I've noticed an amazing increase of views since my last post on "effective theories".
That's cool, seems there's someone out there appreciating the topic (actually I suppose this mighty viewer came here by chance and still regrets the choice... OK, I don't give a shit 'bout y'all anyway).
What I want to discuss is the "difference between classical and quantum theories" and what I mean by that.
First of all Special 'n' General Relativity are classical. Got it?
Since we are doing science and not oiled-walls-climbing theology, what we are interested in is "measurement".
How do we make correspond the measuring process (in real life) to a (mathematical) action on our (mathematical) model (~theory)?
Remember that in real life we measure real scalars, even when you are measuring a "vector quantity" (what's a vector, anyway?) we do measure component after component. Real numbers (with precision related to there measuring process resolution... ok now don't bring uncertainty relation and stuff. Keep it real, bro), real numbers i was saying: never heard of someone measuring a quaternion.
And here it is:
-Classical: measurement is a function f: V-> R that takes an element (which represents the state of the system under inspection) of the configuration space* V and gives us a real number of R.
-Quantum: measurement is the eigenvalue o of an hermitian** operator O applied on a vector (which represents the state of the system under inspection) of an Hilbert space. Note that eigenvalues of hermitian operators can be only real (even if the scalar field of an Hilbert vector space is the complex field) as a measurement must be.
Better formulations of the theories (or just more suitable for applications) measure average values, transition probabilities and so on. Fuck it, I only gave you a conceptual insight that took me several years (no kidding, bro) to appreciate.
You are welcome.
There are all those subleties about quantum effects and fluctuation, yeah I know, but they invented Path Integral and partition functions for some reason (and discovered the connections between the two quantities, which leads to connections between quantum propagators and statistical correlation functions and, man, this is a blog lemme keep it simple!).
Probably I will enrich, modify or delete this stuff.
Or probably I will write something about classical symmetries (invariance of the action/Lagrangian) opposed to symmetries in quantum Mechanics (Linear Unitary or AntiLinear AntiUnitary operators, Wigner Theorem) and maybe I will write a nice intro on Poincaré group representations and then BANG! Quantum Field Theory!
See ya.
* the space of all q's and p's + time parameter t, usually given n parti les it's a 6n+1 manifold blablabla, in case of continuum degrees of freedom things are not that simple, but you know...
**actually there are technicalities for the boundness blablabla and it should be essentially self-adjoint or something like that