They differ from the above only by a phase e ± ± i π π ∑ ∑ k = 1 j − − 1 f k † † f k . The phase is determined by the number of occupied fermionic modes in modes k = 1 , … … , j − − 1 of the field. The phase is equal to + 1 if the number of occupied modes is even, and − − 1 if the number of occupied modes is odd. This phase is often expressed as