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Whenever we measure something, it is usually inexact. For example, the mass of a baseball will never be measured exactly on a scale in any unit of measurement besides "mass in baseballs that are currently being measured" and rational multiples thereof.

Are there any non-arbitrary physical quantities that are "inherently" rational? That is, quantities that can be expressed "exactly" in a non-trivial, fundamental unit of measurement?

As far as I know, there is total charge, because charge can always be represented as an integer multiple of the charge of an electron. And to an extent, photons/light can be described this way too. Are there any others?

Also, are there any inherently "algebraic" quantities that are not inherently rational? That is, quantities that can be expressed closed-form as the solution to an algebraic equation in a non-trivial unit of measurement, but not as an "exact" ratio to that unit of measurement?

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Charge? Intrinsic angular momentum? Or, if you don't want the answers to be every quantized thing in Cohen-Tandouji you might want to explain why those things shouldn't be counted. – dmckee Dec 17 '10 at 1:36
Why the talk about measurement? If you are comfortable with the notion of a measurement having an uncertainty then it should be clear to you that the question doesn't make sense. The only related question which does make sense is whether any dimensionless quantity can be determined in algebraic/closed form (e.g. expressing coupling constants of elementary interactions in terms of some properties of string theory). Is this what you are asking for? – Marek Dec 17 '10 at 1:40
Actually, photons can't be measured like this, if I remember correctly. The photon number operator doesn't commute with the Hamiltonian in a good deal of quantum optical systems, meaning that the number of photons in such systems is subject to the uncertainty principle. – Jerry Schirmer Dec 17 '10 at 2:53
Aren't electron orbital levels naturally quantized, since they are standing waves? I thought that's where the word "quantum" came from. – Mike Dunlavey Oct 25 '11 at 16:52

Are there any non-arbitrary physical quantities that are "inherently" rational?

Absolutely! The conductivity plateaus in the integer and fractional quantum hall effects correspond to conductivities:

$$ \sigma = \nu e^2/h $$

The $\nu$ take on values which are integers (obviously) for the IQHE and fractions for the FQHE. So if you fix the "non-trivial, fundamental unit of measurement", as you put it, for conductance to be $e^2/h$ then the hall conductivities for the IQHE and FQHE are "inherently rational".

The energy levels of the Hydrogen atom also satisfy these criteria. The energy of the $n^\textrm{th}$ orbital is given by:

$$ E_n = - \frac{E_r}{n^2} $$

If you take the Rydberg energy $E_r$ to be a "fundamental unit of measurement" then $E_n$ again satisfy the criteria.

Some might argue that $E_r$ is not fundamental enough. However, the quantum of hall conductance $e^2/h$ certainly is as fundamental as it gets. There are plenty of other examples such as, as mentioned by you and others, charge and spin. I'd like to add magnetic flux to that list. Again, from the quantum hall effect, where $p$ quanta of magnetic flux $\phi_0$ pair up with $q$ electrons to give you a composite fermion with an anyonic exchange phase $pq$.

In general, (almost?) any observable that corresponds to a topological invariant has eigenvalues which are integer or rational.

Also, are there any inherently "algebraic" quantities that are not inherently rational?

Interestingly, that appears to be a harder question to answer. So I'll let someone handle this part ;-)

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Of course that's subject to the limitation that the system has a perfect topolotical invariant that isn't influenced by pesky interaction Hamiltonians – Jerry Schirmer Dec 17 '10 at 2:54
@space_cadet: nothing against your answer. Just that to me the question didn't make any sense, so I wondered why you interpreted it this way. You can always arbitrarily redefine units and then your quantities need not be rational anymore. So for question to make sense it must actually ask for some dimensionless purely mathematical quantities (group rep. numbers, topological invariants, etc.). But it does not ask for it. Instead it talks about measurements and whatnot... – Marek Dec 17 '10 at 16:30
@space_cadet: I am sorry, but again I don't understand the talk about measurement. You will never be able to measure whether or not the system is precisely quantized. Talking about that precision makes sense only on the level of mathematical physics. You have to choose sides eventually :-) – Marek Dec 17 '10 at 17:56
@space_cadet: yeah, so I'll leave you with an exercise of finding how many irrational numbers are there to within "one part in billion" to those predicted ones :-) – Marek Dec 17 '10 at 22:42
@space_cadet: what I meant to point out is that you can never be certain with measurement. Just take that hydrogen atom you yourself mention. First it seemed like the spectrum would have been rational. But then people found out about fine structure and hyper-fine structure, etc. and today we know that those levels actually aren't rational at all. So if you are talking about precise numbers it's always in the context of some theory. Never experiment. That's why I think it's useless mentioning the experiment in the first place. Anyway, let it be, I was half joking anyway :-) – Marek Dec 18 '10 at 7:15

Inherently rational quantities are rare, and inherently algebraic quantities are rarer still. Real numbers are defined by a limit, so you need a large-system of some kind, one of whose dimensionless properties converges to a rational or algebraic number. This occurs in certain thermodynamic systems, and the quantum hall conductivity plateaus are a nontrivial quantum mechanical example.

The critical exponents of phase transitions provide a separate example in purely classical thermodynamics. These quantities have nothing directly to do with quantum mechanics, and they are discrete. In two dimensions, the exponents in two dimensions are rational for the Potts model, which is not free, but is a nontrivial conformal field theory. There are physical realizations of these systems, and the critical exponents can be measured in these. The exponents of physical systems are exactly determined by the models, and this is a statement of universality.

So if you take a two-dimensional surface which is magnetized, and look at zero external field, and vary the temperature near the Curie point, the exponent of the spontaneous magnetization vs. the distance from the critical temperature is 1/8.

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Blockquote Also, are there any inherently "algebraic" quantities that are not inherently rational?

When you say algebraic, do you mean quantities that can be organized as a Group? In my view, we can make measurements only up to an isomorphism which is imposed by your measuring device, so perhaps this answers your second question.

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Algebraic numbers. – Kevin Reid Oct 26 '11 at 0:22

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