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  Quantum Covariance    

There are deep philosophical issues regarding the use of the continuum in physics. Lorentz covariance is expressed in terms of quantities defined on a mathematical continuum. In relational quantum gravity, a physical continuum is not assumed. It is necessary to review, from first principles, the meaning of covariance, and to find the correct expression of the general principle of relativity, that local laws of physics are the same in any reference frame.

Substantivalism v Relationism

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 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.

Measurement

When a human observer seeks to quantify Nature, he chooses some particular matter from which to define a inertial 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.

Discrete Coordinates

QuantumCovariance-1The 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.

Coordinate Rotation

QuantumCovariance-2In 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 eigenstates 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.

Discretisation

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 inertial 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 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 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 × 10−57m, 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, Hilbert space is defined using a cubic lattice, QuantumCovariance-3, 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 QuantumCovariance-4 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.

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.

Momentum Space

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.

The plane wave state, QuantumCovariance-7, is defined strictly as a finite sum,
QuantumCovariance-8
Given the initial state, QuantumCovariance-9, determined by measurement at time x0 = 0 using the discrete coordinate system, QuantumCovariance-3, the momentum space wave function is,
QuantumCovariance-12
This formula defines a continuous function, QuantumCovariance-13, on the 3-torus QuantumCovariance-14. Only a discrete subset, QuantumCovariance-15,
QuantumCovariance-16
is used for inversion by the discrete Fourier transform: for QuantumCovariance-17,
QuantumCovariance-18

If x ≠ y,
QuantumCovariance-20
and if x = y,
QuantumCovariance-22
where in both cases the mathematical integral is used, defined by a limit as distinct from a finite sum. Thus,
QuantumCovariance-23
So, the sum over discrete values of momentum in QuantumCovariance-24 can be replaced with an exact integral over continuum values in QuantumCovariance-25, without altering the theory in any way.

Bounds

In conventional units, the components of momentum are bounded by QuantumCovariance-26, 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 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.

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.

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 inertial 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.

The non-physical periodic property of QuantumCovariance-36 can removed by the substitution QuantumCovariance-37, where QuantumCovariance-38 if QuantumCovariance-39 and QuantumCovariance-40 otherwise. The wave function may then be replaced in flat space with the standard form in relativistic quantum mechanics:
QuantumCovariance-42
At this point it has been shown that quantum theory contains the mathematical structure to describe momentum consistently in the usual way, as a vector defined on a continuum.

It makes little difference whether discreteness is taken into account in practical measurement. For example, the components of the discrete momentum operator are given by
QuantumCovariance-43
where χa is a vector of magnitude χ in the a direction (the stepsize of the lattice). Then
QuantumCovariance-45
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-48eV. This is unrealistic.

Quantum Covariance

Since the points which may be described in physical measurement are a discrete set, we expect to describe physical evolution in a given inertial reference frame with a difference equation. The replacement of differential equations by difference equations is 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 QuantumCovariance-24 can be replaced with an integral over QuantumCovariance-50, the inner product, for QuantumCovariance-17,
QuantumCovariance-52
can be embedded into a continuous function, f : R4 → C, the wave function, defined for QuantumCovariance-54 by
QuantumCovariance-55
QuantumCovariance-56 is found from f(x) simply by restricting to QuantumCovariance-17.

Definition:  Quantum covariance will mean that the wave function is defined on a continuum, while the inner product, is discrete, and that, in a change of reference frame, the lattice and inner product appropriate to one reference frame, or coordinate system, are replaced with the lattice and inner product of another.

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.

Operators

The general form of a linear operator, O on H1, is, for some function O(xy),
QuantumCovariance-59.
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.

Lorentz Transformation

Let QuantumCovariance-60 be a state of a particle at some time x0. Under Lorentz transform the state becomes,
QuantumCovariance-62
Substitute p → Λ'p, observing that the integral is invariant up to the choice of normalisation of plane wave states,
QuantumCovariance-64
Primed coordinates is imposed after transformation by restricting to points x' in a new cubic lattice QuantumCovariance-65. The transformed state is the restriction to the new lattice,
QuantumCovariance-66
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 QuantumCovariance-65 do not commute. The geometry is noncommutative. If no measurement is done, we can transform straight back and recover QuantumCovariance-70,
QuantumCovariance-74
So, there is no issue concerning unitarity under Lorentz transformation using discrete coordinates.

Quantum Covariance ↑Introduction to the Teleconnection →
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