What they don't tell you is that if you write out all the indices (mu and nu represent 4 spacetime coordinates, antisymmetric components (f_abc = -f_bac, etc), the kappas and lambdas I don't even remember!), and their contractions [1], this giant equation becomes a few times larger than its current form.
Physicists like to contract and shorten everything, and while it is fun, you need a dictionary of rules and conventions to write out the full form.
Luckily there are many tricks to use in these shorter representations, but one tends to forget the incredible amount of information within them.
If treated like code, the expression looks almost grotesquely bad: hundreds of cryptic single-letter variables all over the place, no decomposition, no comments, everything crammed into a monstrous, all-encompassing "god function".
Imagine someone opening a pull request with that — it would never pass a code review!
Physics is practiced by hand, pen and paper or chalk and blackboard. There is no IDE or auto-completion.
The short hands, abbreviations, and obscure syntax raise the difficulty to enter into the field but simplify its practice a lot. In fact, it would be nightmarishly verbose if you had to use more explicit terms to the point of making it almost not feasible.
Furthermore, the complexity of learning a few shorthands is incomparable with understanding the underlying concepts. Just because you name things in a more verbose manner and used a more explicit type system you wouldn't be any closer to understanding what any of this means. The effort of learning a short-hand notation to express a concept that takes years of advanced math to grasp is negligible in comparison with the speed up it offers day to day.
The notation is not what is stopping you from understanding it.
These field equations I always see presented as the Lagrangian. But I've had trouble locating any presentation of them as field evolution equations (not sure the right term here, but e.g. how Maxwell's equations are typically presented, as partial differential equations with respect to spacetime dimensions). Deriving this form from the Lagrangian seems a daunting and error-prone task. Does anyone know a reference which presents them in this way?
In QFT, the Lagrangian is usually the form that's most useful, as this is what you use to calculate scattering amplitudes for processes. The Feynman rules for scattering processes come from the path integral formulation, which uses the "action", a quantity that's the integral of the Lagrangian.
My context is I'm (slowly) writing a quantum field simulator as a hobby. I've done this before for EM fields only, and am familiar with how to directly apply Maxwell's equations as a simulation. But the Lagrangian I have no clue how to directly utilize in a simulation. Hence my search for field evolution equations.
The field equations come from differentiating the lagrangian. See https://physics.stackexchange.com/questions/3005/derivation-... for example. Youll need to read a bit on calculus of variations. Usually in university this is gently introduced through the lagrangian formalism of classical mechanics
Ah thank you!! I've been searching for a straightforward derivation like that for a while. Maxwell's equations are familiar ground for me, so that is very helpful.
Well so this is for classical fields, where the standard model is a quantum field theory. So it will effectively follow the Schrodinger equation with the corresponding Hamiltonian from this Lagrangian. There are a lot of complications though, so I don't believe you will be able to just plug the Standard Model Hamiltonian in like this. Renormalization complicates things. You can do numerical simulations for certain parts of the Standard Model, specifically quantum chromodynamics (QCD), using lattice QCD, but that's not something I have experience with, so I can't speak to how easy that is to implement.
Lattice QCD codes may have what you’re looking for. Having said that, I went down that rabbit hole a few years back and found only impenetrable numeric code with little explanation of where the formulas came from.
Worse still, practically all such “codes” use shortcuts, simplifications, or outright non-physical spacetimes to reduce the computer power required.
You and I are looking for the same thing, so if you do find a good reference please reply!
Lattice QCD is a rabbit hole (like all other advanced subjects) but it's IMO a somewhat conceptually more accessible rabbit hole than the canonical QFT formalism with the Feynman diagrams and perturbation theory. LQCD is built around a "clean" approach that is at least possible to visualize - you go back to the basics, which is that you need to do a sum over every possible combination of field configurations (as this is what it seems the universe does). This is obviously intractable even for a small array, but the concept is easy and the QCD sector of the SM is very clean in interactions.
The sum is intractable not because it's big but because it can't be sanely approximated by lesser sums, due to the nature of what you're summing - an exponent of i * the integral of the Lagrangian over each spacetime field configuration. This vary likes crazy so if you skip the some parts you might get a completely different result.
