In quantum field theory, a sum rule is a relation between a static quantity and an integral over a dynamical quantity. Therefore, they have a form such as:
where is the dynamical quantity, for example a structure function characterizing a particle, and is the static quantity, for example the mass or the charge of that particle.
Quantum field theory sum rules should not be confused with sum rules in quantum chromodynamics or quantum mechanics.
Many sum rules exist. The validity of a particular sum rule can be sound if its derivation is based on solid assumptions, or on the contrary, some sum rules have been shown experimentally to be incorrect, due to unwarranted assumptions made in their derivation. The list of sum rules below illustrate this.
Sum rules are usually obtained by combining a dispersion relation with the optical theorem, [1] using the operator product expansion or current algebra. [2]
Quantum field theory sum rules are useful in a variety of ways. They permit to test the theory used to derive them, e.g. quantum chromodynamics, or an assumption made for the derivation, e.g. Lorentz invariance. They can be used to study a particle, e.g. how does the spins of partons make up the spin of the proton. They can also be used as a measurement method. If the static quantity is difficult to measure directly, measuring and integrating it offers a practical way to obtain (providing that the particular sum rule linking to is reliable).
Although in principle, is a static quantity, the denomination of sum rule has been extended to the case where is a probability amplitude, e.g. the probability amplitude of Compton scattering, [1] see the list of sum rules below.
(The list is not exhaustive)
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The light-front quantization of quantum field theories provides a useful alternative to ordinary equal-time quantization. In particular, it can lead to a relativistic description of bound systems in terms of quantum-mechanical wave functions. The quantization is based on the choice of light-front coordinates, where plays the role of time and the corresponding spatial coordinate is . Here, is the ordinary time, is a Cartesian coordinate, and is the speed of light. The other two Cartesian coordinates, and , are untouched and often called transverse or perpendicular, denoted by symbols of the type . The choice of the frame of reference where the time and -axis are defined can be left unspecified in an exactly soluble relativistic theory, but in practical calculations some choices may be more suitable than others. The basic formalism is discussed elsewhere.
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