Lagrangian Quantum Field Theory Research Articles

Solutions in Constructive Field Theory

Constructive field theory aims to rigorously construct concrete, nontrivial solutions to Lagrangians used in particle physics. I examine the relationship of solutions in constructive field theory to both axiomatic and Lagrangian quantum field theory (QFT). I argue that Lagrangian QFT provides conditions for what counts as a successful constructive solution and other information that guides constructive field theorists to solutions. Solutions matter because they describe the behavior of QFT systems and thus what QFT says the world is like. Constructive field theory clarifies existing disputes about which parts of QFT are philosophically relevant and how rigor relates to these disputes.

UNCONSTRAINED HIGHER SPINS IN FOUR DIMENSIONS

The relativistic symmetric tensor fields are, in four dimensions, the right candidates to describe Higher Spin Fields. Their highest spin content is isolated with the aid of covariant conditions, discussed within a group theory framework, in which auxiliary fields remove the lower intrinsic angular momenta sectors. These conditions are embedded within a Lagrangian Quantum Field theory which describes an Higher Spin Field interacting with a Classical background. The model is invariant under a (B.R.S.) symmetric unconstrained tensor extension of the reparametrization symmetry, which include the Fang–Fronsdal algebra in a well defined limit. However, the symmetry setting reveals that the compensator field, which restore the Fang–Fronsdal symmetry of the free equations of motion, is in the existing in the framework and has a relevant geometrical meaning. The Ward identities coming from this symmetry are discussed. Our constraints give the result that the space of the invariant observables is restricted to the ones constructed with the Highest Spin Field content. The quantum extension of the symmetry reveals that no new anomaly is present. The role of the compensator field in this result is fundamental.

Why is $$\mathcal{CPT}$$ Fundamental?

G. L\"uders and W. Pauli proved the $\mathcal{CPT}$ theorem based on Lagrangian quantum field theory almost half a century ago. R. Jost gave a more general proof based on ``axiomatic'' field theory nearly as long ago. The axiomatic point of view has two advantages over the Lagrangian one. First, the axiomatic point of view makes clear why $\mathcal{CPT}$ is fundamental--because it is intimately related to Lorentz invariance. Secondly, the axiomatic proof gives a simple way to calculate the $\mathcal{CPT}$ transform of any relativistic field without calculating $\mathcal{C}$, $\mathcal{P}$ and $\mathcal{T}$ separately and then multiplying them. The purpose of this pedagogical paper is to ``deaxiomatize'' the $\mathcal{CPT}$ theorem by explaining it in a few simple steps. We use theorems of distribution theory and of several complex variables without proof to make the exposition elementary.

In Defence of Naiveté: The Conceptual Status of Lagrangian Quantum Field Theory

I analyse the conceptual and mathematical foundations of Lagrangian quantum field theory (QFT) (that is, the ‘naive’ (QFT) used in mainstream physics, as opposed to algebraic quantum field theory). The objective is to see whether Lagrangian (QFT) has a sufficiently firm conceptual and mathematical basis to be a legitimate object of foundational study, or whether it is too ill-defined. The analysis covers renormalisation and infinities, inequivalent representations, and the concept of localised states; the conclusion is that Lagrangian QFT (at least as described here) is a perfectly respectable physical theory, albeit somewhat different in certain respects from most of those studied in foundational work.

Understanding light quanta: first quantization of the free electromagnetic field

The quantization of the electromagnetic field in vacuum is presented without reference to Lagrangian quantum field theory. The equal-time commutators of the fields are calculated from basic principles. A physical discussion of the commutators suggests that the electromagnetic fields are macroscopic emergent properties of a more fundamental physical system: the photons.

