I'm struggling to understand the physical interpretation behind the field operators $ \phi(\mathbf x)$ and $\phi ^\dagger (\mathbf x)$ in quantum field theory. My understanding is $ \phi ^\dagger (\mathbf x)$ is an operator which creates a particle at a position $\mathbf x$. p37 of Quantum Field Theory for the Gifted Amateur (Lancaster & Blundell, OUP) says that from this definition of $\phi ^\dagger (\mathbf x)$, we can then write $$ \phi ^\dagger (\mathbf x) \propto \sum_{\mathbf{p}} a_{\mathbf p}^\dagger e^{-i \mathbf p \cdot \mathbf x}.$$ However this is different to the approach in Peskin & Schroeder. They write out the mode expansion for $\phi(\mathbf x)$ for the Klein-Gordon field (eqn 2.25) as $$\phi(\mathbf x) \propto \int \frac{1}{\left ( |\mathbf p|^2 + m^2 \right )^{1/4}} \left ( a_{\mathbf p} e^{i \mathbf p \cdot \mathbf x} + a_{\mathbf p} ^\dagger e^{-i \mathbf p \cdot \mathbf x} \right ) d^3 p.$$ I understand that we have two terms since we require $\phi(\mathbf x)$ to be Hermitian. However does this not contradict the definition that $\phi^\dagger (\mathbf x)$ creates a particle at a position $\mathbf x$ above? Is it correct to interpret $\phi(\mathbf x)$ as an operator that creates a particle at $\mathbf x$, or is it better to just ignore the interpretation and only think about the results the theory gives from performing measurements?
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Your second equation is inaccurately copied from P&S (2.25): it has a $(|{\mathbf p}|^2+ m^2 )^{1/4}$ in the denominator, delocalizing the particle created by the second term, only. See this question. The first term destroys a particle! – Cosmas Zachos Dec 14 '20 at 21:48
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@NiharKarve my bad - I forgot to add a dagger on the RHS – Bentley Carr Dec 15 '20 at 11:58
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Also see Problem 2.3 in L&B - the mode expansion for $\phi$ is analogous to the $\hat{x}$ operator – Nihar Karve Dec 15 '20 at 12:04
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One could define the field operator either way. The representation with only creation/annihilation operator $$ \psi(x) \sim \sum_k a_ke^{ikx}, \psi^\dagger(x) \sim \sum_k a_k^\dagger e^{-ikx} $$ has the interpretation of creating/removing a particle at a particular time point, which is a convenient interpretation, e.g., for electrons. Representations with sum or difference of creating and annihilation operators $$ \psi(x)\sim \sum(b_k e^{ikx} + b_k^\dagger e^{-ikx}),\\ \psi(x)\sim \sum i(b_k e^{ikx} - b_k^\dagger e^{-ikx}) $$ are more convenient for bosonic fields, where in classical limit they reduce to the field strength (e.g., $E(x,t)$ for the electric field or polarization for phonons).
Roger V.
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