## Virtual Particles and Conservation of Energy

Virtual particles are never observed directly, so (subject to the limits of experimental error) we don’t actually ever see any violation of conservation of energy.

What virtual particles are is just a part of one particular method for calculating probabilities of events that we do see; but this is not even the same as saying we observe them indirectly, as the various possibilities with different numbers of such particles all contribute to the overall calculation – with no specific numbers ever being required to actually exist.

The use of virtual particles is analogous to Feynman’s path integral approach to quantum mechanics where, as an alternative to solving the Schrodinger equation by traditional methods, Feynman noted that the probability amplitudes predicted from it for going from one event to another could also be calculated by adding up contributions from all conceivable paths between the two events (including unphysical ones). But neither the unphysical paths nor the unphysical particle number histories need to be considered as anything that actually happens.

Another point that is often made in answers to this question is that the contributions from paths or particle histories that violate conservation of energy are inversely proportional to the time durations of those violations in a way that is consistent with Heisenberg’s uncertainty principle \Delta E \Delta t < \frac{h}{4\pi}. But I am not sure how much this helps – other than to explain how (as pointed out in yet a third set of answers) “laws” of physics are not absolute but just expressions of the limits of what, according to current theories, we expect to see – and indeed conservation of energy can appear to be violated if we try to measure things too quickly (though the “violation” can be interpreted as just due to our inability to measure both energy and time with sufficient simultaneous accuracy).

## Waves vs Particles

#### A Quora question asks: Can we measure wave properties of particles or is my contention that ‘waves travel but particles are detected’ correct?

You are right that the actual values of the quantum “wave function” are not generally observable and that the things we can actually measure are usually more like properties of particles.

But there are some ways of getting partial information about the wave function itself. For example chemistry and molecular structure gives us a way of learning about the squared magnitude of the wave function when it is a “standing wave” and a scanning tunneling microscope even gives us a more direct picture of that. And the relationships between phases at different points sometimes lead to observable effects in solid state theory and in the Aharonov-Bohm effect.

## Geometric vs Wave Optics

### A Quora question asks: How can we say that light tends to travel in straight lines, but when we squeeze it to travel through a very narrow slit, it spreads out? Doesn’t this mean the light is not traveling in a straight line anymore?

The key word is “tends”. For the most part that tendency is what dominates our experience – as can be seen for example by constructing a pin-hole camera.

But there are various situations where it does not apply. Most familiar is the change of direction when light encounters a change of refractive index, but there is also a slight tendency to bend around any obstacle. The wave theory that predicts this was demonstrated by Thomas Young’s two slit experiment but an even more impressive demonstration was perhaps the spot of light directly behind a circular barrier that was predicted by Poisson (as a supposedly ridiculous consequence of the wave theory) and then actually observed in a public demonstration by Arago (having actually perhaps been noted much earlier by Maraldi).

The reason we see don’t see these effects more often is because the wavelength of light is very small and only the part of the beam within a wavelength or so of the barrier experiences any noticeable bending, so it requires a very bright source and a special setup to actually see it. In the case of the circular barrier, it is the rotational symmetry that gives constructive interference between the bent beams from different locations on the boundary. But you can actually see a similar effect from periodic symmetry by looking at the sun through a black woven umbrella (if your twirl the umbrella the bright spots just rotate around around the line to the sun rather than the axis of the umbrella so you can see that they aren’t keeping exactly in line with the actual gaps in the weave).

## YM Existence and Mass Gap (in layman’s terms)

“Yang-Mills” is just the name for a class of theories which have a certain kind of symmetry and which include as a special case parts of the “standard model” which physicists use to predict the behaviour and interactions of elementary particles .

The “existence” problem here is that the various procedures used by physicists to make calculations in these theories have never been proved to actually always work. They involve making sequences of successive modifications from some initial guess according to patterns that are known (ie proven mathematically) to work in simpler theories for producing a sequence of numbers that actually converges to a well-defined result (that is independent of the starting point). But proofs of effectiveness have never been found for the theories that are actually used to describe elementary particles. What is therefore not yet known to exist is a set of well-defined final predictions (ie an actual theory defined by the proposed procedures).

The calculations can be done in various ways, and do seem to produce useful approximations to what we actually see in experiments, but we don’t know that the results will actually converge if we keep on going. So we don’t know for sure whether or not we have a well-defined theory. (This applies even to the case of Quantum Electrodynamics, but there is some hope that the more complicated symmetries of a Yang-Mills theory may help to guarantee convergence.)

As an analogy (not to the physics but to the state of our knowledge) imagine coming across a ladder standing up in the middle of a field. It reaches up so far that you cannot see if it is stabilized in any way at the top; but you want to get a better view of what is around you, so you climb up the first few rungs and can see over the nearest hedge (and what you see from the ladder does match what you can see by walking across the ground). But now you want to look over the nearby hill. Perhaps you could climb higher, but what if the ladder is only precariously balanced? If it is infinitely long then it may have enough inertia not to be disturbed by your climbing, but on the other hand it may have enough stretch and flexibility that if you get high enough the part you are on will fall down anyhow. And even if the ladder is infinitely long and stable, on a spherical Earth there is a limit to how far you will actually be able to see (and perhaps there is important stuff happening on the far side that will eventually affect you). So the ladder may never tell you everything you need to know, and if it swings about you may never be sure that your view is ever the “correct” one, so there is no actual final prediction that it tells you.

The “mass gap” issue has to do with whether or not, if we leave out ElectroMagnetism, it is possible to clearly distinguish the vacuum as having strictly less energy than other states, and is also related to having more rapid falloff of non-EM forces such as those between nucleons. (This is actually a much weaker condition than the strict “confinement” that we actually expect for the forces between quarks within nucleons and pions, but proving it might be a first step towards that.)

One reason for combining this more specific “mass gap” issue with the more general and abstract question of “existence” is because, in some simpler cases (of just one or two space dimensions) the techniques used to prove “existence” of a well-defined quantum field theory also prove (and to some extent make use of) the existence of a mass gap.