An Approach to Classical Quantum Field Theory Based on the Geometry of Locally Conformally Flat Space-Time
This paper gives an introduction to certain classical physical theories described in the context of locally Minkowskian causal structures (LMCSs). For simplicity of exposition we consider LMCSs which have locally Euclidean topology (i.e., are manifolds) and hence are Möbius structures. We describe natural principal bundle structures associated with Möbius structures. Fermion fields are associated with sections of vector bundles associated with the principal bundles while interaction fields (bosons) are associated with endomorphisms of the space of fermion fields. Classical quantum field theory (the Dirac equation and Maxwell’s equations) is obtained by considering representations of the structure group of a principal bundle associated with a given Möbius structure where , while being a subset of , is also isomorphic to . The analysis requires the use of an intertwining operator between the action of on and the adjoint action of on and it is shown that the Feynman slash operator, in the chiral representation for the Dirac gamma matrices, has this intertwining property.
It may be appropriate to use the term “classical quantum field theory” to signify the physics described by the Dirac equation (which supercedes the Schrödinger equation), Maxwell’s equations, and Einstein’s general theory of relativity (GR).
The principal concept of GR is that space-time can be represented as a Lorentzian 4-manifold, say, and that gravitation is associated with the metric tensor, say, of . Einstein proposed that is related to a tensor , the energy-momentum tensor, which is generated by the (nongravitational) fields of the universe and that this relationship can be described by the Einstein field equations. Thus in Einstein’s theory gravitation is associated with the geometry of space-time (described by the metric ) and is coupled to the nongravitational fields which, apart from the condition of Einstein’s field equations, have an independent existence. Thus the “data” or given mathematical structure of GR is(1)a smooth 4-manifold, say,(2)a Lorentzian metric for ,(3)electromagnetic (and possibly other) fields together with their field equations,(4)Einstein’s field equations relating the nongravitational fields to the gravitational field.Einstein believed that ultimately GR would evolve into a theory in which all physical fields are associated with the geometry of space-time.
The work described in the present paper models space-time in a fashion following from, or at least in the spirit of, GR. Our work focuses on locally conformally flat space-time and derives classical electrodynamics and quantum mechanics in this context.
It was suggested in  that space-time could be represented as (or has the structure of) an acyclic digraph. Light beams were defined as being maximal sets of interacting events. Light beams can be given the natural topology associated with their total ordering and space-time can be topologized with the topological union of its light beams. Such spaces were called webs  or causal structures .
Minkowski space can be given the structure of a causal structure in a natural way. Two events are considered to be interacting if they both lie on a common null ray. The topology for Minkowski space generated by this causal structure is strictly finer than the Euclidean topology.
Physical space-time is considered to be a causal structure which is locally isomorphic to Minkowski space. Such spaces will be called locally Minkowskian causal structures (LMCSs). If an LMCS is given the atlas of all causal charts then it is not a manifold. If it is given an atlas such that all charts in the atlas have ranges which are Euclidean open sets and such that the overlap isomorphisms are diffeomorphisms then the LMCS is a 4-manifold and will be known as a Euclidean locally Minkowskian causal structure (ELMCS).
Let be the group generated by the Lorentz transformations and the positive dilatations. Then . A diffeomorphism for and open subsets of Minkowski space is conformal if and only ifLet be the pseudogroup  consisting of conformal transformations in Minkowski space. By a (Lorentzian) Möbius structure , will be meant a set , which has an atlas of charts such that () , and is a (Euclidean) open subset of and () .
Any ELMCS is a Möbius structure . A Möbius structure is an ELMCS if and only if the overlap diffeomorphisms are orthochronous and there are no closed polygonal curves for which every side is a future directed light ray.
Any locally conformally flat Lorentzian manifold can be viewed as a Möbius structure. The map from locally conformally flat Lorentzian manifolds to Möbius structures is many to one. However, in the theory described here, it will not be assumed that there is a given distinguished metric. From the physical point of view this means that the theory described here does not have a metric as a “background structure” but relies on an underlying causal structure.
