**...those who don't want to become string theorists...**

If you have 83 spare minutes, here is a July 2013 talk at Princeton's IAS by Edward Witten:

The title was "Feynman diagrams in string theory" and the talk was presented during the "LHC physics" program. Edward Witten wouldn't get to "funny" things like the central charges and critical dimensions but he would get far enough.

The beginning is about the ordinary point-like-particle-based Feynman diagrams. He leads you to think about the (Euclidean signature) propagators in terms of the Schwinger parameters. If you don't know what it is, it is a useful trick (seemingly making things harder but it is an illusion) of rewriting\[

\frac{1}{p^2+m^2} = \int_0^\infty \dd t\, \exp[-t(p^2+m^2)]

\] Each propagator has a real positive Schwinger parameter \(t_i\). The Feynman graph \(\Gamma\) may be thought of as a one-dimensional manifold – well, a singular one because of the vertices. After all, the circle, the infinite line, and the line interval (or their collections) are the only smooth 1-manifolds.

The Schwinger parameters may be interpreted as the proper lengths of the propagators which is why the collection of the Schwinger parameters for all propagators defines the metric on the 1-manifold, up to diffeomorphisms!

Independently of the lengths, there are momenta. The momentum conservation law is imposed at each vertex. The \(\delta\)-function at each vertex may be rewritten as\[Another great popularization from the Institute for Advanced Studies at Princeton:Juan Maldacena explains the ER-EPR correspondence on pages 1, 12, 13 of the new Institute Letter (PDF).

(2\pi)^4 \delta^{(4)}(\sum_i p_i)= \int \dd^4 x\,\exp\zav{ i\sum_i p_i\cdot x }

\] The position-space propagator may be written as an integral over a Schwinger parameter and the momentum as\[

\eq{

G(x,y) &= \int \frac{\dd^4 p}{(2\pi)^4} \frac{ e^{ip\cdot (x-y)} }{ p^2+m^2 }=\\

&= \int_0^\infty \dd t \frac{\dd^4 p}{(2\pi)^4} \exp\zzav{ ip\cdot (x-y) -t(p^2+m^2) }

}

\] Note that the integrand is an exponential of an expression that is at most a quadratic polynomial in the momentum. The Feynman diagram was therefore expressed as an integral over the positions of vertices in the spacetime and the proper lengths of the propagators. The integrand is quadratic and the exponent is bilinear.

But Feynman himself taught us another natural interpretation of the (densitized in \(t\)) propagator \(G(x,y;t)\): it's the path integral for a single particle in "non-relativistic" quantum mechanics expressing the amplitude for the particle to get from one point to another. Normally, the overall time of propagation is fixed; here we integrate over it.

The maths is analogous to the path integral for non-relativistic physics. However, to make it relevant for relativity, \(\vec x\) is now replace by all the spacetime coordinates \(x^\mu\) while the non-relativistic time variable \(t\) from undergraduate quantum mechanics is reinterpreted as a new auxiliary time-like coordinate on the world line (well, it's really a proper time of a sort, it may be shown).

So Witten rewrites the propagators in a new way, as path integrals\[

G(x,y;t) =\\

= \int{\mathcal D}X(t') \exp\zav{ -\int_0^t \dd t' \zav{ \sum_i \zav{ \ddfrac{X^i}{t'} }^2 + m^2 } }

\] over \(X\) variables living on the 1-manifold, the Feynman graph. We are therefore computing a path integral of a 1-dimensional general relativity with a "metric tensor" living on the Feynman graph, the 1-manifold, and some extra fields \(X^\mu(t)\) on it as well. The action for this 1-dimensional theory is\[

I = \int_\Gamma \dd s\sqrt{h}\zav{ h^{-1}\sum_i g_{ij}\ddfrac{X^i}{s}\ddfrac{X^j}{s} + m^2 }.

\] Cool, it is quadratic in the fields \(X^\mu\), i.e. a kind of a general relativity coupled to Klein-Gordon fields in one spacetime (Feynman graph's) dimension. The \(h^{-1}\) is just the inverse of the lower-index metric tensor components; it would become the usual upper-index metric if the dimension were higher. The Einstein-Hilbert action isn't there because \(R_{\alpha\beta\gamma\delta=0}\) identically in one dimension; there is no curvature.

Most people run out of time at the end; on the other hand, it's 1/6 of the talk now and Witten is essentially finished. He plans to use the blackboard more often to get rid of the discrepant factor of six.

