Renormalization

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Under renormalization a field theory refers to the setting of an energy scale , with respect to which the theory is formulated.

Avoidance of divergences

Although renormalization is also possible (and useful) with classical field theories, it is especially inevitable with quantum field theories , since otherwise infinite ( divergent ) expressions occur.

The physical cause of these divergences is that the perturbation evolution of an interacting quantum field theory is an effective theory that is only valid within a certain energy range. This energy range can be very large, but it is definitely finite. In the mathematical elaboration of the theory - depending on the method - energies outside the validity range also contribute, which then deliver infinite (and therefore nonsensical) results.

The mathematical reason for these divergences is that the field operators are distributions , the multiplication of which at the same point in space-time is generally not defined within the framework of a series expansion .

Regularization

An important intermediate step in the course of renormalization is regularization . There are various technical options for regularization, e.g. B. dimensional regularization or a fixed energy scale (“hard cut-off”), which are physically equivalent. The regularization restricts the theory to an energy range around the renormalization scale. Radiation corrections from outside this energy range can be made by redefining the parameters of the theory, e.g. B. masses or coupling constants are taken into account. If a finite number of redefined parameters is sufficient, the theory is called renormalizable . In four spacetime dimensions, the mass dimension of the Lagrangian is four. In four dimensions it can generally be shown that a quantum field theory can only be renormalized if the coupling constants in the interaction terms do not have a negative mass dimension.

Term renormalization

The choice of the energy scale is purely arbitrary, but although different parameters and different radiation corrections result for each energy scale, the physical predictions are identical. The theory was only standardized to one energy value , which was not the case in the original, non-regularized disturbance development.

In this respect, the designation ( re ) normalization for the normalization of the theory and its independent parameters on a reference scale, which in practice corresponds to the selected energy range, is somewhat misleading. In today's parlance, renormalization means the observed energy-dependent variation of these normalized parameters in the energy range (significantly) below the fixed energy scale of the regularization. This energy scale is discussed further below and is a crucial feature of any physical theory with shape-invariant laws on different size scales.

Influence of the energy scale on "constants"

One of the most important new findings in the context of the development of the renormalization group says that some fundamental constants , namely the coupling constants and the particle masses, are not constant, but that their values ​​are always to be understood in relation to a certain energy scale. For example, B. the elementary charge at high energies. Conversely, the coupling of the strong nuclear force decreases at high energies, which is called asymptotic freedom .

The coupling constants are only determined by renormalization on the measured values ​​for the reference energy. Many efforts in modern theoretical physics are therefore aimed at being able to calculate or derive these parameters within the framework of a superordinate theory with an extended range of validity.

additional

The symmetries of the Lagrangian of a quantum field theory and the associated Ward identities can lead to different renormalization constants being the same. This means that certain deviations that occur cancel each other out, thus guaranteeing that the theory can be renormalized.

In axiomatic quantum field theory there is also a mathematically well-defined renormalization procedure with causal perturbation theory. Explicit calculations are very complicated to carry out in this formalism, which is why they are mainly understood as the mathematical underpinning of renormalization theory, but are rarely used for calculations.

A perturbative development of the Einstein-Hilbert effect of the general theory of relativity cannot be renormalized with the methods known so far (2015). This makes it impossible to treat gravity like the other basic forces within the framework of quantum field theory , and is one reason why no generally accepted theory of quantum gravity has yet existed. However, there are methods that quantize and renormalize gravity in a non-perturbative way. The above mentioned Replaced the criterion of asymptotic freedom with the weaker criterion of asymptotic security.

literature

Renormalization is covered in any introductory book on quantum field theory , e.g. B.

A comprehensive treatise can be found in