Most recent edit on 2008-09-20 00:23:50 by CharlesFrancis

Additions:
The construction of qed as a model of particle interactions using finite dimensional Hilbert space is straightforward, but sweeps thorny issues under the carpet. The problems are deep enough that most physicists believe that this approach is fundamentally flawed. I move on from the description of standard theory from a relational viewpoint, and describe my own research into a resolution. Quantum Covariance deals with the issue that manifest covariance requires a continuum model. The Teleconnection shows how quantum theory can be consistently defined when classical spacetime is curved. I conclude that quantum field theory should be interpreted strictly as a theory of discrete particles. A consequence is that the arguments which led to Minkowski metric in special relativity must be adapted and lead to Einstein’s Field Equation in the case for a single gravitating particle.

Quantum Coordinates

Deletions:
The construction of qed as a model of particle interactions using finite dimensional Hilbert space is straightforward, but sweeps thorny issues under the carpet. The problems are deep enough that most physicists believe that this approach is fundamentally flawed. I move on from the description of standard theory from a relational viewpoint, and describe my own research into a resolution. Quantum Covariance deals with the issue that manifest covariance requires a continuum model. The Teleconnection shows how a quantum theory described using Minkowski metric remains consistent when classical spacetime is curved. I conclude that quantum field theory should be interpreted strictly as a theory of discrete particles. Together with A consequence is that the arguments which led to Minkowski metric in special relativity must be adapted and yield Einstein’s Field Equation in the case for a single gravitating particle.
small black diamond

Quantum Coordinates - under construction

Edited on 2008-09-19 10:24:40 by CharlesFrancis

Additions:
small black diamond

Quantum Coordinates - under construction

Edited on 2008-08-29 22:46:10 by CharlesFrancis

Additions:
The construction of qed as a model of particle interactions using finite dimensional Hilbert space is straightforward, but sweeps thorny issues under the carpet. The problems are deep enough that most physicists believe that this approach is fundamentally flawed. I move on from the description of standard theory from a relational viewpoint, and describe my own research into a resolution. Quantum Covariance deals with the issue that manifest covariance requires a continuum model. The Teleconnection shows how a quantum theory described using Minkowski metric remains consistent when classical spacetime is curved. I conclude that quantum field theory should be interpreted strictly as a theory of discrete particles. Together with A consequence is that the arguments which led to Minkowski metric in special relativity must be adapted and yield Einstein’s Field Equation in the case for a single gravitating particle.

Deletions:
The construction of qed as a model of particle interactions using finite dimensional Hilbert space is straightforward, but sweeps thorny issues under the carpet. The problems are deep enough that most physicists believe that this approach is fundamentally flawed. I move on from the description of standard theory from a relational viewpoint, and describe my own research into a resolution. Quantum Covariance deals with the issue that manifest covariance requires a continuum model. The Teleconnection shows how a quantum theory described using Minkowski metric remains consistent when classical spacetime is curved. A consequence is that the arguments which led to Minkowski metric in special relativity must be adapted and yield Einstein’s Field Equation in the case for a single gravitating particle.

Edited on 2008-08-29 22:37:27 by CharlesFrancis

Additions:
The construction of qed as a model of particle interactions using finite dimensional Hilbert space is straightforward, but sweeps thorny issues under the carpet. The problems are deep enough that most physicists believe that this approach is fundamentally flawed. I move on from the description of standard theory from a relational viewpoint, and describe my own research into a resolution. Quantum Covariance deals with the issue that manifest covariance requires a continuum model. The Teleconnection shows how a quantum theory described using Minkowski metric remains consistent when classical spacetime is curved. A consequence is that the arguments which led to Minkowski metric in special relativity must be adapted and yield Einstein’s Field Equation in the case for a single gravitating particle.

Deletions:
The construction of qed as a model of particle interactions using finite dimensional Hilbert space is straightforward, but sweeps thorny issues under the carpet. The problems are deep enough that most physicists believe that this approach is fundamentally flawed. I move on from the description of standard theory from a relational viewpoint, and describe my own research into a resolution. Quantum Covariance deals with the issue that manifest covariance requires a continuum model. The Teleconnection shows how a quantum theory described using Minkowski metric remains consistent when classical spacetime is curved. In Discrete Quantum Electrodynamics the undefined equal point multiplication between fields is removed from the physical model, resulting in the elimination of both the ultraviolet divergence and the Landau pole». A consequence is that the arguments which led to Minkowski metric in special relativity must be adapted and yield Einstein’s Field Equation in the case for a single gravitating particle. The Cosmological Implications of the predicted redshift relations under the teleconnection, together with observational evidence, lead to a closed Friedmann model with no cosmological constant or cold dark matter», and to the resolution of other problems in astrophysics.

