Most recent edit on 2008-08-13 03:56:38 by CharlesFrancis

Additions:

← Quantum Covariance ↑ →

The view that the spacetime has substance, and exists independently of the processes occurring within it, is substantivalism». Substantivalism stands philosophically opposed to Cartesian relationism, on which relational quantum gravity is based. Although most physicists treat general relativity as a substantivalist theory of spacetime, I have given a mathematical treatment preserving relational principles. The matter vexed Einstein greatly during the period of the development of general relativity. The hole argument», which he produced in 1913, challenged the notion that general covariance was even possible for a determinist theory, and, reformulated in a modern context, continues to challenge spacetime substantivalism. The challenge for relationism is to exhibit a mathematical theory preserving fundamental principles and consistent with empirical fact. Quantum theory has shown that determinism in Nature is not a fundamental principle, removing one of the issues which troubled Einstein. Here I will show that a substantive continuum is not required to preserve the general principle of relativity, that local laws of physics are the same in all reference frames, provided that the statement of covariance in the quantum theory takes into account the role of the physical apparatus in determining coordinates.
When a human observer seeks to quantify Nature, he chooses some particular matter from which to define a reference frame or chooses certain matter from which he builds his experimental apparatus. He then observes a defined relationship between this specially, but arbitrarily, chosen reference matter and whatever matter is the subject of study. The measurement is a count of units of a measured quantity, where the definition of the unit of measurement invokes comparison between some aspect of the subject of measurement and a property of the reference matter used to define the unit of measurement. Reference matter is to a large degree arbitrary, and is itself subject to measurement with respect to other matter. Whatever reference matter is used, it includes some form of clock, axes, and some means of determining distance, such as a ruler or radar, and it may include other apparatus used for physical measurement. In all cases, a property is measured relative to other, arbitrarily chosen matter, and the measurement determines a relationship between subject matter and reference matter, rather than an absolute property of the subject of measurement.
In conventional units, the components of momentum are bounded by QuantumCovariance-26

, where the theoretical bound, p_max, depends on the lower bound of small lattice spacing, χ_min, not on the actual lattice appropriate to a given apparatus. It will be shown that the curvature expressed in Einstein’s Field Equation is equivalent to the existence a fundamental discrete unit of proper time, χ, between particle interactions of magnitude twice the Schwarzschild radius for an elementary particle. For an electron of mass m, χ = 4Gm ⁄ c³ = 9.02 × 10⁻⁶⁶s, or χ = 2.70 × 10⁻⁵⁷m, where G is the gravitation constant. This leads to a theoretical bound on momentum of 5.72 × 10⁵¹eV, or 1.02 × 10¹⁴kg for the energy of a single electron, well beyond any reasonable energy level.

Deletions:

← Quantum Covariance →

The view that the spacetime has substance, and exists independently of the processes occurring within it, is substantivalism». Substantivalism stands philosophically opposed to Cartesian relationism, on which relational quantum gravity is based. Although most physicists treat general relativity as a substantivalist theory of spacetime, I have given a mathematical treatment preserving relational principles. The matter vexed Einstein greatly during the period of the development of general relativity. The hole argument», which he produced in 1913, challenged the notion that general covariance was even possible for a determinist theory, and, reformulated in a modern context, continues to challenge spacetime substantivalism. The challenge for relationism is to exhibit a mathematical theory preserving fundamental principles and consistent with empirical fact. Quantum theory has shown that determinism in nature is not a fundamental principle, removing one of the issues which troubled Einstein. Here I will show that a substantive continuum is not required to preserve the general principle of relativity, that local laws of physics are the same in all reference frames, provided that the statement of covariance in the quantum theory takes into account the role of the physical apparatus in determining coordinates.
When a human observer seeks to quantify nature, he chooses some particular matter from which to define a reference frame or chooses certain matter from which he builds his experimental apparatus. He then observes a defined relationship between this specially, but arbitrarily, chosen reference matter and whatever matter is the subject of study. The measurement is a count of units of a measured quantity, where the definition of the unit of measurement invokes comparison between some aspect of the subject of measurement and a property of the reference matter used to define the unit of measurement. Reference matter is to a large degree arbitrary, and is itself subject to measurement with respect to other matter. Whatever reference matter is used, it includes some form of clock, axes, and some means of determining distance, such as a ruler or radar, and it may include other apparatus used for physical measurement. In all cases, a property is measured relative to other, arbitrarily chosen matter, and the measurement determines a relationship between subject matter and reference matter, rather than an absolute property of the subject of measurement.
In conventional units, the components of momentum are bounded by QuantumCovariance-26

Edited on 2008-05-05 02:30:43 by CharlesFrancis

Additions:

Edited on 2008-03-28 02:01:56 by CharlesFrancis

Additions:

← Quantum Covariance →

Quantum Covariance ↑ The Teleconnection in a Friedmann Cosmology →

Deletions:

← Quantum Covariance →

Quantum Covariance ↑ The Teleconnection →

Edited on 2008-03-19 00:37:56 by CharlesFrancis

Additions:
Throughout this treatment I have been at pains to emphasise that, according to their formal definitions in analysis, the derivative is an approximation to a division between finite quantities and the integral is an approximation to a finite sum (more precisely, to a member of a family of finite divisions, and to a member of a family of finite sums). Perhaps it is more usual in physics to think of this the other way around, that the derivative is approximated by a division and the integral is approximated by a sum. This reverses the logical order in which the quantities are defined in mathematics, and brings with it the inference that a continuum has physical existence. A continuum» can be defined in mathematics, but the inference that it has physical analogue is made by inductive», rather than by deductive», reasoning and cannot be rigorously justified» on logical or philosophical grounds.

Deletions:
Throughout this treatment I have been at pains to emphasise that, according to their formal definitions in analysis, the derivative is an approximation to a division between finite quantities and the integral is an approximation to a finite sum (more precisely, to a member of a family of finite divisions, and to a member of a family of finite sums). Perhaps it is more usual in physics to think of this the other way around, that the derivative is approximated by a division and the integral is approximated by a sum. This reverses the logical order in which the quantities are defined in mathematics, and brings with it the inference that a continuum has physical existence. A continuum» can be defined in mathematics, but the inference that it has physical analogue is made by inductive», rather than by deductive», reasoning and cannot be rigorously justified» on logical or philosophical grounds.

