Color Extension of SUSY QM
201
Definition 1 If g admits a bilinear form , : g × g → g satisfying the following
three relations, then g is called a Z n
2 -graded color superalgebra:
1. α , g β ⊆ g α+β ,
2. α , X β = −(−1) α·β β , X α
3. α , β , X γ α·γ + cyclic perm. = 0,
where X α ∈ g α and the third relation is called the graded Jacobi identity.
It is easily recognized that the bilinear form α , X β is realized by commutator
and anticommutator:
α , X β = X α X β − (−1)
α·β X β X α .
(3)
The n = 1 case (Z 2 grading) corresponds to the ordinary Lie superalgebras. The first
non-trivial example is the Z 2
2 -graded color superalgebra consisting of four sectors
labelled by (0, 0), (0, 1), (1, 0), (1, 1).
The Z 2
2 -graded version of SQM considered in [4] (Bruce–Duplij model) is a
realization of Z 2
2 -graded super-Poincaré algebra (Z 2
2 -SPA) in the Z 2
2 -graded Hilbert
space L 2 (R) ⊗ C 4 . Z 2
2 -SPA is spanned by H 00 , Q 01 , Q 10 , Z 11 with the indicated Z 2
2
grading and their non-vanishing relations are given by
{Q 01 , Q 01 } = {Q 10 , Q 10 } = H 00 ,
[Q 01 , Q 10 ] = iZ 11 .
(4)
H 00 is a diagonal matrix operator interpreted as a quantum mechanical Hamiltonian.
Q 01 and Q 10 play the role of supercharges, however, they have different degree. As
a consequence, they close by commutator (instead of anticommutator) into Z 11 , the
central element of the algebra.
3 Extensions of Bruce–Duplij Model
3.1 From Superalgebra to Z 2
2
-Graded Color Superalgebra
As is seen from (4), in the Z 2
2 -graded SQM of [4], each subspaces of degree (0, 1)
and (1, 0) has only one supercharge. So one may say that it is a Z 2
2 -version of N = 1
SQM. We would like to have quantum mechanical models which have more than one
supercharges in each subspace. We also want models of Z 2
2 -graded version of SCM.
These will be done by using the theorem shown below which relates an ordinary
superalgebra to its Z 2
2 -graded version [7].
Let s be an ordinary Lie superalgebra (Z 2 -graded Lie algebra) spanned by the
elements T a
i with a ∈ Z 2 = {0, 1}. The defining relations may be written as
201
Definition 1 If g admits a bilinear form , : g × g → g satisfying the following
three relations, then g is called a Z n
2 -graded color superalgebra:
1. α , g β ⊆ g α+β ,
2. α , X β = −(−1) α·β β , X α
3. α , β , X γ α·γ + cyclic perm. = 0,
where X α ∈ g α and the third relation is called the graded Jacobi identity.
It is easily recognized that the bilinear form α , X β is realized by commutator
and anticommutator:
α , X β = X α X β − (−1)
α·β X β X α .
(3)
The n = 1 case (Z 2 grading) corresponds to the ordinary Lie superalgebras. The first
non-trivial example is the Z 2
2 -graded color superalgebra consisting of four sectors
labelled by (0, 0), (0, 1), (1, 0), (1, 1).
The Z 2
2 -graded version of SQM considered in [4] (Bruce–Duplij model) is a
realization of Z 2
2 -graded super-Poincaré algebra (Z 2
2 -SPA) in the Z 2
2 -graded Hilbert
space L 2 (R) ⊗ C 4 . Z 2
2 -SPA is spanned by H 00 , Q 01 , Q 10 , Z 11 with the indicated Z 2
2
grading and their non-vanishing relations are given by
{Q 01 , Q 01 } = {Q 10 , Q 10 } = H 00 ,
[Q 01 , Q 10 ] = iZ 11 .
(4)
H 00 is a diagonal matrix operator interpreted as a quantum mechanical Hamiltonian.
Q 01 and Q 10 play the role of supercharges, however, they have different degree. As
a consequence, they close by commutator (instead of anticommutator) into Z 11 , the
central element of the algebra.
3 Extensions of Bruce–Duplij Model
3.1 From Superalgebra to Z 2
2
-Graded Color Superalgebra
As is seen from (4), in the Z 2
2 -graded SQM of [4], each subspaces of degree (0, 1)
and (1, 0) has only one supercharge. So one may say that it is a Z 2
2 -version of N = 1
SQM. We would like to have quantum mechanical models which have more than one
supercharges in each subspace. We also want models of Z 2
2 -graded version of SCM.
These will be done by using the theorem shown below which relates an ordinary
superalgebra to its Z 2
2 -graded version [7].
Let s be an ordinary Lie superalgebra (Z 2 -graded Lie algebra) spanned by the
elements T a
i with a ∈ Z 2 = {0, 1}. The defining relations may be written as