Formulation Of Quantum Field Theory Research Articles

Quantum field perturbation theory revisited

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.

A note on the path integral representation for Majorana fermions

Majorana fermions are currently of huge interest in the context of nanoscience and condensed matter physics. Different to usual fermions, Majorana fermions have the property that the particle is its own anti-particle thus, they must be described by real fields. Mathematically, this property makes nontrivial the quantization of the problem due, for instance, to the absence of a Wick-like theorem. In view of the present interest on the subject, it is important to develop different theoretical approaches in order to study problems where Majorana fermions are involved. In this note we show that Majorana fermions can be studied in the context of field theories for constrained systems. Using the Faddeev–Jackiw formalism for quantum field theories with constraints, we derived the path integral representation for Majorana fermions. In order to show the validity of the path integral we apply it to an exactly solvable problem. This application also shows that it is rather simple to perform systematic calculations on the basis of the present framework.

Proof of the quantum null energy condition

We prove the Quantum Null Energy Condition (QNEC), a lower bound on the stress tensor in terms of the second variation in a null direction of the entropy of a region. The QNEC arose previously as a consequence of the Quantum Focussing Conjecture, a proposal about quantum gravity. The QNEC itself does not involve gravity, so a proof within quantum field theory is possible. Our proof is somewhat nontrivial, suggesting that there may be alternative formulations of quantum field theory that make the QNEC more manifest. Our proof applies to free and superrenormalizable bosonic field theories, and to any points that lie on stationary null surfaces. An example is Minkowski space, where any point $p$ and null vector $k^a$ define a null plane $N$ (a Rindler horizon). Given any codimension-2 surface $\Sigma$ that contains $p$ and lies on $N$, one can consider the von Neumann entropy $S_\text{out}$ of the quantum state restricted to one side of $\Sigma$. A second variation $S_\text{out}^{\prime\prime}$ can be defined by deforming $\Sigma$ along $N$, in a small neighborhood of $p$ with area $\cal A$. The QNEC states that $\langle T_{kk}(p) \rangle \ge \frac{\hbar}{2\pi} \lim_{{\cal A}\to 0}S_\text{out}^{ \prime\prime}/{\cal A}$.

Contact interactions between particle worldlines

We construct contact interactions for bosonic and fermionic point particles. We first relate the resulting theories to classical electrostatics by taking functional averages over worldlines whose endpoints are fixed to charged particles. Counting those paths which pass through a space-time point $x^{\mu}$ gives the static electric field at that point, provided we take the limit where the length measured along the worldlines is large. We also investigate corrections to the classical field that arise beyond leading order in this limit before constructing a theory of point particles that interact when their worldlines intersect. We quantise this theory and show that the partition function contains propagator couplings between the endpoints of the particles before discussing how this is related to the worldline formalism of quantum field theory and general action at a distance theories.

Gravity as a four dimensional algebraic quantum field theory

Based on a family of indefinite unitary representations of the diffeomorphism group of an oriented smooth $4$-manifold, a manifestly covariant $4$ dimensional and non-perturbative algebraic quantum field theory formulation of gravity is exhibited. More precisely among the bounded linear operators acting on these representation spaces we identify algebraic curvature tensors hence a net of local quantum observables can be constructed from $C^*$-algebras generated by local curvature tensors and vector fields. This algebraic quantum field theory is extracted from structures provided by an oriented smooth $4$-manifold only hence possesses a diffeomorphism symmetry. In this way classical general relativity exactly in $4$ dimensions naturally embeds into a quantum framework. Several Hilbert space representations of the theory are found. First a "tautological representation" of the limiting global $C^*$-algebra is constructed allowing to associate to any oriented smooth $4$-manifold a von Neumann algebra in a canonical fashion. Secondly, influenced by the Dougan--Mason approach to gravitational quasilocal energy-momentum, we construct certain representations what we call "positive mass representations" with unbroken diffeomorphism symmetry. Thirdly, we also obtain "classical representaions" with spontaneously broken diffeomorphism symmetry corresponding to the classical limit of the theory which turns out to be general relativity. Finally we observe that the whole family of "positive mass representations" comprise a $2$ dimensional conformal field theory in the sense of G. Segal.

Frederic Paugam: “Towards the Mathematics of Quantum Field Theory”

Quantum field theory can be considered as one of the cornerstones of modern physics. Historically, it originates from the problem of quantizing the electromagnetic field, dating back to shortly after the advent of quantum mechanics. This amounts to understanding the quantization of a system with infinitely many degrees of freedom, namely, the values of the electromagnetic field at all spacetime points. It turned out that precisely this property would form the main obstacle in the search for a mathematical formalism for quantum field theories. In the 1950s this led to a separation of the field. On the one hand perturbation theory led to a computationally yet undefeated approach, where in particular Feynman’s graphical method streamlined the analysis of physical probability amplitudes. On the other hand, an elegant axiomatic approach to quantum fields emerged through the work of Haag, Kastler, Wightman and others, leading to the subject known as algebraic quantum field theory. For a historical account we refer to [7]. Even though the two approaches ought to be complementary, it turned out that the separation only became larger and larger towards the end of the last century. This was mainly due to the fact that physically relevant interacting quantum fields were found difficult to be included in the strict algebraic formulation, while at the same time the computational approach of Feynman amplitudes mistified the mathematical structure behind this successful branch of physics.

