Algebraic Quantum Field Theory Research Articles

EPR States and Bell Correlated States in Algebraic Quantum Field Theory

A mathematical rigorous definition of EPR states has been introduced by Arens and Varadarajan for finite dimensional systems, and extended by Werner to general systems. In the present paper we follow a definition of EPR states due to Werner. Then we show that an EPR state for incommensurable pairs is Bell correlated, and that the set of EPR states for incommensurable pairs is norm dense between two strictly space-like separated regions.

On Particle Phenomenology Without Particle Ontology: How Much Local Is Almost Local?

Recently, Clifton and Halvorson have tried to salvage a particle phenomenology in the absence of particle ontology within algebraic relativistic quantum field theory. Their idea is that the detection of a particle is the measurement of a local observable which simulates the measurement of an almost local observable that annihilates the vacuum. In this note, we argue that the measurements local particle detections are supposed to simulate probe radically holistic aspects of relativistic quantum fields. We prove that in an axiomatic (Haag-Araki) quantum field theory on Minkowski spacetime, formulated in a Hilbert space \(\mathcal{H}\), there is no positive observable C, with norm less than or equal to 1, satisfying the conditions: (1) the expectation value of C in the vacuum state Ω is zero, (2) there is at least one vector state Ψ in \(\mathcal{H}\) in which the expectation value of C is different from zero, and (3) there exists at least one spacetime region \(\mathcal{O}\) such that the non-selective measurement of C leaves the expectation values of all observables in the local algebra of that region unaltered regardless of the state the system is in. The result reveals a tension between intuitions regarding localization and intuitions regarding causality: to save “particle phenomena” in the absence of particle ontology, one has to feign “particle” detectors with “good” properties as to locality but “bad” behavior as to causality.

Noncommutative Common Cause Principles in algebraic quantum field theory

States in algebraic quantum field theory “typically” establish correlation between spacelike separated events. Reichenbach's Common Cause Principle, generalized to the quantum field theoretical setting, offers an apt tool to causally account for these superluminal correlations. In the paper we motivate first why commutativity between the common cause and the correlating events should be abandoned in the definition of the common cause. Then we show that the Noncommutative Weak Common Cause Principle holds in algebraic quantum field theory with locally finite degrees of freedom. Namely, for any pair of projections A, B supported in spacelike separated regions VA and VB, respectively, there is a local projection C not necessarily commuting with A and B such that C is supported within the union of the backward light cones of VA and VB and the set {C, C⊥} screens off the correlation between A and B.

Why is the top-quark much heavier than other fermions?

The recent ATLAS and CMS experiments show the first observations of a new particle in the search for the Standard Model Higgs boson at the LHC. We revisit the theoretical inconsistency of the fundamental high-energy cutoff with the parity-violating gauge symmetry of local quantum field theory for the Standard Model. This inconsistency suggests high-dimensional operators of fermion interactions, which are attributed to the quantum gravity. In this Letter, recalling the minimal dynamical symmetry breaking mechanism, we show that it is energetically favorable for the top-quark to acquire its mass via spontaneous symmetry breaking, whereas other fermions acquire their masses via explicit symmetry breaking.

ON NONEQUILIBRIUM PHYSICS AND STRING THEORY

In this paper, we review the relation between string theory and nonequilibrium physics based on our previously published work. First, we explain why a theory of quantum gravity and nonequilibrium statistical physics should be related in the first place. Then, we present the necessary background from the recent research in nonequilibrium physics. The review discusses the relationship of string theory and aging phenomena, as well as the connection between AdS/CFT correspondence and the Jarzynski identity. We also discuss the emergent symmetries in fully developed turbulence and the corresponding nonequilibrium stationary states. Finally, we outline a larger picture regarding the relationship between nonperturbative quantum gravity and nonequilibrium statistical physics. This relationship can be understood as a natural generalization of the well-known Wilsonian relation between local quantum field theory and equilibrium statistical physics of critical phenomena. According to this picture, the AdS/CFT duality is just an example of a more general connection between nonperturbative quantum gravity and nonequilibrium physics. In the appendix of this review, we discuss a new kind of complementarity between thermodynamics and statistical physics which should be important in the context of black hole complementarity.

QUANTIZATION OF LINEARIZED GRAVITY IN COSMOLOGICAL VACUUM SPACETIMES

Linearized Einstein gravity (with possibly non-zero cosmological constant) is quantized in the framework of algebraic quantum field theory by analogy with Dimock's treatment of electromagnetism [Rev. Math. Phys.4 (1992) 223–233]. To achieve this, the classical theory is developed in a full, rigorous and systematic fashion, with particular attention given to the circumstances under which the symplectic product is weakly non-degenerate and to the related question of whether the space of solutions is separated by the classical observables on which the quantum theory is modeled.