Edited on 2008-08-29 21:59:57 by CharlesFrancis

Additions:

The construction of qed as a model of particle interactions using finite dimensional Hilbert space is straightforward, but sweeps thorny issues under the carpet. The problems are deep enough that most physicists believe that this approach is fundamentally flawed. I move on from the description of standard theory from a relational viewpoint, and describe my own research into a resolution. Quantum Covariance deals with the issue that manifest covariance requires a continuum model. The Teleconnection shows how a quantum theory described using Minkowski metric remains consistent when classical spacetime is curved. In Discrete Quantum Electrodynamics the undefined equal point multiplication between fields is removed from the physical model, resulting in the elimination of both the ultraviolet divergence and the Landau pole». A consequence is that the arguments which led to Minkowski metric in special relativity must be adapted and yield Einstein’s Field Equation in the case for a single gravitating particle. The Cosmological Implications of the predicted redshift relations under the teleconnection, together with observational evidence, lead to a closed Friedmann model with no cosmological constant or cold dark matter», and to the resolution of other problems in astrophysics.
← Quantum Electrodynamics Cosmological Implications and Empirical Evidence →

Deletions:
large green circle

The construction of qed as a model of particle interactions using finite dimensional Hilbert space is straightforward, but sweeps thorny issues under the carpet. The problems are deep enough that most physicists believe that this approach is fundamentally flawed. I move on from the description of standard theory from a relational viewpoint, and describe my own research into a resolution. Quantum Covariance deals with the issue that manifest covariance requires a continuum model. The Teleconnection shows how a quantum theory described using Minkowski metric remains consistent when classical spacetime is curved. In Discrete Quantum Electrodynamics the undefined equal point multiplication between fields is removed from the physical model, resulting in the elimination of both the ultraviolet divergence and the Landau pole». A consequence is that the arguments which led to Minkowski metric in special relativity must be adapted and yield Einstein’s Field Equation in the case for a single gravitating particle. The Cosmological Implications of the predicted redshift relations under the teleconnection, together with observational evidence, lead to a closed Friedmann model with no cosmological constant or cold dark matter, and to the resolution of other problems in astrophysics.

Cosmological Implications - under construction

← Quantum Electrodynamics Empirical Evidence →

Edited on 2008-08-26 14:23:52 by CharlesFrancis

Additions:
The construction of qed as a model of particle interactions using finite dimensional Hilbert space is straightforward, but sweeps thorny issues under the carpet. The problems are deep enough that most physicists believe that this approach is fundamentally flawed. I move on from the description of standard theory from a relational viewpoint, and describe my own research into a resolution. Quantum Covariance deals with the issue that manifest covariance requires a continuum model. The Teleconnection shows how a quantum theory described using Minkowski metric remains consistent when classical spacetime is curved. In Discrete Quantum Electrodynamics the undefined equal point multiplication between fields is removed from the physical model, resulting in the elimination of both the ultraviolet divergence and the Landau pole». A consequence is that the arguments which led to Minkowski metric in special relativity must be adapted and yield Einstein’s Field Equation in the case for a single gravitating particle. The Cosmological Implications of the predicted redshift relations under the teleconnection, together with observational evidence, lead to a closed Friedmann model with no cosmological constant or cold dark matter, and to the resolution of other problems in astrophysics.

A Gravitating Particle
small green circle

The Emergence of Spacetime Structure

Deletions:
The construction of qed as a model of particle interactions using finite dimensional Hilbert space is straightforward, but sweeps thorny issues under the carpet. The problems are deep enough that most physicists believe that this approach is fundamentally flawed. I move on from the description of standard theory from a relational viewpoint, and describe my own research into a resolution. Quantum Covariance deals with the issue that manifest covariance requires a continuum model. The Teleconnection shows how a quantum theory described using Minkowski metric remains consistent when classical spacetime is curved. In Discrete Quantum Electrodynamics the undefined equal point multiplication between fields is removed from the physical model, resulting in the elimination of both the ultraviolet divergence and the Landau pole». A consequence is that the arguments which led to Minkowski metric in special relativity must be adapted and yield Einstein’s Field Equation. The Cosmological Implications of the predicted redshift relations under the teleconnection, together with observational evidence, lead to a closed Friedmann model with no cosmological constant or cold dark matter, and to the resolution of other problems in astrophysics.
small blue square

The Physical Origin of Curvature
small blue square

Einstein’s Field Equation - under construction

Edited on 2008-08-13 03:54:59 by CharlesFrancis

Additions:

← Relational Quantum Gravity ↑ →

Deletions:

← Relational Quantum Gravity →

Edited on 2008-06-29 02:19:57 by CharlesFrancis

Additions:
The construction of qed as a model of particle interactions using finite dimensional Hilbert space is straightforward, but sweeps thorny issues under the carpet. The problems are deep enough that most physicists believe that this approach is fundamentally flawed. I move on from the description of standard theory from a relational viewpoint, and describe my own research into a resolution. Quantum Covariance deals with the issue that manifest covariance requires a continuum model. The Teleconnection shows how a quantum theory described using Minkowski metric remains consistent when classical spacetime is curved. In Discrete Quantum Electrodynamics the undefined equal point multiplication between fields is removed from the physical model, resulting in the elimination of both the ultraviolet divergence and the Landau pole». A consequence is that the arguments which led to Minkowski metric in special relativity must be adapted and yield Einstein’s Field Equation. The Cosmological Implications of the predicted redshift relations under the teleconnection, together with observational evidence, lead to a closed Friedmann model with no cosmological constant or cold dark matter, and to the resolution of other problems in astrophysics. small blue square

