Most recent edit on 2009-05-11 01:16:29 by CharlesFrancis

Additions:
A connection defines the notion of parallel in vector spaces defined at nearby points of a manifold. The teleconnection defines the parallel displacement of momentum in quantum mechanics from an initial state to a final state when the reference matter used to describe the initial state is remote from that used to describe the final state. When the initial and final states are determined with respect to nearby reference matter, the teleconnection is equivalent to the Levi-Civita connection.

Absolute Space and Time

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

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.

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 formulates quantum states locally in Hilbert space at time t₁, and defines plane wave states at t₁ using

Cosmological Redshift

Energy Transfer

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).
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
With the substitutions dt' = a(τ)dτ r' = a(τ)ρ, θ' = θ, and φ' = φ, and 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.
Alf formulates quantum states locally in Hilbert space at time t₁, and defines plane wave states at t₁ using

Gravitational Redshift

which is the same as is found by parallel displacement of momentum through a small distance in tangent space when the metric is

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).

Deletions:
A connection defines the notion of parallel in vector spaces defined at nearby points of a manifold. The teleconnection defines the parallel displacement of momentum in quantum mechanics from an initial state to a final state when the reference matter used to describe the initial state is remote from that used to describe the final state. When the initial and final states are determined with respect to nearby reference matter, the teleconnection is equivalent to the Levi-Civita connection.

Absolute Space and Time

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

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.

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. The metric in t-r coordinates is
For ,
Alf formulates quantum states locally in Hilbert space at time t₁, and defines plane wave states at t₁ using

Cosmological Redshift

Energy Transfer

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).
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
With the substitutions dt' = a(τ)dτ r' = a(τ)ρ, θ' = θ, and φ' = φ,
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.
Alf formulates quantum states locally in Hilbert space at time t₁, and defines plane wave states at t₁ using

Gravitational Redshift

which is the same as is found by parallel displacement of momentum through a small distance in tangent space when the metric is

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).

---

Edited on 2009-04-23 06:27:09 by CharlesFrancis

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

Deletions:
Barred vectors are defined in τ−ρ coordinates as for a /www.teleconnection.info/rqg/Teleconnection#TheTeleconnectionInAFriedmannCosmology>Friedmann cosmology.

---

Edited on 2009-04-23 06:25:22 by CharlesFrancis

Additions:
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.

1. When there is a physical calibration between the clocks used to measure initial and final states quantum theory is formulated without expansion.
2. 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.

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.

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.

Definition:  For the displacement, x = (x^τ, x^ρ, x^θ, x^φ), from (τ₁, A), the corresponding barred displacement vector is

Definition:  Barred momentum in quantum theory is
Barred vectors are defined in τ−ρ coordinates as for a /www.teleconnection.info/rqg/Teleconnection#TheTeleconnectionInAFriedmannCosmology>Friedmann cosmology.

Deletions:
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 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.

1. When there is a physical calibration between the clocks used to measure initial and final states quantum theory is formulated without expansion.
2. 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.

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. The view taken here is that quantum theory gives a calculation of probabilities between an initial state and final state, not a description of physical reality between initial and final measured 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.

Definition:  For the displacement, x = (x^τ, x^ρ, x^θ, x^φ), from (τ₁, A), the corresponding barred displacement vector is
Barred momentum is defined in τ−ρ coordinates as for a Friedmann cosmology.

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Edited on 2009-04-10 07:32:01 by CharlesFrancis

Additions:
1) When there is a physical calibration between the clocks used to measure initial and final states quantum theory is formulated without expansion.

1. 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.

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.

Deletions:
1. When there is a physical calibration between the clocks used to measure initial and final states quantum theory is formulated without expansion.

2. 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.

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 (section 7), 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.

---

Edited on 2009-04-10 07:30:38 by CharlesFrancis

Additions:
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 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.

1. When there is a physical calibration between the clocks used to measure initial and final states quantum theory is formulated without expansion.
2. 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.

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 (section 7), 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.

Deletions:
The inner product is defined to generate probabilities. To correctly specify a probability, it is necessary to specify the known conditions to which the probability applies. The teleconnection will be applied to two types of physical situation, distinguished by a difference in conditions. In the case in which 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, cosmological expansion cannot be detected. In this case, the gravitational prediction of the teleconnection is identical to the Levi-Civita connection, at least for gravitational field strengths which can be studied experimentally. When no calibration procedure is available, synchronisation is meaningless and cosmological expansion must be taken into account. In this case, spectroscopic» measurements are predicted to give different results from those of the Levi-Civita connection.

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Edited on 2009-04-10 07:14:07 by CharlesFrancis

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

Deletions:
For a vector x = (x^t, x^r, x^α, x^β) in locally Minkowski, t-r , coordinates at (t₁, A) where ""<span class=math><i>t</i>₁ ↔ τ₁

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Edited on 2009-04-10 07:11:30 by CharlesFrancis

