Finite element approximation in quantum field theory

Chapter 11 - Finite Element Approximation with Splines

Quantum systems governed by non-Hermitian Hamiltonians with symmetry are special in having real energy eigenvalues bounded below and unitary time evolution. We argue that symmetry may also be important and present at the level of Hermitian quantum field theories because of the process of renormalisation. In some quantum field theories renormalisation leads to -symmetric effective Lagrangians. We show how symmetry may allow interpretations that evade ghosts and instabilities present in an interpretation of the theory within a Hermitian framework. From the study of examples -symmetric interpretation is naturally built into a path integral formulation of quantum field theory; there is no requirement to calculate explicitly the norm that occurs in Hamiltonian quantum theory. We discuss examples where -symmetric field theories emerge from Hermitian field theories due to effects of renormalisation. We also consider the effects of renormalisation on field theories that are non-Hermitian but -symmetric from the start.

This book provides a rather self-contained survey of the construction of Hadamard states for scalar field theories in a large class of notable spacetimes, possessing a (conformal) light-like boundary. The first two sections focus on explaining a few introductory aspects of this topic and on providing the relevant geometric background material. The notions of asymptotically flat spacetimes and of expanding universes with a cosmological horizon are analysed in detail, devoting special attention to the characterization of asymptotic symmetries. In the central part of the book, the quantization of a real scalar field theory on such class of backgrounds is discussed within the framework of algebraic quantum field theory. Subsequently it is explained how it is possible to encode the information of the observables of the theory in a second, ancillary counterpart, which is built directly on the conformal (null) boundary. This procedure, dubbed bulk-to-boundary correspondence, has the net advantage of allowing the identification of a distinguished state for the theory on the boundary, which admits a counterpart in the bulk spacetime which is automatically of Hadamard form. In the last part of the book, some applications of these states are discussed, in particular the construction of the algebra of Wick polynomials. This book is aimed mainly, but not exclusively, at a readership with interest in the mathematical formulation of quantum field theory on curved backgrounds.

Rules of quantization and equations of motion for a finite-dimensional formulation of quantum field theory are proposed which fulfill the following properties: (a) Both the rules of quantization and the equations of motion are covariant; (b) the equations of evolution are second order in derivatives and first order in derivatives of the spacetime coordinates; and (c) these rules of quantization and equations of motion lead to the usual (canonical) rules of quantization and the (Schrödinger) equation of motion of quantum mechanics in the particular case of mechanical systems. We also comment briefly on further steps to fully develop a satisfactory quantum field theory and the difficuties which may be encountered when doing so.

We present a formulation of quantum field theory as a path-integral representation for elements of the U or S matrix in the coherent- state basis. These matrix elements are shown to serve as generating functionals for all the usual S-matrix elements between statesof definite particle number. The coherent-state formalism for general bilinear quantum field theories is described and then incorporated into the construction of a path integral such that the formulation is independent of any canonical formalism. We discuss the relationship of this formulation to the usual path-integral formulation and show in what sense they are equivalent for canonical theories. We also discuss how this formulation is more general than the usual one in that it is well defined even for theories having no canonical form and for which a Lagrangian action and the usual path-integral formulation may not apply. Applications of this formulation to specific calculations are exhibited for the cases of a quantized field interacting with a given external source and for the renormalization of the simple quantum field theory model of a scalar meson field interacting with a nonrelativistic nucleon field.

In this thesis we present two geometric theories designed to extend general relativity. It can be seen as one of the aims of such theories to model the observed accelerating expansion of the universe as a gravitational phenomenon, or to provide a mathematical structure for the formulation of quantum field theories on curved spacetimes and quantum gravity. This thesis splits into two parts: In the first part we consider multimetric gravity theories containing N > 1 standard model copies which interact only gravitationally and repel each other in the Newtonian limit. The dynamics of each of the standard model copies is governed by its own metric tensor. We show that the antisymmetric case, in which the mutual repulsion between the different matter sectors is of equal strength compared to the attractive gravitational force within each sector, is prohibited by a no-go theorem for N = 2. We further show that this theorem does not hold for N > 2 by explicitly constructing an antisymmetric multimetric repulsive gravity theory. We then examine several properties of this theory. Most notably, we derive a simple cosmological model and show that the accelerating expansion of the late universe can indeed be explained by the mutual repulsion between the different matter sectors. We further present a simple model for structure formation and show that our model leads to the formation of filament-like structures and voids. Finally, we show that multimetric repulsive gravity is compatible with high-precision solar system data using the parametrized post-Newtonian formalism. In the second part of the thesis we propose a mathematical model of quantum spacetime as an infinite-dimensional manifold locally homeomorphic to an appropriate Schwartz space. This extends and unifies both the standard function space construction of quantum mechanics and the differentiable manifold structure of classical spacetime. In this picture we demonstrate that classical spacetime emerges as a finite-dimensional manifold through the topological identification of all quantum points with identical position expectation value. We speculate on the possible relevance of this geometry to quantum field theory and gravity.

Doubly special relativity (DSR) is usually regarded as a low-energy limit of a quantum gravity theory with testable predictions. On the other hand, non-local quantum field theories have been presented as a solution to the inconsistencies arising when quantizing gravity. Here, we present a new formulation of quantum field theories in DSR with non-local behavior. Our construction restricts the models to those showing linear Lorentz invariance. We derive the deformed Klein–Gordon, Dirac, and electromagnetic Lagrangians, as well as the deformed Maxwell equations. We also discuss the electric potential of a point charge. Finally, we analyze the connection between the nonlocality of field theories and DSR.

