**2.4. The vacuum in quantum field theory**

In this section, we will discuss something that at first glance appears to be entirely unrelated to cosmology: the vacuum in quantum field theory. We will see later, however, that it will in fact be crucially important to cosmology. Let us start with basic quantum mechanics, in the form of the simple harmonic oscillator, with Hamiltonian

(39) |

where and
^{†} are the
lowering and raising operators, respectively, with commutation relation

(40) |

(41) |

The simple harmonic oscillator is pretty much the only problem that physicists know how to solve. Applying the old rule that if all you have is a hammer, everything looks like a nail, we construct a description of quantum fields by placing an infinite number of harmonic oscillators at every point,

(42) |

where the operators
_{k} and
_{k}^{†} have the commutation
relation

(43) |

Here we make the identification that **k** is the momentum of a
particle, and _{k}
is the energy of the particle,

(44) |

for a particle of mass *m*. Taking *m* = 0 gives the familiar
dispersion relation for massless
particles like photons. Like the state kets
|*n*> for the harmonic
oscillator, each momentum vector **k** has an independent ladder of
states, with the associated quantum numbers,
|*n*_{k} ,..., *n*_{k'} >. The
raising and lowering operators are now interpreted as *creation*
and *annihilation* operators, turning a ket with *n*
particles into a ket with *n* + 1 particles, and vice-versa:

(45) |

and we call the ground state |0> the *vacuum*, or zero-particle
state. But there is one small problem: just like the ground state of a
single harmonic oscillator has a nonzero energy
*E*_{0} = (1/2)
, the vacuum state of
the quantum field also has an energy,

(46) |

The ground state energy diverges! The solution to this apparent paradox
is that we expect
quantum field theory to break down at very high energy. We therefore
introduce a cutoff on
the momentum **k** at high energy, so that the integral in Eq. (46)
becomes finite. A reasonable physical scale for the cutoff is the scale
at which we expect
quantum gravitational effects to become relevant, the *Planck
scale* *m*_{Pl}:

(47) |

Therefore we expect the vacuum everywhere to have a constant energy
density, given in units where
= *c* = 1 as

(48) |

But we have already met up with an energy density that is constant
everywhere in space: Einstein's cosmological constant,
_{} =
const.,
*p*_{} = - _{}.
However, the cosmological constant we expect from quantum field theory
is more than a hundred twenty orders of magnitude too big. In order for
_{} to
be less than the critical density,
_{}
< 1, we must have
_{}
< 10^{-120}*m*_{Pl}^{4}! How can we
explain this discrepancy? Nobody knows.