We may go to the gauge (of the diffeomorphisms) with \(h=1\) and the integral over the metric gets reduced to the integral over Schwinger parameters. All of this funny one-dimensional general relativity gives you the same results as the other rules to calculate the Feynman diagrams in quantum field theories!

The integrals over the Schwinger parameters have two extreme limits, \(t\to 0\) and \(t\to\infty\). The latter is essential – it governs the infrared (low \(p\)) part of the physics and must be there to reproduce the singularities in the propagators.

However, the \(t\to 0\) extreme part of the integral over the Schwinger parameter isn't essential (it isn't indisputably determined by doable observations). It is linked to the high-energy, short-distance physics and and the right theory may modify it. In fact, it should better modify it because the \(t\to 0\) region of the integral is the source of the ultraviolet divergences in quantum field theories.

But you shouldn't just truncate the \(t\to 0\) part of the integral in an arbitrary, bull-in-china way. You would spoil the locality (essentially because you would declare that there is a minimum proper time – almost like if you do so in the spacetime: too bad) and probably the Lorentz invariance or some unitarity as a consequence. You may only get rid of it in a smart way.

Another subtlety is that spins, masses, and other labels making particle species different and special were all neglected. The Feynman vertices are not just constants; they are tensors of coefficients that depend on the particles' labels. All those bells and whistles are what gives individual quantum field theories their identity and charm.

Now, when quantum field theories were rewritten as general relativity in 1 dimension of the Feynman graph (connected world lines), it is equally natural to consider similar general relativity in 2 dimensions (the world sheet). Well, that's the perturbative approach to string theory. The 1-dimensional manifolds had to be singular (at vertices); the 2-dimensional ones may be completely smooth.

Also, the bells and whistles automatically arise from the internal excitations of the strings (the propagators become tubes and the excitations around the circumference – the closed string in this case – give you different spins and other labels of the particles; smoothened pants-like vertices automatically give you some tensor structure which is realistic but fully determined by the theory).

In the 2D case, the three components \(h_{11},h_{12},h_{22}\) may be locally eliminated by 2 degrees of freedom in the diffeomorphisms and 1 overall (Weyl) scaling of the 2D metric so no infinite-dimensional path integral is left! In the 1D case, you had a remaining integral over a few Schwinger parameters; in string theory, you have a finite-dimensional integral over the shape of the Riemann surfaces (world sheets). In higher dimensions, there would be an infinite-dimensional (path) integral left, a problematic quantized gravity at the world volume itself, and it's no good. The string theory case is the only one that cures the problems coming from a low diagram's dimension and those from a high dimension, too!

The \(t\to 0\) region is eliminated naturally along with the ultraviolet divergences, too. (It's because a too thin torus may be "rotated by 90 degrees" and reinterpreted as a very thick one, reinterpreting UV divergences as "already counted" and "possibly cancelling" IR divergences along the way.)

Quantum field theories don't have to include gravity; string theory does unavoidably describe spacetime gravity as well, essentially due to the state-operator correspondence which exists in string theory but not field theory. The last 20 minutes of Witten's talk tell you something about it.

All things get better and you are invited to watch the whole talk by Witten.

Without viewing in detail, just seeing flat diagrams turn into balloons is something I didn't realize existed so simply.

ReplyDeleteOff-topic: it is from yesterday, but surely deserving a post in TRF ;-) http://xxx.lanl.gov/pdf/1312.5415.pdf

ReplyDeleteMotl, Witten made some pretty strong comments about locality in string theory in the questions section.

ReplyDeleteThey were somewhat vague but I must say that they were almost identical, in spirit, details, and vagueness, to the comments that I always say about locality in string theory. ;-)

ReplyDeleteWow Lumo, thanks for explaining very nice article :-)

ReplyDeleteI have not seen Feynman diagrams considered from this point of view, but I think it is cute!

Will have to reread it and take a bit more time when I'm at home (now at work) ...

And of course watch the talk :-)

Cheers

" perturbative approach" Perturbative techniques exclude emergent symmetries. Rigorous derivation can be empirically irrelevant. Physics suffers a terrible dimensional addend problem "solved" by denying the problem. CFT are inescapably chiral in 2D. 3D and higher have supersymmetry (empirical hornswoggle from mathematical convenience).

ReplyDeleteThat is why you fail to be predictive.