Edited on 2008-03-18 04:45:23 by CharlesFrancis

Additions:
The view that the spacetime has substance, and exists independently of the processes occurring within it, is substantivalism». Substantivalism stands philosophically opposed to Cartesian relationism, on which relational quantum gravity is based. Although most physicists treat general relativity as a substantivalist theory of spacetime, I have given a mathematical treatment preserving relational principles. The matter vexed Einstein greatly during the period of the development of general relativity. The hole argument», which he produced in 1913, challenged the notion that general covariance was even possible for a determinist theory, and, reformulated in a modern context, continues to challenge spacetime substantivalism. The challenge for relationism is to exhibit a mathematical theory preserving fundamental principles and consistent with empirical fact. Quantum theory has shown that determinism in nature is not a fundamental principle, removing one of the issues which troubled Einstein. Here I will show that a substantive continuum is not required to preserve the general principle of relativity, that local laws of physics are the same in all reference frames, provided that the statement of covariance in the quantum theory takes into account the role of the physical apparatus in determining coordinates.
When a human observer seeks to quantify nature, he chooses some particular matter from which to define a reference frame or chooses certain matter from which he builds his experimental apparatus. He then observes a defined relationship between this specially, but arbitrarily, chosen reference matter and whatever matter is the subject of study. The measurement is a count of units of a measured quantity, where the definition of the unit of measurement invokes comparison between some aspect of the subject of measurement and a property of the reference matter used to define the unit of measurement. Reference matter is to a large degree arbitrary, and is itself subject to measurement with respect to other matter. Whatever reference matter is used, it includes some form of clock, axes, and some means of determining distance, such as a ruler or radar, and it may include other apparatus used for physical measurement. In all cases, a property is measured relative to other, arbitrarily chosen matter, and the measurement determines a relationship between subject matter and reference matter, rather than an absolute property of the subject of measurement.
The practical difference between the measurement of a vector quantity in one coordinate system and its measurement in another is usually small enough to be neglected, but the fact of discreteness, both in measurement and in the numerical statement of position coordinates, raises issues of principle for the mathematical statement of covariance. The broad meaning of covariance is that it refers to something which varies with something else, so as to preserve certain mathematical relations. The standard definition of Lorentz covariance assumes that vectors exist, independently of coordinate systems, and that the discrete values given in measurement are approximations to the true coordinates of the vector. For this to be true, it must be possible in principle to define coordinate systems with arbitrarily fine resolution. In practice this is impossible because of the finite resolution of any given apparatus and because quantum uncertainty comes into play. If covariance is not now to be interpreted as applicable to the components of classical vectors, then a new form of covariance, quantum covariance, is required to express the principle of general relativity. Quantum covariance will require that local laws of physics have the same form in any reference frame but not that the same physical process may be described identically in different reference frames, since the reference frame, i.e. the choice of apparatus, can affect both the process under study and the description of that process.
In practice there is a much lower bound since an interaction between a sufficiently high energy electron and any background electromagnetic field leads to pair creation. It follows from conservation of energy that the total energy of a system is bounded provided that energy has been bounded at some time in the past. This is true whenever an energy value is known since a measurement of energy creates an eigenstate with a definite value of energy. Then momentum is also bounded, by the mass shell condition. The probability of finding a momentum above the bound is zero, and we assume that for physically realisable states QuantumCovariance-31

vanishes above the bound on each component of momentum. The bound depends on the system under consideration, but without needing to specify a least bound, we may reasonably assume that momentum is always much less than π ⁄ 4.
So the eigenvalue of momentum is sinp ≈ p for p much less than the bound of QuantumCovariance-47

. A 0.1% difference between p and sinp for an electron would require an energy, QuantumCovariance-48

eV. This is unrealistic.

Deletions:
The view that the spacetime has substance, and exists independently of the processes occurring within it, is substantivalism». Substantivalism stands philosophically opposed to Cartesian relationism, on which relational quantum gravity is based. Although most physicists treat general relativity as a substantivalist theory of spacetime, I have given a mathematical treatment preserving relational principles. The matter vexed Einstein greatly during the period of the development of general relativity. The hole argument», which he produced in 1913, challenged the notion that general covariance was even possible for a determinist theory, and reformulated in a modern context, continues to challenge spacetime substantivalism. The challenge for relationism is to exhibit a mathematical theory preserving fundamental principles and consistent with empirical fact. Quantum theory has shown that determinism in nature is not a fundamental principle, removing one of the issues which troubled Einstein. Here I will show that a substantive continuum is not required to preserve the general relativity principle of relativity, that local laws of physics are the same in all reference frames, provided that the statement of covariance in the quantum theory takes into account the role of the physical apparatus in determining coordinates.
When a human observer seeks to quantify nature, he chooses some particular matter from which to define a reference frame or chooses certain matter from which he builds his experimental apparatus. He then observes a defined relationship between this specially, but arbitrarily, chosen reference matter and whatever matter is the subject of study. The measurement is a count of units of a measured quantity, where the definition of the unit of measurement invokes comparison between some aspect of the subject of measurement and a property of the reference matter used to define the unit of measurement. Reference matter is to a large degree arbitrary, and is itself subject to measurement with respect to other matter. Whatever reference matter is used it includes some form of clock, axes, and some means of determining distance, such as a ruler or radar, and it may include other apparatus used for physical measurement. In all cases, a property is measured relative to other, arbitrarily chosen matter, and the measurement determines a relationship between subject and reference matter, rather than an absolute property of the subject of measurement.
The practical difference between the measurement of a vector quantity in one coordinate system and its measurement in another is usually small enough to be neglected, but the fact of discreteness, both in measurement and in the numerical statement of position coordinates, raises issues of principle for the mathematical statement of covariance. The broad meaning of covariance is that it refers to something which varies with something else, so as to preserve certain mathematical relations. The standard definition of Lorentz covariance assumes that vectors exist, independently of coordinate systems, and that the discrete values given in measurement are approximations to the true coordinates of the vector. For this to be true, it must be possible in principle to define coordinate systems with arbitrarily fine resolution. In practice this is impossible because of the finite resolution of any given apparatus and because quantum uncertainty comes into play. If covariance is not now to be interpreted as applicable to the components of classical vectors, then a new form of covariance, quantum covariance, is required to express the principle of general relativity. Quantum covariance will require that local laws of physics have the same form in any reference frame but not that the same physical process may be described identically in different reference frames, since the reference frame, i.e. the choice of apparatus, can affect both the process under study and the description of that process.
In practice there is a much lower bound since an interaction between a sufficiently high energy electron and any background electromagnetic field leads to pair creation. It follows from conservation of energy that the total energy of a system is bounded provided that energy has been bounded at some time in the past. This is true whenever an energy value is known since a measurement of energy creates an eigenstate with a definite value of energy. Then momentum is also bounded, by the mass shell condition. The probability of finding a momentum above the bound is zero, and we assume that for physically realisable states QuantumCovariance-31

vanishes above the bound on each component of momentum. The bound depends on the system under consideration, but without needing to specify a least bound, we may reasonably assume that, momentum is always much less than π ⁄ 4.
So the eigenvalue of momentum is sinp ≈ p for p much less than the bound of QuantumCovariance-47

. A 0.1% difference between p and sinp for an electron would require an energy, QuantumCovariance-48

eV. This is unrealistic.

Edited on 2008-03-18 03:23:12 by CharlesFrancis

Additions:

x = y, In conventional units, the components of momentum are bounded by <img title="momentum bound" alt="QuantumCovariance-26" src="images/quantumcovariance/QuantumCovariance-26.gif" align="texttop" vspace="1">, where the theoretical bound, pmax, depends on the lower bound of small lattice spacing, χmin, not on the actual lattice appropriate to a given apparatus. It will be shown that the curvature expressed in [[EinsteinsFieldEquation Einstein’s Field Equation]] is equivalent to the existence a fundamental discrete unit of proper time, χ, between particle interactions of magnitude twice the Schwarzschild radius for an elementary particle. For an electron of mass m, χ = 4Gm ⁄ c3 = 9.02 × 10−66 s, or χ = 2.70 × 10−57 m, where G is the gravitation constant. This leads to a theoretical bound on momentum of 5.72 × 1051 eV, or 1.02 × 1014 kg for the energy of a single electron, well beyond any reasonable energy level. A theoretical bound on momentum might introduce a problem of principle for Lorentz transformation. If a high energy electron were boosted beyond the bound it might appear after the boost with a low energy, or with opposite direction of momentum. However, realistic Lorentz transformation means that macroscopic matter (the reference frame) is physically boosted by the amount of the transformation. For example, for a cubic lattice with spacing equal to the Schwarzschild radius of an electron, a boost in the order of π ⁄ 4 would require an energy of 2 × 1014 solar masses per kilogram of matter to be boosted. It is therefore reasonable to assume that in any reference frame determined by physical matter there is no other matter with sufficient energy to define a reference frame boosted from the first by more than π ⁄ 4, so that momentum remains bounded by π ⁄ 2"". Thus, in practice, Lorentz transformation cannot boost momentum beyond the level for which it is consistently defined.

