Are Natural Units Truly Dimensionless?

[Bobhawke]

This is perhaps a stupid question but:

When we use natural units and set h=c=1, do we choose appropriate units so that their value is one but these constants still have dimensions, or are we somehow choosing h=c=1 to be a pure number with no dimensions?

 

[Meir Achuz]

There are two ways to consider c=1. One is to use units like light years and years so that
speed has the dimensions of LY/Y and c happens to equal 1 in those units.
A more fundamental use of c=1 is that space and time are in the same units so that c=1 and is dimensionless. This is accomplished by giving space and time each the unit of second or each having the unit cm with a conversion between cm and seconds of
1=3X10^10 cm/sec. This is similar to the conversion 1 mile=5280 feet in an American topological map. Similarly with hbar. When hbar and c are 1, a useful conversion is
1=197 MeV-fm.

 

[HallsofIvy]

The answer to your question is, yes, they still have units. "c" would be in "light years per year" or, equivalently, "light seconds per second".

 

[Vanadium 50]

Note that there is a difference between "units" and "dimensions". Something can have units of meters per meter (e.g. describing an incline) even though that's dimensionless.

 

[marcus]

There are two ways to consider c=1. One is to use units like light years and years so that
speed has the dimensions of LY/Y and c happens to equal 1 in those units.
A more fundamental use of c=1 is that space and time are in the same units so that c=1 and is dimensionless. This is accomplished by giving space and time each the unit of second or each having the unit cm with a conversion between cm and seconds of
1=3X10^10 cm/sec. This is similar to the conversion 1 mile=5280 feet in an American topological map. Similarly with hbar. When hbar and c are 1, a useful conversion is
1=197 MeV-fm.

Hi clem,
this is how I thought it was before I read some of the other answers. I had the impression that in some research they choose units so that the numerical values of hbar and c expressed in those units is unity.

And in other writings they go all the way and actually declare hbar and/or c to be equal to 1 as a pure number. And then someone giving a seminar talk may say something like "blah blah...which you can see if you put the units back in..."

The way I understood it was that in the first case there really was some everpresent concrete system of units, like for example some version of Planck units in which certain quantities had specified values, like one or (8 pi)^-1 or whatever, and you respected them as physical quantities and treated them as such.

But in the second case the system of units and associated conventions had been to some extent collapsed and the suckers really were pure dimensionless numbers.

So I would have agreed with you. But after reading some of the other posts I'm no longer sure. Do you have any more to say? Do you have some way you tell which case a given paper belongs to?

 

[blechman]

The way I always thought of it was as follows:

In high school/college physics, we are taught the "MKS" (or "CGS") system of units. The idea being that there are three "fundamental" units:

1. length
2. mass
3. time

ALL other units (coulombs, pascals, Newtons, tesla, etc) all can be expressed in terms of products or divisions of these units.

But why should we choose these three? What about the following choice:

1. velocity
2. angular momentum
3. energy

This is an equally good choice of "unit system", and as you might now imagine, it is certainly not the only one.

Now, let us chose some units to describe these "dimensions"

1. velocity: let's measure all velocities in units of c.
2. angular momentum: let's measure all a.m. in units of hbar.
3. energy: let's measure energy in units of GeV.

This is the choice of units used by particle physicists. And with this choice, ALL derived units can be expressed as powers of hbar, c, and GeV. Furthermore, we might as well just drop the hbar and c, since it is ALWAYS clear from the context what you mean. (If you don't believe that, then consider this: if I tell you I'm using MKS units, and the velocity is 5, would you think that this was "5 grams.meters^2"?). With this choice, ALL dimensionful quantities can be expressed as a power of energy. This is how particle physicists like to quote numerical results.

You can do this with any unit system you feel like. Sometimes, GR people like to work in units of Newton's gravity constant and c. If you do condensed matter, you probably don't want units of c, but you might want units of Boltzmann's constant. etc.

As to marcus's question: which unit system you use is set by the conventions of the community. you just have to know what those conventions are. i don't think you'll find them published anywhere, but you can always ask someone in the field for clarification if you're not sure.

Hope that helps!

 

[Meir Achuz]

I agree with much of Blechman's post, but it really is not just arbitrary.
Let me address Marcus's questions a bit also.
I'll start with the American topographical map. If the x and y coordinates are in miles and the z coordinate is also in miles, the ratio z/x is dimensionless, and there is no "conversion" between them.
However if (as American's do) the z coordinate is feet, then the ratio z/x has dimensions,
and there is a "universal constant" C=5,280 feet/mile. Then, someone notices (perhaps by flying a plane) that there is a rotational symmetry that can rotate z into x, etc. Then the choice of miles and feet is seen to mask this symmetry and the constant C is seen to be just an artifact caused by measuring the same quantity, length, by two different methods. The same reasoning applies to c, ever since Minkowski showed that time and space have a four dimensional symmetry that transforms one into the other. c=1, and being dimensionless, is the acknowledgment of space-time symmetrhy. In fact c should not appear in the first place and then be "set equal to one". It shouldn't be there at all in any relativistic formula, in the same way as C need not appear in 3D rotation. These dimensioned constants only appear as a mismatch of units for the same dimension. A similar statement can be made for hbar once one realizes that, in QM, p and x are Fourier transform transform variables which means that p and 1/x should have the same units. Then hbar never appears.

 

[hamster143]

clem is correct.

First you have to realize that space and time are the same "thing", they are measured in different units because our ancestors did not have the knowledge of Special Relativity and they independently picked an arbitrary unit of length (1/40,000,000'th of Earth's circumference) and an arbitrary unit of time (1/86,400th of Earth's period of rotation). That's easy.

It's slightly harder to understand that mass (energy) is the same thing as frequency (inverse time). Again, our ignorant ancestors picked a random unit of mass (equal to the weight of 1 cm^3 of water at the temperature of melting ice) and we ended up needing a "constant" (h) to translate mass into frequency.

 

[blechman]

I agree with everything clem and hamster143 said.

In defense of my "arbitrary" statement: I only mean that this is not the only choice of "natural units" used in physics. In particle physics, where QM is important and everything is relativistic, then everything said is right. However, if you are a GR researcher (not quantum gravity but classical GR), then you are foolish to use hbar=1 units! You COULD, but you don't. Instead, since the relevant constants are c and G_N, you use units where they are =1. Then instead of everything being in units of energy, everything's in units of length. That's all I meant by the "arbitrary" statement: you have to decide what the "relevant dimensions" of your problem are, and choose units accordingly. But a solid-state physicist, a chemist, and GR expert and a particle physicist will not generally agree on those conventions, and so will use different "units" by setting different constants equal to 1. That's all I meant to say.

 

[Meir Achuz]

I agree. I wouldn't use MeV with hbar=1 to weigh myself.
But, so many elementary texts use Kg for the mass of the proton.
It amuses me (along with some pain) that physicists make things easy for themselves,
but complicated for beginning students.

 

[blechman]

well put.

i guess it's for the "benefit" of the engineers among us!