Japanese string theorist Judži Tačikawa (100% racially create clean Czech nationalist Mr Tomio Okamura got over 10% inward Czech elections so, as Luboshi Nakamotl, I excogitation to conduct keep over Nippon amongst my Japanese nationalist political party as a revenge) gave a talk somewhere at IPMU where he mentioned quantum plain theories that don't conduct keep a Lagrangian – meet the final page – patch many courses piece of work obsolete textbooks that pretend otherwise, namely that the Lagrangians are enough. This has attracted some involvement of the TRF readers. The involvement is fun but I am disappointed that something that I consider basic conceptual lore of modern physics is even so so utterly unknown inward the broader community of people who are interested inward physics.
Lagrange. I acknowledge it was my nickname during the introduction summertime military camp preparing in addition to baptizing the soon-to-be freshmen at Faculty of Mathematics in addition to Physics at the Charles University. The camps conduct keep house inward Albeř, a village inward Czech Canada on the Austrian border (not to survive confused amongst Czech Switzerland on the Saxon i.e. East German linguistic communication border LOL).
Five days ago, I discussed some general in addition to maybe surprising insights virtually symmetry inward Nature that theoretical physicists made inward recent decades. In quantum gravity, exact symmetries conduct keep to survive estimate symmetries patch estimate symmetries may survive emergent in addition to their identity to a greater extent than ofttimes than non depends on the signal inward the configuration space. Everyone who hypes his theory of quantum gravity amongst global symmetries is belike an incompetent amateur. Everyone who assumes ane special estimate grouping for his theory of everything, including quantum gravity, is an incompetent amateur, in addition to so on.
It's similar amongst the Lagrangian. Theoretical physicists who are upwards to their labor know that Lagrangians are extremely useful but they only cannot assume that the theories constructed from Lagrangians exhaust the listing of all interesting, relevant, promising, or possible quantum theories of Nature. And fifty-fifty to a greater extent than obviously, they know that the methods in addition to descriptions based on the Lagrangian aren't the exclusively ones they conduct keep to master copy in addition to piece of work – only because they're using unlike methods than the Lagrangian ones all the time.
So everyone who assumes that every quantum theory – it doesn't conduct keep to survive ane amongst gravity – must start amongst a Lagrangian is patently non an goodness who understands the lessons from recent decades inward physics.
OK, what is the Lagrangian \(L\)? It's the density of the activity \(S\) inward time. In other words, it's the integrand inward the integral over fourth dimension that defines the action:\[
south = \int dt\,L
\] In plain theory, nosotros integrate non exclusively over fourth dimension but over the whole spacetime. Equivalently, \(L\) may survive written as an integral over (e.g. three-dimensional) space. So nosotros terminate upwards with\[
south = \int dt\,dx\,dy\,dz\,{\mathcal L}.
\] Here, the integrand \({\mathcal L}\) is known as the Lagrangian density although many of us ofttimes telephone telephone it just the "Lagrangian", too.
This link betwixt the activity in addition to the Lagrangian is inseparable. For this reason, whatsoever approach to a employment based on the activity is precisely the same thing as the approach based on the Lagrangian. The 2 concepts exclusively differ past times having or omitting an integral over fourth dimension (or spacetime)! Physics students tend to piece of work the term "Lagrangian" for these discussions because "action" is reserved for discussions virtually Hollywood activity movies. ;-)
Fine. What is the activity or, equivalently, what is the Lagrangian? The activity \(S\) is something that is extremized when the dynamical equations of displace are satisfied. It way that all the laws of physics that conduct keep the shape of differential equations for positions \(x(t)\) or fields \(\phi(x,y,z,t)\) may survive written only as\[
\delta south = 0.
\] The activity is extremized which way that its variation is zero. Using the differential calculus in addition to a formula for \(S\) as a component of degrees of liberty such as \(x(t)\) or \(\phi(x,y,z,t)\), nosotros tin sack derive the differential, "Euler-Lagrange" equations for the degrees of liberty that are obeyed inward Nature classically. Many key classical theories may survive written using this "principle of to the lowest degree action", some can't. When y'all add together things similar friction or other irreversible terms, it becomes harder or impossible to piece of work the action. But fundamentally, there's no friction inward the laws of physics in addition to the activity ofttimes industrial plant great.
In mechanics, the Lagrangian – the integrand inward the activity – is \(L=T-U\), the difference betwixt the kinetic in addition to potential energy, at to the lowest degree inward the simplest examples. This \(L\) is related to the Hamiltonian \(H\) inward some other description (using the Poisson brackets) past times the Legendre transform. Note that \(H=T+U\) inward the simplest examples. I don't desire to extend this basic give-and-take of actions also far. You're supposed to know at to the lowest degree something virtually the actions, otherwise halt reading because this isn't supposed to survive a text helping tutors amongst troubled schoolkids at a high school.
