package jsci.maths.algebras;
import jsci.maths.*;
import jsci.maths.fields.ComplexField;
import jsci.maths.matrices.AbstractComplexMatrix;
import jsci.maths.matrices.AbstractComplexSquareMatrix;
import jsci.maths.matrices.ComplexSquareMatrix;
import jsci.maths.vectors.AbstractDoubleVector;
import jsci.maths.vectors.DoubleVector;
/**
* The su3Dim3 class encapsulates su(3) algebras using
* the 3 dimensional (fundamental) representation.
* Elements are represented by vectors with a matrix basis.
* @version 1.2
* @author Mark Hale
*/
public final class su3Dim3 extends LieAlgebra {
/**
* Useful complex constants.
*/
private final static Complex T8_1=new Complex(0.5/Math.sqrt(3.0),0.0);
private final static Complex T8_2=new Complex(-1.0/Math.sqrt(3.0),0.0);
private final static Complex t1[][]={
{Complex.ZERO,ComplexField.HALF,Complex.ZERO},
{ComplexField.HALF,Complex.ZERO,Complex.ZERO},
{Complex.ZERO,Complex.ZERO,Complex.ZERO}
};
private final static Complex t2[][]={
{Complex.ZERO,ComplexField.MINUS_HALF_I,Complex.ZERO},
{ComplexField.HALF_I,Complex.ZERO,Complex.ZERO},
{Complex.ZERO,Complex.ZERO,Complex.ZERO}
};
private final static Complex t3[][]={
{ComplexField.HALF,Complex.ZERO,Complex.ZERO},
{Complex.ZERO,ComplexField.MINUS_HALF,Complex.ZERO},
{Complex.ZERO,Complex.ZERO,Complex.ZERO}
};
private final static Complex t4[][]={
{Complex.ZERO,Complex.ZERO,ComplexField.HALF},
{Complex.ZERO,Complex.ZERO,Complex.ZERO},
{ComplexField.HALF,Complex.ZERO,Complex.ZERO}
};
private final static Complex t5[][]={
{Complex.ZERO,Complex.ZERO,ComplexField.MINUS_HALF_I},
{Complex.ZERO,Complex.ZERO,Complex.ZERO},
{ComplexField.HALF_I,Complex.ZERO,Complex.ZERO}
};
private final static Complex t6[][]={
{Complex.ZERO,Complex.ZERO,Complex.ZERO},
{Complex.ZERO,Complex.ZERO,ComplexField.HALF},
{Complex.ZERO,ComplexField.HALF,Complex.ZERO}
};
private final static Complex t7[][]={
{Complex.ZERO,Complex.ZERO,Complex.ZERO},
{Complex.ZERO,Complex.ZERO,ComplexField.MINUS_HALF_I},
{Complex.ZERO,ComplexField.HALF_I,Complex.ZERO}
};
private final static Complex t8[][]={
{T8_1,Complex.ZERO,Complex.ZERO},
{Complex.ZERO,T8_1,Complex.ZERO},
{Complex.ZERO,Complex.ZERO,T8_2}
};
/**
* Basis.
*/
private final static AbstractComplexSquareMatrix basisMatrices[]={
new ComplexSquareMatrix(t1),
new ComplexSquareMatrix(t2),
new ComplexSquareMatrix(t3),
new ComplexSquareMatrix(t4),
new ComplexSquareMatrix(t5),
new ComplexSquareMatrix(t6),
new ComplexSquareMatrix(t7),
new ComplexSquareMatrix(t8)
};
private final static su3Dim3 _instance = new su3Dim3();
/**
* Constructs an su(3) algebra.
*/
private su3Dim3() {
super("su(3) [3]");
}
/**
* Singleton.
*/
public static final su3Dim3 getInstance() {
return _instance;
}
/**
* Returns an element as a matrix (vector*basis).
