Simplification is perhaps the most common operation performed in a computer algebra system. In Physics, this typically entails simplifying tensorial expressions, or expressions involving noncommutative operators that satisfy certain commutator/anticommutator rules, or sums and integrals involving quantum operators and Dirac delta functions in the summands and integrands. Relevant enhancements were introduced in Maple 18 for all these cases.
Examples
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![[mathematicalnotation = true]](/products/maple/new_features/images18/Physics_3.gif) |
(1) |
Simplification of sums when the summand is linear in KroneckerDeltas:
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![Sum(`+`(`*`(sqrt(`+`(n, 1)), `*`(KroneckerDelta[m, `+`(n, 1)])), `*`(sqrt(n), `*`(KroneckerDelta[m, `+`(n, `-`(1))]))), n = 0 .. infinity)](/products/maple/new_features/images18/Physics_4.gif) |
![Sum(`+`(`*`(`^`(`+`(n, 1), `/`(1, 2)), `*`(Physics:-KroneckerDelta[m, `+`(n, 1)])), `*`(`^`(n, `/`(1, 2)), `*`(Physics:-KroneckerDelta[m, `+`(n, `-`(1))]))), n = 0 .. infinity)](/products/maple/new_features/images18/Physics_5.gif) |
(2) |
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![Simplify(Sum(`+`(`*`(`^`(`+`(n, 1), `/`(1, 2)), `*`(Physics:-KroneckerDelta[m, `+`(n, 1)])), `*`(`^`(n, `/`(1, 2)), `*`(Physics:-KroneckerDelta[m, `+`(n, `-`(1))]))), n = 0 .. infinity))](/products/maple/new_features/images18/Physics_6.gif) |
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(3) |
Simplification of tensorial expressions. To facilitate typing, set the spacetime indices to be lowercaselatin:
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![[spacetimeindices = lowercaselatin]](/products/maple/new_features/images18/Physics_9.gif) |
(4) |
Define a tensor
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![Define(F[a])](/products/maple/new_features/images18/Physics_11.gif) |
![{Physics:-Dgamma[mu], F[a], Physics:-Psigma[mu], Physics:-d_[mu], Physics:-g_[mu, nu], Physics:-KroneckerDelta[mu, nu], Physics:-LeviCivita[alpha, beta, mu, nu]}](/products/maple/new_features/images18/Physics_13.gif) |
(5) |
The following tensorial expression,
has various terms with contracted indices. In each term, {a,b,c} are free indices:
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![Check(`+`(`-`(`*`(1010921950548, `*`(F[d], `*`(F[`~d`], `*`(F[f], `*`(F[`~f`], `*`(F[j], `*`(F[`~j`], `*`(F[a], `*`(F[b], `*`(F[c]))))))))))), `-`(`*`(2023081095744, `*`(F[d], `*`(F[`~d`], `*`(F[f], `...](/products/maple/new_features/images18/Physics_31.gif) |
![[{d, f, j}, {d, f}, {d, e, f, h, i, j, l, n}, {d, e, f, h, i, j, k}, {e, h}, {h, i, n}, {e, h, i, n}, {d, e, g, h, i, k, m}, {d}, {g, h}, {d, e, i, n}, {d, e}, {h}, {d, e, j}, {g, h, k}, {h, n}], {a, ...](/products/maple/new_features/images18/Physics_33.gif)
![[{d, f, j}, {d, f}, {d, e, f, h, i, j, l, n}, {d, e, f, h, i, j, k}, {e, h}, {h, i, n}, {e, h, i, n}, {d, e, g, h, i, k, m}, {d}, {g, h}, {d, e, i, n}, {d, e}, {h}, {d, e, j}, {g, h, k}, {h, n}], {a, ...](/products/maple/new_features/images18/Physics_34.gif) |
(7) |
Taking into account Einstein's sum rule for contracted (repeated) indices, the symmetry properties of
and
, this tensorial expression is equal to zero:
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![Simplify(`+`(`-`(`*`(1010921950548, `*`(F[d], `*`(F[`~d`], `*`(F[f], `*`(F[`~f`], `*`(F[j], `*`(F[`~j`], `*`(F[a], `*`(F[b], `*`(F[c]))))))))))), `-`(`*`(2023081095744, `*`(F[d], `*`(F[`~d`], `*`(F[f]...](/products/maple/new_features/images18/Physics_37.gif) |
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(8) |
The simplification of integrals and sums involving quantum operators that satisfy algebra rules is now more powerful, both in the continuous and discrete case. Consider a field,
, and its expansion in terms in a basis of functions,
using operators,
and
, that satisfy:
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(9) |
The expansion of terms
and
is given by:
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![Typesetting:-mprintslash([psi(r) = Int(`*`(phi(n, r), `*`(a(n))), n = `+`(`-`(infinity)) .. infinity)], [psi(r) = Int(`*`(phi(n, r), `*`(a(n))), n = `+`(`-`(infinity)) .. infinity)])](/products/maple/new_features/images18/Physics_50.gif) |
(10) |
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(11) |
The commutator
is equal to:
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(12) |
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(13) |
The products of integrals on the right-hand side can both be combined into double integrals, then recombined into a single integral and simplified taking into account the algebra rule stated:
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(14) |
The step involving only the combination of the integrals can now also be performed separately:
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(15) |
The extended capabilities in Simplify regarding integration also work in the discrete case, over sums. Redo the algebra rule now considering the same relations but in the discrete case.