What happens amongst the Lagrangian in addition to Hamiltonian inward quantum mechanics? The Hamiltonian becomes an operator. Does the similar Lagrangian conk an operator? The respond is No. Instead, the approach that uses the activity inward quantum mechanics is the Feynman meat over histories – or path integral. We straight calculate the complex transition probability amplitudes using the formula\[
{\mathcal A} = \int {\mathcal D}^n\phi_i(x,y,z,t)\,\exp[iS(\phi_j)/\hbar]
\] where some extra factors – the so-called "insertions" – may survive added to the integrand if nosotros calculate some expectation values or other things. This is an integral over the classical histories – the variables \(\phi_j\) nosotros integrate over are classical degrees of liberty – but the final result is interpreted quantum mechanically, as a probability amplitude. The integrand is a hugely oscillating complex number on the unit of measurement circle inward the complex plane, because of that \(\exp(iS/\hbar)\) which is the exponential of a huge imaginary number. This huge number exclusively matters "modulo \(2\pi\)" in addition to this residuum of the partition past times \(2\pi\) is a "random number", ane that is uniformly distributed over the circle.
Because these phases are so random, they mostly average out in ane lawsuit nosotros integrate (i.e. meat many of them). The exclusively places inward the infinite-dimensional configuration infinite parameterized past times \(\phi_i(x,y,z,t)\) where they don't average out inward the classical boundary are places where \(\delta S=0\). Near such places, the signal on the unit of measurement circle is almost constant, so lots of nearby histories back upwards each other in addition to constructive interference makes them important. And that's why quantum mechanics inward Feynman's path integral description reduces to the regulation of to the lowest degree action: the histories unopen to the history amongst the to the lowest degree activity cooperate in addition to constructively interfere, instead of cancelling each other – which is why they plough over a large collective contribution to the transition probabilities! That's why a quantum theory inward the classical boundary predicts that "the probability is unopen to one" that a particle volition displace along a trajectory that is rather unopen to the solution of classical equations.
So y'all should meet that the fates of the Lagrangian in addition to the Hamiltonian inward quantum mechanics are real different. The sometime remains a classical component of classical variables patch the other is an operator. That's why e.g. Cesaruliana's comment that ane only relates the Hamiltonian in addition to Lagrangian past times the Legendre transform is utterly naive. This transform is a concept inward classical physics. In quantum mechanics, it is exclusively useful as long as the quantum mechanical theory is "directly determined" past times a classical theory, i.e. past times its classical limit. That's ofttimes non the case.
As inward so many cases, y'all tin sack meet that lots of the people don't really sympathise quantum mechanics. They are thinking classically in addition to they assume that everything they know remains basically the same inward quantum mechanics. But it doesn't. The quantum revolution has changed many things – qualitative features as good as quantitative predictions – dramatically. The classical theories are exclusively goodness plenty proxies close the classical boundary but many phenomena predicted past times quantum mechanics conduct keep house real far from whatsoever such limit.
Great. So in that location may survive some full general Hamiltonian operator which is a component of some key non-commuting degrees of freedom. But because their commutators may survive real large in addition to inward no way "tiny", they're non similar to whatsoever classical theory where the Hamiltonian is a component of commuting variables. So ane can't lay this quantum mechanical theory amongst whatsoever classical theory which is why he can't piece of work the Legendre transform. That way that he won't survive able to uncovering whatsoever Lagrangian description. That may patently happen. It's mutual sense. An intelligent reader amongst an undergraduate marking inward physics – or someone rightfully considering himself to survive equivalent to that – must sympathise it. This paragraph is meant to survive the total proof that the intelligent reader should survive able to verify in addition to confirm.
It makes no feel to sell the "theories without a Lagrangian" as some questionable heresy or miracle. It's totally obvious why y'all can't rely on the existence of the Lagrangian inward quantum mechanics – because the Lagrangian in addition to activity are classical concepts in addition to they're exclusively usable inward quantum mechanical theories that conduct keep some of import classical theory to build upon.
How does it piece of work inward reality? Do nosotros conduct keep theories without Lagrangians etc.? In quantum plain theory, nosotros may write the Lagrangian for the Standard Model etc. (see the mug above) so nosotros may acquire rather far. But in that location are lots of analogous quantum plain theories that only don't allow y'all a similar straightforward treatment. If they're non weakly coupled inward whatsoever limit, e.g. if it's impossible to laid whatsoever \(g\ll 1\) for the coupling \(g\), your Lagrangian strategy may survive inward trouble. So strongly coupled in addition to emergent theories may neglect to conduct keep a goodness Lagrangian. This may include lots of situations inward condensed affair physics.
