Answer by A Quora admin:
Here's an equation that breaks down in a similar way:
Here's the analogy:
- The left hand side is like the exact answer (what the correct theory of everything would predict)
- The right hand side is the kind of answer we get from current theories – a series expansion (and we actually don't know all the coefficients)
- Small x is like low energies (most of observed physics)
- Large x is like high energies (relevant to black holes and the Big Bang)
Try plugging in a few small values, like 0.1, 0.0345, etc. You'll find that the formula works very well. Now try some bigger numbers like 1 or 1000. Not only does the equation not work, but the right hand side doesn't even make sense. This is what we mean by an equation "breaking down at high energies."
Edit: Yes I am aware that the radius of convergence of the above Taylor series is 1 and thus the equality doesn't hold for x>1. That is the whole point of my answer. The energy expansion in effective field theory also has a finite radius of convergence and thus can't be used at high energies.