Are there physical theories in use, which don't fit into the frameworks of either Thermodynamics, Classical Mechanics (including General Relativity and the notion of classical fields) or Quantum Mechanics (including Quantum Field Theory and friends)?
Answer
The proposed partition of physics into Thermodynamics, Classical Mechanics, and Quantum Mechanics is quite arbitrary. To take just one conspicuous example, statistical mechanics does not fit, as it is the discipline that mediates between these three areas of physics.
The Physics and Astronomy Classification Scheme (PACS) http://www.aip.org/pacs/pacs2010/individuals/pacs2010_regular_edition/index.html , ''an internationally adopted, hierarchical subject classification scheme, designed by the American Institute of Physics (AIP)'', partitions physics instead into
- The physics of elementary particles and fields
- nuclear physics
- atomic and molecular physics
- electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics
- physics of gases, plasmas, and electric discharges
- condensed matter: structural, mechanical and thermal properties
- condensed matter: electronic structure, electrical, magnetical, and optical properties
- interdisciplinary physics and related areas of science and technology
- geophysics, astronomy, and astrophysics.
It would be quite meaningless to put each of these general containers under the hood of either Thermodynamics, Classical Mechanics, or Quantum Mechanics. In many cases, there is an interplay between thermodynamical, classical, and/or quantum aspects that bear on a given physical problem.
But let me respond to the challenge by proposing a systematic view of physics not by its phenomena but by classifying it in terms of 7 orthogonal criteria.
The first criterion is methodological, and distinguishes between
- applied physics (AP), didactical physics (DP), experimental physics (EP), theoretical physics (TP), and mathematical physics (MP).
The other six criteria are defined in terms of the six limits that play an important role in physics:
- the classical limit ($\hbar\to 0$) distinguishes between classical physics (Cl), in which $\hbar$ is negligible, and quantum physics (Qu) where it is not.
- the nonrelativistic limit ($c\to \infty$) distinguishes between nonrelativistic physics (Nr), in which $c^{-1}$ is negligible, and relativistic physics (Re) where it is not.
- the thermodynamic limit ($N\to\infty$) distinguishes between macroscopic physics (Ma), in which microscopic details are negligible, and microscopic physics (Mi) where they are not.
- the eternal limit ($t\to\infty$) distinguishes between stationary physics (St), in which time is negligible, and nonequilibrium physics (Ne) where it is not.
- the cold limit ($T\to 0$) distinguishes between conservative physics (Co), in which entropy is negligible, and thermal physics (Th) where it is not.
- the flat limit ($G\to 0$) distinguishes between physics in flat space-time (Fl), in which curvature is negligible, and general relativistic physics (Gr) where it is not.
A particular subfield is characterized by a signature consisting of choices of labels (or double arrows between labels) in some categories.
A few examples:
- Thermodynamics: Ma ,Th
- Equilibrium thermodynamics: Ma, Th, St
- Classical Mechanics: Cl, Co
- Classical field theory: Cl, Co, Ma
- General relativity: Cl, Re, Ma, Gr
- Quantum mechanics: Qu, Nr
- Relativistic quantum field theory: TP, Qu, Re, Mi
- Statistical mechanics: TP, Mi$<->$Ma, Th
- Precision tests of the standard model: TP$<->$EP, Qu, Re, Mi, St, Co
- The empty signature is simply the field of physics itself.
In each category, one can choose no label, a single label, or an arrow between two labels, giving $1+5+5*4/2=16$ cases for the first category, and $1+2+1=4$ cases in the six other categories. Thus the classification splits physics hierarchically into $16*4^6=65536$ potential subfields with different signatures, of which of course only the most important ones carry conventional names.
Let me give what I think is a particularly useful subhierarchy of the complete hierarchy. This subhierarchy splits the whole physics recursively into quadrangles of subfields.
On the highest first level, we split physics according to the cold limit and the flat limit. This gives a quadrangle of first level theories of
- thermal physics in curved spacetime (Th Cu)
- thermal physics in flat spacetime (Th Fl)
- conservative physics in curved spacetime (Co Cu)
- conservative physics in flat spacetime (Co Fl) together with two first level interface theories
- statistical physics (Th<->Co)
- geometrization of physics (Cu<->Fl)
These first level theories describe very general principles on the theoretically most fundamental level of physics.
On the second level, we split each first level theory according to the eternal limit and the thermodynamic limit. This gives in each case a quadrangle of theories of
- nonequilibrium particle physics (Ne Mi)
- nonequilibrium thermodynamics (Ne Ma)
- physics of bound states and scattering (St Mi)
- equilibrium thermodynamics (St Ma) together with two second level interface theories
- long time asymptotics (Ne<->St)
- thermodynamic limits (Ma<->Mi)
These second level theories describe physics on a level already close to many applications, especially outside physics, though still lacking detail.
On the third, lowest level, we split each second level theory according to the nonrelativistic limit and the classical limit. This gives in each case a quadrangle of theories of
- relativistic quantum physics (Re Qu)
- relativistic classical physics (Re Cl)
- nonrelativistic quantum physics (Nr Qu)
- nonrelativistic classical physics (Nr Cl) together with two third level interface theories
- nonrelativistic limit (Re<->Nr)
- quantization and classical limit; quantum-classical systems (Qu<->Cl)
These third level theories describe physics on the usual textbook and research level.
(Maybe someone who likes to do graphics can illustrate this hierarchy with appropriate diagrams.)
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