8 ms·

Why is symmetry so important in particle physics?

121 points by mfn 4y ago

adamnemecek 4y ago
This idea shows up in essentially all scientific fields. It’s the idea of adjointness. Together with norm, they give you the idea of fixed points, (invariants, spectra, embeddings, braids etc).I'm Lawvere's fixed point theorem is I think the best formulation of the idea https://ncatlab.org/nlab/show/Lawvere%27s+fixed+point+theorem https://ncatlab.org/nlab/show/Lawvere%27s+fixed+point+theore... I've been putting together a brain dump on the topic https://github.com/adamnemecek/adjoint/ https://github.com/adamnemecek/adjoint/ Join the discord https://discord.gg/mr9TAhpyBW https://discord.gg/mr9TAhpyBW
- VirusNewbie 4y ago
  Wouldn’t adjointness be distinctly not “symmetry”, given it relies on a forgetful functor?
  - adamnemecek 4y ago
    The beauty is that under certain circumstances you can infer the lost things lost due to the forgetfulness of the forgetful function.
behnamoh 4y ago
I wonder how much of this can be ELI5'ed with diagrams and intuitions instead of algebra.
- jiggawatts 4y ago
  Most of it. Or conversely, a lot of it can be explained without having to resort to category theory or similarly dense terminology. I've found that Mathematicians like to come up with the most general, tersest definitions possible, where every symbol is overloaded with layer upon layer of meaning. You end up with language that looks like: a = b c For some absurd percentage of the statements. Sometimes the equal sign is an arrow, and the various symbols have superscripts and subscripts on them, but all meaning is lost unless you read the text, which then becomes an exponentially expanding set of hyperlinks to definitions that you need to unpack the definitions. This is never useful as a method of pedagogy, yet this is about the only type of content you will ever find online in places like Wikipedia or nLab. Any attempt to clarify with an example is resisted, because it's not "general", or "not the definition" of the concept. In the end, all practicality is erased, leaving definitions so pure that they could be referring to almost anything.
  - paulpauper 4y ago
    Because to expand the whole expression, such as the Einstein field equations, would fills many lines or pages.
    - teaearlgraycold 4y ago
      As someone that also doesn't like the status quo - I would love for there to be a better mathematical notation. If we could get some kind of functional programming-like syntax it would be nice and consistent. Also what's up with the acceptance of single letter variables in math?
      - kirkules 4y ago
        I think maybe you mean what's up with the dearth of multi letter variables, and if so IMO the answer is, satisfying or not, that juxtaposition as an operator is too hard to give up. Edit: juxtaposition without some sort of brackets, of course
      - jasonwatkinspdx 4y ago
        Interesting bit of history that's relatively unknown: APL was originally an attempt by Iverson to create a uniform and systematic mathematical notation. It was only after he'd designed the notation that it was turned into a programming language. This helps explain APLs quirks, and perhaps the limit of popularity it hit.
      - db48x 4y ago
        Find yourself a copy of The Structural Interpretation of Classical Mechanics. The primary author of this textbook on mechanics was the inventor of Scheme. It uses two notations throughout. The first will be very familiar to mathematicians, but with some changes to disambiguate things like derivatives, while the other notation is Scheme. The Mechanics package for MIT-Scheme adds a whole computer algebra system so that it can compute the derivatives of ordinary Scheme functions, symbolically evaluate them, etc. https://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/book.html https://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/bo...
- irjustin 4y ago
  PBS Spacetime has a very good video[0] that helps visualize what's going on in the equation, recent too! https://www.youtube.com/watch?v=PHiyQID7SBs https://www.youtube.com/watch?v=PHiyQID7SBs
  - nyc111 4y ago
    Great video, thanks. But I don’t understand why they call this thing “beautiful”. To me it looks as ugly as tax law.
- photochemsyn 4y ago
  Feynman's Caltech lectures (Symmetry in Physical Law plus a few others) tackle the problem in this manner (at first) before going into vector analysis and vector algebra. For example with respect to rotation: > "Another example in which the laws are not symmetrical, that we know quite well, is this: a system in rotation at a uniform angular velocity does not give the same apparent laws as one that is not rotating. If we make an experiment and then put everything in a space ship and have the space ship spinning in empty space, all alone at a constant angular velocity, the apparatus will not work the same way because, as we know, things inside the equipment will be thrown to the outside, and so on, by the centrifugal or Coriolis forces, etc. In fact, we can tell that the earth is rotating by using a so-called Foucault pendulum, without looking outside." https://www.feynmanlectures.caltech.edu/I_52.html https://www.feynmanlectures.caltech.edu/I_52.html
nyc111 4y ago
“Let’s say we have some complex field [. . .] Just to keep things as simple as possible, let’s say that the field is only a function of time.” This is not simplifying things. First they assume that spacetime is not space and time separately but a different entity then reduce this entity to time only. Then we are not dealing with spacetime. This is like assuming a cube then reducing the cube to one dimension but calling the line a cube. It’s not.
- mfn 4y ago
