Based on a commonly used model for the time evolution of extensive variables and on the basis of the analogies between the nonlinear stochastic theory and imaginary time quantum mechanics in curved space, we set up a canonical operator formulation of non-equilibrium thermodynamics which emphasizes the fundamental role played by the Boltzmann constant k in fluctuations. The k to 0 of this theory yields the classical description of non-equilibrium thermodynamics. The new formulation is technically useful and provides new insights which have important consequences, namely the thermodynamic uncertainty principle and the quantization of the entropy production rate for stationary states.
The spread of the wave-function, or quantum uncertainty, is a key notion in quantum mechanics. At leading order, it is characterized by the quadratic moments of the position and momentum operators. These evolve and fluctuate independently from the position and momentum expectation values. They are extra degrees of quantum mechanics compared to classical mechanics, and encode the shape of wave-packets. Following the logic that quantum mechanics must be lifted to quantum field theory, we discuss the quantization of the quantum uncertainty as an operator acting on wave-functions over field space and derive its discrete spectrum, inherited from the sl2 Lie algebra formed by the operators x^2, p^2 and xp. We further show how this spectrum appears in the value of the coupling of the effective conformal potential driving the evolution of extended Gaussian wave-packets according to Schrödinger equation, with the quantum uncertainty playing the same role as an effective intrinsic angular momentum. We conclude with an open question: is it possible to see experimental signatures of the quantization of the quantum uncertainty in non-relativistic physics, which would signal the departure from quantum mechanics to a QFT regime?