But ane subject that has been increasingly obvious since the 1970s is that key theoretical particle physics in addition to condensed affair physics are unopen relatives in addition to some "surprising in addition to of import phenomena" that are discovered inward ane of them are extremely probable to survive relevant inward the other, too. We know that the Mexican chapeau potential is of import to analyze stage transitions inward condensed affair physics in addition to thermodynamics; in addition to it's of import for the Higgs machinery inward particle physics. The renormalization grouping is also of import inward both, in addition to so on in addition to so on.
And indeed, genuine theoretical particle physicists only know that all these full general lessons that could survive considered typical for condensed affair physics are truthful inward key particle physics in addition to quantum plain theory, too. We ofttimes meet lots of quasiparticles. Whether a particle is a "real key particle" or a "quasiparticle" may ofttimes survive a subtle question. The Lagrangian description may ofttimes survive absent.
String theory is total of such descriptions in addition to theories that don't start amongst a classical Lagrangian. The (2,0) superconformal theory inward 5+1 dimensions is a cracking representative of a non-gravitational theory (well, it's a boundary CFT dual for some AdS background of [gravitational] M-theory, too). And M-theory inward special offers us other examples inward which y'all can't rely on the Lagrangian. But I wanted something much to a greater extent than elementary. It doesn't conduct keep precisely the properties I demand – but I retrieve that the reader should survive able to acquire the same lessons from the simpler example, anyway.
Take the two-dimensional Ising model, the scaling boundary of it which is a conformal plain theory. First, is it some "super recent" theory created final summer? No, it's not. The Ising model began amongst Ising's thesis inward 1924. It's been just about for 93 years. Various extra insights were added decades afterwards but most of the genuinely of import ones – fifty-fifty virtually the conformal boundary – conduct keep been known for to a greater extent than than xxx years. It's only no cutting-edge scientific discipline anymore.
OK, does this Ising model conduct keep a Lagrangian? For most people who report the model using the "Zamolodčikov" methods, it's far from obvious. In the "Zamolodčikov" methods, we're really interested inward the operators, their products, expectation values, in addition to what happens when ane operator's location rotates just about some other ane inward the 2D Euclideanized bird where the two-dimensional conformal plain theory lives.
And this Ising model CFT has three primary fields: the identity \(1\) of dimension null – every sane CFT has it; the spin plain whose chiral dimension is \(\Delta=1/16\); in addition to the fermion plain \(\psi\) whose dimension is \(\Delta=1/2\), or some composite fields similar the unloose energy density that are made of it. The details depend on "primary fields nether what" nosotros discuss.
Does this Ising model CFT conduct keep a Lagrangian description? Well, yes. You may write it as a theory of a Majorana (real or Hermitian) fermion \(\psi\) living inward the 2D space. Or y'all may write it as the bosonic \(\phi^4\) model. The Lagrangians are "straightforward" given these full general words. The kinetic term for the bosonic or the fermionic field; decorated past times the appropriate potential inward the bosonic case.
Now, this is curious. We don't conduct keep ane Lagrangian description but 2 Lagrangian descriptions. And they're real different. One of them says that the CFT is a theory of a fermion, some other ane says that it's a theory of a boson. Those facial expression different. Why they can't survive a unmarried answer? Is it a boson, or a fermion? Well, both are as correct. The fermion may survive written schematically as \(\psi=\exp(\phi)\) patch \(\partial\phi=\psi\partial \bar\psi\) etc. These identifications demand to survive clarified in addition to all the renormalization in addition to ordering issues conduct keep to survive discussed in addition to refined.
Imagine that in that location are precisely 2 Lagrangian descriptions for this CFT – which isn't really the case. Why two? Clearly, the police that in that location is "one" has been violated. The number of Lagrangian descriptions seems to survive a random integer. If it tin sack survive 2 instead of one, it tin sack for certain survive zero, right? And indeed, inward other cases, it's zero.
But fifty-fifty inward this instance where some Lagrangian descriptions exist, nosotros may meet how weak in addition to insufficient the Lagrangian description is. I mentioned that a primary plain of the Ising model is the spin plain \(\sigma(z)\) whose dimension is \(\Delta=1/16\). This contestation really way that the plain \(\sigma\) has the units of "mass to the one-sixteenth-th" mightiness or what's the right way to attach "th" from the 2 sources (fraction; in addition to describing word to count). Well, if nosotros conduct keep a non-chiral CFT that contains both left-movers in addition to right-movers, the relevant primary plain is the production \(\sigma\bar\sigma\) of the left-moving in addition to right-moving spin fields. At whatsoever rate, this spin plain \(\sigma\) is as "fundamental" as the plain \(\psi\).
