Analysis in Higher Gauge Theory
golem.ph.utexas.edu
golem.ph.utexas.edu
One can ask, how can I compare field values at 2 distinct points a and b? They are in distinct vector spaces V_a vs V_b. Well, gauge theory says you must imagine the vector v_a being moved along a path between a and b, and see how the translated vector value compares to the vector v_b over b. This is called "parallel transport." A rule about how all vectors move along all paths during this translation is called a "connection." These connections encode distinct force-carrying particle types like photons and gluons. Sort of like how an object naturally rotates along a curvy surface. It's naively possible to describe this transport, this curvyness, but the natural notation is such that distinct descriptions are actually the same connection. Dealing with this ambiguity is key to properly computing probabilities using path integrals.
By the way, each gauge theory is required to have an underlying "gauge group," meaning a set of possible symmetries. In the standard model these groups are called U(1), SU(2) and SU(3). U(1) is 1-dimensional, so only 1 kind of electromagnetic force particle, the photon. SU(2) is 3-dimensional so 3 types of weak force particles (W-,W+, and Z) and SU(3) is 8 dimensional so 8 types of gluons.
Eric Weinstein [1], managing director of Thiel Capital, has been advocating upgrading the current economic calculus with a quantitative model based on gauge theory that can better adapt to changing behavior and changes in consumer preferences across time (for example).
Eric has intimated that James Simons [2] discovered a secret (in the Peter Thiel sense of the word) related to gauge theory during the time Simons and C.N. Yang [3] (as in Yang-Mills Theory [4]) discovered the correspondences in what has become known as the "Wu-Yang Dictionary" [5], which later led to Simons founding Renaissance Technologies [6] in 1982 and is part of the secret-sauce in their mathematical models that has enabled them to dominate quantitative trading ever since. A quote from the 1975 paper:
...The mathematics of these results is in fact well
known to the mathematicians in fiber bundle theory.
An identification table of terminologies is given in
Sec. V. We should emphasize that our interest in this
paper does not lie in the beautiful, deep, and general
mathematical development in fiber bundle theory.
Rather we are concerned with the necessary concepts to
describe the physics of gauge theories. It is
remarkable that these concepts have already been
intensively studied as mathematical constructs.
[0] Gauge Theory https://en.wikipedia.org/wiki/Gauge_theory
[1] Eric Weinstein https://en.wikipedia.org/wiki/Eric_Weinstein
[2] James Simons https://en.wikipedia.org/wiki/James_Harris_Simons
[3] Nobel Laureate C.N. Yang https://en.wikipedia.org/wiki/Yang_Chen-Ning
[4] Yang-Mills Theory https://en.wikipedia.org/wiki/Yang–Mills_theory
[5] Concept of nonintegrable phase factor and global formulation of gauge fields (AKA the Wu-Yang Dictionary) [pdf] http://www.indiana.edu/~jpac/QCDRef/1970s/Concept%20of%20non...
[6] Renaissance Technologies https://en.wikipedia.org/wiki/Renaissance_Technologies
[7] The conceptual origins of Maxwell's equations and gauge theory, by C.N. Yang (2014) [pdf] http://www.physics.umd.edu/grt/taj/675e/OriginsofMaxwellandG...
Previous Discussion: https://news.ycombinator.com/item?id=16325605
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.
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.
(For the physics layman - you might enjoy my game https://omnisplore.wordpress.com/2016/04/25/learning-quantum... )
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.
I'm likely still missing something.
It also seems I got several downvotes for my comment, trying to reach moderator dang, which I've seen other HNers do in the comments section, for example, when a title should be altered, etc.