Re: Setting constants like hbar to unity
From: J. J. Lodder (nospam_at_de-ster.demon.nl)
Date: 08/16/04
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Date: 16 Aug 2004 13:56:00 -0400
Erik <erite423@yahoo.se> wrote:
> Many systems of units are often defined by setting assorted constants
> to unity:
>=20
> Atomic units: m =3D e =3D hbar =3D 1
> Natural units: hbar =3D c =3D 1
> Geometrized units: G =3D c =3D 1
> Planck units: hbar =3D G =3D c =3D 1
More precisely: by choosing units such that they are 1
(Meaning have the value 1)
> When I first saw equations like the ones above, I thought that they
> were just lazy ways of writing
>=20
> m =3D 1 mass unit,
> e =3D 1 electric charge unit,
>=20
> et cetera. However, judging from Appendix F in R.M. Wald's "General
> Relativity" or Appendix F in Burcham & Jobes' "Nuclear and Particle
> Physics", particle physicists and general relativists really do mean
> exactly what they write.=20
Of course, what else did you expect? This is physics.
> Appendices like those discuss which
> dimensions become equal when constants are set to unity. Thus, the
> above equations really do mean that the entire physical constant, as
> opposed to just its numerical value, is set to 1.
Indeed, an often made choice is to have c =3D 1 and dimensionless.
> Or so I thought until I, less than an hour ago, reread Duff, Okun &
> Veneziano's discussion of how many truly fundamental constants there
> are.
>=20
> http://www.arxiv.org/abs/physics/0110060
>=20
> There, in section "5. The art of putting c =3D 1, hbar =3D 1, G =3D 1",=
L.B.
> Okun claims that these expressions are not genuine equalities and that
> they should not be taken too literally.
Okun doesn't understand this game.
> So what is the status of the above equations?
They define a system of units.
> What do they really mean?
They have no 'real' meaning, beyond defining a unit system.
Reality (by definition) doesn't depend on the unit system
we -choose- to describe it with.
> Is Okun wrong?
Yes.
> (I understand that constants can be made to disappear from fundamental
> equations through clever changes of variables. E.g., a cleverly chosen
> factor b will transform the Schr=F6dinger equation for psi(x,t) to an
> equation in terms of the field phi(s,t) =3D psi(bs,t). That equation
> looks exactly like the simplified version of the Schr=F6dinger equation
> that could obtained if the equation m =3D e =3D hbar =3D 1 were taken
> literally. Perhaps m =3D e =3D hbar =3D 1 is just the instruction "make=
the
> changes of variables that make m, e, and hbar disappear!", rather than
> a statement that m, e, and hbar are all equal to each other and to 1?
That amounts to the same thing, in practice.
> But then it's difficult to make sense of Wald's appendix F and Burcham
> & Jobes' appendix F...)
The key to understanding this subject is that dimensions
do not have an objective existence.
They are human constructs, without relation to reality,
to be chosen in any convenient way,
subject only to the requirement of consistency.
Any consistent system of dimensions is as good as any other,
although perhaps less useful for certain applications.
In particular, one can give c value 1, and dimension [I],
or value 1, and dimension [L]/[T],
whichever may be convenient for a particular application.
A often used compromise is to have c =3D 1,
and to write it nevertheless in final results,
as in for example m_{top} =3D 175 GeV/c^2.
Best,
Jan
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