5 ms·

I’d like to learn more about this symmetry. Does anyone recommend a good article (or paper) about this? > Researchers note that electromagnetism, the weak forc

I’d like to learn more about this symmetry. Does anyone recommend a good article (or paper) about this? > Researchers note that electromagnetism, the weak force and the strong force each follow directly from a specific kind of symmetry — a change that doesn’t change anything overall (the way rotating a square by 90 degrees gives us back the same square).

srl 5y ago
I'm _pretty_ sure that this is referring to gauge symmetry, so that's the term to search for. I dunno of a good article as an introduction. As a grad student, I found Terry Tao's explanation [1] rather helpful, but of course it has a strongly mathematical flavor. [1] https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/ https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/
- floatingatoll 5y ago
  Thanks! It sounds like the next step is to look for curvature transformations associated with terms used for each of those three. (If someone knows more, always open to hear that too.) Also, a textbook is revealed on what seems to be this exact topic. Does this seem right? https://physicstoday.scitation.org/doi/pdf/10.1063/PT.3.2421 https://physicstoday.scitation.org/doi/pdf/10.1063/PT.3.2421
  - srl 5y ago
    Yeah... I'd be hesitant to buy a whole book. This is one of the topics where some expositions are really bad, some are really good, and some people just can't agree on which are bad and which are good. I've always had best luck just pulling up 8 different articles and reading them in parallel, hoping it eventually sticks. (Which I realize is the least helpful possible reply.) The textbook Peskin&Schroeder is standard in this field. "If" you can find a PDF online, there's a chapter introducing gauge theories (the chapter on yang-mills) that has a reasonably clear explanation near the beginning, I think.
- gaugewat 5y ago
  Wow... > “Gauge theory” is a term which has connotations of being a fearsomely complicated part of mathematics – for instance, playing an important role in quantum field theory, general relativity, geometric PDE, and so forth. But the underlying concept is really quite simple: Then goes on to write an incomprehensible novella length explanation. I like Tao, but this is a problem in 'pure' maths. I feel the issue is exacerbated by the fact that if you can follow along with Terry then you likely will be unable to recognise the issue. A good counter point is this series by Timothy Gowers developing a proof of "Pingala's Determinant". When Gowers finds the proof he states: > Since this clearly has determinant 1 and this clearly has determinant 1, so does their products, ie. Pascal's[sic] determinant is 1. Could have done that, but that would have been uh, well depends on what you find interesting, but that would have been really sort of rabbit out of the hat, look what a billiant clever mathematician I am. I can just sort of produce this fabulous identity out of nowhere and it's got a nice simple proof that also came out of nowhere. What I want to emphasize is these sort of proofs don't come out of nowhere. Timestamped to quotation: https://youtu.be/m8R9rVb0M5o?t=886 https://youtu.be/m8R9rVb0M5o?t=886 The series is great and will show anyone outside the field of pure maths that even pure mathematicians use numerical reasoning while trying to reason about a subject.
  - srl 5y ago
    I think you're underrating heterogeneity. Not all people will agree that Tao's exposition is unclear. I linked it precisely because I found it more clear than other, "more physical" explanations (as a graduate student in physics, not math). When I was trying to understand gauge symmetry, it was that article that finally made things click for me. And, of course, many people I know would disagree, and point to other expositions as superior. That's the point: different people will not agree on what is well-written and what is incomprehensible. You state "even pure mathematicians use numerical reasoning while trying to reason about a subject". Yes, some pure mathematicians do. Others use graphical reasoning. Others tend to lean purely on algebraic manipulations. Others tend to "reason" by analogy. And I'm sure there are other modes of thinking with which I'm too poorly acquainted to name. The result is that it's perfectly reasonable for people to disagree, quite strongly, about what is comprehensible and what is not, particularly people coming from different intellectual traditions. It's a tangential nitpick, but I can't resist: > [...] novella length explanation As an undergrad I wanted to learn differential geometry, so I went to the library to pick a diff geo textbook. Well, I didn't want a difficult one, so I decided to pick the shortest book I could find --- surely that would be the easiest textbook to read? Haha. As far as I'm concerned, the length of Tao's explanation is a strong merit.
wrycoder 5y ago
I'd recommend Physics From Symmetry by Jakob Schwichtenberg Also, (though I've not read it) this appears to be a longer, more detailed text in the same vein: Quantum Theory, Groups and Representations: An Introduction by Peter Woit This one gets to the EM field on page 573 :-/