Simple recursion relations for general field theories

On-shell methods offer an alternative definition of quantum field theory at tree-level, replacing Feynman diagrams with recursion relations and interaction vertices with a handful of seed scattering amplitudes. In this paper we determine the simplest recursion relations needed to construct a general four-dimensional quantum field theory of massless particles. For this purpose we define a covering space of recursion relations which naturally generalizes all existing constructions, including those of BCFW and Risager. The validity of each recursion relation hinges on the large momentum behavior of an n-point scattering amplitude under an m-line momentum shift, which we determine solely from dimensional analysis, Lorentz invariance, and locality. We show that all amplitudes in a renormalizable theory are 5-line constructible. Amplitudes are 3-line constructible if an external particle carries spin or if the scalars in the theory carry equal charge under a global or gauge symmetry. Remarkably, this implies the 3-line constructibility of all gauge theories with fermions and complex scalars in arbitrary representations, all supersymmetric theories, and the standard model. Moreover, all amplitudes in non-renormalizable theories without derivative interactions are constructible; with derivative interactions, a subset of amplitudes is constructible. We illustrate our results with examples from both renormalizable and non-renormalizable theories. Our study demonstrates both the power and limitations of recursion relations as a self-contained formulation of quantum field theory.

An introduction to the general boundary formulation of quantum field theory

We give a brief introduction to the so-called general boundary formulation (GBF) of quantum theory. This new axiomatic formulation provides a description of the quantum dynamics which is manifestly local and does not rely on a metric background structure for its definition. We present the basic ingredients of the GBF, in particular we review the core axioms that assign algebraic structures to geometric ones, the two quantisation schemes so far developed for the GBF and the probability interpretation which generalizes the standard Born rule. Finally we briefly discuss some of the results obtained studying specific quantum field theories within the GBF.

Friction forces on atoms after acceleration

The aim of this paper is to revisit the calculation of atom–surface quantum friction in the quantum field theory formulation put forward by Barton (2010 New J. Phys. 12 113045). We show that the power dissipated into field excitations and the associated friction force depend on how the atom is boosted from being initially at rest to a configuration in which it is moving at constant velocity (v) parallel to the planar interface. In addition, we point out that there is a subtle cancellation between the one-photon and part of the two-photon dissipating power, resulting in a leading order contribution to the frictional power which goes as v4. These results are also confirmed by an alternative calculation of the average radiation force, which scales as v3.

Form factor for a two-particle system within a relativistic quasipotential approach: Case of arbitrary masses and of a vector current

A new relativistic form factor for a bound two-particle system was obtained for the case of a vector current. The present consideration was performed within the relativistic quasipotential approach based on the covariant Hamiltonian formulation of quantum field theory by going over to the three-dimensional relativistic configuration representation for the case of interaction between two relativistic spinless particles of arbitrary mass.

The origins of Schwinger׳s Euclidean Green׳s functions

This paper places Julian Schwinger׳s development of the Euclidean Green׳s function formalism for quantum field theory in historical context. It traces the techniques employed in the formalism back to Schwinger׳s work on waveguides during World War II, and his subsequent formulation of the Minkowski space Green׳s function formalism for quantum field theory in 1951. Particular attention is dedicated to understanding Schwinger׳s physical motivation for pursuing the Euclidean extension of this formalism in 1958.

Holographic entropy production

The suspicion that gravity is holographic has been supported mainly by a variety of specific examples from string theory. In this paper, we propose that such a holography can actually be observed in the context of Einstein's gravity and at least a class of generalized gravitational theories, based on a definite holographic principle where neither is the bulk space-time required to be asymptotically AdS nor the boundary to be located at conformal infinity, echoing Wilson's formulation of quantum field theory. After showing the general equilibrium thermodynamics from the corresponding holographic dictionary, in particular, we provide a rather general proof of the equality between the entropy production on the boundary and the increase of black hole entropy in the bulk, which can be regarded as strong support to this holographic principle. The entropy production in the familiar holographic superconductors/superfluids is investigated as an important example, where the role played by the holographic renormalization is explained.

