We measure real particles in the laboratory, and model their existence and behavior using quantum field theory and the Feynman diagrams that give the values for measurable quantities: crossections, decays, angular distributions... .
Quantum field theories are useful in many domains of physics, from nuclear models to condensed matter to particle physics.
In particle physics it is posited that each particle ( and implied antiparticle) in the elementary particles table generates a field that fills up all space time
an electron field, a positron field an electron neutrino field, etc for the ennumerated particles.
On these fields , which mathematically are the plane wave solutions ( no potential) of the corresponding quantum mechanical equation ( Dirac, Klein Gordon, quantized Maxwell) act creation and annihilation operators. Where they act the corresponding particle is created, or destroyed.
The Feynman diagrams show this at the vertices,
This diagram is for computing the interaction of e- e- scattering. At the vertices the electrons go in time in the y direction, by consecutive creation and annihilation , and at the vertices a photon is created which is exchanged between them, and is anihilated , in a virtual mathematical space, as everything is under an integral.
This integral is important because it allows for the existence of real electrons input and output. Remember a plane wave covers from -infinity to infinity, and cannot be a useful representation of a particle. The uncertainty principle mathematically leads to wave packets representing the real paticles , in this case electrons in electrons out after scatter. The mathematical formalism of calculating measurable quantities using feynman diagrams is shown here , clarifying the need for wavepackets for modeling measureable quantities of real particles.