Cadabra

"Cadabra is a sym­bolic com­puter al­ge­bra sys­tem (CAS) de­signed specif­i­cally for the so­lu­tion of prob­lems en­coun­tered in field the­ory. It has ex­ten­sive func­tion­al­ity for ten­sor com­puter al­ge­bra, ten­sor poly­no­mial sim­pli­fi­ca­tion in­clud­ing multi-term sym­me­tries, fermi­ons and anti-com­mut­ing vari­ables, Clif­ford al­ge­bras and Fierz trans­for­ma­tions, com­po­nent com­pu­ta­tions, im­plicit co­or­di­nate de­pen­dence, mul­ti­ple index types and many more. The input for­mat is a sub­set of TeX. Both a com­mand-line and a graph­i­cal in­ter­face are avail­able.

The features:

• De­signed for field-the­ory prob­lems, with han­dling of anti-com­mut­ing and non-com­mut­ing ob­jects with­out spe­cial no­ta­tions for their prod­ucts, gamma ma­trix al­ge­bra, Fierz iden­ti­ties, Dirac con­ju­ga­tion, viel­beine, flat and curved, co­vari­ant and con­travari­ant in­dices, im­plicit de­pen­dence of ten­sors on co­or­di­nates, par­tial and co­vari­ant de­riv­a­tives...
• Pow­er­ful ten­sor sim­pli­fi­ca­tion al­go­rithms, not just for mono-term sym­me­tries but also for multi-terms sym­me­tries like the Bianchi iden­tity, or di­men­sion­ally-de­pen­dent sym­me­tries like the Schouten iden­tity.
• Al­lows for both ab­stract and com­po­nent com­pu­ta­tions.
• A stand­alone graph­i­cal note­book in­ter­face which can also be used as a front-end for SymPy.
• Input and out­put using a sub­set of LaTeX no­ta­tion.
• Pro­gram­ma­ble using Python. 

http://cadabra.science/

http://www.sympy.org/en/index.html