Additions:
The inner product is defined to generate probabilities. To correctly specify a probability, it is necessary to specify the known conditions to which the probability applies. The teleconnection will be applied to two types of physical situation, distinguished by a difference in conditions. In the case in which 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, cosmological expansion cannot be detected. In this case, the gravitational prediction of the teleconnection is identical to the Levi-Civita connection, at least for gravitational field strengths which can be studied experimentally. When no calibration procedure is available, synchronisation is meaningless and cosmological expansion must be taken into account. In this case, spectroscopic» measurements are predicted to give different results from those of the Levi-Civita connection.
With the substitutions dt' = a(τ)dτ r' = a(τ)ρ, θ' = θ, and φ' = φ,
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,
For a vector x = (x^t, x^r, x^α, x^β) in locally Minkowski, t-r , coordinates at (t₁, A) where t₁ ↔ τ₁ 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, <span class=math><i>t</i>. 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 [[http://en.wikipedia.org/wiki/Foliation foliation]]. At each stage of the motion, quantum theory is formulated in a space with a non-physical metric <img alt="Teleconnection-93" title="non-physical metric" src="images/teleconnection/Teleconnection-93.gif" align=top vspace=0>. Classical motion is determinate and may be described as an ordered sequence, <img alt="Teleconnection-94" title="sequence of effectively measured states" src="images/teleconnection/Teleconnection-94.gif" align=top vspace=0>, of <a href=http://www.teleconnection.info/rqg/Observables#Measurement>effectively» measured</a> states at instances <span class=math><i>t<sub>i such that <span class=math>0 < <i>t</i>_<i>i</i>+1 - <i>t</i>_<i>i</i> < δ where <span class=math>δ is sufficiently small that there is negligible alteration in predictions in the limit <span class=math>δ → 0. Each state, <img alt="Teleconnection-98" title="state at t_i" src="images/teleconnection/Teleconnection-98.gif" align=top vspace=0>, is a multiparticle state in [[Multiparticle Fock space]], which has been relabelled using mean properties determined by an effective classical measurement. For the motion between times <span class=math><i>t<sub>i and <span class=math><i>t</i><sub><i>i</i>+1, <img alt="Teleconnection-101" title="state at t_i" src="images/teleconnection/Teleconnection-101.gif" align=top vspace=0> may be regarded as the initial state and <img alt="Teleconnection-102" title="state at t_i+1" src="images/teleconnection/Teleconnection-102.gif" align=top vspace=0> may be regarded as the final state. Since the time evolution of mean properties is determinate, there is no collapse and <img alt="Teleconnection-103" title="state at t_i+1" src="images/teleconnection/Teleconnection-103.gif" align=top vspace=0> is the initial state for the motion to <span class=math><i>t</i><sub><i>i</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 <a href=http://www.teleconnection.info/rqg/Curvature#ParallelTransport>parallel» transport</a>"". 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).

Deletions:
The inner product is defined to generate probabilities. To correctly specify a probability, it is necessary to specify the known conditions to which the probability applies. The teleconnection will be applied to two types of physical situation, distinguished by a difference in conditions. In the case in which 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, cosmological expansion cannot be detected. In this case, the gravitational prediction of the teleconnection is identical to the Levi-Civita connection, at least for gravitational field strengths which can be studied experimentally. When no calibration procedure is available, synchronisation is meaningless and cosmological expansion must be taken into account. In this case, spectroscopic» measurements are predicted to give different results from those of the Levi-Cevitica connection.
With the substitutions dt' = a(τ)dτ and r' = a(τ)ρ,
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,
For a vector x = (x^t, x^r, x^α, x^β) in locally Minkowski t-r , coordinates at (τ₁, A),
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).

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Edited on 2008-09-20 01:42:48 by CharlesFrancis

Additions:
The Teleconnection ↑ Illusory Velocity →

Deletions:
The Teleconnection ↑ Quantum Coordinates →

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Edited on 2008-09-19 10:21:34 by CharlesFrancis

Additions:

←  The Teleconnection  ↑  →

Using τ−ρ coordinates for a Penrose diagram, in a Friedmann cosmology the metric is
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.
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 .

The Teleconnection:  For the barred momentum , the plane wave state at time τ is given in τ−ρ coordinates by
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.
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
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,
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.
Barred momentum is defined in τ−ρ coordinates as for a Friedmann cosmology.
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,
The Teleconnection ↑ Quantum Coordinates →

Deletions:

←  The Teleconnection  ↑  →

Using τ - ρ coordinates for a Penrose diagram, in a Friedmann cosmology the metric is
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.
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 .

The Teleconnection:  For the barred momentum , the plane wave state at time τ is given in τ-ρ coordinates by
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.
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
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,
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.
Barred momentum is defined in τ-ρ coordinates as for a Friedmann cosmology.
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,
The Teleconnection ↑ Particles or Fields? →

---

Edited on 2008-09-12 08:05:43 by CharlesFrancis

Additions:
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.

Deletions:
This definition preserves Newton’s first law and the constancy of the speed of light in non-physical coordinates with 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.

---

Edited on 2008-08-14 01:11:00 by CharlesFrancis

Additions:
Using τ - ρ coordinates for a Penrose diagram, in a Friedmann cosmology the metric is

Deletions:
Using τ - ρ coordinates, as described in Large Scale Structure, in a Friedmann cosmology the metric is

---

Edited on 2008-08-13 04:01:49 by CharlesFrancis

Additions:

←  The Teleconnection  ↑  →

A connection defines the notion of parallel in vector spaces defined at nearby points of a manifold. The teleconnection defines the parallel displacement of momentum in quantum mechanics from an initial state to a final state when the reference matter used to describe the initial state is remote from that used to describe the final state. When the initial and final states are determined with respect to nearby reference matter, the teleconnection is equivalent to the Levi-Civita connection.

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 correctly specify a probability, it is necessary to specify the known conditions to which the probability applies. The teleconnection will be applied to two types of physical situation, distinguished by a difference in conditions. In the case in which 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, cosmological expansion cannot be detected. In this case, the gravitational prediction of the teleconnection is identical to the Levi-Civita connection, at least for gravitational field strengths which can be studied experimentally. When no calibration procedure is available, synchronisation is meaningless and cosmological expansion must be taken into account. In this case, spectroscopic» measurements are predicted to give different results from those of the Levi-Cevitica connection.
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. The view taken here is that quantum theory gives a calculation of probabilities between an initial state and final state, not a description of physical reality between initial and final measured 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, as described in Large Scale Structure, 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. The metric in t-r coordinates is

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.

Definition:  For a vector x = (x^τ, x^ρ, x^θ, x^φ) at (τ₁, A), the corresponding barred vector is

Definition:  For the displacement, x = (x^τ, x^ρ, x^θ, x^φ), from (τ₁, A), the corresponding barred displacement vector is

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:

Definition:  Barred momentum is

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 .

