How to correctly represent a quantum field? As something single that fills all space (and particles are excitations of the field just like waves are excitations of a single ocean), or as a set of particles, each of which creates its own field (and when there are no particles, there is no field)?
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In quantum field theory, whether for particle physics or other frames , (for example nuclear physics,) the fields are the plane wave solutions, i.e. without a potential, of the corresponding quantum equation, filling all points (x,y,z,t), a kind of coordinate system, whether particles exist or not. Creation and annihilation differential operators create or destroy a particle according to the problem at hand. Feynman diagrams are used as a shorthand describing the interactions.
The field conceptually always exists, for the standard model of particle physics: each particle in the table and their antiparticles is expressed as a field over all space time, electron field,neutrino field etc.
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