*/
public AbstractComplexSquareMatrix getElement(final AbstractDoubleVector v) {
AbstractComplexMatrix m=basisMatrices[0].scalarMultiply(v.getComponent(0));
m=m.add(basisMatrices[1].scalarMultiply(v.getComponent(1)));
m=m.add(basisMatrices[2].scalarMultiply(v.getComponent(2)));
m=m.add(basisMatrices[3].scalarMultiply(v.getComponent(3)));
m=m.add(basisMatrices[4].scalarMultiply(v.getComponent(4)));
m=m.add(basisMatrices[5].scalarMultiply(v.getComponent(5)));
m=m.add(basisMatrices[6].scalarMultiply(v.getComponent(6)));
m=m.add(basisMatrices[7].scalarMultiply(v.getComponent(7)));
return (AbstractComplexSquareMatrix)m.scalarMultiply(Complex.I);
}
/**
* Returns the Lie bracket (commutator) of two elements.
*/
public AbstractDoubleVector multiply(final AbstractDoubleVector a, final AbstractDoubleVector b) {
double array[]=new double[8];
array[0]=b.getComponent(1)*a.getComponent(2)-b.getComponent(2)*a.getComponent(1)+0.5*(
b.getComponent(3)*a.getComponent(6)-b.getComponent(6)*a.getComponent(3)+
b.getComponent(5)*a.getComponent(4)-b.getComponent(4)*a.getComponent(5));
array[1]=b.getComponent(2)*a.getComponent(0)-b.getComponent(0)*a.getComponent(2)+0.5*(
b.getComponent(3)*a.getComponent(5)-b.getComponent(5)*a.getComponent(3)+
b.getComponent(4)*a.getComponent(6)-b.getComponent(6)*a.getComponent(4));
array[2]=b.getComponent(0)*a.getComponent(1)-b.getComponent(1)*a.getComponent(0)+0.5*(
b.getComponent(3)*a.getComponent(4)-b.getComponent(4)*a.getComponent(3)+
b.getComponent(6)*a.getComponent(5)-b.getComponent(5)*a.getComponent(6));
array[3]=Math.sqrt(0.75)*(b.getComponent(4)*a.getComponent(7)-b.getComponent(7)*a.getComponent(4))+0.5*(
b.getComponent(6)*a.getComponent(0)-b.getComponent(0)*a.getComponent(6)+
b.getComponent(5)*a.getComponent(1)-b.getComponent(1)*a.getComponent(5)+
b.getComponent(4)*a.getComponent(2)-b.getComponent(2)*a.getComponent(4));
array[4]=Math.sqrt(0.75)*(b.getComponent(7)*a.getComponent(3)-b.getComponent(3)*a.getComponent(7))+0.5*(
b.getComponent(0)*a.getComponent(5)-b.getComponent(5)*a.getComponent(0)+
b.getComponent(6)*a.getComponent(1)-b.getComponent(1)*a.getComponent(6)+
b.getComponent(2)*a.getComponent(3)-b.getComponent(3)*a.getComponent(2));
array[5]=Math.sqrt(0.75)*(b.getComponent(6)*a.getComponent(7)-b.getComponent(7)*a.getComponent(6))+0.5*(
b.getComponent(4)*a.getComponent(0)-b.getComponent(0)*a.getComponent(4)+
b.getComponent(1)*a.getComponent(3)-b.getComponent(3)*a.getComponent(1)+
b.getComponent(2)*a.getComponent(6)-b.getComponent(6)*a.getComponent(2));
array[6]=Math.sqrt(0.75)*(b.getComponent(7)*a.getComponent(5)-b.getComponent(5)*a.getComponent(7))+0.5*(
b.getComponent(0)*a.getComponent(3)-b.getComponent(3)*a.getComponent(0)+
b.getComponent(1)*a.getComponent(4)-b.getComponent(4)*a.getComponent(1)+
b.getComponent(5)*a.getComponent(2)-b.getComponent(2)*a.getComponent(5));
array[7]=Math.sqrt(0.75)*(b.getComponent(3)*a.getComponent(4)-b.getComponent(4)*a.getComponent(3)+
b.getComponent(5)*a.getComponent(6)-b.getComponent(6)*a.getComponent(5));
return new DoubleVector(array);
}
/**
* Returns the basis used to represent the Lie algebra.
*/
public AbstractComplexSquareMatrix[] basis() {
return basisMatrices;
}
}
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