In what ways can energy transform into matter and vice versa? Annihilation is one way to tranform matter to energy. Fission is another (when splitting and atom, what happens to its two parts?)
Are quantum fluctuations one way to transform energy to matter?
In what ways can energy transform into matter and vice versa?
Energy and matter are connected according to special relativity and this has been experimentally demonstrated . It is the famous formula:
$E=mc^2$ , where $m$ is the relativistic mass and $c$ the velocity of light. or
$E^2=m_0^2c^4 +p^2c^2$ , for a particle with rest mass $m_0$ moving with momentum $p$.
The rules of transformation follow Quantum Mechanical solutions of kinematic and potential problem equations .
Annihilation is one way to transform matter to energy.
Yes
Fission is another (when splitting and atom, what happens to its two parts?)
In the quantum mechanical description of nuclei they are represented by potential wells with energy levels, some filled. The number of baryons ( protons and neutrons) bound in this potential well characterize the nucleus. Nucleus A that is struck by a neutron ( for example) becomes a nucleus B higher up in baryons by absorbing it into an energy level of this potential well. In fission this higher up nucleus is unstable and falls into a lower energy state, giving up part of its mass in energy according to the relativistic formulae, and breaking into smaller nuclei and free neutrons which go on to sustain the fission on another original nucleus. Generally a form of fission happens if a nucleus is unstable.
There is also fusion, two deuterium nuclei adhering at a lower energy level and giving up energy. The binding energy curve shows whether nucleons can fuse or fission and give up as energy a part of their mass.
Are quantum fluctuations one way to transform energy to matter?
No, quantum fluctuations are virtual . If you mean tunneling, yes.
Related note:
Fission isn't exactly turning matter into energy. It just releases the binding energy of the nucleus. This binding energy is part of the measured mass pf the nucleus, but if you want to separate "matter" and "energy" (not really possible), then it counts as energy. $\newcommand{\a}[3]{\mathrm{^{#1}_{#2}#3}}$ $$\a{235}{92}{U}+\a10n\to\a{236}{92}{U}^*\to\a{144}{56}{Ba}+\a{89}{36}{Kr}+3\a10n+|\Delta H|\approx177\:\rm{MeV}$$
Note that initially, we have 93 protons and 142 neutrons; and in the end this number does not change. From this POV, where particles count as "mass", we can say that no mass was created or destroyed, and the nuclear binding energy was released.
Why do we call this a conversion from mass to energy if its just a converseion of types of energy? Well, that's because mass is energy.
The fact is, if you "weighed" $\a{235}{92}{U}+\a10n$, it would weigh more than $\a{144}{56}{Ba}+\a{89}{36}{Kr}+3\a10n$. Actually, $\a{235}{92}{U}$ weighs less than $92\a11p+141\a10n$. That's because the binding energy of the nucleus is "negative" energy, and thus "annihilates" some mass (since mass is energy). It turns out that due to this, the fission products are lighter than the reactants, even if the number of nucleons is the same. And this loss of "mass" is converted into energy.
So really, there's a bit of fuzziness on the border of "energy" and "mass". Anything with an energy density will have extra mass, and you won't be able to tell the difference between a body with mass $m$ and a body with mass $m-\frac{U}{c^2}$ and internal energy $U$.
Wood burning is an example of converting mass into energy. Another is a seed which takes energy from the sun (and water, air etc.) and converts it into matter.
As well as the other answers, in particular, Anna V's comprehensive Answer, I would like to capture Ben Crowell's comment for permanence in this dicusssion:
There's nothing special about nuclear reactions. Chemical reactions also result in a change in mass due to the energy released. It's just that the energy scale for chemical reactions is about $10^6$ times smaller. [My italics]
and urge you to think of matter and energy to be different states of the same essential thing. People still get overwrought by the conversion of one into the other, but now physics and the physics culture has moved on to such a degree that the word "matter" has well and truly passed its use-by date. We (physicists) have kept the word "energy" for meaning the quantity that is conserved by dint of Noether's theorem applied to time-shift invariance of physical laws - and this word comprises everything that might be considered "stuff", i.e. all matter and energy in the old usage - more precisely: it comprises anything that constributes to the $T_{0\,0}$ term of the relativistic stress energy tensor. You might, for example, want to use the word "matter" for anything that has nonzero rest mass, but even this doesn't work properly as my writeup of the light-in-a-box thought experiment here shows that confined light has a rest mass.
So, at the risk of sounding too colloquial for scientific discussion, I think simply of the word "stuff" for meaning anything that contributes to the stress energy tensor in the way described above, the word "energy" for quantifying the amount of "stuff" and if you need more precision than this, then you must specify the exact class of "stuff" in terms of the precise particle / quantum field names from the standard model and chemical reactants / products. To try to otherwise partition "stuff" into matter and energy is to grope for what is now a thoroughly artificial, imprecise and outdated dichotomy.
Are quantum fluctuations one way to transform energy to matter?
Yes, at least in theory.
This is how Hawking Radiation is predicted to work. In this case the gravitational energy of a black hole 'boosts' the energy of a quantum fluctuation to create an actual particle/antiparticle pair, one of which gets sucked into the black hole and one of which escapes.
Have a look at this excerpt from the Wikipedia page on Gamma rays talking about a gamma ray turning into an electron-position pair.
By interaction with the electric field of a nucleus, the energy of the incident photon is converted into the mass of an electron-positron pair. Any gamma energy in excess of the equivalent rest mass of the two particles (totaling at least 1.02 MeV) appears as the kinetic energy of the pair and in the recoil of the emitting nucleus.
Essentially creation of matter from energy (and vice versa) needs to follow Einstein's famous equation...
E=mc2
In what ways can energy transform into matter and vice versa?
I am sure in special relativistic theory there is no such transformation. Why?
Energy is an abstract mathematical quantity obeying local conservation law.
Matter is a basic thing the world is made of. It is not a mathematical concept. One can quantify one aspect of it, say introduce inertial mass $m$. That ignores all the other things about matter we know - much of chemistry and physics.
Obviously, the basic thing the world is made of does not change into abstract mathematical quantities.
There is a real idea behind that quoted statement, but it needs to be stated differently. The idea is Einstein's conclusion that loss of energy $L$ from a body is accompanied by decrease of body's mass by $L/c^2$ (and vice versa).
Based on this conclusion, he introduced the formula
$$ E = mc^2 $$
as definition of total energy of body at rest. When someone says "mass can change into radiation energy " he really means "part of energy of massive body associated with mass $m$ has changed form and location from energy in the body to energy in the EM field".
One cannot obtain "clean" energy which is completely free off momentum, and cannot obtain "clean" matter which is free off momentum and potential energy.
So question is ill-posed, there is no "clean" states which can be described as "energy into matter". That just cannot happen.
When we consider reactions of elementary particles, the most common scenario is fission of one big particle, which is unstable by interactions which govern its stability. In this "one into many" scenario, you have energy released, because your momentum could be easily preserved.
But mostly discussed electron-positron annihilation is very unprobable in "common random occurence". Because momentums of motion should satisfy $p_1+p_2<\delta$. In common scenario these two particles will just scatter, without any annihilation!