  Yeah I wasn't sure how to set things up there - if I kept a single space dimension + a time dimension, then I'd have to explain the negative sign on one of the terms, and probably also talk about the Einstein summation convention to keep things clean. Whereas with a single time dimension, it's not really 'spacetime' as you pointed out. What motivated this post was that I wanted to give a concrete example of what it really means for some symmetry to 'dictate' the structure of a physical theory, but do so in the simplest way possible - i.e. not deal with spinors, gamma matrices, quantum fields - and the rest of the actual machinery of the standard model. The core idea is so profound that I felt like there has to be a way to get a taste of it across in a way that's accessible. Turned out to be a lot harder than I thought - I had to skip quite a few steps in the post to keep it from becoming too long, but I'm hoping the model still conveys the essence of how a symmetry + action principle can 'predict' particles.
  - nyc111 4y ago
    I think what you did is standard. But what I question is, if we can solve our problem without assuming spacetime, why do we need the abstraction called spacetime? Spacetime looks like a historical quirk that physicists feel obligated to carry. For instance, bending of the light experiment is not done in spacetime but in space and time.
    - kryptiskt 4y ago
      But you can't rip them apart in our physical theories, in relativity different observers will have different ideas about space and time, but will agree about certain invariants if you take both space and time into account. Then a few years later, Minkowski formulated special relativity very elegantly using a four dimensional space-time. That view is basic to general relativity, where the foundation is the spacetime metric and the energy-momentum tensor.
teleforce 4y ago
This is an excellent book on the subject by Jakob Schwichtenberg [1]. [1]Physics From Symmetry: http://physicsfromsymmetry.com/ http://physicsfromsymmetry.com/
- mfn 4y ago
  Yes, that's an excellent book, along with his book on QFT. I also can't recommend this course enough, Susskind has done a remarkably good job at making advanced physics concepts accessible: https://theoreticalminimum.com/ https://theoreticalminimum.com/ Also, "Symmetry and the Standard Model: Mathematics and Particle Physics" by Matthew Robinson does a great job of developing the group theory needed before diving into the physics.
ranger207 4y ago
I don't like to be critical, but I've been wanting to understand symmetry in physics for a while, so here's a few points of confusion I have > The principle, then, is that the particles and fields that were used to build up the theory will move in a way that minimizes or maximizes the sum of L over the path taken by the system The next couple of examples only minimize the Lagrangian; are there any systems in this article that maximize it? > a collection of objects and a recipe to build a Lagrangian from those objects, with the movement of those objects determined by a path that minimizes the Lagrangian What in this case are the objects? Just particles? Is mass in the first example (of classical motion) an object? I'm trying to figure out what kind of objects to use to build an equation, or basically, what type (in the programming sense) an object is > This is a surprising fact Why? Are there other theories, maybe from earlier in the development of physics, that used a different approach? > We could use particles as the building blocks, and represent each particle by its position and velocity. However, fields turn out to be a much more useful way of representing the way particles behave. A field ϕ(x,t) is a function that takes a point in spacetime and spits something out for each point. I assume that in this passage "building blocks" is equivalent to "objects" in the last passage? Why are fields more useful? Is there an example of what using particles as objects would look like? In particular, a field looks to me like a function; if you used particles as an object, would you represent a particle as a function using its position and velocity? Would that function have time as a parameter like fields do? Typing that out I can kind of see why you'd use fields > For example, a field could take a position (in spacetime) and spit out a number (which could be real or complex) What does the output represent? Anything in particular? If not, then it seems like you could define the field function to be anything since the output doesn't represent anything, then when you feed the field function into the Lagrangian eventually you'd get massively different results > The simplest Lagrangian we can write for this field is: L=δtϕ⋆δtϕ How is the Lagrangian constructed? This "simplest" Lagrangian is the derivative of the field with respect to time, along with the derivative of the field's complex conjugate with respect to time, but how'd you know to do that? What makes this the simplest possible Lagrangian? Calling this the "simplest Lagrangian" hints that there are other equally valid ways to create a Lagrangian; is that correct? What are the rules for that? Why would you make a more complex Lagrangian? > One interesting observation about Lagrangians is that any term of the form V(ϕ) represents potential energy. What is V(ϕ)? My initial assumption would be velocity, but how do you take velocity of a field? Actually, I can see what they're doing: velocity of a particle is the derivative of it's position with respect to time, so I guess V(ϕ) aka velocity of the field is the derivative of the field with respect to time. That could've stood to have been spelled out > Since there are no time or space derivatives involved Ok I guess V(ϕ) doesn't represent the velocity of the field. I've got no clue what it is > Now let’s take this a step further. In the previous example, we rotated the field at all points by some angle. But why do we need to rotate the field the same way everywhere? What if we measure things at one point with one coordinate system, but measure them at a