Tunnelling of the 3rd kind: A test of the effective non-locality of quantum field theory

Integrating out virtual quantum fluctuations in an originally local quantum field theory results in an effective theory which is non-local. In this letter we argue that tunnelling of the 3rd kind —where particles traverse a barrier by splitting into a pair of virtual particles which recombine only after a finite distance— provides a direct test of this non-locality. We sketch a quantum-optical setup to test this effect, and investigate observable effects in a simple toy model.

The Einstein-Jordan conundrum and its relation to ongoing foundational research in local quantum physics

We demonstrate the extraordinary modernity of the 1924/25 "Einstein-Jordan fluctuation conundrum", a Gedankenexperiment which led Jordan to his quantization of waves published as a separate section in the famous Born-Heisenberg-Jordan 1926 "Dreim\"annerarbeit". The thermal nature of energy fluctuations caused by the restriction of the QFT vacuum to a subvolume remained unnoticed mainly because it is not present in QM. In order to understand the analogy with Einstein's fluctuation calculation in a thermal black body system, it is important to expose the mechanism which causes a global vacuum state to become impure on a localized subalgebra of QFT. The present work presents the fascinating history behind this problem which culminated in the more recent perception that "causal localization" leads to thermal manifestations. The most appropriate concept which places this property of QFT into the forefront is "modular localization". These new developments in QFT led to a new access to the existence problem for interacting quantum fields whose solution has remained outside the range of renormalized perturbation theory. It also clarifies open problems about the relation of particles and fields in particular about the incompletely understood crossing property. Last not least it leads to a constructive understanding of integrable versus non-integrable QFTs..

Quantum Steering and Spacelike Separation

In nonrelativistic quantum mechanics, measurements performed by separate observers are modeled via tensor products. In algebraic quantum field theory, though, local observables corresponding to spacelike separated parties are just required to commute. The problem of determining whether these two definitions of separation lead to the same set of bipartite correlations is known in nonlocality as Tsirelson's problem. In this article, we prove that the analog of Tsirelson's problem in steering scenarios is false. That is, there exists a steering inequality that can or cannot be violated depending on how we define spacelike separation at the operator level.

Bohrification of local nets

Recent results by Spitters et. al. suggest that quantum phase space can usefully be regarded as a ringed topos via a process called Bohrification. They show that quantum kinematics can then be interpreted as classical kinematics, internal to this ringed topos. We extend these ideas from quantum mechanics to algebraic quantum field theory: from a net of observables we construct a presheaf of quantum phase spaces. We can then naturally express the causal locality of the net as a descent condition on the corresponding presheaf of ringed toposes: we show that the net of observables is local, precisely when the presheaf of ringed toposes satisfies descent by a local geometric morphism.

Renormalization in the three-body problem with resonantp-wave interactions

Resonant P-wave interactions can be described by a minimal zero-range model defined by a truncated effective range expansion, so that the only 2-body interaction parameters are the inverse scattering volume 1/a_P and the P-wave effective range r_P. This minimal model can be formulated as a local quantum field theory with a P-wave interaction between atom fields and a molecular field. In the two-atom sector, the model is renormalizable, but it has unphysical behavior at high energies, because there are negative-probability states with momentum scale r_P. In the sector with three atoms, two of which are identical, renormalization in some parity and angular-momentum channels involves an ultraviolet limit cycle, indicating asymptotic discrete scale invariance. The Efimov effect occurs in the unitary limit a_P^(-1/3), r_P \to 0, but this limit is unphysical because there are low-energy states with negative probability. The minimal model can be of physical relevance only at energies small compared to the energy scale set by r_P, where the effects of negative-probability states are suppressed.

Entropy bound with generalized uncertainty principle in general dimensions

In this letter, the entropy bound for local quantum field theories (LQFT) is studied in a class of models of the generalized uncertainty principle (GUP) which predicts a minimal length as a reflection of the quantum gravity effects. Both bosonic and fermionic fields confined in an arbitrary spatial dimension d ⩾ 4 ball are investigated. It is found that the GUP leads to the same scaling A(d−3)/(d−2)d−2 correction to the entropy bound for bosons and fermions, although the coefficients of this correction are different for each case. Based on our calculation, we conclude that the GUP effects can become manifest at the short distance scale. Some further implications and speculations of our results are also discussed.

Black holes, quantum information, and unitary evolution

The unitary crisis for black holes indicates an apparent need to modify local quantum field theory. This paper explores the idea that quantum mechanics and in particular unitarity are fundamental principles, but at the price of familiar locality. Thus, one should seek to parameterize unitary evolution, extending the field theory description of black holes, such that their quantum information is transferred to the external state. This discussion is set in a broader framework of unitary evolution acting on Hilbert spaces comprising subsystems. Here, various constraints can be placed on the dynamics, based on quantum information-theoretic and other general physical considerations, and one can seek to describe dynamics with "minimal" departure from field theory. While usual spacetime locality may not be a precise concept in quantum gravity, approximate locality seems an important ingredient in physics. In such a Hilbert space approach an apparently "coarser" form of localization can be described in terms of tensor decompositions of the Hilbert space of the complete system. This suggests a general framework in which to seek a consistent description of quantum gravity, and approximate emergence of spacetime. Other possible aspects of such a framework -- in particular symmetries -- are briefly discussed.

