Quantum Fluctuations of Fields: Vacuum Explained

• Lapidus
In summary, Fields do fluctuate in the vacuum, however they are considered as 'mathematical fiction' and not really significant.
Lapidus
Do fields have quantum fluctuations and are non-zero in the vacuum?

Non-zero in the same way that for a harmonic oscillator the wave function is non-zero for values of the position of the harmonic oscillator?

Or are they considered as 'mathematical fiction' just as 'virtual' particles?

thanks

As far as I know the term "vacuum fluctuation" is assigned just to virtual particles spontaneously coming to "existence".

Let's take a case from standard non-relativistic QM:

Consider two noble gas atoms in vacuum, separated by a significant distance. They interact via the Coulomb potential (so it's instantaneous, no fields or mediation going on here). So the total wave function of the system can't be the sum of the wave functions of the two atoms, since there's an interaction going on. It's basically an entangled state, the state of atom 1 is not independent of the state of atom 2.

This gives rise to the London dispersion force - the total energy of the true system is lower than the system of two non-mutually-interacting atoms. You can describe this as being due to the quantum uncertainty or 'fluctuations' in electron density since the contribution is by definition the contribution from the instantaneous, position-dependent interaction between the electrons. But it's the ground state and so it's time-independent, so there are no actual measurable temporal 'fluctuations' going on.

So how do you calculate that interaction? Again, assume you know the wave functions of the isolated individual atoms. So you can use that as a basis and use the Ritz variational method or perturbation theory. As I already said, the ground states of the two isolated atoms is not the true ground state of the interacting pair of atoms. So if I describe my interacting wave function in that basis, I'll have contributions from the "virtual" excited states of those non-interacting atomic wave functions. That does not mean that those atoms are actually in an excited state, because those states are not eigenstates of the true Hamiltonian. These "excitations" only exist because of the choice of basis. The electrons aren't in excited states - which should be obvious: If you have two atoms in their electronic ground states and bring them into proximity, lowering their mutual ground state energy, how could that lead to a real excitation?

Now realize that the electromagnetic field is itself quantized and subject to the same kind of quantum behavior. If you take that into account in the above situation, you have an additional contribution to the dispersion force called the Casimir effect. But if you like, you can also simply consider a single atom and its interaction with the field, rather than another atom. Again you have contributions to its behavior because of that interaction. And again, you can describe those contributions in terms of 'virtual' excited states, which in the case of the field is what you call 'virtual particles'. And the reason is the same, you're starting from a non-interacting description.

Thanks for the anwers, especially alxm, nice one!

1. What are quantum fluctuations of fields?

Quantum fluctuations of fields refer to the random and unpredictable variations in the strength and direction of a field, which is a region of space that has a specific physical property, such as electromagnetic field or gravitational field. These fluctuations occur at the smallest scale of subatomic particles and are a fundamental aspect of quantum mechanics.

2. How are quantum fluctuations of fields related to the vacuum state?

According to quantum field theory, the vacuum state is not truly empty, but is instead filled with a sea of virtual particles and antiparticles that constantly pop in and out of existence. These virtual particles contribute to the quantum fluctuations of fields, causing them to fluctuate in strength and direction.

3. What is the significance of quantum fluctuations of fields?

Quantum fluctuations of fields are essential for our understanding of the behavior of particles at the quantum level. They play a crucial role in phenomena such as the Casimir effect and the Lamb shift, and are also important in cosmology, as they are thought to have played a role in the evolution of the early universe.

4. Can quantum fluctuations of fields be observed?

Since quantum fluctuations occur at such a small scale, they cannot be directly observed. However, their effects can be observed through experiments and calculations, such as in the measurement of the Casimir effect. Additionally, the existence of virtual particles and their contribution to quantum fluctuations has been confirmed through the effects of Hawking radiation.

5. How do quantum fluctuations of fields relate to the uncertainty principle?

The uncertainty principle states that it is impossible to simultaneously know the exact position and momentum of a particle. Quantum fluctuations contribute to this uncertainty, as the particles that pop in and out of existence in the vacuum state can have unpredictable effects on the position and momentum of particles. This is a fundamental aspect of quantum mechanics and is related to the Heisenberg uncertainty principle.

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