Renormalization

Among a renorm field theory refers to the definition of an energy scale, with respect to which the theory is formulated.

Although renormalization even in classical field theories is possible ( and useful), it is inevitable especially in quantum field theories, otherwise infinite (divergent ) expressions occur. The physical cause of these differences is that the perturbation expansion of interacting quantum field theory is an effective theory, which is valid only within a certain energy range. Although this energy can be very large, but is definitely finite. In the mathematical elaboration of the theory carry - related methods - even energy out of scope at that deliver then infinite (and therefore meaningless ) results. The mathematical reason for this divergence is that the field operators are distributions whose multiplication is not defined on the same space-time point in a series development in general.

An important intermediate step in the course of renormalization is the regularization. For the regularization, there are various technical options, such as dimensional regularization or a fixed energy scale ( "hard cut- off"), but which are physically equivalent. The regularization restricts the theory to a an energy to the renormalization scale. Radiative corrections from outside this energy range can be accounted for by redefining the parameters of the theory, such as masses or coupling constants. If a finite number of parameters redefined sufficient refers to the theory renormalizable. In four space-time dimensions is the mass dimension of the Lagrangian four. In four dimensions, it can be shown generally that a quantum field theory is renormalizable only if the coupling constants in the interaction terms do not have negative mass dimension.

The choice of the energy scale is purely arbitrary, but even should other parameters and other radiative corrections for each energy scale, the physical predictions are identical. The theory was normalized only on an energy value, in so far explains the name renormalization. In practical use to choose the energy scale which corresponds to the subject region.

One of the most important new discoveries in the development of the renormalization group stating that some physical constants, namely the coupling constants and particle masses are not constant, but their values ​​are always to be understood in terms of a certain energy scale. So, for example, takes the elementary charge to be at high energies. Conversely, from the coupling of the strong force at high energy, which is referred to as asymptotic freedom.

The coupling constants are determined by the renormalization only the measured values ​​for the reference power. Many efforts of modern theoretical physics, therefore, aim to calculate these parameters in a higher-level theory with extended scope and can be derived.

The symmetries of the Lagrangian of a quantum field theory and the associated Ward identities can cause various Renormierungskonstanten are equal. This means that certain occurring deviations cancel each other and thus the renormalization of the theory is ensured.

In the axiomatic quantum field theory, there is the causal perturbation theory also a mathematically well-defined renormalization. Explicit calculations are very complicated to carry out in this formalism, which is why it is seen primarily as a mathematical underpinning of renormalization theory, but hardly used for calculations.

A perturbative development of the Einstein - Hilbert action of general relativity is known with the date (2011) methods are not renormalizable. This makes it impossible gravitation in the framework of quantum field theory to be treated like the other fundamental forces, and is one reason that so far no generally accepted theory of quantum gravity exists.