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MathGroup Archive 2005

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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.


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