Standing Waves: Preserving Isotropy

Equation (6.70) tells us the frequency of
oscillation of the mode , but the question still remains
whether we should use the plus or minus sign in the exponential.
The answer is that we must use both. This fact arises from a
simple property of Fourier transforms, namely that the Fourier
transform of a real field must obey the symmetry

(6.73) |

(6.74) |

(6.75) |

(6.76) |

In practice the signs you use for the exponential time dependence
of different modes has a negligible effect on the evolution once
preheating begins. Even if every mode is initialized to be
left-moving, the total momentum this imparts to the field is
unnoticeable by the late stages of the evolution in every problem
we have considered. Nonetheless it is presumably desirable to
enforce Lorentz invariance, at least in an averaged sense. You
could do this by randomly initializing each mode with either a
plus or a minus sign. Instead, we choose to set up both left and
right moving waves with equal amplitude at each value of . In
other words the initial conditions correspond to standing waves.
Thus the final form of the initial fluctuations is

(6.77) |

(6.78) |

By now it may have struck you that we seem to be determining these initial conditions based on issues of convenience, symmetry, and so on. What about whatever is the physically correct form for vacuum fluctuations, as given by their quantum mechanical probability distributions? Shouldn't those distributions provide an answer to all of these questions as to the correct form of the equations? The answer is no. Although equation (6.65) gives the correct quantum distribution for the mode amplitudes, it is not correct to use this distribution and then use equation (6.72) to set the values of the field derivatives. The problem is that quantum mechanically and are noncommuting operators and can not be simultaneously set. Although this uncertainty presents a problem in principle it is unimportant in practice. Once parametric resonance begins the occupation numbers of the modes become large and their quantum uncertainty becomes irrelevant. Moreover the rapid growth that occurs during this resonance effectively destroys all information about the initial values of the modes so that the final simulation results are insensitive to the details of how the initial conditions are set. In our experience runs that use the probability distribution of equation (6.65) give essentially the same results as ones that use the exact value of equation (6.68) for each mode, and likewise all qualitative results are unchanged by the use of left-moving waves, right-moving waves, or any combination of the two.

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This documentation was generated on 2008-01-21