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Measure and probability theory



Where a probability density function (on a measure space) is a real-valued function ρ\rho (satisfying certain conditions), a probability amplitude is a complex-valued function ψ\psi such that its pointwise absolute value squared

ρ(x)ψ *(x)ψ(x) \rho(x) \coloneqq \psi^\ast(x) \psi(x)

is a probability density.

In quantum physics


Probability amplitudes appear as pure states of quantum mechanical systems in the form of wave functions ψ\psi on the phase space of a corresponding classical system. The Born rule of quantum physics says that the probability density ρ=ψ *ψ\rho = \psi^\ast \psi describes the probability to find the physical system in a given classical state (in a given region of its phase space).

The fact that probability amplitudes are complex-valued means that under addition (“superposition”) they exhibit quantum interference (the addition of their complex phases) in stark contrast to the addition of probability densities, for which this cannot happen.

For instance on some probability space (X,μ)(X,\mu) there are the probability amplitudes exp(iπ/2)μ \exp(i \pi/2) \mu and exp(iπ/2)μ\exp(-i \pi/2)\mu whose associated probability densities are both just μ\mu itself again. But the sum of these two probability amplitudes vanishes, in contrast to the sum of their associated probability densities. This is known as “destructive” quantum interference.

Scattering amplitudes

In perturbative quantum field theory the key probability amplitudes considered are scattering amplitudes, which encode the probability for a given configuration of field quanta to come in from the far past, interact with each other and hence “scatter off” each other, and then re-emerge as some other set of field quanta in the far future.

These amplitudes are collected in the scattering matrix of the perturbative quantum field theory.

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Last revised on July 13, 2022 at 16:44:24. See the history of this page for a list of all contributions to it.