We describe the free Dirac field in a four-dimensional spacetime as a locally covariant quantum field theory in the sense of Brunetti, Fredenhagen and Verch, using a representation independent construction. The freedom in the geometric constructions involved can be encoded in terms of the cohomology of the category of spin spacetimes. If we restrict ourselves to the observable algebra, the cohomological obstructions vanish and the theory is unique. We establish some basic properties of the theory and discuss the class of Hadamard states, filling some technical gaps in the literature. Finally, we show that the relative Cauchy evolution yields commutators with the stress-energy-momentum tensor, as in the scalar field case.

The locally covariant Dirac field

The main motivation for this book lies in the breadth of applications
in which a statistical model is used to represent small departures
from, for example, a Poisson process. Our approach uses information geometry to provide
a common context but we need only rather elementary material from differential geometry, information theory and mathematical statistics. Introductory
sections serve together to help those
interested from the applications side in making use of our methods
and results.

Reported in this monograph is a body of results, and
computer-algebraic methods that seem to have quite general
applicability to statistical models admitting representation
through parametric families of probability density functions. Some
illustrations are given from a variety of contexts for geometric
characterization of statistical states near to the three important standard
basic reference states: (Poisson) randomness, uniformity, independence. The
individual applications are somewhat heuristic
models from various fields and we incline more to terminology and notation
from the applications rather than from formal statistics. However, a
common thread is a geometrical representation for statistical perturbations of the
basic standard states, and hence results gain qualitative stability.
Moreover, the geometry is controlled by a metric structure that
owes its heritage through maximum likelihood to information theory
so the quantitative features---lengths of curves, geodesics,
scalar curvatures etc.---have some respectable authority. We
see in the applications simple models for galactic
void distributions and galaxy clustering, amino acid clustering
along protein chains, cryptographic protection, stochastic fibre
networks, coupled geometric features in hydrology and quantum chaotic behaviour.

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Information Geometry: Near Randomness and Near Independence

A generalized symplectic structure on the bundle of connections 
$p\colon C(P)\rightarrow M$
 of an arbitrary principal G-bundle 
$\pi\colon P\rightarrow M$
 is defined by means of a 
$p^{\ast}\mathrm{ad}P$
 -valued differential 2-form 
$\Omega_{2}$
 on C(P), which is related to the generalized contact structure on 
$J^{1}(P)$
 . The Hamiltonian properties of 
$\Omega_{2}$
 are also analyzed.

The geometry of the bundle of connections

This paper is aimed at introducing an algebraic model for physical scales and units of measurement. This goal is achieved by means of the concept of “positive space” and its rational powers. Positive spaces are “semi-vector spaces” on which the group of positive real numbers acts freely and transitively through the scalar multiplication. Their tensor multiplication with vector spaces yields “scaled spaces” that are suitable to describe spaces with physical dimensions mathematically. We also deal with scales regarded as fields over a given background (e.g., spacetime).

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An Algebraic Approach to Physical Scales

We analyse the Schrodinger operator for a quantum scalar particle in a curved spacetime which is fibred over absolute time and is equipped with given spacelike metric, gravitational field and electromagnetic field. We approach the Schrodinger operator in three independent ways: in terms of covariant differentials induced by the quantum connection, via a quantum Lagrangian and directly by the only requirement of general covariance. In particular, in the flat case, our Schrodinger operator coincides with the standard one.

Covariant Schrödinger operator

Quantum mechanics of a spin particle in a curved spacetime with absolute time

I present an integrated approach to tetrad gravity and Maxwell–Dirac fields, which is not singular in the case when the tetrad is degenerate. All field equations are linear. The geometric data consist only of a complex vector bundle S→M with two-dimensional fibers, where M is a real 4-manifold. All needed structures are functorially derived from S. Any considered object which is not a functorial construction is assumed to be a dynamical field.

Possibly degenerate tetrad gravity and Maxwell–Dirac fields

A partly new approach to 2-spinor geometry, recently developed, turns out to yield a naturally integrated formulation of Einstein–Cartan and Maxwell–Dirac fields, and to be suitable for describing several topics in field theories which are relevant to covariant quantization on curved spacetime. Here, first a revised version of the above mentioned geometric tools and field theory is presented, in which the geometric data consist only of a complex vector bundle S → M with two-dimensional fibres (all the needed structures are functorially derived from S; any considered object which is not a functorial construction is assumed to be a dynamical field). Then various developments are considered, concerning some features of the electromagnetic gauge and a partly new covariant description of electroweak geometry and fields. Finally, a 2-spinor treatment of optical geometry and geometric optics in curved spacetime is examined, leading to a description of the photon field as a section of the vertical bundle of the bundle of Riemann spheres associated with S.

Two-Spinors, Field Theories and Geometric Optics in Curved Spacetime

The first three sections contain an updated, not-so-short account of a partly original approach to spinor geometry and field theories introduced by Jadczyk and myself [3–5]; it is based on an intrinsic treatment of 2-spinor geometry in which the needed background structures do not need to be assumed, but rather arise naturally from a unique geometric datum: a vector bundle with complex 2-dimensional fibers over a real 4-dimensional manifold. The following two sections deal with Dirac algebra and 4-spinor groups in terms of two spinors, showing various aspects of spinor geometry from a different perspective. The last section examines particle momenta in 2-spinor terms and the bundle structure of 4-spinor space over momentum space.

Daniel Canarutto

Papers

Quantum mechanics of a spin particle in a curved spacetime with absolute time

Possibly degenerate tetrad gravity and Maxwell–Dirac fields

Two-Spinors, Field Theories and Geometric Optics in Curved Spacetime

"Minimal geometric data" approach to Dirac algebra, spinor groups and field theories

Choice of template delivery mitigates the genotoxic risk and adverse impact of editing in human hematopoietic stem cells