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![Setup(redo, quantumoperators = {a, psi}, algebrarules = {%Commutator(a[n], a[p]) = 0, %Commutator(a[n], %Dagger(a[p])) = KroneckerDelta[n, p]}, spacetimeindices = greek)](/products/maple/new_features/images18/Physics_63.gif)
![Setup(redo, quantumoperators = {a, psi}, algebrarules = {%Commutator(a[n], a[p]) = 0, %Commutator(a[n], %Dagger(a[p])) = KroneckerDelta[n, p]}, spacetimeindices = greek)](/products/maple/new_features/images18/Physics_64.gif) |
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(16) |
The following sum can now be simplified by combining the sums and taking into account the new (discrete) algebra rules, or just performing the combination step:
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, `*`(a[n])), n = `+`(`-`(infinity)) .. infinity)](/products/maple/new_features/images18/Physics_66.gif) |
, `*`(a[n])), n = `+`(`-`(infinity)) .. infinity)](/products/maple/new_features/images18/Physics_67.gif) |
(17) |
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, `*`(a[n])), n = `+`(`-`(infinity)) .. infinity)))](/products/maple/new_features/images18/Physics_68.gif) |
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(18) |
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, `*`(a[n])), n = `+`(`-`(infinity)) .. infinity), Physics:-Dagger(psi(s)) = Sum(`*`(conjugate(phi[p](s)), `*`(Physics:-Dagger(a[p]))), p = `+`(`-`(infinity...](/products/maple/new_features/images18/Physics_70.gif) |
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(19) |
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, `*`(a[n])), n = `+`(`-`(infinity)) .. infinity), S...](/products/maple/new_features/images18/Physics_72.gif) |
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(20) |
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, `*`(a[n])), n = `+`(`-`(infinity)) .. infinity), Su...](/products/maple/new_features/images18/Physics_74.gif) |
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(21) |
Improvements in the simplification of annihilation and the creation of fermionic operators, as well as the related occupation number operator:
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![[anticommutativeprefix = {_lambda, psi}]](/products/maple/new_features/images18/Physics_77.gif) |
(22) |
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(23) |
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(24) |
The related occupation number operator:
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(25) |
Consider the application of these fermionic operators to a related state vector:
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(26) |
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(27) |
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(28) |
Increasing the occupation number,
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(29) |
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(30) |
In other words, powers of annihilation and creation fermionic operators are equal to zero:
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(31) |
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(32) |
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(33) |
The occupation number operator is also idempotent:
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(34) |
These expressions can now be simplified:
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(35) |
The simplification of vectorial expressions was also enhanced. For example:
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![Typesetting:-mprintslash([`.`(B0_, Physics:-Vectors:-`&x`(v_, `+`(B0_, B1_)))], [Physics:-Vectors:-`.`(B0_, Physics:-Vectors:-`&x`(v_, `+`(B0_, B1_)))])](/products/maple/new_features/images18/Physics_106.gif) |
(36) |
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![Typesetting:-mprintslash([`.`(B0_, Physics:-Vectors:-`&x`(v_, B1_))], [Physics:-Vectors:-`.`(B0_, Physics:-Vectors:-`&x`(v_, B1_))])](/products/maple/new_features/images18/Physics_108.gif) |
(37) |
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