The Teleconnection:  For the barred momentum , the plane wave state at time τ is given in τ-ρ coordinates by

where the barred dot product uses the non-physical metric, .
This definition preserves Newton’s first law and the constancy of the speed of light in non-physical coordinates with 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.

Theorem:  Let a₁ = a(t₁) and a₀ = a(t₀) . Light emitted at time t₁ with wavelength λ₁ from a distant point A, and detected at B, at time t₀ with wavelength λ₀ is redshifted according to

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

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 and r' = a(τ)ρ,

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 momentum is defined in τ-ρ coordinates as for a Friedmann cosmology.

Definition:  For a vector x = (x^τ, x^ρ, x^θ, x^φ), at (τ₁, A), the corresponding barred vector is

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

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:

Definition:  Barred momentum is

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 ↑ Particles or Fields? →

Deletions:

←  The Teleconnection in a Friedmann Cosmology  ↑  →

To introduce the idea of the teleconnection I will here describe it diagramatically for a Friedmann Cosmology. In this case the analysis is particularly simple. The result is that the cosmological redshift of light from a distant galaxy is proportional to the square of the expansion parameter, not linear with it as predicted by the Levi-Civita connection. A more general, and more mathematical, treatment will be given on the next page.

The Status Quo

In non-Euclidean geometry, an affine connection is a rule defining parallel vectors, when the vectors are defined in tangent space at points separated by a small displacement. An affine connection allows us to parallel transport a vector from one place to another along a path in such a way that we may say that it remains parallel to itself in each part of the motion. Sliding a ruler using parallel transport is clearly sound for coordinate systems local to the Earth, but we cannot physically transport rulers indefinitely far into space, and certainly not from distant stars. Instead we must rely on the transmission of light. If we do slide a ruler, for example on a spacecraft like Pioneer, even a short way into space, we cease to be in direct contact with it and have to rely on electromagnetic transmissions to read the ruler. In practice, we deduce the properties of distant coordinate systems from properties we find in light from distant stellar objects. We can do this only if we know the laws governing the transmission of light over large distances.
In standard general relativity it is assumed that photon momentum is parallel transported through large distances, using the Levi-Civita connection in a manner analogous to sliding a ruler across a curved surface while maintaining parallellism in each part of the motion. It is sometimes argued that, since the the laws of physics are locally the same in all coordinates systems, Maxwell’s equations, which govern the propagation of light, must hold locally in empty space, and hence that the Levi-Civita connection holds in empty space. The argument relies on induction», which cannot be rigorously justified, and it ignores the fact that we cannot define local coordinates in empty space, since empty space contains neither reference matter nor matter whose position can be measured. More seriously still, it ignores the known inconsistency between classical electromagnetism and general relativity. It was pointed out by Einstein that the Levi-Civita connection appears to be wrong. In particular, it leads to an inconsistency in the paths of light transmitted by a distant object, seen in the mismatch of Alf’s and Beth’s coordinates. Moreover, the application of the Levi-Civita connection takes no account of the propagation of a photon wave function in a curved spacetime, which would imply that a photon created with precise momentum in a distant galaxy would not have precise momentum when detected on Earth.
In the early part of the 20th century, attempts were made, notably by Einstein himself, and by Herman Weyl, to unify general relativity with classical electromagnetism. Those attempts failed and it seems that theoretical physicists have lost interest. The Levi-Civita connection is almost universally accepted, and its inconsistency is largely ignored. This is dangerous. Our knowledge of stellar kinematics and of cosmology is almost entirely dependent on spectroscopy» (doppler shift»). There is no direct empirical evidence that the standard Doppler» law holds for the transmission of light across astronomical distances. At the same time, observations appear to reveal a universe with bizarre and paradoxical properties. There is still no sound theoretical basis for the cosmological constant», for exotic cold dark matter», for cosmic inflation» or for modifying Newton’s law of gravity (MOND»). A significant minority of cosmologists recognise that there are serious problems in standard cosmology, but, in the main, empirical evidence which plainly contradicts the standard model, such as galaxy lensing profiles, and the rotation curves of globular clusters», is ignored.

Distant Parallelism

A number of Einstein’s attempts at unification with classical electromagnetism were centred on the investigation of distant parallelism». Relational quantum gravity uses as similar, but not identical, notion, and seeks unification between general relativity and quantum electrodynamics. In relational quantum gravity, general relativity is regarded as correct in the classical correspondence, but must be extended for the treatment of quantum phenomena. Since light consists of photons, light is treated quantum mechanically, even in the description of its transmission over astronomical distances. This is done using a form of distant parallelism which I have called the teleconnection to distinguish it from other teleparallel theories of gravity. I will show that the teleconnection reduces to the Levi-Civitaconnection in the classical correspondence, and that, when cosmological expansion can be neglected, predicted redshifts are identical to those of general relativity, but it turns out that cosmological expansion affects the calculation of redshift even, in the case of the Pioneer anomaly», within the environs of the solar system.
The teleconnection is an extension of classical general relativity and applies to the treatment of quantum mechanics. It was proposed to reconcile the propagation of the wave function in standard quantum theory using Minkowski metric with curvature in classical spacetime. Instead of defining a connection between vectors at nearby points, the teleconnection is defined remotely, between vectors used to describe the initial and final states when the reference matter used to describe the initial state is remote from that used to describe the final state.
To define the teleconnection, it is observed that quantum theory provides rules for calculating the probability of the measurement of a final state, given the measurement of a initial state. According to the rules, imaginary wave motions are used to calculate the probability that a given initial state will lead to a given final state. Only the measured, i.e. initial and final, states are physically meaningful. The intermediate wave motion should not be thought of as physically real, but as a device for calculating probabilities.
Since the calculation of wave evolution is carried out by an observer at a particular place and time, it is natural that the calculation should be done using the metric at that place and time. In other words, wave evolution is determined in a tangent space. More strictly this generalises the notion of tangent space (it can be regarded as a fibre»), but for our purpose the important properties are those of a tangent space, and I will refer to it as such. As for the wave function itself, no ontology is attached to tangent space. 3-momentum has been defined from the Fourier transform of the probability amplitude, and found to be a well defined, locally conserved, property. Now, if momentum is a well defined property for Alf on a space craft or another planet, then it is also a well defined property for Beth on Earth, because Alf can communicate to Beth his value of momentum. Alf's value of momentum does not have to be the same as Beth's, but we seek a way of converting his value to the value Beth will use.