srl 5y ago

I'm _pretty_ sure that this is referring to gauge symmetry, so that's the term to search for. I dunno of a good article as an introduction. As a grad student, I found Terry Tao's explanation [1] rather helpful, but of course it has a strongly mathematical flavor. [1] https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/ https://terrytao.wordpress.com/2008/09/27/what-is-a-gauge/

floatingatoll 5y ago
Thanks! It sounds like the next step is to look for curvature transformations associated with terms used for each of those three. (If someone knows more, always open to hear that too.) Also, a textbook is revealed on what seems to be this exact topic. Does this seem right? https://physicstoday.scitation.org/doi/pdf/10.1063/PT.3.2421 https://physicstoday.scitation.org/doi/pdf/10.1063/PT.3.2421
- srl 5y ago
  Yeah... I'd be hesitant to buy a whole book. This is one of the topics where some expositions are really bad, some are really good, and some people just can't agree on which are bad and which are good. I've always had best luck just pulling up 8 different articles and reading them in parallel, hoping it eventually sticks. (Which I realize is the least helpful possible reply.) The textbook Peskin&Schroeder is standard in this field. "If" you can find a PDF online, there's a chapter introducing gauge theories (the chapter on yang-mills) that has a reasonably clear explanation near the beginning, I think.
gaugewat 5y ago
Wow... > “Gauge theory” is a term which has connotations of being a fearsomely complicated part of mathematics – for instance, playing an important role in quantum field theory, general relativity, geometric PDE, and so forth. But the underlying concept is really quite simple: Then goes on to write an incomprehensible novella length explanation. I like Tao, but this is a problem in 'pure' maths. I feel the issue is exacerbated by the fact that if you can follow along with Terry then you likely will be unable to recognise the issue. A good counter point is this series by Timothy Gowers developing a proof of "Pingala's Determinant". When Gowers finds the proof he states: > Since this clearly has determinant 1 and this clearly has determinant 1, so does their products, ie. Pascal's[sic] determinant is 1. Could have done that, but that would have been uh, well depends on what you find interesting, but that would have been really sort of rabbit out of the hat, look what a billiant clever mathematician I am. I can just sort of produce this fabulous identity out of nowhere and it's got a nice simple proof that also came out of nowhere. What I want to emphasize is these sort of proofs don't come out of nowhere. Timestamped to quotation: https://youtu.be/m8R9rVb0M5o?t=886 https://youtu.be/m8R9rVb0M5o?t=886 The series is great and will show anyone outside the field of pure maths that even pure mathematicians use numerical reasoning while trying to reason about a subject.
- srl 5y ago
  I think you're underrating heterogeneity. Not all people will agree that Tao's exposition is unclear. I linked it precisely because I found it more clear than other, "more physical" explanations (as a graduate student in physics, not math). When I was trying to understand gauge symmetry, it was that article that finally made things click for me. And, of course, many people I know would disagree, and point to other expositions as superior. That's the point: different people will not agree on what is well-written and what is incomprehensible. You state "even pure mathematicians use numerical reasoning while trying to reason about a subject". Yes, some pure mathematicians do. Others use graphical reasoning. Others tend to lean purely on algebraic manipulations. Others tend to "reason" by analogy. And I'm sure there are other modes of thinking with which I'm too poorly acquainted to name. The result is that it's perfectly reasonable for people to disagree, quite strongly, about what is comprehensible and what is not, particularly people coming from different intellectual traditions. It's a tangential nitpick, but I can't resist: > [...] novella length explanation As an undergrad I wanted to learn differential geometry, so I went to the library to pick a diff geo textbook. Well, I didn't want a difficult one, so I decided to pick the shortest book I could find --- surely that would be the easiest textbook to read? Haha. As far as I'm concerned, the length of Tao's explanation is a strong merit.

wrycoder 5y ago

I'd recommend Physics From Symmetry by Jakob Schwichtenberg Also, (though I've not read it) this appears to be a longer, more detailed text in the same vein: Quantum Theory, Groups and Representations: An Introduction by Peter Woit This one gets to the EM field on page 573 :-/