The trick to anyhow get anything out of this is to restrict yourself to observables that can be calculated from the ground state, which turn out to be accessible by doing the field integrals over imaginary time instead. This is a trick of calculus, it's not really "non-physical", although it is a pretty crazy method :) This has allowed practitioners to calculate hadron masses for example, it's getting better and better over the decades.
> Shouldn't there be an equation in "Schroedinger form" with some relativistic Hamiltonian?
Writing it this way goes against the basic idea of QFT, which is that, in a relativistic context, quantum systems can no longer be described as "wave functions evolving in time", which is what the Schrodinger/Hamiltonian formulation describes.
> non-relativistic field equations of the Standard Model
If you're using the non-relativistic approximation, most of the Standard Model is irrelevant since you're limited to interaction energies much less than the rest mass of the lightest particle involved. You're basically looking at the low energy regime of QED, or straightforward non-relativistic QM with an appropriate potential in the Hamiltonian and no pretense of even trying to derive things from an underlying QFT model.
https://en.wikipedia.org/wiki/Mathematical_formulation_of_th... I think at least provides the field evolution equations for the free fields. But I can't find the equivalent for the interaction terms. E.g. the path integral formation I can only find Lagrangians for.
The original TeX representation of the formula was written by Thomas D. Gutierrez in 1999 [1]. It was discussed many times on HN, initially in 2016 a day after the post of this article on symmetrymagazine [2].
What they don't tell you is that if you write out all the indices (mu and nu represent 4 spacetime coordinates, antisymmetric components (f_abc = -f_bac, etc), the kappas and lambdas I don't even remember!), and their contractions [1], this giant equation becomes a few times larger than its current form.
Physicists like to contract and shorten everything, and while it is fun, you need a dictionary of rules and conventions to write out the full form.
Luckily there are many tricks to use in these shorter representations, but one tends to forget the incredible amount of information within them.
[1] https://en.m.wikipedia.org/wiki/Einstein_notation
a few times? more like hundreds of pages
Indeed, you could fill a book with it.
I wrote this just before bed, my estimation skill was already asleep.
If treated like code, the expression looks almost grotesquely bad: hundreds of cryptic single-letter variables all over the place, no decomposition, no comments, everything crammed into a monstrous, all-encompassing "god function".
Imagine someone opening a pull request with that — it would never pass a code review!
What you are missing...
Physics is practiced by hand, pen and paper or chalk and blackboard. There is no IDE or auto-completion.
The short hands, abbreviations, and obscure syntax raise the difficulty to enter into the field but simplify its practice a lot. In fact, it would be nightmarishly verbose if you had to use more explicit terms to the point of making it almost not feasible.
Furthermore, the complexity of learning a few shorthands is incomparable with understanding the underlying concepts. Just because you name things in a more verbose manner and used a more explicit type system you wouldn't be any closer to understanding what any of this means. The effort of learning a short-hand notation to express a concept that takes years of advanced math to grasp is negligible in comparison with the speed up it offers day to day.
The notation is not what is stopping you from understanding it.
This is like seeing a function that explains a Deep neural network…
>the expression looks almost grotesquely bad:
"at some point the learning stops and the pain begins"
- S. Rao Kosaraju, Professor Emeritus
These field equations I always see presented as the Lagrangian. But I've had trouble locating any presentation of them as field evolution equations (not sure the right term here, but e.g. how Maxwell's equations are typically presented, as partial differential equations with respect to spacetime dimensions). Deriving this form from the Lagrangian seems a daunting and error-prone task. Does anyone know a reference which presents them in this way?
In QFT, the Lagrangian is usually the form that's most useful, as this is what you use to calculate scattering amplitudes for processes. The Feynman rules for scattering processes come from the path integral formulation, which uses the "action", a quantity that's the integral of the Lagrangian.
My context is I'm (slowly) writing a quantum field simulator as a hobby. I've done this before for EM fields only, and am familiar with how to directly apply Maxwell's equations as a simulation. But the Lagrangian I have no clue how to directly utilize in a simulation. Hence my search for field evolution equations.
The field equations come from differentiating the lagrangian. See https://physics.stackexchange.com/questions/3005/derivation-... for example. Youll need to read a bit on calculus of variations. Usually in university this is gently introduced through the lagrangian formalism of classical mechanics
Ah thank you!! I've been searching for a straightforward derivation like that for a while. Maxwell's equations are familiar ground for me, so that is very helpful.