I assume you are not talking about me. I absolutely support what Churchill did--but that has zero to do with libertarian views. A govt can be strong, even warlike and truculent. As long as it allows personal rights and liberties and doesn't butt into peoples lives if those people are not annoying others, then that is libertarian. Nothing I support will destroy liberty. The route the current US govt is going is destroying personal liberty.

ReplyDeleteLubos, thanks a lot for the Maldacena link! Unfortunately, he even put in quotes what I consider the most puzzling feature of quantum mechanics, or the most puzzling thing in all of nature that humans have pondered upon. "decide" Maldacena writes. That's it, and I just won't ever get over this. If I could get over this, I would almost completely stop worrying about quantum mechanics. How does the particle "decide", and how does it know it's time to "decide"? There has to be a mysterious connection, a connection I'm afraid is outside the scope of human understanding.

ReplyDeleteHi Lubos; I have been unable to write down the words for the following letters. D,K,M,T,U,Z. Please ask the 4 year old to enlighten me. Thanks.

ReplyDeleteLOL, it's deštník, kolo, mrkev, tužka, ucho, and zmrzilna – umbrella, bike, carrot, pencil, ear, and icecream.

ReplyDeleteDear Justin, what's wrong with you? I have explained these things about 50 times to you already. Won't waste another pre-Christmas hour with you, sorry.

ReplyDeleteLOL Motl. No need for such hatred. I've never used such language on you. I can at least rest easy knowing I'm a decent person who doesn't hate a pacifist like myself.

ReplyDeleteBy the way, my Uncle has the same views on QM, and in fact is the one who influenced my thinking, and he has a full professorship in chemistry at a major university. So, no I'm not retarded.

As a pacifist, I would like to wish you a Merry Christmas!

Interesting Post, Could you give me some more updates about Auto repair shop

ReplyDeleterochester.

“That's it, and I just won't ever get over this. If I could get over this, I would almost completely stop worrying about quantum mechanics. How does the particle "decide", and how does it know it's time to "decide"? There has to be a mysterious connection, a connection I'm afraid is outside the scope of human understanding.”

ReplyDeleteUnfortunately your question has no answer that you will accept as such. Throughout most of human intellectual history it was thought that a true “explanation” of any phenomenon had to be deterministic and that nothing that could not be explained in this way could really be understood. There were very few exceptions to this way of thinking, probably the most remarkable (I am pleased to say ☺) being the Roman poet Lucretius, who in “De Rerum Natura” presented a standard version of epicurean atomistic picture of the world with what seems to be his only one original contribution: his atoms can “swerve” in a totally unpredictable way. This appears to anticipate quantum mechanics :

http://plato.stanford.edu/entries/lucretius/

So, if something could be found “understandable” to a mere poet 2000 years ago, why isn’t it to someone whose uncle is a professor of chemistry today? Well, unlike Lubos, I am only joking ;-)

There are lots of people, even some Nobel prize winners in physics and not all of them pacifists, who have the same difficulty. However, if they have any sense, they accept that all the scientific evidence is against them, and it appears that “this damned quantum jumping” (as Schroedinger described it) is almost certainly here to stay. So I think a reasonable person has three choices.

1) Accept it and learn to like it. If you do, you will find you will soon “understand it”.

2). Accept it grudgingly, which means you stop calling it “nonsense” (annoying Lubos and others), and resign yourself to the fact that Nature has turned out to be pretty appalling after all. Since following the advice Lubos once gave to someone like you to “ask for asylum in another Nature” does not at present seem realistic, you can still secretly hope that someone will one day come up with a “super-quantum” theory that will vindicate your instinctive preferences. If you have enough invention and courage and don’t mind being called the c word, you could try constructing such a theory yourself , although you should be aware that doing so will almost certainly turn out a waste of time.

3). The last fairly sensible choice (if you really can’t live in a non-deterministic universe) is to decide that science does not describe everything there is. Science deals only with the empirical aspects of existence and all the evidence supports the idea that all the empirical phenomena involve irremovable indeterminism. There is nothing however that prevents one believing in a deterministic but unknowable reality “behind” it. But should not confuse this ultimate reality with any kind of “hidden variable” theory. Unknowable must mean really unknowable, which means not part of science at all. As far as scientific understanding of the world this idea is completely useless but if it gives you a psychological comfort and makes it possible to reconcile with those “damned jumps”, it’s not such a bad choice.

