arXiv:hep-th/9304011 v1 Apr 5 1993

By upper triangular transformations of the basis of operators, analogous to the change
of basis in sec. 10.3 relating σ j to ˆσ j , we obtain the full operator expansion of the macroscopic
loop [91]:
W in (l, p) = Ṽp Γ(|p| + 1) µ −|p|/2 I |p| (2 √ µl) −
∞∑
r=1
ˆB r,p
2(−1) r r
r 2 − p 2 µ−r/2 I r (2 √ µl) . (11.65)
12. Fermi Sea Dynamics and Collective Field Theory
12.1. Time dependent Fermi Sea
Another source of intuition, very different from the macroscopic loop approach, comes
from the motions of the Fermi sea of the upside-down oscillator and the associated collective
field theory, also known as the Das–Jevicki–Sakita theory [76,77,93]. As we saw in our
analysis of the tree-level free energy (sec. 11.2), one is naturally lead to think about the
fermionic phase space. This point of view leads to a very beautiful description of tree-level
c = 1 dynamics [95].
In sec. 11.2, we studied the ground state from the point of view of a fluid in phase
space. When describing dynamics, we have to perturb the system so we are now looking at
time-dependent Fermi seas resulting from the disturbances produced by various operators.
The possible dynamical solutions of the system can be described in the following way [95].
Consider the generating functional for correlators in the theory,
∫
Z[J] =
dψ e − ∫ ψ † (id/dt + d 2 /dλ 2 + λ 2 )ψ + Jψ † ψ , (12.1)
where we imagine J has been turned on and off during a finite time interval. (In this section,
the c = 1 coordinate t will always be taken to be a Minkowskian time coordinate.). The
source J acts as an external force on the fermions. After it has been turned off, the state
evolves as some time-dependent solution of the system. It is clear that the points simply
move along trajectories in phase space appropriate to the upside-down oscillator, that is,
they move along lines of constant p 2 − λ 2 .
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