Functional Properties of Hörmander’s Space of Distributions Having a Specified Wavefront Set

The space \({\mathcal{D}_\Gamma^\prime}\) of distributions having their wavefront sets in a closed cone \({\Gamma}\) has become important in physics because of its role in the formulation of quantum field theory in curved spacetime. In this paper, the topological and bornological properties of \({\mathcal{D}_\Gamma^\prime}\) and its dual \({\mathcal{E}_\Lambda^\prime}\) are investigated. It is found that \({\mathcal{D}_\Gamma^\prime}\) is a nuclear, semi-reflexive and semi-Montel complete normal space of distributions. Its strong dual \({\mathcal{E}_\Lambda^\prime}\) is a nuclear, barrelled and (ultra)bornological normal space of distributions which, however, is not even sequentially complete. Concrete rules are given to determine whether a distribution belongs to \({\mathcal{D}_\Gamma^\prime}\) , whether a sequence converges in \({\mathcal{D}_\Gamma^\prime}\) and whether a set of distributions is bounded in \({\mathcal{D}_\Gamma^\prime}\) .

Lorentz invariance and gauge equivariance

Trying to place Lorentz and gauge transformations on the same foundation, it turns out that the first one generates invariance, the second one equivariance, at least for the abelian case. This similarity is not a hypothesis but is supported by and a consequence of the path integral formalism in quantum field theory.

Hadrons on the worldline, holography, and Wilson flow

Holographic principles have impacted the way we look at strong coupling phenomena in quantum chromodynamics, strongly interacting extensions of the standard model, and {condensed-matter} physics. In real world settings, however, we still lack understanding of why and when such an approach is justified. Therefore, here, without invoking any such principle a priori, we demonstrate how such a picture arises in the worldline formulation of quantum field theory. Among other connections to holographic models, a warped AdS5 geometry, a quantum mechanical picture, and hidden local symmetry emerge, as well as a Wilson flow (gradient flow), which extends the four-dimensional sources to five-dimensional fields and a link to the Gutzwiller trace formula. The worldline formulation also reproduces the non-relativistic case, which is important for condensed-matter physics.

Form factor of a relativistic two-particle system within the relativistic quasipotential approach: Case of an arbitrary masses and a scalar current

A new relativistic form factor for a bound two-particle system was obtained for the case of a scalar current. The respective analysis was performed within the relativistic quasipotential approach based on a covariant Hamiltonian formulation of quantum field theory by going over to a three-dimensional relativistic configuration representation for the case of the interaction of two relativistic spinless particles that have arbitrary masses.

Metropolis Monte Carlo integration on the Lefschetz thimble: Application to a one-plaquette model

We propose a new algorithm based on the Metropolis sampling method to perform Monte Carlo integration for path integrals in the recently proposed formulation of quantum field theories on the Lefschetz thimble. The algorithm is based on a mapping between the curved manifold defined by the Lefschetz thimble of the full action and the flat manifold associated with the corresponding quadratic action. We discuss an explicit method to calculate the residual phase due to the curvature of the Lefschetz thimble. Finally, we apply this new algorithm to a simple one-plaquette model where our results are in perfect agreement with the analytic integration. We also show that for this system the residual phase does not represent a sign problem.

Monte Carlo simulations on the Lefschetz thimble: Taming the sign problem

We present the first practical Monte Carlo calculations of the recently proposed Lefschetz thimble formulation of quantum field theories. Our results provide strong evidence that the numerical sign problem that afflicts Monte Carlo calculations of models with complex actions can be softened significantly by changing the domain of integration to the Lefschetz thimble or approximations thereof. We study the interacting complex scalar field theory (relativistic Bose gas) in lattices of size up to 8^4 using a computationally inexpensive approximation of the Lefschetz thimble. Our results are in excellent agreement with known results. We show that - at least in the case of the relativistic Bose gas - the thimble can be systematically approached and the remaining residual phase leads to a much more tractable sign problem (if at all) than the original formulation. This is especially encouraging in view of the wide applicability - in principle - of our method to quantum field theories with a sign problem. We believe that this opens up new possibilities for accurate Monte Carlo calculations in strongly interacting systems of sizes much larger that previously possible.

Quantum field theory on timelike hypersurfaces in Rindler space

The general boundary formulation of quantum field theory is applied to a massive scalar field in two-dimensional Rindler space. The field is quantized according to both the Schr\"odinger-Feynman quantization prescription and the holomorphic one in two different spacetime regions: a region bounded by two Cauchy surfaces and a region bounded by one timelike curve. An isomorphism is constructed between the Hilbert spaces associated with these two boundaries. This isomorphism preserves the probabilities that can be extracted from the free and the interacting quantum field theories, proving the equivalence of the $S$-matrices defined in the two settings, when both apply.

Interparticle potentials in a scalar quantum field theory with a Higgs-like mediating field

We study the interparticle potentials for few-particle systems in a scalar theory with a nonlinear mediating field of the Higgs type. We use the variational method, in a reformulated Hamiltonian formalism of quantum field theory, to derive relativistic three- and four-particle wave equations for stationary states of these systems. We show that the cubic and quartic nonlinear terms modify the attractive Yukawa potentials but do not change the attractive nature of the interaction if the mediating fields are massive.