# Are certain fields of physics axiomatized?

Everything from Quantum mechanics can be derived from six (?) postulates. Similarly classical electrodynamics can be reduced to Maxwell's equations and Lorentz force law, and special relativity is based on two postulates.

Are there similar sets of postulates for Newtonian mechanics/gravity or other fields of physics, like quantum field theory or string theory?

• Can you offer a source for the "six postulates" statement? – rob Feb 26 '18 at 15:51
• Both ergodicity studies in statistical mechanics and constructive quantum, field theory have been invested with sufficient mathematical rigor and axiomatic strictures to have safely transcended physics and, by now, irreversibly reside in mathematics. – Cosmas Zachos Feb 26 '18 at 16:36
• Newton's laws and energy conservation might be the key fundamentals here – Steeven Feb 26 '18 at 16:42
• Related: physics.stackexchange.com/q/87239 , physics.stackexchange.com/q/94560/2451 , physics.stackexchange.com/q/14939/2451 and links therein. – Qmechanic Feb 26 '18 at 18:02
• I find the "six postulates" statement very questionable and have never seen a proof that these (whatever number of) postulates stated in most QM textbooks actually are sufficient to cover all seen phenomena. Take for a concrete example arrival times. There is no time operator, so what do? There is a whole book on different approaches, none are accepted (Time in Quantum Mechanics by Muga). But even to cover the phenomena which have been explained by QM, it would have to be checked that all these phenomena actually follow from only the postulates without any additional "intuitive" arguments. – doublefelix Feb 14 '20 at 13:45

Many fields of physics are axiomatized, either completely or to some extent.

First of all, for something that will be important for almost all fields, both Lagrangian mechanics and Hamiltonian mechanics are rooted in formal math, via calculus on the jet bundle for the Lagragian, and the Legendre bundle for the Hamiltonian (or for something less complex, Gâteaux derivatives on functionals for the Lagrangian and the Legendre transform for the Hamiltonian). You can check this for instance, as well as Henneaux for all constraint-related matters on the topic.

Special relativity has quite a variety of axiom system, either based on the fairly straightforward theory of Lorentz affine spaces, such as described in Gourgoulhon, or through awful first-order axiom systems such as $\text{Basax}$, $\text{Reich}$ or variations. You can learn more about such axiom systems here for instance. It's also possible to axiomatize it via its causal structure, as done by Zeeman, Carter, Penrose and Kronheimer.

General relativity is also based on axiomatic rules. Basically a spacetime is a tuple $(\mathcal M, \mathcal A, g, \nabla)$, with $\mathcal M$ an $n$-dimensional ($n \geq 2$), Hausdorff, paracompact manifold, $\mathcal A$ a smooth structure on that manifold, $g$ a section of the metric bundle of signature $(-++...)$, and $\nabla$ the Levi-Civitta connection. It's also often assumed to be $(\mathcal M, \mathcal A, g, \nabla, \uparrow, \varepsilon)$, with $\uparrow$ a time-orientation and $\varepsilon$ a measure form. Then you can define the matter content and dynamics on it via sections of vector bundles and the Lagrangian formalism.

Quantum mechanics is usually defined via the Dirac-von Neumann axioms, as a theory of operators acting on a Hilbert space (a good review is in Hall, with a nice overview of the probabilistic shenanigans reviewed in Streater), or via path integrals using the Wick-rotated Wiener functional integral on the configuration space of the system. It is also possible to axiomatize it on less popular mathematical (equivalent) formalisms such as fractional quantum mechanics (where particles are described by stochastic processes), or deformation quantization.

Quantum field theory is harder to axiomatize, but there are a variety of attempts, more or less successful.

1. The Osterwalder–Schrader axiomatization is the equivalent to the QM path integral axiomatization for field theories.
2. The Wightman axiomatization is the equivalent of the Dirac-von Neumann axiomatization.
3. The Haag-Kastler axiomatization is a presheaf between open sets of the spacetime and $C^*$ algebras.

All of these are described to some degree in Glimm and Jaffe, as well as Wightman and Streater and Haag. There's a handful of other axiomatizations, such as functorial quantum field theory.

Most of those work fine only really for the free case. There are some attempts at extending those systems to the interacting case as well involving a lot of really awful microlocal analysis and Wick polynomials.

Classical mechanics isn't terribly hard to axiomatize. The kinematic part is usually simply axiomatized by Newtonian space (a vector space $\Bbb R \times \Bbb R^3$ with an inner product on $\Bbb R^3$ and so forth), although you can model it as a manifold using the Newton-Cartan theory. The dynamic can then be done a variety of ways, either using the Newton equation directly, or via Lagrangian mechanics (a bundle approach is sometimes used for this as well), or Hamiltonian. You might want to check Arnold for more fun details on the topic. Nothing too complicated although regularity conditions are important to specify to avoid weird edge cases such as Norton's dome or the space invader. I also can't fail to mention the really stupid geometric axiomatization, which is absolutely not fit for any calculations but has the merit to exist.

EM, and by extension gauge theory in general, is done by the formalism of principal connections. You can find out more informations about it for instance in Topology, geometry and gauge fields.

Thermodynamics can be axiomatized using the headache-inducing contact manifolds, in a hellish formalism called geometrothermodynamics.

Those are about all the fields that have a really formal axiomatization that I can think of, but there are probably others.

• Can you provide a source that any of these supposed existing axiomizations actually logically imply all of the phenomena which physicists derive from that respective theory? This is a non-trivial question; writing down postulates of a theory from which physicists base their reasoning is very different from actually having axioms that can be used to prove all major results of the theory, without intuitive reasoning to "fill the gaps". In my studies I have seen plenty of intuitive physical arguments that don't clearly follow from the postulates of that theory. – doublefelix Feb 14 '20 at 13:51
• I mean if you want the proof of experimental data for all of physics, it's gonna take a little while – Slereah Feb 14 '20 at 13:52
• Even just the major results - standard in textbooks - do not clearly follow from only the postulates of each theory. Until this happens, the theory has not been axiomized, but rather only some subset of the theory which doesn't require intuition to derive, has been axiomized. But I believe OP is not interested in that subset. One would not consider calculus to be axiomized just because the chain and product rule, and maybe a handful of theorems, follow from the definitions. – doublefelix Feb 14 '20 at 13:57
• For a properly axiomatized theory, the experimental results follow from those axioms (what is called the formalism), and the rules of correspondance, which map the real life measurements to abstract quantities of the theory. The rules of correspondance are usually a bit messier (real devices don't usually map perfectly to the basic quantities of a theory), and short of omniscience it's hard to make proper predictions without some other assumptions, but they do exist, yes. You will just require to be a bit more specific on which theory you would like to know more about. – Slereah Feb 14 '20 at 14:02
• Classical mechanics is one of the most interesting cases. Starting from some set of postulates, one should be able to prove the phenomenological laws, for example Hooke's law, friction. But where does one begin? Is the fundamental constituent a point particle, or an object occupying some volume with properties like density and temperature? Macroscopic theories are especially difficult because they require separate characterization of so many different phenomena. What, exactly, is a spring, and how do you know these things follow Hooke's law? – doublefelix Feb 14 '20 at 14:27