←  The Teleconnection  ↑  →

Absolute Space and Time

The purpose of the teleconnection is to retain the fundamental arguments leading from special relativity to quantum electrodynamics and to general relativity, and to use them to show how apparent fundamental incompatibilities can be reconciled. The teleconnection adapts the probabilistic structure described in quantum theory to the situation studied in general relativity, in which a clock at the position of the emission of a quantum particle does not measure time at the same rate as an identical clock at the position at which the particle is detected.

In a Friedmann cosmology the existence of cosmic time, or time on a geodesic from the big bang conflicts with Einstein's precepts in which only local law, and local time as measured on a clock, are physically important or meaningful. The teleconnection assumes that since cosmic time makes sense on the large scale in a Friedmann cosmology, it should also make sense on the small scale, even after taking local variations in geometry into account. Cosmic time gains a greater importance than in standard general relativity, and plays a role very much like Newton's absolute time. To describe the transmission of photons over large distances, the inner product in quantum theory is defined using an integration over all space, and, in an expanding cosmology, only makes complete sense on a synchronous slice. Synchronous slices for given cosmic time yield a "preferred" frame, in which the teleconnection is defined, and play the role of Newton's absolute space. Nonetheless the fundamental elements of the model are particles of matter, and in contrast to Newton’s ideas, time and space appear as emergent organisational principles, not as physical background.

Suppression of Expansion in a Neighbourhood

The inner product is defined to generate probabilities. To specify a probability, it is necessary to specify the known conditions to which the probability applies. We may interpret the collapse of the wave function as the change in probability when the known condition changes. The quantum state describes knowledge of the particle. “Knowledge” in this context refers to information which is available in principle from the physical situation, whether or not it is known in practice by a particular observer. The knowledge about a particle which is possible in principle depends on the physical relationship of that particle with other matter and, in the context of an expanding universe, will have empirical consequences which cannot be reconciled with a view of the wave function as a physical field.
The teleconnection will be applied to two types of physical situation, depending on whether the clock used to determine the final state can be calibrated to the clock used to determine the initial state, for example by Einstein’s calibration procedure.
This should be not be understood as a change in physical behaviour, but as a change in the mathematical formulation of quantum theory as a theory of probabilities, reflecting the difference in information available to the observer depending on whether synchronisation is possible between the clocks used to determine the initial and final states. Because a factor of the expansion parameter is removed in the calculation of energy/momentum, both formulations will give the same classical quantities, but there will be a difference in predictions for spectral shifts, determined, for example, by diffraction. This is legitimate for interpretations of quantum theory in which the wave function is a device for the calculation of probabilities, but excludes interpretations which attribute any form of ontological existence to the wave function.

Thus, the rule regarding the evolution of the wave function from an initial state at A to a remote final state at B depends on whether the coordinates at B can be physically calibrated to those at A, for example by Einstein’s synchronisation procedure. When there is a physical calibration, quantum theory is formulated without expansion. When no calibration is possible, as is the case for light from a distant stellar object, the quantum theory must be formulated taking into account expansion between coordinates used for initial and final states. Wave evolution takes place in coordinates defined from the initial state, so that interference effects depend on wavelengths modified by a factor of the expansion parameter. This factor is removed in the calculation of energy-momentum from the wavelength, because physical quantities are defined with respect to current coordinates. When a synchronisation procedure exists between clocks used to describe the initial and final states, expansion is suppressed. This is not a change in the physical behaviour of light, but a change in the mathematical formulation of quantum theory as a theory of probabilities, and reflects the different information available to the observer.

At the present time, it is not possible to write down conditions under which Einstein’s synchronisation procedure is possible. It is known that the anomalous Pioneer blue shift appeared after radar lock was lost. After radar lock was lost with Pioneer, it was no longer possible to calibrate processes on Pioneer to processes on Earth. Synchronisation became impossible in practice, creating the conditions under which the anomalous shift was observed. The shift is seen not as an indication of a change in motion, but as a consequence of a change in the conditions under which quantum theory is formulated. There are indications of an anomalous shift during planetary flybys, when radar lock is also lost, but no indication of a shift for planets in the outer solar system, whose distance and orbital periods are known to good accuracy and together give information equivalent to clock synchronisation. It remains to be established whether the anomaly appears in consequence of some deep reason dependent on local geometry and motion according to which maintenance of radar lock is impossible in principle, or whether it is sufficient that a synchronisation procedure cannot be carried out in practice. Hopefully future missions will cast greater light on the onset of the anomaly.

