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.Comment: 8 pages, LaTeX + PoS, contribution to the proceedings of the conference "Frontiers of Fundamental Physics 14" (Marseille, 2014

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.Comment: 42 pages, LaTeX + AMS; v2: expanded to improve readability, new sections 3.1 (geometric data) and 3.3 (core axioms), minor corrections, update of references; v3: further update of reference

S-matrix at spatial infinity

We provide a new method to construct the S-matrix in quantum field theory. This method implements crossing symmetry manifestly by erasing the a priori distinction between in- and out-states. It allows the description of processes where the interaction weakens with distance in space, but remains strong in the center at all times. It should also be applicable to certain spacetimes where the conventional method fails due to lack of temporal asymptotic states.Comment: 4 pages, LaTeX + revtex4; v2: normalization factors corrected; v3: two paragraphs added, minor corrections and enhancements, reference list updated; v4: references corrected/update

Coherent states in fermionic Fock-Krein spaces and their amplitudes

We generalize the fermionic coherent states to the case of Fock-Krein spaces, i.e., Fock spaces with an idefinite inner product of Krein type. This allows for their application in topological or functorial quantum field theory and more specifically in general boundary quantum field theory. In this context we derive a universal formula for the amplitude of a coherent state in linear field theory on an arbitrary manifold with boundary.Comment: 20 pages, LaTeX + AMS + svmult (included), contribution to the proceedings of the conference "Coherent States and their Applications: A Contemporary Panorama" (Marseille, 2016); v2: minor corrections and added axioms from arXiv:1208.503

Renormalization of Discrete Models without Background

Conventional renormalization methods in statistical physics and lattice quantum field theory assume a flat metric background. We outline here a generalization of such methods to models on discretized spaces without metric background. Cellular decompositions play the role of discretizations. The group of scale transformations is replaced by the groupoid of changes of cellular decompositions. We introduce cellular moves which generate this groupoid and allow to define a renormalization groupoid flow. We proceed to test our approach on several models. Quantum BF theory is the simplest example as it is almost topological and the renormalization almost trivial. More interesting is generalized lattice gauge theory for which a qualitative picture of the renormalization groupoid flow can be given. This is confirmed by the exact renormalization in dimension two. A main motivation for our approach are discrete models of quantum gravity. We investigate both the Reisenberger and the Barrett-Crane spin foam model in view of their amenability to a renormalization treatment. In the second case a lack of tunable local parameters prompts us to introduce a new model. For the Reisenberger and the new model we discuss qualitative aspects of the renormalization groupoid flow. In both cases quantum BF theory is the UV fixed point.Comment: 40 pages, 17 figures, LaTeX + AMS + XY-pic + eps; added subsection 4.3 on relation to spin network diagrams, reference added, minor adjustment

Structure Theorem for Covariant Bundles on Quantum Homogeneous Spaces

The natural generalization of the notion of bundle in quantum geometry is that of bimodule. If the base space has quantum group symmetries one is particularly interested in bimodules covariant (equivariant) under these symmetries. Most attention has so far been focused on the case with maximal symmetry -- where the base space is a quantum group and the bimodules are bicovariant. The structure of bicovariant bimodules is well understood through their correspondence with crossed modules. We investigate the ``next best'' case -- where the base space is a quantum homogeneous space and the bimodules are covariant. We present a structure theorem that resembles the one for bicovariant bimodules. Thus, there is a correspondence between covariant bimodules and a new kind of ``crossed'' modules which we define. The latter are attached to the pair of quantum groups which defines the quantum homogeneous space. We apply our structure theorem to differential calculi on quantum homogeneous spaces and discuss a related notion of induced differential calculus.Comment: 7 pages; talk given at QGIS X, Prague, June 2001; typos correcte

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.Comment: 30 pages, 5 figures, LaTeX; v2: typos corrected, footnote and remark added, references added/updated; v3: more typos corrected; v4: with corrections of the published versio

Schr\"odinger-Feynman quantization and composition of observables in general boundary quantum field theory

We show that the Feynman path integral together with the Schr\"odinger 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.Comment: 47 pages, LaTeX + AMS; v2: minor corrections, references update

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 K\"ahler 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