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I was reading some QFT notes and there is one point that I don't understand, they are justifying why we need QFT saying that the number of particles is not preserved once we consider special relativity.

For example, if a particle is in a box of side $L$, then by Heisenberg's principle: $\Delta p \geq h/L$ and then $\Delta E \geq hc/L$. But when $\Delta E > 2mc^2$ creation of particle anti-particle out of vacuum may happen.

I don't understand why the uncertainty of the knowledge of energy of the particle inside the box implies the creation of particles by the vacuum. I mean why particle and vacuum are related somehow?

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A state with an energy $E$ is generally a multi-particle state with particles and anti-particles. Only if this energy is small enough, it can be reduced to one-particle or no-particle (vacuum) state. –  Vladimir Kalitvianski Oct 7 '12 at 14:09
    
The number of particles cannot be preserved because causal behavior requires microlocality, which cannot be implemented without antiparticles. Conserved is the lepton number (number of electrons minus number of positrons). But this has nothing to do with the vacuum. –  Arnold Neumaier Oct 7 '12 at 17:01

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Measurable particle anti-particle pairs are never created out of vacuum, but only out of other particles or [edit] out of externally applied fields. (The cross section for creating anything from the vacuum state but the vacuum itself is exacly zero.)

The reference to creation out of the vacuum is for unmeasurable virtual particles only, which are visual mnemonics for lines in a so-called Feynman diagram, encoding terms in a perturbation series for computing real scattering processes. See http://physics.stackexchange.com/a/22064/7924 and Chapter A8: Virtual particles and vacuum fluctuations of my theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html

An electron loop in a Feynman diagram may be interpreted as the creation of a virtual electron-antielectron pair from vacuum, and later decaying into the vacuum again. Those who believe that virtual particles have some sort of real existence then add virtual verbal imagery involving vacuum fluctuations and the uncertainty relation to ''explain'' such unphysical behavior.

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The creation of virtual particles is required when you have a box, since you are perturbing the vacuum with external source. This is not a scattering experiment where you can put everything on shell. This is not an opinion, and you should stop calling it an "interpretation" or "verbal imagery", it is the intermediate states in a quantum description of the field, it is Fock space. –  Ron Maimon Oct 7 '12 at 14:16
    
@RonMaimon: If the vacuum is perturbed by an external sourse, the particles are created from that source, not from the vacuum. - Intermediate states integrated over in a perturbative calculation are off-shell, hence not in Fock space. They don't have an associated wave function. You'd have to prove the opposite by giving a reference. –  Arnold Neumaier Oct 7 '12 at 15:15
    
you are confusing covariant and old perturbation theory. In the old perturbation theory, the virtual particles are on shell, but their energy is not conserved. This is the Hamiltonian picture, the picture of time-dependent perturbation theory. It is mathematically asymptotically equivalent to covariant perturbation theory, where the particles are off shell. I find the intuitive discussion of virtual stuff easier in old perturbation theory, although the calculations are easier in the modern covariant form. –  Ron Maimon Oct 7 '12 at 16:05
    
@RonMaimon: You forgot to say how you associate a Fock state with an off-shell particle, which you claimed is not an opinion, hence should be a mathematical statement. –  Arnold Neumaier Oct 7 '12 at 16:12
    
The Fock space description and field hamiltonians never have an off shell particle. They have on shell virtual particles. The intermediate states are on shell, but don't conserve energy. This is because the Hamiltonian is defined over all space, but between two nearby slices in time. The covariant formalism uses local particle paths, and has energy and momentum conserved, but introduces off-shell condition. –  Ron Maimon Oct 7 '12 at 17:44

This heuristic argument is just noting that in a small box, you can't say for sure that you have just one particle, because the walls of the box will allow the particle to fluctuate into different particle numbers, so the idea of "just one particle" is inconsistent with the idea "I have a box which confines the particles".

It is the box that is creating new particles, because the particle is bouncing off the walls. You can't have a particle bouncing off a wall sharper than the Compton wavelength (the inverse mass in natural units) and guarantee that it stays one particle. The interaction of the particle and the external potential always leads to pair creation.

The reason for this is that particle propagation is always faster than light, because energy is always positive. There are no functions which are zero outside the future lightcone whose Fourier transforms are only nonzero on positive energy. This mathematical fact is explained by Dyson, and popularized in Feynman's 1986 Dirac Lectures--- a particle always must be allowed to travel faster than light.

This means that if you have a wall that deflects the particle, and you put a particle far away, it will start deflecting the particle before the particle has time to get to the wall! But in order to not violate causality, this requires that the wall is already doing deflection of something else when there is no extra particle, and this something-else deflection is the virtual particle-antiparticle stuff that is present in the vacuum. If you measure what's going on at the wall for times less than the time to propagate to the wall, you won't be able to tell whether it is the particle that is reflecting, or the vacuum fluctuations, this is how causality is restored.

The mathematical statement that encodes this is microcausality, that field operators commute at spacelike separation.

The states of a quantum field theory are as follows: you have a collection of different positions or momenta. In each position, you can have 0,1,2,3,4... particles (imagine the positions/momenta are discrete and finite volume) in the case of Bosons, or 0,1 particle in the case of Fermions. When the quantum amplitudes for this state fluctuate over time (as they always do in a Hamiltonian formalism) you say that you have old-fasioned virtual particles. This can be rewritten in a covariant formalism, and then the virtual particles acquire a different relativistically invariant language, but the old-fasioned perturbation theory is the source of the virtual particle picture.

Then the statement that there are virtual particle is the statement that you can have mixtures of states of different particle numbers. This is unavoidable when you have a potential box localizing particles, because a potential that can deflect particles can also pair create.

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So the particles are created from the box, not from the vacuum! - Field operators only act on on-shell states, not on virtual particles. Microcausality guarantees that nothing propagates faster than light. Mixtures of physical states of different particles are still fully on-shell, and contain no virtual (off-shell) particles! –  Arnold Neumaier Oct 7 '12 at 15:21
    
@ArnoldNeumaier: The question is about a box--- of course particles are not created out of a vacuum, but that's only because a vacuum is a nonlocally defined object with no particles (by definition). When people say "particles pop out of a vacuum" they mean that a local measurement by external field capable of detecting a particle also necessarily is capable of misdetecting a particle when it is in an otherwise-vacuum. –  Ron Maimon Oct 7 '12 at 16:03

A paper by Colosi and Rovelli that I read may have another interesting answer to the question.

Consider space partitioned off (arbitrarily) into two regions $R_1$ and $R_2$. The global Fock vacuum is found by solving for the global Hamiltonian which includes the possible correlations between the regions. A local detector in a region $R_1$ is governed by a local Hamiltonian $H_1$ and thus has a local vacuum $|0\rangle_1$ which need not be the same as the global Fock vacuum.

What Colosi and Rovelli show is that the two vacuua are distinct and that you can actually detect a local particle when you're actually in a (global) Fock vacuum. So to piggyback off of the last comment by Ron Maimon (10/7/12), a local measuring apparatus governed by a local Hamiltonian can measure particles when you would've otherwise thought that you were in a vacuum

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