# Dimensionful vs dimensionless constants

1. Dec 13, 2009

### apeiron

In describing physical reality, we have two classes of constants, the dimensionless and the dimensionful.

http://en.wikipedia.org/wiki/Physical_constant

Which is the more fundamental and why? And exactly what is the meaning of their difference?

I have my own take of course. My feeling is that a constant is always a measurement of an equilibrium resulting from two processes or actions "in balance". In some sense, it is where a dynamic system settles.

All the constants seem to have this basic duality. So c is a dimensionful constant that relates distance and time. G is a dimensional constant that relates distance and mass.

Dimensionless constants also seem to be constructed from other things in interaction.

Wiki says: "Such a number is typically defined as a product or ratio of quantities that have units individually, but cancel out in the combination."
http://en.wikipedia.org/wiki/Dimensionless

So the essential difference between the two classes of constants would seem to be that the dimensionful are the result of asymmetries - you need measurements of two orthogonal axes of description like distance and time - while the dimensionless are the result of what appear to be symmetries. Measurements are made in the one coin (even if more than one axes is required) and so can be cancelled out.

So I think this is an interesting difference. One requires pairs of measurements that remain orthogonal and cannot be reduced further, the other requires pairs which do allow reduction to a naked number.

Is this the right way to look at it and what does it mean?

The fine structure constant alpha is a complicated and key example of a dimensionless constant, which may in fact create problems for my simple analysis.

http://en.wikipedia.org/wiki/Fine_structure_constant

And there may be other good challenges lurking in the considerable list of dimensionless constants given by Wiki.

http://en.wikipedia.org/wiki/Dimensionless

2. Dec 13, 2009

### Pythagorean

imo, dimensionless constants still have dimension. Otherwise we'd be able to mix different dimensionless constants without any logic breakdown.

Dimensionless constants, in many cases, are ratios of one dimension to itself (like the ratio of the top edge of as rectangle to the side edge of a rectangle). We may call it dimensionless, but we can still look at it as meters per meter.

Mass was a "dimensionless" quantity at one time. You put two objects on a balance and compare their relative distances from the center. The ratio of the distances is dimensionless. I have a professor who insists that the only fundamental units are space and time with this argument.

Anyway, my final point is basically that distance this way per distance that way is not equivalent to my tame taken per your time taken. They both have dimensions even if the units cancel out mathematically.

3. Dec 13, 2009

### Pythagorean

I agree with my professor (mentioned above) that space and time are the two fundamental constants. Everything else can be framed from there (considering what I said about mass).

Thus, dimensional constants are more fundamental (but not by definition, just seems to be that way given your definitions).

Is asymmetry necessarily associated with orthogonality? I'm not sure I see asymmetry and symmetry here.

4. Dec 13, 2009

### apeiron

Yes, they would have to define some dimension, or exist in some dimension.

Which would be the difference between physical and mathematical constants?

But wouldn't they be the fundamental dimensions and the constants would then be the various way the two aspects of reality are related.

So, for example, c would be the rate at which coherent change can evolve? h or planck constant seems to describe the fundamental grain of spacetime. G seems to be a measure of the global flatness of spacetime (or rather it scales the local departures from flatness).

So treating spacetime as the fundamental stuff is attractive. But the constants would arise as ways of scaling the relationship between space and time. Or are their irreducible constants which are just spatial or temporal that you can think of?

Orthogonality would be the definite requirement. Two unrelated axes of measurement.

I do see this as related to symmetry and asymmetry. A symmetry is where you can go back and forth in the same dimension. An asymmetry is where that symmetry is broken. In some sense, another whole new direction now exists.

But this is not essential to the question. The distinction between orthogonal pairs of measurements and sets of measurements where the units cancel is sufficient.

5. Dec 13, 2009

### Pythagorean

I think the simplest way to put it would be observability. The observability associated with a constant, the more physical it is.

Yes, I mis-spoke. I meant fundamental units, not constants. Units are more associated with dimensionality.

