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Definition of Separable

  1. Apr 4, 2009 #1

    Q_Goest

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    What is your definition of separable? For a physical system to be separable, what rules should be applied to define whether a given system is separable or not? Note that if a system is non-separable, we can claim it is ‘holistic’ and therefore capable of producing some unique (emergent) phenomenon which can not be produced by a system which is separable.

    As a starter, I’d suggest that if a system can be broken up into individual volumes of space; and if one can then apply boundary conditions on each of the individual volumes of space, thereby duplicating everything which occurs within that volume, then those volumes are separable and any system composed of those volumes is separable. Using this definition, classical physics is generally considered separable while quantum mechanical systems are often considered non separable.

    Note that in the above definition of separable, there are 2 key issues.
    1. Boundary conditions on a given volume of space can in principal be determined and duplicated. Note this isn’t generally possible for molecules, although there are often ‘classical’ ways of modeling molecules (ex: bond energy) which allows some knowledge to be appied to specific bits of the molecule which is independent of the molecule in general, and in that sense, bond energy is ‘separable’. Other molecular properties are not separable however, so generally, molecules can not be broken up and put into boxes which have boundary conditions at the surface.
    2. If 1 above is met (ie: if boundary conditions can be identified on individual volumes of space), then for a nonseparable system - those duplicate boundary conditions must produce phenomena within said volume which are different than the phenomena produced inside the volumes of a separable system.

    Over the past few decades, there’s been considerable debate over whether or not chaotic systems are separable, especially nonlinear systems. Take for example, the three body problem which can be described by the Hamiltonian energy function which is a summation of kinetic and potential energy. It seems clear (to me) that what occurs within any given volume of space in any system of 3 or more gravitationally attracting bodies, can be duplicated by duplicating boundary conditions. By that, I mean that the gravitational field within any given volume of space is simply the summation of numerous individual bodies, and those individual bodies can in principal, be replaced by a different set of gravitationally attracting bodies that produce a gravitational field inside the volume of space in question which exactly duplicates the original, thereby producing the same motions within that volume of space. If this were true, then such a system by my definition above would be non separable.*

    Alwyn Scott was one of the most vocal supporters that “nonlinear dynamics at each level of description generate emergent structures, and nonlinear interactions among these structures provide a basis for the dynamics at the next higher level. (Scott, 2003)” Here, his mention of levels reflects the levels as given by Phil Anderson (Anderson, 1972); for example, from lower level down to higher level:
    solid state or many-body physics (referring to elementary particle physics)
    chemistry
    molecular biology
    cell biology
    .
    .
    .
    psychology
    social sciences
    We’ll get back to these levels in a minute…

    Scott points out that nonlinear systems are “those for which the whole is greater than the sum of its parts.” He states (Scott, 2004):
    One might ask, what kind of systems are nonlinear as Scott suggests? He gives numerous examples, of which I’ll choose a few key ones to emphasize the “levels” he wants to apply this philosophy to: chemical molecules, nerve impulses, lynch mobs, [fluid] turbulence, tornadoes, the Gulf Stream, Jupiter’s Great Red Spot, black holes. The list of nonlinear phenomena is enormous.

    Next, one might wonder how he defines these emergent phenomena and how nonlinear dynamics plays a part. To understand this, one has to understand how Kim (Kim, 2000), breaks down nature. Kim is a reductionist who provides an argument in favor of bottom up causation. In short, if there is a physical basis for a phenomena P1, and that physical basis gives rise to a higher level physical (or mental) state M1, and if there is a physical basis for a phenomena P2, and that physical basis gives rise to a higher level physical (or mental) state M2, then if P2 follows causally from P1, and if physical (or mental) state M2 follows causally from M1, then it is in error to suggest that M1 caused M2 since there is a lower level physical cause, P1, which is the cause of P2. We can not claim two physical causes for P2. I apologize for the ten second snapshot of the work by Kim. Hopefully people here are already familiar with his work.