different point using a different coordinate system that’s rotated. Although it’s hard to imagine why anyone would want to do this, one would still expect that this shouldn’t affect the actual physics predicted by the theory... Multiplying these together, we see that the Lagrangian is different. This is not what we wanted - rotating the complex plane in has affected the results of our theory. I have no idea what's going on here. Why would you measure different points of the field with different coordinate systems and expect sensical results? I'm imagining a surveyor walking in a line starting from the origin: he takes a measurement at the origin, then at (1,0), then at (2,0), then at (3,0), etc. (Imagine that the underlying field is frozen in time so we aren't dealing with the Lagrangian yet.) Since we know the field equations we can predict what he'll measure at each of those points in the line. But if the coordinate system changes with every step, he's still moving in a straight line as seen from a bird flying overhead, but at his first step he's at (1,0), then at the next step (2,0) turns into (5,1), then at the next step (6,1) (aka (3,0)) turns into (12,-3), etc, because the coordinate system changes each step. It's still (1,0),(2,0),(3,0) if you measure in the original coordinate system. But the underlying field wouldn't change in that case. Sure, if you put (5,1) into the field equation you'll get a different result than if you put in (2,0), but if you're only changing the coordinate system then that has to be compensated for in the field equation itself and you're not going to get different results for the same physical point. I mean, you should get the same result if you do f((2,0), coordinate system a) as if you did f((5,1), coordinate system b) Edit: I think the core of my confusion is that in order for f((2,0), coordinate system a) to equal f((5,1), coordinate system b) then you need knowledge of how the coordinate system changes, and I don't see how that gets incorporated into the function > Note that the issue here is that when we take the derivative of the field, we get an extra iϕδθδt term proportional to the derivative Proportional to the derivative of what? > This property - that a theory is not affected by changing some symmetry parameter throughout spacetime - is called gauge invariance Why is it called that? I assume someone chose that name because it made sense to them for good reasons > Also note that this new term, iAϕ, looks like a potential from the perspective of our field, with V(ϕ)=iAϕ There's V(ϕ) again. I still don't know what it represents > our theory now predicts some type of force involved in the interaction between our field and this new field Wait, "our field" and "the new field"? What fields are those? We were talking about a field defined by the function ϕ(x,t) and thinking about its Lagrangian. We added a term A to the Lagrangian and that was it. What's the "new field"? Why does ϕ(x,t) have an interaction with it? Is A the new field? > This mechanism of introducing an additional field to make an existing theory gauge invariant is exactly what gives rise to photons in the Standard Model! ‘Rotations’ of the electron field correspond to an additional field, called a gauge field, that behaves exactly the way photons do. Ok, I guess A is a new field. I can see how it arises, but I'm not sure how you actually get to it > This derivative doesn’t really make sense if we aren’t using the same coordinate system everywhere you don't say > The way we measure our field at x is different than the way we measure it at x+δx, so to get the actual difference, we need to make the field comparable by fixing it up before subtracting it What is "the way we're measuring it"? I think it's, basically, the coordinate system of the surveyor changes each step, so that's a different "way" of measuring it? I still don't see why changing the coordinate system makes new stuff pop out of the equation > As a first step, we can expand it: W(x,x+δx)=1−iδxA(x)+O(δx2) How do you expand it? Are you giving the definition of W(x, x+δx)? How'd you get that? What's O(δx^2)? > A group is a set of elements associated with some operation... What’s important here is that these two sets - the set of rotations, with the operation being composition, and the set of 1×1 complex matrices with the operation being multiplication - have the exact same behavior Where'd the second set come from? Wait, I see, it's just saying that you can say that "multiplying a number by a 1x1 matrix" is the same thing as saying "you can compose a number with a rotation". It's literally the same thing, just said in a less clear manner. Does the new terminology get us anything useful? > U(1) invariance of the electron field gives rise to the photon field. > SU(2) invariance across lepton fields (such as the electron and electron neutrino) leads to W+, W−, and Z bosons. SU(2) has two generators, so there are three gauge bosons. > SU(3) invariance across quark fields leads to eight gluons, since SU(3) has eight generators. What? How do you know how many generators there are? Why does SU(2) have two generators but three gauge bosons? > It’s remarkable how the observation that an equation doesn’t change under some operation, which seems quite trivial, can have deep consequences, dictating the nature of forces and interactions in the theory. Yeah, I think the part I'm not getting is how changing coordinate systems affects the equation. I think I can see that if you insist on doing something ridiculous like this you'd need some math to correct for it and if the new correction functions are fields then it looks like new particles popping out, but I don't see how that doesn't result in an infinite number of new particles. Like, I can add a function f(x) = x^2 to the Lagrangian, then a g(x) = -x^2 to compensate for it, but those don't represent new particles do they? Why do those cancel out but A doesn't? I just don't see how changing coordinate systems results in different results Despite my questions, I think I have a better idea of what's going on. You have a function; it should spit out the same numbers when you rotate it; you need a function to correct for the rotation; in physics the new function looks like a particle. I can kinda sorta see how it works now. Thanks for the article! Edit: Ok, I think I've narrowed down my confusion to the θ(t). I can see that if you want to measure the same (x,y) over time as the coordinate systems change even though that (x,y) represents a different physical point every t, then you'd need to take the change in coordinate system over time into account in the derivative. But I'm not sure how that would be useful, nor how that would result in new physics over the case of a fixed θ
