LOG#054. Barrow units.

In a nice paper titled Dimensionality, see here http://rsta.royalsocietypublishing.org/content/310/1512/337.short , the physicist John Barrow studied the rôle of known fundamental constants and their links to the dimensionality of space-time.

I will review the main ideas appearing in that paper, and some interesting conclusions we can obtain from it.

Firstly, Barrow points out the connection between some equations and dimensionality. For instance, the Poisson-Laplace equation in any dimension D>2 provides a potential function

    \[\phi(r)\propto r^{2-D}\]

and a force

    \[ F(r)\propto r^{1-D}\]

whenever spherical symmetry holds \forall D>2. A short-range modification of such a potential (force) is given by screening potentials, also called Yukawa potentials. In the simplest case, D=3, we get:

    \[ \phi (r)\propto \dfrac{e^{-\lambda r}}{r}\]

The atomic stability can be read from a really simple condition for the energy with a Coulomb potential:

    \[ E(r)=\dfrac{h^2}{2mr^2}-\dfrac{e^2}{r}\]

Imposing E=0, we obtain

    \[ r_0=\dfrac{h^2}{2me^2}\]

Indeed, Ehrenfest proved that for atoms in D>5 dimensions, those conditions generalize into:

    \[ r_L(D)\approx \left(me^2L^{-2}h^{-2}\right)^{\frac{1}{D-4}}\]

    \[ E\approx \dfrac{h^2}{2mr^2}-\dfrac{e^2}{r^{D-2}}\]

    \[ \forall D>5\]

This fact means that only 3d space (4D spacetime) appears to have the beautiful and nice properties necessary for transmissions of high fidelity signal due to the simultaneous realization of reverberationless and distortionless propagation of information.

Barrow realized that we can build a dimensionless basic and fundamental system of units, that we can call Barrow units system (or Barrow system). The dimensionless quantity of this system is made of 4 constants:

1st. The electric charge e.

Its physical units are \left[ e^2\right]=ML^DT^{-2} or \left[ e\right]=M^{1/2}L^{D/2}T^{-1}

2nd. The Planck constant h. Barrow used the pure Planck constant but we will use instead the rationalized Planck constant \hbar.

Its physical units are \left[ \hbar\right]=ML^2T^{-1}

3rd. The Newton gravitational constant G_N.

Its physical units are \left[ G_N\right]=M^{-1}L^DT^{-2}

4th. The speed of light c.

Its physical units are \left[ c\right]=LT^{-1}

From these 4 fundamental constants, the Barrow system formed by (e,\hbar, G_N,c) allows us to create a dimensionless constant of Nature \forall D. It is given by the combination:

    \[ \alpha=e^{D-1}\hbar^{2-D}G_N^{\frac{3-D}{2}}c^{D-4}\]


1) D=1,2,3,4 are special dimensions since, in those dimensions, we observe the absence of

i) The electric charge/electromagnetic coupling constant in D=1. e^{D-1}

ii) Planck’s constant/The quantum theory fundamental constant in D=2. \hbar^{2-D}

iii) Gravitational Newton constant/gravitational Newton coupling constant in D=3. G_N^{3-D}

iv) The speed of light/The fundamental constant of special relativity in D=4. c^{D-4}

2) Dimensionless units for all D>4 probably will be redefined if some new fundamental constant arises. It could be due to the structure of GUT (Gran Unified Theory) or Quantum Gravity (QG).

3) Fractal dimensions with non-integer values are not allowed in this classical setting. We note that strange attractors provide objects with non-integer dimensions d<N.

4) A fundamental length L can emerge from quantum gravity (QG). In the most general case, it would imply

L\approx g_\star^{-1/2}L_p

where L_p^2=G\hbar/c^3\sim 10^{-70}m^2 and g_\star is some gauge coupling. If the gauge coupling is “weak”, it could be about 10^{-1}, 10^{-2} or smaller, and thus, the fundamental length could be larger than Planck length. This result can be obtained from basic electromagnetic, gravitational, relativistic and quantum theories. We begin from

    \[ G\dfrac{M^2}{R^2}=\dfrac{e^2}{R^2}\]

From special relativity E=Mc^2 and from quantum theory ER=\hbar c or equivalently

    \[ R=\dfrac{\hbar c}{E}\]


    \[ G\dfrac{M^2}{R^2\hbar c}=\dfrac{e^2}{\hbar cR^2}\]

We introduce \alpha or the electromagnetic fine structure constant \alpha=e^2/\hbar c. Then,

    \[ G\dfrac{M^2}{R^2\hbar c}=\dfrac{\alpha}{R^2}\]

    \[ G\dfrac{M^2 c^4}{R^2\hbar c}=\dfrac{\alpha c^4}{R^2}\]

    \[ G\dfrac{E^2}{R^2\hbar c}=\dfrac{\alpha c^4}{R^2}\]

    \[ \dfrac{GE^2R^2}{\hbar c}=\alpha c^4 R^2\]

    \[ G\hbar c=\alpha c^4 R^2\]

    \[ R^2=\dfrac{1}{\alpha}\dfrac{G\hbar c}{c^4}\]

    \[ R^2=\dfrac{1}{\alpha}\dfrac{G\hbar }{c^3}\]

    \[ R^2=\dfrac{1}{\alpha}L_p^2\]

    \[ R=L=\sqrt{\dfrac{1}{\alpha}}L_p\]


5) Fractal dimension theory will not be considered in this post, but it deserves further study.

6) We will check that the dimensionless fundamental constant for the Barrow system is \alpha=e^{D-1}\hbar^{2-D}G_N^{\frac{3-D}{2}}c^{D-4}. Using the physical dimensions of the 4 constants above, we get:

    \[ M^{\frac{D-1}{2}}L^{\frac{D(D-1)}{2}}T^{-(D-1)}M^{2-D}L^{2(2-D)}T^{D-2}L^{D-4}T^{-(D-4)}M^{\frac{D-3}{2}}L^{\frac{(3-D)D}{2}}T^{D-3}\]

Thus, for M, L and T, we deduce:

    \[ M: 2-D+\dfrac{D-1}{2}+\dfrac{D-3}{2}=0\]

    \[ L: 4-2D+\dfrac{D^2}{2}-\dfrac{D}{2}+\dfrac{3D}{2}-\dfrac{D^2}{2}+D-4=0\]

    \[ T: 1-D+D-2+4-D+D-3=0\]


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LOG#054. Barrow units. — 3 Comments

  1. It is not implied, but not obvious, in your discussion that alpha is also a magnitude of SR, of relativistic mechanics. If I recall correctly, at least in 3+1 dimensions it is the maximum angular momentum, measured as a multiple of Planck constant, for a particle in a circular orbit, around a charge 1 particle. Probably the requirement of circular orbit can be relaxed to any rotating ellipse, as described in the articles of Sommerfeld.

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