arXiv subjects

Robert Oeckl

Publications and source records attributed to Robert Oeckl.

At least 37 records · Page 2Linked to original sources

arXiv2016

Towards state locality in quantum field theory: free fermions

Hilbert spaces of states can be constructed in standard quantum field theory only for infinitely extended spacelike hypersurfaces, precluding a more local notion of state. In fact, the Reeh-Schlieder Theorem prohibits the localization of states on pieces of hypersurfaces in the standard formalism. From the point of view of geometric quantization the problem lies in the non-locality of the complex structures associated to hypersurfaces in standard quantization. We show that using a weakened version of the positive formalism puts this problem into a new perspective. This is a local TQFT type formalism based on super-operators and mixed state spaces rather than on amplitudes and pure state spaces as the one of Atiyah-Segal. In particular, we show that in the case of purely fermionic degrees of freedom the complex structure can be dispensed with when the notion of state is suitably generalized. These generalized states do localize on compact hypersurfaces with boundaries. For the simplest case of free fermionic fields we embed this in a rigorous and functorial quantization scheme yielding a local description of the quantum theory. Crucially, no classical data is needed beyond the structures evident from a Lagrangian setting. When the classical data is augmented with complex structures on hypersurfaces, the quantum data correspondingly augment to the full positive formalism. This scheme is applicable to field theory in curved spacetime, but also to field theories without metric background.

hep-th↗

arXiv2015

Schrödinger-Feynman quantization and composition of observables in general boundary quantum field theory

We show that the Feynman path integral together with the Schrödinger representation gives rise to a rigorous and functorial quantization scheme for linear and affine field theories. Since our target framework is the general boundary formulation, the class of field theories that can be quantized in this way includes theories without a metric spacetime background. We also show that this quantization scheme is equivalent to a holomorphic quantization scheme proposed earlier and based on geometric quantization. We proceed to include observables into the scheme, quantized also through the path integral. We show that the quantized observables satisfy the canonical commutation relations, a feature shared with other quantization schemes also discussed. However, in contrast to other schemes the presented quantization also satisfies a correspondence between the composition of classical observables through their product and the composition of their quantized counterparts through spacetime gluing. In the special case of quantum field theory in Minkowski space this reproduces the operationally correct composition of observables encoded in the time-ordered product. We show that the quantization scheme also generalizes other features of quantum field theory such as the generating function of the S-matrix.

hep-th↗

arXiv2015

Complex structures for an S-matrix of Klein-Gordon theory on AdS spacetimes

While the standard construction of the S-matrix fails on Anti-de Sitter (AdS) spacetime, a generalized S-matrix makes sense, based on the hypercylinder geometry induced by the boundary of AdS. In contrast to quantum field theory in Minkowski spacetime, there is not yet a standard way to resolve the quantization ambiguities arising in its construction. These ambiguities are conveniently encoded in the choice of a complex structure. We explore in this paper the space of complex structures for real scalar Klein-Gordon theory based on a number of criteria. These are: invariance under AdS isometries, induction of a positive definite inner product, compatibility with the standard S-matrix picture and recovery of standard structures in Minkowski spacetime under a limit of vanishing curvature. While there is no complex structure that satisfies all demands, we emphasize two interesting candidates that satisfy most: In one case we have to give up part of the isometry invariance, in the other case the induced inner product is indefinite.

hep-th↗

arXiv2015

Coherent states in the fermionic Fock space

We construct the coherent states in the sense of Gilmore and Perelomov for the fermionic Fock space. Our treatment is from the outset adapted to the infinite-dimensional case. The fermionic Fock space becomes in this way a reproducing kernel Hilbert space of continuous holomorphic functions.

math-ph↗

arXiv2014

A first-principles approach to physics based on locality and operationalism

Starting from the guiding principles of spacetime locality and operationalism, a general framework for a probabilistic description of nature is proposed. Crucially, no notion of time or metric is assumed, neither any specific physical model. Remarkably, the emerging framework converges with a recently proposed formulation of quantum theory, obtained constructively from known quantum physics. At the same time the framework also admits statistical theories of classical physics.

