# Small distance behaviour in field theory and power counting

@article{Symanzik1970SmallDB, title={Small distance behaviour in field theory and power counting}, author={Kurt Symanzik}, journal={Communications in Mathematical Physics}, year={1970}, volume={18}, pages={227-246} }

For infinitesimal changes of vertex functions under infinitesimal variation of all renormalized parameters, linear combinations are found such that the net infinitesimal changes of all vertex functions are negligible relative to those functions themselves at large momenta in all orders of renormalized perturbation theory. The resulting linear first order partial differential equations for the asymptotic forms of the vertex functions are, in quantum electrodynamics, solved in terms of one… Expand

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#### References

SHOWING 1-10 OF 23 REFERENCES

Renormalizable models with simple symmetry breaking

- Physics
- 1970

If to a Lagrangian density with invariance under a continuous group of linear transformations of the fields a term linear or bilinear in the fields is added, the symmetry is in general reduced and… Expand

High-Energy Behavior in Quantum Field Theory

- Mathematics
- 1960

An attack is made on the problem of determining the asymptotic behavior at high energies and momenta of the Green's functions of quantum field theory, using new mathematical methods from the theory… Expand

Partial summation of the perturbation expansion in high-energy quantum electrodynamics

- Physics
- 1963

SummaryThe renormalization group method is applied to the perturbation expansion of electromagnetic scattering cross-sections at high momentum transfer. It is shown that the limit of vanishing… Expand

QUANTUM ELECTRODYNAMICS AT SMALL DISTANCES

- Physics
- 1954

The renormalized propagation functions DFC and SFC for photons and electrons, respectively, are investigated for momenta much greater than the mass of the electron. It is found that in this region… Expand

Gauge invariance in quantum electrodynamics

- Physics
- 1969

Abstract A formulation of quantum electrodynamics based on finite local field equations is employed in order to prove and discuss the gauge invariance of the theory in a meaningful and rigorous way.… Expand

Remarks concerning the possibility of finite charge renormalization in spinor electrodynamics

- Physics
- 1968

Abstract The analysis of spinor electrodynamics, performed by Johnson, Willey and Baker, is examined by use of the renormalization group. It is concluded that the vanishing of the coefficient of the… Expand

On the Green's Functions of Quantized Fields: I.

- Mathematics, Medicine
- Proceedings of the National Academy of Sciences of the United States of America
- 1951

These notes are a preliminary account of a general theory of Green’s functions, in which the defining property is taken to be the representation of the fields of prescribed sources. Expand

Quantum Electrodynamics with Auxiliary Fields

- Physics
- 1953

A consistent scheme of quantum electrodynamics with auxiliary fields of photons and electrons of infinitely large masses is developed. The auxiliary fields are introduced from the very beginning in… Expand

Feynman rules and Feynman integrals for system with higher-spin fields

- Mathematics
- 1965

SummaryIt is known, that Feynman rules for many systems may be expressed by replacing the interaction Hamiltonian in the Dyson formula by minus the interaction Lagrangian, provided we also make… Expand

Proof of the Bogoliubov-Parasiuk theorem on renormalization

- Mathematics
- 1966

A new proof is given that the subtraction rules ofBogoliubov andParasiuk lead to well-defined renormalized Green's distributions. A collection of the counter-terms in “trees” removes the difficulties… Expand