Well so this is for classical fields, where the standard model is a quantum field theory. So it will effectively follow the Schrodinger equation with the corresponding Hamiltonian from this Lagrangian. There are a lot of complications though, so I don't believe you will be able to just plug the Standard Model Hamiltonian in like this. Renormalization complicates things. You can do numerical simulations for certain parts of the Standard Model, specifically quantum chromodynamics (QCD), using lattice QCD, but that's not something I have experience with, so I can't speak to how easy that is to implement.
Lattice QCD codes may have what you’re looking for. Having said that, I went down that rabbit hole a few years back and found only impenetrable numeric code with little explanation of where the formulas came from.
Worse still, practically all such “codes” use shortcuts, simplifications, or outright non-physical spacetimes to reduce the computer power required.
You and I are looking for the same thing, so if you do find a good reference please reply!
Lattice QCD is a rabbit hole (like all other advanced subjects) but it's IMO a somewhat conceptually more accessible rabbit hole than the canonical QFT formalism with the Feynman diagrams and perturbation theory. LQCD is built around a "clean" approach that is at least possible to visualize - you go back to the basics, which is that you need to do a sum over every possible combination of field configurations (as this is what it seems the universe does). This is obviously intractable even for a small array, but the concept is easy and the QCD sector of the SM is very clean in interactions.
The sum is intractable not because it's big but because it can't be sanely approximated by lesser sums, due to the nature of what you're summing - an exponent of i * the integral of the Lagrangian over each spacetime field configuration. This vary likes crazy so if you skip the some parts you might get a completely different result.
The trick to anyhow get anything out of this is to restrict yourself to observables that can be calculated from the ground state, which turn out to be accessible by doing the field integrals over imaginary time instead. This is a trick of calculus, it's not really "non-physical", although it is a pretty crazy method :) This has allowed practitioners to calculate hadron masses for example, it's getting better and better over the decades.
Thanks for the lead! That definitely sounds like the direction I'm looking in.
I've had the same question for ages. Shouldn't there be an equation in "Schroedinger form" with some relativistic Hamiltonian?
> Shouldn't there be an equation in "Schroedinger form" with some relativistic Hamiltonian?
Writing it this way goes against the basic idea of QFT, which is that, in a relativistic context, quantum systems can no longer be described as "wave functions evolving in time", which is what the Schrodinger/Hamiltonian formulation describes.
For those of us interested in non-relativistic field equations of the Standard Model, what are the right keywords to chase?
> non-relativistic field equations of the Standard Model
If you're using the non-relativistic approximation, most of the Standard Model is irrelevant since you're limited to interaction energies much less than the rest mass of the lightest particle involved. You're basically looking at the low energy regime of QED, or straightforward non-relativistic QM with an appropriate potential in the Hamiltonian and no pretense of even trying to derive things from an underlying QFT model.
Ah thank you for the clear answer! My Wikipedia excursions have left me missing all of this context.
You're welcome! :-)
https://en.wikipedia.org/wiki/Mathematical_formulation_of_th... I think at least provides the field evolution equations for the free fields. But I can't find the equivalent for the interaction terms. E.g. the path integral formation I can only find Lagrangians for.
The original TeX representation of the formula was written by Thomas D. Gutierrez in 1999 [1]. It was discussed many times on HN, initially in 2016 a day after the post of this article on symmetrymagazine [2].
[1] https://www.tdgutierrez.com/
[2] https://news.ycombinator.com/item?id=12182230
Are there (SymPy,) CAS versions of the Standard Model Lagrangians and corollaries? Maybe latex2sympy2 (antlr)?
Mathematical formulation of the Standard Model: https://en.wikipedia.org/wiki/Mathematical_formulation_of_th...
I liked the shade cast on dilettantes in the footnote:
"In Gutierrez’s dissemination of the transcript, he noted a sign error he made somewhere in the equation. Good luck finding it!"
Where's the equality sign? Shouldn't every equation have one?
It starts with "Lagrangian standard model =" or "fancy L sub SM ="
In picture form, the full beast looks like: https://upload.wikimedia.org/wikipedia/commons/b/bc/Formula_...