Examples Worksheet #1:Basic Rules, Canonical Example, 4-fold SU(2)--shorthand
Special Note:
It does not matter whether this worksheet or the TabProcs worksheet is opened first. After opening one of these, ensure that the Kernal mode is set to shared before opening the second. Also ensure that the Procedures have been loaded into memory by clicking on any of the red code (in that Worksheet) and pressing enter.
<Text-field layout="Heading 1" style="Heading 1">Basic Instructions and Rules</Text-field>Entering Irreps and Tensor Products: 1. The number of non-zero entries in Tensors must equal the number of entries in Irrep--add zeros as needed.
2. The sum of all enties in Tensors must equal the sum of all entries in Irrep. A constant can be added to all entries of the irrep without changing the irrep.
Entering Casimir Operators--Indice Notation: An operator Li,j is entered as list, as follows:
L := [i,j]:
In Maple, lists always are surrounded by square brackets, with it's entries seperated by commas.
Products of operators are entered by creating a list of several duple list, as in:
C := [[1,2],[2,1]]:
Likewise, addition of several products is emplied by enclosing products in a list. The previous operator squared is:
C := [[[1,2],[2,1]], [[1,2],[2,1]]]:
A fully specified sum of products of L-operators is then a three-fold list--the innermost lists being the duples of indices, the middle level implying products, and the outermost level implying sums.
Entering Casimir Operators--Coupling notation:
Rather than entering all the indices necessary to specify a Casimir Operator, one can also specify which tensors in the product the Casimir should go over, as well as to what power the operator should be raised. In Maple, a set is defined similarly to a list, but is surrounded by braces { }. For example, a quadratic Casimir over the 1st and 2nd spaces in the Tensor Product would be written as:
C := [{1, 2}, 2]:
Likewise the cubic Casimir over just the 2nd, 4th and 5th spaces in a Tensor product would be written as:
C := [{2, 4, 5}, 3]:
Note that such an operator would be quite complicated to write out in indice notation.
<Text-field layout="Heading 1" style="Heading 1">How to Use Examples</Text-field> Press enter when the cursor is on red code to have it interpreted by Maple. Start with the first red code in each example and press enter to see the results.
<Text-field layout="Heading 1" style="Heading 1">Canonical Example</Text-field>The Irrep and Tensor Product is entered as follows:
Irrep := [3,2,1,0]:
Tensors := [[2,1,0],[2,1,0]]:
Map the Highest Weight Vector in Terse Mode
MapHW( Terse );
Apply a degenerate Generalized Casimir Operator
Casimir[first]( [[1,2],[2,1]] );
Apply a non-degenerate Generalized Casimir Operator
Casimir[first]( [[1,2],[2,2],[2,1]] );
<Text-field layout="Heading 1" style="Heading 1">A 4-fold SU(2)</Text-field>Choose another Irrep and Tensor Product
Irrep := [5,3,0,0]:
Tensors := [[2,0],[2,0],[2,0],[2,0]]:
Map the Hightest Weight Vector in Verbose Mode
#MapHW( Verbose );
MapHW( Irrep, Tensors):
Export_MapHW("Su2");Import_MapHW("Su2");Apply a quadratic Casimir Operator coupling 1 & 2
lambda1, P1 := Casimir[first]( {1,2}, 2 ):
lambda1;
Combine with a quadratic Casimir coupling 3 & 4
lambda1, P1 := Casimir[additional]( {3,4}, 2 ):
lambda1;
Apply a new scheme, starting with a quadratic Casimir Operator coupling 1 & 3
lambda2, P2 := Casimir[first]( {1,3}, 2 ):
lambda2;
Combine with a quadratic Casimir coupling 2 & 4
lambda2, P2 := Casimir[additional]( {2,4}, 2 ):
lambda2;
Compute the Racah Coefficients betweent the coupling schemes:
R := Racah( P1, P2 );
Choose Basis for direct product and direct sum basesIrrepTableau := [[5,3],[4]];
ProductTableau := [[[2,0],[2]], [[2,0],[1]], [[2,0],[1]], [[2,0],[0]] ];
Compute Clebsch-Gordan Coefficient in coupling scheme #1CGC:-Compute(P1);Compute the Clebsch-Gordan Coefficient for couping scheme #2CGC:-Rotate(P2);ExportMatrix("temp", P2);"hi"||" there";TTdSMApJNlJUQUJMRV9TQVZFLzEzODc3NjE0NFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyciJyIiIiIpIiM/RigiIzdGKUYpRiYKTTdSMApJNlJUQUJMRV9TQVZFLzEzOTY3NDY5NlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIy0iJyIjIiM/RiciIikiIzdGKUYpRilGJ0YpRidGCihGKUYmCg==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTMxODM3MlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyciJyIiIiIpIiM3RihGKCIjP0YpRiYKTTdSMApJNlJUQUJMRV9TQVZFLzE0MDAwNjUzNlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIy0iJyIjIiIpIiM3RihGKCIjP0YpRihGKUYnRihGCilGKEYmCg==TTdSMApJNlJUQUJMRV9TQVZFLzEzODQ3MjcyNFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhI0UiJyInLCQqJCIiJiMiIiIiIiMjISIiIiInLCQqCiQiIzpGKiNGK0YvI0YrIiIkLCRGKEYzI0YuRjUsJCokRjVGKkYtRjcsJEY5RjNGJ0Y0RjZGMEY2RjBGN0YnRjRGOCwkRjFGLSIiIUY4RjsKRjhGPEY4RjxGMEY4RjBGPEY3RjhGJ0Y0RjZGO0YmCg==TTdSMApJNlJUQUJMRV9TQVZFLzEzNTM4MTMwOFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiciJywkKiQiI0kjIiIiIiIjI0YrIiM/LCQqJCIjCjVGKiMhIiJGLiIiIUYnRjQsJCokRixGKiNGMyIiJUYmCg==TTdSMApJNlJUQUJMRV9TQVZFLzEzNTcyMjg4OFgqJSlhbnl0aGluZ0c2IjYiW2dsISMlISEhIiciJyIiISwkKiQiI0kjIiIiIiIjIyEiIiIjP0YnCiwkKiRGLUYrI0YvIiIlLCQqJCIjNUYrRi4sJEYpI0YsRjBGJgo=