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I found a webpage that just kind of mentions the uncertainty principle lightly but doesn't really go into detail as to why we need it in the first place when considering quantum fluctuations and particles/anti-particles.

I want to understand why we care about this equation as it is related to the creation and annihilation of virtual particles.

I would guess that it helps us answer the question: "Well, if these particles are being created and destroyed in a really small time interval, then we can estimate that the energy they create must be relatively large." But then this just gives me another question: in the full time interval (from $t_i$ to $t_f$), wouldn't $\Delta E=0$?

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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.

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:

feynmdiag

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:

virtual loops

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.

vacuum fluctuations

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.

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  • $\begingroup$ I'm confused by your answer. You say that virtual particles are a mathematical concept, yet still say that they are ''allowed'' to exist via the HUP in higher order diagrams. If they're not physically real, then what causes Hawking radiation? $\endgroup$ – whatwhatwhat Apr 22 '16 at 2:37
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    $\begingroup$ @whatwhatwhat interactions. The horizon is full with real particles falling in. These cause interactions between particles and with the fields in the horizon. From these, at the horizon, some loops may get enough energy for one of the components to become real and escape and the other to disappear into the horizon. $\endgroup$ – anna v Apr 22 '16 at 3:27
  • $\begingroup$ @whatwhatwhat One can write down integrals for a loop diagram with only virtual particles, but it is just mathematics because the particles are off mass shell, and the integral is over the available phase space and even this off mass shell changes under the integral. Only when some real outgoing or incoming line exists will there be a physically observable effect. An off shell line from a field at the horizon to the loop can provide enough energy for one of the particles to escape and the other to fall in, depleting the energy of the black hole. $\endgroup$ – anna v Apr 22 '16 at 3:51

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