Is the Planck length a proper length?

[jcap]

The reciprocal of the Planck length, $\Lambda=1/l_P$, is used as a high-frequency cutoff in the particle-physics estimation of the vacuum energy.

For example in "the cosmological constant problem" Steven Weinberg says that summing the zero-point energies of all normal modes of some field of mass $m$ up to a wave number cutoff $\Lambda >> m$ yields a vacuum energy density (with $\hbar=c=1$)
$$<\rho>=\int_0^\Lambda \frac{4 \pi k^2 dk}{(2\pi)^3}\frac{1}{2}\sqrt{k^2+m^2}\simeq \frac{\Lambda^4}{16\pi^2}.$$
He goes on to say that "if we believe general relativity up to the Planck scale" then
$$<\rho> \approx \frac{1}{16\pi^2}\frac{1}{l_P^4}.$$
It seems that in this calculation the Planck length $l_P$ is taken to be the size of the smallest interval of space that can be described by general relativity.

But the FRW metric implies that the length of any interval of space expands with the scale factor $a(t)$.

Therefore should the Planck length actually be a proper length so that
$$l_P=a(t)\ l_{P0}$$
where $l_{P0}$ is a constant representing the Planck length at the present time $t_0$?

 

[PeterDonis]

It seems that in this calculation the Planck length $l_P$ is taken to be the size of the smallest interval of space that can be described by general relativity.

Yes, but this in itself assumes that GR stops being valid at length scales smaller than that.

But the FRW metric implies that the length of any interval of space expands with the scale factor $a(t)$.

Yes, but that in itself assumes that GR does not stop being valid at any length scale, no matter how small.

should the Planck length actually be a proper length

No. You just need to not mix apples and oranges. Either GR is valid at all length scales, no matter how small, or it isn't. If it isn't, then the FRW metric is only an approximation and you can't apply it on length scales as small as the Planck length.

 

[jcap]

To me it seems reasonable to assume that QFT and GR are correct for any spatial interval whose size is greater than or equal to the Planck length. GR, through the FRW expanding metric, seems to imply that the smallest size of that interval, i.e. the Planck length, must expand with the scale factor $a(t)$.

 

[vanhees71]

The Planck Length is just a scale, and nobody knows what's going on when you start to resolve distances at this scale. It's natural to expect that at this scale some quantum effect of the gravitational field should become relevant, and since in GR the gravitational field is closely connected with the space-time manifold itself, one expects quantum effects of spacetime itself. Since there is no consistent quantum theory of gravitation yet, nobody can tell you with certainty what to expect, let alone conduct experiments to resolve the scale since you need particle energies also of the typical Planck scale (in this case Planck mass).

 

[PeterDonis]

GR, through the FRW expanding metric, seems to imply that the smallest size of that interval, i.e. the Planck length, must expand with the scale factor $a(t)$.

It might seem that way to you, but it's not correct. Again, if GR is not valid on length scales as small as the Planck length, then you can't use it to say anything about how such a length interval behaves.

 

[vanhees71]

It might seem that way to you, but it's not correct. Again, if GR is not valid on length scales as small as the Planck length, then you can't use it to say anything about how such a length interval behaves.

How do you know that? As far as I know there's no complete theory of quantum gravitation. So how can you say that GR doesn't work on Planck scales?

 

[PeterDonis]

how can you say that GR doesn't work on Planck scales?

I didn't say it doesn't, categorically. I said if it doesn't, then we can't use it to say anything about length intervals on those scales.