mu}(x^{\nu}_1 x_{1,\nu}) = 2 x_{1,\mu} \delta_{k ~ 1} \, . \end{aligned} \end{equation} $$ Site counting indices are never raised, they are always downstairs as they only serve for counting.

There is a wonderful approach here on how to implement derivatives on four-vectors, when the pattern matching is only in one variable. A sample code is given below

First we implement some rules:

(*---addtn---*)
a : pD[_Plus, ___] := Thread[Unevaluated[a], Plus, 1]
(*---mltplctn---*)
pD[a_Times, x___] := 
 Plus @@ (MapAt[pD[#, x] &, a, #] & /@ Range[Length[a]])
(---prs---)
pD[a_^b_, x___] := b a^(b - 1) pD[a, x]
(---cnstnts---)
pD[a_, fvd[var_, idx_List]] := 0
(---sclr prdcts---)
pD[fvd[k1_Symbol, k2_Symbol], fvd[k_Symbol, derividx_List]] := 
 Module[{i}, 
  pD[fvd[k1, {i}], fvd[k, derividx]] fvd[k2, {i}] + 
    fvd[k1, {i}] pD[fvd[k2, {i}], fvd[k, derividx]] /. 
   fvd[a_Symbol, 
      j_List] (fvd[j_List, l_List] | fvd[l_List, j_List]) :> fvd[a, l]]
(---mtrc drvtv---)
pD[fvd[idx1_List, idx2_List], fvd[var_, derividx_]] := 0;
(---4-vctrs drvtvs---)
pD[fvd[notk_Symbol, idx_List], fvd[k_Symbol, derividx_List]] := 
 0 /; FreeQ[notk, k]
pD[fvd[k_Symbol, idx_List], fvd[k_Symbol, derividx_List]] := 
 fvd[idx, derividx]
(---dspl---)
(---4-vctrs---)
Format[fvd[k_Symbol, {i_}]] := Subscript[k, i]
(---mtrc---)
Format[fvd[{i_}, {j_}]] := Subscript[[Delta], {i, j}]
(---sclr prdct---)
Format[fvd[k1_Symbol, k2_Symbol]] := DisplayForm@(k1 k2)

And the above works fine in cases where we have only the spacetime index. Examples:

1. 4-vector display $x_{\alpha}$:

for which we write

fvd[x, {α}]

2. Euclidean metric display $\delta_{\alpha ~ \beta}$:

we need to run

fvd[{α}, {β}]

3. $\partial_b x_a = \delta_{a ~ b}$

for which we write

pD[fvd[x, {α}], fvd[x, {β}]]

4. $\partial_{\mu} x^2 = 2 x_{\mu}$

for which we run:

pD[fvd[x, x], fvd[x, {μ}]]

5. Sums and differences $\partial_{\mu}(x_{\alpha} \pm x_{\beta})=\delta_{\alpha \mu} \pm \delta_{\beta \mu}$

for which we have

pD[fvd[x, {α}] + fvd[x, {β}], fvd[x, {μ}]]
pD[fvd[x, {α}] - fvd[x, {β}], fvd[x, {μ}]]

And it works fine. How can I generalize the above to include the second index?

After replacing SubscriptBox by Subscript and getting rid of the RowBox, which was suggested in the comments, the formatting is just fine; see below.

6. $x_{\alpha,i}$ and $x_{\alpha,1}$

are given by:

fvd[x, {α}, {i}]
fvd[x, {α}, {1}]

Then I tried to implement the derivative rule as follows, in order to account for both indices:

pD[fvd[k_Symbol, idx1_List, idx2_List], 
  fvd[k_Symbol, derividx1_List, derividx2_List]] := 
 fvd[idx1, derividx1] fvd[idx2, derividx2]

The above works correctly in a simple example:

7. $\partial_{2,\beta} x_{1, \alpha} = \delta_{1,2} \delta_{\alpha,\beta}$

we run

pD[fvd[x, {1}, {α}], fvd[x, {2}, {β}]]

and when I perform the sum

8. $\partial_{1,\beta}(x_{1,\alpha}+x_{0,\alpha})=\delta_{0,1} \delta_{\alpha, \beta} + \delta_{1,1} \delta_{\alpha, \beta}$

it works without any issues

pD[(fvd[x, {1}, {α}] + fvd[x, {0}, {α}]), 
 fvd[x, {1}, {β}]]

However, when I tried to compute the difference it returned an un-evaluated expression:

9. $\partial_{1,\beta}(x_{1,\alpha}-x_{0,\alpha})=-\delta_{0,1} \delta_{\alpha, \beta} + \delta_{1,1} \delta_{\alpha, \beta}$

which should be computable by:

pD[(fvd[x, {1}, {α}] - fvd[x, {0}, {α}]), 
 fvd[x, {1}, {β}]]

causes errors.

Question: what is the proper way of implementing the derivative rules (sums, differences, products) in the case of these two parameters?

Final question: The main quantity of interest would be the following:

$$ \begin{equation} \partial_{k,\nu}(x^2_{1,0}) = 2 (\delta_{k,1}-\delta_{k,0})x_{1,0,\nu} \end{equation} $$

where $x_{1,0,\nu}=x_{1,\nu}-x_{0,\nu}$ with its inverse that would be:

$$ \begin{equation} \partial_{k,\nu}(\frac{1}{x^2_{1,0}}) = - \frac{2}{x^4_{1,0}} (\delta_{k,1}-\delta_{k,0})x_{1,0,\nu} \end{equation} $$

Answer 1

It seems to me that the issue with the difference is a matter of not being able to handle scalar multiples.

Adding an additional rule for scalar products for an arbitrary number of indices is not too difficult. I used:

pD[(k_ /; FreeQ[k, fvd[_]]) v_fvd, l__] := k pD[v, l]

In this case, k_ is a pattern to capture multiples which do not contain an fvd[_] pattern, v_ is a pattern capture groups which are 4-vectors, and l__ is a pattern capture of all subsequent indices of derivation.

The resulting pattern moves the non-4-vector terms outside of the derivative. You can be more precise if you only want to consider specific types of scalar values (perhaps k_?NumericQ instead?).

Evaluating:

pD[(fvd[x, {1}, {α}] - fvd[x, {0}, {α}]), fvd[x, {1}, {β}]]

$-\delta_{(0,1)} \delta_{(\alpha,\beta)} + \delta_{(1,1)} \delta_{(\alpha,\beta)}$

Seems to be reasonable.