One might think that Möbius structures do not have a rich enough structure to model the diversity and complexity of the universe from microscopic to macroscopic scales. In fact they do have a rich and variable structure. The simplest (4D) Möbius structure is itself and the next most simple way to construct Möbius structures is to take an arbitrary open subset of . Thus, for example, one can consider a Möbius structure , where are closed subsets, where the boundary of any set may be connected by a “wormhole” to the boundary of another such set within the given copy of or to a subset of another copy of .
Many of the space-time types of general relativity are locally conformally flat and hence are Möbius structures, notably the de Sitter and anti-de Sitter space-times.
It is known that many Riemann surfaces can be viewed as a quotient of a discrete subgroup of acting on the complex upper half plane. Since the space of such Riemann surfaces can be imbedded in the category of Möbius structures. There is a large literature on conformal field theory in 2 dimensions in the context of statistical field theory and string theory.
The category of Möbius structures has at least the richness of the category of 3-manifolds since, through the Thurston “geometrization program” [5, 6] most 3-manifolds can be represented as a prime decomposition of model geometries where a model geometry is defined by the action of a discrete subgroup of a Lie group acting on a manifold. Lie groups involved in the model geometries are subgroups of . Therefore the category of 3-manifolds can essentially be imbedded in the category of Möbius structures.
For the rest of this paper we will assume that the Möbius structures considered are orientable and orthochronous; that is, in (1) is the group generated by the orthochronous special Lorentz transformations and the dilatations.
The results described in this paper can be readily generalized from ELMCSs to LMCSs but are presented in the former context because of its familiarity.
Our work is related to the work of Dirac  who considered the derivation of physics in the context of 4-dimensional hypersurfaces in 5D projective space, generalizing the de Sitter universe. Our work is more general than that of Dirac. It is also related to the work of Cartan  who considered natural connections associated with homogeneous spaces. Our work is distinct to that of Cartan because while some of the spaces studied by Cartan are Möbius structures there exist many Möbius structures which are not of this form.
Brozos-Vázquez et al.  point out that locally conformally flat space-times have not been studied much except the Schwarzschild interior solution and the Einstein static universe and also that Robertson-Walker space-times are conformally flat for any possible warping function. They obtain necessary and sufficient conditions for a static space-time to be locally conformally flat. Cabral and Lobo  discuss the connection between electrodynamics and the geometry and causal structure of space-time. Schwarz  discusses the AdS/CFT correspondence and generalizations to dimensions.
In the Riemannian case (as opposed to the pseudo-Riemannian case) an important class of locally conformally flat Riemannian -manifolds arises when the developing map (conformal immersion into ) is injective in which case the manifold is the quotient of an open subset of by a Kleinian group . Such locally conformally flat manifolds are called Kleinian manifolds and have been extensively studied by many mathematicians including Mostow, Thurston, Kulkarni, Goldman, Kamishima, Nayatami, and Maier. Izeki  studied the interplay between the group cohomology of a Kleinian group and the topology of the associated locally conformally flat Riemannian manifold.
Locally conformally flat pseudo-Riemannian manifolds can be obtained in a similar fashion by taking open subsets of quotients of spaces such as on which acts by Möbius automorphisms, by discrete subgroups of .
Conformal transformations have been utilised in geometry and physics for more than a century . The twistor program of Penrose  views space-time points as collections of null geodesics and obtains results concerning zero rest mass particles and fields. ’t Hooft  considers a class of elementary particle models involving conformally flat space-times from the point of view of the conventional quantum gravity approach.
“It may be necessary, for example, to reformulate classical geometry in a way that makes causal relations more fundamental, or to somehow ‘quantize’ spacetime” . Bombelli et al.  defined “causal sets” as locally finite partially ordered sets and proposed that space-time could be modeled as a causal set and, in particular, as a discrete structure. This has led to a considerable amount of research appearing in the literature (the causal set program).
In this paper some properties of Möbius structures are described. As mentioned above Möbius structures are closely related to locally conformally flat (psuedo-)Riemannian manifolds. Five principal bundle structures may be naturally associated with a Möbius structure. The first has a structure group isomorphic to , the second has as structure group the conformal group. We consider a general class of principal bundles associated with Möbius structures where the transition functions are determined by valued transition functions. Reduction in structure group from to a subgroup , say, is carried out. A natural epimorphism from to a subgroup results in a principal bundle , say.
