There appears to be an apparent dichotomy between the interpretation of second quantized operators in condensed matter and quantum field theory proper. For example, if we look at Peskin and Schroeder, the Hamiltonian for the quantized Dirac field Hamiltonian is given by eqn. 3.104 on p. 58:
$$H =\int \frac{d^3 p}{(2\pi)^3}\sum_s E_p (a_p^{s\dagger}a_p^s+b_p^{s\dagger}b_p^s). \tag{3.104}$$
Here, $a_p^{s\dagger}$ and $a_p^s$ are the creation and annihilation operators for fermions, while $b_{p}^{s\dagger}$ and $b_p^s$ are the creation and annihilation operators for anti-fermions. Note that particles and antiparticles are described by the same field, which we will call $\psi$. Also note that these operators satisfy the relations
$$a_p^{s\dagger}=b_{p^*}^s,\qquad a_p^s=b_{p^*}^{s\dagger}$$
In condensed matter notation, we might similarly have creation and annihilation operators for some many-body state, which we will call $c_i^\dagger$ and $c_i$ (we consider spinless fermions for simplicity). In numerous texts and in my classes, I have learned that $c_i$ might be interpreted as the creation operator of a hole, which might be thought of as the condensed matter equivalent of an antiparticle.
Why do we need separate creation and annihilation operators for fermions and anti-fermions in quantum field theory, but in condensed matter we can simply treat the annihilation operator as the creation operator of the "antiparticle"? It is my understanding that there is a fundamental difference between the Dirac field second quantized operators and the condensed matter second quantization operators, but I can't find any references that discuss this. Any explanation would be appreciated, as well as any references at the level of Peskin and Schroeder.