The teleconnection defines the way in which Beth converts the value of momentum calculated by Alf to the value found in her own coordinate system, preserving parallelism. Wave evolution (red) takes place in a tangent space. Discontinuity is associated with wave function collapse, at which point the teleconnection describes the projection of the evolved initial state to the final state (blue). The teleconnection does not apply when there is an intermediate measured (or classical) state. The same recipe applies whether Alf and Beth are separated by astronomical distances, or are in the same laboratory, and reduces to the Levi-Civita connection when the initial and final states are separated by small times. The classical correspondence is conceived as a limit of many quantum motions in which curved spacetime (green) emerges as the envelope of flat spaces (pale blue), and in which parallel transport is found in the limit of small teleparallel displacements.

The Teleconnection in a Friedmann Cosmology

Let Alf be an observer at A, on a space craft or a distant planet, and let Beth be an observer at B, such that Alf can signal to Beth and Alf and Beth are moving with the cosmic fluid, i.e. on geodesics from the big bang as described in Weyl’s postulate. At cosmic time, t₁, of emission of a photon passing from Alf to Beth, Alf defines synchronous co-ordinates at cosmic time t₁ in 3 dimensions, with the origin at B, using proper distance as the radial coordinate. For definiteness and clarity, I will describe a closed cosmos. The universe can then be mapped onto a 3-dimensional finite space, which will be called Alf’s map. Beth defines Beth’s map in exactly the same way, also with the origin at B, at cosmic time t₀ when the photon is detected, and to the same scale, i.e. using units of measurement defined by identical physical processes. For a closed universe Alf’s and Beth’s maps are each contained in a sphere. Let a(t) be the scale factor and let a₀ = a(t₀) and a₁ = a(t₁). If the universe expands during the time of travel of the photon from Alf to Beth, then Beth’s map is larger than Alf’s map by a factor a₀ ⁄ a₁.

Alf and Beth’s maps are each defined at constant cosmic time with radial coordinate equal to proper distance. The metric is as described in Large Scale Structure of the Universe. Alf and Beth are here drawn in different galaxies. For convenience Beth’s galaxy is taken as the origin of both maps. The maps are spherical in three dimensional space with one dimension suppressed. Identical galaxies are shown on geodesics emanating from the big bang in the x-t plane. As drawn, each galaxy has the same radial dimension but the galaxies acquire an increasing angular stretch towards the edge of the map. The scaling distortions are physically meaningless and can be removed by embedding the x-y plane onto the surface of a sphere with B at the North pole. The circles on the maps correspond to lines of latitude at 15° and 30° intervals for Beth’s and Alf’s respective maps. The region corresponding to the Southern hemisphere is shaded blue. The outermost circumference is infinitely stretched and represents a point, the South pole. The galaxy at the South pole is so stretched that it surrounds the map. The radial path of light is shown in yellow, with the signal from Alf to Beth shown in red. The region outside Beth’s past light cone is shaded grey. A space-like radial vector has the same apparent and proper length at any point (green).
Beth defines new coordinates, maintaining her own local distance scale, but rescaling Alf’s map so that it is equal in size with Beth’s map, and rescaling the time axis so that light speed is constant in radial directions in the in the new coordinates. The metric is as described in Large Scale Structure of the Universe. Together with non-physical Minkowski metric h, the new coordinates are tangent chart at Beth’s position, B at cosmic time t₀.
Beth defines tangent space vectors in these coordinates (red), denoted by a bar, and which can be translated radially in tangent space and are equal to physical vectors at her own origin. She defines plane wave states obeying Newton’s first law using barred vectors,

The wave function,

is strictly a function on an abstract tangent space, and is defined at the origin for each observer. This reflects the orthodox interpretation of quantum theory, as used in relational quantum gravity, in which wave functions do not represent a physical wave property, and are merely used by an observer to calculate probabilities.

Similarly Alf defines a tangent chart at A at cosmic time t1, so as to maintain his own local distance scale. Alf’s and Beth’s tangent charts are identical up to a scale factor, a1 ⁄ a0. The teleconnection defines the relationship between Alf’s and Beth’s tangent charts, and makes possible the definition of an inner product beween states in Alf’s and Beth’s formulations of quantum theory. To define the teleconnection, Beth first enlarges Alf’s map by the factor a0 ⁄ a1 (equivalently Alf reduces Beth’s map).

Definition:  The teleconnection is such that photon momentum is represented by a vector (red) of equal magnitude and direction on Beth’s map and on Alf’s enlarged map.
The definition of a teleconnection assumes that if momentum has a precise value at one place and time then it also has a precise value other places and times and is justified empirically in so far as observation yields precise values for cosmological redshift after allowing for dispersion due to dust or other known factors. This is a fundamental assumption in this model, of equal importance to the assumption of the constancy of the speed of light in special relativity. Like that assumption, if it were dropped we would be left, not with a different theory, but with no known consistent theory.
In practice, Beth can compare the scale of her map to that of Alf’s map by studying redshift. There are two scaling effects. First Alf’s map has been enlarged by a factor a₀ ⁄ a₁. In addition, the scaling on the map changes in time, giving another factor a₀ ⁄ a₁ (as may be calculated from the physical metric). Thus, the model predicts a net factor for cosmological redshift which varies with the square of the expansion parameter:

The square redshift law applies to spectroscopic» measurements of the wavelength of light, and takes into account that, as viewed by Beth, Alf’s coordinate axes are enlarged by a factor a₀ ⁄ a₁. Alf’s measurement of energy-momentum is relative to his own coordinate axes. This factor must be removed in the calculation of energy/momentum of light from a distant source. Thus, the energy transferred by a photon passing from Alf to Beth is inverse to expansion, not inverse to the square of expansion, giving the same relation as is given by parallel transport under the Levi-Civita connection. The square redshift law affects spectroscopic measurements of light from distant sources, but does not affect energy transfer.

