*source*of reality. I'm not there yet, but we're getting close. This fall.

Quantum Field Theory combines the Lagrangian formalism of Classical Mechanics--with the elegance of tensor notation from General Relativity--with the power of gauge potentials and gauge transformations from Electromagnetism--with the mysterious operator algebra of Quantum Mechanics. Blend them all smoothly together into a single coherent picture and you get Quantum Field Theory... the language in which the Standard Model of particle physics is written.

I'm writing this entry mainly to give those who have never heard of Quantum Field Theory, or not much about it, an idea of its importance and a small appreciation for what it means to me (and to many physicists). All this past year I took classes in four subjects: Classical Mechanics, Electromagnetism, Quantum Mechanics, and Statistical Mechanics (a more sophisticated version of Thermodynamics). These are the 4 required subjects for getting any advanced degree in physics. All next year I will take Quantum Field Theory which combines them all together in one shot as described above. The only thing I didn't mention is how statistical mechanics fits in. SM is not as fundamentally basic to the QFT language as all the others, but there is an amazing number of dualities and connections between the two: symmetry breaking, renormalization, applying partition functions and thermodynamic potentials to path integration, among others.

In one sweeping blow, QFT connects all the branches of physics together and provides a common language to express everything in. Even the parts of physics which the Standard Model can't describe, such as gravity or dark matter, are still being attempted in the language of QFT, by things like string theory and loop quantum gravity. The hope is that one day these will be subsumed as well. That to me, is exciting!

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## Comments

mike_b"All science is either physics or stamp collecting." -- Rutherford

firmamentspoonlessfirmamentgustavolacerdaIt guess not, since by incorporating relativity, you should get relativistic mechanics.

So is your point that QFT combines the languages and notations of those theories, rather than combining the theories themselves into one big general theory?

spoonlessVery cool. Perhaps you could explain this in more detail. So is QFT a generalization of CM, general relativity and EM?

I'd say it's best called a synthesis (or a generalization) of CM, special relativity, QM, and EM.

In fact, in my description above I should probalby not have said "general relativity" since it really only incorporates special relativity (high speeds, but no gravity). I only said that because I was thinking about notation.

I guess not, since by incorporating relativity, you should get relativistic mechanics.

It is a relativistic mechanics, in particular a relativistic

quantummechanics.So is your point that QFT combines the languages and notations of those theories, rather than combining the theories themselves into one big general theory?

Combining the notations into one language is one of the cool parts about it, however it's a lot more than that. In many ways, it doesn't just synthesize these fields it provides a justification for them.

For instance, classical mechanics can be summed up by saying "everything follows the path of least action". But you're left always wondering

whyit's that particular path and why this mysterious quantity called "action" ends up being minimized. There's no explanation until you get to the deeper level at which point it becomes obvious. In QFT, everything followsallpaths, but they interfere with each other in such a way that paths near the path of least action end up constructively interfering and hence it picks out that path naturally.EM is essentially the study of light. However, you study it as if it's simply a classical wave (as if it's not quantized). In QM, you study wave-particles that are quantized like electrons, but you never really do anything with light unless it's way toward the end of the graduate level treatment. Quantizing the electromagnetic field is called "second quantization" and it opens the doors to all kinds of things like virtual photon exchange and in many ways it justifies how everything in EM works but from a deeper level. In the same step it puts matter (like electrons) on the same footing as energy (like light, the electromagnetic field). By treating them both with the same notation, you get to see better what they are and how they're connected.

gustavolacerdaIn QFT, everything follows all paths, but they interfere with each other in such a way that paths near the path of least action end up constructively interfering and hence it picks out that path naturally.is this because most paths (or the bulk of the wavefunction) are close to the "least action" path?

I guess not, since by incorporating relativity, you should get relativistic mechanics.It is a relativistic mechanics, in particular a relativistic quantum mechanics.

And doesn't relativistic mechanics => non-classical mechanics?

spoonlessI actually don't know what you mean by "action". Is "least action" another way of formulating Newton's laws?