bigbacaloa 4y ago
When we don't have symmetry we don't know what to do and can't compute anything. Fortunately many situations can be modeled as near a symmetric one. We solve the symmetric one and study it's asymmetric perturbations. The prevalence of symmetry reflects our inability to do anything in its absence.
- psychphysic 4y ago
  That's one way to consider it. The truth is there are many ways to skin a cat and we've found quite effective ways to do it. The real cause of the unreasonable effectiveness of mathematics is humans ability to constantly rephrase tasks within the wheelhouse of our mathematical tools. Eventually... There's long periods of humans being stuck and then getting unstuck and history compresses that to appear like constant smooth progression. But you're right, many people have a bizarre view of the world painted by the Schrödinger equation. That the world is made up for snap shots of fixed particle number defined energy states. Really peculiar if you think about it and quite clearly incorrect (we know it doesn't apply to 'collpase' which is really the way the world is experienced by us). And compared to QFT.
moring 4y ago
I find it incredibly frustrating that again and again, the Lagrangian gets introduced and then said it should be minimized without ever explaining the motivation behind doing so. What is the Lagrangian and why should it be minimized? I totally get how L gets defined mathematically, how it is derived from Newton's laws (this part is typically well explained by textbooks), and why in the case of point particles, a curve that violates Newton's laws does not minimize L. But there is no understanding at all, just saying "okay, it checks out" on a math level. It doesn't help at all that on a math level, L isn't actually minimized but its derivative set to 0, which isn't even equivalent to "minimized or maximized". Why doesn't a single textbook explain why maximizing L is also okay when they first stated "minimized"? Or why derivative=0 is sufficient? As a reader, I always get the impression that "well, of course they cannot explain that, because they don't even know why L should be minimized in the first place". It's just all formulas that are easy to verify but don't convey a single bit of understanding. Just for comparison, I found quantum mechanics based on the Schrödinger equation and the Hamiltonian rather easy to grasp, because every piece of it has an easy-to-understand meaning, that also gets explained really well. Why is this seemingly impossible for the Lagrangian?
- elashri 4y ago
  It is not minimum or maximum when you take derivative L = 0. You get sudden points which could be minimum or maximum or not. In cases of classical Lagrangian with V is the gravitional Field. It would be minimum. If you want to derive optics laws (snell/reflection) you will find that the path it not a minimum path. Actually you will find that the light will take the path that minimize the time it takes from point a to b. For more complicated theorie. You always tty to start from one point (usually symmetric or equilibrium) and try to build your theory's Lagrangian. Usually this involve some inputs from experiments (i.e. Standard Model Lagrangian). If we are Lucky enough then the true theory wouldn't be too far.
- yccs27 4y ago
  Starting from quantum mechanics, the Lagrangian describes the phase change per unit of time. For most evolutions of a system, similar evolutions have very different phases and cancel out. But when the Lagrangian is stationary (derivative 0), they interfere constructively. The stationary point is often a minimum, but it could also be a maximum or saddle point.
psychoslave 4y ago
Let me guess, because symmetry is far more easier to handle from a human cognitive point of view?
- Koshkin 4y ago
  Symmetry is “handled” by Group Theory which is rather involved.
misja111 4y ago
After reading to half way into the article, there is this disappointing text: > It turns out that there are a range of particles in nature that exhibit this kind of symmetry. There isn’t any easy way to argue why this kind of symmetry must exist, but it does. And then the article continues to show some implications and predictions that follow when one assume that this symmetry must exist. But I opened the article expecting to find the why ..
xeonmc 4y ago
When being "here" and "there" is exactly the same, you move spontaneously. When being "here" and "there" is almost-but-not-quite the same, you move easily. The degree of "not-the-same-ness" is called the Lagrangian. In other words, symmetry is "fungibility of states". Things happen because the before and after is not very different, the transaction of different-ness is the energy involved. In a classical system it's nigh impossible to encounter an "exactly the same" situation, because there are just too damn many participants to rule out every possible interaction. In a quantum system you encounter "exactly the same" situations frequently because there is only a tiny number of participants interacting.
deleted 4y ago
[deleted]