quant-ph↗

arXiv2013

A positive formalism for quantum theory in the general boundary formulation

We introduce a new "positive formalism" for encoding quantum theories in the general boundary formulation, somewhat analogous to the mixed state formalism of the standard formulation. This makes the probability interpretation more natural and elegant, eliminates operationally irrelevant structure and opens the general boundary formulation to quantum information theory.

quant-ph↗

arXiv2013

Free Fermi and Bose Fields in TQFT and GBF

We present a rigorous and functorial quantization scheme for linear fermionic and bosonic field theory targeting the topological quantum field theory (TQFT) that is part of the general boundary formulation (GBF). Motivated by geometric quantization, we generalize a previous axiomatic characterization of classical linear bosonic field theory to include the fermionic case. We proceed to describe the quantization scheme, combining a Fock space quantization for state spaces with the Feynman path integral for amplitudes. We show rigorously that the resulting quantum theory satisfies the axioms of the TQFT, in a version generalized to include fermionic theories. In the bosonic case we show the equivalence to a previously developed holomorphic quantization scheme. Remarkably, it turns out that consistency in the fermionic case requires state spaces to be Krein spaces rather than Hilbert spaces. This is also supported by arguments from geometric quantization and by the explicit example of the Dirac field theory. Contrary to intuition from the standard formulation of quantum theory, we show that this is compatible with a consistent probability interpretation in the GBF. Another surprise in the fermionic case is the emergence of an algebraic notion of time, already in the classical theory, but inherited by the quantum theory. As in earlier work we need to impose an integrability condition in the bosonic case for all TQFT axioms to hold, due to the gluing anomaly. In contrast, we are able to renormalize this gluing anomaly in the fermionic case.

math-ph↗

arXiv2012

Reverse Engineering Quantum Field Theory

An approach to the foundations of quantum theory is advertised that proceeds by "reverse engineering" quantum field theory. As a concrete instance of this approach, the general boundary formulation of quantum theory is outlined.

quant-ph↗

arXiv2012

The Schrödinger representation and its relation to the holomorphic representation in linear and affine field theory

We establish a precise isomorphism between the Schrödinger representation and the holomorphic representation in linear and affine field theory. In the linear case this isomorphism is induced by a one-to-one correspondence between complex structures and Schrödinger vacua. In the affine case we obtain similar results, with the role of the vacuum now taken by a whole family of coherent states. In order to establish these results we exhibit a rigorous construction of the Schrödinger representation and use a suitable generalization of the Segal-Bargmann transform. Our construction is based on geometric quantization and applies to any real polarization and its pairing with any Kähler polarization.

math-ph↗

arXiv2012

Affine holomorphic quantization

We present a rigorous and functorial quantization scheme for affine field theories, i.e., field theories where local spaces of solutions are affine spaces. The target framework for the quantization is the general boundary formulation, allowing to implement manifest locality without the necessity for metric or causal background structures. The quantization combines the holomorphic version of geometric quantization for state spaces with the Feynman path integral quantization for amplitudes. We also develop an adapted notion of coherent states, discuss vacuum states, and consider observables and their Berezin-Toeplitz quantization. Moreover, we derive a factorization identity for the amplitude in the special case of a linear field theory modified by a source-like term and comment on its use as a generating functional for a generalized S-matrix.

hep-th↗

arXiv2012

Observables in the General Boundary Formulation

We develop a notion of quantum observable for the general boundary formulation of quantum theory. This notion is adapted to spacetime regions rather than to hypersurfaces and naturally fits into the topological quantum field theory like axiomatic structure of the general boundary formulation. We also provide a proposal for a generalized concept of expectation value adapted to this type of observable. We show how the standard notion of quantum observable arises as a special case together with the usual expectation values. We proceed to introduce various quantization schemes to obtain such quantum observables including path integral quantization (yielding the time-ordered product), Berezin-Toeplitz (antinormal ordered) quantization and normal ordered quantization, and discuss some of their properties.

