This is what it means to be linear operator A map from a vector space to another (in this case the same vector space) The definition of r symmetry acting on supercharges acting is made in a classical field theory (in super space) Promoting this classical field theory to a quantum field theory involves computing the dirac braket in the standard fashion and promoting it to a (anti. However, susy representations furnish reducible poincaré representations, so supermultiplets in general correspond to multiple particles having the same mass, which are related by supersymmetry transforms In this context, the broader term multiplet is used interchangeably with supermultiplet.
I think i figured out the meaning of this after some research so, i am posting an answer to my own question The answer is there is nothing called $\mathcal {n}= (1,1)$ superalgebra The superalgebra is always named by $\mathcal {n}$ with integers The $\mathcal {n}= (1,1)$ actually means a supergravity multiplet so my original question was wrong We get this multiplet as the massless level of. 4 supergravity by daniel z freedman and antoine van proeyen is quite excellent for illustrating clifford algebra techniques and calculations in the classical susy/sugra in general (in the component formalism)
If so, standard treatments (e.g Numerical ones) to solve the qm system as an eigenvalue problem would totally miss this effect because they would start from a diagonal hamiltonian! In modern supersymmetry by john terning, page 13, the states of the massless supermultiplets of $n=4$ susy are labelled by the helicity and representation of r.
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