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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 ...
According to the correspondence principle, in certain limits the quantum equations of states must approach Hamilton's equations of motion.The latter state the following relation between the generalized coordinate q (e.g. position) and the generalized momentum p: {˙ = = {,}; ˙ = = {,}.
After integrating the resulting delta functions, what will remain of the LSZ reduction formula is merely a Fourier transformation operation where the integration is over the internal point positions that the external leg propagators were attached to. In this form the reduction formula shows that the S-matrix is the Fourier transform of the ...
The equation itself usually refers to the position space form, where it can be written in terms of separated space and time components ( , ) or by combining them into a four-vector = ( , ) . By Fourier transforming the field into momentum space, the solution is usually written in terms of a superposition of plane waves whose energy and momentum ...
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).
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.
Integrating over all different values of φ(x) is equivalent to integrating over all Fourier modes, because taking a Fourier transform is a unitary linear transformation of field coordinates. When you change coordinates in a multidimensional integral by a linear transformation, the value of the new integral is given by the determinant of the ...
Realize that for caustics the description diverges and use the insight by Maslov (approximately Fourier transforming to momentum space (stationary phase approximation with h a small parameter) to avoid such points and afterwards transforming back to position space can cure such a divergence, however gives a phase factor).