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.

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.

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. |

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 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. |

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

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

The plane wave state, , is defined strictly as a finite sum,

Given the initial state, , determined by measurement at time

This formula defines a continuous function, , on the 3-torus . Only a discrete subset, ,

is used for inversion by the discrete Fourier transform: for ,

If

and if

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 can be replaced with an exact integral over continuum values in , without altering the theory in any way.

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 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 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 × 10

The non-physical periodic property of can removed by the substitution , where if and otherwise. The wave function may then be replaced in flat space with the standard form in relativistic quantum mechanics:

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

Specifically, because the sum over discrete values of momentum in can be replaced with an integral over , the inner product, for ,

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.

.

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.

So, there is no issue concerning unitarity under Lorentz transformation using discrete coordinates.

Quantum Covariance ↑ Introduction to the Teleconnection →

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