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1.4 Physical theories and the program of theoretical physics.\
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"What is a physical theory? It turns out that the answer to this question is \
a little counterintuitive from the point of view of the general public. A ",
StyleBox["scientific theory",
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" is a body of work leading to a self-consistent idea that is considered to \
be a fact. In most cases there is no controversy about the theory in \
question. So, once again, scientists use a word that public at large thinks \
is one thing in a different way, thus a miscommunication has developed. The \
program of theoretical physics is all about developing physical theories. \
Unfortunately there is more than one such program.\n\tWe begin with the \
modeling approach to theoretical physics. Another way of calling this would \
be the phenomena-centered approach, whose goal is to understand a specific \
phenomena by developing model of it, and then subjecting the model to \
different situations and derermining its properties. The process begins by \
forming primitive, intuitive, and ill-defined notions about what you are \
studying. You choose an approach to representing the phenomena; can you \
represent it as particle? a field? or some continuous distribution of matter? \
of some combination? From a previous section we can choose either a particle \
theory, a field theory, or a theory of matter. From this beginning you \
construct precise ideas and give them symbolic representation. Often the \
symbolic representations are stated in the form, ",
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\[CloseCurlyDoubleQuote]",
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" Then you choose a mathematical formulation. Examples of mathematical \
formulations are Newtonian mechanics, Maxwell\[CloseCurlyQuote]s equations, \
Lorentz transformations, the Maxwell-Boltzmann distribution, etc. You then \
adapt your mathematical formulation to the specific phenomena you are \
studying, thus developing a mathematical representation of your phenomena. By \
manipulating your mathematical representation, making physical arguments and \
calculations for specific situations, you can make predictions with these \
activituies\[LongDash]often in the form of tables, formulas, and plots. This \
might involve deriving new principles and performing computational \
simulations. This type of prediction is called a ",
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". By studying the results in different circumstances you can extend our \
understanding of the phenomena. A body of models linked by physical argument, \
derivation methods, ",
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" computer simulation is a physical theory. This is the most direct method \
of doing theoretical physics, it is a straight application of mathematical or \
computational methods. It is certainly the most structured way of doing \
theoretical physics. Such formulations constitute much of the material of \
most textbooks and courses on physics.\n\tAnother program is the constructive \
approach to theoretical physics. This can be thought of as the method to \
develop a new formulation of a physical theory. Examples are the Lagrangian \
formulation of mechanics, the Lagrangian formulation of electrodynamics, the \
Eulerian formulation of fluid dynamics, the path-integral formulation of \
quantum mechanics, and so on. You begin by choosing how you represent objects \
in your developing theory. Then you choose some quantity, or set of \
quantities to base your construction on. Then you choose an argument to base \
your construction on. Are you seeking to find symmetries? Are you arguing \
from some conserved quantity? Are you assuming that your quantity is \
minimized? For example, in the Lagrangian formulation you choose to create a \
new quantity called the Lagrangian and then you work out the consequences \
when some quantity based on the Lagrangian\[LongDash]the action, for example\
\[LongDash]is minimized. This leads to the Euler-Lagrange equations of \
motion, a new formulation of classical mechanics. This is a much more \
difficult, but powerful method\[LongDash]you build the formulation. The \
difficulty stems from the lack of structural guidelines in creating a new \
formulation. Once you have the new formulation, it is actually easier to use \
in most situations.\n\tA third approach is that of abstraction. Here you take \
a number of specific cases and generalize their results. For example, knowing \
that when a rate-of-change is 0 a quantity is unchanged; you take the \
zero-rates-of-change of momentum in many cases and generalize that into the \
statment that the quantity of momentum doesn\[CloseCurlyQuote]t change\
\[LongDash]a statement of the law of conservation of momentum. This sort of \
activity is very difficult since there are few guidelines for how to proceed \
beyond what is already known.\n\tA fourth approach is to simply play with \
ideas. Here we are on new ground, there are few guidelines for such play. We \
can ask, \[OpenCurlyDoubleQuote]What happens if we introduce a higher \
dimension? A lower dimension? Multiple bodies? Fewer bodies? and so on.\
\[CloseCurlyDoubleQuote] You can even note the similarity in words describing \
things. Scientific discoveries have been made with all of these.\n\tThe last \
case we will examine here is that of unification. Unification is the idea \
that different phenomena are governed by a single\[LongDash]higher-level\
\[LongDash]theory instead of a theory for each phenomena. There is no real \
reason to believe that this is true generally, and this is one difficulty \
with practical application. Another difficulty is that all of our equations \
are, to one degree or another, an approximation of reality. So the fact that \
equations in different fields look alike is another way of saying that the \
approximations are similar. Does that mean the phenomena are also similar? \
Sometimes. Isaac Newton unified gravity at the surface of the Earth and \
gravity away from the Earth. James Maxwell unified electricity, magnetism, \
and light. Abdus Salam, Sheldon Glashow and Steven Weinberg unified \
electromagnetism and the weak nuclear force. The work of unifying electroweak \
theory with the strong interaction force is a work in progress. Even less \
success has been made in unifying gravity."
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