COMPUTER ALGEBRA FOR THEORETICAL PHYSICS

Generally speaking, physicists still experience that computing with paper and pencil is in most cases simpler than computing on a Computer Algebra worksheet. On the other hand, recent developments in the Maple system implemented most of the mathematical objects and mathematics used in theoretical physics computations, and dramatically approximated the notation used in the computer to the one used in paper and pencil, diminishing the learning gap and computersyntax distraction to a strict minimum. In connection, in this talk the Physics project at Maplesoft is presented and the resulting Physics package illustrated tackling problems in classical and quantum mechanics, general relativity and field theory. In addition to the 10 a.m lecture, there will be a handson workshop at 1pm in the Alice Room.


... Why computers?


We can concentrate more on the ideas instead of on the algebraic manipulations
We can extend results with ease
We can explore the mathematics surrounding a problem
We can share results in a reproducible way


Representation issues that were preventing the use of computer algebra in Physics


Notation and related mathematical methods that were missing:
coordinate free representations for vectors and vectorial differential operators,
covariant tensors distinguished from contravariant tensors,
functional differentiation, relativity differential operators and sum rule for tensor contracted (repeated) indices
Bras, Kets, projectors and all related to Dirac's notation in Quantum Mechanics
Inert representations of operations, mathematical functions, and related typesetting were missing:
inert versus active representations for mathematical operations
ability to move from inert to active representations of computations and viceversa as necessary
handlike style for entering computations and texbooklike notation for displaying results
Key elements of the computational domain of theoretical physics were missing:
ability to handle products and derivatives involving commutative, anticommutative and noncommutative variables and functions
ability to perform computations taking into account customdefined algebra rules of different kinds
(problem related commutator, anticommutator, bracket, etc. rules)
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Computer algebra systems were not originally designed to work with this compact notation, having attached so dense mathematical contents, active and inert representations of operations, not commutative and customizable algebraic computational domain, and the related mathematical methods, all this typically present in computations in theoretical physics.

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This situation has changed. The notation and related mathematical methods are now implemented.



Tackling examples with the Physics package



General Relativity


Given the spacetime metric,
a) Compute the trace of
where is some function of the radial coordinate, is the Ricci tensor, is the covariant derivative operator and is the stressenergy tensor
b) Compute the components of the traceless part of of item a)
c) Compute an exact solution to the nonlinear system of differential equations conformed by the components of obtained in b)
Background: paper from February/2013, "Withholding Potentials, Absence of Ghosts and Relationship between Minimal Dilatonic Gravity and f(R) Theories", by P. Fiziev.



The Physics Project


"Physics" is a software project at Maplesoft that started in 2006. The idea is to develop a computational symbolic/numeric environment specifically for Physics, targeting educational and research needs in equal footing, and resembling as much as possible the flexible style of computations used with paper and pencil. The main reference for the project is the Landau and Lifshitz Course of Theoretical Physics.
A first version of "Physics" with basic functionality appeared in 2007. Since then the package has been growing every year, including now, among other things, a searcheable database of solutions to Einstein equations and a new dedicated programming language for Physics.
Since August/2013, weekly updates of the Physics package are distributed on the web, including the new developments related to our plan as well as related to people's feedback.

