Forget particles, and even forget quantum mechanics for a moment, and think in terms of a classical field. A classical field has a configuration—there are “field values” and the way they vary from point to point in space defines the configuration—and an equation of motion. If the equation of motion is nice, then the field can be understood as a sum of modes that are only weakly coupled to each other, e.g. Fourier modes.
Now introduce quantum mechanics. The states of a quantum field will be superpositions of these classical field configurations. A generic quantum field state will associate a probability amplitude with every possible classical field configuration, no matter how crazy.
The simplest way for a quantum field to become analytically tractable is to analyze the classical configurations via Fourier modes. Each Fourier mode is a harmonic oscillator, and the quantum mechanics of a harmonic oscillator is fairly simple. You have discrete energy levels. Each energy level is described by a wavefunction for all the possible displacements of the oscillator from what would be its classical resting place. The absolute values of the amplitudes drop away to zero as the possible displacements go to infinity, expressing the fact that the further away from the classical resting place you look, the less likely it is that the oscillator is displaced that far from home. The energy levels are distinguished by the number of peaks and troughs in the wavefunction.
Notably, even at the lowest energy level, the oscillator has a finite probability of being found away from the classical resting point. This is the zero-point energy and it’s a consequence of the uncertainty principle.
We can think of a classical field as an infinite number of harmonic oscillators (the Fourier modes), and a quantum field as an infinite number of entangled quantum harmonic oscillators. So each field mode is oscillating, it’s in a superposition of energy levels, and then all the field modes are entangled with each other.
“Energy level” isn’t just a buzzword; the higher the energy level of a field mode, the more energy there is in it. In fact we say that a field mode in the nth energy level is the same thing as the existence of n particles with a momentum equal to hbar times k, the “wave vector” of the field mode. The ground state, the zeroth energy level, corresponds to the presence of no particles with that momentum.
So we can define a quantum field state by stipulating that all the field modes are in the lowest energy level. Remember that a quantum field state is a superposition of classical field configurations; to find the amplitude for a particular configuration, given this quantum state, you would list the coefficients of all the Fourier modes, i.e. how big the displacement is in each mode, and then you multiply the amplitudes for those displacements in the zero-level harmonic oscillator wavefunction.
By definition, this is a zero-particle state, but it’s not a zero-energy state, because each field mode has a zero-point energy (since it is a quantum oscillator and always has some amplitude to be away from rest). And it’s definitely not a state in which all the quantum amplitude is concentrated on the classical configuration where the field is zero everywhere.
So the vacuum state of a quantum field is indeed one in which the field value has an amplitude to be nonzero, and this shows up e.g. in the Casimir effect. In fact, because the quantum field state is supposed to be a superposition over all classical field configurations, even those which aren’t differentiable, there is always supposed to be an amplitude for an arbitrarily violent vacuum fluctuation, because the classical field configuration space includes arbitrarily spiky variations in field value. This is one of the major causes of divergent integrals in quantum field theory, and what you do is impose an upper bound on the Fourier-mode frequencies that you allow yourself to care about, i.e. we won’t allow contributions to the quantum state from classical field configurations that vary on space-time scales smaller than the cutoff. The physical rationale for this is that in the true theory of everything, we expect there to be a cutoff scale, such as the length of a superstring.
Anyway, bottom line, the vacuum state of a quantum field certainly contains a finite probability amplitude for the field to be deviating from zero, and this is responsible for various measurable effects.
Forget particles, and even forget quantum mechanics for a moment, and think in terms of a classical field. A classical field has a configuration—there are “field values” and the way they vary from point to point in space defines the configuration—and an equation of motion. If the equation of motion is nice, then the field can be understood as a sum of modes that are only weakly coupled to each other, e.g. Fourier modes.
Now introduce quantum mechanics. The states of a quantum field will be superpositions of these classical field configurations. A generic quantum field state will associate a probability amplitude with every possible classical field configuration, no matter how crazy.
The simplest way for a quantum field to become analytically tractable is to analyze the classical configurations via Fourier modes. Each Fourier mode is a harmonic oscillator, and the quantum mechanics of a harmonic oscillator is fairly simple. You have discrete energy levels. Each energy level is described by a wavefunction for all the possible displacements of the oscillator from what would be its classical resting place. The absolute values of the amplitudes drop away to zero as the possible displacements go to infinity, expressing the fact that the further away from the classical resting place you look, the less likely it is that the oscillator is displaced that far from home. The energy levels are distinguished by the number of peaks and troughs in the wavefunction.
Notably, even at the lowest energy level, the oscillator has a finite probability of being found away from the classical resting point. This is the zero-point energy and it’s a consequence of the uncertainty principle.
We can think of a classical field as an infinite number of harmonic oscillators (the Fourier modes), and a quantum field as an infinite number of entangled quantum harmonic oscillators. So each field mode is oscillating, it’s in a superposition of energy levels, and then all the field modes are entangled with each other.
“Energy level” isn’t just a buzzword; the higher the energy level of a field mode, the more energy there is in it. In fact we say that a field mode in the nth energy level is the same thing as the existence of n particles with a momentum equal to hbar times k, the “wave vector” of the field mode. The ground state, the zeroth energy level, corresponds to the presence of no particles with that momentum.
So we can define a quantum field state by stipulating that all the field modes are in the lowest energy level. Remember that a quantum field state is a superposition of classical field configurations; to find the amplitude for a particular configuration, given this quantum state, you would list the coefficients of all the Fourier modes, i.e. how big the displacement is in each mode, and then you multiply the amplitudes for those displacements in the zero-level harmonic oscillator wavefunction.
By definition, this is a zero-particle state, but it’s not a zero-energy state, because each field mode has a zero-point energy (since it is a quantum oscillator and always has some amplitude to be away from rest). And it’s definitely not a state in which all the quantum amplitude is concentrated on the classical configuration where the field is zero everywhere.
So the vacuum state of a quantum field is indeed one in which the field value has an amplitude to be nonzero, and this shows up e.g. in the Casimir effect. In fact, because the quantum field state is supposed to be a superposition over all classical field configurations, even those which aren’t differentiable, there is always supposed to be an amplitude for an arbitrarily violent vacuum fluctuation, because the classical field configuration space includes arbitrarily spiky variations in field value. This is one of the major causes of divergent integrals in quantum field theory, and what you do is impose an upper bound on the Fourier-mode frequencies that you allow yourself to care about, i.e. we won’t allow contributions to the quantum state from classical field configurations that vary on space-time scales smaller than the cutoff. The physical rationale for this is that in the true theory of everything, we expect there to be a cutoff scale, such as the length of a superstring.
Anyway, bottom line, the vacuum state of a quantum field certainly contains a finite probability amplitude for the field to be deviating from zero, and this is responsible for various measurable effects.