Perturbation Theory of Transformed Quantum Fields
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Perturbation Theory of Transformed Quantum Fields Paul-Hermann Balduf1 Received: 4 June 2019 / Accepted: 19 August 2020 / © The Author(s) 2020
Abstract We consider a scalar quantum field φ with arbitrary polynomial self-interaction in perturbation theory. If the field variable φ is repaced by a global diffeomorphism φ(x) = ρ(x) + a1 ρ 2 (x) + . . ., this field ρ obtains infinitely many additional interaction vertices. We propose a systematic way to compute connected amplitudes for theories involving vertices which are able to cancel adjacent edges. Assuming tadpole graphs vanish, we show that the S-matrix of ρ coincides with the one of φ without using path-integral arguments. This result holds even if the underlying field has a propagator of higher than quadratic order in the momentum. The diffeomorphism can be tuned to cancel all contributions of an underlying φ t -type self interaction at one fixed external offshell momentum, rendering ρ a free theory at this momentum. Finally, we mention one way to extend the diffeomorphism to a non-diffeomorphism transformation involving derivatives without spoiling the combinatoric structure of the global diffeomorphism. Keywords Diffeomorphism of quantum fields · Propagator cancellation · Bell polynomials · Diffeomorphism invariance of the S-matrix · Connected perspective Mathematics Subject Classification (2010) 81T18
1 Introduction 1.1 Motivation and Content A quantum field theory can be defined via a Lagrangian density L, simply called Lagrangian hereafter. In perturbative computations in this theory, the monomials of
The author thanks Dirk Kreimer and Karen Yeats for helpful discussion. Paul-Hermann Balduf
[email protected] 1
Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, Newtonstrasse 15, 12489 Berlin, Germany
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Math Phys Anal Geom
(2020) 23:33
degree n > 2 in L correspond to n-valent interaction vertices in Feynman diagrams. The monomials of degree two define the propagator of the field. We only consider scalar quantum fields in this paper. A free scalar quantum field theory has no selfinteraction and is defined via the Lagrangian 1 1 ∂μ φ(x)∂ μ φ(x) − m2 φ 2 (x), (1.1) 2 2 where m ≥ 0 is the mass of a φ-particle. We allow m = 0 for the massles theory. x is a point in spacetime which we take to be 4-dimensional Minkowski space for concreteness even if our results do not depend on the dimensionality. We express the field φ(x) as a global diffeomorphism Lφ =
φ(x) =
∞
aj ρ j +1 (x),
a0 = 1
(1.2)
j =0
of another field ρ(x). The term global stresses that {aj }j ≥1 are constants with respect to spacetime, i.e. the diffeomorphism is the same transformation globally. This is common physics nomenclature (e.g. global vs local gauge symmetries), alternatively one could call (1.2) local since it only involves ρ(x) at that very point x (that is the perspective of [17]). The constraint a0 = 1 means the diffeomorphism is tangent to identity such that the fields ρ and φ coincide at leading order. Equation 1.2, when applied to the
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