Comment by adrian_b

Comment by adrian_b 3 days ago

The Heisenberg uncertainty principle is completely irrelevant for metrology and it certainly does not have any relationship whatsoever with the algebraic structure of the set of values of a physical quantity.

The Heisenberg uncertainty principle is just a trivial consequence of the properties of the Fourier transform. It just states that there are certain pairs of physical quantities which are not independent (because their probability densities are connected by a Fourier transform relationship), so measuring both simultaneously with an arbitrary precision is not possible.

The Heisenberg uncertainty principle says absolutely nothing about the measurement of a single physical quantity or about the simultaneous measurement of a pair of independent physical quantities.

There is no such thing as a quantity that cannot be measured, i.e. the set of its values does not have the required Archimedean group algebraic structure. If it cannot be measured, it is not a quantity (there are also qualities, which can only be compared, but not measured, so the sets of their values are only ordered sets, not Archimedean groups; an example of a physical quality, which is not a physical quantity, is the Mohs hardness, whose numeric values are just labels attached to certain values, a Mohs hardness of "3" could have been as well labeled as "Ktcwy" or with any other arbitrary string, the numeric labels have been chosen only to remember easy the order between them).

btilly 3 days ago

The Heisenberg uncertainty principle says that basic physical quantities do not have unique values. They have ranges of values. And therefore it is not always possible to compare two physical properties. For example you can't always compare two particles to discover which is farther away from you.

Various proposals exist in which the fundamental physical structure of space form a kind of "quantum foam". And so, somewhere near the Planck length, it doesn't even make sense to talk about distance.

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adrian_b 3 days ago

I am sorry, but you have forgotten what the Heisenberg uncertainty principle really says.
What you have in mind about "ranges of values" has nothing to do with Heisenberg, but with the formalism of quantum mechanics as developed by Schroedinger, Dirac and others (not counting the matrix mechanics of Heisenberg, which was an inferior mathematical method, soon forgotten after superior alternatives were developed, and which is something completely else than the Heisenberg principle of uncertainty).
In the various variants of the formalism of quantum mechanics, the correspondent of a single value in classic dynamics is a function, the so-called wave function, but even those functions cannot be called "ranges of values". There are several interpretations of what the "wave" functions mean, but the most common is that they are probability densities for the values of the corresponding physical quantity.
Depending on the context, in quantum mechanics the value of a physical quantity may be unknown, when only the probability of it having various values is known, but the value may also be known with absolute certainty, usually in some stationary states.
In both cases the value of the physical quantity can be measured, but in the former only an average value can be measured, which nonetheless may have unlimited precision, while in the latter case the exact value can be measured, exactly like in classical dynamics.

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- btilly 3 days ago
  
  You are being unnecessariky rude. I have not forgotten what it says.
  There are pairs of physical properties, connected by a Fourier transform, that cannot both be known at once. Therefore each property is, in general, only known to be in a range. And therefore we cannot always compare these properties between different particles.
  As for measurement, in the Everett interpretation the values are not known after measurement either. All that is knowable are correlations due to entanglement.
  But now we are far afield. You made an a priori statement about physics. Physics is not required to oblige you. My understanding of physics says that it does not. But even if I am wrong, and it currently does oblige you, it is still not required to.
  A priori assertions do not have a good history. For an example, look at Kant's assertion that reality must be described by Euclidean geometry. He was not in a position to understand the ways in which he would prove to be wrong.
  
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