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
> 


(1) 
Simplification of sums when the summand is linear in KroneckerDeltas:
> 


(2) 
> 


(3) 
Simplification of tensorial expressions. To facilitate typing, set the spacetime indices to be lowercaselatin:
> 


(4) 
Define a tensor
> 


(5) 
The following tensorial expression,
has various terms with contracted indices. In each term, {a,b,c} are free indices:
> 


(7) 
Taking into account Einstein's sum rule for contracted (repeated) indices, the symmetry properties of and , this tensorial expression is equal to zero:
> 


(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:
> 


(9) 
The expansion of terms and is given by:
> 


(10) 
> 


(11) 
The commutator is equal to:
> 


(12) 
> 


(13) 
The products of integrals on the righthand side can both be combined into double integrals, then recombined into a single integral and simplified taking into account the algebra rule stated:
> 


(14) 
The step involving only the combination of the integrals can now also be performed separately:
> 


(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.
> 


(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:
> 


(17) 
> 


(18) 
> 


(19) 
> 


(20) 
> 


(21) 
Improvements in the simplification of annihilation and the creation of fermionic operators, as well as the related occupation number operator:
> 


(22) 
> 


(23) 
> 


(24) 
The related occupation number operator:
> 


(25) 
Consider the application of these fermionic operators to a related state vector:
> 


(26) 
> 


(27) 
> 


(28) 
Increasing the occupation number,
> 


(29) 
> 


(30) 
In other words, powers of annihilation and creation fermionic operators are equal to zero:
> 


(31) 
> 


(32) 
> 


(33) 
The occupation number operator is also idempotent:
> 


(34) 
These expressions can now be simplified:
> 


(35) 
The simplification of vectorial expressions was also enhanced. For example:
> 

> 


(36) 
> 


(37) 

See All Examples >