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Conversely, the inverse Fourier transform of a momentum space function is a position space function. These quantities and ideas transcend all of classical and quantum physics, and a physical system can be described using either the positions of the constituent particles, or their momenta, both formulations equivalently provide the same ...
This operator occurs in relativistic quantum field theory, such as the Dirac equation and other relativistic wave equations, since energy and momentum combine into the 4-momentum vector above, momentum and energy operators correspond to space and time derivatives, and they need to be first order partial derivatives for Lorentz covariance.
Note that because the initial Fourier transformation contained Lorentz invariant quantities like = only, the last expression is also a Lorentz invariant solution to the Klein–Gordon equation. If one does not require Lorentz invariance, one can absorb the 1 / 2 E ( p ) {\displaystyle 1/2E(\mathbf {p} )} -factor into the coefficients A ( p ...
Now, position and momentum are Fourier transforms of each other, and the Fourier transform of a square-integrable nonzero function with a compact support is entire and hence does not have non-isolated zeroes. Therefore, there is no wave function that is both normalizable in momentum space and vanishes on precisely x ≥ 0.
In the generalized version of the Bloch theorem, the Fourier transform, i.e. the wave function expansion, gets generalized from a discrete Fourier transform which is applicable only for cyclic groups, and therefore translations, into a character expansion of the wave function where the characters are given from the specific finite point group.
By Fourier transforming the propagator, the Feynman rules can be reformulated from position space into momentum space. [ 1 ] : 91–94 In order to compute the n -point correlation function to the k -th order, list all valid Feynman diagrams with n external points and k or fewer vertices, and then use Feynman rules to obtain the expression for ...
The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 3-dimensional 'position space' to a function of 3-dimensional momentum (or a function of space and time to a function of 4-momentum).
The Fourier transform of the position space propagators can be thought of as propagators in momentum space. These take a much simpler form than the position space propagators. They are often written with an explicit ε term although this is understood to be a reminder about which integration contour is appropriate (see above).