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Watching Discovery's first episode of the first season of Curiosity (entitled "Did God Create the Universe?" by Stephen Hawking), I heard this information:

[...] you enter a world where conjuring something out of nothing is possible (at least, for a short while). That's because at this scale particles, such as protons, behave according to the laws of nature we call "quantum mechanics", and they really can appear at random, stick around for a while, and then vanish again to reappear somewhere else.

... and this isn't the only time I've heard this. I imagine countless billions (trillions!) of particles popping into existence all the time in the smallest of spaces for the shortest periods of time.

If subatomic particles pop into existence all the time and in all locations, why doesn't the weight of my body (or anything) change?

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It does. Very slightly. Negligibly small. But the effect is negligible even compared to many other processes, such as water molecules condensing and evaporating on the object. – centralcharge Jun 10 '13 at 7:50
Very similar – Dilaton Jun 10 '13 at 11:44
do even protons appear and disappear (as the above question states) or is this phenomena observed only for virtual particles? – good_ole_ray Jun 10 '13 at 17:40
Don't they also disappear at similar rates, canceling out much of the gains that might otherwise accumulate? I'd certainly prefer to think of this as a way to lose weight than another reason to worry about weight gain ;) – Nick Stauner Jul 21 '14 at 21:59
up vote 3 down vote accepted

There is Einstein's famous formula $$ E = m c^2$$ that governs the relationship between mass and energy. The subatomic particles (actually particle-antiparticle pairs) are created using energy that has already been there. The Energy from which they were created also has its weight (in fact, most of the proton's and neutron's mass is kinetic energy, not mass of their constituents) and this is exactly the same weight as the particle-antiparticle pair has.

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but this phenomena appears even in empty space. so how can you say the energy is already there? – good_ole_ray Jun 10 '13 at 9:24
If the energy is not sufficient to create the rest mass of the particle-antiparticle pair, the particles are "virtual" and can only exist as quantum fluctuations. On such scales, their mass can not contribute to weight. Furthermore even empty space contains a certain ammount of energy, which is why the universe is expanding. – Neuneck Jun 10 '13 at 10:37

This$^1$ has to do with the uncertainty relation for energy and time:

$$\Delta E \Delta t \approx \hbar$$

where $\hbar\sim10^{-15}\,\mathrm{eV\cdot s}$ is Dirac's constant (or Planck's constant divided by $2\pi$). This means conservation of energy can apparently be broken for very short time scales. The emphasis is on the very. A particle can therefore be created from the vacuum, as long as it disappears again within some time scale governed by the above approximate equality. This phenomenon is called a quantum fluctuation.

For example, if you wanted to create a proton (mass of approximately $10^9\,\mathrm{eV}$) in a vacuum, it would only be around for about

$$\frac{10^{-15}\,\mathrm{eV\cdot s}}{10^9\,\mathrm{eV}} = 10^{-24}\,\mathrm{s}$$

which is obviously not very long. And a proton has a negligible mass compared to your entire body.

Of course, if your body is around, it's likely not happening in a vacuum. So then Neuneck's answer is probably more appropriate. The particles are created using energy available in the environment. In this case however, there's no reason for them to necessarily disappear since there was no breaking of energy conservation. So although this is probably the situation more in tune with your question, it is most likely not the situation that is meant in the quote. (quantum fluctuations are more likely)

$^1$ In the vacuum of space at least, and I think that's what they're talking about in this particular quote. In presence of other matter, it is (or should be) more conceivable to people that it is possible to create particles using some energy from the matter in the environment. Though upon re-reading the quote, perhaps they're talking about quantumtunneling as well.

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there is a listing of conservation laws on wikipedia

one of them is conservation of mass-energy which states that both the total mass and the total energy inside an isolated system remain constant over time, as seen by any single observer in a given inertial frame. the above link says this is an "exact law, or more precisely has never been shown to be violated"

hence, according to this the reason you dont gain weight is because the total mass-energy stays constant (except for these transient "noise" fluctuations which add up to zero),

and it seems Hawking is misleading the public, by extrapolating conclusions from phenomena which has never been observed to add up to anything

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To be clear: there is ample experimental evidence for the "existence" of virtual particles in the sense that they must be included in the calculations of obervable phenomena to get the right answer. There is no doubt or confusion about this. The problem in science communication is trying to attach a common sense physical meaning to the terms appearing in a particular calculational technique (perturbative QFT) that isn't really there. When we say virtual particles pop into existence, what we mean is that we need to calculate higher order Feynman diagrams. But that doesn't sell well. – Michael Brown Jun 10 '13 at 12:57

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