Deletions:

Edited on 2008-03-18 03:06:12 by CharlesFrancis

Additions:
The view that the spacetime has substance, and exists independently of the processes occurring within it, is substantivalism». Substantivalism stands philosophically opposed to Cartesian relationism, on which relational quantum gravity is based. Although most physicists treat general relativity as a substantivalist theory of spacetime, I have given a mathematical treatment preserving relational principles. The matter vexed Einstein greatly during the period of the development of general relativity. The hole argument», which he produced in 1913, challenged the notion that general covariance was even possible for a determinist theory, and reformulated in a modern context, continues to challenge spacetime substantivalism. The challenge for relationism is to exhibit a mathematical theory preserving fundamental principles and consistent with empirical fact. Quantum theory has shown that determinism in nature is not a fundamental principle, removing one of the issues which troubled Einstein. Here I will show that a substantive continuum is not required to preserve the general relativity principle of relativity, that local laws of physics are the same in all reference frames, provided that the statement of covariance in the quantum theory takes into account the role of the physical apparatus in determining coordinates.
When a human observer seeks to quantify nature, he chooses some particular matter from which to define a reference frame or chooses certain matter from which he builds his experimental apparatus. He then observes a defined relationship between this specially, but arbitrarily, chosen reference matter and whatever matter is the subject of study. The measurement is a count of units of a measured quantity, where the definition of the unit of measurement invokes comparison between some aspect of the subject of measurement and a property of the reference matter used to define the unit of measurement. Reference matter is to a large degree arbitrary, and is itself subject to measurement with respect to other matter. Whatever reference matter is used it includes some form of clock, axes, and some means of determining distance, such as a ruler or radar, and it may include other apparatus used for physical measurement. In all cases, a property is measured relative to other, arbitrarily chosen matter, and the measurement determines a relationship between subject and reference matter, rather than an absolute property of the subject of measurement.

<a href=http://www.teleconnection.info/rqg/FoundationsOfQuantumTheory#SentencesDescribingHypotheticalResults>Hilbert» space</a>. The range and resolution of the lattice may be greater than that of our particular measurement apparatus. Actual measurement results (represented by blue dots) will be a subset of the lattice, and need not be rectilinear. States describing the actual positions recorded as measurement results are represented by a sum of basis states defined on the lattice, as indicated by the magenta dividers. A displacement vector (green) may be described in the usual way, as the difference between discrete points of the lattice. Only displacements between blue dots may be physically determined. <table width=100% cellspacing=0 cellpadding=0><td><img class=right title="rotation of Discrete coordinates" alt="QuantumCovariance-2" src="images/quantumcovariance/QuantumCovariance-2.gif" >In relational quantum gravity it is recognised that coordinates are defined with respect to a particular apparatus or system of measurement. Coordinates may be rotated either mathematically, using the formulae for coordinate transformation and rounding decimals to the new lattice, or physically, by rotating the apparatus. Clearly rotation of the apparatus also affects the <a href=http://www.teleconnection.info/rqg/Observables#ObservableOperators>eigenstates</a»> of measurement of position. In either case, it is not expected that a displacement vector in the new coordinates will correspond precisely to a displacement vector in the old. </td></table> Any method of measuring coordinates may be used, calibrated to the [[FoundationsOfSpecialRelativity radar method]], which gives the fundamental definition of distance in relational quantum gravity. It would be natural to use synchronous spherical coordinates with time as a parameter as in non-relativistic quantum mechanics. In practice the resolution of measurement is much greater than [[http://en.wikipedia.org/wiki/Planck_length Planck length]], sometimes suggested as the scale of fundamental discreteness in Nature. Planck length is 1.6 × 10−35 m, or about 10−20 times the diameter of a proton. Relational quantum gravity will suggest an even smaller distance, twice the Schwarzschild radius of an electron, or 2.7 × 10−57 m, as the scale on which discreteness becomes important. This being so, errors in coordinate transformation may be neglected compared to errors in measurement. For convenience, Cartesian coordinates will be chosen. This simplifies certain formulae, but makes no fundamental difference to the treatment. For definiteness, <a href=http://www.teleconnection.info/rqg/FoundationsOfQuantumTheory#SentencesDescribingHypotheticalResults>Hilbert» space</a> is defined using a cubic lattice, <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="1">, defined to be of sufficient size that it contains the light cone for any particle described in the initial state, so that the probability of finding the particle on or outside the boundary of the lattice will be zero. <Definition: ; The discrete coordinate system <img title="The discrete coordinate system" alt="QuantumCovariance-4" src="images/quantumcovariance/QuantumCovariance-4g.gif" align="texttop" vspace="1"> is described by a cubic lattice, with lattice spacing χ and containing 2ν + 1 points per side, with the origin at the centre of the lattice. It is natural that a physical change of coordinates should modify displacement vectors, because position is affected in measurement. The treatment of momentum is more subtle. Momentum is calculated, not directly measured. Since it is conserved in interaction, momentum should not be directly affected by a rotation of the apparatus. I will show that it is possible to define momentum vectors consistently on a continuum, even when coordinate space is discrete. Given the initial state, <img title="intitial ket" alt="QuantumCovariance-9" src="images/quantumcovariance/QuantumCovariance-9.gif" align="texttop" vspace="0">, determined by measurement at time x0 = 0 using the discrete coordinate system, <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="1">, the momentum space wave function is, This formula defines a continuous function, <img title="The momentum space wave function is defined on a mathematical continuum" alt="QuantumCovariance-13" src="images/quantumcovariance/QuantumCovariance-13.gif" align="texttop" vspace="1">, on the 3-torus <img title="The 3-torus" alt="QuantumCovariance-14" src="images/quantumcovariance/QuantumCovariance-14.gif" align="texttop" vspace="0">. Only a discrete subset, <img title="the discrete subset" alt="QuantumCovariance-15" src="images/quantumcovariance/QuantumCovariance-15.gif" align="texttop" vspace="1">, is used for inversion by the discrete Fourier transform: for <img title="x is a discrete coordinate" alt="QuantumCovariance-17" src="images/quantumcovariance/QuantumCovariance-17.gif" align="texttop" vspace="1">, So, the sum over discrete values of momentum in <img title="Discrete momentum space" alt="QuantumCovariance-24" src="images/quantumcovariance/QuantumCovariance-24.gif" align="texttop" vspace="1"> can be replaced with an exact integral over continuum values in <img title="Continuum momentum space" alt="QuantumCovariance-25" src="images/quantumcovariance/QuantumCovariance-25.gif" align="texttop" vspace="1">, without altering the theory in any way. In conventional units, the components of momentum are bounded by <img title="momentum bound" alt="QuantumCovariance-26" src="images/quantumcovariance/QuantumCovariance-26.gif" align="texttop" vspace="0">, where the theoretical bound, pmax, depends on the lower bound of small lattice spacing, χmin, not on the actual lattice appropriate to a given apparatus. It will be shown that the curvature expressed in [[EinsteinsFieldEquation Einstein’s Field Equation]] is equivalent to the existence a fundamental discrete unit of proper time, χ, between particle interactions of magnitude twice the Schwarzschild radius for an elementary particle. For an electron of mass m, χ = 4Gm ⁄ c3 = 9.02 × 10−66 s, or χ = 2.70 × 10−57 m, where G is the gravitation constant. This leads to a theoretical bound on momentum of 5.72 × 1051 eV, or 1.02 × 1014 kg for the energy of a single electron, well beyond any reasonable energy level. The non-physical periodic property of <img title="the momentum space wave function with bounded support" alt="QuantumCovariance-36" src="images/quantumcovariance/QuantumCovariance-36.gif" align="texttop" vspace="0"> can removed by the substitution <img title="the momentum space wave function" alt="QuantumCovariance-37" src="images/quantumcovariance/QuantumCovariance-37.gif" align="texttop" vspace="0">, where <img title="step function" alt="QuantumCovariance-38" src="images/quantumcovariance/QuantumCovariance-38.gif" align="texttop" vspace="1"> if <img title="step function" alt="QuantumCovariance-39" src="images/quantumcovariance/QuantumCovariance-39.gif" align="texttop" vspace="1"> and <img title="step function" alt="QuantumCovariance-40" src="images/quantumcovariance/QuantumCovariance-40.gif" align="texttop" vspace="1"> otherwise. The wave function may then be replaced in flat space with the standard form in relativistic quantum mechanics: So the eigenvalue of momentum is sinp ≈ p for p much less than the bound of <img title="bound on momentum" alt="QuantumCovariance-47" src="images/quantumcovariance/QuantumCovariance-47.gif" align="texttop" vspace="3">. A 0.1% difference between p and sinp for an electron would require an energy, <img title="electron energy to give a difference of 0.1% between p and sinp" alt="QuantumCovariance-48" src="images/quantumcovariance/QuantumCovariance-48.gif" align="texttop" vspace="1">eV. This is unrealistic. Since the points which may be described in physical measurement are a discrete set, we expect to describe physical evolution in a given reference frame with a difference equation. The replacement of [[http://en.wikipedia.org/wiki/Differential_equation differential equations]] by [[http://en.wikipedia.org/wiki/Recurrence_relation difference equations]] is [[http://en.wikipedia.org/wiki/discretisation discretisation]]. While the description of an initial state is determined by measurement, and is therefore reference frame dependent, physical evolution thereafter takes place independently of reference matter, and is unaffected by an observer’s choice of coordinate system, up until the final measurement. It follows that an identical law of evolution must apply in all coordinate systems. We observe that systems of linear first order differential equations, can be discretised exactly, and conclude that the same differential equation, i.e. the Dirac equation, will have the same solution as the appropriate difference equation in any given coordinate system. Specifically, because the sum over discrete values of momentum in <img title="Discrete momentum space" alt="QuantumCovariance-24" src="images/quantumcovariance/QuantumCovariance-24.gif" align="texttop" vspace="1"> can be replaced with an integral over <img title="Continuum momentum space" alt="QuantumCovariance-50" src="images/quantumcovariance/QuantumCovariance-50.gif" align="texttop" vspace="1">, the inner product, for <img title="x is a discrete coordinate" alt="QuantumCovariance-17" src="images/quantumcovariance/QuantumCovariance-17.gif" align="texttop" vspace="0">, <img title="the discrete wave function" alt="QuantumCovariance-56" src="images/quantumcovariance/QuantumCovariance-56.gif" align="texttop" vspace="0"> is found from f(x) simply by restricting to <img title="x is a discrete coordinate" alt="QuantumCovariance-17" src="images/quantumcovariance/QuantumCovariance-17.gif" align="texttop" vspace="1">. According to quantum covariance this expression has an invariant form under a change of reference frame. This is important for the definition of quantum fields, since these are operators and are invariant, not covariant as is usually assumed. The invariance of operators under rotations is perhaps at first a little surprising, particularly when one considers the presumed importance of manifest covariance in axiomatic quantum field theory. It may be clarified a little with a nautical analogy. On a boat the directions fore, aft, port and starboard are invariant because they are defined with respect to the boat. Similarly operators are necessarily defined with respect to chosen reference matter and have an invariant form with respect to reference matter. Primed coordinates is imposed after transformation by restricting to points x' in a new cubic lattice <img title="Primed coordinates" alt="QuantumCovariance-65" src="images/quantumcovariance/QuantumCovariance-65.gif" align="texttop" vspace="1">. The transformed state is the restriction to the new lattice, The transform of a position ket, <img title="The transform of a position ket" alt="QuantumCovariance-67" src="images/quantumcovariance/QuantumCovariance-67.gif" align="texttop" vspace="0">, is not an eigenstate of position in <img title="Primed coordinates" alt="QuantumCovariance-65" src="images/quantumcovariance/QuantumCovariance-65.gif" align="texttop" vspace="1">. If a measurement of position were done and we were then to transform back to <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="1">, the state would no longer be <img title="A position ket" alt="QuantumCovariance-70" src="images/quantumcovariance/QuantumCovariance-70.gif" align="texttop" vspace="0">. In other words, the operators for position in <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="1"> and in <img title="Primed coordinates" alt="QuantumCovariance-65" src="images/quantumcovariance/QuantumCovariance-65.gif" align="texttop" vspace="1"> do not commute. The geometry is [[http://en.wikipedia.org/wiki/Noncommutative_geometry noncommutative]]. If no measurement is done, we can transform straight back and recover <img title="A position ket" alt="QuantumCovariance-70" src="images/quantumcovariance/QuantumCovariance-70.gif" align="texttop" vspace="0">"",

