# Behavior of particles at very small scale of spacetime

If I act $$\phi(x)$$ on $$|0\rangle$$ I get a particle created at $$x$$. Now if I let my system evolve for some time and now I do the measurement on this system at $$y$$ and my $$(x-y)^2$$ is small let say of the order of Planck length scale what kind of observations I'm bound to get? Few things that I want to point out:

1. I have done the very rudimentary calculation that if you try to probe lengths below Planck length it becomes hidden due to the formation of event horizon. This calculation is very perplexing for me because of this event horizon there is upper limit to which you can see the continuum structure (if it even exists at that length scale) then doesn't this problem reduces to study of interior of black hole?

2. In QM I have the evolution operator but there seems to be no such thing in QFT so I don't even know how to evolve this system and situation will get complicated because we can other particles popping out and producing some observable effects

3. Does the basic principles for this question, I have following ones in my mind Causality, energy conservation, Lorentz symmetry can be violated for this measurement and yes I have heard and seen the argument for energy conservation violation using Heisenberg uncertainty principle but the relation $$\Delta E \Delta t > \hbar$$ to begin with is not correct. The usual derivation is for $$\hat{x}$$, $$\hat{p}$$ and $$t$$ is not an operator in QM. I am assuming $$\phi$$ satisfy Klein Gordon equation but you are free to go for higher spin Fields as well if there is any illuminating effect due to it. I am also assuming my background field can be curved as well so my question is not just limited to Minkowski spacetime.

## 1 Answer

If I act ϕ(x) on |0⟩ I get a particle created at x. Now if I let my system evolve for some time and now I do the measurement on this system at y and my (x−y)2 is small let say of the order of Planck length scale what kind of observations

Quantum mechanical, i.e.the probability of finding the particle at (x+Δx,y+Δy,z+Δz) That you use Planck units makes no difference to a quantum mechanical wavefunction, it is a function of (x,y,z,t) which are continuous variables, much closer than Planck length,which is one in a set of units used i cosmological scales. The context here is not cosmological.

Your 1. is in classical general relativity, which is not definitively quantized yet. You cannot use general relativity concepts on quantum mechanical entities, until gravity is quantized.