Story
I can't talk about an anime or movie or other content without talking about the plot. :P Toaru Kagaku no Railgun plays in a scientifically-driven city called 'Academy City' in which pupils or people, in general, are taught to be espers. The main plot focuses on a student called Misaka Mikoto who is considered a level 5 esper and she can control electricity.
That should give you an idea on the scope of the anime without revealing anything that's happening.
Personal Reality
In some of the teachings, the anime keeps repeating the idea of finding your personal reality or your own reality. This goes well with a principle in magick that I and a few people in my contact circle often like to call bubble worlds. The basic idea is everyone lives in their own world or in their own bubble. Everything they believe in is real in this bubble. If two people get near each other those bubbles merge to a certain degree. If both bubbles are the same (same beliefs) nothing changes. However, if those bubbles differ the intersected area can be considered to be in a superstate as first. Which means it is at first not clear what it's gonna be like. What it will actually be like is later determined by the probability of which of both bubbles matches better.
Magick User vs Magick User
Even if both magick users or espers believe in magick to work differently it will still exist in both worlds depending on how open-minded they are both systems may work or both systems will work to a lesser effect.
Magick User vs Skeptiker
Assuming the magick user knows about the real world and sees it as one of many realities there's a higher chance for the normal world or real world as we know it to be the one the superstate will result in.
Skeptiker vs Skeptiker
The probability of magick to exist is nearly zero. The result of the superstate will be the normal world as we - and probably them too - know it.
In Context with the Anime
So, if we take this back to the context of the anime the idea is for your esper ability to improve you need to find your own reality, the reality in which your ability exists to a high degree. Unfortunately, this is not so simple as it sounds and it may be even more difficult in real life. The issue is that there are more than just millions of people on the world and each of their bubbles almost completely intersects with another. There's hardly any space that does not intersect. Hence why magick users create their own room or area to perform magick. At the same time, this is also the reason why magick users form groups to perform rituals as multiple bubbles with the same beliefs overlap resulting in a higher success on magick. Additionally with an increase of people believing in magick or simply being open-minded or not denying it would increase the probability of successful rituals as well. Hence why academy city houses mostly espers and almost only people who believe in it.
Outro
Lastly, it's really interesting to see another form of artwork to base itself on a concept that is used in modern magick. It makes one think if the author was inspired by those beliefs had the same thought process of maybe it's something we all unconsciously know and believe in? Whatever it is it's our best bet to explain magick scientifically. Whether or not this explanation is correct is something we will figure out as we continue to understand quantum physics and as we dive deeper into the lowest levels of physics. Funnily enough, people already believe in the placebo and have proven which amazing feats it can do and they keep on studying it. "Glaube versetzt Berge" (Faith can move mountains) is a German saying that matches this too well.