Deletions:
The view that the spacetime has substance, and exists independently of the processes occurring within it, is substantivalism». Substantivalism stands philosophically opposed to Cartesian relationism, on which relational quantum gravity is based. Although most physicists treat general relativity as a substantivalist theory of spacetime, I have given a mathematical treatment preserving relational principles. The matter vexed Einstein greatly during the period of the development of general relativity. The hole argument», which he produced in 1913, challenged the notion that general covariance was even possible for a determinist theory, and reformulated in a modern context, continues to challenge spacetime substantivalism. The challenge for relationism is to exhibit a mathematical theory preserving fundamental principles and consistent with empirical fact. Quantum theory has shown that determinism in nature is not a fundamental principle, removing one of the issues which troubled Einstein. Here I will show that an substantive continuum is not required to preserve the general relativity principle of relativity, that local laws of physics are the same in all reference frames, provided that the statement of covariance in the quantum theory takes into account the role of the physical apparatus in determining coordinates.
When a human observer seeks to quantify nature, he chooses some particular matter from which to define a reference frame or chooses certain matter from which he builds his experimental apparatus. He then observes a defined relationship between this specially, but arbitrarily, chosen reference matter and whatever matter is the subject of study. The measurement is a count of units of a measured quantity, where the definition of the unit of measurement invokes comparison between some aspect of the subject of measurement and a property of the reference matter used to define the unit of measurement. Reference matter is to a large degree arbitrary, and is itself subject to measurement with respect to other matter. Whatever reference matter is used it includes some form of clock, axes, and some means of determining distance, such as a ruler or radar, and it may include other apparatus used for physical measurement. In all cases a property is measured relative to other, arbitrarily chosen matter, and the measurement determines a relationship between subject and reference matter, rather than an absolute property of the subject of measurement.