    Understanding causation in this way, Scott quotes Emmeche (Emmeche et al., 2000) and breaks up “emergence” into three categories depending on the type of downward causation which the emergent structure should entail.
    - Strong downward causation (SDC) in which upper level phenomena can act as efficient causal agents in the dynamics of lower levels. In other words, a higher level configuration (such as M1 or M2) causes the physical change at a lower level (such as from P1 to P2). Here, Scott agrees with Kim (as should everyone) pointing out that, “Presently, there is no empirical evidence for the downward action of efficient causation, so SDC is almost universally rejected by biologists.” (Scott is a biologist.)
    - Weak downward causation (WDC) in which there are ‘attractors’ (physical states that are in some way more stable and more likely to be trespassed through). Certainly, there is nothing too controversial here. But there is also nothing particularly interesting about WDC. This is essentially the same kind of “weak emergence” described by Bedau (Bedau, 1997) in which the Game of Life produces weakly emergent “gliders” for example. I’m in favor of WDC and weak emergence, and the example of volumes of space being dependant on boundary conditions certainly supports WDC.
    - Medium downward causation (MDC) which goes further than WDC “in supposing that higher-level dynamics (e.g., the emergence of a higher-level structure) can modify the local features of an organisms lower-level phase space through the downward actions of formal causes.” Scott continues, “In modern biology, MDC is a key aspect of evolutionary theory, and in neuroscience, the phenomenon of learning is an example of MDC, in which higher level experiences (or training) of an organism alter the ways that neurons interact, changing its behavioral spectrum.” Here again, Scott resorts to ‘levels’ in nature which have attractors and thus the equations and/or physical laws that apply to such levels have some sort of causal influence over the lower level constituents.

    I’m going to leave it there before getting too far off track. Bottom line is that Scott argues in favor of nonlinear phenomena being nonseperable. This is extremely important in deciding if a given physical system can produce any kind of downward causation. It is important because the concept I’ve provided of separable might be sidestepped if boundary conditions acting on a given volume of space are insufficient as an explanandum for what occurs within said volume.

    I would argue Scott’s perspective of nature is problematic, as is any paradigm of classical mechanics which doesn’t follow the definition of separable I’ve provided above. Specifically, if I apply the concept of spatial volumes and what occurs within those volumes to be a function ONLY of boundary conditions on the volume, then nothing can occur within ANY volume that is a function of some higher level of nature. The paradigm I propose allows for only two levels in nature which IMHO is in keeping with the mathematical rules that differentiate between classical and quantum mechanics.

    Such a view is also argued for by physicist Henry Stapp who states “Nothing in classical physics can create something that is essentially more than an aggregation of its parts (Stapp, 1993).” In other words, Stapp believes that classical mechanics / classical physics is separable, regardless of any nonlinearities. In fact, I’m fairly confident that Stapp would agree with the definition of separability I’ve provided above.

    Ok, I know you’ve been reading for a long time now, but I have a few more papers to mention before you nod off! First is one by Kronz (Kronz, Tiehen, 2002). In this paper, Kronz forcefully supports the view that quantum mechanics is nonseparable. From the abstract:
    The paper is rather heavy on the quantum mechanical side, so I can’t recommend it unless you’re a physicist (which I’m not). However, Kronz makes it clear that there is a formal, mathematical split between classical and quantum mechanics. Of course, Kronz isn’t the only one who’s made this split, but he combines that split with the notion of emergence and separability which is unique.

    In that paper he also states, “Because the direct sum is used in classical mechanics to define the states of a composite system in terms of its components, rather than the tensor product operation as in quantum mechanics, there are no nonseparable states in classical mechanics.” Curiously, Kronz then states, “There are non separable Hamiltonians in classical mechanics – the Hamiltonian corresponds to the total energy of the system and is related to the time evolution of the system. This type of nonseparability is the result of nonlinear terms in the equations of motion. Perhaps a kind of emergence can be associated with it. Some measure of plausibility is given to this claim since a classical system can exhibit chaotic behavior only if its Hamiltonian is nonseparable.” (emphasis mine) When quizzed on this statement, his response is that a 3 body system is non separable since its Hamiltonian is non separable. (ie: three body problem)

