Ladder operators vs. conjugate variables

Question

In the book  Introduction to Many-Body Physics by Piers Coleman, it states on page 12 that

... the particle field and its complex conjugate are
conjugate variables.

In other words, the particle field $psi(x)$ and its complex conjugate $psi^dagger(x)$ obey the canonical commutation relation $ [psi(x), psi^dagger(y) ]_pm = delta(x-y) $ as given in (2.8) and (2.10), and can therefore be viewed as ladder operators which create and annihilate particles. This does not seem to be a general result, for example

Consider the complex Klein-Gordon field $phi(x)$. From the Lagrangian $mathcal{L} = partial_mu phi^* partial^mu phi$, the canonically conjugate variables are $phi(x)$ and $pi(x) = partial_0 phi^*(x) neq phi^*(x) $. If we work with a real scalar field, then there isn't even a notion of the complex conjugate field either.
Consider Maxwell's equations. From the Lagrangian $mathcal{L} = -frac{1}{4} F_{mu nu} F^{mu nu}$, we have the conjugate variables $A_mu(x)$ and $pi^mu(x) = - F^{0 mu} neq A_mu^*(x)$. Again, this field is real too.

The only fields this seems to be the case for is the Schrodinger and Dirac fields with Lagrangians that contain the term $ipsi^dagger partial_0 psi$.
My question
Does Coleman's statement about position space quantum fields being ladder operators for particles only apply for the Schrodinger and Dirac fields? For other fields that have particle-like excitations, this does not seem to be the case, as one finds the ladder operators appear only in momentum space, i.e., the ladder operators $a^dagger(p), a(p)$. In particular for real fields, there does not seem to be a notion of a creation/annihilation pair in position space as the operators are Hermitian. It seems that many-body and high energy physics treatments of QFT define fields in different ways.

mike stone · Accepted Answer

I imagine that Coleman is restricting himself to non-relatvistic fields. For fields obeying the Schroedinger equation with action
$$
S= int d^dx dt ]left{ihbar psi^dagger partial_t psi + frac{hbar^2}{2m} (nabla psi)^dagger cdot nablapsi +{rm interactions}right}
$$
the conjugate field to $psi$ is $pi= ipsi^dagger$ so the commutation relations are
$$
[psi(x,t),pi(x',t)]_pm = ihbar delta^d(x-x')
$$
or
$$
[psi(x,t),psi^dagger(x',t)]_pm = hbar delta^d(x-x').
$$
This result is even simpler than the Dirac case because there are no antiparticles. Instead $psi(x)$ simply annihilates a particle at $x$ and $psi^dagger(x)$ creates one. For Dirac $psi$ can either annihilate a particle or create an antiparticle.  So for a Schroedinger field in a static potential $V(x)$ we have a mode expansion
$$
psi(x,t)= sum_n a_n u_n(x)e^{-iE_nt/hbar }
$$ where the $u_n(x)$ are normalized wavefunctions of enegy $E_n$, and $a_n$ with $[a_n, a_m^dagger]_pm = delta_{nm}$  are the the corresponding annihilation operators.

Michele Grosso · Answer

In second quantization the quantum fields are promoted to operators and defined as integrals over creation and annihilation operators for each momentum $vec p$. For instance a real scalar field $phi (vec x)$ in the Schroedinger picture shows
$phi (vec x) = int frac{d^3p}{(2 pi)^3} frac{1}{sqrt{2 omega_p}} left(a (vec p) e^{i vec p cdot vec x} + a^dagger (vec p) e^{-i vec p cdot vec x}right)$
where $a (vec p)$ is the annihilation operator, $a^dagger (vec p)$ is the creation operator and $[a_p, a^dagger_k] = (2 pi)^3 delta^3 (vec p - vec k)$.
The conjugate operator is defined as $pi (vec x) = frac{partial mathcal L}{partial (partial_t phi (vec x))} = partial_t phi (vec x)$ and $[phi (vec x), pi (vec y)] = i delta^3 (vec x -vec y)$.
As for a Dirac field $psi (vec x)$ the conjugate operator is $frac{partial mathcal L}{partial (partial_t psi (vec x))} = i psi^dagger (vec x)$ and $[psi (vec x), i psi^dagger (vec y)] = i delta^3 (vec x - vec y)$.
The quantum fields and their conjugate are position space ladder operators. In second quantization the Hilbert space is promoted to a Fock space, which is defined as a direct sum of Hilbert spaces of physical $n$-particles states
$mathcal F = oplus_n mathcal H_n$
That is the many-body description, so it is consistent with QFT.

Ladder operators vs. conjugate variables

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