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The Plenum

Dirac has pointed out, quite clearly, that values of observable quantities, such as position, do not make sense except when they can be measured. Einstein based special relativity on the observation that coordinates in time and space are defined from physical processes, namely those processes which can be used in measurement procedures. As developed on this site, quantum theory is seen as the mathematical structure appropriate to physical situations in which only probabilistic results are possible for measurements of position at given time. Spacetime is the mathematical structure obeyed by measurements of coordinates for time and space calibrated to the radar method, and only makes sense when such measurements necessarily give definite results. These ideas strongly support the notion that only measured position actually makes sense, and that an absolute background has no part in the fundamental structures of matter.

In typical field theoretic treatments of quantum electrodynamics, Feynman diagrams have meaning only as aids to calculation. In contrast, in relational quantum gravity, Feynman diagrams describe possible configurations of small sections of a material plenum consisting of interactions between physical electrons and physical photons (the model can be extended to include additional particles, but for simplicity I consider only these two). Lines represent particles and vertices represent interactions. The whole represents the material plenum. The diagrams we use to calculate probabilities for events (e.g., scattering) represent possibilities for small sections of the plenum. We cannot say what combination of particle interactions takes place between measurements and we form the sum of possible diagrams, where sum means quantum logical OR.

Mathematically Feynman diagrams are graphs. Only the relationships between lines and vertices are important. The paper on which the diagram is drawn has no meaning for a graph. Spacetime does not appear at a fundamental level in a Feynman diagram. It is an emergent quantity, describing an organisational principle resulting from the calculations of probabilities for observed results. Stable configurations (with probability 1) in Feynman diagrams lead to the structures of matter we observe (I do not here consider the manner of the binding between the quarks in the proton of a hydrogen atom). Distance scales defined from these diagrams depend on the exchange of photons between charged particles, the same physical process as we have used to define distance using the radar method.

Quantum Field Theory

I have disagreements with the field theoretic approach on both philosophical and mathematical grounds. Mathematically, a field is a function on manifold, the mechanical equivalent is a machine with an indefinitely large number of moving parts. Mathematicians have always recognised that complex numbers are an abstraction and cannot represent anything physically real, but wave functions are complex. I have never found it reasonable to think that such an elaborate structure should model the fundamantal constituents of matter. The definition of field quantities as fundamental constituents of matter requires the prior definition of a background, and is modelled on Newton’s view, modified to replace absolute space with spacetime. Philosophically it seems to me that this is opposed both to the principle of relativity and to the evidence of quantum theory.

The fact that fields are defined everywhere runs counter to physical experience that particles interact at particular points, at least whenever they are observed. Field theory does not explain what the fields actually are, or why they should manifest pointlike behaviour. Instead it starts with a classical field and quantises it by summarily replacing numerical quantities with operators, but without explanation as to why physical matter should have the properties of a quantised field. Such questions are usually discarded through a philosophy which says that physics does not interpret Nature, but merely produces formulae to predict its behaviour. If one follows such a philosophy consistently, it does not even permit the interpretation that the fundamental objects of Nature are fields.

It is thought that to generate a theory of interacting particles one must start with a theory of interacting fields. It is thought that the perturbation expansion, expressed in Feynman diagrams using non-interacting fields, is only an approximation to the true perturbation expansion which should be expressed with interacting fields. Haag’s theorem shows that for interacting fields the interaction picture does not exist. This seriously undermines the canonical development of perturbative quantum field theory, including quantum electrodynamics, which uses the interaction picture throughout.

Constructive quantum field theory attempts to give quantum field theory a rigorous basis in terms of well defined mathematical concepts, but has only been able to do so for unrealistic models with fewer than three space dimensions. The central problem is the assumption that fields are covariant as required by relativity. This assumption becomes an axiom in any of the proposed formalisations, namely the Wightman axioms, the Osterwalder-Schrader axioms, and the Haag-Kastler axioms. The objective is to construct field operators obeying one of these axiom sets from the axioms of quantum mechanics as described by Von Neumann. The assumption of covariance means that fields must be defined on spacetime, not on space. There is then no way to avoid the equal point multiplication between fields, φ^†(x)φ(x) = ∞. One attempts to define the fields as operator valued distributions, meaning generalised functions for which integrals are defined, but the integrals break down both in the ultraviolet divergence and at the Landau pole. No construction has been successfully carried out. Mathematically, it is said that quantum electrodynamics does not “exist”.

Particle Theory

The classical notion of a particle places a point-like object at a particular position in a background space. A classical particle always has a position in space, whether or not its position is known. This is not the notion described in quantum mechanics. For a quantum particle, a numerical value of position only exists either when a measurement of position is carried out (or when the interactions of the particle with other matter are such as to define a numerical value of position which may or may not be known). Photons do not have a measurable position at all. We can (within the limitations of quantum theory) talk of the position of an electron when it creates or annihilates a photon, but not the position of the photon itself. This is entirely in keeping with Descartes’ original description of relationism, according to which the position of an object only exists by dint of contact (interaction) with other objects.

The absence of a position observable for photons and the lack of positive definite norm for photon states have lead many physicists to reject the idea that photons are particles. Relational quantum gravity takes a different view of “particle”. Particles are sizeless objects. A position observable is only required in so far as position can be measured. Positive definite norm is only required to give probabilities for those measurements which can actually be performed, and is satisfied since there is no observable polarisation as described in the observable polarisation between longitudinal and timelike photon states.

In the particle theory, interactions are modeled by an operator, the interaction Hamiltonian on the Fock space of particles at time t, and is necessarily a map to Fock space at a subsequent time, t + Δt, for some small Δt, H_int(t) : F(t) → F(t + Δt). The time evolution of operators, as well as states, is naturally given in the interaction picture. There is no requirement to construct “interacting fields”, and no issue over Haag’s theorem. It would be meaningless to set Δt = 0, since this would describe the result of an interaction which has not yet taken place. The equal point multiplication between fields is not a physical problem in the theory; physically it cannot occur. The problem is to describe the theory in such a way that the equal point multiplication also does not occur in the mathematical model. When this is done rigorously using the method of Epstein and Glaser and limits are taken in the correct order, the ultraviolet divergence is excluded.

Similarly lattice regularisation excludes the equal point multiplication between fields from the outset. As in relational quantum gravity, lattice field theory uses discrete coordinates, but differs in philosophy. Discrete coordinates eliminate the equal point multiplication and with it the ultraviolet divergence, but violate Lorentz covariance. Lorentz covariance is restored in the limit in which lattice spacing tends to zero. Relational quantum gravity replaces Lorentz covariance with quantum covariance, reflecting the fact that a coordinate system is set up by a particular observer using physical measurement procedures with finite resolution. Quantum covariance requires that lattice spacing is smaller than the practical resolution of measurement, but not that the limit is taken in which lattice spacing goes to zero

Causal perturbation theory and lattice field theory deal with the ultraviolet divergence, but neither eliminate the Landau Pole. Relational quantum gravity introduces a minimal effective interval of proper time between the interactions of a particle. Such a small scale modification sets a lower bound on the lattice spacing which can be used in principle, and thereby eliminates the Landau pole. In the following sections, it will be seen that when an effective time interval between interactions is introduced in a simplified model (neglecting spin), Minkowski metric must be replaced with a metric for curved spacetime obeying Einstein’s field equation.

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