Charge conjugation C is also an antiunitary operator which acts on α α c i as C α α c i C − − 1 = α α ∗ ∗ ( U C † † ) j i c j , and can be written as C = U C K where U C is unitary. Again either C 2 = 1 or C 2 = − − 1 depending on what U C is. A Hamiltonian with particle hole symmetry satisfies C H ^ ^ C − − 1 = − − H ^ ^ , or on the level of first-quantised Hamiltonian matrices, U C H ∗ ∗ U C − − 1 = − − H , for some choice of U C .