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The use of arbitrary parameters in theoretical models

  1. Oct 30, 2011 #1
    Hello. I have three questions that I would like to ask.

    1. Could someone provide me with a resource containing examples of arbitrary parameters in theoretical physics. E.g. a modern example, yet to be varified, is dark energy. A historical example, one which has since been discredited, is the luminiferous ether.

    2. What is your opinion on the relatively frequent invoking of arbitrary parameters within theoretical physics? Naturally they will be necessary when theories are first being developed, but what about when a theory has been around for many years, or even decades, and yet still contains arbitrary parameters that have not been observed in nature or experimentally verified. Any personal opinions on this?

    3. In your opinion, is it justifiable that many scientific theories, with their arbitrary parameters, can be used as support for related theories that are developed further down the line. What becomes of these related theories if and when said arbitrary parameters are discredited? Do foundations crumble? Or is it merely a case of patching things up with more arbitrary parameters?

    I apologise if I come across as a little cynical here, that is not my intention. On the contrary, I myself am hoping to work my way through the formal scientific education system in the coming years, with the intention of one day being a scientist. So for me, this is more of an area of concern for which I would like some clarification, rather than an attempt to sneer at the scientific community.

    Thank you.
    Last edited: Oct 30, 2011
  2. jcsd
  3. Oct 31, 2011 #2


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    Lots of good questions.

    I'll pass for now on Q1, but depending on the system/theory, the physics is rather complex, and at times its necessary to determine an approximate solution with empirical models that allow us to obtain a solution.

    On Q2, we observe, and we build models based on observation, or we fit data with simple models. As we better understand the physics, we can build better models. Please realize that there were real limitations on scientists 30 or 40 years ago with regard to computational capability. We now have computing power that is orders of magnitude greater than way back in the days for the internet.

    On Q3, a theory's form (including use of empirical models/constants) is only justifiable if it matches observation or enables prediction of future behavior of a system - with reasonable fidelity.

    The OP reminded me of somthing I heard the other day regarding the precession of equinoxes and the observations of Hipparchus.

  4. Oct 31, 2011 #3
    Please note that I am not an astrophysicist, so hopefully one will step in. The dark energy is one of the most baffling unexplained parameter in physics. We just don't know what it is. When measured in terms of total energy content of the Universe, it comes out enormous (say 70%). When trying to predict it from quantum physics, it is just so incredibly tiny that we have to fine-tune out theories so that it is not exactly zero (it may come out 70 orders of magnitude too big if we really do a good job at trying to predict it).

    Please note that there are many ideas to try to improve the situation. One I am more familiar with is
    Standard-Model Condensates and the Cosmological Constant
    Essence of the vacuum quark condensate

    On the general question, I cannot provide you with a complete list, and I am not sure anybody can. Some have tried. Depending how one counts (i.e. what one will include), the standard models of particle physics and cosmology have about 26 arbitrary constants

    How Many Fundamental Constants Are There?

    In my view, we should not expect that fundamental theories must have zero or even just one arbitrary constant. Since we do have theories with a finite number of parameters, the onus should be on those who claim it should be otherwise. From what I can understand, the idea is that a unified theory of all interactions should be able to predict "everything". It should be able to prove that our world is the only possible world. Well, I see that as too restrictive. In the standard theories we have (the so-called standard model of particle physics, and the LambdaCDM model of cosmology), there are adjustable parameters which would accomodate different universes.

    In my opinion, what matters for a scientific theory is the predictive power. So I have four points together with their error bars, four observations, and say I hypothesize a linear relation between them. I can use three points and fit them with a straight line. The quality of the fit tells me whether my law is appropriate at all. Then I can use the last point to make a prediction with my toy "theory" and compare with experiment. If down the line I come up with a more fundamental explanation of "why" there should be such a linear law, I believe that yes, this is a worthwhile scientific endeavor.

    In general, the fewer the free parameters for a given set of observations, the better the theory. If as of today we accept 25 or so parameters, that's the job we have accomplished to explain all known experimental observations. It may not be as good as we would like, but it is also pretty darn hard to do better.

    What is really, seriously, occupying some physicists is the following. Our standard models can accommodate different universes. But, we may not be able to live in those universes. Then what matters is "how generic is our universe ?". If it turns out that some of the free parameters must be incredibly fine tuned (to hundreds of decimal places) to accommodate our existence, then we may say that our theories are not very satisfactory.
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