Theory of incomplete measurements

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The Theory of Incomplete Measurements (TIM) is an attempt to unify Quantum Mechanics and General Relativity by focusing on physical measurement processes. In that theory, general relativity is a continuous approximation of discrete measurements, and quantum mechanics axioms are derived from properties of measurements.

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[edit] Introduction

The theory of incomplete measurements observes that the laws of physics apply to symbolic results of measurements, not to real numbers. Properties such as continuity or linearity result from an adequate choice of calibration for the measurements. When distinct physical methods are used to measure the same physical property, we choose to calibrate them so that they match, but this artificial identity does not necessarily exist at all scales, for all physical systems.

The theory identifies common sense characteristics for measurements and calls them "measurement postulates". From these postulates, several axioms of quantum mechanics can be deduced by inference. Similarly, the structure of general relativity is a consequence of the observation that distance and time measurements do not obey Euclidean geometry.

[edit] Measurement postulates

Physical measurements are identified using six postulates:

  1. Measurements are physical processes
  2. Their input and output are known in advance
  3. They give consistent (repeatable) results
  4. They depend only on their input
  5. They impact only their output
  6. The change in the output can be given a symbolic or numeric interpretation

[edit] State vector

In general, the state of a system as far as a measurement is concerned can be represented by a vector of probabilities for each possible measurement result. Since probabilities are positive and since their sum is 1, each state can be represented by a unit vector.

The number of dimensions for the state vector is the number of possible symbolic outcomes for the measurement. Unlike quantum mechanics, this dimension is always finite, because physical measurements only give discrete results, limited by the number of graduation markings or precision of the instruments.

Physical experiments are described by an operator on this state vector. They transform one set of probabilities to another. The operators are not necessarily linear. In fact, measurements are in general represented by non-linear operators. However, under specific linearity conditions, it is possible to identify the operator with a linear operator corresponding precisely to a quantum-mechanics observable.

[edit] Quantum Mechanics Implications

The wavefunction collapse is a consequence of the requirement that measurements give repeatable results. If the measurement is repeatable, immediately after a measurement, the state vector must indicate a certainty for the measurement that was observed, and an impossibility for the others.

Eigenvalues are not necessarily real numbers. Instead, they correspond to individual representations for each possible symbolic measurement result. However, it is common to choose a numeric representation for the measurement, in which case the eigenvalues are numbers. The theory derives the form of the eigenvalue equations from the measurement postulates.

A particular case is that of a 2-valued measurement. A unit vector in that case can be represented as a unit complex number. If the measurement being performed tests is there a particle, which has two possible results (yes/no), then at every point where the measurement is performed, probabilities can be represented as a unit complex number.

Furthermore, if there is a single particle in the entire experiment, then a second normalization condition exists, to ensure that the particle is only found in one place. In that case, the entire state if the system can be represented as a field of complex numbers. That field can be shown to have the properties normally associated with the wavefunction of the particle.

Quantum mechanics is interpreted as a linear approximation of the more general case.

[edit] General Relativity Implications

Individual space and time measurements give discrete results. However, it is reasonable to calibrate different physical measurements using a single scale. In that case, one is led to interpolate the physical measurements with an idealized function. This gives rise to the notion of spacetime continuum.

Associating a geometry to the results of space and time measurement is possible, but experimental evidence shows that this cannot be an Euclidean geometry. Many equations of general relativity are of geometric nature, and apply irrespective of the measurement being done. However, space and time measurements are experimentally not invariant by scaling, as in doubly-special relativity.

General relativity is interpreted as an approximation where continuity is postulated and dependency on scale or measurement are ignored.

[edit] See also

[edit] References