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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 th

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?

d_tr 2y ago
I've had the same question for ages. Shouldn't there be an equation in "Schroedinger form" with some relativistic Hamiltonian?
- colanderman 2y ago
  https://en.wikipedia.org/wiki/Mathematical_formulation_of_the_Standard_Model#Free_fields 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.
- pdonis 2y ago
  > 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.
  - colanderman 2y ago
    For those of us interested in non-relativistic field equations of the Standard Model, what are the right keywords to chase?
    - pdonis 2y ago
      > 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.
      - colanderman 2y ago
        Ah thank you for the clear answer! My Wikipedia excursions have left me missing all of this context.
        pdonis 2y ago
        You're welcome! :-)
DangerBird 2y ago
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.
- colanderman 2y ago
  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.
  - deleted 2y ago
    [deleted]
  - shwouchk 2y ago
    The field equations come from differentiating the lagrangian. See https://physics.stackexchange.com/questions/3005/derivation-of-maxwells-equations-from-field-tensor-lagrangian 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
    - colanderman 2y ago
      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.
      - DangerBird 2y ago
        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.
jiggawatts 2y ago
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!
- colanderman 2y ago
  Thanks for the lead! That definitely sounds like the direction I'm looking in.
- l33tman 2y ago
  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.

d_tr 2y ago

I've had the same question for ages. Shouldn't there be an equation in "Schroedinger form" with some relativistic Hamiltonian?

colanderman 2y ago
https://en.wikipedia.org/wiki/Mathematical_formulation_of_the_Standard_Model#Free_fields 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.
pdonis 2y ago
> 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.
- colanderman 2y ago
  For those of us interested in non-relativistic field equations of the Standard Model, what are the right keywords to chase?
  - pdonis 2y ago
    > 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.
    - colanderman 2y ago
      Ah thank you for the clear answer! My Wikipedia excursions have left me missing all of this context.
      - pdonis 2y ago
        You're welcome! :-)

DangerBird 2y ago

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.

colanderman 2y ago
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.
- deleted 2y ago
  [deleted]
- shwouchk 2y ago
  The field equations come from differentiating the lagrangian. See https://physics.stackexchange.com/questions/3005/derivation-of-maxwells-equations-from-field-tensor-lagrangian 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
  - colanderman 2y ago
    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.
    - DangerBird 2y ago
      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.

jiggawatts 2y ago

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!

colanderman 2y ago
Thanks for the lead! That definitely sounds like the direction I'm looking in.
l33tman 2y ago
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.