Classical Spacetime

When Beth performs a measurement, the wave function collapses. Her measurement determines both the final state of a quantum motion of a particle travelling from Alf to Beth, and the initial state for the next part of the motion. At the point of the measurement quantum theory is reformulated with coordinate axes in scale by a factor a₁ ⁄ a₀.
The teleconnection applies to a quantum motion between an initial state and a final state when it is not meaningful to describe the intervening state in terms of measured physical quantities. In the classical correspondence, time is taken as a continuous variable and motion may be described as a sequence of measured states at instances t = t_n < t_n−1 < … < t₁ < t₀  in the limit in which max(t_i−1 − t_i) → 0. The state at any instant, t_i, may be regarded as an initial state, and the state at the next instant, t_i−1, may be regarded as a final state for that part of the motion. Quantum theory is redefined and the final state then becomes the initial state for the next part of the motion. At each instant, t_i, when quantum theory is redefined, the coordinate system is rescaled, removing a factor a(t_i−1) ⁄ a(t_i). The paths of light and of the galaxies become continuous in the limit, and the classical notion of continuous expansion is restored.

It follows that, for a classical motion, momentum is parallel transported from t_i to t_i−1 in the normal way. The result is parallel transport. An overall factor,

is removed from cosmological redshift, and the usual linear law,

is recovered for a classical ray of light which is observable in each part of the motion. For example, the cosmic background radiation» is continuously observable and obeys the linear redshift law, not the square law applicable to the observation of a photon from a distant source, and in which the detection of the photon takes place after diffraction or refraction of the quantum wave function.
The Teleconnection in a Friedmann Cosmology ↑ The Teleconnection →

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Edited on 2008-08-13 04:00:40 by CharlesFrancis

Additions:

←  The Teleconnection in a Friedmann Cosmology  ↑  →

To introduce the idea of the teleconnection I will here describe it diagramatically for a Friedmann Cosmology. In this case the analysis is particularly simple. The result is that the cosmological redshift of light from a distant galaxy is proportional to the square of the expansion parameter, not linear with it as predicted by the Levi-Civita connection. A more general, and more mathematical, treatment will be given on the next page.

The Status Quo

In non-Euclidean geometry, an affine connection is a rule defining parallel vectors, when the vectors are defined in tangent space at points separated by a small displacement. An affine connection allows us to parallel transport a vector from one place to another along a path in such a way that we may say that it remains parallel to itself in each part of the motion. Sliding a ruler using parallel transport is clearly sound for coordinate systems local to the Earth, but we cannot physically transport rulers indefinitely far into space, and certainly not from distant stars. Instead we must rely on the transmission of light. If we do slide a ruler, for example on a spacecraft like Pioneer, even a short way into space, we cease to be in direct contact with it and have to rely on electromagnetic transmissions to read the ruler. In practice, we deduce the properties of distant coordinate systems from properties we find in light from distant stellar objects. We can do this only if we know the laws governing the transmission of light over large distances.
In standard general relativity it is assumed that photon momentum is parallel transported through large distances, using the Levi-Civita connection in a manner analogous to sliding a ruler across a curved surface while maintaining parallellism in each part of the motion. It is sometimes argued that, since the the laws of physics are locally the same in all coordinates systems, Maxwell’s equations, which govern the propagation of light, must hold locally in empty space, and hence that the Levi-Civita connection holds in empty space. The argument relies on induction», which cannot be rigorously justified, and it ignores the fact that we cannot define local coordinates in empty space, since empty space contains neither reference matter nor matter whose position can be measured. More seriously still, it ignores the known inconsistency between classical electromagnetism and general relativity. It was pointed out by Einstein that the Levi-Civita connection appears to be wrong. In particular, it leads to an inconsistency in the paths of light transmitted by a distant object, seen in the mismatch of Alf’s and Beth’s coordinates. Moreover, the application of the Levi-Civita connection takes no account of the propagation of a photon wave function in a curved spacetime, which would imply that a photon created with precise momentum in a distant galaxy would not have precise momentum when detected on Earth.
In the early part of the 20th century, attempts were made, notably by Einstein himself, and by Herman Weyl, to unify general relativity with classical electromagnetism. Those attempts failed and it seems that theoretical physicists have lost interest. The Levi-Civita connection is almost universally accepted, and its inconsistency is largely ignored. This is dangerous. Our knowledge of stellar kinematics and of cosmology is almost entirely dependent on spectroscopy» (doppler shift»). There is no direct empirical evidence that the standard Doppler» law holds for the transmission of light across astronomical distances. At the same time, observations appear to reveal a universe with bizarre and paradoxical properties. There is still no sound theoretical basis for the cosmological constant», for exotic cold dark matter», for cosmic inflation» or for modifying Newton’s law of gravity (MOND»). A significant minority of cosmologists recognise that there are serious problems in standard cosmology, but, in the main, empirical evidence which plainly contradicts the standard model, such as galaxy lensing profiles, and the rotation curves of globular clusters», is ignored.

