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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)?

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Your classification is very crude, and not particularly appropriate. You may look at publish.aps.org/PACS to see how professionals sort physics into fields. –  Arnold Neumaier Mar 5 '12 at 5:48
    
For example Complex systems. –  Piotr Migdal May 8 '12 at 11:17
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Perfectly valid question. Removed the ridiculous -1 by adding a +1. Biological systems are very complicated and does not seem to fit into your framework. For example studying bulk properties of membranes by analyzing ionic currents seems to take more of an engineering/mathematical approach. –  Antillar Maximus May 8 '12 at 13:28
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What is this question all about? I interpreted it as follows: is there a physics which is based not on deterministic systems, probabilistic systems, or quantum systems (aside from quantum probabilistic systems, of course). The answer for physics is not really, but one could try to dream another option up. If this isn't your question, this should be a separate question. –  Ron Maimon May 11 '12 at 18:57

7 Answers 7

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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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    wow. Actually, I have been wanting to see similar diagrams that are used to classify books (e.g. $|000\rangle$ for NR single particle classical mechanics eigenstate, even though I don't think we need phase coherence to label each topics). Also, how useful would it be to choose a phenomenological eigenbasis like length/energy scale? –  pcr May 12 '12 at 19:21
        
    @pcr: Of course, one can refine each linit by introducing an explicit scale. For some of the limits, this may be appropriate. But I didn't wnat to complicate the classification too much.... –  Arnold Neumaier May 13 '12 at 7:59

    The answer to your question depends a lot on the what you think what fitting into a framework means.

    In some aspects condensed matter physics does not fit it any of your categories. While quantum mechanics is used heavily in condensed matter theory there are proponents that it is a different field. So it is not applied QM.

    P.W. Anderson phrased it elegantly:

    The elementary entities of science X obey the laws of science Y. But this hierarchy does not imply that science X is "just applied Y". At each stage entirely new laws, concepts and generalizations are necessary, requiring inspiration and creativity just as great a degree as in the previous one.

    So taking "more is different" literally, condensed matter does not fit in neither classical or quantum mechanics nor thermodynamics.

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    I like the "more is different" paper, but I never really understood it. To me it seems as if these "entirely new laws" should be obtainable be the right renormalization from the deeper theory to the other. Phenomena are emerging, but you can see them coming from the more detailed theory. Of course, nobody can learn everything and so it's certainly a field of its own - its "not just application" only in the practical sense, because of that. –  NikolajK May 8 '12 at 21:59
        
    @NickKidman: I think that is not yet decided if you can see the emergent behaviour coming from the more underlying theory but that is an interesting question. –  Alexander May 8 '12 at 23:33

    However, it is a strong belief among many physicists that Quantum mechanics is providing the framework for everything under, and inside, and beyond the sun. The reason that some particular theories appear not to "fit" in this framework is because at this stage we can not prove them YET, either due to computation limitation or some unsolved problems of Quantum mechanics itself.

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    It should be noted that hydrodynamics cannot be derived from Newtons principles of molecular interaction. Is hydrodynamics a physical theory that doesn't fit in the others?

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    hydrodynamics is applies thermodynamics, and can be derived from statistical mechnaics. This is done, e.g., in the statistical physics book by Reichl. –  Arnold Neumaier Mar 5 '12 at 5:46
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    I think it is better thought of as classical mechanics, since V.A. Arnold teaches us it's the Euler motion on the diffeomorphism group, so it's like the rotation of an infinite dimensional top (with infinite dimensional friction). –  Ron Maimon May 8 '12 at 22:16
        
    @RonMaimon: It is of course both classical mechanics and thermodynamics. But hydrodynamics is more than the Euler equations. I don't think that one can regard the full Navier-Stokes equation in a bounded domain as motion on the diffeomorphism group. –  Arnold Neumaier May 13 '12 at 19:20
        
    @ArnoldNeumaier: You can always do it, since the integral lines are a diffemorphism. The question is whether it is a productive identification. For NS, there is friction, but the simple Euler equation is just geodesic motion on an energy metric, which makes a nice point of view geometrically. –  Ron Maimon May 13 '12 at 21:21

    In contrast to your classification I would define as framework a complete system as far as possible, which has a thorough self consistent mathematical and physical description starting with "axioms" and ending with differential equations that predict and describe experimental results.

    In my opinion frameworks in physics are hierarchical, i.e. a type of meta levels, one morphing into the others given certain magnitudes of the basic variables.

    Here goes :

    There is and underlying General Relativity level ( large curvatures) leading to a Special Relativity (flat space) level which at the limit becomes classical mechanics ( velocities much smaller than velocity of light)

    Classical mechanics is one level which leads to statistical mechanics at the many body formulation which leads to thermodynamics as continuum. .

    In parallel there is classical electromagnetic theory

    String formulation in Quantum mechanics ( GR included) leads to Quantum electrodynamics and quantum field theory which goes into quantum statistical mechanics which will also go into thermodynamics at the continuum level.

    In this type of framework classification, in self consistent levels, it is easy to see whether one is mixing up two frameworks which use different underlying mathematical formulations and physical modeling.

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    Statistical physics and kinetics. To avoid misunderstanding, thermodynamics is a purely phenomenological theory.

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    What makes a theory more phenomenological than another? –  NikolajK Mar 6 '12 at 8:22
        
    Every theory has its basic notions and concepts, on which the theory builds up. If the notions are considered foundamental, the theory is foundamental as well, whereas if the notions are of empirical nature, the theory is phenomenological. So, the answer is experimental accessibility. –  Alexey Bobrick Mar 6 '12 at 10:52
        
    @AlexBobrick: But by this definition, the phenomenological laws, which are experimentally accessible are more fundamental. On the other side, statistical mechanics is a way to thermodynamics. –  NikolajK May 8 '12 at 22:03
        
    Are statistical machanics and QFT not connected by the Wick rotation? And thermodynamics is the makroscopic limit of statistical machanics whereas kinetics is just a part of nonequilibrium statistical mechanics ? –  Dilaton May 8 '12 at 23:09
        
    @Nick: Foundamental commonly means theoretically basic, not following from other concepts/statements/definitions/... . –  Alexey Bobrick May 13 '12 at 1:14

    Here you go:

    enter image description
  here

    From Wikipedia

    A rather crude framework map; it doesn't include theories "in between", like Kaluza-Klein, and supergravity.

    And "? Quantum Gravity ?" should be reoplaced with a "Theory of Everything", maybe.

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