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The Deconstructed Standard Model Equation

symmetrymagazine.org

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Re: The Deconstructed Standard Model Equation

#51

Post by andrepd »

alanbernstein wrote: post #5
andrepd wrote: post #3

I've been banging my head on QFT and the Standard Model for enough time now that I can actually glimpse more or less what those terms actualy mean , as in , I can translate them into words. It's almost like learning a new language! Pretty awesome. Still, despite all this complexity, we still have several minor and major theoretical gaps to fill. Exciting times!

I, as a layman with basic college physics knowledge, and passing knowledge of the standard model, want to understand the meaning of those terms. Do you know how I can go about that? Ideally, with something a little less dense than a textbook.

I'm afraid there really isn't another way around. This stuff is just very difficult. If you want to understand the standard model you need to understand Quantum Field Theory, and to understand Quantum Field Theory you need to understand Quantum Mechanics, and to understand that you need to understand classical hamiltonian and lagrangian mechanics, etc etc, and to understand all that you need lots of mathematical notions, so... There really isn't a fast way but to study the standard texts.

Re: The Deconstructed Standard Model Equation

#52

Post by psi-squared »

nonbel wrote: post #38

"To clean up these redundancies, theorists use virtual particles they call ghosts. This part of the equation describes how matter particles interact with Higgs ghosts, virtual artifacts from the Higgs field. [...] This last part of the equation includes more ghosts. These ones are called Faddeev-Popov ghosts, and they cancel out redundancies that occur in interactions through the weak force." So the second half this…

The two types of "ghosts" here are very different. In both cases, though, they're mathematical artefacts rather than anything "physical", but I'll try to explain them as well as I know. Disclaimer: The most advanced physics I've done was a first course in this stuff, so I might be wrong about some things.

For what they're calling "Higgs ghosts": There are two different ways of describing the electromagnetic and weak forces, and which one is best depends on how much energy the particles you're dealing with have. At really high energies, it makes the most sense to talk about a combined "electroweak force", described in terms of four fields with 2 components each (often called W1, W2, W3 and B), and one 4-component field (the Higgs field).

In contrast, at low energies, it makes more sense to talk about the electromagnetic force, with one 2-component field (y), and the weak force, with three 3-component fields (W+, W-, Z0) and a 1-component field (the Higgs field, again). So, where did the other three components of the Higgs field go? Well, we just rearranged things - if you check, the total number of components stayed the same. There are various names for this rearrangement, and I haven't seen this one before, but I guess they're calling these "missing" components "ghosts".

As for the other type, the Faddeev-Popov ghosts, those are more obviously mathematical artefacts. Normally, you'd start by writing down a "physical" Lagrangian ("physical" here meaning something like "written in terms of actual physical fields").

But it turns out that you can't actually calculate with the physical Lagrangian. So you have to rewrite it in a (mostly) mathematically-equivalent way, which involves extra fields. These fields come along with extra rules which basically say "no state you can actually measure involves the ghost fields in any way". Really, they're just there as a calculational aid and aren't physically "real", and they're called "ghosts" to reflect that.

Hope that's at least vaguely comprehensible, it's difficult to explain this stuff without assuming a lot of background knowledge.

Re: The Deconstructed Standard Model Equation

#53

Post by auntienomen »

psi-squared wrote: post #52
nonbel wrote: post #38

"To clean up these redundancies, theorists use virtual particles they call ghosts. This part of the equation describes how matter particles interact with Higgs ghosts, virtual artifacts from the Higgs field. [...] This last part of the equation includes more ghosts. These ones are called Faddeev-Popov ghosts, and they cancel out redundancies that occur in interactions through the weak force." So the second half this…

The two types of "ghosts" here are very different. In both cases, though, they're mathematical artefacts rather than anything "physical", but I'll try to explain them as well as I know. Disclaimer: The most advanced physics I've done was a first course in this stuff, so I might be wrong about some things. For what they're calling "Higgs ghosts": There are two different ways of describing the electromagnetic and weak…

Mostly right, but one correction:

You actually _can_ calculate with the physical Lagrangian. This is what lattice gauge theory simulations do. But it's inconvenient and difficult in perturbation theory, so physicists use the Fadeev-Popov ghost trick instead. The resulting computations are _entirely_ mathematically equivalent, not "(mostly)".

