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It's an assumption. You can skim the details here: https://en.wikipedia.org/wiki/Mathematical_formulation_of_the_Standard_Model#Neutrino_masses https://en.wikip

by ajkjk 5y ago

It's an assumption. You can skim the details here: https://en.wikipedia.org/wiki/Mathematical_formulation_of_the_Standard_Model#Neutrino_masses https://en.wikipedia.org/wiki/Mathematical_formulation_of_th... The standard model in its normal form assumes neutrinos to be massless (and that right-handed neutrinos don't exist). Neutrino oscillation implies that they have mass (because to oscillate, they must have an 'internal clock', meaning that they experience proper time, thus are not massless). In light of the fact that they are not massless, some sort of additional term needs to be added to the standard model, but experiment has not provided a way to distinguish between (or invalidate) the candidate theories... yet. The standard model has two parts -- a general schema for theory building (Lagrangians with terms corresponding to each symmetry field and each coupling between fields, plus symmetry breaking and the Higgs mechanism for conferring mass to (some of) the gauge bosons) -- and a particular set of choices within this schema, which is the specific Lagrangian + choices of constants that predict essentially all of the physics we've seen in particle colliders. As far as I know, massive Neutrinos can be solved 'within' the SM schema, although additional Lagrangian terms are required. What's unclear is whether dark matter, gravity, fermionic masses, etc can be resolved within the same framework. But massive neutrinos don't _necessarily_ challenge it (although if the existing theories are ruled out, they might start to). Everyone is pretty much on board that the exact choice of Lagrangian terms can't completely describe reality, and needing to add terms doesn't mean invalidating the theory.

denton-scratch 5y ago
I will have to find out what "Lagrangian" means. It seems to be important in SM.
- ajkjk 5y ago
  Unfortunately it is very hard to find a lucid explanation of the idea .. anywhere.. that isn't a bunch of 'just so' explanations. The 'short version' is that a Lagrangian is a device for book-keeping the interactions in a physical theory. It consists of terms like aBa, where 'a' is some physical quantity, and (Ba) is the 'cost' to change it. B in general is an operator that can depend on all of the other physical quantities around, possibly including derivatives. So you end up with something like: L = aBa + cDc + eFe + gHg + ... Derivatives of this function with respect to a given variable -- for instance, a particular value `a` or `c` -- tell you the 'physical law' for that field. The simplest example in classical mechanics is L = 1/2mv^2 - V(x), and the derivative rule is dL/dx - d/dt dL/dv = - V_x - m dv/dt = (F - ma) = 0. In general including a term like ... + cD(a)c +... causes the c field to couple to the a field, which is how interactions between different fields (represented by the lower-case variables a, c, etc here) are encoded in field theory. I haven't found out a cogent explanation of why the terms have the form aBa, though, or why this is the 'simplest' way to express the concepts that it encodes, but it seems to be. But it works well in part because of it expresses physical laws in terms of scalar functions (instead of a vector-equality like F=ma, which becomes a complicated differential equation in practice), which somehow means it is more naturally encoding the underlying symmetries of the system (because, I guess, one equation is simpler than three).
  - denton-scratch 5y ago
    I'm grateful for your attempt at explaining! I've read it twice, and I'll read it again. Your explanation relies on calculus; at school, I grokked differentiation, but I got a huge mental block on integration. My maths teacher was very good, he wrote famous maths textbooks, but he had no insight into my difficulties. He couldn't teach me, because he was too clever. That's why I never tackled graduate-level maths. I resent that old maths teacher; he meant well, but being super-clever wasn't his job. His job was to teach me.
    - ajkjk 5y ago
      It's not going to be useful to understand Lagrangians in detail unless you're actually doing the physics yourself -- especially because the whole framework is super problematic, pedagogically (I hope it gets 'refactored' someday into something more intuitive). The important part is: a Lagrangian is a function that encodes the relationship between all of objects in a physical model, in such a way that you can 'add terms' to the function to add new interactions in your model. That much, at least, is quite elegant, and it's kinda miraculous that such a thing exists at all. For what it's worth, integration is really not that tricky. If I had to guess what really deterred you was the obsession with annoying proofs about dividing functions into little rectangles and summing them up and taking their limit. That stuff is totally unnecessary, imo. It's easier to think of integration as the inverse of differentiation. If differentiation tells you how fast a function is changing, then differentiating your position gives your velocity. Integration is the opposite: integrating your velocity over a length of time tells you how the position changes. That's a pretty intuitive idea: if I told you the velocity vector for the whole duration of a road trip, you can probably figure out where I ended up, if I also tell you where I started. That's all integration is.
      - denton-scratch 5y ago
        What annoyed me about integral calculus was that it all seemed to be a bunch of heuristic rules. They showed me how to differentiate "from scratch", as it were, like from first principles; but integration seemed to be working backwards from the results of differentiation (which is roughly what you just said). That is, it didn't look like maths to me. It didn't seem to have the rigour of, say, trig. I expected maths to be deterministic, and I hated having heuristics in there. It was a gut reaction, it happened when I was about 17, and I'm sure that integration isn't heuristic in that way; I wonder if my mental block might not have ocurred if it had been presented to me differently.
raattgift 5y ago
I have a puzzle for you. How would you distinguish, in hackernews-level informal (but still reasonably accurate) prose, flavour-changing oscillation and wave frequency in the sense of E=hf? Doesn't the latter frequency imply an internal clock even for photons? (For fun we could attribute an extremely tiny mass to photons and repeat the question, bearing in mind this necessarily ousts the photon from its (affinely-parametrizable but not proper-time-parametrizable) null worldline.)
- ajkjk 5y ago
  I don't know much about this off the top of my head, but roughly, the idea is that a neutrino we observe with a definite mass is a mixture of the three flavors, and a neutrino with a definite lepton flavor is a mixture of those with different masses. The different masses mean that as a neutrino of a certain flavor travels its different constituent states evolve at different speeds, so what it is measured at have different chances of being measured based on when the measurement takes place. Photons have no effect like this because they are not mixtures of different particles and have no concept of 'flavor'. Massive photons would be the same.

