# Violation of the weak equivalence principle via the Born-Jordan quantized TOA operator

### Abstract

In classical physics, the weak equivalence principle (WEP) states that bodies fall with the same acceleration regardless of their composition and mass. On the contrary, studies on the quantum image of a particle undergoing free-fall has shown a violation of the WEP. Here, we show that these violation is manifested through quantum corrections to the classical time of arrival (TOA). We start by constructing a TOA operator by quantizing the classical TOA function* t _{class}* using Born-Jordan ordering rule. The expectation value of the TOA operator

*τ*is then calculated and expressed as an asymptotic expansion of

_{quant}*ℏ*. The leading term of the expansion is shown to be equal to

*t*while the rest are quantum correction terms. However, these quantum corrections vanish in the large momentum limit thereby recovering the WEP. We trace this quantum violation of the WEP to fact that there is no velocity operator in quantum mechanics.

_{class}*Proceedings of the Samahang Pisika ng Pilipinas*

**37**, SPP-2019-2G-05 (2019). URL: https://paperview.spp-online.org/proceedings/article/view/SPP-2019-2G-05.