In quantum field theory (QFT), the inability to define a position operator for relativistic particles is related to the nature of the theory and the principles of relativity.
In non-relativistic quantum mechanics, position operators are well-defined and play a fundamental role in describing the position of particles. However, when extending quantum mechanics to incorporate special relativity, new challenges arise.
One of the key principles of special relativity is that information cannot be transmitted faster than the speed of light. This implies that measurements of position and momentum in relativistic systems cannot be performed with arbitrary precision simultaneously. The uncertainty principle, which relates the uncertainties in position and momentum, becomes more stringent in the relativistic context.
In QFT, particles are described as excitations of quantum fields that permeate spacetime. These fields are inherently relativistic, and particles are interpreted as localized disturbances or excitations of these fields. Unlike in non-relativistic quantum mechanics, where particles are treated as separate entities with definite positions, QFT treats particles as field excitations that can exist over a range of spacetime.
Due to the relativistic nature of QFT, the concept of a position operator for individual particles loses its meaning. Instead, QFT focuses on operators associated with the field itself, such as creation and annihilation operators, which describe the creation and annihilation of particles at different spacetime points.
In QFT, observables related to position are typically described in terms of scattering processes and correlation functions involving fields at different spacetime points. These observables provide information about the probabilistic distribution of particle positions and their correlations in the context of the quantum field.
Therefore, while QFT does not have a well-defined position operator for individual particles, it still provides a comprehensive framework for describing relativistic quantum phenomena by focusing on the behavior of quantum fields and their interactions.