Functional integral equation for the complete effective action in quantum field theory

Based on a methodological analysis of the effective action approach certain conceptual foundations of quantum field theory are reconsidered to establish a quest for an equation for the effective action. Relying on the functional integral formulation of Lagrangian quantum field theory a functional integral equation for the complete effective action is proposed which can be understood as a certain fixed point condition. This is motivated by a critical attitude towards the distinction artificial from an experimental point of view between classical and effective action. While for free field theories nothing new is accomplished, for interacting theories the concept differs from the established paradigm. The analysis of this new concept is concentrated on gauge field theories treating QED as the prototype model. An approximative approach to the functional integral equation for the complete effective action of QED is exploited to obtain certain nonperturbative information about the quadratic kernels of the action. As particular application the approximative calculation of the QED coupling constant $\alpha$ is explicitly studied. It is understood as one of the characteristics of a fixed point given as a solution of the functional integral equation proposed. Finally, within the present approach the vacuum energy problem is considered and possible implications on the induced gravity concept are contemplated.

Feynman path integral for fermions.

With Lagrangian quantum field theory as a starting point, the general quantum transition amplitude is derived as a sum of action exponentials over all paths connecting the initial and final fermion states. For bookkeeping convenience in intermediate stages anticommuting parameters are introduced. All integrals are determined by completeness and unitarity. Grassmann calculus is not postulated.

Renormalization over asymptotics

In connection with the new achievements of Lagrangian quantum field theory, there is currently a resurgence of interest in the old problem of renormalization. This paper describes a new renormalization procedure for the generating functional of the Green's functions. The scheme is based on renormalization of the integral over the loop momentum. Ward identities for the renormalized Green's functions are obtained from the condition of ''quantum invariance of the action.''

A generalized quantum field theory

An expression in quantum-field-theoretic language of the four-space formulation (FSF), especially the FSF group properties, is derived by generalizing Schwinger's formulation of Lagrangian quantum field theory (LQFT). The resulting theoretical framework includes a mass operator in addition to the energy-momentum and angular momentum operators. It also contains LQFT as a special case. Broad conclusions regarding conservation laws (of rest mass, energy-momentum, and angular momentum) are obtained from the general formalism. Many mathematical details concerning the FSF group and FSF transformations are presented.

Functional techniques and a non-formal proof of the equivalence theorem in Lagrangian quantum field theory

We give a proof of the equivalence theorem in Lagrangian quantum field theory employing functional techniques and a regularization prescription which yields finite results at each order of the perturbation expansion.

Higher-order corrections in nonrenormalizable quantum field theories

Within the framework of Lagrangian quantum field theory, the perturbative expansion of the S-matrix leads to a power series in the coupling constant with diverging coefficients. These infinities can be properly removed, through the renormalization program, only in a few cases, while in all the other cases a general reliable prescription is not yet available. However, it is possible that finite S-matrix elements exist also for nonrenormalizable theories, even if the single terms of the perturbative expansion are diverging. This assumption is supported by some specific field theoretical models (1-3) and enables us to infer that nonrenormalizable theories may be physically meaningful, if we are able to sum the whole perturbative series. For instance, in the intermediate boson theory of weak interactions, it has been shown that the sum of an infinite subset of Feynman graphs, the so-called (( uncrossed ladder graphs ~>, can be obtained by solving an integral equation and gives finite results. This procedure, however, is not completely justified because the contributions coming from the missed diagrams are not, a pr ior i , negligible. In this paper we want to propose a general summation procedure for perturbative series with divergent terms, arising from nonrenormalizable Lagrangians. Let us start from the complete set of Feynman graphs relative to some physical process and let us regularize the divergent integrals by means of a proper cut-off parameter A s in order to make the series finite term by term. We are then faced with the infinite sum

On Polynomial Systems in a Banach Ring

We define and discuss equations on Banach rings (algebras) which are of polynomial form. We prove a local uniqueness theorem for the homogeneous case, and an existence and local uniqueness theorem for the nonhomogeneous case. In order to apply these results to the equations of Lagrangian quantum field theory we find it necessary to extend the concept of a ring to that of an n-ring. The resulting theory is applied to a simple model equation arising in quantum field theory.