<table width=100% cellspacing=0 cellpadding=0><td><img class=right title="rotation of Discrete coordinates" alt="QuantumCovariance-2" src="images/quantumcovariance/QuantumCovariance-2.gif" >In relational quantum gravity it is recognised that coordinates are defined with respect to a particular apparatus or system of measurement. Coordinates may be rotated either mathematically, using the formulae for coordinate transformation and rounding decimals to the new lattice, or physically, by rotating the apparatus. Clearly rotation of the apparatus also affects the <a href=http://www.teleconnection.info/rqg/Observables#ObservableOperators>eigenstates</a»> of measurement of position. In either case it is not expected that a displacement vector in the new coordinates will correspond precisely to a displacement vector in the old. </td></table> Any method of measuring coordinates may be used, calibrated to the [[FoundationsOfSpecialRelativity radar method]], which gives the fundamental definition of distance in relational quantum gravity. It would be natural to use synchronous spherical coordinates with time as a parameter as in non-relativistic quantum mechanics. In practice the resolution of measurement is much greater than [[http://en.wikipedia.org/wiki/Planck_length Planck length]], sometimes suggested as the scale of fundamental discreteness in Nature. Planck length is 1.6 × 10−35 m, or about 10−20 times the diameter of a proton. Relational quantum gravity will suggest an even smaller distance, twice the Schwarzschild radius of an electron, or 2.7 × 10−57 m, as the scale on which discreteness becomes important. This being so, errors in coordinate transformation may be neglected compared to errors in measurement. For convenience, Cartesian coordinates will be chosen. This simplifies certain formulae, but makes no fundamental difference to the treatment. For definiteness, <a href=http://www.teleconnection.info/rqg/FoundationsOfQuantumTheory#SentencesDescribingHypotheticalResults>Hilbert» space</a> is defined using a cubic lattice, <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="0">, defined to be of sufficient size that it contains the light cone for any particle described in the initial state, so that the probability of finding the particle on or outside the boundary of the lattice will be zero. <Definition: ; The discrete coordinate system <img title="The discrete coordinate system" alt="QuantumCovariance-4" src="images/quantumcovariance/QuantumCovariance-4g.gif" align="texttop" vspace="0"> is described by a cubic lattice, with lattice spacing χ and containing 2ν + 1 points per side, with the origin at the centre of the lattice. It is natural that a physical change of coordinates should modify displacement vectors, because position is affected in measurement. The treatment of momentum is more subtle. Momentum is calculated, not directly measured. Since it is conserved in interaction, it should not be directly affected by a rotation of the apparatus. I will show that it is possible to define momentum vectors consistently on a continuum, even when coordinate space is discrete. Given the initial state, <img title="intitial ket" alt="QuantumCovariance-9" src="images/quantumcovariance/QuantumCovariance-9.gif" align="texttop" vspace="0">, determined by measurement at time x0 = 0 using the discrete coordinate system, <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="0">, the momentum space wave function is, This formula defines a continuous function, <img title="The momentum space wave function is defined on a mathematical continuum" alt="QuantumCovariance-13" src="images/quantumcovariance/QuantumCovariance-13.gif" align="texttop" vspace="0">, on the 3-torus <img title="The 3-torus" alt="QuantumCovariance-14" src="images/quantumcovariance/QuantumCovariance-14.gif" align="texttop" vspace="0">. Only a discrete subset, <img title="the discrete subset" alt="QuantumCovariance-15" src="images/quantumcovariance/QuantumCovariance-15.gif" align="texttop" vspace="0">, is used for inversion by the discrete Fourier transform: for <img title="x is a discrete coordinate" alt="QuantumCovariance-17" src="images/quantumcovariance/QuantumCovariance-17.gif" align="texttop" vspace="0">, So, the sum over discrete values of momentum in <img title="Discrete momentum space" alt="QuantumCovariance-24" src="images/quantumcovariance/QuantumCovariance-24.gif" align="texttop" vspace="0"> can be replaced with an exact integral over continuum values in <img title="Continuum momentum space" alt="QuantumCovariance-25" src="images/quantumcovariance/QuantumCovariance-25.gif" align="texttop" vspace="0">, without altering the theory in any way. In conventional units, the components of momentum are bounded by <img title="momentum bound" alt="QuantumCovariance-26" src="images/quantumcovariance/QuantumCovariance-26.gif" align="texttop" vspace="0">, where the theoretical bound on momentum depends on the lower bound of small lattice spacing, χmin, not on the actual lattice appropriate to a given apparatus. It will be shown that the curvature expressed in [[EinsteinsFieldEquation Einstein’s Field Equation]] is equivalent to the existence a fundamental discrete unit of proper time, χ, between particle interactions of magnitude twice the Schwarzschild radius for an elementary particle. For an electron of mass m, χ = 4Gm ⁄ c3 = 9.02 × 10−66 s, or χ = 2.70 × 10−57 m, where G is the gravitation constant. This leads to a theoretical bound on momentum of 5.72 × 1051 eV, or 1.02 × 1014 kg for the energy of a single electron, well beyond any reasonable energy level. The non-physical periodic property of <img title="the momentum space wave function with bounded support" alt="QuantumCovariance-36" src="images/quantumcovariance/QuantumCovariance-36.gif" align="texttop" vspace="0"> can removed by the substitution <img title="the momentum space wave function" alt="QuantumCovariance-37" src="images/quantumcovariance/QuantumCovariance-37.gif" align="texttop" vspace="0">, where <img title="step function" alt="QuantumCovariance-38" src="images/quantumcovariance/QuantumCovariance-38.gif" align="texttop" vspace="0"> if <img title="step function" alt="QuantumCovariance-39" src="images/quantumcovariance/QuantumCovariance-39.gif" align="texttop" vspace="0"> and <img title="step function" alt="QuantumCovariance-40" src="images/quantumcovariance/QuantumCovariance-40.gif" align="texttop" vspace="0"> otherwise. The wave function may then be replaced in flat space with the standard form in relativistic quantum mechanics: So the eigenvalue of momentum is sinp ≈ p for p much less than the bound of <img title="bound on momentum" alt="QuantumCovariance-47" src="images/quantumcovariance/QuantumCovariance-47.gif" align="texttop" vspace="3">. A 0.1% difference between p and sinp for an electron would require an energy, <img title="electron energy to give a difference of 0.1% between p and sinp" alt="QuantumCovariance-48" src="images/quantumcovariance/QuantumCovariance-48.gif" align="texttop" vspace="3">eV. This is unrealistic. Since the points which may be described in physical measurement are a discrete set, we expect to describe physical evolution in a given reference frame with a difference equation. The replacement of [[http://en.wikipedia.org/wiki/Differential_equation differential equations]] by [[http://en.wikipedia.org/wiki/Recurrence_relation difference equations]] is [[http://en.wikipedia.org/wiki/discretisation discretisation]]. While the description of an initial state is determined by measurement, and is therefore reference frame dependent, physical evolution thereafter takes place independently of reference matter, and is unaffected by an observer’s choice of coordinate system, up until the final measurement. An identical law of evolution must apply in all coordinate systems. We observe that systems of linear first order differential equations, can be discretised exactly, and conclude that the same differential equation, i.e. the Dirac equation, will have the same solution as the appropriate difference equation in any given coordinate system. Specifically, because the sum over discrete values of momentum in <img title="Discrete momentum space" alt="QuantumCovariance-24" src="images/quantumcovariance/QuantumCovariance-24.gif" align="texttop" vspace="0"> can be replaced with an integral over <img title="Continuum momentum space" alt="QuantumCovariance-50" src="images/quantumcovariance/QuantumCovariance-50.gif" align="texttop" vspace="0">, the inner product, for <img title="x is a discrete coordinate" alt="QuantumCovariance-17" src="images/quantumcovariance/QuantumCovariance-17.gif" align="texttop" vspace="0">, <img title="the discrete wave function" alt="QuantumCovariance-56" src="images/quantumcovariance/QuantumCovariance-56.gif" align="texttop" vspace="0"> is found from f(x) simply by restricting to <img title="x is a discrete coordinate" alt="QuantumCovariance-17" src="images/quantumcovariance/QuantumCovariance-17.gif" align="texttop" vspace="0">. According to quantum covariance this expression has an invariant form under a change of reference frame. This is important for the definition of quantum fields, since these are operators and are invariant, not covariant as is usually assumed. The invariance of operators under rotations is perhaps at first a little surprising, particularly when one considers the presumed importance of manifest covariance in axiomatic quantum field theory. It may be clarified a little with a nautical analogy. On a boat the directions fore, aft, port and starboard are invariant because they are defined with respect to the boat. Similarly operators are necessarily defined with respect to chosen reference matter and have an invariant form with respect to reference matter. Primed coordinates is imposed after transformation by restricting to points x' in a new cubic lattice <img title="Primed coordinates" alt="QuantumCovariance-65" src="images/quantumcovariance/QuantumCovariance-65.gif" align="texttop" vspace="0">. The transformed state is the restriction to the new lattice, The transform of a position ket, <img title="The transform of a position ket" alt="QuantumCovariance-67" src="images/quantumcovariance/QuantumCovariance-67.gif" align="texttop" vspace="0">, is not an eigenstate of position in <img title="Primed coordinates" alt="QuantumCovariance-65" src="images/quantumcovariance/QuantumCovariance-65.gif" align="texttop" vspace="0">. If a measurement of position were done and we were then to transform back to <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="0">, the state would no longer be <img title="A position ket" alt="QuantumCovariance-70" src="images/quantumcovariance/QuantumCovariance-70.gif" align="texttop" vspace="0">. In other words, the operators for position in <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="0"> and in <img title="Primed coordinates" alt="QuantumCovariance-65" src="images/quantumcovariance/QuantumCovariance-65.gif" align="texttop" vspace="0"> do not commute. The geometry is [[http://en.wikipedia.org/wiki/Noncommutative_geometry noncommutative]]. If no measurement is done, we can transform straight back and recover <img title="A position ket" alt="QuantumCovariance-70" src="images/quantumcovariance/QuantumCovariance-70.gif" align="texttop" vspace="0">"",