Distant Parallelism

A number of Einstein’s attempts at unification with classical electromagnetism were centred on the investigation of distant parallelism». Relational quantum gravity uses as similar, but not identical, notion, and seeks unification between general relativity and quantum electrodynamics. In relational quantum gravity, general relativity is regarded as correct in the classical correspondence, but must be extended for the treatment of quantum phenomena. Since light consists of photons, light is treated quantum mechanically, even in the description of its transmission over astronomical distances. This is done using a form of distant parallelism which I have called the teleconnection to distinguish it from other teleparallel theories of gravity. I will show that the teleconnection reduces to the Levi-Civitaconnection in the classical correspondence, and that, when cosmological expansion can be neglected, predicted redshifts are identical to those of general relativity, but it turns out that cosmological expansion affects the calculation of redshift even, in the case of the Pioneer anomaly», within the environs of the solar system.
The teleconnection is an extension of classical general relativity and applies to the treatment of quantum mechanics. It was proposed to reconcile the propagation of the wave function in standard quantum theory using Minkowski metric with curvature in classical spacetime. Instead of defining a connection between vectors at nearby points, the teleconnection is defined remotely, between vectors used to describe the initial and final states when the reference matter used to describe the initial state is remote from that used to describe the final state.
To define the teleconnection, it is observed that quantum theory provides rules for calculating the probability of the measurement of a final state, given the measurement of a initial state. According to the rules, imaginary wave motions are used to calculate the probability that a given initial state will lead to a given final state. Only the measured, i.e. initial and final, states are physically meaningful. The intermediate wave motion should not be thought of as physically real, but as a device for calculating probabilities.
Since the calculation of wave evolution is carried out by an observer at a particular place and time, it is natural that the calculation should be done using the metric at that place and time. In other words, wave evolution is determined in a tangent space. More strictly this generalises the notion of tangent space (it can be regarded as a fibre»), but for our purpose the important properties are those of a tangent space, and I will refer to it as such. As for the wave function itself, no ontology is attached to tangent space. 3-momentum has been defined from the Fourier transform of the probability amplitude, and found to be a well defined, locally conserved, property. Now, if momentum is a well defined property for Alf on a space craft or another planet, then it is also a well defined property for Beth on Earth, because Alf can communicate to Beth his value of momentum. Alf's value of momentum does not have to be the same as Beth's, but we seek a way of converting his value to the value Beth will use.

The teleconnection defines the way in which Beth converts the value of momentum calculated by Alf to the value found in her own coordinate system, preserving parallelism. Wave evolution (red) takes place in a tangent space. Discontinuity is associated with wave function collapse, at which point the teleconnection describes the projection of the evolved initial state to the final state (blue). The teleconnection does not apply when there is an intermediate measured (or classical) state. The same recipe applies whether Alf and Beth are separated by astronomical distances, or are in the same laboratory, and reduces to the Levi-Civita connection when the initial and final states are separated by small times. The classical correspondence is conceived as a limit of many quantum motions in which curved spacetime (green) emerges as the envelope of flat spaces (pale blue), and in which parallel transport is found in the limit of small teleparallel displacements.

The Teleconnection in a Friedmann Cosmology

Let Alf be an observer at A, on a space craft or a distant planet, and let Beth be an observer at B, such that Alf can signal to Beth and Alf and Beth are moving with the cosmic fluid, i.e. on geodesics from the big bang as described in Weyl’s postulate. At cosmic time, t₁, of emission of a photon passing from Alf to Beth, Alf defines synchronous co-ordinates at cosmic time t₁ in 3 dimensions, with the origin at B, using proper distance as the radial coordinate. For definiteness and clarity, I will describe a closed cosmos. The universe can then be mapped onto a 3-dimensional finite space, which will be called Alf’s map. Beth defines Beth’s map in exactly the same way, also with the origin at B, at cosmic time t₀ when the photon is detected, and to the same scale, i.e. using units of measurement defined by identical physical processes. For a closed universe Alf’s and Beth’s maps are each contained in a sphere. Let a(t) be the scale factor and let a₀ = a(t₀) and a₁ = a(t₁). If the universe expands during the time of travel of the photon from Alf to Beth, then Beth’s map is larger than Alf’s map by a factor a₀ ⁄ a₁.

Alf and Beth’s maps are each defined at constant cosmic time with radial coordinate equal to proper distance. The metric is as described in Large Scale Structure of the Universe. Alf and Beth are here drawn in different galaxies. For convenience Beth’s galaxy is taken as the origin of both maps. The maps are spherical in three dimensional space with one dimension suppressed. Identical galaxies are shown on geodesics emanating from the big bang in the x-t plane. As drawn, each galaxy has the same radial dimension but the galaxies acquire an increasing angular stretch towards the edge of the map. The scaling distortions are physically meaningless and can be removed by embedding the x-y plane onto the surface of a sphere with B at the North pole. The circles on the maps correspond to lines of latitude at 15° and 30° intervals for Beth’s and Alf’s respective maps. The region corresponding to the Southern hemisphere is shaded blue. The outermost circumference is infinitely stretched and represents a point, the South pole. The galaxy at the South pole is so stretched that it surrounds the map. The radial path of light is shown in yellow, with the signal from Alf to Beth shown in red. The region outside Beth’s past light cone is shaded grey. A space-like radial vector has the same apparent and proper length at any point (green).
Beth defines new coordinates, maintaining her own local distance scale, but rescaling Alf’s map so that it is equal in size with Beth’s map, and rescaling the time axis so that light speed is constant in radial directions in the in the new coordinates. The metric is as described in Large Scale Structure of the Universe. Together with non-physical Minkowski metric h, the new coordinates are tangent chart at Beth’s position, B at cosmic time t₀.
Beth defines tangent space vectors in these coordinates (red), denoted by a bar, and which can be translated radially in tangent space and are equal to physical vectors at her own origin. She defines plane wave states obeying Newton’s first law using barred vectors,

The wave function,

is strictly a function on an abstract tangent space, and is defined at the origin for each observer. This reflects the orthodox interpretation of quantum theory, as used in relational quantum gravity, in which wave functions do not represent a physical wave property, and are merely used by an observer to calculate probabilities.

Similarly Alf defines a tangent chart at A at cosmic time t1, so as to maintain his own local distance scale. Alf’s and Beth’s tangent charts are identical up to a scale factor, a1 ⁄ a0. The teleconnection defines the relationship between Alf’s and Beth’s tangent charts, and makes possible the definition of an inner product beween states in Alf’s and Beth’s formulations of quantum theory. To define the teleconnection, Beth first enlarges Alf’s map by the factor a0 ⁄ a1 (equivalently Alf reduces Beth’s map).