by ajkjk 5y ago

denton-scratch 5y ago

I will have to find out what "Lagrangian" means. It seems to be important in SM.

ajkjk 5y ago
Unfortunately it is very hard to find a lucid explanation of the idea .. anywhere.. that isn't a bunch of 'just so' explanations. The 'short version' is that a Lagrangian is a device for book-keeping the interactions in a physical theory. It consists of terms like aBa, where 'a' is some physical quantity, and (Ba) is the 'cost' to change it. B in general is an operator that can depend on all of the other physical quantities around, possibly including derivatives. So you end up with something like: L = aBa + cDc + eFe + gHg + ... Derivatives of this function with respect to a given variable -- for instance, a particular value `a` or `c` -- tell you the 'physical law' for that field. The simplest example in classical mechanics is L = 1/2mv^2 - V(x), and the derivative rule is dL/dx - d/dt dL/dv = - V_x - m dv/dt = (F - ma) = 0. In general including a term like ... + cD(a)c +... causes the c field to couple to the a field, which is how interactions between different fields (represented by the lower-case variables a, c, etc here) are encoded in field theory. I haven't found out a cogent explanation of why the terms have the form aBa, though, or why this is the 'simplest' way to express the concepts that it encodes, but it seems to be. But it works well in part because of it expresses physical laws in terms of scalar functions (instead of a vector-equality like F=ma, which becomes a complicated differential equation in practice), which somehow means it is more naturally encoding the underlying symmetries of the system (because, I guess, one equation is simpler than three).
- denton-scratch 5y ago
  I'm grateful for your attempt at explaining! I've read it twice, and I'll read it again. Your explanation relies on calculus; at school, I grokked differentiation, but I got a huge mental block on integration. My maths teacher was very good, he wrote famous maths textbooks, but he had no insight into my difficulties. He couldn't teach me, because he was too clever. That's why I never tackled graduate-level maths. I resent that old maths teacher; he meant well, but being super-clever wasn't his job. His job was to teach me.
  - ajkjk 5y ago
    It's not going to be useful to understand Lagrangians in detail unless you're actually doing the physics yourself -- especially because the whole framework is super problematic, pedagogically (I hope it gets 'refactored' someday into something more intuitive). The important part is: a Lagrangian is a function that encodes the relationship between all of objects in a physical model, in such a way that you can 'add terms' to the function to add new interactions in your model. That much, at least, is quite elegant, and it's kinda miraculous that such a thing exists at all. For what it's worth, integration is really not that tricky. If I had to guess what really deterred you was the obsession with annoying proofs about dividing functions into little rectangles and summing them up and taking their limit. That stuff is totally unnecessary, imo. It's easier to think of integration as the inverse of differentiation. If differentiation tells you how fast a function is changing, then differentiating your position gives your velocity. Integration is the opposite: integrating your velocity over a length of time tells you how the position changes. That's a pretty intuitive idea: if I told you the velocity vector for the whole duration of a road trip, you can probably figure out where I ended up, if I also tell you where I started. That's all integration is.
    - denton-scratch 5y ago
      What annoyed me about integral calculus was that it all seemed to be a bunch of heuristic rules. They showed me how to differentiate "from scratch", as it were, like from first principles; but integration seemed to be working backwards from the results of differentiation (which is roughly what you just said). That is, it didn't look like maths to me. It didn't seem to have the rigour of, say, trig. I expected maths to be deterministic, and I hated having heuristics in there. It was a gut reaction, it happened when I was about 17, and I'm sure that integration isn't heuristic in that way; I wonder if my mental block might not have ocurred if it had been presented to me differently.

raattgift 5y ago

I have a puzzle for you. How would you distinguish, in hackernews-level informal (but still reasonably accurate) prose, flavour-changing oscillation and wave frequency in the sense of E=hf? Doesn't the latter frequency imply an internal clock even for photons? (For fun we could attribute an extremely tiny mass to photons and repeat the question, bearing in mind this necessarily ousts the photon from its (affinely-parametrizable but not proper-time-parametrizable) null worldline.)

ajkjk 5y ago
I don't know much about this off the top of my head, but roughly, the idea is that a neutrino we observe with a definite mass is a mixture of the three flavors, and a neutrino with a definite lepton flavor is a mixture of those with different masses. The different masses mean that as a neutrino of a certain flavor travels its different constituent states evolve at different speeds, so what it is measured at have different chances of being measured based on when the measurement takes place. Photons have no effect like this because they are not mixtures of different particles and have no concept of 'flavor'. Massive photons would be the same.