Affine Field Theory Research Articles

Locally Covariant Quantum Field Theory with External Sources

We provide a detailed analysis of the classical and quantized theory of a multiplet of inhomogeneous Klein-Gordon fields, which couple to the spacetime metric and also to an external source term; thus the solutions form an affine space. Following the formulation of affine field theories in terms of presymplectic vector spaces as proposed in [Annales Henri Poincare 15, 171 (2014)], we determine the relative Cauchy evolution induced by metric as well as source term perturbations and compute the automorphism group of natural isomorphisms of the presymplectic vector space functor. Two pathological features of this formulation are revealed: the automorphism group contains elements that cannot be interpreted as global gauge transformations of the theory; moreover, the presymplectic formulation does not respect a natural requirement on composition of subsystems. We therefore propose a systematic strategy to improve the original description of affine field theories at the classical and quantized level, first passing to a Poisson algebra description in the classical case. The idea is to consider state spaces on the classical and quantum algebras suggested by the physics of the theory (in the classical case, we use the affine solution space). The state spaces are not separating for the algebras, indicating a redundancy in the description. Removing this redundancy by a quotient, a functorial theory is obtained that is free of the above mentioned pathologies. These techniques are applicable to general affine field theories and Abelian gauge theories. The resulting quantized theory is shown to be dynamically local.

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.

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.

Tuning up mind's pattern to nature's own idea: Eddington's early twenties case for variational derivatives

This paper sets out to show how Eddington's early twenties case for variational derivatives significantly bears witness to a steady and consistent shift in focus from a resolute striving for objectivity towards “selective subjectivism” and structuralism. While framing his so-called “Hamiltonian derivatives” along the lines of previously available variational methods allowing to derive gravitational field equations from an action principle, Eddington assigned them a theoretical function of his own devising in The Mathematical Theory of Relativity (1923). I make clear that two stages should be marked out in Eddington's train of thought if the meaning of such variational derivatives is to be adequately assessed. As far as they were originally intended to embody the mind's collusion with nature by linking atomicity of matter with atomicity of action, variational derivatives were at first assigned a dual role requiring of them not only to express mind's craving for permanence but also to tune up mind's privileged pattern to “Nature's own idea”. Whereas at a later stage, as affine field theory would provide a framework for world-building, such “Hamiltonian differentiation” would grow out of tune through gauge-invariance and, by disregarding how mathematical theory might precisely come into contact with actual world, would be turned into a mere heuristic device for structural knowledge.

Classical gravity and quantum matter fields in unified field theory

The Einstein-Schrodinger purely affine field theory of the non-symmetric field provides canonical field equations without constraints. These equations imply the Heisenberg-Pauli commutation rules of quantum field theory. In the Schrodinger gauging of the Einstein field coordinatesUkli=Γkli−δliΓkmm, this unified geometric field theory becomes a model of the coupling between a quantized Maxwellian field in a medium and classical gravity. Therefore, independently of the question as to the physical truth of this model, its analysis performed in the present paper demonstrates that, in the framework of a quantized unified field theory, gravity can appear as a genuinely classical field.

An electrodynamic connection

Starting from a first principles basis deformation of a Riemannian manifold, a new affine connection is derived that contains both gravitational and electromagnetic fields on equal footing term for term. Using this connection, it is shown that parallel transport of the tangent vector on a curve yields the Lorentz force equation for a charged particle, while transport of the normal vector yields the equation of motion for a classical spin including the covariant Thomas precession. Furthermore, the new electrodynamic connection exactly satisfies the form and the compatibility criteria for the general non-symmetric connection sought by Schrödinger for his affine field theory. The results of the present work are shown to be equivalent to introducing an electrodynamic torsion on the manifold thus rendering it non-Riemannian. The new connection may serve to illuminate the geometry of electrodynamics.