The Teleconnection in a Friedmann Cosmology

Using τ−ρ coordinates for a Penrose diagram, in a Friedmann cosmology the metric is
where f(ρ) = sinρ, ρ, or sinhρ for space with positive, zero or negative curvature respectively. Coordinate time, τ, is related to cosmic time, t, by a(τ)dτ = dt. An observer, Alf, at A at cosmic time t₁ and coordinate time τ₁, defines radial unprimed locally Minkowski coordinates, (t, r, θ, φ) based on cosmic time, t, such that dt = a(τ)dτ and r = a(τ)ρ. For cross terms in the metric can be neglected,
For ,
Alf defines a non-physical metric, , using τ−ρ coordinates, such that,
where A and B are real numbers, whose values are to be determined. Alf defines barred vectors in τ−ρ coordinates.


This definition will apply to classical energy-momentum, but not to quantum energy-momentum which appears in the wave function. Barred quantum energy-momentum will be defined separately.


It follows that for a vector (including a displacement vector) x = (x^t, x^r, x^θ, x^φ), in locally Minkowski coordinates at (t₁, A),
and that, for barred vectors and at (τ₁, A), the barred dot product, evaluated with non-physical metric , satisfies

Alf formulates quantum states locally in Hilbert space at time t₁, and defines plane wave states at t₁ using
Quantum theory is then reformulated in terms of barred quantities, under the requirement that the inner product is preserved.
This requires that , and we have the definition:


It follows that the momentum space wave function,
is preserved when it is given that vanishes outside a sphere of radius πa, and when the wave function is normalised such that .

The teleconnection defines the inner product between the initial state of a particle emitted at (τ₁, A) and the final state of that particle detected some later time, by translating barred momentum from in τ−ρ coordinates with constant non-physical metric .


This definition preserves Newton’s first law and the constancy of the speed of light in τ−ρ coordinates with non-physical metric . It is required to retain the formal structure of relativistic quantum theory, and preserves the momentum space wave function as a constant of the motion. A plane wave is sometimes regarded as a physical field in a substantive spacetime. Here, in accordance with the orthodox interpretation, it is seen as a mathematical construction in a non-physical space defined by an observer, Alf, at A. The evolution of the wave function is determined in τ−ρ coordinates, using non-physical metric , and is such that it corresponds to the standard wave function in locally Minkowski coordinates.

Cosmological Redshift

A second observer, Beth, at (τ₀, B), remote from (τ₁, A) and with τ₀ > τ₁, also defines quantum states in Hilbert space.


Proof:  One period of light in locally Minkowski coordinates, (t, r, θ, φ), with an origin at A at cosmic time t₁ is represented by a timelike vector of magnitude λ₁. In τ−ρ coordinates, its timelike component is λ₁ ⁄ a₁. The corresponding barred quantity is λ₁ ⁄ Aa₁. The barred vector is translated to (τ, B). The corresponding unbarred quantity is found from the physical metric, g, and has magnitude a₀λ₁ ⁄ a₁². On transforming to primed locally Minkowski coordinates (t', r', θ', φ'), with an origin at B, we find

It follows that, for small r,
Thus recession velocity due to expansion is half the value calculated from Doppler, and coordinates in which radial distance from Earth is calculated from redshift exhibit a stretch of factor half in the radial direction. So A = 2. The time taken for a pulse of light at this distance to traverse a small angular distance dθ is . So B = ½ (a stretch of factor two in angular directions gives 4π in a circle, which may be related to the spin states of Fermions). Thus, the non-physical metric, ,is

Energy Transfer

The squared redshift law appears at first to be at odds with the claim that parallel displacement under the teleconnection reduces to parallel transport in the classical correspondence. This is due to the manner in which expansion is treated in the quantum theory and is resolved at the point of the collapse of the wave function. The square redshift law applies to spectroscopic measurements of light from distant sources, when the detection of the photon takes place after diffraction or refraction of the quantum wave function. It does not affect energy transfer because, in order to formulate the inner product between an initial state at time t₀ and a final state at time t₁, Beth enlarges the coordinate axes at time t₀ by a factor a₀ ⁄ a₁. This affects spectroscopic measurements, but, since the initial measurement of energy-momentum is relative to the coordinate axes at time t₀. this factor does not appear in the calculation of energy/momentum of light from a distant source. The energy measured locally by Beth at time t₀ of a photon emitted at time t₁ with energy from a from a distant source at time t₁ is given by
Thus we have the same relation for energy transferred as is given by parallel transport under the Levi-Civita connection.

Geometries with Expansion

In general relativity, time is determined from a clock locally. Although cosmic time is defined globally, its definition, based on Weyl’s postulate depends on the local time of many galaxies on geodesics from the Big Bang. The Friedmann models are based on the particular assumptions that the distribution of matter is homogeneous and isotropic. This is observably true in reasonable approximation when the matter distribution is averaged over large enough distances. So, we expect these models to give a reasonable description of the universe at large scales. But these models can only be approximate because they take no account of local mass distributions or of peculiar motions of galaxies and the orbits of stars within a galaxy. On their own, the Friedmann models say nothing about what happens at smaller scales, at which matter is clearly not homogeneous. However, it is natural to think that local fluctuations in geometry due to the inhomogeneous local matter distribution can be treated as perturbations to a Friedmann model (at least for points where the gravitational field is not large).