121 points by mfn 4y ago

This idea shows up in essentially all scientific fields. It’s the idea of adjointness. Together with norm, they give you the idea of fixed points, (invariants, spectra, embeddings, braids etc).I'm Lawvere's fixed point theorem is I think the best formulation of the idea https://ncatlab.org/nlab/show/Lawvere%27s+fixed+point+theorem https://ncatlab.org/nlab/show/Lawvere%27s+fixed+point+theore... I've been putting together a brain dump on the topic https://github.com/adamnemecek/adjoint/ https://github.com/adamnemecek/adjoint/ Join the discord https://discord.gg/mr9TAhpyBW https://discord.gg/mr9TAhpyBW

VirusNewbie 4y ago
Wouldn’t adjointness be distinctly not “symmetry”, given it relies on a forgetful functor?
- adamnemecek 4y ago
  The beauty is that under certain circumstances you can infer the lost things lost due to the forgetfulness of the forgetful function.

behnamoh 4y ago

I wonder how much of this can be ELI5'ed with diagrams and intuitions instead of algebra.

jiggawatts 4y ago
Most of it. Or conversely, a lot of it can be explained without having to resort to category theory or similarly dense terminology. I've found that Mathematicians like to come up with the most general, tersest definitions possible, where every symbol is overloaded with layer upon layer of meaning. You end up with language that looks like: a = b c For some absurd percentage of the statements. Sometimes the equal sign is an arrow, and the various symbols have superscripts and subscripts on them, but all meaning is lost unless you read the text, which then becomes an exponentially expanding set of hyperlinks to definitions that you need to unpack the definitions. This is never useful as a method of pedagogy, yet this is about the only type of content you will ever find online in places like Wikipedia or nLab. Any attempt to clarify with an example is resisted, because it's not "general", or "not the definition" of the concept. In the end, all practicality is erased, leaving definitions so pure that they could be referring to almost anything.
- paulpauper 4y ago
  Because to expand the whole expression, such as the Einstein field equations, would fills many lines or pages.
  - teaearlgraycold 4y ago
    As someone that also doesn't like the status quo - I would love for there to be a better mathematical notation. If we could get some kind of functional programming-like syntax it would be nice and consistent. Also what's up with the acceptance of single letter variables in math?
    - kirkules 4y ago
      I think maybe you mean what's up with the dearth of multi letter variables, and if so IMO the answer is, satisfying or not, that juxtaposition as an operator is too hard to give up. Edit: juxtaposition without some sort of brackets, of course
    - jasonwatkinspdx 4y ago
      Interesting bit of history that's relatively unknown: APL was originally an attempt by Iverson to create a uniform and systematic mathematical notation. It was only after he'd designed the notation that it was turned into a programming language. This helps explain APLs quirks, and perhaps the limit of popularity it hit.
    - db48x 4y ago
      Find yourself a copy of The Structural Interpretation of Classical Mechanics. The primary author of this textbook on mechanics was the inventor of Scheme. It uses two notations throughout. The first will be very familiar to mathematicians, but with some changes to disambiguate things like derivatives, while the other notation is Scheme. The Mechanics package for MIT-Scheme adds a whole computer algebra system so that it can compute the derivatives of ordinary Scheme functions, symbolically evaluate them, etc. https://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/book.html https://groups.csail.mit.edu/mac/users/gjs/6946/sicm-html/bo...
irjustin 4y ago
PBS Spacetime has a very good video[0] that helps visualize what's going on in the equation, recent too! https://www.youtube.com/watch?v=PHiyQID7SBs https://www.youtube.com/watch?v=PHiyQID7SBs
- nyc111 4y ago
  Great video, thanks. But I don’t understand why they call this thing “beautiful”. To me it looks as ugly as tax law.
photochemsyn 4y ago
Feynman's Caltech lectures (Symmetry in Physical Law plus a few others) tackle the problem in this manner (at first) before going into vector analysis and vector algebra. For example with respect to rotation: > "Another example in which the laws are not symmetrical, that we know quite well, is this: a system in rotation at a uniform angular velocity does not give the same apparent laws as one that is not rotating. If we make an experiment and then put everything in a space ship and have the space ship spinning in empty space, all alone at a constant angular velocity, the apparatus will not work the same way because, as we know, things inside the equipment will be thrown to the outside, and so on, by the centrifugal or Coriolis forces, etc. In fact, we can tell that the earth is rotating by using a so-called Foucault pendulum, without looking outside." https://www.feynmanlectures.caltech.edu/I_52.html https://www.feynmanlectures.caltech.edu/I_52.html

nyc111 4y ago

“Let’s say we have some complex field [. . .] Just to keep things as simple as possible, let’s say that the field is only a function of time.” This is not simplifying things. First they assume that spacetime is not space and time separately but a different entity then reduce this entity to time only. Then we are not dealing with spacetime. This is like assuming a cube then reducing the cube to one dimension but calling the line a cube. It’s not.