Hamiltonian dynamics of purely affine fields (EINSTEIN-SCHRÖDINGER Theory)

The Lagrangian of the general-relativistic affine field theory of the non-symmetric connection field Γikl is Schrödinger scalar density \documentclass{article}\pagestyle{empty}\begin{document}$ H = \frac{2}{\lambda }\sqrt { - \det [R} _{ik} ] $\end{document}, and the field variables (canonical coordinates) are Einstein's affine tensors Umnl = Γlmn - δnlΓrmn. The field equations are the Einstein-Schrödinger equations The minors give by definiton gmn = λ−1 Rmn, and λ becomes the cosmological constant. The Hamiltonian density is the V00-component of the Einstein energy-momentum complex and the tensor-density components are the canonically conjugated momentum densities of the field coordinates Ulmn. The canonical equations are , and we have no constraints. The affine field theory is invariant with respect to all transformations which preserve the Levi-Civita parallelism (Einstein's unified T-A group), and the field equations possess transposition invariance: With Ũlmn = Ulnm we get R̃mn = Rnm, g̃mn = gnm, and Ñmnl = Unml. The symmetry conditions Γimn = Γinm reduce the space to the general-relativistic Einstein spaces with Rik = Rki. The equation Rik = λgik yields Γikl = {ikl}, and the pathes of test particles define geodesic world lines of the Einstein spaces.

THE AFFINE N = 4 YANG–MILLS THEORY

An affine field theory may be constructed by gauging an affine algebra. The momentum integrals of the affine N = 4 Yang–Mills theory are ultraviolet finite but diverge because the sum over states is infinite. If the affine symmetry is broken by postulating a nonvanishing vacuum expectation value for that component of the scalar field lying in the L0 direction, then the theory acquires a linear mass spectrum. This broken theory is ultraviolet finite too, but the mass spectrum is unbounded. If it is also postulated that the mass spectrum has an upper bound (say, the Planck mass), then the resulting theory appears to be altogether finite. The influence of the exotic states has been estimated and, according to the proposed scenario, is negligible below energies at which gravitational interactions become important. The final effective theory has the symmetry of a compact Lie algebra augmented by the operator L0.

Gravitational coupling of affine field theories

The coupling of a N=4 affine gauge theory to N=1 supergravity is investigated. For the unbroken Kac-Moody theory the known sum rule for finite Lie multiplets is extended to cover the infinite dimensional Kac-Moody multiplet. For the broken Kac-Moody theory, the infinite Kac-Moody multiplet splits into an infinite set of finite multiplets representing a finite Lie symmetry * the U(1) generated by L0; in this case there is a separate sum rule for each Lie multiplet. The gravitationally coupled theory is more restricted but at the level of this investigation is not qualitatively different from the corresponding global theory.

Dyons in affine field theories

By postulating that the gauge vector field and the Higgs field both lie in an affine algebra rather than a Lie algebra, an affine field theory with an expanded soliton sector is obtained. An infinite family of exact, special solutions of the new equations corresponding to the original ’t Hooft–Polyakov solitonic solutions, is now found. A perturbation method for an arbitrary solution of the new equation is also proposed.

Spontaneous symmetry breaking of affine field theory

It is possible to construct non-Abelian field theories by gauging Kac-Moody algebras. Here we discuss the spontaneous symmetry breaking of such theories via the Higgs mechanism. If the Higgs particle lies in the Cartan subalgebra of the Kac-Moody algebra, the previously massless vectors acquire a mass spectrum that is linear in the Kac-Moody index and has additional fine structure depending on the associated Lie algebra.

The torsionless affine field theory of Einstein

The affine field theory of a symmetric connection is based on a more general variational principle than was used by Einstein. The action density is a function of the Ricci tensor and a covariant vector field Ai, which is determined by the connection. The terms in the field equations which depend explicitly on Ai are interpreted as charge- current 4-vectors. The field equations give a generalization of Mie's theory of electrodynamics and in a special case they correspond to Born's non-linear equations of electrodynamics. A connection with Maxwell's equations is also established in this case. The static spherically symmetric field of an electron is described, and an interpretation of the solutions, which ascribes a structure to the electron, is briefly discussed.

Integral Invariants of the Affine Field

In this paper the method of infinitesimal transformations of coordinates, used by Weyl to determine conditions that a function of the tensors gik and φi, and certain of their derivatives, should be a scalar density, is applied (with certain modifications so as to give tensor relations) to functions of and . It is known that in order that such a function should be a scalar density it must be a homogeneous function, of degree ½n, of , and this must of course be deducible from the equations found by the infinitesimal transformations. In view of the part which these equations may play, as “equations of energy,” etc., in purely affine field theories, it seems desirable that the connection should be explicitly shown, and this is done in § 3.

Schrödinger-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.