Analogies in the formulation of quantum field theory (QFT) for elementary particles and of many-body theory for condensed matter systems are briefly discussed. One one side, photons, gravitons, etc. are identified with the quanta of radiation fields; on the other side, phonons, plasmons, etc. appear from quantization of collective waves in condensed matter.Both in elementary particle and in solid state physics, a fundamental role is played by symmetry. Specifically, we consider the mechanism of spontaneous symmetry breaking, which takes place when the ground state of a system possesses a lower symmetry than the equations of motion. In the case of continuous symmetry, the breaking is accompanied by the appearance of Nambu-Goldstone modes, which are massless bosons in QFT and elementary excitations without energy-gap in non-relativistic many-body theory. A relevant example is the case of superfluidity.When long-range interactions are present, such as those mediated by massless vector fields, the would-be Goldstone mode acquires a non-vanishing mass, and it could be interpreted as the longitudinal component of the vector field which becomes massive. These considerations are applied to the case of superconductivity.

The results of Chapter 1 provide a consistent and developed picture of local quantum theory conceived as the ‘axiomatic algebraic approach’ in quantum field theory. The picture, however, does not look finished since the information available on many properties of local nets (such as types of the algebras R (O), their modular structure, duality, etc.) is only fragmentary. Another unsatisfactory point concerns applications of the formalism. The whole complex of the results of Chapter 1 is still too poor for the needs of describing concrete models and processes of interaction of elementary particles. It lacks a wide range of notions and properties inherent in quantum field systems but not reflected in Axioms I-VI (cf. Introduction). As a result, the Haag-Araki or Haag-Kastler theory, like other axiomatic approaches, provides an essentially incomplete formulation of quantum field theory and should not be considered as a self-contained ‘axiomatic theory’ in the strict sense of axiomatic theories in mathematics. It is rather a starting ground, a base set of firmly established facts, which still needs to be expanded and complemented (obviously, on some other principles, not purely axiomatic any more).

Schwinger's formalism in quantum field theory can be easily implemented in the case of scalar theories in $D$ dimension with exponential interactions, such as $\mu^D\exp(\alpha\phi)$. In particular, we use the relation $$ \exp\big(\alpha{\delta\over \delta J(x)}\big)\exp(-Z_0[J])=\exp(-Z_0[J+\alpha_x]) $$ with $J$ the external source, and $\alpha_x(y)=\alpha\delta(y-x)$. Such a shift is strictly related to the normal ordering of $\exp(\alpha\phi)$ and to a scaling relation which follows by renormalizing $\mu$. Next, we derive a new formulation of perturbation theory for the potentials $V(\phi)={\lambda\over n!}:\phi^n:$, using the generating functional associated to $:\exp(\alpha\phi):$. The $\Delta(0)$-terms related to the normal ordering are absorbed at once. The functional derivatives with respect to $J$ to compute the generating functional are replaced by ordinary derivatives with respect to auxiliary parameters. We focus on scalar theories, but the method is general and similar investigations extend to other theories.

We study the electromagnetic structure of the pion in terms of the quantum cromodynamic (QCD) model on the Breit-frame. We calculated the observables, such as the electromagnetic form factor. The priori to have a calculation covariant need to get the valence term of the eletromagnetic form factor. We use the usual formalism in quantum field theory (QFT) and light-front quantum field theory (LFQFT) in order to test the properties of form factor in nonperturbative QCD. In this particular case, the form factor can be obtained using the pion Light-Front (LF) wave function including self-energy from Lattice-QCD. Specifically, these calculations was performed in LF formalism. We consider a quark-antiquark vertex model having a quark self-energy. Also we can use other models to compare the pion electromagnetic form factor with different wave function and to observe the degree of agreement between them.

A gauge-invariant continuation of the S-matrix generating functional outside the mass shell is proposed. A generating functional of gauge-invariant Green functions is obtained. Using the path integral formalism, collective gauge-invariant bilocal field variables are introduced. With the aid of the bilocal formalism, an integral equation is obtained for the gauge-invariant spinor propagator. It is shown that the propagator of the bilocal field corresponds to the gauge-invariant wave function of a two-particle system in ladder approximation.

The connection between the Hamilton and the standard Lagrange formalism is established for a generic quantum field theory with vanishing vacuum expectation values of the fundamental fields. The effective actions in both formalisms are the same if and only if the fundamental fields and the momentum fields are related by the stationarity condition. These momentum fields in general differ from the canonical fields as defined via the effective action. By means of functional methods a systematic procedure is presented to identify the full correlation functions, which depend on the momentum fields, as functionals of those usually appearing in the standard Lagrange formalism. Whereas Lagrange correlation functions can be decomposed into tree diagrams, the decomposition of Hamilton correlation functions involves loop corrections similar to those arising in n-particle effective actions. To demonstrate the method we derive for theories with linearized interactions the propagators of composite auxiliary fields and the ones of the fundamental degrees of freedom. The formalism is then utilized in the case of Coulomb gauge Yang–Mills theory for which the relations between the two-point correlation functions of the transversal and longitudinal components of the conjugate momentum to the ones of the gauge field are given.

This chapter is based in great part on the author’s paper which appeared in 2015 in the European Physical Journal ( Dittrich, W.: The cofounder of quantum field theory: Pascal Jordan. The Eur. Phy. J. H 40, 241–260 (2015)).Before we start with the most general covariant formulation of quantum field theory, we want to go back to Jordan, who contributed more than anybody else to the birth of quantum field theory in 1925, the year of the “Dreimännerarbeit” by Born, Heisenberg and Jordan ( Born, M., Heisenberg, W., Jordan, P.: Zur Quantenmechanik II. Z. Phys. 30, 558 (1925)).

It is demonstrated how the one-time formulation of quantum field theory leads to the quasipotential equation for three-particle system.