mfn 4y ago
Yeah I wasn't sure how to set things up there - if I kept a single space dimension + a time dimension, then I'd have to explain the negative sign on one of the terms, and probably also talk about the Einstein summation convention to keep things clean. Whereas with a single time dimension, it's not really 'spacetime' as you pointed out. What motivated this post was that I wanted to give a concrete example of what it really means for some symmetry to 'dictate' the structure of a physical theory, but do so in the simplest way possible - i.e. not deal with spinors, gamma matrices, quantum fields - and the rest of the actual machinery of the standard model. The core idea is so profound that I felt like there has to be a way to get a taste of it across in a way that's accessible. Turned out to be a lot harder than I thought - I had to skip quite a few steps in the post to keep it from becoming too long, but I'm hoping the model still conveys the essence of how a symmetry + action principle can 'predict' particles.
- nyc111 4y ago
  I think what you did is standard. But what I question is, if we can solve our problem without assuming spacetime, why do we need the abstraction called spacetime? Spacetime looks like a historical quirk that physicists feel obligated to carry. For instance, bending of the light experiment is not done in spacetime but in space and time.
  - kryptiskt 4y ago
    But you can't rip them apart in our physical theories, in relativity different observers will have different ideas about space and time, but will agree about certain invariants if you take both space and time into account. Then a few years later, Minkowski formulated special relativity very elegantly using a four dimensional space-time. That view is basic to general relativity, where the foundation is the spacetime metric and the energy-momentum tensor.

teleforce 4y ago

This is an excellent book on the subject by Jakob Schwichtenberg [1]. [1]Physics From Symmetry: http://physicsfromsymmetry.com/ http://physicsfromsymmetry.com/

mfn 4y ago
Yes, that's an excellent book, along with his book on QFT. I also can't recommend this course enough, Susskind has done a remarkably good job at making advanced physics concepts accessible: https://theoreticalminimum.com/ https://theoreticalminimum.com/ Also, "Symmetry and the Standard Model: Mathematics and Particle Physics" by Matthew Robinson does a great job of developing the group theory needed before diving into the physics.

