In order to satisfy the Heisenberg equation of motion, the quark fields must satisfy an equal time anticommutator algebra where and all other anticommutators are zero. Note that we have restored implicit spinor indices j , k and used the property ( γ $^{0}$) 2 = 1 as well as the Heisenberg equation of motion for ¯ Q . It is important to note that the Heisenberg equation of motion can also be satisfied by operators that have a commutator algebra. However, the spin-statistics theorem requires that fermions satisfy an anticommutator algebra. Since Dirac fields are fermions, we must use the algebra (1.8). This issue is discussed in many standard texts, see for instance Ref. [100]. The Dirac fields can be expanded in a basis of plane wave states, where p 0 = E$_{p}$ and s denotes the spin state. Using these expressions and the algebra (1.8) it is easy to show that and all other anticommutators involving these operators are zero. The solutions (1.9) are linear combinations of the basis vectors of the Hamiltonian (1.6). In quantum field theory, the operators (1.9) are interpreted as creating and annihilating field quanta by acting on the vacuum (ground) state | 0 〉 , which contains no field quanta. This requires that a r j ( p ) | 0 〉 = 0. Particles and antiparticles are understood as field quanta and are represented by momentum eigenstates with an associated spin state s . For later convenience it is useful to define the normal-ordering operator. When acting on a product of creation and annihilation operators, 13 { a r j ( p ) , a s † k ( q ) } = { b r j ( p ) , b s † k ( q ) } = (2 π ) $^{3}$δ$^{3}$( p − q ) δ $^{rs}$δ$_{jk}$ (1.10) Q$_{j}$ ( x ) = ∫ d $^{3}$p (2 π ) 3 1 $^{√}$2 E$_{p}$ ∑ s $^{(}$a $^{s}$( p ) u s j ( p ) e − ip · x + b s $^{†}$( p ) v s j ( p ) e ip · x ) ¯ Q$_{k}$ ( x ) = ∫ d $^{3}$p (2 π ) 3 1 $^{√}$2 E$_{p}$ ∑ s $^{(}$b $^{s}$( p ) ¯ v s k ( p ) e − ip · x + a s $^{†}$( p ) ¯ u s k ( p ) e ip · x ) (1.9) { Q α j ( x ) , ¯ Q β k ( y ) } = δ $^{3}$( x − y ) γ 0 jk , (1.8)