Since it does not make sense to talk of expansion locally, a(τ) is a global parameter. We may define space-like hypersurfaces with a(τ) = const and define τ to be a global time parameter with τ = const on any surface with scale factor a(τ) = const. The teleconnection postulates that we may define a non-physical metric, , as for a Friedmann cosmology in which the speed of light is constant. In τ−ρ coordinates,
This seems reasonable for points which are not hidden by an event horizon. Points inside an event horizon will be considered again later.

The calculation of the general form of the metric for observers at constant ρ goes through as for stationary observers, but now we have an additional factor of the expansion parameter. Thus the physical metric has the form
where the factor k describes gravitational redshift. While the calculation of parallel displacement in τ−ρ coordinates is straightforward, the determination of the relationship between τ−ρ coordinates and inertial locally Minkowski coordinates depends on both the motion of the observer and on local variations in geometry, summarised in the factor k. The calculation is complicated by the fact that, in the general case, motion with ρ = const is not inertial. The consequence is that an an inertial observer will need to modify wave functions in such a way that they are subjected to apparent accelerations, or Doppler shifts which do not reflect acceleration in the classical domain.

With the substitutions dt' = a(τ)dτ r' = a(τ)ρ, θ' = θ, and φ' = φ, and for
For ,
in agreement with the calculation of the form of the metric for stationary observers. Locally Minkowski coordinates at the point x with r = 0 are found by substituting dt = k⁻¹dt' and dr = kdr', rdα = r'dθ', r sinα dβ = r' sinθ' dφ'. For small r,

Barred vectors are defined in τ−ρ coordinates as for a Friedmann cosmology.


For a vector x = (x^t, x^r, x^α, x^β) in locally Minkowski, t-r , coordinates at (t₁, A) where t₁ ↔ τ₁

Alf formulates quantum states locally in Hilbert space at time t₁, and defines plane wave states at t₁ using
Quantum theory is reformulated globally using barred quantities, under the requirement that the inner product is preserved.
This requires that , and we have the definition:


We will be interested in weak fields and the transmission of photons over astronomical distances, for which momentum may be taken as radial up to the bending of light by a lens. It will be sufficient to consider the approximation

Gravitational Redshift

Let A and B be sufficiently close that gravitational lensing effects may be ignored. Then, in coordinates with an origin at A, the momentum of a photon passing from A at t₁ to B at t₂ is radial,
Let the redshift factor at A at time t₁ be k_A, and let the redshift factor at B at t₂ be k_B, and let a₁ = a(t₁) and a₂ = a(t₂). Then, barred momentum is
Barred momentum is translated to (τ₂, B) in τ−ρ coordinates. Beth converts to locally Minkowski t'-r' coordinates, and finds, with the removal of a factor of the expansion,
which is the same as is found by parallel displacement of momentum through a small distance in tangent space when the metric is
In locally Minkowski coordinates at A, gravitational redshift is given by a factor (g₀₀)^−½, in agreement with standard general relativity.

The Levi-Civita Connection

The classical correspondence is found when there is sufficient information that states may be treated as being continuously measured. The quantum state is defined on a synchronous slice using quantum time, t. In order to describe a continuous classical motion, we must take a collection of synchronous slices, and treat each slice as the final state of one quantum step and as the initial state of the next quantum step, then allow the step size to go to zero, to obtain a foliation. At each stage of the motion, quantum theory is formulated in a space with a non-physical metric . Classical motion is determinate and may be described as an ordered sequence, , of effectively measured states at instances t_i such that 0 < t_i+1 - t_i < δ where δ is sufficiently small that there is negligible alteration in predictions in the limit δ → 0. Each state, , is a multiparticle state in Fock space, which has been relabelled using mean properties determined by an effective classical measurement. For the motion between times t_i and t_i+1, may be regarded as the initial state and may be regarded as the final state. Since the time evolution of mean properties is determinate, there is no collapse and is the initial state for the motion to t_i+2. Classical formulae are recovered by considering a sequence of initial and final effectively measured states in the limit as the maximum time interval between them goes to zero. In each stage of the motion momentum is parallel displaced using the teleconnection, which gives the same result as parallel displacement in tangent space. The cumulative effect of such infinitesimal parallel displacements is parallel transport. So the teleconnection between initial and final states in quantum theory leads to parallel transport in the classical domain. The same argument holds for a classical beam of light, in which each photon wave function is localised within the beam at any time, and for a classical field which has a measurable value at each point where it is defined (up to the possible addition of an arbitrary gauge function).

The Teleconnection ↑ Illusory Velocity →