    Needless to say, I have problems with Kronz’s position on classical systems. Turning again to the definition of separable I've provided, I find all 3 body systems should, in principal, be reducible to gravitational forces (ie: boundary conditions) on any volume of space. Any volume of space must have a gravitational field within it which is only a function of the contribution from each body in the given system. Regardless of how chaotic a given body’s motion may be, and regardless of how impossible it may be to write a single equation of motion, and regardless of the fact that a constructionist will never be capable of exactly determining the body’s motion (for all the reasons mentioned by Anderson for example), the motion is still a function of the gravitational field to which it is subjected, and the gravitational field is a simple summation of the individual contributions from all other bodies. There are never any nonlocal, downward causes which the system as a whole imposes on any portion of that system. And if this is true, then any body in a 3 body system is separable because its motion within that volume is strictly a function of the local gravitational field, and it will react accordingly to that field regardless of the locations of other bodies.

    If three body systems are too abstract, we might also discuss Bishop’s (Bishop 2008) paper in which he supposes that Rayleigh-Benard convection provides a real life example of downward causation. He states,
    For his part, Bishop also points to nonlinear equations but also provides considerable mathematical treatment of heat transport equations, Navier-Stokes equations and the continuity equation. Basically, he claims that it is this combination of affects which produce the non separable, emergent behavior.

    This comes full circle now. Just as Scott wants nonlinear systems to be non separable and emergent, so does Kronz and Bishop. What this states in simple terms is this. In the case of [fluid] turbulence, tornadoes, the Gulf Stream, Jupiter’s Great Red Spot … Rayleigh-Benard convection, such phenomena are ‘levels’ of nature which some philosophers and scientists are suggesting, are nonseparable and in some way ontologically emergent.

    For there to be levels in physics, I would contend that we must be able to locate identical volumes of space on which there are identical boundary conditions, but these volumes must act differently depending on what system they are associated with! If they don’t act differently, they are separable. And if they are separable, we are left with ‘weak emergence’. If they are nonseparable, and these two volumes DON’T act the same, then emergence and downward causation requires non-local causal actions! Think about it… for there to be levels in nature, an arbitrarily selected volume of space with a given set of boundary conditions on it, must act DIFFERENTLY than an identical volume with an identical set of boundary conditions on it in order for there to be ANY kind of downward causation – medium or not. This is exactly the kind of thing Kim is trying to dissuade us from. Supporting the Anderson POV in which levels of nature create new and unique physical laws on the constituent levels that make up a given system is a horrific injustice to science. Even these philosophers and scientists whom would like to accept levels in nature will agree that such levels are entirely arbitrary. Frontal systems and thunderstorms may be a great way for a meteorologist to examine the weather, but no one should be suggesting that the atmosphere doesn’t obey the same heat transport equations, Navier-Stokes equations and continuity equations that every other fluid system obeys.

    ~

    *On the flip side, people have made the argument that energy is more fundamental to such a classical system than are forces, and since the Hamiltonian is nonseparable, the physical system is nonseparable.


    References:
    Anderson, P. W., 1972, “More is different: Broken symmetry and the nature of hierarchical structure of science”
    Bedau, M. A., 1997, “Weak Emergence”
    Bishop, R. C., 2008, “Downward Causation in Fluid Mechanics”
    Emmeche, C. et al, 2000, “Levels, emergence, and three versions of downward causation”
    Kim, J, 2000, “Mind in a Physical World”
    Kronz, F. M. et al, 2002, “Emergence and Quantum Mechanics”
    Scott, A. C., 2003, “Nonlinear Science: Emergence and Dynamics of Coherent Structures”
    Scott, A. C., 2004, “Reductionism Revisited”
    Scott, A. C., 1996, “On Quantum Theories of the Mind”
    Stapp, H. P., 1993, “Mind, Matter and Quantum Mechanics”
    Stapp, H. P., 1997, “On Quantum Theories of the Mind”
     
    Last edited by a moderator: Apr 4, 2009
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