A field theory is a theory that describes a physical phenomenon by fields and how they evolve and interact over time. A field is just a function that takes a point in space and time and gives you a value. You can model temperature as a function that given a point in space and instant in time gives you the temperature at that specific point in space and time. Or the wind velocity which will assign a vector indicating the direction and speed of the wind to each point in space and time instead of a single number as in the case of temperature - this is the difference between a scalar field and a vector field and there are other types of fields like tensor fields. A field theory then provides you with the equations that explain how fields change over space and time and how they interact with each other in case the theory has multiple interacting fields.
A gauge theory then is a special kind of field theory, one which is invariant under some symmetry transformation. You could for example come up with the idea to describe temperature with complex numbers instead of real numbers. The equations wouldn't really change, they would essentially just ignore the imaginary part of the temperature making 20°C, 20 + 42i°C, and 20 - 7i°C three different representations of the same temperature. In consequence you are free to adjust the imaginary parts of all the temperatures without this affecting your predictions.
As it turns out if you want to use field theory to describe quantum physics and if you want your field theory to have certain nice properties, for example making it obvious that the theory respects locality, then you are forced to use gauge theories, i.e. you are unable to write a field theory without those redundancies where several different values correspond to the same real physical state. In the example from above, the real temperature is of course 20°C but because of the way you would like to formulate your theory, you are forced to use complex numbers for the temperature and because the resulting theory is so nice to work with you accept this and just declare that all temperatures with the same real component are the same regardless of the imaginary part.
And while you often hear that the world is fundamentally made out of fields, that is not true, or at least it does not have to be true. Quantum field theory is one of our best tools to describe our world but the fields are just a nice and convenient mathematical tools, they are not in one to one correspondence with actually existing things.
Came back here to see this question answered. Was not disappointed. +10 for clarity :)
So, the phase of a wavefunction in QM makes it a gauge theory, then? It doesn't matter what the actual phase is at any point, but it does matter how it varies between one point and the next.
So, the phase of a wavefunction in QM makes it a gauge theory, then?
There are two kinds of symmetries, global symmetries and local symmetries. In case of a global symmetry the same transformation is applied everywhere, for example moving all particles of a system to the right by one meter. In case of a local symmetry the transformation can vary over space and time, so you could move different particles by different distances, for example double the x coordinate.
Classical mechanics is invariant under global translations, it does not matter where the system is, here or one meter to the left, what matters are just differences between positions, the distances between particles. On the other hand classical mechanics is not invariant under local translations because it will alter the distances between particles. Local symmetries are a much stronger constraints than global symmetries which are essentially just a special case of local symmetries where the transformation parameters are fixed across space and time.
The wave function has a global symmetry, you can rotate the phase, but phase differences are important and you can therefore not apply different phase rotations to different parts of the wave function. On the other hand the relevant symmetries in gauge theories are local symmetries, so the global symmetry of the wave function is not what makes a theory of quantum physics a gauge theory. In case of electrodynamics it is for example the electromagnetic four-potential that has a gauge freedom, i.e. there are many different electromagnetic four-potentials that give rise to identical physically observable magnetic and electric fields.
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.
A gauge theory then is a special kind of field theory, one which is invariant under some symmetry transformation. You could for example come up with the idea to describe temperature with complex numbers instead of real numbers. The equations wouldn't really change, they would essentially just ignore the imaginary part of the temperature making 20°C, 20 + 42i°C, and 20 - 7i°C three different representations of the same temperature. In consequence you are free to adjust the imaginary parts of all the temperatures without this affecting your predictions.
As it turns out if you want to use field theory to describe quantum physics and if you want your field theory to have certain nice properties, for example making it obvious that the theory respects locality, then you are forced to use gauge theories, i.e. you are unable to write a field theory without those redundancies where several different values correspond to the same real physical state. In the example from above, the real temperature is of course 20°C but because of the way you would like to formulate your theory, you are forced to use complex numbers for the temperature and because the resulting theory is so nice to work with you accept this and just declare that all temperatures with the same real component are the same regardless of the imaginary part.
And while you often hear that the world is fundamentally made out of fields, that is not true, or at least it does not have to be true. Quantum field theory is one of our best tools to describe our world but the fields are just a nice and convenient mathematical tools, they are not in one to one correspondence with actually existing things.