Definition:  The teleconnection is such that photon momentum is represented by a vector (red) of equal magnitude and direction on Beth’s map and on Alf’s enlarged map.
The definition of a teleconnection assumes that if momentum has a precise value at one place and time then it also has a precise value other places and times and is justified empirically in so far as observation yields precise values for cosmological redshift after allowing for dispersion due to dust or other known factors. This is a fundamental assumption in this model, of equal importance to the assumption of the constancy of the speed of light in special relativity. Like that assumption, if it were dropped we would be left, not with a different theory, but with no known consistent theory.
In practice, Beth can compare the scale of her map to that of Alf’s map by studying redshift. There are two scaling effects. First Alf’s map has been enlarged by a factor a₀ ⁄ a₁. In addition, the scaling on the map changes in time, giving another factor a₀ ⁄ a₁ (as may be calculated from the physical metric). Thus, the model predicts a net factor for cosmological redshift which varies with the square of the expansion parameter:

The square redshift law applies to spectroscopic» measurements of the wavelength of light, and takes into account that, as viewed by Beth, Alf’s coordinate axes are enlarged by a factor a₀ ⁄ a₁. Alf’s measurement of energy-momentum is relative to his own coordinate axes. This factor must be removed in the calculation of energy/momentum of light from a distant source. Thus, the energy transferred by a photon passing from Alf to Beth is inverse to expansion, not inverse to the square of expansion, giving the same relation as is given by parallel transport under the Levi-Civita connection. The square redshift law affects spectroscopic measurements of light from distant sources, but does not affect energy transfer.

Classical Spacetime

When Beth performs a measurement, the wave function collapses. Her measurement determines both the final state of a quantum motion of a particle travelling from Alf to Beth, and the initial state for the next part of the motion. At the point of the measurement quantum theory is reformulated with coordinate axes in scale by a factor a₁ ⁄ a₀.
The teleconnection applies to a quantum motion between an initial state and a final state when it is not meaningful to describe the intervening state in terms of measured physical quantities. In the classical correspondence, time is taken as a continuous variable and motion may be described as a sequence of measured states at instances t = t_n < t_n−1 < … < t₁ < t₀  in the limit in which max(t_i−1 − t_i) → 0. The state at any instant, t_i, may be regarded as an initial state, and the state at the next instant, t_i−1, may be regarded as a final state for that part of the motion. Quantum theory is redefined and the final state then becomes the initial state for the next part of the motion. At each instant, t_i, when quantum theory is redefined, the coordinate system is rescaled, removing a factor a(t_i−1) ⁄ a(t_i). The paths of light and of the galaxies become continuous in the limit, and the classical notion of continuous expansion is restored.

It follows that, for a classical motion, momentum is parallel transported from t_i to t_i−1 in the normal way. The result is parallel transport. An overall factor,

is removed from cosmological redshift, and the usual linear law,

is recovered for a classical ray of light which is observable in each part of the motion. For example, the cosmic background radiation» is continuously observable and obeys the linear redshift law, not the square law applicable to the observation of a photon from a distant source, and in which the detection of the photon takes place after diffraction or refraction of the quantum wave function.
The Teleconnection in a Friedmann Cosmology ↑ The Teleconnection →

Deletions:

←  The Teleconnection  →

A connection defines the notion of parallel in vector spaces defined at nearby points of a manifold. The teleconnection defines the parallel displacement of momentum in quantum mechanics from an initial state to a final state when the reference matter used to describe the initial state is remote from that used to describe the final state. When the initial and final states are determined with respect to nearby reference matter, the teleconnection is equivalent to the Levi-Civita connection.

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 correctly specify a probability, it is necessary to specify the known conditions to which the probability applies. The teleconnection will be applied to two types of physical situation, distinguished by a difference in conditions. In the case in which 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, cosmological expansion cannot be detected. In this case, the gravitational prediction of the teleconnection is identical to the Levi-Civita connection, at least for gravitational field strengths which can be studied experimentally. When no calibration procedure is available, synchronisation is meaningless and cosmological expansion must be taken into account. In this case, spectroscopic» measurements are predicted to give different results from those of the Levi-Cevitica connection.
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. The view taken here is that quantum theory gives a calculation of probabilities between an initial state and final state, not a description of physical reality between initial and final measured 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, as described in Large Scale Structure, 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. The metric in t-r coordinates is

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.

Definition:  For a vector x = (x^τ, x^ρ, x^θ, x^φ) at (τ₁, A), the corresponding barred vector is

Definition:  For the displacement, x = (x^τ, x^ρ, x^θ, x^φ), from (τ₁, A), the corresponding barred displacement vector is

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:

Definition:  Barred momentum is

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 .

The Teleconnection:  For the barred momentum , the plane wave state at time τ is given in τ-ρ coordinates by

where the barred dot product uses the non-physical metric, .
This definition preserves Newton’s first law and the constancy of the speed of light in non-physical coordinates with 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.

Theorem:  Let a₁ = a(t₁) and a₀ = a(t₀) . Light emitted at time t₁ with wavelength λ₁ from a distant point A, and detected at B, at time t₀ with wavelength λ₀ is redshifted according to

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

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 and r' = a(τ)ρ,

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 momentum is defined in τ-ρ coordinates as for a Friedmann cosmology.

Definition:  For a vector x = (x^τ, x^ρ, x^θ, x^φ), at (τ₁, A), the corresponding barred vector is

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

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:

Definition:  Barred momentum is

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 ↑ Particles or Fields? →

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Edited on 2008-06-01 01:44:48 by CharlesFrancis

Additions:
The Teleconnection ↑ Particles or Fields? →

Deletions:
The Teleconnection ↑ Particles or Fields →

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Edited on 2008-05-31 04:59:48 by CharlesFrancis

Additions:

←  The Teleconnection  →

Deletions:

←  The Teleconnection  →

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Edited on 2008-05-31 04:57:02 by CharlesFrancis

Additions:
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.

