Linearity of quantum mechanics and nonlinearity of macroscopic physics

Question

We live in a world where almost all macroscopic physical phenomena are non-linear, while the description of microscopic phenomena is based on quantum mechanics which is linear by definition. What are the physics points of connection between the two descriptions?

Answer 1

There is an all too common misconception that because the Schrodinger equation is linear, non-linear phenomena (like chaos) are only classical. The wavefunction does obey a linear equation, the Schrodinger equation, but it is not directly related to observable physics. Observables quantities, like expectation values of operators, obey non-linear equations. In fact, many times the same equations as their classical counterparts, with small corrections.

Answer 2

Assuming you mean "linear" in the mathematical sense of "the sum of two solutions to the relevant equation is also a solution," there's no particular reason why macroscopic objects are inherently non-linear. In fact, there is a large body of work in the quantum foundations community on ways to have macroscopic objects behave in a linear manner but look non-linear. That's the whole point of things like the Many-Worlds interpretation of quantum mechanics, and the research into decoherence by people like W. Zurek. There may be a scale above which it is impractical to see superposition states, but that doesn't mean that they can't exist.

If that's not what you mean, then I don't know how to answer you.

Answer 3

Mean field dynamics, describing the effective evolution of a particle in a system with a very large number of particles, is non-linear, even if the quantum dynamics is linear. Convergence towards mean field dynamics has been rigorously proved for quantum systems of many particles (and even quantum fields), and is nowadays a thoroughly studied subject in mathematical physics. In this sense there are solid foundations on the connection between the linear quantum dynamics and the nonlinear effective evolution of macroscopic systems.

The idea is that time-evolved reduced density matrices of the quantum system converge, in the limit $N\to\infty$, to the projector on the solution of mean field nonlinear equations (at least for some particular quantum states, e.g. coherent states, with general states the picture gets more complicated, but the nonlinear dynamics governs the evolution in the limit).

Answer 4

There is other "domain of linearity"; $\ddot{x}=-x$ is a linear equation with solutions nonlinear in time.

Answer 5

Linear in the quantum mechanics has nothing to do with its complexity. A two-state spin can be described by a simple 2-by-2 matrix; however, 30 interacting spin, in general, must be described by a 1 billion by 1 billion matrix. It grows exponentially as the number of spins increases, for $10^{23}$ spin, you may need a matrix of size $2^{10^{23}}$. It is not easy to understand and not simple in most sense. If you learn some statistical mechanics, you will know that this number is large enough to have new emergent phenomenon.