Apologies if this has been asked before (I did check and I believe it wasn't). I have a question about the particle/wave duality of photons (or other particles). Depending on what and how we measure the photon turns out to be either a wave or a particle. Recently I saw some web page (and I can't remember where) that maybe neither is true. Both the wave perception and the particle perception are just that: perceptions, strengthened by our perhaps limited ability to observe. What if the reality of the photon is something else, something "on top" of our two perceptions?? Could someone direct me to a site that could tell me more about it (or maybe debunk the whole idea?)?
This is the idea: when you see the high school equations that describe a parabolic motion, you consequently visualize a flying stone in your mind. Physics does not care about what you imagine, but rather it deals with the equations and their ability to tell you the distance the stone will reach. Because the equations were invented and tuned for that purpose, the numeric results of that mathematical tricks are equal to what happens in nature.
Around the 1930s, some men eventually managed to build a consistent sets of mathematical tricks that correctly give the same results as nature, when dealing with a wide range of problems for which the known theories had failed. It is known as Quantum Mechanics. But, unlike the high school parabolic shot, the equations of QM are there but nobody can imagine a graphical representation.
Whatever it may be happening down there, there is no possible way of depicting it in your human brain. It is just so. You may spend your life thinking about a wave as a transparent difusse entity that suddently feels observed and transforms into a tiny hard ball... But the result will be nothing but a headache. With Quantum Mechanics you see the equations and check the result, but there is never a parallel mental image of a flying thing or whatever.
There is however good news, because the most basic of the theory is not difficult to grasp. But bear in mind that it consists on maths (by the way, Physics may be well defined as the science that invents mathematical models that resemble the behavior of nature). Google for a small collections of video lectures called "Quantum Entanglements 1" by Susskind. After that, the third volume of the Feynman Lectures is a good choice (but that requires more work).
Of course, there is room for much more than pure maths alone, but that room is only accessible in a mentally healthy way after you know the maths. With no knowledge of the maths, words about Quantum Mechanics are meaningless, and intuitive ideas are unavoidable wrong. Whatever meaning you give to the words "wave" and "particle", there is no way to combine them in a satisfactory explanation without at least some elementary knowledge of Quantum Mechanics.
Assuming you believe quantum field theory, specifically quantum electrodynamics, a photon is neither a particle nor a wave, so your web page is correct in this respect though I hesitate to give it too much credit without seeing its reasoning.
A photon is an excitation in the photon quantum field. It can interact in ways that resemble a particle and it can interact in ways that resemble a wave, but is incorrect to say it is a particle or a wave or some mixture of the two. It is an excitation in a quantum field.
The statement "Depending on what and how we measure the photon turns out to be either a wave or a particle." is quite old-fashioned, and unhelpful, in my opinion.
Let's consider an electromagnetic wave of frequence $\nu$. Classically, this wave can carry an arbitrarily small amount of energy, i.e., the electromagnetic field can have arbitrarily small wiggles in it, with frequency $\nu$. But this classical picture is wrong: quantum mechanics tells us that a wave of frequency $\nu$ can only have energies $n h \nu$, where $n$ is a non-negative integer and $h$ is Planck's constant. We say that "$n$ photons are present".
Suppose we pass the electromagnetic wave through a narrow slit. Classically, we expect diffraction, and quantum mechanically... exactly the same! All that changed with quantum mechanics is the possible values of the energy contained in the wave. So our $n$ photons 'behave like a wave'.
Now imagine we have an apparatus (perhaps a photographic film) set up to detect the electromagnetic wave. Classically, any amount of energy could be deposited anywhere on the film, but quantum mechanically, we know that absorption occurs via interactions with single atoms/molecules/whatever. Since the electromagnetic field can only carry discrete amounts of energy, and energy is conserved, it must be absorbed in 'units' of $h\nu$. In this sense, it looks like individual particles impinging on the apparatus, each of energy $h\nu$.
Photons, electrons, quarks,... are always particles [*] and never are waves. As the CERN site correctly notes everything in the universe is made of particles. To be concrete the photon is a boson particle.
The source of the wave-particle duality misconception is pointed out in Ballentine recent textbook on quantum mechanics:
A more detailed discussion is given in the Klein site.
[*] Particle here means quantum particle not Newtonian particle. Quantum particles do not behave as a tiny ping-pong balls, but follow the laws of quantum mechanics.