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As I understand it in QFT interactions are generally modeled as being from the exchange of virtual particles. If I was to think of how to simulate a classical analog I would model two spheres A and B, that each can only change velocity by emitting or absorbing an exchange sphere C. I would use a random number generator to decide whether or not A might at some point emit C and in what direction C might be emitted in, then calculate whether C and B would intercept, and only model C as actually being emitted if it was to intercept B and then model the velocities of A and B changing if they exchange C. This type of modeling wouldn't require using a differential equation in order to model the motion of A and B.

I understand that the quantum case of A, B, and C would be different from the classical case as even if A and B don't interact their equations of motion would not be the same as equations of motion for classical bodies moving with constant velocity, and C would be a virtual particle. Still I'm wondering if it would be possible to create a simulation for an interaction in QFT that doesn't use differential equations or at least wouldn't require knowing what differential equations would be involved. If so what would be the methods for modeling an interaction without using differential equations in QFT?

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    $\begingroup$ Look at the meaning of the words "Quantum Field Theory". You need both the field and the quantum and the theory is the differential equations. What you ask is not possible. $\endgroup$
    – anna v
    Commented Aug 29, 2021 at 5:32
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    $\begingroup$ To expound upon anna v's comment: the language about "virtual particles" and even "particles" themselves is very far down a very long road. The technical meaning of these words in the context of QFT is not what a layman would imagine upon hearing them. Concisely: the picture you describe corresponds to a misunderstanding of the meaning of words in this technical context. $\endgroup$
    – Richard Myers
    Commented Aug 29, 2021 at 7:31
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    $\begingroup$ If you want to try a simulation based on so-called virtual particles, try it in classical field theory first. Yes, so-called virtual particles also arise in classical field theory, for exactly the same reason as in QFT. This is explained in Helling's "Solving Classical Field Equations" (link to pdf). If nothing else, this exercise will help build appreciation for just how misleading the name "virtual particle" really is. $\endgroup$
    – Chiral Anomaly
    Commented Aug 29, 2021 at 14:02

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There's no QFT without Green functions and much more, but something like your question is addressed by Mattuck's "drunken man propagator", followed by "the classical quasi particle propagator", in the introductory chapters of his book "A Guide To Feynman Diagrams In The Many-Body Problem".

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One particle is a source of an external (retarded) field for another one, and vice versa. Retarded fields are described with differential equations.

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