# Wave function renormalization

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In quantum field theory wave function renormalization is a rescaling (or renormalization) of quantum fields to take into account the effects of interactions. For a noninteracting or free field the field operator creates or annihilates a single particle with probability 1. Once interactions are included, however, this probability is modified in general to Z${\displaystyle \neq }$ 1. This appears when one calculates the propagator beyond leading order; e.g. for a scalar field,

In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics and is used to construct physical models of subatomic particles and quasiparticles.

Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of quantities to compensate for effects of their self-interactions. However, even if it were the case that no infinities arise in loop diagrams in quantum field theory, it can be shown that renormalization of mass and fields appearing in the original Lagrangian is necessary.

In physics a free field is a field without interactions, which is described by the terms of motion and mass.

${\displaystyle {\frac {i}{p^{2}-m_{0}^{2}+i\varepsilon }}\rightarrow {\frac {iZ}{p^{2}-m^{2}+i\varepsilon }}}$

(The shift of the mass from m0 to m constitutes the mass renormalization.)

One possible wave function renormalization, which happens to be scale independent, is to rescale the fields so that the Lehmann weight (Z in the formula above) of their quanta is 1. For the purposes of studying renormalization group flows, if the coefficient of the kinetic term in the action at the scale Λ is Z, then the field is rescaled by ${\displaystyle {\sqrt {Z}}}$. A scale dependent wave function renormalization for a field means that that field has an anomalous scaling dimension.

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