Edited on 2008-03-18 02:31:27 by CharlesFrancis

Additions:
Throughout this treatment I have been at pains to emphasise that, according to their formal definitions in analysis, the derivative is an approximation to a division between finite quantities and the integral is an approximation to a finite sum (more precisely, to a member of a family of finite divisions, and to a member of a family of finite sums). Perhaps it is more usual in physics to think of this the other way around, that the derivative is approximated by a division and the integral is approximated by a sum. This reverses the logical order in which the quantities are defined in mathematics, and brings with it the inference that a continuum has physical existence. A continuum» can be defined in mathematics, but the inference that it has physical analogue is made by inductive», rather than by deductive», reasoning and cannot be rigorously justified» on logical or philosophical grounds.
The view that the spacetime has substance, and exists independently of the processes occurring within it, is substantivalism». Substantivalism stands philosophically opposed to Cartesian relationism, on which relational quantum gravity is based. Although most physicists treat general relativity as a substantivalist theory of spacetime, I have given a mathematical treatment preserving relational principles. The matter vexed Einstein greatly during the period of the development of general relativity. The hole argument», which he produced in 1913, challenged the notion that general covariance was even possible for a determinist theory, and reformulated in a modern context, continues to challenge spacetime substantivalism. The challenge for relationism is to exhibit a mathematical theory preserving fundamental principles and consistent with empirical fact. Quantum theory has shown that determinism in nature is not a fundamental principle, removing one of the issues which troubled Einstein. Here I will show that an substantive continuum is not required to preserve the general relativity principle of relativity, that local laws of physics are the same in all reference frames, provided that the statement of covariance in the quantum theory takes into account the role of the physical apparatus in determining coordinates.
When a human observer seeks to quantify nature, he chooses some particular matter from which to define a reference frame or chooses certain matter from which he builds his experimental apparatus. He then observes a defined relationship between this specially, but arbitrarily, chosen reference matter and whatever matter is the subject of study. The measurement is a count of units of a measured quantity, where the definition of the unit of measurement invokes comparison between some aspect of the subject of measurement and a property of the reference matter used to define the unit of measurement. Reference matter is to a large degree arbitrary, and is itself subject to measurement with respect to other matter. Whatever reference matter is used it includes some form of clock, axes, and some means of determining distance, such as a ruler or radar, and it may include other apparatus used for physical measurement. In all cases a property is measured relative to other, arbitrarily chosen matter, and the measurement determines a relationship between subject and reference matter, rather than an absolute property of the subject of measurement.

The result of measurement of position at given time is always three numbers. We assume measurement to a level of accuracy limited only by physical law and the ingenuity of the makers of the apparatus. In practice measurement results can always be expressed as terminating decimals. A lattice is chosen at some bounding range and resolution, and is used to define a basis for a finite dimensional Hilbert space. The range and resolution of the lattice may be greater than that of our particular measurement apparatus. Actual measurement results (represented by blue dots) will be a subset of the lattice, and need not be rectilinear. States describing the actual positions recorded as measurement results are represented by a sum of basis states defined on the lattice, as indicated by the magenta dividers. A displacement vector (green) may be described in the usual way, as the difference between discrete points of the lattice. Only displacements between blue dots may be physically determined. The practical difference between the measurement of a vector quantity in one coordinate system and its measurement in another is usually small enough to be neglected, but the fact of discreteness, both in measurement and in the numerical statement of position coordinates, raises issues of principle for the mathematical statement of covariance. The broad meaning of covariance is that it refers to something which varies with something else, so as to preserve certain mathematical relations. The standard definition of Lorentz covariance assumes that vectors exist, independently of coordinate systems, and that the discrete values given in measurement are approximations to the true coordinates of the vector. For this to be true, it must be possible in principle to define coordinate systems with arbitrarily fine resolution. In practice this is impossible because of the finite resolution of any given apparatus and because quantum uncertainty comes into play. If covariance is not now to be interpreted as applicable to the components of classical vectors, then a new form of covariance, quantum covariance, is required to express the principle of general relativity. Quantum covariance will require that local laws of physics have the same form in any reference frame but not that the same physical process may be described identically in different reference frames, since the reference frame, i.e. the choice of apparatus, can affect both the process under study and the description of that process. Any method of measuring coordinates may be used, calibrated to the [[FoundationsOfSpecialRelativity radar method]], which gives the fundamental definition of distance in relational quantum gravity. It would be natural to use synchronous spherical coordinates with time as a parameter as in non-relativistic quantum mechanics. In practice the resolution of measurement is much greater than [[http://en.wikipedia.org/wiki/Planck_length Planck length]], sometimes suggested as the scale of fundamental discreteness in Nature. Planck length is

