4 ms·

Would anyone on HN care to explain to physics laymen what Gauge theory is about in the first place?

figure8 8y ago
Gauge theory is the collection of techniques used to describe forces in quantum field theory (electro-magnetism, weak force, and strong force). In field theory you have many vector spaces (1-d, 2-d, ... n-d) defined for every point in space time. So a specific field requires a distinct vector value for each space-time point. These fields encode certain particle types like electrons, quarks, etc. 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.
- espeed 8y ago
  One emerging realization is that the essence of gauge theory [0] has revealed itself as an insight to some whom have learned to apply it beyond the realm of physics. 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. See "Gauge Theory and Inflation: Enlarging the Wu-Yang Dictionary to a unifying Rosetta Stone for Geometry in Application - Eric Weinstein" [video] https://www.youtube.com/watch?v=h5gnATQMtPg https://www.youtube.com/watch?v=h5gnATQMtPg [0] Gauge Theory https://en.wikipedia.org/wiki/Gauge_theory https://en.wikipedia.org/wiki/Gauge_theory [1] Eric Weinstein https://en.wikipedia.org/wiki/Eric_Weinstein https://en.wikipedia.org/wiki/Eric_Weinstein [2] James Simons https://en.wikipedia.org/wiki/James_Harris_Simons https://en.wikipedia.org/wiki/James_Harris_Simons [3] Nobel Laureate C.N. Yang https://en.wikipedia.org/wiki/Yang_Chen-Ning https://en.wikipedia.org/wiki/Yang_Chen-Ning [4] Yang-Mills Theory https://en.wikipedia.org/wiki/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%20nonintegrable%20phase%20factor%20and%20global%20formulation%20of%20gauge%20fields%20-%20Wu,%20Yang%20-%201975.pdf http://www.indiana.edu/~jpac/QCDRef/1970s/Concept%20of%20non... [6] Renaissance Technologies https://en.wikipedia.org/wiki/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/OriginsofMaxwellandGauge.pdf http://www.physics.umd.edu/grt/taj/675e/OriginsofMaxwellandG... Previous Discussion: https://news.ycombinator.com/item?id=16325605 https://news.ycombinator.com/item?id=16325605
  - balsam 8y ago
    Uh Eric told you this himself? Rentech is famously secretive it wouldnt surprise me if they went after anyone who even leaks the existence of specific sauce.
danbruc 8y ago
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.
- sideshowb 8y ago
  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. (For the physics layman - you might enjoy my game https://omnisplore.wordpress.com/2016/04/25/learning-quantum-mechanics-the-easy-way/ https://omnisplore.wordpress.com/2016/04/25/learning-quantum... )

Gauge theory is the collection of techniques used to describe forces in quantum field theory (electro-magnetism, weak force, and strong force). In field theory you have many vector spaces (1-d, 2-d, ... n-d) defined for every point in space time. So a specific field requires a distinct vector value for each space-time point. These fields encode certain particle types like electrons, quarks, etc. 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.

espeed 8y ago
One emerging realization is that the essence of gauge theory [0] has revealed itself as an insight to some whom have learned to apply it beyond the realm of physics. 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. See "Gauge Theory and Inflation: Enlarging the Wu-Yang Dictionary to a unifying Rosetta Stone for Geometry in Application - Eric Weinstein" [video] https://www.youtube.com/watch?v=h5gnATQMtPg https://www.youtube.com/watch?v=h5gnATQMtPg [0] Gauge Theory https://en.wikipedia.org/wiki/Gauge_theory https://en.wikipedia.org/wiki/Gauge_theory [1] Eric Weinstein https://en.wikipedia.org/wiki/Eric_Weinstein https://en.wikipedia.org/wiki/Eric_Weinstein [2] James Simons https://en.wikipedia.org/wiki/James_Harris_Simons https://en.wikipedia.org/wiki/James_Harris_Simons [3] Nobel Laureate C.N. Yang https://en.wikipedia.org/wiki/Yang_Chen-Ning https://en.wikipedia.org/wiki/Yang_Chen-Ning [4] Yang-Mills Theory https://en.wikipedia.org/wiki/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%20nonintegrable%20phase%20factor%20and%20global%20formulation%20of%20gauge%20fields%20-%20Wu,%20Yang%20-%201975.pdf http://www.indiana.edu/~jpac/QCDRef/1970s/Concept%20of%20non... [6] Renaissance Technologies https://en.wikipedia.org/wiki/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/OriginsofMaxwellandGauge.pdf http://www.physics.umd.edu/grt/taj/675e/OriginsofMaxwellandG... Previous Discussion: https://news.ycombinator.com/item?id=16325605 https://news.ycombinator.com/item?id=16325605
- balsam 8y ago
  Uh Eric told you this himself? Rentech is famously secretive it wouldnt surprise me if they went after anyone who even leaks the existence of specific sauce.

danbruc 8y ago

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.

sideshowb 8y ago
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. (For the physics layman - you might enjoy my game https://omnisplore.wordpress.com/2016/04/25/learning-quantum-mechanics-the-easy-way/ https://omnisplore.wordpress.com/2016/04/25/learning-quantum... )