A photon is the quantum of the EM force, so it cannot be divided, but a photon can give rise to two photons carrying less energy.
Be careful here, let me elaborate.
A photon like all elementary particles was thought not to divid at all because of feature of renomalization of a quantum filed called Ward–Takahashi identity (
Ward-Takahashi identity ), In laymans terms it's completely balanced. So for a long time it was thought it could not and would not split ... ever.
In 1970 an observation of splitting a photon was first observed by passing a laser thru a crystal lattice. The process is called Spontaneous parametric down-conversion (
SPDC description).
That work lead to the realization that while light is passing thru a medium you can play with QM statistics in the waveform and once it exits the medium when it reforms it can have new characteristics.
You may remember the photonic molecules of 2013 which in the science magazines seemed to cause excitement because someone thought they resemble the starwars lightsabre. (
http://www.gizmag.com/photonic-molecules-pave-the-way-for-quantum-computers-and-lightsabers/29924/)
The reason for the caution is you need a medium in which the photons interact there is no situation they are ever known to do it in a vacuum (Ward–Takahashi holds).
A graviton would be the quantum of gravity. Could a graviton (hypothetically) divide to become two gravitons, each carrying less energy? The quote “Gravitons participate in the gravitational field, which means that a graviton can emit gravitons”, suggests that this might be the case.
So now when we look at this suggestion, if we are making a graviton a spin 2 quantum field particle we have to assume the same rules apply. So gravitons as per photons are never going to split in the vacuum of space but possibly could in the presence of matter or other suitable mediation layer.
If gravity creates gravity, and gravitons participate in this, does one graviton create another? If so, the process of creating more gravity must mean that there is more gravitational energy in the system after graviton A has created graviton B than there was before. Where would the additional energy come from?
You are looking at that, wrong the Quantum Statistics of the wave are being effected by the mediating layer. As per SPDC of a photon splitting the energy and indeed all QM statistics are preserved and that is why the two photons are actually known to be entangled.
SPDC is used often as a source of entangled photons for exactly that reason that the entanglement is guaranteed. So if a QM graviton existed you would expect the same behaviour and the split graviton pair should be entangled.
