Overview
The creation/annihilation operators are operators on the Second Quantized Hilbert Space. That is, they are functions that take a vector in the Hilbert space as input, and returns a vector in the Hilbert space as output. These operators are constructed in analogy to the creation/annihilation operators in the quantum oscillator problem.
Creation Operators
The creation operators are denoted {% a_p ^\dagger %}. It operates on a vector in the Hilbert space and returns the vector in the Hilbert space with an additional particle with momentum {% p %} added to it. When operating on the vacuum state, we get
{% a_p^\dagger |0 \rangle = |p \rangle %}
At times a constant factor is included in the operator, such as
{% a_p^\dagger |0 \rangle = \frac{1}{\sqrt{2 \omega_p}}|p \rangle %}
When operating on the state with a single particle with momentum {% p_1 %}, it produces a two particle state
with momentums {% p_1 %} and {% p_2 %}
{% a_{p2}^\dagger |p_1 \rangle = |p_1,p_2 \rangle %}
Annihilation Operator
The annihilation operator removes the particle created by the corresponding creation operator.
{% a_p |p \rangle = |0 \rangle %}
An annihilation operator applied to the ground state returns zero.
{% a_p |0 \rangle = 0 %}
Commutation Relations
The following is the commuation relation for the creation and annihilation operators:
{% [a_k, a_p^\dagger] = (2\pi)^3 \delta^3(\vec{p} - \vec{k}) %}
Representing State by Operators
Any state in the Second Quantized Hilbert space can then be represented as a set of creation/annihilation operators applied to the vacuum state. That is, we flip the equations above as
{% |p_1,p_2 \rangle = a_{p2}^\dagger |p_1 \rangle %}
Quantum Field
A field is just an integral over the creation and annihilation operators. As an example
{% \phi(\vec{x}) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2 \omega_p}} (a_p e^{i\vec{p}\vec{x}} + a_p^{\dagger} e^{-i\vec{p}\vec{x}} ) %}
That is, the field is a function that takes a point in space and returns and integral over the operators.
Note, an integral (think sum) of operators results in an operator.
{% \langle p | \phi(x) | x \rangle = \langle 0 | \sqrt{2 \omega_p} a_p \int \frac{d^3 k}{(2\pi)^3} \frac{1}{\sqrt{2 \omega_k}}(a_k e^{i\vec{k}\vec{x}} + a_k^\dagger e^{-i \vec{k} \vec{x}}) | 0 \rangle %}
{% = \int \frac{d^3 k}{(2\pi)^3} \sqrt{\frac{\omega_p}{\omega_k}} [e^{i \vec{k}\vec{x}} \langle 0 | a_pa_k | 0 \rangle + e^{-i \vec{k}\vec{x}} \langle 0 | a_pa_k^\dagger | 0 \rangle ] %}
{% = e^{-i \vec{p} \vec{x}} %}
which from single particle quantum mechanics, we know that {% \langle \vec{p} | \vec{x} \rangle = e^{-i \vec{p} \vec{x}} %}
That is, the operator produced by this field (integral) creates a particle at {% \vec{x} %}.
(see Schwartz)