Re: Grassmann Calculus / Indexed Objects / Simplify

*To*: mathgroup at smc.vnet.net*Subject*: [mg60839] Re: Grassmann Calculus / Indexed Objects / Simplify*From*: Robert Schoefbeck <schoefbeck at hep.itp.tuwien.ac.at>*Date*: Fri, 30 Sep 2005 03:57:08 -0400 (EDT)*References*: <dhdank$8ae$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

sorry for being rather unclear in my initial post. david park told me about his tensorial package to do index calculations, i hope i can learn how to handle dummies properly in simplifiactions from his code.. but id also like to rephrase my problem in a more understandable way: (Using Einstein conventions all along) For a short version of the actual mathematica stuff scroll down to (XXX). I have indexed objects theta^a, lambda^a ... where ^ means an upper index and _ means a lower index. The (two-value)indices a,b,... are pulled with an anit(!)-symmetric matrix eps^{a,b}, eps^{1,1} = eps^{2,2} = 0, eps^{1,2} = - eps^{2,1} = 0 theta^a =eps^{a,b} theta_b theta_a =eps_{a,b} theta^b and by consistency eps_{a,b}eps^{b,c} = delta^a_c. Furthermore, theta^1 and theta^2 anit-commute, that is any element squares to zero theta^1*theta^1 = 0,... theta^1*theta^2 = -theta^2*theta^1 (theta could be replaced by lambda,...) note that theta^a theta_a = theta^1 theta_1 + theta^2 theta_2 = -2 theta^1 theta^2 A Superfield is an object SF = phi + lambda^a theta_a +F theta^a theta_a where phi , lambda and F are x-dependent fields and theta is constant. Furthermore, theta and lambda are odd as above (that is, exchanging two odd objects produces a minus, i.e. theta_1 lambda_2 = -lambda_2 theta_1 and son on) Since any two thetas (with the same index in the same position) square to zero one finds that powers of Superfields always have a finite decomposition in the theta variables. due to the limited index range (a = 1,2) one has theta^a theta^b theta^c = 0 (XXX) My problem is as follows: I want to compute arbitrary powers of superfields (that already works, it is MyTimes[CSF,CSF] in my code in the initial post) and i want to simplify them with all due care on the dummies. For simplifications i have to tell mathematica for example that X^a Y_a = - X_a Y^a for any two indices on any objects X and Y (due to the antisymmetry of the metric) and that in X^a Y_a U^b V_b it is IMportant that X,Y as well as U, V have the same index but it is UNimportant that they are called a and b but that it is IMportant that a and b are different. In my code i use Unique[] to generate the indices because it must not happen that an index occurs more than twice in an expression. Unfortunately there is then a problem with products of objects generated in this way: It frequently happens that terms like theta^a$1 theta_a$1 + theta_a$2 theta^a$2 appear. Due to the rule above, this expression equals zero. Mathematica would for example have to take out an eps^{a$2, a$3} in theta^a$2, flip the indices of eps, therby acquire a minus, contract the eps with theta_a$2 and then realize that theta^a$1 theta_a$1 - theta^a$2 theta_a$2 is zero since the sums are the same, nevermind the dummy. in terms of formulas: problem: theta^a theta_a + theta^b theta_b Mathematica:"lets try this" theta^a theta_a + theta^b delta_b^c theta_c Mathematica:"now that was not enough, lets split delta in two eps's" theta^a theta_a + theta^b eps_b^c eps_c^d theta_d Mathematica:"that reminds me of something..." theta^a theta_a - eps^c_b theta^b eps_c^d theta_d Mathematica:"Jesus!.." theta^a theta_a - theta^c eps_c^d theta_d Mathematica:"My brain hurts..." theta^a theta_a - theta^c theta_c Mathematica:"If only c was a... dammit. Wait! I WAS TOLD HOW TO TREAT DUMMIES!!!" Out[1]: 0 Me:"Jesus!.." I don't yet know how to do that though this problem is addressed in david parks package; although there seems (im not sure) to be a problem with that package 'tensorial 3' as index flipping always comes with a plus sign. I'd hence also be interested to know if that can be changed in the code.