How do we picture/imagine a quantum field ? How is it different from a classical field ? How do we picture a particle arising out of an excitation of its respective quantum field ?
Please give an accurate and not pop-sci kind of answer. Even if the QF's aren't so simple so as to give them a mental picture, it is fine. Please explain as to what degree the picture you give for it is correct.
How is it different from a classical field ?
For starters, a quantum field is operator valued rather than scalar, vector, tensor, etc. valued. A quantum field assigns an operator to each event in spacetime.
How do we picture a particle arising out of an excitation of its respective quantum field ?
We have to be careful to distinguish the quantum (operator valued) field from the entity that is operated on and its quanta. See, for example, this answer.
Also, from the first chapter of "Student Friendly Quantum Field Theory", section 1.8:
When the word “field” is used classically, it refers to an entity, like fluid wave amplitude, E, or B, that is spread out in space, i.e., has different values at different places.
By that definition, the wave function of ordinary QM, or even the particle state in QFT, is a field. But, it is important to realize that in quantum terminology, the word “field” means an operator field, which creates and destroys particle states.
States (= particles = wave functions = kets) are not considered fields in that context.
Honestly, I do not know how to picture or even if it is possible to picture a quantum field and the quanta they create and destroy.
I will be very interested in other's answers as I too have long struggled to find a satisfying picture.
As Alfred Centauri mentioned in QFT a field can be thought of as an operator. This holds in the canonical approach to QFT. However, in the equivalent path integral approach, one deals with QFT by assigning an amplitude to a particular trajectory through field space. So if we measure a field at time 0 and then again at time T then we evaluate the amplitude for any particular measurement result at T by evaluating the amplitude for all trajectories through field space that go from the configuration we measured at time 0 and the proposed configuration at time T. The total amplitude is given by the sum of all these amplitudes. The probability is obtained by taking the square of the absolute value of the total amplitude. If you want a particle at time 0 at position x one puts in a delta function in the potential which is zero everywhere except at time 0 and position x. Alternatively, a plane wave will give a particle with a definite momentum. So particles are just fluctuations in the field.
