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## III. SPACETIME FORMALISM 

A (Heisenberg picture) quantum field is an operator-valued function ˆ ψ on spacetime satisfying the Klein-Gordon equation 

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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 

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<img src='content_image/368535.jpg'>

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 

<img src='content_image/368533.jpg'>

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 

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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 

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