Matter fields (fermion fields) are defined in the context of Möbius structures to be sections of vector bundles associated with . The case of the fundamental representation of the structure group of is considered. Interaction fields are defined to be differential endomorphisms of the space of matter fields. Fundamental to this analysis is the use of an intertwining operator between a representation of on and its adjoint representation on . It is shown that the Feynman slash operator (in the chiral representation of the Dirac gamma matrices) has just this intertwining property. It is shown how interaction fields can be generated by potential fields. Dirac’s equation is shown to be an eigenvector equation associated with an interaction field endomorphism. The space of relative fields (relative to a reference field and an object which we call a gravitational gauge) is shown to have the structure of a bundle of operator algebras.
Maxwell’s equations are derived by considering the canonical differential forms representing alternating multilinear forms which are obtained from the standard representation by the Hodge star operator. Distinguished alternating multilinear forms, representing physical electromagnetic fields, are obtained by considering the de Rham cohomology of forms.
2. The Conformal Group
In this section we summarize some properties of the conformal group [19, p. 64].
Let denote the set of maximal domain conformal transformations in Minkowski space. By Liouville’s theorem is (up to removable discontinuities) the set of transformations generated by the translations, the Lorentz transformations, the conformal inversion, and the positive dilatations. can be made into a group by defining, for conformal transformations , , the product to be the unique such that (where a function is identified with its graph).
Minkowski space can be identified with by means of the Pauli algebra as follows:where are the Pauli matricesand . The map has the property thatwhere is the Minkowski space quadratic form.
is defined to be the set of all matrices such thatwhere is a Hermitian form of signature and is defined byIf is considered in the representation in which the Hermitian form is defined by the matrixthen, for each the map defined in Minkowski space bywhereis an element of . Furthermore, the map is a homomorphism from to . The map has kernel and its image contains the proper orthochronous conformal group .
The map has the property that
3. The Principal Bundle Structures for a Möbius Structure
In this section we describe five principal bundles which may be associated with a Möbius structure. The first has as structure group the Lorentz group times the positive dilatations. The second has as structure group the conformal group. We assume that the transition functions for this bundle can be obtained as images of the transition functions of a bundle with structure group . The fourth is obtained from the third by reduction of structure group and the fifth is a homomorphic image of the fourth.
The first way that a Möbius structure can be considered as a principal bundle is as follows. Let , wherebe defined byLet be the principal bundle obtained by taking as transition functions. The structure group of this bundle is . The vector bundle associated with the fundamental representation of is isomorphic to the tangent bundle of . Define functions and by
Given a conformal transformation defined in Minkowski space let denote the unique element such that . has the property thatLet be a Möbius structure. Suppose that, for each , is defined byIf we write then can be thought of as a (constant) valued function over . It is straightforward to show that the cocycle conditionis satisfied. Therefore define transition functions for a principal bundle with typical fiber .
is locally isomorphic to . For the rest of this paper we will investigate the consequences of the assumption that there are valued (constant) transition functions such thatThis assumption is analogous to the assumption of orientability that we have made above which is frequently made in differential geometric investigations. Thus we are considering a subclass of the class of Möbius structures. A consequence of this assumption is that the Lie groups that will be considered in this work are matrix Lie groups.
Let and be the principal fiber bundle obtained by taking as transition functions.
One would like to determine a natural principal bundle structure for a given Möbius structure. An indication of what this principal bundle structure might be can be obtained by considering the Möbius structure . In the representation for in which the Hermitian form is defined by the matrix acts on byConsider the Cayley transform , defined byand its inverse, with domain an open subset of , defined byIf is given the atlas consisting of the inverse Cayley transform and its images as a result of being acted on by elements of (where, as usual, Minkowski space has been identified with ) then acts by Möbius automorphisms. The isotropy subgroup at is the groupTherefore . It follows that can be viewed as the total space of a principal fiber bundle with base space and typical fiber .
With respect to the representation for where the metric is defined by (7) is given by
3.1. Reduction of Structure Group from to
We take the point of view that geometric objects at a point in space-time are things which have values in all coordinate systems about that point and which transform covariantly. In special relativistic physics “covariantly” is taken to be with respect to the group of Lorentz transformations while in general relativity one considers “general covariance” which is covariance with respect to the group of (germs of) diffeomorphisms between neighbourhoods of the point in space-time under consideration. In our work we restrict the diffeomorphisms to be conformal transformations.