ranger207 4y ago

I don't like to be critical, but I've been wanting to understand symmetry in physics for a while, so here's a few points of confusion I have > The principle, then, is that the particles and fields that were used to build up the theory will move in a way that minimizes or maximizes the sum of L over the path taken by the system The next couple of examples only minimize the Lagrangian; are there any systems in this article that maximize it? > a collection of objects and a recipe to build a Lagrangian from those objects, with the movement of those objects determined by a path that minimizes the Lagrangian What in this case are the objects? Just particles? Is mass in the first example (of classical motion) an object? I'm trying to figure out what kind of objects to use to build an equation, or basically, what type (in the programming sense) an object is > This is a surprising fact Why? Are there other theories, maybe from earlier in the development of physics, that used a different approach? > We could use particles as the building blocks, and represent each particle by its position and velocity. However, fields turn out to be a much more useful way of representing the way particles behave. A field ϕ(x,t) is a function that takes a point in spacetime and spits something out for each point. I assume that in this passage "building blocks" is equivalent to "objects" in the last passage? Why are fields more useful? Is there an example of what using particles as objects would look like? In particular, a field looks to me like a function; if you used particles as an object, would you represent a particle as a function using its position and velocity? Would that function have time as a parameter like fields do? Typing that out I can kind of see why you'd use fields > For example, a field could take a position (in spacetime) and spit out a number (which could be real or complex) What does the output represent? Anything in particular? If not, then it seems like you could define the field function to be anything since the output doesn't represent anything, then when you feed the field function into the Lagrangian eventually you'd get massively different results > The simplest Lagrangian we can write for this field is: L=δtϕ⋆δtϕ How is the Lagrangian constructed? This "simplest" Lagrangian is the derivative of the field with respect to time, along with the derivative of the field's complex conjugate with respect to time, but how'd you know to do that? What makes this the simplest possible Lagrangian? Calling this the "simplest Lagrangian" hints that there are other equally valid ways to create a Lagrangian; is that correct? What are the rules for that? Why would you make a more complex Lagrangian? > One interesting observation about Lagrangians is that any term of the form V(ϕ) represents potential energy. What is V(ϕ)? My initial assumption would be velocity, but how do you take velocity of a field? Actually, I can see what they're doing: velocity of a particle is the derivative of it's position with respect to time, so I guess V(ϕ) aka velocity of the field is the derivative of the field with respect to time. That could've stood to have been spelled out > Since there are no time or space derivatives involved Ok I guess V(ϕ) doesn't represent the velocity of the field. I've got no clue what it is > Now let’s take this a step further. In the previous example, we rotated the field at all points by some angle. But why do we need to rotate the field the same way everywhere? What if we measure things at one point with one coordinate system, but measure them at a different point using a different coordinate system that’s rotated. Although it’s hard to imagine why anyone would want to do this, one would still expect that this shouldn’t affect the actual physics predicted by the theory... Multiplying these together, we see that the Lagrangian is different. This is not what we wanted - rotating the complex plane in has affected the results of our theory. I have no idea what's going on here. Why would you measure different points of the field with different coordinate systems and expect sensical results? I'm imagining a surveyor walking in a line starting from the origin: he takes a measurement at the origin, then at (1,0), then at (2,0), then at (3,0), etc. (Imagine that the underlying field is frozen in time so we aren't dealing with the Lagrangian yet.) Since we know the field equations we can predict what he'll measure at each of those points in the line. But if the coordinate system changes with every step, he's still moving in a straight line as seen from a bird flying overhead, but at his first step he's at (1,0), then at the next step (2,0) turns into (5,1), then at the next step (6,1) (aka (3,0)) turns into (12,-3), etc, because the coordinate system changes each step. It's still (1,0),(2,0),(3,0) if you measure in the original coordinate system. But the underlying field wouldn't change in that case. Sure, if you put (5,1) into the field equation you'll get a different result than if you put in (2,0), but if you're only changing the coordinate system then that has to be compensated for in the field equation itself and you're not going to get different results for the same physical point. I mean, you should get the same result if you do f((2,0), coordinate system a) as if you did f((5,1), coordinate system b) Edit: I think the core of my confusion is that in order for f((2,0), coordinate system a) to equal f((5,1), coordinate system b) then you need knowledge of how the coordinate system changes, and I don't see how that gets incorporated into the function > Note that the issue here is that when we take the derivative of the field, we get an extra iϕδθδt term proportional to the derivative Proportional to the derivative of what? > This property - that a theory is not affected by changing some symmetry parameter throughout spacetime - is called gauge invariance Why is it called that? I assume someone chose that name because it made sense to them for good reasons > Also note that this new term, iAϕ, looks like a potential from the perspective of our field, with V(ϕ)=iAϕ There's V(ϕ) again. I still don't know what it represents > our theory now predicts some type of force involved in the interaction between our field and this new field Wait, "our field" and "the new field"? What fields are those? We were talking about a field defined by the function ϕ(x,t) and thinking about its Lagrangian. We added a term A to the Lagrangian and that was it. What's the "new field"? Why does ϕ(x,t) have an interaction with it? Is A the new field? > This mechanism of introducing an additional field to make an existing theory gauge invariant is exactly what gives rise to photons in the Standard Model! ‘Rotations’ of the electron field correspond to an additional field, called a gauge field, that behaves exactly the way photons do. Ok, I guess A is a new field. I can see how it arises, but I'm not sure how you actually get to it > This derivative doesn’t really make sense if we aren’t using the same coordinate system everywhere you don't say > The way we measure our field at x is different than the way we measure it at x+δx, so to get the actual difference, we need to make the field comparable by fixing it up before subtracting it What is "the way we're measuring it"? I think it's, basically, the coordinate system of the surveyor changes each step, so that's a different "way" of measuring it? I still don't see why changing the coordinate system makes new stuff pop out of the equation > As a first step, we can expand it: W(x,x+δx)=1−iδxA(x)+O(δx2) How do you expand it? Are you giving the definition of W(x, x+δx)? How'd you get that? What's O(δx^2)? > A group is a set of elements associated with some operation... What’s important here is that these two sets - the set of rotations, with the operation being composition, and the set of 1×1 complex matrices with the operation being multiplication - have the exact same behavior Where'd the second set come from? Wait, I see, it's just saying that you can say that "multiplying a number by a 1x1 matrix" is the same thing as saying "you can compose a number with a rotation". It's literally the same thing, just said in a less clear manner. Does the new terminology get us anything useful? > U(1) invariance of the electron field gives rise to the photon field. > SU(2) invariance across lepton fields (such as the electron and electron neutrino) leads to W+, W−, and Z bosons. SU(2) has two generators, so there are three gauge bosons. > SU(3) invariance across quark fields leads to eight gluons, since SU(3) has eight generators. What? How do you know how many generators there are? Why does SU(2) have two generators but three gauge bosons? > It’s remarkable how the observation that an equation doesn’t change under some operation, which seems quite trivial, can have deep consequences, dictating the nature of forces and interactions in the theory. Yeah, I think the part I'm not getting is how changing coordinate systems affects the equation. I think I can see that if you insist on doing something ridiculous like this you'd need some math to correct for it and if the new correction functions are fields then it looks like new particles popping out, but I don't see how that doesn't result in an infinite number of new particles. Like, I can add a function f(x) = x^2 to the Lagrangian, then a g(x) = -x^2 to compensate for it, but those don't represent new particles do they? Why do those cancel out but A doesn't? I just don't see how changing coordinate systems results in different results Despite my questions, I think I have a better idea of what's going on. You have a function; it should spit out the same numbers when you rotate it; you need a function to correct for the rotation; in physics the new function looks like a particle. I can kinda sorta see how it works now. Thanks for the article! Edit: Ok, I think I've narrowed down my confusion to the θ(t). I can see that if you want to measure the same (x,y) over time as the coordinate systems change even though that (x,y) represents a different physical point every t, then you'd need to take the change in coordinate system over time into account in the derivative. But I'm not sure how that would be useful, nor how that would result in new physics over the case of a fixed θ