This is an interesting perspective and it makes sense intuitively. What would define as "coherent change" though? And would we agree that we must restrict "change" specifically to the linear ratio of distance per time (or it's reciprocal depending on your perspective)?

Well, since we're taking care to look at the "history" of a constant (so that we know a dimensionless constant is really a ratio of dimensions) even temporal constants may themselves include hidden ratios in them that have canceled out mathematically.

For instance, the time constant of an RC circuit gives you physical information relevant to the circuit, and it is in units of seconds, but you can express the time constant, tau as

$$\tau = RC$$

Where R and C are electrical ratios that come, ultimately, from classical electromagnetic theory, whose units (Ohms and Farads) ultimately come from space, time, and mass. (Of course, we can argue that mass is reducible to space and time itself, but I'm not completely sure if that's the case. I have no formal exposure to GR, for instance, which may have more to say about that, and seems to agree conceptually.)

Yes, that seems to be the case to me. Somehow, though, I feel like time is more independent of space then space is of time. From a human perspective, this makes time merciless. We can go back and forward in space at different rates, but we acknowledge a universe in which we only travel forward through time (albeit at different rates, which we somehow control with our velocity).

But then this makes me wonder if time itself is simply an operation on space. There is no absolute time frame we all operate in. Our velocity can only be compared to another object that's "fixed" in our frame. There is no ultimate origin in the universe. There's instead, boundaries imposed on the rate of change of distances between objects.

what do you mean by "you can go back and forth"? Where's an example where you can't go back? In one sense, time is a dimension you can't go back in. It may be convenient to talk about particles that go back in time (Feynman diagrams), but I'm not sure that's something we've actually observed or not. This might be a good question for the particle physics subforum.

6. Dec 13, 2009

### Freeman Dyson

Is this correct?

"Is this correct?

"C is a speed, so it has dimensions of distance per time. For example, it has a value of about 186,000 miles per second, or 300,000,000 meters per second.

A dimensionless constant like Pi is different. Pi is 3.14159... regardless of whether you are measuring in miles, or in meters, or in carob seeds."

7. Dec 13, 2009

### Staff: Mentor

8. Dec 13, 2009

### Pythagorean

pi can also be seen as the ratio of a circle's circumference to its diameter, so you could argue that it has units of distance around per distance across.

9. Dec 14, 2009

### apeiron

I am thinking of light cones. The inside of a light cone would be coherent in that all parts have the change to exchange signals.

c also seems a good example of the dynamism of constants. We may treat time and space as linear dimensions that can be counted out in standard units. But relativity instead identified a constant that relates the units of these different dimensions in a "constant" fashion.

And doesn't it introduce an asymmetry? Give meaning to inverse and reciprocals, as you say?

The constant has the effect of shrinking the scale in one direction (space) as you increase the scale in the other (time). And vice versa.

Without c, the relationship between space and time would be symmetric, or perhaps more accurately, vague. Signalling would have no speed limit. But with c, there is always one that is being made smaller if the other is being made larger. This is asymmetric.

Good point. But how does it bear on the general nature of constants?

I would say time seems different because it is asymmetric. There is a thermodynamic arrow pointed the one way.

But to model spacetime in GR, time is given an artificial symmetry (travel both ways is allowed) so a time "dimension" can be related to the spatial dimensions via a constant.

Yes time is an example of something that seems actually an asymmetry but which we find easier to model if we force it into the garb of a symmetry.

10. Dec 14, 2009

### apeiron

Mathematical constants are actually similar in their construction I would say - the equilbrium balance points of a pair of opposed processes.

So pi can be geometrically constructed as the limit of an infinite sided polygon. And you can either build up from the inside of a circle or shrink the polygon to fit from the outside. And pi is arithmetically constructed as a harmonic series, like the classic ¼ pi = 1 – 1/3 + 1/5 – 1/7.... (as well as a host of others). You can see the "opposed process" on the alternate pluses and minuses of the series.

It is also worth noting that pi is 3.14159... only in flat space. Its value varies from 2 on a hypersphere to infinity with hyperbolic curved space.