Deletions:
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 this 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.

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Edited on 2008-05-31 04:55:58 by CharlesFrancis

Additions:
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 this 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.

Deletions:
The purpose of the teleconnection is to retain the fundamental arguments leading from special relativity to quantum electrodynamics and to general relativity. Spacetime is simply 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. Quantum theory is simply the mathematical structure appropriate to physical situations in which only probabilistic results are possible for measurements of position at given time. The teleconnection adapts this probabilistic structure 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.

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Edited on 2008-05-31 04:50:32 by CharlesFrancis

Additions:
The Teleconnection ↑ Particles or Fields →

Deletions:

Newton's Views on Space, Time, and Motion “Isaac Newton founded classical mechanics on the view that space is something distinct from body and that time is something that passes uniformly without regard to whatever happens in the world. For this reason he spoke of absolute space and absolute time, so as to distinguish these entities from the various ways by which we measure them (which he called relative spaces and relative times). From antiquity into the eighteenth century, contrary views which denied that space and time are real entities maintained that the world is necessarily a material plenum. Concerning space, they held that the idea of empty space is a conceptual impossibility. Space is nothing but an abstraction we use to compare different arrangements of the bodies constituting the plenum. Concerning time, they insisted, there can be no lapse of time without change occurring somewhere. Time is merely a measure of the cycles of change within the world.” - the Stanford Encyclopedia of Philosophy^».

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“In the general case we cannot speak of an observable having a value for a particular state, but we can … speak of the probability of its having a specified value for the state, meaning the probability of this specified value being obtained when one makes a measurement of the observable” - Dirac, Quantum Mechanics, Clarendon Press, Oxford 1958.

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. Relational quantum gravity develops these ideas from first principles. Mathematicians have always recognised that complex numbers are an abstraction and cannot represent anything physically real. A complex wave function arises naturally in a mathematical structure designed to yields probabilistic results of measurement, constrained by relativity. In relational quantum gravity, Feynman diagrams describe the possible configurations of a plenum consisting of interactions between physical electrons and photons. Spacetime does not appear in the fundamental structure described by a Feynman diagram, but only as an organisational principle, in the resulting calculations of probabilities for observational results.
The Teleconnection ↑ Discrete Quantum Electrodynamics →

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Edited on 2008-05-22 01:52:38 by CharlesFrancis

Additions:

Definition:  For a vector x = (x^τ, x^ρ, x^θ, x^φ) at (τ₁, A), the corresponding barred vector is

Definition:  For the displacement, x = (x^τ, x^ρ, x^θ, x^φ), from (τ₁, A), the corresponding barred displacement vector is

Geometries with Expansion

This seems reasonable for points which are not hidden by an event horizon». Points inside an event horizon will be considered again later.
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,

Definition:  For a vector x = (x^τ, x^ρ, x^θ, x^φ), at (τ₁, A), the corresponding barred vector is
In locally Minkowski coordinates at A, gravitational redshift is given by a factor (g₀₀)^−½, in agreement with standard general relativity.

Deletions:

Definition:  For a vector x = (x^τ, x^ρ, x^θ, x^φ) at (τ₁, A), the barred vector, , is

Definition:  For the displacement, x = (x^τ, x^ρ, x^θ, x^φ), from (τ₁, A), the barred displacement vector, , is

Other Geometries with Expansion

This seems reasonable for points which are not hidden by an event horizon». The meaning of points inside an event horizon will be considered again later.
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 = dt' ⁄ k and dr = kdr', rdα = r'dθ', r sinα dβ = r' sinθ dφ'. For small r,

Definition:  For a vector x = (x^τ, x^ρ, x^θ, x^φ), at (τ₁, A) the barred vector, , is
In locally Minkowski coordinates at A, gravitational redshift is given by a factor (g₀₀)^−½, in agreement with standard general relativity.

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Oldest known version of this page was edited on 2008-05-05 09:01:32 by CharlesFrancis []

←  The Teleconnection  →

Absolute Space and Time

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. Relational quantum gravity develops these ideas from first principles. Mathematicians have always recognised that complex numbers are an abstraction and cannot represent anything physically real. A complex wave function arises naturally in a mathematical structure designed to yields probabilistic results of measurement, constrained by relativity. In relational quantum gravity, Feynman diagrams describe the possible configurations of a plenum consisting of interactions between physical electrons and photons. Spacetime does not appear in the fundamental structure described by a Feynman diagram, but only as an organisational principle, in the resulting calculations of probabilities for observational results.

The purpose of the teleconnection is to retain the fundamental arguments leading from special relativity to quantum electrodynamics and to general relativity. Spacetime is simply 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. Quantum theory is simply the mathematical structure appropriate to physical situations in which only probabilistic results are possible for measurements of position at given time. The teleconnection adapts this probabilistic structure 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 correctly specify a probability, it is necessary to specify the known conditions to which the probability applies. The teleconnection will be applied to two types of physical situation, distinguished by a difference in conditions. In the case in which 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, cosmological expansion cannot be detected. In this case, the gravitational prediction of the teleconnection is identical to the Levi-Civita connection, at least for gravitational field strengths which can be studied experimentally. When no calibration procedure is available, synchronisation is meaningless and cosmological expansion must be taken into account. In this case, spectroscopic» measurements are predicted to give different results from those of the Levi-Cevitica connection.

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. The view taken here is that quantum theory gives a calculation of probabilities between an initial state and final state, not a description of physical reality between initial and final measured 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, as described in Large Scale Structure, 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. The metric in t-r coordinates is
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.


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 non-physical coordinates with 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.

Other 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». The meaning of 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 and r' = a(τ)ρ,
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 = dt' ⁄ k and dr = kdr', rdα = r'dθ', r sinα dβ = r' sinθ dφ'. For small r,

Barred momentum is 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 (τ₁, A),

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 ↑ Discrete Quantum Electrodynamics →