The Physical Origin of Curvature
small blue square

Einstein’s Field Equation - under construction
small blue square

Cosmological Implications - under construction

Deletions:
The construction of qed as a model of particle interactions using finite dimensional Hilbert space is straightforward, but sweeps thorny issues under the carpet. The problems are deep enough that most physicists believe that this approach is fundamentally flawed. I move on from the description of standard theory from a relational viewpoint, and describe my own research into a resolution. Quantum Covariance deals with the issue that manifest covariance requires a continuum model. The Teleconnection shows how a quantum theory described using Minkowski metric remains consistent when classical spacetime is curved. In Discrete Quantum Electrodynamics the undefined equal point multiplication between fields is removed from the physical model, resulting in the elimination of both the ultraviolet divergence and the Landau pole». A consequence is that the arguments which led to Minkowski metric in special relativity must be adapted and yield Einstein’s Field Equation. The Cosmological Implications of the predicted redshift relations under the teleconnection, together with observational evidence, lead to a closed Friedmann model with no cosmological constant or cold dark matter, and to the resolution of other problems in astrophysics.
small blue square

Einstein’s Field Equation - under construction
small blue square

Cosmological Implications- to be written

Edited on 2008-05-31 05:17:15 by CharlesFrancis

Additions:
small green circle

Particles Or Fields?

Edited on 2008-05-26 10:31:25 by CharlesFrancis

Additions:

← Relational Quantum Gravity →

Deletions:

← Relational Quantum Gravity →

Edited on 2008-05-26 10:30:41 by CharlesFrancis

Additions:
small blue square

Einstein’s Field Equation - under construction

Deletions:
small blue square

Discrete Quantum Electrodynamics - to be written small blue square

Einstein’s Field Equation - to be written

Edited on 2008-05-05 00:26:39 by CharlesFrancis

Additions:
large green circle

← Relational Quantum Gravity →

The construction of qed as a model of particle interactions using finite dimensional Hilbert space is straightforward, but sweeps thorny issues under the carpet. The problems are deep enough that most physicists believe that this approach is fundamentally flawed. I move on from the description of standard theory from a relational viewpoint, and describe my own research into a resolution. Quantum Covariance deals with the issue that manifest covariance requires a continuum model. The Teleconnection shows how a quantum theory described using Minkowski metric remains consistent when classical spacetime is curved. In Discrete Quantum Electrodynamics the undefined equal point multiplication between fields is removed from the physical model, resulting in the elimination of both the ultraviolet divergence and the Landau pole». A consequence is that the arguments which led to Minkowski metric in special relativity must be adapted and yield Einstein’s Field Equation. The Cosmological Implications of the predicted redshift relations under the teleconnection, together with observational evidence, lead to a closed Friedmann model with no cosmological constant or cold dark matter, and to the resolution of other problems in astrophysics.

Quantum Covariance
small green circle

The Teleconnection in a Friedmann Cosmology
small black diamond

The Teleconnection
small blue square

Discrete Quantum Electrodynamics - to be written
small blue square

Einstein’s Field Equation - to be written
small blue square

Cosmological Implications- to be written

← Quantum Electrodynamics Empirical Evidence →

Deletions:
large black diamond

← 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

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^».

“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 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, Teleconnection-11

, 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 barred vector, Teleconnection-14g

, is

Definition: For the displacement, x = (x^τ, x^ρ, x^θ, x^φ), from (τ₁, A), the barred displacement vector, Teleconnection-17g

, 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 Teleconnection-21

and

at (τ₁, A), the barred dot product, evaluated with non-physical metric Teleconnection-23

, 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 Teleconnection-29

, and we have the definition:

Definition: Barred momentum is

It follows that the momentum space wave function,

is preserved when it is given that Teleconnection-32

vanishes outside a sphere of radius πa, and when the wave function is normalised such that Teleconnection-33

.
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 Teleconnection-34

The Teleconnection: For the barred momentum Teleconnection-35g

, the plane wave state at time τ is given in τ-ρ coordinates by

where the barred dot product uses the non-physical metric, Teleconnection-37g

This definition preserves Newton’s first law and the constancy of the speed of light in non-physical coordinates with metric Teleconnection-38

. 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 Teleconnection-39

, 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 Teleconnection-53

. 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, Teleconnection-55

,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 Teleconnection-58

measured locally by Beth at time t₀ of a photon emitted at time t₁ with energy Teleconnection-59

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, Teleconnection-61

, 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.

Definition: For a vector x = (x^τ, x^ρ, x^θ, x^φ), at (τ₁, A) the barred vector, Teleconnection-75g

, 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 Teleconnection-83

, 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 Teleconnection-93

. Classical motion is determinate and may be described as an ordered sequence, Teleconnection-94

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 Teleconnection-98

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, Teleconnection-101

may be regarded as the initial state and Teleconnection-102

may be regarded as the final state. Since the time evolution of mean properties is determinate, there is no collapse and Teleconnection-103

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 →

Edited on 2008-05-04 12:24:50 by CharlesFrancis

Additions:
large black diamond