What is the shape and volume of a photon?

Photons are point particles with no substructure as far as we have been able to determine. Photons have momentum and spin, no mass or charge.  The spin is manifested in its polarization.  You can think of this as two different photons that can be interchanged by rotating them 90 degrees around the axis that they are going.

I think you're imagining the sine curve a bit too literally. A photon has no mass. A photon has a wave function (quantum) - which is the probabilistic distribution of the presence of the photon at a given position (probability amplitude).  Something like the electron has mass and wave like properties. Each  electron "vibrates" (good analogy) with its wave function. Quantum  Mechanics seems to suggest almost everything in the universe has such a wave  function. For all we know, the property of matter called mass could just be a certain frequency of oscillation.

A photon is an elementary particle, the quantum of light and all other forms of electromagnetic radiation, and the force carrier for the electromagnetic force, even when static via virtual photons. A photon, one of the smallest particles, is yet to be seen by someone. Our eyes, just like any other optical instrument, can detect photons..may not be able to see them. They are tooo small and tooo fast to be seen and spotted (less than 0.1 fm in size and as fast as the fastest). Just like every other sub-atomic or elementary particles, they are studied by their effect and not by how they appear. Another trouble in seeing them is that we know their velocity precisely (299,792,458 metres/sec). This would bring infinite uncertainty in their position..thanks to Heisenberg and his uncertainty principle..and not to mention Born's probability interpretation. From the applications of quantum mechanics, we could say it wont be of any of the shapes described. It has a spin of 1, which means it would look similar as before only after a 360-degree spin (because most of the shapes mentioned above have spin 2, as they look exactly same after a 180-degree turn). We even don't know if all the photons look alike, or are their shape influenced by their frequencies. We prove their existence and detect them whenever we experiment at the microscopic level where light interacts with particles..as in Photoelectric effect (and excitation of electrons yet another example). Though there are claims that photons are visualized by some, those are yet to be confirmed and accepted by the scientific community. But things are not always to be seen. We can feel them. Isn't that wonderful enough..? The brilliant spread of beauty around us is conveyed to our brains by them.. Sometimes things are to enjoyed and not worried about understanding.. that's the plain truth of life.

A photon in QFT is like money. If you're in the US you probably have some dollars in the bank. You know how many you have. But where are they? They don't really have a physical location. You can talk about the location of data on hard drives or in RAM on some servers somewhere, but that's not really where the money is. When you draw money out of an ATM does money travel down the wires? And if you deposit a dollar in one city, how do you make sure you get the same one back when you make a withdrawal in another city? Money doesn't have the full spectrum of properties you associate with ordinary objects. When you say you have $5 you aren't saying there are 5 objects sitting somewhere physically in space. You might be saying that there is some information stored on servers somewhere and that if you go to a bank they'll be prepared to give you five dollar bills if you ask nicely and so on. And you can talk loosely about the location of money in the sense that you could say it's somewhere on Earth, and not on Mars. Similarly, talk about photons should't be taken literally as talk about shiny little spheres. When someone says a system has 5 photons in it, they're saying something about how energy is distributed through modes of oscillation in the system. (Think of a mode as a way something can oscillate. Pluck the E string on a guitar and there will be waves of a given frequency corresponding to the note E. But there will also be higher frequency oscillations present, the overtones. These are all modes.) It turns out that quantum mechanics only allows discrete amounts of energy in each individual mode. So it's convenient to think about energy coming in lumps. And it's convenient to give these lumps names. Like photon. But if you wanted to you could completely ignore photons and talk about states with this or that amount of energy in each mode.

Photons have no shape - they are massless particles. They are just discrete packets of electromagnetic energy. Anything that travels at the speed of light has zero length. This makes it even harder to predict the photon's shape (even if it had one). Cheers :)

A photon is a corpuscle of light or similar radiation. It is the most basic 3D matter-particle. It has a disc-shaped (segmented spheroid) 3D matter-core, surrounded by EM wave-like distortions in surrounding universal medium. 3D matter-core of a photon spins about one of its diameter. Spin speed of a photon is proportional to its 3D matter-content and they are moved at the highest possible linear speed by universal medium. Radial sizes of 3D matter-cores of all photons are identical. Peripheral thickness of segments (in equatorial region) are proportional to their 3D matter-content. Shape and volume of distorted regionof universal medium, surrounding photon's 3D matter-core,  differ to suit parameters of 3D matter-core.  see: Chapter 4 of 'MATTER) (Re-exmined)' and  http://vixra.org/abs/1312.0130