  The formalism of Quantum Mechanics is based by definition on conserving
probabilities and thus there is no room for the description of dissipative
systems in Quantum Mechanics. The treatment of time-irreversible evolution (the
arrow of time) is therefore ruled out by definition in Quantum Mechanics. In
Quantum Field Theory it is, however, possible to describe time-irreversible
evolution by resorting to the existence of infinitely many unitarily
inequivalent representations of the canonical commutation relations (ccr). In
this paper I review such a result by discussing the canonical quantization of
the damped harmonic oscillator (dho), a prototype of dissipative systems. The
irreversibility of time evolution is expressed as tunneling among the unitarily
inequivalent representations. The exact action for the dho is derived in the
path integral formalism of the quantum Brownian motion developed by Schwinger
and by Feynman and Vernon. The doubling of the phase-space degrees of freedom
for dissipative systems is related to quantum noise effects. Finally, the role
of dissipation in the quantum model of the brain and the occurrence that the
cosmological arrow of time, the thermodynamical one and the biological one
point into the same direction are shortly mentioned.