Extended analytic QCD model with perturbative QCD behavior at high momenta

In contrast to perturbative QCD, the analytic QCD models have running coupling whose analytic properties correctly mirror those of spacelike observables. The discontinuity (spectral) function of such running coupling is expected to agree with the perturbative case at large timelike momenta; however, at low timelike momenta, it is not known. In the latter regime, we parametrize the unknown behavior of the spectral function as a sum of (two) delta functions; while the onset of the perturbative behavior of the spectral function is set to be $1.0--1.5\text{ }\text{ }\mathrm{GeV}$. This is in close analogy with the ``minimal hadronic ansatz'' used in the literature for modeling spectral functions of correlators. For the running coupling itself, we impose the condition that it basically merges with the perturbative coupling at high spacelike momenta. In addition, we require that the well-measured nonstrange semihadronic ($V+A$) tau decay ratio value be reproduced by the model. We thus obtain a QCD framework which is basically indistinguishable from perturbative QCD at high momenta ($Q>1\text{ }\text{ }\mathrm{GeV}$), and at low momenta, it respects the basic analyticity properties of spacelike observables as dictated by the general principles of the local quantum field theories.

Distinguishing causality principles

We distinguish two sub-types of each of the two causality principles formulated in connection with the Common Cause Principle in Henson (2005) and raise and investigate the problem of logical relations among the resulting four causality principles. Based in part on the analysis of the status of these four principles in algebraic quantum field theory we will argue that the four causal principles are non-equivalent.

Radiation Entropy Bound from the Second Law of Thermodynamics

It has been suggested heuristically by Unruh and Wald, and independently by Page, that at given energy and volume, thermal radiation has the largest entropy. The suggestion leads to the corresponding universal bound on entropy of physical systems. Using a gedanken experiment we show that the bound follows from the Second Law of Thermodynamics if the CPT symmetry is assumed and a general condition on matter holds. The experiment suggests that a wide class of Lorentz invariant local quantum field theories obeys a bound on the density of states.

Infraparticles with Superselected Direction of Motion in Two-Dimensional Conformal Field Theory

Particle aspects of two-dimensional conformal field theories are investigated, using methods from algebraic quantum field theory. The results include asymptotic completeness in terms of (counterparts of) Wigner particles in any vacuum representation and the existence of (counterparts of) infraparticles in any charged irreducible product representation of a given chiral conformal field theory. Moreover, an interesting interplay between the infraparticle’s direction of motion and the superselection structure is demonstrated in a large class of examples. This phenomenon resembles the electron’s momentum superselection expected in quantum electrodynamics.

Causal posets, loops and the construction of nets of local algebras for QFT

We provide a model independent construction of a net of C*-algebras satisfying the Haag-Kastler axioms over any spacetime manifold. Such a net, called the net of causal loops, is constructed by selecting a suitable base K encoding causal and symmetry properties of the spacetime. Considering K as a partially ordered set (poset) with respect to the inclusion order relation, we define groups of closed paths (loops) formed by the elements of K. These groups come equipped with a causal disjointness relation and an action of the symmetry group of the spacetime. In this way the local algebras of the net are the group C*-algebras of the groups of loops, quotiented by the causal disjointness relation. We also provide a geometric interpretation of a class of representations of this net in terms of causal and covariant connections of the poset K. In the case of the Minkowski spacetime, we prove the existence of Poincar\'e covariant representations satisfying the spectrum condition. This is obtained by virtue of a remarkable feature of our construction: any Hermitian scalar quantum field defines causal and covariant connections of K. Similar results hold for the chiral spacetime $S^1$ with conformal symmetry.

Towards a Construction of Inclusive Collision Cross-Sections in the Massless Nelson Model

The conventional approach to the infrared problem in perturbative quantum electrodynamics relies on the concept of inclusive collision cross-sections. A non-perturbative variant of this notion was introduced in algebraic quantum field theory. Relying on these insights, we take first steps towards a non-perturbative construction of inclusive collision cross-sections in the massless Nelson model. We show that our proposal is consistent with the standard scattering theory in the absence of the infrared problem and discuss its status in the infrared-singular case.

Reconstructing Bohr’s Reply to EPR in Algebraic Quantum Theory

Halvorson and Clifton have given a mathematical reconstruction of Bohr's reply to Einstein, Podolsky and Rosen (EPR), and argued that this reply is dictated by the two requirements of classicality and objectivity for the description of experimental data, by proving consistency between their objectivity requirement and a contextualized version of the EPR reality criterion which had been introduced by Howard in his earlier analysis of Bohr's reply. In the present paper, we generalize the above consistency theorem, with a rather elementary proof, to a general formulation of EPR states applicable to both non-relativistic quantum mechanics and algebraic quantum field theory; and we clarify the elements of reality in EPR states in terms of Bohr's requirements of classicality and objectivity, in a general formulation of algebraic quantum theory.