bigbacaloa 4y ago

When we don't have symmetry we don't know what to do and can't compute anything. Fortunately many situations can be modeled as near a symmetric one. We solve the symmetric one and study it's asymmetric perturbations. The prevalence of symmetry reflects our inability to do anything in its absence.

psychphysic 4y ago
That's one way to consider it. The truth is there are many ways to skin a cat and we've found quite effective ways to do it. The real cause of the unreasonable effectiveness of mathematics is humans ability to constantly rephrase tasks within the wheelhouse of our mathematical tools. Eventually... There's long periods of humans being stuck and then getting unstuck and history compresses that to appear like constant smooth progression. But you're right, many people have a bizarre view of the world painted by the Schrödinger equation. That the world is made up for snap shots of fixed particle number defined energy states. Really peculiar if you think about it and quite clearly incorrect (we know it doesn't apply to 'collpase' which is really the way the world is experienced by us). And compared to QFT.

moring 4y ago

I find it incredibly frustrating that again and again, the Lagrangian gets introduced and then said it should be minimized without ever explaining the motivation behind doing so. What is the Lagrangian and why should it be minimized? I totally get how L gets defined mathematically, how it is derived from Newton's laws (this part is typically well explained by textbooks), and why in the case of point particles, a curve that violates Newton's laws does not minimize L. But there is no understanding at all, just saying "okay, it checks out" on a math level. It doesn't help at all that on a math level, L isn't actually minimized but its derivative set to 0, which isn't even equivalent to "minimized or maximized". Why doesn't a single textbook explain why maximizing L is also okay when they first stated "minimized"? Or why derivative=0 is sufficient? As a reader, I always get the impression that "well, of course they cannot explain that, because they don't even know why L should be minimized in the first place". It's just all formulas that are easy to verify but don't convey a single bit of understanding. Just for comparison, I found quantum mechanics based on the Schrödinger equation and the Hamiltonian rather easy to grasp, because every piece of it has an easy-to-understand meaning, that also gets explained really well. Why is this seemingly impossible for the Lagrangian?

elashri 4y ago
It is not minimum or maximum when you take derivative L = 0. You get sudden points which could be minimum or maximum or not. In cases of classical Lagrangian with V is the gravitional Field. It would be minimum. If you want to derive optics laws (snell/reflection) you will find that the path it not a minimum path. Actually you will find that the light will take the path that minimize the time it takes from point a to b. For more complicated theorie. You always tty to start from one point (usually symmetric or equilibrium) and try to build your theory's Lagrangian. Usually this involve some inputs from experiments (i.e. Standard Model Lagrangian). If we are Lucky enough then the true theory wouldn't be too far.
yccs27 4y ago
Starting from quantum mechanics, the Lagrangian describes the phase change per unit of time. For most evolutions of a system, similar evolutions have very different phases and cancel out. But when the Lagrangian is stationary (derivative 0), they interfere constructively. The stationary point is often a minimum, but it could also be a maximum or saddle point.

psychoslave 4y ago

Let me guess, because symmetry is far more easier to handle from a human cognitive point of view?

Koshkin 4y ago
Symmetry is “handled” by Group Theory which is rather involved.

misja111 4y ago

After reading to half way into the article, there is this disappointing text: > It turns out that there are a range of particles in nature that exhibit this kind of symmetry. There isn’t any easy way to argue why this kind of symmetry must exist, but it does. And then the article continues to show some implications and predictions that follow when one assume that this symmetry must exist. But I opened the article expecting to find the why ..

xeonmc 4y ago

When being "here" and "there" is exactly the same, you move spontaneously. When being "here" and "there" is almost-but-not-quite the same, you move easily. The degree of "not-the-same-ness" is called the Lagrangian. In other words, symmetry is "fungibility of states". Things happen because the before and after is not very different, the transaction of different-ness is the energy involved. In a classical system it's nigh impossible to encounter an "exactly the same" situation, because there are just too damn many participants to rule out every possible interaction. In a quantum system you encounter "exactly the same" situations frequently because there is only a tiny number of participants interacting.

deleted 4y ago

[deleted]