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## Main Question or Discussion Point

In short, the question is, how is the position operator related to the position-parameters of a quantum field

For instance, consider a quantum-mechanical state of two particles |Ψ>. This can be expanded in terms of the position eigenstates

In quantum field theory, the position eigenstates are derived from the field

Within this field-theoretic approach, how to show that the state

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 operator

*ψ(x)*?For instance, consider a quantum-mechanical state of two particles |Ψ>. This can be expanded in terms of the position eigenstates

*|x*to give the position representation wavefuntion Ψ_{1},x_{2}>*(x*. The position eigenstates are eigenstates of_{1},x_{2})__two__distinct operators*X*and_{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 field

*ψ(x)*, where*|x*is proportional to_{1},x_{2}>*ψ*where |0> is the vacuum state.^{+}(x_{1})ψ^{+}(x_{2})|0>Within this field-theoretic approach, how to show that the state

*ψ*is a actually a^{+}(x_{1})ψ^{+}(x_{2})|0>*position*eigenstate, 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 operator

*a*. What is the relationship between the^{+}(p)*parameter p*in*a*and the^{+}(p)*observable P*?
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