We study the previously conjectured form of the physical electron propagator
and its allegedly Luttinger type of behavior in the theory of the pseudogap
phase of high-temperature copper-oxide superconductors and other effective
QED-like models. We demonstrate that, among a whole family of seemingly
gauge-invariant functions, the conjectured "stringy ansatz" for the electron
propagator is the only one that is truly invariant. However, contrary to the
results of the earlier works, it appears to have a negative anomalous
dimension, which makes it a rather poor candidate to the role of the physical
electron propagator. Instead, we argue that the latter may, in fact, feature a
"super-Luttinger" behavior characterized by a faster than any power-law decay:
G(x) ~ exp(-const ln^2|x|).Comment: Latex, 10 pages, no figure