FQFT

The finite quantum field theory (causal perturbation theory) is an attempt with the classic problems of quantum field theory deal (QFT).

One of these classic difficulties is the UV catastrophe , which in the classic theory is treated by renormalization . Problems are of a conceptual and mathematical nature: On the one hand, you get a theory in which many elements have to be used ad hoc or from experimental experience; on the other hand, many of the theory's intrinsic symmetries are lost, which are reconstructed "by hand" after renormalization Need to become. The difficulties are mainly due to mathematical facts. This includes, for example, the fact that, unlike functions, distributions in general do not form algebra (for example, a delta distribution should not be raised to the power).

The FQFT avoids this problem with the so-called causal perturbation theory, which was developed by Ernst Carl Gerlach Stueckelberg , Nikolai Nikolajewitsch Bogoljubow , the physicist Henri Epstein and Wladimir Glaser . The S-matrix is constructed order by order:

${\ displaystyle S (h) = 1 + \ sum _ {n = 1} ^ {\ infty} {\ frac {1} {n!}} \ int d ^ {4} x_ {1} \ ldots d ^ { 4} x_ {n} T_ {n} (x_ {1}, \ ldots, x_ {n}) h (x_ {1}) \ cdot \ ldots \ cdot h (x_ {n})}$

where is a tempered test function and they are operator-valued distributions. The first order specifies the model. All higher orders are now constructed inductively, with causality playing an essential role. The method of H. Epstein and W. Glaser consists in splitting distributions with a carrier on a generalized forward and backward light cone in a causally correct manner (which can be carried out in momentum space for theories with massive fields by a dispersion integral). This causal splitting of the operator-valued distributions occurring in the inductive construction essentially corresponds to the time order of operator products, and if the problem is handled correctly, no UV divergences occur. In general, however, the construction is not unambiguous: local operator-valued distributions whose carriers lie on the so-called diagonal can be added to the if necessary. The form of these local distributions is limited by their scale behavior and general symmetry conditions ( e.g. Poincaré symmetry). The FQFT was successfully based on various model theories, but also on (massive) gauge theories such as B. applied the standard model of elementary particle physics. ${\ displaystyle h (x)}$ ${\ displaystyle T_ {n} (x_ {1}, \ ldots, x_ {n})}$ ${\ displaystyle T_ {1} (x_ {1})}$ ${\ displaystyle x_ {1} = \ ldots ... = x_ {n}}$ ${\ displaystyle T_ {n} (x_ {1}, \ ldots, x_ {n})}$

During the calculation of the perturbation-theoretical S-matrix, they are left in general, and at the end the adiabatic limit can be carried out, during which the infrared divergences can be controlled and effective cross-sections can finally be calculated. ${\ displaystyle h (x)}$ ${\ displaystyle h (x) \ rightarrow 1}$

Development opportunities

The FQFT is a very general theory that can now be further developed in different directions. Worth mentioning is the approach in the not classical fields and the corresponding Lagrangians is understood but general scalar - vector - and Tensor - fields are quantized. Together with suitable geometric calibration conditions, for example, the theory of the electroweak interaction can be constructed. The existence of the Higgs boson or a Higgs sector follow automatically from the theory and do not have to be introduced "by hand" into the theory.

Furthermore, the FQFT can be generalized to curvilinear coordinates, which may enable the connection to quantum gravity.

evaluation

The main advantage of FQFT is that the theory is well defined from a mathematical point of view as a perturbative theory. An interesting field of application of this theory could be the assessment of higher-dimensional theories such as string theory , since these should lead back to a (mathematically correct) QFT in a suitable reduction of the dimensions.

The methods that are used are so far less familiar to experts. In the professional world, many calculations are carried out with the help of path integrals ( Feynman diagrams ), which are not used in FQFT.

literature

H. Epstein, V. Glaser, Ann. Inst. Poincaré, A 29 (1973) 211
Günter Scharf, Finite Quantum Electrodynamics: The Causal Approach, Springer Texts and Monographs in Physics, ISBN 3-540-60142-2
Günter Scharf, Quantum Gauge Theories: A True Ghost Story, Wiley-Interscience, ISBN 0-471-41480-8
Günter Scharf, Michael Dütsch, Perturbative Gauge Invariance: The Elektroweak Theory, Ann. Phys 8 (1999) 5, 359-387