19.6 The Planck Era and Quantum Physics
293
accelerator larger than the solar system. Even the highest energy cosmic rays ever
observed have much less than Planck energy.
It is believed by many cosmologists that there was an initial Planck era before
inflation in which quantum effects were important and the universe was governed
by Planck scale phenomena, that is quantum gravity. Much work has gone into
constructing a quantum theory of gravity appropriate to the Planck scale, but with
little or no predictive success (Rovelli 2008; Frignanni 2011). For example the strings
of superstring theory are of Planck size. We cannot go into theories such as string
theory or loop quantum gravity here, but will instead content ourselves with the
more modest task of obtaining a generalized uncertainty principle and using it to
show that the Planck length arises naturally as a minimum meaningful distance
when we combine the ideas of quantum mechanics and basic ideas of gravity and
general relativity.
We will first generalize the uncertainty principle (UP) of quantum mechanics to
include gravitational effects and obtain a generalized uncertainty principle (GUP).
Our argument will be rough order of magnitude and largely based on the Heisenberg
uncertainty principle of quantum theory, which we will now recall. General principles
of optics and quantum mechanics tell us that if we measure the position of a particle,
such as an electron, with a photon of wavelength λ we cannot expect better precision
than about λ, which we express as
x H λ.
(19.22)
A photon of wavelength λ has a momentum of p = 2π /λ, and when it interacts
and scatters from the particle a significant fraction of this momentum will generally
be given to the particle p ≈ p = /λ. This makes its momentum uncertain to
roughly
p ≈ /λ.
(19.23)
Combining these last two equations we obtain
x H p , or x H //p,
(19.24)
which is the well-known Heisenberg uncertainty principle. Figure 19.6 shows a
picture of the process of measuring the particle position.
But this illustration of the uncertainty principle ignores gravity. The particle will
also interact gravitationally with the photon which produces spacetime curvature,
and this should produce an additional uncertainty in the position of the particle. If
the wavelength of the photon is small and its momentum and energy are large this
interaction can become too large to ignore. We can estimate the effect by a heuristic
dimensional argument. Let us include the gravitational effect and call the extra term
x g . This gravitational term should obviously be proportional to the gravitational
293
accelerator larger than the solar system. Even the highest energy cosmic rays ever
observed have much less than Planck energy.
It is believed by many cosmologists that there was an initial Planck era before
inflation in which quantum effects were important and the universe was governed
by Planck scale phenomena, that is quantum gravity. Much work has gone into
constructing a quantum theory of gravity appropriate to the Planck scale, but with
little or no predictive success (Rovelli 2008; Frignanni 2011). For example the strings
of superstring theory are of Planck size. We cannot go into theories such as string
theory or loop quantum gravity here, but will instead content ourselves with the
more modest task of obtaining a generalized uncertainty principle and using it to
show that the Planck length arises naturally as a minimum meaningful distance
when we combine the ideas of quantum mechanics and basic ideas of gravity and
general relativity.
We will first generalize the uncertainty principle (UP) of quantum mechanics to
include gravitational effects and obtain a generalized uncertainty principle (GUP).
Our argument will be rough order of magnitude and largely based on the Heisenberg
uncertainty principle of quantum theory, which we will now recall. General principles
of optics and quantum mechanics tell us that if we measure the position of a particle,
such as an electron, with a photon of wavelength λ we cannot expect better precision
than about λ, which we express as
x H λ.
(19.22)
A photon of wavelength λ has a momentum of p = 2π /λ, and when it interacts
and scatters from the particle a significant fraction of this momentum will generally
be given to the particle p ≈ p = /λ. This makes its momentum uncertain to
roughly
p ≈ /λ.
(19.23)
Combining these last two equations we obtain
x H p , or x H //p,
(19.24)
which is the well-known Heisenberg uncertainty principle. Figure 19.6 shows a
picture of the process of measuring the particle position.
But this illustration of the uncertainty principle ignores gravity. The particle will
also interact gravitationally with the photon which produces spacetime curvature,
and this should produce an additional uncertainty in the position of the particle. If
the wavelength of the photon is small and its momentum and energy are large this
interaction can become too large to ignore. We can estimate the effect by a heuristic
dimensional argument. Let us include the gravitational effect and call the extra term
x g . This gravitational term should obviously be proportional to the gravitational