He's already had one-and-a-half Good Cop lectures, one from Gene Day that was overflowing with the milk of human kindness and one half from the boss before he became exasperated. You think another Good Cop talk, from you, will make a difference? I doubt it.

ReplyDeleteUnlike Linus (of "Peanuts" fame), Justin clings to not one but

twosecurity blankets. One is his insistence that his classical intuition is sacrosanct and not to be questioned, the other that he is smart and that it is never his fault if he does not understand something with minimal effort in five minutes. Good luck trying to rip those two blankets out of his fingers.In my humble opinion the importance of understanding quantum mechanics is over-rated. It won't turn you into a wiser human being. Just look at this guy whose mastery of the subtleties of QM and accurate defenses of its verities against determined challengers have earned him the rarely bestowed Seal of Approval from the boss, if my memory does not deceive me.

Yet the poor fellow is also

non compos mentis, describing himself as a psychopathic robot and predicting collective suicide as the only rational (and desirable) course open to humanity. (Don't bother posting critical comments to his page, he deletes them.)Again interesting battle between Lubos and someone who

ReplyDeletebelieves there is some philosophical problem with the way QM works! Actually I

can understand both sides and I do not have any problem with either side. Lubos has already decided that it is a waste

of time to even discuss these questions. I can see his point. He wants to do theoretical physics and explain it to people like us. This is very worthy objective and I appreciate it. On the other hand I can see also why such great scientists as

Schrodinger and (more recently) t’Hooft and others regard(ed) it as something

worthwhile to pursue. There is an endless debate about interpretations of QM on

other blogs and books. I think Weinberg is also not happy with any of the

current interpretations of QM. If Lubos does not want to spend his time on it, it is not our place to force him!!!

I agree that Justin is not a master of logic. His evidence for not being “retarded” amounts to the assertion that he does not hate himself (or perhaps that he does not hate pacifists) and that he has an uncle who is a professor of chemistry and also doesn’t understand quantum mechanics. Since we don’t know whether pacifism and non understanding of quantum mechanics are correlated or whether they are hereditary (not to mention the question whether a “differently-abled” person could become a chemistry professor), it’s impossible to rationally judge the strength of his argument.

ReplyDeleteHerr Doktor Vongher suffers from the same problem as his (presumed) compatriot philosopher Herr Schopenhauer, who also argued for similar ideas although without the benefit of knowledge of QM. Rather than try to go into this myself I will just quote Bertrand Russell’s well known criticism (in “History of Western Philosophy”) which can also be found in the Wikipedia entry for Schoppenhauer):

'He habitually dined well, at a good restaurant; he had many trivial love-affairs, which were sensual but not passionate; he was exceedingly quarrelsome and unusually avaricious. ... It is hard to find in his life evidences of any virtue except kindness to animals ... In all other respects he was completely selfish. It is difficult to believe that a man who was profoundly convinced of the virtue of asceticism and resignation would never have made any attempt to embody his convictions in his practice.'

As long as Herr Vongher remains alive his opinions suffer from a serious lack of credibility.

I agree that a TRF topic about this would be nice. As a student of mathematics moving to physics, let me guess some math useful for field theorists.

ReplyDeleteI think that functional and complex analysis are pretty useful to QM, together with some abstract algebra, group theory and representations (things like spinors). Also, as an additional material - a bonus -, a bit of Clifford algebras would be very nice, since it contains a lot of algebraic structure employed in field theory.

But generally, I guess that differential geometry is very helpful - if not fundamental - to do some research work in field theory and in string theory. Not only because it is the natural setting of GR, but because fiber bundles became central in the study of gauge theories. [ And also, many branches of mathematical physics has been geometrizied in the last decades, like the relation between classical and quantum mechanics and symplectic geometry ].

Finally, let me say that the Cartan formalism of differential geometry is also very useful in field theory, like differential forms, Hodge duality, codifferentials and etc. For instance, Maxwell equations in such formalism are just

dF = 0 and d*F=-*J

where F is the EM 2-form and J the current. ;)

Of course that this is far too incomplete, and a pedagogical problem is that many mathematical books can make you waste a lot of time with excessive rigor useless to a physicist. Good textbooks which overcome this are Nakaraha's and Frankel's. If you have patience with mathematical rigor, you can learn a lot about this tools in Thrring's course of mathematical physics. Another text (with much more material) is Göckeler & Schücker, which I bought last week in Amazon...