Physical Interpretation of Field Operator in Quantum Field Theory and Mode Expansion

Question

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?

Answer 1

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).