One formulates a theory that is supposed to describe a natural phenomenon (and therefore have some kind of predictive power, I presume).
Only ... that theory has too many degrees of freedom and because of that, it can produce states that can't possibly exist in the real world.
Or maybe, it produces states that have extraneous variables (like in your complex temperature example) that must be "projected away" to obtain something meaningful. Helper variables, so to speak.
Therefore, one needs to add constraints (i.e. remove degrees of freedom), until the initial theory "behaves" as expected and starts to fit the real world.
That prompts two questions:
Where do these additional constraints come from? Does one just pull them out of thin air, or are they somehow derived from an intuitive understanding of what actually happens at the physical level? Or are these additional constraints simply a refinement of the initial model that was just too broad to fit the physical world?
In what way does following that mental path actually help obtain the final result (general theory + addt'l constraints). Isn't adding constraints to a too-broad model a standard enough way of conducting business that it deserves a special name?
All the physics I've been exposed to in my life describes phenomena as a more or less complicated system of partial differential equations in N variables, with many of these variables found both in the numerator or the denominator of the differentiation operator.
The way I'm understanding gauge theory after reading your explanation is that you have a model with more variables than partial diffeqs and that in order for the system to "compute" one has to add additional partial diffeqs until number of equations matches number of variables. That's a pretty general way of operating, and I guess what I'm still not seeing is what makes gauge theory special in that regard.
One formulates a theory that is supposed to describe a natural phenomenon (and therefore have some kind of predictive power, I presume).
Only ... that theory has too many degrees of freedom and because of that, it can produce states that can't possibly exist in the real world.
Or maybe, it produces states that have extraneous variables (like in your complex temperature example) that must be "projected away" to obtain something meaningful. Helper variables, so to speak.
Therefore, one needs to add constraints (i.e. remove degrees of freedom), until the initial theory "behaves" as expected and starts to fit the real world.
That prompts two questions:
Where do these additional constraints come from? Does one just pull them out of thin air, or are they somehow derived from an intuitive understanding of what actually happens at the physical level? Or are these additional constraints simply a refinement of the initial model that was just too broad to fit the physical world?
In what way does following that mental path actually help obtain the final result (general theory + addt'l constraints). Isn't adding constraints to a too-broad model a standard enough way of conducting business that it deserves a special name?
All the physics I've been exposed to in my life describes phenomena as a more or less complicated system of partial differential equations in N variables, with many of these variables found both in the numerator or the denominator of the differentiation operator.
The way I'm understanding gauge theory after reading your explanation is that you have a model with more variables than partial diffeqs and that in order for the system to "compute" one has to add additional partial diffeqs until number of equations matches number of variables. That's a pretty general way of operating, and I guess what I'm still not seeing is what makes gauge theory special in that regard.
I'm likely still missing something.