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.(adsbygoogle = window.adsbygoogle || []).push({});

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##?

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# I Is the Planck length a proper length?

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