1.6 × 10−35 m, or about 10−20 times the diameter of a proton. Relational quantum gravity will suggest an even smaller distance, twice the Schwarzschild radius of an electron, or 2.7 × 10−57 m, as the scale on which discreteness becomes important. This being so, errors in coordinate transformation may be neglected compared to errors in measurement. For convenience, Cartesian coordinates will be chosen. This simplifies certain formulae, but makes no fundamental difference to the treatment. For definiteness, <a href=http://www.teleconnection.info/rqg/FoundationsOfQuantumTheory#SentencesDescribingHypotheticalResults>Hilbert» space</a> is defined using a cubic lattice, <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="0">, defined to be of sufficient size that it contains the light cone for any particle described in the initial state, so that the probability of finding the particle on or outside the boundary of the lattice will be zero. The space bound of the coordinate system is chosen to extend beyond the light cone of any process under study, so that the wave function vanishes outside a region described by all coordinate sytems under consideration. In particularly the wave function vanishes on the boundary of the lattice. The definition of the inner product identifies points on the boundary of the lattice, so that the dimension of Hilbert space is N = 8ν3. Given the initial state, <img title="intitial ket" alt="QuantumCovariance-9" src="images/quantumcovariance/QuantumCovariance-9.gif" align="texttop" vspace="0">, determined by measurement at time x0 = 0 using the discrete coordinate system, <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="0">, the momentum space wave function is, where in both cases the mathematical integral is used, defined by a limit as distinct from a finite sum. Thus, So, the sum over discrete values of momentum in <img title="Discrete momentum space" alt="QuantumCovariance-24" src="images/quantumcovariance/QuantumCovariance-24.gif" align="texttop" vspace="0"> can be replaced with an exact integral over continuum values in <img title="Continuum momentum space" alt="QuantumCovariance-25" src="images/quantumcovariance/QuantumCovariance-25.gif" align="texttop" vspace="0">, without altering the theory in any way. A theoretical bound on momentum might introduce a problem of principle for Lorentz transformation. If a high energy electron were boosted beyond the bound it might appear after the boost with a low energy, or with opposite direction of momentum. However, realistic Lorentz transformation means that macroscopic matter (the reference frame) is physically boosted by the amount of the transformation. For example, for a cubic lattice with spacing equal to the Schwarzschild radius of an electron, a boost in the order of π ⁄ 4 would require an energy of 2 × 1014 solar masses per kilogram of matter to be boosted. It is therefore reasonable to assume that in any reference frame determined by physical matter there is no other matter with sufficient energy to define a reference frame boosted from the first by more than π ⁄ 4. Thus, in practice, Lorentz transformation cannot boost momentum beyond the level for which it is consistently defined. So the eigenvalue of momentum is sinp ≈ p for p much less than the bound of <img title="bound on momentum" alt="QuantumCovariance-47" src="images/quantumcovariance/QuantumCovariance-47.gif" align="texttop" vspace="3">. A 0.1% difference between p and sinp for an electron would require an energy, <img title="electron energy to give a difference of 0.1% between p and sinp" alt="QuantumCovariance-48" src="images/quantumcovariance/QuantumCovariance-48.gif" align="texttop" vspace="3">eV. This is unrealistic. Since the points which may be described in physical measurement are a discrete set, we expect to describe physical evolution in a given reference frame with a difference equation. The replacement of [[http://en.wikipedia.org/wiki/Differential_equation differential equations]] by [[http://en.wikipedia.org/wiki/Recurrence_relation difference equations]] is [[http://en.wikipedia.org/wiki/discretisation discretisation]]. While the description of an initial state is determined by measurement, and is therefore reference frame dependent, physical evolution thereafter takes place independently of reference matter, and is unaffected by an observer’s choice of coordinate system, up until the final measurement. An identical law of evolution must apply in all coordinate systems. We observe that systems of linear first order differential equations, can be discretised exactly, and conclude that the same differential equation, i.e. the Dirac equation, will have the same solution as the appropriate difference equation in any given coordinate system. Thus, the continuum appears not as the result of induction or other non-empirical limiting procedure, but by using the association between a difference equation and a differential equation. The continuum equations are exact in the restriction to discrete coordinates, and remove any dependency on a specific measurement apparatus and resolution because they contain embedded within them the solutions for all discrete coordinate systems possible in principal or in practice. Substitute p → Λ'p, observing that the integral is invariant up to the choice of <a href=http://www.teleconnection.info/rqg/TheDiracEquation#DiracParticles>normalisation» of plane wave states</a>, Primed coordinates is imposed after transformation by restricting to points x' in a new cubic lattice <img title="Primed coordinates" alt="QuantumCovariance-65" src="images/quantumcovariance/QuantumCovariance-65.gif" align="texttop" vspace="0">. The transformed state is the restriction to the new lattice, The transform of a position ket, <img title="The transform of a position ket" alt="QuantumCovariance-67" src="images/quantumcovariance/QuantumCovariance-67.gif" align="texttop" vspace="0">, is not an eigenstate of position in <img title="Primed coordinates" alt="QuantumCovariance-65" src="images/quantumcovariance/QuantumCovariance-65.gif" align="texttop" vspace="0">. If a measurement of position were done and we were then to transform back to <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="0">, the state would no longer be <img title="A position ket" alt="QuantumCovariance-70" src="images/quantumcovariance/QuantumCovariance-70.gif" align="texttop" vspace="0">. In other words, the operators for position in <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="0"> and in <img title="Primed coordinates" alt="QuantumCovariance-65" src="images/quantumcovariance/QuantumCovariance-65.gif" align="texttop" vspace="0"> do not commute. The geometry is [[http://en.wikipedia.org/wiki/Noncommutative_geometry noncommutative]]. If no measurement is done, we can transform straight back and recover <img title="A position ket" alt="QuantumCovariance-70" src="images/quantumcovariance/QuantumCovariance-70.gif" align="texttop" vspace="0">"",
So, there is no issue concerning unitarity under Lorentz transformation using discrete coordinates.

Deletions:
Throughout this treatment I have been at pains to emphasise that, according to their formal definitions in analysis, the derivative is an approximation to a division between finite quantities and the integral is an approximation to a finite sum (more precisely, to any member of a family of finite divisions, and to any member of a family of finite sums). Perhaps it is more usual in physics to think of this the other way around, that the derivative is approximated by a division and the integral is approximated by a sum. This reverses the logical order in which the quantities are defined in mathematics, and brings with it the inference that a continuum has physical existence. A continuum» can be defined in mathematics, but the inference that it has physical analogue is made by inductive», rather than by deductive», reasoning and cannot be rigorously justified» on logical or philosophical grounds.
The view that the spacetime has substance, and exists independently of the processes occurring within it, is substantivalism», and stands philosophically opposed to Cartesian relationism, on which relational quantum gravity is based. Although most physicists treat general relativity as a substantivalist theory of spacetime, I have given a mathematical treatment preserving relational principles. The matter vexed Einstein greatly during the period of the development of general relativity. The hole argument», which he produced in 1913, challenged the notion that general covariance was even possible for a determinist theory, and reformulated in a modern context, continues to challenge spacetime substantivalism. The challenge for relationism is to exhibit a mathematical theory preserving fundamental principles and consistent with empirical fact. Quantum theory has shown that determinism in nature is not a fundamental principle, removing one of the issues which troubled Einstein. Here I will show that an substantivalist continuum is not required to preserve the general relativity principle of relativity, that local laws of physics are the same in all reference frames, provided that the statement of covariance in the quantum theory takes into account the role of the physical apparatus in determining coordinates.
When a human observer seeks to quantify nature, he chooses some particular matter from which to define a reference frame or chooses certain matter from which he builds his experimental apparatus. He then observes a defined relationship between this specially, but arbitrarily, chosen reference matter and whatever matter is the subject of study. The measurement is a count of units of a measured quantity, where the definition of the unit of measurement invokes comparison between some aspect of the subject of measurement and a property of the reference matter used to define the unit of measurement. Reference matter is to a large degree arbitrary, and is itself subject to measurement with respect to other matter. Whatever reference matter is used it includes some form of clock, axes, and some means of determining distance, such as a ruler or radar, and it may include any form of apparatus used for physical measurement. In all cases a property is measured relative to other, arbitrarily chosen matter, and the measurement determines a relationship between subject and reference matter, rather than an absolute property of the subject of measurement.

The result of measurement of position at given time is always three numbers. We assume measurement to a level of accuracy limited only by physical law and the ingenuity of the makers of the apparatus. In practice measurement results can always be expressed as terminating decimals. A lattice is chosen at some bounding range and resolution, and is used to define a basis for a finite dimensional Hilbert space. The range and resolution of the lattice may be greater than that of our particular measurement apparatus. Actual measurement results (represented by blue dots) will be a subset of the lattice, and need not be rectilinear. States describing the actual positions recorded as measurement results are represented by a sum of basis states in the lattice, as indicated by the magenta dividers. A displacement vector (green) may be described in the usual way, as the difference between discrete points of the lattice. Only displacements between blue dots may be physically determined. The practical difference between the measurement of a vector quantity in one coordinate system and its measurement in another is usually small enough to be neglected, but the fact of discreteness both in measurement and in the numerical statement of position coordinates raises issues of principle for the mathematical statement of covariance. The broad meaning of covariance is that it refers to something which varies with something else, so as to preserve certain mathematical relations. The standard definition of Lorentz covariance assumes that vectors exist, independently of coordinate systems, and that the discrete values given in measurement are approximations to the true coordinates of the vector. For this to be true, it must be possible in principle to define coordinate systems with arbitrarily fine resolution. In practice this is impossible because of the finite resolution of any given apparatus and because quantum uncertainty comes into play. If covariance is not now to be interpreted as applicable to the components of classical vectors, then a new form of covariance, quantum covariance, is required to express the principle of general relativity. Quantum covariance will require that local laws of physics have the same form in any reference frame but not that the same physical process may be described identically in different reference frames, since the reference frame, i.e. the choice of apparatus, can affect both the process under study and the description of that process. Any method of measuring coordinates may be used, calibrated to the [[FoundationsOfSpecialRelativity radar method]], which gives the fundamental definition of distance in relational quantum gravity. It would be natural to use synchronous spherical coordinates with time as a parameter as in non-relativistic quantum mechanics. In practice the resolution of measurement is much greater than [[http://en.wikipedia.org/wiki/Planck_length Planck length]],