So the constant itself is a constant only at an exact balance point between all possible states of curvature.

11. Dec 14, 2009

### Freeman Dyson

I read a good book on this by John Barow. The history of finding the constants and making them as fundamental as posslbe. The idea was, what constant could you communicate to an alien intelligence that they would immediately understand. It would be universal. Reduce it to its most basic form.

Einstein thought things like Pi were the only real type of constants.

I cant copy and paste google books but here is him talking about it

http://books.google.com/books?id=es...e question of the universal constants&f=false

Last edited: Dec 14, 2009
12. Dec 14, 2009

### Freeman Dyson

I tried to copy and past some of it:

What Einstein is saying is that there are some apparent constants which are created by our habit of measuring things in particular units.

Why cant I copy pdf files from my hard drive? Anyway, that is just Einstein's view. The book shows many other views and the history and philosophy of the search for constants.

13. Dec 15, 2009

### apeiron

Good cite. The thing that has puzzled me is that the dimensionful constants like c, h and G have the look of being basic. They appear to scale the container - spacetime. While the dimensionless constants of the standard theory are "just" the values of particular elements of the system - the local contents inside the global container. But Baez like many others is more excited by the dimensionless constants. They are the ones that seem more basic.

Here Baez says just this...

So QM and GR describe the container - the larger stage that has axes of description that need to be related. I think Baez is missing a point in that the planck constants create a scale relationship between the axes (as with c and spacetime). That does not seem trivial.

So the ratio of the observed masses to the theoretical boundary state for mass (the planck mass) gives us a set of "constants". Fundamental particles always have the same (at rest) size. The planck mass itself would be a dimensionful constant I believe (relating hc to G). So in a roundabout way, these particle masses are all slices off a dimensionful constant. And if particle masses could be tied back to some gauge symmetry principle, then perhaps they too would be "opened up" - relating the planck mass to some local resonance or spacetime dimensional principle. So no longer mass compared to mass, but mass compared to "another direction".

Same deal with the bosons as with the fermions. But now substituting a fundamental scale for electromagnetic fields or strong force fields for planck mass, or upper bound on gravitational curvature?

The story continues. All the dimensionless constants are local values - slices off the naked strength of the four fundamental forces. Gravity seems easy to relate to a dimensionful view - so much curvature over so much spacetime, two directions that need to be measured and related - but the other three forces are not so easily broken down into a dimensional view. Though the strength of each may be relate via the dimensional structure implied by gauge symmetry perhaps.

So far we only have a naked measurement and no mechanism that might "open up" this constant as an interaction.

Is the Higgs mechanism then a candidate for making the connection to dimensionful constants? With mixing angles, for example, it seems that the Higgs is a way to relate internal gauge symmetries to external spacetime environments. Literally an opening up.

Considering the pure constants of maths, these do all seem to be a story of two processes or directions of action in interaction. You converge on a point because you are coming towards it from two directions. And this could be from either side in the same dimension - in which case the units would cancel to a dimensionless ratio (though I can't think of examples of "same dimension" maths constants off hand) - or it would be coming from two orthogonal directions and the units of the two directions would become related in a fixed value way.

In fundamental physics, constants seem to arise either as ways to relate the different aspects of the "container" - fixing how variation in one forces a matched variation in the other. Or they fix the size of the contents. They take an interaction, a coupling, like gravity, strong or EW, and cut off a constant slice of the naked value.

Makes some kind of sense to me anyway.

14. Dec 15, 2009

### Pythagorean

From the other replies here, I've realized I've been harboring an assumption in my replies that I didn't quite realize until now. I've been assuming:

a) there's an irreducible sized chunk of spacetime and that spacetime is not continuous.

I believe this is why I'm speaking about units as if they were constants, because I've somehow deduced that a) implies:

b) there are fundamental units pertaining to distance and time themselves, (which I would consider the fundamental constants).

But I have no logical reasoning for why a) would imply b), really, given that a) were actually true.

