[HN Gopher] The deconstructed Standard Model equation
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       The deconstructed Standard Model equation
        
       Author : thorum
       Score  : 19 points
       Date   : 2024-10-05 22:46 UTC (5 days ago)
        
 (HTM) web link (www.symmetrymagazine.org)
 (TXT) w3m dump (www.symmetrymagazine.org)
        
       | colanderman wrote:
       | 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 wrote:
         | I've had the same question for ages. Shouldn't there be an
         | equation in "Schroedinger form" with some relativistic
         | Hamiltonian?
        
           | colanderman wrote:
           | 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 wrote:
           | _> 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.
        
         | DangerBird wrote:
         | 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 wrote:
           | 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.
        
         | jiggawatts wrote:
         | 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!
        
       | qrios wrote:
       | 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
        
       | readthenotes1 wrote:
       | 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!"
        
       | lnauta wrote:
       | 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
        
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