1.6 × 10−35 metres, or about 10−20 times the diameter of a proton. Relational quantum gravity will suggest an even smaller distance, twice the Schwarzschild radius of an electron, or 2.7 × 10−57 m, as the scale on which discreteness becomes important. This being so, errors in coordinate transformation may be neglected compared to errors in measurement. For convenience, Cartesian coordinates will be chosen. This simplifies certain formulae, but makes no fundamental difference to the treatment. For definiteness, <a href=http://www.teleconnection.info/rqg/FoundationsOfQuantumTheory#SentencesDescribingHypotheticalResults>Hilbert» space</a> is defined using a cubic lattice, <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="0">, defined to be of sufficient size that it contains the light cone for any particle described by the quantum theory in the initial state, so that the probability of finding the particle on or outside the boundary of the lattice will be zero. The space bound of the coordinate system is chosen to extend beyond the light cone of any process under study, so that the wave function vanishes outside a region described by all coordinate sytems under consideration. In particularly the wave function vanishes on the boundary of the lattice. The definition of the inner product on Hilbert space identifies points on the boundary of the lattice, so that the dimension of Hilbert space is N = 8ν3. Given the initial state, <img title="intitial ket" alt="QuantumCovariance-9" src="images/quantumcovariance/QuantumCovariance-9.gif" align="texttop" vspace="0">, determined by measurement at time x0 = 0 in the discrete coordinate system, <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="0">, the momentum space wave function is, where in both cases the mathematical integral is used, defined by a limit as distinct from a finite sum appropriate to the empirical description of physics. Thus, So the sum over discrete values of momentum in <img title="Discrete momentum space" alt="QuantumCovariance-24" src="images/quantumcovariance/QuantumCovariance-24.gif" align="texttop" vspace="0">can be replaced with an exact integral over continuum values in <img title="Continuum momentum space" alt="QuantumCovariance-25" src="images/quantumcovariance/QuantumCovariance-25.gif" align="texttop" vspace="0">, without altering the theory in anyway. A theoretical bound on momentum might introduce a problem of principle for Lorentz transformation. If a high energy electron were boosted beyond the bound in momentum it might appear after the boost with a low energy, or with opposite direction of momentum. However, A realistic Lorentz transformation means that macroscopic matter has been physically boosted by the amount of the transformation. For example, for a cubic lattice with spacing equal to the Schwarzschild radius of an electron, a boost in the order of π ⁄ 4 would require an energy of 2 × 1014 solar masses per kilogram of matter to be boosted. It is therefore reasonable to assume that in any reference frame determined by physical matter there is no other matter with sufficient energy to define a reference frame boosted from the first by more than π ⁄ 4. Thus, in practice, Lorentz transformation cannot boost momentum beyond the level for which it is consistently defined. So the eigenvalue of momentum is sinp ≈ p for p much less than the bound of <img title="bound on momentum" alt="QuantumCovariance-47" src="images/quantumcovariance/QuantumCovariance-47.gif" align="texttop" vspace="3">. An electron of 0.1% difference between p and sinp would have energy eV, which is unrealistic. Since the points which may be described in physical measurement are a discrete set, we might expect to describe physical evolution in a given reference frame with a difference equation. The replacement of [[http://en.wikipedia.org/wiki/Differential_equation differential equations]] by [[http://en.wikipedia.org/wiki/Recurrence_relation difference equations]] is [[http://en.wikipedia.org/wiki/Discretization discretisation]]. While the description of an initial state is determined by measurement, and is therefore reference frame dependent, physical evolution thereafter takes place independently of reference matter, and is unaffected by an observer’s choice of coordinate system, up until the final measurement. An identical law of evolution must apply in all coordinate systems. We observe that systems of linear first order differential equations, can be discretized exactly, and conclude that the same differential equation, i.e. the Dirac equation, will have the same solution as the appropriate difference equation in any given coordinate system. Thus the continuum appears not as the result of induction or other non-empirical limiting procedure, but by using the association between a difference equation and a differential equation. The continuum equations remove any dependency on a specific measurement apparatus and resolution because they contain embedded within them the solutions for all discrete coordinate systems possible in principal or in practice. Substitute p → Λ'p, observing that the integral is invariant up to the <a href=DiracParticles>normalisation of plane wave states</a>, Primed coordinates is imposed after transformation by restricting to points x' in a new cubic lattice <img title="Primed coordinates" alt="QuantumCovariance-65" src="images/quantumcovariance/QuantumCovariance-65.gif" align="texttop" vspace="0">. The transformed state is the restriction to the new lattice, i.e. The transform of a position ket, <img title="The transform of a position ket" alt="QuantumCovariance-67" src="images/quantumcovariance/QuantumCovariance-67.gif" align="texttop" vspace="0">, is not an eigenstate of position in <img title="Primed coordinates" alt="QuantumCovariance-65" src="images/quantumcovariance/QuantumCovariance-65.gif" align="texttop" vspace="0">; if a measurement of position were done and we were then to transform back to <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="0"> the state would no longer be <img title="A position ket" alt="QuantumCovariance-70" src="images/quantumcovariance/QuantumCovariance-70.gif" align="texttop" vspace="0">. In other words, the operators for position in <img title="The discrete coordinate system" alt="QuantumCovariance-3" src="images/quantumcovariance/QuantumCovariance-3.gif" align="texttop" vspace="0"> and in <img title="Primed coordinates" alt="QuantumCovariance-65" src="images/quantumcovariance/QuantumCovariance-65.gif" align="texttop" vspace="0"> do not commute. But if no measurement is done, we can transform straight back and recover <img title="A position ket" alt="QuantumCovariance-70" src="images/quantumcovariance/QuantumCovariance-70.gif" align="texttop" vspace="0">"",
So there is no issue concerning unitarity under Lorentz transformation using discrete coordinates.

Edited on 2008-03-17 09:30:00 by CharlesFrancis

No differences.

Edited on 2008-03-17 09:26:39 by CharlesFrancis

Additions:
QuantumCovariance-42

can be embedded into a continuous function, f : R⁴ → C, the wave function, defined for QuantumCovariance-54

by
The transform of a position ket, QuantumCovariance-67

, is not an eigenstate of position in QuantumCovariance-65

; if a measurement of position were done and we were then to transform back to QuantumCovariance-3

the state would no longer be QuantumCovariance-70

. In other words, the operators for position in QuantumCovariance-3

and in

do not commute. But if no measurement is done, we can transform straight back and recover QuantumCovariance-70

,

Deletions:
QuantumCovariance-42

.
can be embedded into a continuous function f : R⁴ → C, the wave function, defined for QuantumCovariance-54

by
The transform of a position ket, QuantumCovariance-66

, is not an eigenstate of position in QuantumCovariance-65

; if a measurement of position were done and we were then to transform back to QuantumCovariance-3

the state would no longer be QuantumCovariance-70

. In other words, the operators for position in QuantumCovariance-3

and in

do not commute. But if no measurement is done, we can transform straight back and recover QuantumCovariance-70

Edited on 2008-03-17 09:22:01 by CharlesFrancis

Additions:
The plane wave state, QuantumCovariance-7

, is defined strictly as a finite sum,

Given the initial state, QuantumCovariance-9

, determined by measurement at time x⁰ = 0 in the discrete coordinate system, QuantumCovariance-3

, the momentum space wave function is,
This formula defines a continuous function, QuantumCovariance-13

, on the 3-torus QuantumCovariance-14

. Only a discrete subset, QuantumCovariance-15

,
If x ≠ y,
and if x =; y,
where in both cases the mathematical integral is used, defined by a limit as distinct from a finite sum appropriate to the empirical description of physics. Thus,
In conventional units, the components of momentum are bounded by QuantumCovariance-26

, where the theoretical bound on momentum depends on the lower bound of small lattice spacing, χ_min, not on the actual lattice appropriate to a given apparatus. It will be shown that the curvature expressed in Einstein’s Field Equation is equivalent to the existence a fundamental discrete unit of proper time, χ, between particle interactions of magnitude twice the Schwarzschild radius for an elementary particle. For an electron of mass m, χ = 4Gm ⁄ c³ = 9.02 × 10⁻⁶⁶s, or χ = 2.70 × 10⁻⁵⁷m, where G is the gravitation constant. This leads to a theoretical bound on momentum of 5.72 × 10⁵¹eV, or <

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