hep-th↗

arXiv2012

Holomorphic Quantization of Linear Field Theory in the General Boundary Formulation

We present a rigorous quantization scheme that yields a quantum field theory in general boundary form starting from a linear field theory. Following a geometric quantization approach in the Kähler case, state spaces arise as spaces of holomorphic functions on linear spaces of classical solutions in neighborhoods of hypersurfaces. Amplitudes arise as integrals of such functions over spaces of classical solutions in regions of spacetime. We prove the validity of the TQFT-type axioms of the general boundary formulation under reasonable assumptions. We also develop the notions of vacuum and coherent states in this framework. As a first application we quantize evanescent waves in Klein-Gordon theory.

hep-th↗

arXiv2012

Probabilities in the general boundary formulation

We give an introductory account of the general boundary formulation of quantum theory. We refine its probability interpretation and emphasize a conceptual and historical perspective. We give motivations from quantum gravity and illustrate them with a scenario for describing gravitons in quantum gravity.

hep-th↗

arXiv2011

S-Matrix for AdS from General Boundary QFT

The General Boundary Formulation (GBF) is a new framework for studying quantum theories. After concise overviews of the GBF and Schrödinger-Feynman quantization we apply the GBF to resolve a well known problem on Anti-deSitter spacetime where due to the lack of temporally asymptotic free states the usual S-matrix cannot be defined. We construct a different type of S-matrix plus propagators for free and interacting real Klein-Gordon theory.

hep-th↗

arXiv2011

On Unitary Evolution in Quantum Field Theory in Curved Spacetime

We investigate the question of unitarity of evolution between hypersurfaces in quantum field theory in curved spacetime from the perspective of the general boundary formulation. Unitarity thus means unitarity of the quantum operator that maps the state space associated with one hypersurface to the state space associated with the other hypersurface. Working in Klein-Gordon theory, we find that such an evolution is generically unitary given a one-to-one correspondence between classical solutions in neighborhoods of the respective hypersurfaces. This covers the case of pairs of Cauchy hypersurfaces, but also certain cases where hypersurfaces are timelike. The tools we use are the Schroedinger representation and the Feynman path integral.

hep-th↗

arXiv2009

States and amplitudes for finite regions in a two-dimensional Euclidean quantum field theory

We quantize the Helmholtz equation (plus perturbative interactions) in two dimensions to illustrate a manifestly local description of quantum field theory. Using the general boundary formulation we describe the quantum dynamics both in a traditional time evolution setting as well as in a setting referring to finite disk (or annulus) shaped regions of spacetime. We demonstrate that both descriptions are equivalent when they should be.

hep-th↗

arXiv2009

General boundary quantum field theory: Foundations and probability interpretation

We elaborate on the proposed general boundary formulation as an extension of standard quantum mechanics to arbitrary (or no) backgrounds. Temporal transition amplitudes are generalized to amplitudes for arbitrary spacetime regions. State spaces are associated to general (not necessarily spacelike) hypersurfaces. We give a detailed foundational exposition of this approach, including its probability interpretation and a list of core axioms. We explain how standard quantum mechanics arises as a special case. We include a discussion of probability conservation and unitarity, showing how these concepts are generalized in the present framework. We formulate vacuum axioms and incorporate spacetime symmetries into the framework. We show how the Schroedinger-Feynman approach is a suitable starting point for casting quantum field theories into the general boundary form. We discuss the role of operators.

hep-th↗

arXiv2008

Hopf algebras and the combinatorics of connected graphs in quantum field theory

In this talk, we are concerned with the formulation and understanding of the combinatorics of time-ordered n-point functions in terms of the Hopf algebra of field operators. Mathematically, this problem can be formulated as one in combinatorics or graph theory. It consists in finding a recursive algorithm that generates all connected graphs in their Hopf algebraic representation. This representation can be used directly and efficiently in evaluating Feynman graphs as contributions to the n-point functions.

math-ph↗