Symmetries and excited states

1 Sparsity of the hlock-symmetry operators Suppose that 0 is a symmetry operator of the system block (for example, particle-hole or spin-flip symmetry). Then it can be shown that 

Equation (H.IO) has two significant consequences. To explain these, we modify the notation for the system blocks states from ns))( ) to where 

Q denotes the conserved quantum numbers of the block and is now a supplementary index. Then 

If Q = Q then commutes with pq and the eigenstates of pq are simultaneously eigenstates of Thus, 

If g 7 g then all the states with complementary quantum numbers may be defined as 

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