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I'm building a world where my protagonist invents a theory of quantum gravity. Since I don't want the world to be much changed than it is now.
Is it possible that the theory wouldn't have any practical purpose?
I mean something that would allow scientist to understand how the nature works, but it wouldn't be able to use that knowledge to create new products & services.
It is not only possible but likely that practical applications of a full theory of quantum gravity will take 50 or 100 years, or even longer to reduce to a novel application. If you look over the history of physics, the practical application of a theory -- where knowledge of the theory leads to some new application -- is usually long delayed. Maxwell developed a complete theory of electromagnetism around 1850, but it was the early 1900s before his theory was used in any significant practical application. General Relativity was discovered in 1916, but its first practical application was probably the GPS satellites 75 years later. (The development of the atomic bomb, while explained by Relativity and e=mc2, was developed based on experiment and classical physics.) Quantum Mechanics was developed in the 20s and 30s, and was first applied in the 40s and 50s, which is probably the fastest theory has moved to practice.
A second reason an full theory of quantum gravity (QG) may not have any immediate practical effect is that we already have an excellent theory of gravity -- General Relativity (GR) -- and it is very likely that the deviations from the predictions of GR that QG successfully explains will be at very high energies and very small distances which we many not have the technology to exploit for a long time.
Of course it's possible. Most theories don't have any practical application.
Right now we're using just a few % of major discoveries since WWII, and less than 1% of new scientific knowledge discovered since the 1970s. That number goes further down if you start counting all the "discoveries" rather than just bona fide ones.
It's hard to think of any discovery since the Cold War that is now in use, and it's been almost 3 decades. It takes a separate layer of highly talented people - not simple engineers, but scientists-inventors (and they have to be both at once!) - to turn advanced theory into something with practical applications.
Without an inventor that understands the theory (important), has the resources and dedication, and (most important) a lot of luck, the discovery will go unused.
Just a practical example illustrating the the excellent arguments above:
His theory explains gravity, and tells us how we can build an anti-gravity device, but at current levels of technology, it would take a CERN-sized accelerator and enough electicity to power a small country, just to levitate a few atoms of hydrogen.
The simple answer is to make the minimum requirements of use very very high, if not unattainable. From what I understand, quantum computing for example requires near absolute temperatures so far and wont exactly be everyday use. Having even higher requirements in terms of machinery, precision and energy to make use of quantum gravity solves it.
Is it possible that the theory wouldn't have any practical purpose?
This is not only possible, but extremely likely. And, indeed, discovery of a perfectly correct theory of quantum gravity is an area of active research that could easily happen in our lifetimes.
Einstein's theory of general relativity is extremely accurate over a wide domain of applicability. A theory of quantum gravity would reproduce and explain the successes of general relativity in this domain of applicability, but would not materially change it in circumstances where it works and certainly in all circumstances where it has engineering relevance (e.g. GPS satellites).
There are basically only two areas where a quantum gravity theory would differ in its phenomenology (i.e. predictions for the real world) from general relativity:
The details of what is going on in extremely strong gravitational fields near black holes and the Big Bang.
An explanation for phenomena describes as "dark matter" and "dark energy" in extremely weak gravitational fields at distances on the order of the size of galaxies or larger.
A correct theory of quantum gravity might allow for the engineering of better gravitational telescopes (like LIGO today), and might allow us to better understanding of what our telescopes see in the sky, but that is about it.
A quantum gravity theory might also help guide us towards a "theory of everything" revealing the deeper structure of the laws of the universe and perhaps calculating some physical constants exactly from first principles rather than simply measuring them, but this would be an additional step beyond quantum gravity and the engineering benefit there would be mostly in the high energy physics involved in short lived particles made up of quarks (called QCD) where physical constants are currently known least accurately.
Pretty much all of the fundamental laws of physics with engineering relevance have already been discovered to the full precision at which they have significant engineering implications, outside some very narrow areas like practical nuclear fusion reactors where more precision in existing physical laws would have engineering benefits.
