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I have read these

and found out that quantum fluctuations (or virtual particles) are just tools of QFT theories in the perturbative context. Nothing more than mathematical tools. However, I have also read that these quantum fluctuations are the cause of some observable phenomena such as

  • The Casimir effect
  • Lamb Shift
  • And others

This part is very confusing to me. How can something that does not exist in reality (just math), but still influence the real world?

Edit

As discussed in the comments, I will rephrase the overused term "quantum fluctuation" as the ground state energy of a quantum field that has some energy.

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    $\begingroup$ @Tachyon you may want to check also this: "Is the term "quantum fluctuation" an aide to understanding? " physics.stackexchange.com/q/441144/226902 and links therein. $\endgroup$
    – Quillo
    Apr 27, 2023 at 0:22
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    $\begingroup$ What's wrong with math describing real phenomena? The book of nature is written in math, as Galileo observed... $\endgroup$ Apr 27, 2023 at 0:23
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    $\begingroup$ Would it amuse you if I chime in to object that "a small, positive energy that every quantum field has" is (1) going to be causing problems with GR and (2) is false for fermions, which has a negative version of the half $\hslash\omega$? $\endgroup$ Apr 27, 2023 at 2:12
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    $\begingroup$ @Tachyon I would not even want to claim that the QHO ground state is or is not a good picture to have in mind, because I just do not want to bias either way. The reason why is because the first step in QFT is to renormalise the Hamiltonian by normal ordering, so that the infinite $\frac12\hslash\omega$ goes away, and that kinda also means we are not taking the Gaußian literally (for the fundamental fields). It is not like we have a composite system, e.g. a diatomic molecule, where the vibrational ground state energy actually exists and does not cause trouble. $\endgroup$ Apr 28, 2023 at 2:38
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    $\begingroup$ As for writing answer, I am also incredibly uncomfortable. I am currently learning causal perturbation theory, which should totally change my understanding. But my current understanding of QFT by the standard stuff, it really looks like the whole thing is about taking the interacting field's full path integral Lagrangian and unnaturally breaking it into two parts, the interaction and renormalisation part, and the free particle part. Then we break the exponential of the interaction part into a Dyson series, and this is the source of all the virtual particles nonsense. Absent in original scheme $\endgroup$ Apr 28, 2023 at 2:44

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Explaining what virtual particles and quantum fluctuations are in intuitive terms is difficult, because they are inherently complicated mathematically. But I can maybe address your question "How can something that does not exist in reality (just math), but still influence the real world?" without getting into the physics, because this is common to much simpler mathematical problems.

Say we want to solve the quadratic equation $x^2-6x+5=0$. The numbers in the equation represent real physical quantities in our problem, it doesn't matter what. We initially only know how to solve equations of the form $x^2-2a+a^2=(x-a)^2=b$, with solution $x-a=\pm \sqrt{b}$. Our given equation doesn't fit this pattern, but if we insert a $+9-9$ in the middle, it does. $x^2-6x+9-9+5=0$ goes to $(x-3)^2-9+5=0$ goes to $(x-3)^2=4$ so $x-3=\pm \sqrt{4}$.

What are these $+9-9$ terms we inserted in the middle? They add up to zero, so they seemingly "popped out of nothing". The $5$ in the original equation represents something physically real, but these $9$'s are entirely fictional (and may even be physically impossible, e.g. if the constant terms represent some quantity required to be positive). It's "just math". But this fiction allowed us to turn a situation we don't know how to solve into one we already understand.

We have a similar problem in quantum field theory. The theory of a charged particle is a non-linear differential equation - the motion of the charged particle's field affects the electromagnetic field, which affects the charged field, which affects the electromagnetic field, and so on. We need to know how the charge moves to figure out how the electromagnetic field will behave, and we need to know the electromagnetic field to figure out how the charge moves. We can't generally solve that sort of problem. (In linear problems we can find a convenient basis of solutions, and then add combinations of them together to solve the equation. But in nonlinear problems superposition of basis solutions doesn't work.)

So we do it a step at a time. We start with a free electron moving without any electromagnetic field, then we add in the effect of the particle on the electromagnetic field (inserting virtual photons representing disturbances of the electromagnetic field), then we add the effect of the modified electromagnetic field on the particle field (by adding new virtual particles), and so on. Each stage of the calculation is formally incorrect - we are always neglecting the highest-order effects of one field on the other. We use Feynman diagrams to keep track of them. Each step involves a separate multidimensional integral - but each one is wrong (and may even diverge) because it represents a physically impossible, inconsistent situation. However, when we add them all up, the wrongnesses and divergences cancel out (hopefully!) and we get the right answer.

So what we are doing is taking a simplified equation we know how to solve, and adding an infinite series of unreal, "just math" terms to turn it into the real problem we have been given. Like our $+9-9$ terms above, these represent modelled physical entities that "pop out of nowhere". But if we think of the problem perturbatively, as an infinite series of simple linear interactions bouncing back and forth rather than two fields that simply interact nonlinearly together, then it is as if these intermediate states and particles really existed, interacting with the real particles. The $+5$ term is physically real, the $+9-9$ terms are not, and taken together cancel out, but they all appear together in the equation on the same footing, and the virtual bits "influence the real world" in the sense that the calculation process works by treating them as if they do.

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  • $\begingroup$ Plasma equilibrum is a nonlinear equation, for the reason you state. But a charged particle doesn't feel its own electric field, which is why the schrodinger equation for the hydrogen atom is not a nonlinear equation $\endgroup$
    – AXensen
    Apr 27, 2023 at 8:43
  • $\begingroup$ If I am not mistaken, I heard that the vector potential $B = \nabla \times A$ in classical EM theory was just a mathematical convenience, but was later shown to play a direct role in the Aharonov-Bohm Effect. But I could be wrong. $\endgroup$
    – Tachyon
    Apr 27, 2023 at 21:33
  • $\begingroup$ So the Heisenberg uncertainty principle (energy and time) states that there are particles coming in and out of the vacuum all the time during interactions. So in reality, there are no such particles, but rather, something much more difficult to visualize. So it is not particles popping into and out of nothing, but some interaction that we cannot visualize that follows the uncertainty principle? Is that right? $\endgroup$
    – Tachyon
    Apr 27, 2023 at 21:38
  • $\begingroup$ However, could you at least describe the vacuum energy as a sum of $\hbar\omega$? I have seen it here: physics.stackexchange.com/a/22663/231892. So then, since $\hbar\omega$ describes the energy of a wave, then you could think of the vacuum as containing waves of energy filling all of spacetime, no? $\endgroup$
    – Tachyon
    Apr 27, 2023 at 21:53

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