III. SPACETIME FORMALISM i ∫ Σ d $^{3}$x ( | x$_{+}$ 〉$^{↔}$ ∂$_{0}$ 〈 x$_{+}$ | − | x$_{−}$ 〉$^{↔}$ ∂$_{0}$ 〈 x$_{−}$ | ) = 1 . 〈 x ′ ± | x$_{±}$ 〉 = Δ$_{±}$( x ′ − x ) . | x$_{±}$ 〉 = ˆ ψ † $_{±}$( x ) | 0 〉 . (11) Δ$_{±}$( x ′ − x ) ≡ (2 π ) 3 ∫ d $^{3}$p e i[ ∓ ω$_{p}$ ( x ′ $^{0}$− x $^{0}$)+ p · ( x $^{′}$− x )] 2 ω$_{p}$ , [ ˆ ψ$_{±}$ ( x $^{′}$) , ˆ ψ † $_{±}$( x )] = Δ$_{±}$( x ′ − x ) , (10) ( ∂ 2 ∂x 2 − m $^{2}$) ˆ ψ ( x ) = 0 . (9) A (Heisenberg picture) quantum field is an operator-valued function ˆ ψ on spacetime satisfying the Klein-Gordon equation The operator ˆ ψ ( x ) acts on the Fock space of particle position states, destroying a particle at position x , while its adjoint ˆ ψ $^{†}$( x ) acts to create a particle at x [36–38]. Distinguish particle fields ˆ ψ$_{+}$ from antiparticle fields ˆ ψ$_{−}$ . Each kind of field has a specific nonzero commutator with its adjoint: where with ω$_{p}$ ≡ √ p 2 + m $^{2}$. (For simplicity I will only consider scalar fields here. The general- ization to non-scalar fields and fermionic anticommutation rules is straightforward and does not substantially effect the following discussion.) Let | 0 〉 be the vacuum state of the Fock space. Then single particle and antiparticle position states are given by While these states represent a particle or antiparticle localized at at a specific position x , they are not orthogonal and states for different positions overlap. This is due to the fact that they are constrained to be on-shell by Eq. (9). Indeed, the commutation relations from Eq. (10) imply that They do obey a completeness relation, but only over any spacelike hypersurface Σ, not over all of spacetime [18]: These properties make the | x$_{±}$ 〉 states inconvenient for constructing projection operators. Instead, one would like to have an orthogonal basis for spacetime position states, analo- gous to the three-dimensional spacial position states familiar from non-relativistic quantum 8