If is a smooth manifold with atlas and is a principal bundle with structure group and is any representation of as automorphisms of an object then there is an associated bundle with typical fiber and elements can be thought of as maps , where , such thatwhere are the transition functions for and denotes the value of in coordinate system .
Theorem 1. Let be a manifold with atlas , be a Lie group, be valued transition functions for a principal bundle on , and be a Lie subgroup of . Then if are smooth for all and are such thatthen form transition functions for a principal bundle with structure group which can be obtained from by reduction of structure group. Conversely if is a principal bundle and is a reduction of structure group from to then there exist smooth such that (27) holds and are the transition functions for .
A proof of this theorem can be found in . Let
Theorem 2. Let be a Möbius structure. Then there is a reduction of structure group from to where is a principal fiber bundle with structure group . Also there exists a reduction of structure group from to , where is a principal bundle with structure group . The transition functions for are related to the transition functions for by
Proof. Let for be the translation operator defined byDefineEach is an element of such that . Therefore , and . NowTherefore by Theorem 1 define transition functions for a principal bundle which can be obtained from by reduction of structure group from to .
Furthermorewherewhere, as usual, the point has been identified with its corresponding point in . ThereforewhereNow (viewing an element as being the map defined in a subset of ) we have , , , . Thus and so , , . ThusHence , , .
Therefore by Theorem 1 form transition functions for a principal bundle which can be obtained from by a reduction of structure group from to .
This reduction of structure group from to will be called the standard reduction from to .
In a subsequent paper it will be shown that there is a map from to for all which transforms under the adjoint action of under a change of coordinate system. Therefore has the structure of a bundle of Lie algebras and can be made into a pseudo-Riemannian manifold when equipped with the metric induced by the Killing form in .
3.2. The Principal Bundle Resulting from a Natural Epimorphism from to a Subgroup
LetIt is straightforward to show that .
is isomorphic to .
If then is a Lorentz transformation. This is because is a linear map from to andfor all , whereNow is a homomorphism from to . Thus the map is a homomorphism and defines an action of on Minkowski space. We will call the representation of on Minkowski space induced by this map the standard representation of on Minkowski space. If then denote by the Lorentz transformation associated with .
If thenThereforeThusHence we may define byIt is straightforward to show that is a well defined homomorphism. For each and defineThen form transition functions for a principal bundle with structure group .
Theorem 3. If have been obtained as described above using the standard reduction then the induced transformation of Minkowski space (identified with ) is .
Proof. Let . ThenNowbecauseThereforewhereThereforeIt can be shown that the map has derivativeTherefore and, comparing with (14), .
3.3. Some Other Principal Bundles Associated with a Möbius Structure
By Kobayashi and Nomizu [3, p. 59] the structure group of is reducible to any of its maximal compact subgroups. Such maximally compact subgroups are locally isomorphic to .
4. Fermion Fields and Interaction Fields
We have discussed Möbius structures and some of their properties. In particular, there is, with any Möbius structure associated, in a natural way, a principal bundle . has structure group which is isomorphic to .
One may describe a general paradigm for generating physical theories as follows. Take any space (e.g., a vector space) and a representation of in the space of endomorphisms (linear maps) in Construct the (vector) bundle associated with the representation and consider the space of sections of the bundle.
Physical fields and particles (fermions) are identified with elements of . We propose that interaction fields (bosons) can be identified with endomorphisms of , that is, mappings from to which are linear.
Our approach does not derive physics using the commonly used variational principles (involving Hamiltonians, Lagrangians, gravitational actions, etc.) but focuses on mathematically “natural,” and, in particular, well defined, constructions and could be readily axiomatized from the starting point of LMCSs with few axioms.
Nevertheless, the action principles remain true since while the action principles imply the field equations, the field equations imply the action principles. We just do not take the action principles as the starting point from which physics is derived.
