Historical background

The H-principle has its conceptual background in the discussions of the 1920s concerning description of physical systems. The optimism of scientists resulted in statements of the kind that 'nature is deterministic' or 'all physical systems can be described by appropriate differential equations'. But Heisenberg pointed out that there are certain limitations concerning our description of the physical world. The discussions of Heisenberg, Niels Bohr and other physicists were centered around the two questions:

Ø 1) How well can we expect the measurement equipment to measure the experimental situation?

Ø 2) How 'close' can the measurement equipment come to the phenomenon in question? In other words, when is the measurement equipment interfering with the phenomenon in question?

These questions were studied in the light of examples from quantum mechanics. As a result of the numerous discussions, Heisenberg formulated his theory on the Uncertainty Inequality and Bohr his theory of complementarity. Heisenberg's theory concerns 'conjugate magnitudes', which cannot be observed exactly at the same time. An example of such conjugate magnitudes is the position and momentum (speed) of an elementary particle. He formulated his famous Uncertainty Inequality as

D(position) • D(momentum) > constant,

where the constant is related to the Planck's constant of light. Bohr went in a way one step further and said that in quantum mechanic situations there are generally observables that cannot be separated, but belong together. Bohr often used the Chinese symbols 'Yin and Yang' of the male and female concept to illustrate that in the measurement situation we always have two aspects of the physical system, which cannot be separated, but both are needed in order to obtain a proper description of the physical system.

Although Heisenberg formulated his uncertainty inequality in terms of micro physical systems, it has been transferred to 'macro' physical systems. The basic measurement questions are of general importance, not only for micro physical systems. In philosophy and methodology of sciences it has often been formulated by stating that the description of a physical system and the associated precision are mutually conjugate magnitudes. The uncertainty inequality takes here the form

D(Description) • D(Associated Precision) > constant.

The H-principle is concerned with a mathematical formulation of this theory. The basic idea of the H-principle is to carry out the computations in steps, where at each step we seek to balance the improvement in description, we are looking for, with the associated precision, we in fact can obtain. The balance is obtained by minimizing the left-hand side of the inequality.

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