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**These are excerpts and elaborations from my book "The Nature of Consciousness"**

In classical
Physics, a quantity (such as the position or the mass) is both an attribute of
the state of the system and an observable (a quantity that can be measured by
an observer). Quantum Theory makes a sharp distinction between states and
observables. If the system is in a given state, an observable can assume a
range of values (so called “eigenvalues”), each one with a given
probability. The evolution over time of
a system can be viewed as due (according to Heisenberg) to time evolution of the
observables or (according to Schroedinger) to time evolution of the
states. An observer can measure
at the same time only observables that are compatible. If the observables are
not compatible, they stand in a relation of mutual indeterminacy: the more
accurate a measurement of the one, the less accurate the measurement of the
other. Position and momentum are, for example, incompatible. This is a direct
consequence of the wave-particle dualism: only one of the two natures is
"visible" at each time. One can choose which one to observe (whether
the particle, that has a position, or the wave, that has a momentum), but
cannot observe both aspects at the same time. Precisely,
Heisenberg’s famous
"uncertainty principle" states that there is a limit to the precision
with which we can measure, at the same time, certain pairs of quantities,
notably the momentum and the position of a particle. If one measures the
momentum, then it cannot measure the position, and viceversa. Technically speaking: the product of
uncertainties in position and in momentum
must be greater than Planck’s constant. This is actually a
direct consequence of Einstein's equation that related the
wavelength and the momentum (or the frequency and the energy) of a light wave:
if coordinates (wavelength) and momentum are related, they are no longer
independent quantities. Einstein never believed in this principle, but he was
indirectly the one who discovered it. A similar
principle applies to other incompatible observables, for example between time
and energy: one cannot measure energy precisely at a precise instant in time.
Either the time or the amount of energy has to be imprecise. Hence, in theory,
violations of energy conservation can occur… but we cannot observe them. The
more the energy missing (unaccounted for), the faster it will be returned (the
shorter the period of time before it is accounted for). If you try to measure
energy at a precise time, then no information is known on how much energy is
there. The wave
function contains the answers to all the questions that can be asked about a
system, but not all those questions can be asked simultaneously. If they are
asked simultaneously, the replies will not be precise. The degree of
uncertainty is proportional to the Planck constant. This implies that there is a limit to how small a
physical system can be, because, below a quantity proportional to the Planck
constant and called "Planck length", the physical laws of Quantum
Theory stop working altogether. The Planck scale (10 Note that
Heisenberg does not forbid precise measurements of
"compatible observables", for example of position, charge and spin.
It only applies to "incompatible observables", which are couples:
position/momentum, energy/time, electric field/magnetic field, angle/angular
momentum, etc. The uncertainty
predicted by Quantum Theory (and verified by countless experiments in countless
laboratories) has been sometimes interpreted as a consequence of the fact that,
at the microscopic level, one cannot pretend that a measurement is “objective”
at all: a measurement is an interaction between two systems, which, like all
interactions, affects both. But that is not quite where Heisenberg’s calculations came from. They
originate, as everything else, from Planck’s constant. For the record,
there had been other “principles of uncertainty” in Physics, and an important
one in Mathematics, the one discovered by Joseph Fourier in the 19th century that a signal cannot be simultaneously
localized both in time and in frequency: for example, there is a limit to the
precision of the simultaneous measurement of the duration and frequency of a
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