This work is devoted to incorporating into QFT the notion that particles and hence the particle states should be localizable in space. It focuses on the case of the Dirac field in 1+1 dimensional flat spacetime, generalizing a recently developed formalism for scalar fields. This is achieved exploiting again the non-uniqueness of quantization process. Instead of elementary excitations carrying definite amounts of energy and momentum, we construct the elementary excitations of the field localized (at some instant) in a definite region of space. This construction not only leads to a natural notion of localized quanta, but also provides a local algebra of operators. Once constructed, the new representation is confronted to the standard global (Fock) construction. In spite of being unitarily inequivalent representations, the localized operators are well defined in the conventional Fock space. By using them we dig up the issues of localization in QFT showing how the globality of the vacuum state is responsible for the lack of a "common sense localization" notion for particles in the standard Fock representation of QFT and propose a method to meet its requirements.
This work is devoted to incorporating into QFT the notion that particles and hence the particle states should be localizable in space. It focuses on the case of the Dirac field in 1+1 dimensional flat spacetime, generalizing a recently developed formalism for scalar fields. This is achieved exploiting again the non-uniqueness of quantization process. Instead of elementary excitations carrying definite amounts of energy and momentum, we construct the elementary excitations of the field localized (at some instant) in a definite region of space. This construction not only leads to a natural notion of localized quanta, but also provides a local algebra of operators. Once constructed, the new representation is confronted to the standard global (Fock) construction. In spite of being unitarily inequivalent representations, the localized operators are well defined in the conventional Fock space. By using them we dig up the issues of localization in QFT showing how the globality of the vacuum state is responsible for the lack of a "common sense localization" notion for particles in the standard Fock representation of QFT and propose a method to meet its requirements.
Non-locality and quantum appear together entitling scientific projects and papers quite usually.These are two hardly detachable concepts and modern quantum theories are built up taking into account this aspect explicitly.However, that characteristic turns challenging from a fundamental point of view and can be unhelpful for practical purposes.Here, we construct quantum field theory with deliberately local aspects for a 1+1 dimensional Dirac field in flat spacetime generalizing a recently developed formalism for scalar fields.That purpose is achieved exploiting the non-uniqueness of quantization process.This local construction leads to a natural notion of local quanta and provides a local algebra of operators.As an application, it is shown that the standard vacuum state is a boiling soup of local particles and this fact is connected with a still Gedankenexperiment consisting in slamming down and removing a mirror in an empty cavity. I. MOTIVATION AND BACKGROUNDNon-locality is a feature as intuitively challenging as genuine in the quantum world.The so-successful ordinary representation of quantum field theory (QFT) is based on the information carried by global operators (creation-annihilation operators) related to field excitations with well defined momentum, and therefore, completely delocalized in space.This picture accounts for an innumerable amount of phenomena with an astonishing accuracy, as particle physics or solid state physics shows.However, this conventional formalism can hardly deal with tasks related to local properties of quantum field states.Simple questions as "where is that particle?"or "how are these particles distributed in space?" are unsuccessfully answered in this background.What is more, it is difficult to reconcile the common understanding of a particle as a minute entity in space with the particle notion arising from the ordinary Fock construction, which rather describes excitations spreading in whole space.Furthermore, no localized states can be constructed in such QFT where occupation number representation of a quantum field can be understood as a "momentum" representation in the sense that the number of particles informs about the momentum carried by the field.The problem at this point is that it does not exist a reciprocal "position" representation, we have no local (in space) creation-annihilation operators.This result can be understood as a consequence of Hegerfeldt's theorem.
In quantum time (QT) schemes, time is promoted to a degree of freedom, allowing Lorentz covariance to be made explicit for single particles. We ask whether this can be lifted to QFT, so that Lorentz covariance becomes manifest at the Hilbert-space level, rather than being hidden as in the standard canonical formulation. We address this question by proposing a second-quantized approach in which the elementary particle is the QT particle itself, leading naturally to the notion of spacetime field algebras and of quantum action. We show, however, that a naive many-body construction runs into inconsistencies. To pinpoint their origin we introduce a classical counterpart of the second-quantized formalism, spacetime classical mechanics (SCM), and prove a no-go theorem: Dirac quantization of SCM collapses back to standard QFT and therefore hides covariance. We circumvent this problem by presenting a quantum-action-based quantization that yields a spacetime version of quantum mechanics (SQM), making covariance manifest for (interacting) QFTs. Finally, we show that this resolution is tied to a genuine spacetime generalization of the notion of quantum state, required by causality and closely connected to recent ``states over time'' proposals and, in dS/CFT-motivated settings, to microscopic notions of timelike entanglement and emergent time.
There are significant problems involved in determining the ontology of quantum field theory (QFT). An ontology involving particles seems to be ruled out due to the problem of defining localized position operators, issues involving interactions in QFT, and, perhaps, the appearance of unitarily inequivalent representations. While this might imply that fields are the most natural ontology for QFT, the wavefunctional interpretation of QFT has significant drawbacks. A modified field ontology is examined where determinables are assigned to open bounded regions of spacetime instead of spacetime points.
The diffculties of relativistic particle theories formulated my means of canonical quantization, such as Klein-Gordon and Dirac theories, ultimately led theoretical physicists to turn on quantum field theory to model elementary particle physics. The aim of the present work is to pursue a method alternative to canonical quantization that avoids these dfficulties. In order to guarantee this result, the present approach is constrained to be developed deductively from physical principles. The physical principles assumed for a free particle consist of the symmetry properties of the particle with respect to the Poincaré group and of the transformation properties of the position observable, expressed by means of a suitably conceived notion of quantum transformation. In so doing, the effectiveness of group theoretical methods is exploited. Our work has pointed out the necessity of new classes of irreducible representations of the Poincaré group the theory can be based on. For spin 0 particle, four inequivalent theories are completely determined, that differ from Klein-Gordon theory for the physical interpretation but also for the mathematical structures. In particular, two of these theories requires irreducible representations in the new classes. The present lack of the explicit transformation properties of position with respect to boosts forbids the complete determination of non zero spin particle theories. A particular form of these transformation properties was in past adopted by Jordan and Mukunda. We check its consistency within the present approach; it is found that for spin 1/2 particles there is only one consistent theory; it is unitarily related to Dirac theory, but, once again, it requires the new classes of irreducible representations.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.
We construct the algebra of operators acting on the Hilbert spaces of Quantum Mechanics for systems of N identical particles from the field operators acting in the Fock space of Quantum Field Theory by providing the explicit relation between the position and momentum operators acting in the former spaces and the field operators acting on the latter.This is done in the context of the non-interacting Klein-Gordon field.It may not be possible to extend the procedure to interacting field theories since it relies crucially on particle number conservation.We find it nevertheless important that such an explicit relation can be found at least for free fields.It also comes out that whatever statistics the field operators obey (either commuting or anticommuting), the position and momentum operators obey commutation relations.The construction of position operators raises the issue of localizability of particles in Relativistic Quantum Mechanics, as the position operator for a single particle turns out to be the Newton-Wigner position operator.We make some clarifications on the interpretation of Newton-Wigner localized states and we consider the transformation properties of position operators under Lorentz transformations, showing that they do not transform as tensors, rather in a manner that preserves the canonical commutation relations.