There is large literature on the action of conformal vector fields on spinor bundles (e.g., [20, 21]) and conformally invariant differential operators on Minkowski space . Our work is different to this work; we do not construct differential operators with respect to given vector fields on the manifold (Lie derivatives). Rather, we start with an intertwining operator between the standard representation of on and its adjoint representation in and construct endomorphisms of . We use the fact, here and in all the following work, that acts on , , and in natural ways. We show that, remarkably, the Feynman slash operator (in its chiral representation) has just the intertwining properties that we require (and, in particular, the Dirac gamma matrices, when multiplied by the imaginary unit , are all elements of ). Dirac’s equation is shown to be the eigenvector equation for the endomorphisms that we have constructed.
The space of interaction fields is shown to have an affine structure; given a reference field and an object which we call a gravitational gauge, there is a natural mapping between the space of interaction fields and the space of sections of a bundle of operator algebras.
5. Analytic Properties of Fermion Fields and Interaction Fields on Möbius Structures
In this section we define matter fields and interaction fields in the context of the associated vector bundle to the principle bundle through the fundamental representation of . Central to this section are operators which will be later related to the Dirac gamma matrices in the chiral representation but which are here defined by their property as intertwining operators. The properties of these intertwining operators together with the properties of the Möbius structure coordinate transformations lead to simple transformation properties of differential operators constructed using them. Interaction fields allow for the construction of differential operators which have natural transformation properties and which are, up to multiplication by a (gravitational) gauge, differential endomorphisms of the space of matter fields.
Let denote the space of sections of . Elements of will be called fermion fields, matter fields, or simply fields. Elements of can be thought of as collections which satisfy is a module over the algebra .
Let be the Lie algebra of . Suppose that is an intertwining operator between the standard representation of in and the adjoint representation of in . An explicit example of such an intertwining operator will be given in a later section. Then for all where is the Lorentz transformation of Minkowski space corresponding to . ThereforeFrom this it follows thatwhere for and is the standard basis for . Let be obtained from by raising of indices. That is,where is the Minkowski space metric. ThenHenceTherefore
Theorem 4. Matter fields have the following differential transformation property:
Proof. where the last equality occurs by virtue of the identity Equation (60) which results in the disappearance of from the expression for .
Define an interaction field to be a collection of quantities which transform according to
Let denote the set of all interaction fields.
Theorem 5. Suppose that . Then the map , defined on byhas the following transformation property:
Define a gauge (gravitational gauge) to be a section of the principal bundle obtained by taking as transition functions. Since is diffeomorphic to a Euclidean space it follows  that there exists at least one gauge. Let the set of gauges be denoted by . Gauges can be thought of as collections of functions which transform according toWe have the following.
Theorem 6. Let and . Then the map , defined on byis a linear endomorphism of .
Proof. Hence is a matter field, by the defining property of (53). Therefore . The required result follows from the fact that is linear.
Given an interaction field and a gauge it is natural to consider eigenvectors of the linear operator , and such eigenvectors have a distinguished status in . If then is an eigenvector of (with positive eigenvalue) if there exists an such that . This is equivalent to the equation
6. Potential Fields
In this section we give a definition of a potential field on . We show how a potential field on gives rise to an interaction field. A potential field on may be thought of as a collection of quantities such that transforms in a natural way for all . A natural transformation law is as follows:
Define a potential field on to be a collection of quantities which have the following transformation properties:
Theorem 7. Let , , . Then transforms according to (71) if and only if is a potential field.
Proof. We recall from (53) thatThereforeHence the transformation law of (71) is satisfied if and only ifThus, after making a cancellation, we see that transformation law (71) is satisfied if and only ifThis is true for all if and only ifwhich is equivalent to the condition that be a potential field.
In a subsequent paper it will be shown that potential fields for are induced by linear connections on the total space for the tangent bundle for .
Theorem 8. Suppose that is a potential field on . DefineThen is an interaction field.
Proof. Using (60) we have
A special case with and commuting is when the potential field components are scalar valued quantities. In this case the potential field components transform according to
7. A Natural Intertwining Operator between the Standard Representation of in Minkowski Space and the Adjoint Representation of in
In this section we define an intertwining operator between the standard representation of in Minkowski space and the adjoint representation of in where is the Lie algebra of . We show that this operator is times the Feynman slash operator with the chiral representation for the Dirac gamma matrices.
Consider in the representation in which the metric is given by (7); that is,Define bywhere denotes the parity operator defined byand are the Pauli matrices. Then, for all Also , . Therefore .
Let , where and . Then for all