localized state as given by the uncertainty principle. When viewed through this formalism, the measurement of the wave function will randomly lead to wave function collapse to a sharply peaked function at some location. For particles with mass, the likelihood of detecting the particle at any particular location is equal to the squared amplitude of the wave function there. The measurement will return a well-defined position, and is subject to Heisenberg's uncertainty principle. Following the development of quantum field theory the ambiguity disappeared. The field permits solutions that follow the wave equation, which are referred to as the wave functions. The term particle is used to label the irreducible representations of the Lorentz group that are permitted by the field. An interaction as in a Feynman diagram is accepted as a calculationally convenient approximation where the outgoing legs are known to be simplifications of the propagation and the internal lines are for some order in an expansion of the field interaction. Since the field is non-local and quantized, the phenomena that previously were thought of as paradoxes are explained. Within the limits of the wave–particle duality the quantum field theory gives the same results. Visualization There are two ways to visualize the wave-particle behaviour: by the standard model and by the de Broglie–Bohm theory. Below is an illustration of wave–particle duality as it relates to de Broglie's hypothesis and Heisenberg's Uncertainty principle, in terms of the position and momentum space wavefunctions for one spinless particle with mass in one dimension. These wavefunctions are Fourier transforms of each other. The more localized the position-space wavefunction, the more likely the particle is to be found with the position coordinates in that region, and correspondingly the momentum-space wavefunction is less localized so the possible momentum components the particle could have are more widespread. Conversely, the more localized the momentum-space wavefunction, the more likely the particle is to be found with those values of momentum components in that region, and correspondingly the less localized the position-space wavefunction, so the position coordinates the particle could occupy are more widespread. Alternative views Wave–particle duality is an ongoing conundrum in modern physics. Most physicists accept wave–particle duality as the best explanation for a broad range of observed phenomena; however, it is not without controversy. Alternative views are also presented here. These views are not generally accepted by mainstream physics, but serve as a basis for valuable discussion