Re: The Deconstructed Standard Model Equation

#54

Post by jwmerrill »

T0T0R0 wrote: post #20

Given the world we live in, right now, why does mathematics continue to insist on minified expressions? Given the option, most development teams would choose to read and write against verbose source code, rather than scrape obfuscated variables and method signatures out of a minified, transpiled, compressed package. So why do we continue this archaic practice of obscure, inscrutable symbols in mathematics? Cultural i…

A lot of mathematical notation is optimized for hand-written algebra.

One of the main things you "do" with expressions like these is to transform them into equivalent expressions that have some desirable form, or examine special cases that allow simplifications, or examine the result of applying some additional operation.

Compact symbols and single letter variables reduce the amount of writing you need to do for each step of such transformations, but they also make the "structure" of expressions more visually obvious.

Programmers also spend time transforming programs into equivalent programs (aka refactoring), but refactoring isn't quite as important to programmers as algebra is to mathematicians and physicists, so (popular) programming languages don't optimize for it to quite the same degree.

Re: The Deconstructed Standard Model Equation

#55

Post by adrianratnapala »

cygx wrote: post #40
adrianratnapala wrote: post #23

Earlier quoted context omitted.

True, but with nitpick that it is the Lagrangian density so nature minimises its total over a region of 4-space. And with the more important nitpick that this Lagrangian is a quantum operator rather than a number. In some sense nature does try to minimise it even so, but I never got an intuitive grasp on what that sense is.

And with the more important nitpick that this Lagrangian is a quantum operator rather than a number. I don't think that's true when dealing with the formulation in terms of path integrals as we do here...

It's the focus on path integrals that obscures the fact that the Lagrangian is an operator. Except maybe for convenience, there is nothing about such equations that is unique to the path-integral formulation.

The fields, the Langragian, and the scattering matrix are all operators; but of course their matrix elements are numbers. The path-integral formulation is a good -- and I think physically well motivated -- trick for calculating those matrix elements in terms of merely fields and their associated Largrangians.

Re: The Deconstructed Standard Model Equation

#56

Post by namin »

abecedarius wrote: post #8
drauh wrote: post #6

Earlier quoted context omitted.

I have a PhD in physics, though not particle physics. QFT was required, since the methods appear all over in condensed matter physics. Unfortunately, there is no real shortcut to being able to understand those terms except by studying one of the standard texts. And, physics being what it is, you can expect a pretty hard slog because the texts will assume you know first quantization back and forth. Add to that, it's n…

A related question: Feynman's popular book QED claims to explain quantum electrodynamics enough that you could almost do calculations with it, just ridiculously inefficiently. After reading it, I can't: the details left out, I can't easily fill in from the grad-level texts, even with a pretty decent undergrad physics background. Shouldn't it be possible to explain QED to a programmer using a literate program? (Again,…

Apart the character of physical laws and the easy pieces, what's the gateway book to understand feynann more advanced work?

Re: The Deconstructed Standard Model Equation

#57

Post by Confusion »

nonbel wrote: post #38

"To clean up these redundancies, theorists use virtual particles they call ghosts. This part of the equation describes how matter particles interact with Higgs ghosts, virtual artifacts from the Higgs field. [...] This last part of the equation includes more ghosts. These ones are called Faddeev-Popov ghosts, and they cancel out redundancies that occur in interactions through the weak force." So the second half this…

It's the same as noting you can more quickly calculate 19x21 by noting it's equivalent to 20x20 - 1. This ghostly term is the -1

Re: The Deconstructed Standard Model Equation

#58

Post by abecedarius »

namin wrote: post #56
abecedarius wrote: post #8

Earlier quoted context omitted.

A related question: Feynman's popular book QED claims to explain quantum electrodynamics enough that you could almost do calculations with it, just ridiculously inefficiently. After reading it, I can't: the details left out, I can't easily fill in from the grad-level texts, even with a pretty decent undergrad physics background. Shouldn't it be possible to explain QED to a programmer using a literate program? (Again,…

Apart the character of physical laws and the easy pieces, what's the gateway book to understand feynann more advanced work?

I don't really know, because I don't understand the more advanced work like QED. But http://www.feynmanlectures.caltech.edu/I_toc.html is one of my favorite books ever.
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