# How to understand After quantum mechanics, nature itself suddenly became linear?

How to understand Freeman Dyson's Saying:

After quantum mechanics, nature itself suddenly became linear. • See Freeman Dyson, Birds and Frogs. – Mauro ALLEGRANZA Oct 11 '18 at 10:46
• You tagged the post with linear algebra so you already know about the Hilbert space and matrix formulation (historically as per Heisenberg, as opposed to wave mechanics) and all that. What is your real question? – Lee David Chung Lin Oct 11 '18 at 14:55
• why nature itself became linea? – aircraft Oct 11 '18 at 14:57
• Cum grano salis – sand1 Oct 11 '18 at 16:02
• Equations of classical mechanics are non-linear ODE, equations of quantum mechanics are linear PDE, so metaphorically the non-linear classical nature "became" linear at a more fundamental level. – Conifold Oct 11 '18 at 21:54

## 1 Answer

Until quantum mechanics, linear differential equations arising in physics were the result of approximation (linearization) of non-linear ones. So it was believed that the fundamental laws of nature are described by non-linear differential equations, while linear ones are only used to approximate them.

With the invention of quantum mechanics we know that the most fundamental equation of physics (Schrodinger equation) is a linear differential equation. And that the space of states of a physical system is a vector space (in classical mechanics it is a manifold, again a non-linear object). As QM is the most fundamental theory that we currently possess about the real world, the statement of Dyson is well justified.

QM is indeed the most fundamental theory: most physical theories are thought to be approximate. This is the only one that seems to be exact, at the present state of our knowledge.

• General relativity also seems to be exact. It is true that Quantum Mechanics and General Relativity contradict each other, but until we can study black holes or the like better both theories are holding up. – Rory Daulton Oct 12 '18 at 0:03
• @Rory Daulton: you yourself give the reason why GR must be considered only an approximation: it is not compatible with quantum mechanics. Quantum mechanics has more fundamental status and is much better verified. – Alexandre Eremenko Oct 12 '18 at 0:08
• QM is not the "most fundamental" theory that we currently possess, QFT is, including the Standard Model, nor does QM seem to be exact, it is non-relativistic. And even QFT does not subsume general relativity. So Schrodinger equation is definitiely not the "most fundamental" equation of physics. – Conifold Oct 12 '18 at 0:08
• @Conifold: When I say QM I include the relativistic one and QED. Speaking of all other theories you mention, they use QM, they just build over it. And I maintain that this building block is the most well-verified, and most fundamental one. – Alexandre Eremenko Oct 12 '18 at 0:15
• There is no Schrodinger equation in QED, and existence of soft photons introduces explicit corrections to QM predictions. So this building block is an already revised approximation. General relativity does not build over QM either, the two are incompatible, which is why people are looking for quantum gravity. – Conifold Oct 12 '18 at 0:20