Swedish Foundation for Strategic Research Activity Report 2006

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20 – The Swedish Foundation for Strategic Research • Activity Report 2006Following an initial imperfect understanding, futurework often leads to a more and more complex modelthat can intuitively be regarded as unsatisfying even byits proponents, and that is later replaced by a simpleror at least quite different model, typically describedas “more fundamental”. The first model can typically,at least with hindsight, be described as retaining toomuch of an “old” way of thinking. This is the situationwhen people talk about changes in paradigms.During this second phase there is already a potentialfor a large flow of information between experimentand theory. Typically, many models can be tried out,and most of them fall by the wayside as time goes by.This is the standard way science progresses: theorieshave to be backed up by observations and experimentsor they have to be discarded. Sometimes an influentialscientist can keep a personal favourite model alive inthe field until he or she dies, after which it is quietlydropped when nobody takes over the torch. 2The most dramatic change of paradigm in physicsduring the 20th century was arguably the introductionof quantum mechanics to replace the old Newtonianmechanistic view of the world. Some characteristics ofthis shift are: new concepts that take the fundamentalroles (“the state of a system”, “the uncertainty of aposition measurement”, “the tunnelling probability”),and new types of questions that replace some old types(“What is the probability of …?” but no longer “Alongwhich path does the electron… ?” or “Which of theelectrons…?”).In the third phase most observed phenomena areexplained by some theory and it becomes unclearwhether there are really any fundamentally unexplainedphenomena at all. In this phase, either anincreased level of understanding is hampered by alack of reliable experimental data that go beyondthe realms of the theory, or the unclarity is removedwhen tentative new phenomena do not materialise asexperiments are improved. The latter seems to havebeen more common in the case of physics. However,while this situation prevails, theorists can still literally2”It is a good morning exercise for a research scientist todiscard a pet hypothesis every day before breakfast.” (KonradLorenz)fill the room with speculative mathematical constructs,all more or less consistent with known observations, butalso more or less totally different from each other withregard to the outcomes of hypothetical new experiments.In some other cases they may remain, for allpractical purposes, indistinguishable but for their morephilosophical differences. Then again, occasionally thesituation is eventually resolved only at a new level oftheoretical understanding.Today, with the emerging role played by mathematicalsimulations, it is also possible to substitute “simulateddata” for experimental data in the descriptionof this third phase. Indeed, the scope of the modelsutilised in the simulations may well be as fundamentalas the analytical constructs.In the field of elementary particle physics, which isone of few major fields where we seem to have spentconsiderable time in the third phase, and where wecan simultaneously address both the smallest and thelargest structures in the world, it is conceivable that, forpurely financial reasons, we will reach a point wherewe will eventually have to be content with merelyobserving the ultimate experiment: the universe itself,of which only one copy is currently available to us.This is available to us more or less for free, and it ofcourse continuously exerts a considerable influence ontheoretical work.All elementary particles known today have a model basedon the so-called Standard Model. This includes quarks,electrons and other material particles together with theinteractions known as strong, weak and electromagnetic.It is ad hoc supplemented by general relativity as thetheory of gravitation. However, our view of this entireconstruction is contingent on the existence of an as-yetundiscovered particle, the Higgs boson, without whichthis model loses most of its coherence. The search for thisparticle is an important argument for the next generationof particle accelerators.In retrospect, it is interesting to note how relativelysuccessful models that steer away from complexityand toward simplicity, and accompanying mathematicalbeauty, have been in the actual development ofphysical theory during the 20th century. However,it is also true that much of the real world still seemstoo complex to ever be treated by analytical models.
The Swedish Foundation for Strategic Research • Activity Report 2006 – 21The substitution of quantum mechanics for classicalmechanics has not fundamentally altered this situation.And beauty doesn’t always work: at least the expansionof the universe was considered to be an ugly theoreticalpossibility before it was discovered. So perhaps it isan unreliable guiding principle on the path to understanding.String theory is an example of a theory without any specific experimentalsupport today, although it is presumably consistentwith all known experiments. However, there is a first-rate theoreticalreason for its acceptance by many physicists, namelythat it offers the only known way to reconcile the mathematicalproblems encountered when trying to combine two of themost celebrated theoretical breakthroughs of the 20th century– general relativity as a theory for gravitation, and quantummechanics – to describe all established knowledge concerningelementary particles and the remaining physical properties ofNature within a single framework.Most particle physicists seem to take this theoretical argumentseriously, although there are some sceptics. However, theentire scientific enterprise is based on imagination on someconceptual level. The scientific machinery – including framingof hypotheses, testing, making reproducible observations, etc.– will play its part as well, but it is human imagination that propelsscience forward.n What kind of help do mathematicalmodels offer?Science can be transformed into technology only whenit’s underlying parameters are defined and understoodand can be controlled. This requires some kind of mathematicaldescription. Even a partial understandingcan be very useful when the remaining uncertaintiesare irrelevant, for example in a particular setting.To start with, mathematical models can often yieldpartial answers even in cases where most of the completetheoretical structure is unknown. An example isthermodynamics, which is a body of knowledge aboutsystems that can be characterised by a small set ofmacroscopic quantities (like pressure, volume, temperature,energy and entropy). Many questions aboutsuch macroscopic systems can indeed be formulatedand answered independently of any further detailedmicroscopic information about the systems. Similarly,we can have a very good understanding of how genetictraits are inherited in spite of mounting difficulties inprecisely defining what a gene really is. And mathematicalmodels are of value as soon as they offer some understanding,since they support the process of structuringavailable information and formulating more preciseneeds for further information.Because mathematicians always know exactly whatthey are talking about, working painstakingly from precisedefinitions, the mathematical knowledge producedtends to be very robust and does not suffer from laterreinterpretations or paradigm shifts.Scientists usually look for deterministic models,where the state of the system at hand can be determinedfrom a defined initial state by calculation. We wereforced to back off a bit and reconsider our positionwhen quantum mechanics turned out to limit us tothe calculation of probabilities. But at least we couldargue that these probabilities could be deterministicallycalculated.The types of equations that are of value in one fieldmay differ quite a bit from those used in another field.It is nevertheless quite common that the same equationsemerge in what might at first have appeared tobe very different settings. Since the same equations dohave the same solutions, this creates opportunities for alot of cross-fertilization.An interesting question with a bearing on muchof the terminology used in the natural sciences, inparticular when we deal with objects that are literallytoo small to be seen, is the following: When does somethingexist?In practical terms, objects that are eventuallyconsidered to exist are those that are the objects of thefundamental theories (to be interpreted as widely applicablemathematical models in their particular contexts)and vice versa – once we believe we have them. This ismore or less what the word fundamental means. Thus,the quarks of physics are considered to exist althoughthey are particles that are essentially unable to take onidentities as well-defined isolated objects, but hide insidemore complex structures. In this way, mathematicsoffers necessary credence to the very concepts we haveintroduced to describe the systems we study in science.While the proper choice of variables – and correspondingfundamental objects – is normally ofcrucial importance in the successful application of
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Swedish Foundation for Strategic Research Activity Report 2006

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