Can the conservation of energy law be violated? This is a very interesting question and to be truthful, widely debated.
Let us first write out the law:
In a closed system, the amount of energy is fixed. You can't create any more energy inside the system or destroy any of the energy that's already in there. But you can convert the energy you have from one form to another (and sometimes back again).
So it says all energy is converted into other forms of energy it is never lost and never destroyed. In fact if you can see this in one of my past posts, where you stretch an elastic band and the potential energy in the band is stored as mass. The overall energy of the system stays the same.
The short answer is that no, there is no PROVEN theory that allows the violation of the conservation of energy over a LONG time.
I will break this down through the post.
We have a lot of things that APPEAR to violate the conservation of energy. To use an example we are going to look at Dark energy. Now this rather miss understood so we will go down a little road of knowledge.
So theory of dark energy arose from the observations of the expanding universe. Our universe is expanding, by that i mean the physical space is getting larger, Its is not the adding of extra space into gaps. You can picture this by drawing 2 spots on a balloon, as you blow up the balloon the space between the spots expands.
Now this is not in any way a violation on the conservation of energy, however not only is the universe expanding but it is expanding at an increasing rate. Now we can prove this by looking at the cosmological scale factor and supernovae explosions. So how is the universe expanding at this increasing rate? The answer is dark energy and by dark energy we mean..
We have absolutely no idea, so we create value and make it sound mysterious.
Literally the only thing we know for sure is that energy must fuel this expansion.
So does dark energy violate the law? Well its hard to say, lets create an example. Lets say i have a little experimental universe of size 1 meter. This is increasing in size by using some energy to expand it. So in this system we have an ever increasing space and as the space increases we have more energy working on that space. This would imply that we are getting more and more energy to expand the expanding space. There are hundred of solutions to this such as negative work being done to expand the universe and energy used to maintain the same energy density in the expanded space. There are also hundreds of ways to look at this problem.
Its important to make the point that its not really believed that dark energy is a in violation of the conservation of energy, its just not proven that it is in accordance with conservation of energy. I'm just trying to get across the idea that even though conservation of energy cant be violated in our current theories, it doesn't mean that something might come across that can change that.
Moving onto another point, which is the uncertainty principle. This principle states that you cannot measure the position and velocity of a particle exactly. If you measure the exact position of a particle then you cannot know its velocity and vise versa.
Now it is important to stress that this is not due to measuring errors and that in a few years we will have equipment that removes this. It is in fact a law of nature, it actually has physical effects in our world. For example an electron in orbit of a nucleus. Now you would expect a positive nucleus and a negative electron to pull each other together. However if the electron gets too close to the nucleus we would know its position too accurately, which in turn means that its velocity would be too uncertain and could have enough velocity to escape the atom. There are other things such as quantum tunnelling which is explained with the uncertainty principle.
Anyway lets try get back on track, so we know what the principle means, now this seems strange but there is another equation the "time energy uncertainty relation" (TEUR) which is very similar to the uncertainty principle and gives us an uncertainty in measurements of energy with time (i had to explain the uncertainty principle so you can understand what i mean by this). So on a very short time scales the uncertainty means that energy can be "borrowed" to create things such as virtual particles.
This source describes how virtual particles are made and what they are very well (much better than i can).
https://profmattstrassler.com/articles-and-posts/particle-physics-basics/virtual-particles-what-are-they/
Basically virtual particles are excitation in a fields that exist for extremely short times. The energy for their creation has to come from somewhere, the answer its "borrowed".
Now as a disclaimer. Quantum field theory uses virtual particles as an analogy which resembles underlying phenomena. It is not an actual particle, but it is a real thing. So virtual particles are allowed to be created on time scales that are so small that it is uncertain that they were ever there. But we can still measure there interactions.
So there you have it.
Your head hurting? If its not then have another read, mine is and i had 4 years studying this shit.
Any questions let me know
The Surf Graduate