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|Standard Model of Particle Physics
|Scalar 1+1 Model
The best model for the behaviour and interactions of elementary particles is known as the Standard Model of Particle Physics. It consists of Quantum Chromodynamics, which describes strong interactions, and the Electroweak Model, which describes electroweak interactions. A crucial part of the Electroweak Model is Quantum Electrodynamics.
Newtonian Model Types
Some models are not intended to describe nature accurately. Instead, they are toy models. Toy models are theoretical laboratories that help us to understand aspects of more complicated theories in a simplified setup. Examples are
The simplest kind of toy models where we have essentially isolated one elementary degree of freedom are called basic models. Examples for basic models are
Definition of a Model
In addition, there can be multiple Lagrangians for one and the same model. This is known as a duality.
The traditional approach of theoreticians, going back to the foundation of quantum mechanics, is to run to Schrödinger’s equation when confronted by a problem in atomic, molecular or solid state physics! One establishes the Hamiltonian, makes some (hopefully) sensible approximations and then proceeds to attempt to solve for the energy levels, eigenstates and so on. However, for truly complicated systems in what, these days, is much better called “condensed matter physics,” this is a hopeless task; furthermore, in many ways it is not even a very sensible one!
The modern attitude is, rather, that the task of the theorist is to understand what is going on and to elucidate which are the crucial features of the problem. For instance, if it is asserted that the exponent [β] depends on the dimensionality, d, and on the symmetry number, n, but on no other factors, then the theorist’s job is to explain why this is so and subject to what provisos. If one had a large enough computer to solve Schrödinger’s equation and the answer came out that way, one would still have no understanding of why this was the case!
Thus the need is to gain understanding, nut just numerical answers: that does not necessarily mean going back to Schrödinger's equation which, in any case, should be really regarded just as an approximation to some sort of gauge field theory. So the crucial change of emphasis of the last 20 or 30 years that distinguishes the new era from the old one is that when we look at the theory of condensed matter nowadays we inevitably talk about a "model". As a matter of fact even Schrödinger's equation and gauge field theories themselves are just models of the physical world, albeit pretty good ones as far as we can presently judge.
We should be prepared to look even at rather crude models, and, in particular, to study the relations between different models. We may well try to simplify the nature of a model to the point where it represents a ‘mere caricature’ of reality. But notice that when one looks at a good political cartoon one can recognise the various characters even though the artist has portrayed them with but a few strokes. Those well chosen strokes tell one all one really needs to known about the individual, his expression, his intentions and his character. So, accepting Frenkel’s guidance, [. . . ] a good theoretical model of a complex system should be like a good caricature: It should emphasise those features which are most important and should downplay the inessential details. Now the only snag with this advice is that one does not really know which are the inessential details until one has understood the phenomena under study. Consequently, one should investigate a wide range of models and not stake one’s life (or one’s theoretical insight) on one particular model alone. Nevertheless, one model which, historically, has been of particular importance and which has given us a great deal of confidence in the phenomenological descriptions of critical exponents and scaling presented earlier deserves special attention: this is the so-called Ising model. Even today its study continues to provide us with new insights.
Michael E. Fisher. Scaling, universality and renormalization group theory. In F.J.W. Hahne, editor, Critical Phenomena, volume 186 of Lecture Notes in Physics, Berlin, 1983. Summer School held at the University of Stellenbosch, South Africa; January 18–29, 1982, Springer-Verlag.
Scientific thinking is grounded in the evolved human ability to freely create and manipulate mental models in the imagination. This modeling ability enabled early humans to navigate the natural world and cope with challenges to survival. Then it drove the design and use of tools to shape and control the environment. Spoken language facilitated the sharing of mental models in cooperative activities like hunting and in maintaining tribal memory through storytelling. The evolution of culture accelerated with the invention of written language, which enabled creation of powerful symbolic systems and tools to think with. That includes deliberate design of mathematical tools that are essential for physics and engineering . A mental model coordinated with a symbolic representation is called a conceptual model. Conceptual models provide symbolic expressions with meaning. This essay proposes a Modeling Theory of cognitive structure and process. Basic definitions, principles and conclusions are offered. Supporting evidence from the various cognitive sciences is sampled. The theory provides the foundation for a science pedagogy called Modeling Instruction, which has been widely applied with documented success and recognized most recently with an Excellence in Physics Education award from the American Physical Society.
David Hestenes Conceptual Modeling in physics, mathematics and cognitive science In D. Hestenes, SemiotiX, November 2015. http://semioticon.com/semiotix/2015/11/.