15. Dec 15, 2009

### apeiron

But I believe this too, for a different reason.

You are taking the view that reality is constructed from fundamental components (by the sounds of it). And so the irreducible chunks of spacetime. It would be the same if you had said reality is constructed of information.

I see it as a top-down issue of resolution - what an observer can hope to see. So there is an ultimate limit placed on our resolution of the very small, or very dense, or very energetic, etc. This is the origin of the planck scale constants. They are limits because when you focus all your attention on making one dimension of measurement as small and local as possible, you in effect completely lose sight of the other direction to which it is related. So that direction goes to infinity - the infinitely large.

Spacetime is two directions of measurement. So the constant arises as the dynamic balance of measurements across these two directions. The units within each dimension flexes to reflect this.

So yes, the units with which we measure distance and duration are not constant in some independent newtonian fashion. They are dependent in a relativistic fashion. A constant can only arise as a balance point that equilibrates action.

A top-down resolution approach then says that spacetime would become grainy on the most local view. Or at least break down radically at a limiting scale.

16. Dec 15, 2009

### Staff: Mentor

I agree with Baez on this. The dimensionless ones are really the more fundamental ones. It is an enjoyable and valuable exercise to go through, it turns out that if the dimensionful physical constants (e.g. c, h, G) change in such a way that the dimensionless physical constants (e.g. fine structure constant) are unchanged then none of our measurements will change, but if the dimensionless constants are changed then we can measure something different. In other words, physics is not sensitive to changes in dimensionful constants, only dimensionless constants. So the dimensionless constants are the ones that describe physics. See:
https://www.physicsforums.com/showpost.php?p=2011753&postcount=55
https://www.physicsforums.com/showpost.php?p=2015734&postcount=68
It sounds like you are thinking of geometrized units where mass is given dimensions of length: http://en.wikipedia.org/wiki/Geometrized_unit_system

17. May 22, 2010

### SdogV

18. May 23, 2010

### apeiron

19. May 25, 2010

### SdogV

Trying, but little follow up so far. Printed out their revised Feb 2, 2008 (arXiv:0711.4276v2 [physics.class-ph] 4 Dec 2007) version. Clearly it's possible significance lies in unveiling a clue to a relation between inertial and "active" gravitational mass. (Note # 14 in references)

So, the juggling required to make the ratio of units in masses, G to h protocols, equal to 1 is quite interesting. Not ready to report it....yet.

20. May 27, 2010

I find the dimensional constants most interesting, because they relate to what I think is the most fundamental question about the physical world, i.e. why its structure has the specific set of parameters that it has. Physics has a large and very disparate set of parameters, all of which are defined in terms of the others through its fundamental equations.

So far, it seems to me, physical theory has not really tried to understand the structure of this complicated web of inter-reference. Instead, in searching for a “unified” theory, the goal has been to reduce the number of parameters to a minimum – ideally, the “final” theory would reveal a single mathematical pattern involving as few parameters as possible. I suppose this preference for a purely mathematical foundation would be a reason for looking to the dimensionless parameters as more fundamental.

But this assumes that the differences between physical parameters are inessential. In other threads I’ve argued that the profound structural differences among these parameters – space and time, mass and charge, energy and momentum, etc. etc. – are required in order that any of them be observable / measurable.

If we assume there is a reason why the universe is organized so that every observable has a set of other observables in terms of which it can be defined and measured, then a logical place to begin an analysis of this inter-referential structure would be with the dimensional constants – each of which seems to have a unique character.

For example, Planck’s constant describes a minimal unit of action – a kind of “atom of happening” between things. Is there any other irreducible unit in physics? The speed of light defines a maximal velocity, which also defines null-vectors in spacetime. And the gravitational constant says something still quite mysterious about the connection between mass and the spacetime metric.

It’s not the numeric value of these constants that’s significant – as noted above, their values depend entirely on the units we choose to express them in. And they’re not necessarily more basic to the structure of the physical world than the dimensionless constants. But I think they must represent the key interdependencies that we ultimately need to understand.