Yes, it's another way of formulating Newton's laws. A lot of people confuse "Classical Mechanics" with "Newtonian Mechanics". They are studying the same physical thing, but classical mechanics came afterwards. It's a different way of looking at things which is a lot more abstract, but it is equivalent to newtonian mechanics in terms of the motion that results.

The most basic object in Classical Mechanics is called the "Lagrangian". Integrating it over a path gives you the action, and minimizing the action is the central axiom of CM. Almost all problems in CM are finding the path which minimizes the action, given some initial set of conditions.

is this because most paths (or the bulk of the wavefunction) are close to the "least action" path?

There are an infinite number of paths, and most of them are far away from the path of least action. However, the ones far away tend to interfere with each other in a quantum way and cancel each other out. Which leaves the ones nearby as the important ones. For large enough distances, it leaves only the single path which is followed classically.

And doesn't relativistic mechanics => non-classical mechanics?

As an adjective, classical usually means "non-quantized". Rarely, you might find a textbook use classical to mean "non-relativistic" but it's usually shunned in order to avoid confusion. Classical mechanics as a subject usually includes some coverage of special relativity, but most of it is done in the non-relativistic (low speeds) limit.

Quantum field theory is definitely

nota classical theory, since it is quantized. However, all quantum theories reduce to classical theories in the large-distance limit. So any quantum theory is a generalization of the corresponding classical theory. The main thing QFT inherits from CM is the Lagrangian notation... it's more general, but set up in the same way.spinemasherBut to be more precise historically, Field Theory was completely modeled after Statistical Mechanics/Thermodynamics. In that sense it is

morefundamental to QFT than any other branch of physics. The likenesses are no mistake but rather deliberate (or stolen if you will though high energy theorists love to try and sweep that part of history under the rug)The thing is that when I look at QFT I do not see a beautifully finely crafted blend of physics. I see a hodge podge of mathematical atrocities and a fly-by-the-seat-of-your-pants-do-anythin

The reason we have to go trotting off into theories like string theory is because QFT does not generalize nicely at all. Also don't be mistake by the notation they use in QFT, it is not strictly speaking properly covariant. Even in the lesser case, while the expressions themselves are Lorentz Invariant as they appear on paper you don't ever actually evaluate them as such. You always deconstruct the whole symmetry and add it all back together in pieces. Every time you see the appearance of measures like d

^{3}por d^{3}xyou can rest assure that what you are writing down is crap for more than one reason! Firstly because it is not Lorentz Invariant but secondly because it is most certainly divergent! That's because the energy term in the denominator is always quadratic. Said more generally, you cannot ever integrate a (linear; though not even necessary here) second order differential equation (the Fourier transformed Green's Function always gives a quadratic denominator) over an infinite range of measure of dimension 3(2) or more, that is if you would like an answer other than the big fat ∞!spoonlessBut to be more precise historically, Field Theory was completely modeled after Statistical Mechanics/Thermodynamics. In that sense it is more fundamental to QFT than any other branch of physics. The likenesses are no mistake but rather deliberate (or stolen if you will though high energy theorists love to try and sweep that part of history under the rug)

Ah, now that is pretty interesting. I had sort of hoped, because it seems to me that the 4 subjects taught in first-year graduate school hold up QFT like 4 pillars of a platform. But I wasn't sure how much statmech really played into it.

I was under the impression that some of it, like renormalization, was done by QFT first and then borrowed later by statmech/(condensed matter) theorists. Apparently, the two are inseparable.

I take offense to you calling Feynman a used car salesman, but I admit that I'll have to actually learn how to use this stuff in order to get a feeling for how smooth or hodge-podge it really is. Either way, I eagerly await plunging into it.

spinemasherBut that's for you to decide. I am curious as to what you conclude as you delve into QFT yourself.

nibotspoonlessbtw, do you know kathy cooksey?

No, should I?

nibotah, actually she's astrophysics. i guess that might be why.

nibotAny idea what kind of physics you might want to do?Quantum optics, perhaps..

I have this idea that I want to stay away from "big physics" for awhile and work on something smaller.. but we'll see.

spoonlessquantum optics is pretty cool, I was pretty interested in that type of stuff too... but I decided to focus on other stuff for the time being.