Non Local Theories: New Rules for Old Diagrams

Abstract

Abstract:We show that a general variant of the Wick theorems can be used to reduce the time ordered products in the Gell-Mann & Low formula for a certain class on non local quantum field theories, including the case where the interaction lagrangian is defined in terms of twisted products. The only necessary modification is the replacement of the Stueckelberg-Feynman propagator by the general propagator (the "contractor" of Denk and Schweda)where the violations of locality and causality are represented by the dependence of τ, τ on other points, besides those involved in the contraction. This leads naturally to a diagrammatic expansion of the Gell-Mann & Low formula, in terms of the same diagrams as in the local case, the only necessary modification concerning the Feynman rules. The ordinary local theory is easily recovered as a special case, and there is a one-to-one correspondence between the local and non local contributions corresponding to the same diagrams, which is preserved while performing the large scale limit of the theory.

Quantum Spacetime and Algebraic Quantum Field Theory

“…The common feature of all approaches is that, due to the quantum nature of spacetime at the Planck scale, locality is broken (even at the level of free fields, for explicit estimates see [1]); in perturbation theory, its breakdown manifests itself in a non local kernel, which spreads the interaction vertices [1,12,9] ; this forces on us the appropriate modifications of Feynman rules [13].…”

DFR Perturbative Quantum Field Theory on Quantum Space Time, and Wick Reduction

“…There is, however, a profound reason why such a clever rearrangement of terms leads to the desired result: indeed, one can instead reproduce exactly the same line of reasoning which can be found in any standard textbook on local quantum field theory, since the Wick reduction of time ordered products of Wick monomials can be performed in the nonlocal setting considered here, too [1]. In other words, the second Wick theorem is not local; this is so deeply true that even the original proof of Wick does not rely on locality [20].…”

“…Both these tools are completely unrelated to locality, hence this result should have not come out as a surprise. Feynman diagrams require an additional tool, the Gell-Mann & Low formula [21], which also has nothing to do with locality; indeed, it was shown in [1] that the dear old Feynman diagrams also arise naturally in the reduction of the nonlocal Green functions of the DFR perturbative model. Actually, we did not introduce any really new argument, everything was already virtually contained in the papers of Dyson, Wick, and Gell-Mann & Low.…”

An anthology of non-local QFT and QFT on non-commutative spacetime

“…This is precisely what the D-F-R theory [21][22] [23] achieves; it does so by building a QFT over a non-commutative version of fully Poincaré-invariant Minkowski spacetime. In a more colorful artistic terminology one may say that the static ether is rendered dynamic i.e.…”

“…Note that in the C-P direct particle interaction theory the non-locality also prevented the existence of covariant tensor objects corresponding to Heisenberg fields 23 . But thanks to the scattering equivalences this, did not spoil the existence of a clustering unitary S-matrix.…”