In short, the question is, how is the position operator related to the position-parameters of a quantum field(adsbygoogle = window.adsbygoogle || []).push({}); ψ(x)?

For instance, consider a quantum-mechanical state of two particles |Ψ>. This can be expanded in terms of the position eigenstates|xto give the position representation wavefuntion Ψ_{1},x_{2}>(x. The position eigenstates are eigenstates of two distinct operators_{1},x_{2})Xand_{1}X. (the identical nature of the particles is found in the symmetry of the wave-function, which in quantum mechanics has to be added ad hoc.)_{2}

In quantum field theory, the position eigenstates are derived from the fieldψ(x), where|xis proportional to_{1},x_{2}>ψwhere |0> is the vacuum state.^{+}(x_{1})ψ^{+}(x_{2})|0>

Within this field-theoretic approach, how to show that the stateψis a actually a^{+}(x_{1})ψ^{+}(x_{2})|0>positioneigenstate, that is, that

X?_{1}ψ^{+}(x_{1})ψ^{+}(x_{2})|0> = x_{1}ψ^{+}(x_{1})ψ^{+}(x_{2})|0>

I'm thinking that this relationship is itself an assumption, that it is itself the mathematical statement that the parameters of the field are to be interpreted as spatial position. Comments?

The question could of course be rephrased in terms of the momentum operators with respect to the creation operatora. What is the relationship between the^{+}(p)parameter pinaand the^{+}(p)observable P?

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# Position and momentum operators in quantum field theory

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