In the four dimensional spacetime of field theory, which is where virtual particles live, Energy is one of the four momentum components: p_x, p_y,p_z,E . In this fourvector space , actually a pseudovector space because a minus sign enters between the fourth and the rest of the coordinates:

the dot product of the vector with itself is its scalar measure, analogous to the length of the three dimensional vector in the normal three dimensional space. This scalar number in "real" space is fixed and unchangeable. In "virtual" space it can change according to the mathematical formulations of the Feynman diagrams describing the interaction.

Virtual space is a mathematical tool for complicated calculations and is constrained only by mathematics, not physics laws like energy conservation. This is because one can never do an experiment in virtual space, by construction.

As @DavidH comments to you the HUP is another story, it is a general uncertainty in the possibility of measuring some pairs of variables, well defined, as for example energy and time. If you measure the energies before an interaction well the energy should be conserved after the interaction. The HUP would just constrain the knowledge of how long the interaction took within the uncertainty limits. It is not the HUP that is generating virtual particles, but the specific calculations of interactions.

You may be thinking of vacuum fluctuations, but these are virtual, unless there is an interaction, i.e. energy is supplied by an interacting particle. Otherwise we would be able to get energy out of the vacuum :).

The Heisenberg uncertainty principle is a basic foundation stone of quantum mechanics, and is derivable from the commutator relations of the quantum mechanical operators describing the pair of variables participating in the HUP.

You are discussing the energy time uncertainty, .

For an individual particle, it describes a locus in the time versus energy space, within which the quantum mechanical solution for the existence of the particle is undefined by these variables, it is only bounded.

Now let us attack virtual particles:

In physics, a virtual particle is an explanatory conceptual entity found in mathematical calculations in quantum field theory. It visualizes, usually in perturbation theory, mathematical terms that have some appearance of representing particles inside a subatomic process such as a collision. Virtual particles, however, do not appear directly amongst the observable and detectable input and output quantities of these calculations, which refer only to actual, as distinct from virtual, particles. Virtual particle terms correspond to notional "particles" that are said to be "off mass shell".

These mathematical representations are called "particles" because they carry the quantum numbers of the named particle, except their mass is a variable in the total integration for the process under calculation. Here is an example:

The neutron and the proton are of order of GeV, they are real particles with their on shell masses in the calculations. The same is true for the outgoing antineutrino and electron. The W- is virtual, very much off shell, that's why the neutron does not decay immediately, as the on shell mass of the W is in the denominator of the propagator in the integral and together with the weak coupling constant the decay of a free neutron takes minutes.

Where does the HUP enter in this diagram?

If we take delta(t) the 16 minutes of the lifetime it tells us that multiplied with delta(e) the energy of the interaction, order of 2GeV, multiplied should be larger than h_bar/2 . This is of course fulfilled as h_bar is such a small number.

Now by themselves, virtual particle loops have no meaning , because there are no input and output legs entering the diagram. Loops exist in higher order perturbative expansion calculations of real diagrams, the red circle:

As far as Hawking radiation goes, the logic is that : for a real particle to come out an interaction line of a virtual particle with the fields of the horizon has to have taken place, the energy necessary for reality picked up from the field of the black hole.

Here is a diagram ( not a feynman one) of the picture of hawking radiation from vacuum fluctuations next to the horizon.

It is not a feynman diagram, because it does not have the interaction vertex with a field at the horizon which will provide the energy for the particle to become real and the absorption of the second one by the black hole. But it is on the lines that due to the HUP a locus in energy and time exists which is not measurable but can be described by virtual loops of the type existing in higher order diagrams.

How does the uncertainty principle relate to quantum fluctuations?

The uncertainty principle defines a locus in the relevant phase space, energy-time, or momentum-space where virtual particles, i.e. mathematical constructs with off shell mass, can exist. In general, vacuum fluctuations can be imagined, but their expectation value has to be zero if there is no energy input. It is an imaginative stretching of the mathematics of perturbation theory and the HUP , imo.

## Best Answer

You asked for a qualitative picture, so here goes.

Consider a simplified example: the quantum harmonic oscillator.

Its ground state is given by

$$ \Psi(x) = \text{const} \cdot \exp \left( - m \omega_0 x^2 / 2 \hbar \right). $$

Now suppose that we are measuring the position of this oscillator in the ground state. We could get any real value, with probability density $|\Psi|^2$. In reality, because of the exponential decay, most of the values are distributed within the window of width

$$ \Delta x \sim \sqrt{\frac{2 \hbar}{m \omega_0}}, $$

with the mean concentrated at $x = 0$.

Because measuring an individual oscillator is a complicated process which results in it getting entangled with the measurement device, let's simplify the problem – say we have an ensemble of non-interacting oscillators all in ground states, and we measure them all independently. The distribution of values $\{x_i\}$ is expected to mostly lie within the mentioned above window, but the actual values are unknown. We usually say that those are due to

quantum fluctuationsof the position operator.The same thing happens with the quantum field, which upon inspection is nothing more than a collection of weakly interacting harmonic oscillators. If we take an ensemble of vacuum quantum field configurations (say, independent experiments at a particle accelerator), and we measure a value of the field at a point, we will see that it is not equal to zero (as it would be in the classical theory), but instead the values are distributed within an error window and are otherwise random. This are quantum fluctuations of the QFT vacuum.

These fluctuations are sometimes attributed to "virtual particles", or "virtual pairs", which are said to be "born from the vacuum". Sometimes it is also said that they can "borrow energy from vacuum for a short period of time". AFAIK these are just analogies, appealing to the consequence of Erenfest's theorem (the so-called time-energy uncertainty relation).

But the fluctuations undisputably have very real, measurable effects. Qualitatively, those effects come from a difference between the physical picture of the same thing painted by classical fields and quantum fields. You can say that quantum fields reproduce classical fields on certain scales (measured in the field value), which are much greater than the size of the error window. But once the precision with which you measure field values becomes comparable to the size of the error window, quantum effects kick in. Those who like painting intuitive pictures in their heads say that this is caused by quantum fluctuations, or virtual particles.

UPDATEBelief that observed Casimir effect has something to do with vacuum fluctuations of the

fundamentalQFT is misguided. In fact, in the calculation of the Casimir force we use an effective field theory – free electromagnetism in the 1D box, bounded by the two plates. Then we look at theeffective vacuumstate of this effective QFT, and we interpret the Casimir force as a consequence of the dependence of its properties on the displacement between the plates, $d$.From the point of view of the fundamental QFT however (Standard Model, etc.)

there is no external conducting plates in the first place. If there were, it would violate Lorentz invariance. Real plates used in real experiments are made of the same matter described by the fundamental QFT, thus the state of interest is extremely complicated. What we observe as Casimir force is really just a complicated interaction of the fundamental QFT, which describes the time evolution of the complicated initial state (which describes the plates + electromagnetic field in between).It is hopeless to try to calculate this in the fundamental QFT, just like it is hopeless to calculate the properties of the tennis ball by studying directly electromagnetic interactions holding its atoms together. Instead, we turn to the effective description, which captures all the interesting properties of our setup. In this case it is free electromagnetic effective QFT in the 1D box.

So to summarize: we are looking at the vacuum state of the effective QFT and the dependence of its properties on $d$. Alternatively, we are observing an extremely complicated fundamental system in a state which we can't hope to describe.