If y'all remove heed it for the kickoff time, you're belike shocked past times the claim that its dimension is \(\Delta=1/16\). Why is it ane over sixteen? Well, it may survive derived using some – non-Lagrangian – methods of two-dimensional plain theory. For physics types, this is a real of import share of quantum plain theory that conceptually goes beyond the "textbook cloth of perturbative QFT" in addition to I recommend y'all e.g. Volume I of Polchinski's "String Theory" for an fantabulous handling of 2D conformal plain theory for physicists. "Less physically" oriented readers may selection "the large yellowish book" past times Di Francesco in addition to other authors, Conformal Field Theory.
The dimension \(\Delta=1/16\) may incidentally survive expressed as\[
\eq{
2\Delta &= \zav{ \frac 12 + \frac 32 + \frac 52 + \dots }-(1+2+3+\dots) =\\
&= \frac{1}{24} + \frac{1}{12} = \frac{1}{8}.
}
\] This is non just a random identity involving random seemingly divergent sums; it's a way to really calculate the number for the kickoff time. Whining directed against the key identities that were used to a higher house won't survive tolerated inward the comments, I conduct keep wasted also much fourth dimension amongst these clueless critics.
Influenza A virus subtype H5N1 funny thing virtually \(\Delta = 1/16\) is that it is non fifty-fifty a multiple of \(1/2\). Because the dimension of \(\psi\) is \(1/2\) in addition to the derivatives add together \(1\), y'all can't build the plain \(\sigma\) of dimension \(1/16\) as a polynomial of the "basic Majorana fermion" in addition to the derivatives at all! So fifty-fifty though the Ising model "is" formally some quantum plain theory of a Majorana fermion which does follow from a Lagrangian, ane of the 3 most important, primary operators inward the theory isn't fifty-fifty a component of the "fundamental degrees of freedom" used inward the Lagrangian (and their derivatives)!
And inward some proper counting, the whole dimension \(\Delta=1/16\) is some sort of a "quantum correction". Classically, fields conduct keep dimensions that are integer or half-integer (the units – the kilogram to the integral or half-integral mightiness – may survive determined past times dimensional analysis involving the Lagrangian, from the noesis that the activity has to survive dimensionless in addition to the derivatives conduct keep the dimension of mass). Quantum mechanically, the truthful dimensions of the corresponding operators may survive unlike – they are corrected past times quantum corrections which are basically proportional to powers of \(\hbar\). The Lagrangian-based dimensional analysis no longer industrial plant so only because fields at nearby or coincident points atomic number 82 to short-distance effects in addition to ultraviolet (UV) divergences in addition to those demand to survive renormalized away etc. The quantum corrections to dimensions are known as "anomalous dimensions" etc. And the Ising model is a simple representative of a theory where the total dimension \(\Delta = 1/16\) – basically a value comparable to ane that can't survive made "very small" – is a quantum correction. The spin plain may survive schematically written as \(\sigma=\exp(\phi/2)\) inward the bosonic description ("square rootage of the fermion \(\psi\)") in addition to naively classically, the exponential of anything should survive dimensionless! But quantum mechanically, \(\sigma\) isn't dimensionless because \(1/16\) isn't zero.
If y'all acquire some basics of the CFT methods from Polchinski's textbook, for example, y'all volition concur that the reasons are obvious. Even to create upwards one's heed the existence of the plain \(\sigma\) in addition to its dimension \(\Delta=1/16\), y'all demand to piece of work clever methods such as the "operator production expansions" (OPEs) or the "state-operator correspondence" based on the conformal mapping of a cylinder to the whole complex plane, in addition to other things. These methods don't piece of work whatsoever Lagrangian – in addition to hence don't depend on the existence of a Lagrangian – at all.
There is a lot to acquire in addition to I don't desire this weblog postal service to compete amongst Polchinski's or Di Francesco's volume – I would conduct keep no adventure in addition to Polchinski wrote his volume after a decade of perfectionist work, anyway, patch I exclusively wanted to pass less than an sixty minutes of recreational plenty Sabbatum piece of work amongst this text today. ;-) But my signal is that the Japanese speaker is inward no way the kickoff individual who suggests that quantum plain theories may piece of work without a Lagrangian. And the full general suggestion is inward no way novel or recent.
Instead, it's something else – a component of the basic lore or conceptual cloth that folks who are interested inward theoretical physics know already as students. Folks just shouldn't acquire a "formal" (not LHC analysis-oriented etc.) theoretical physics PhD without existence aware of the things above. I exclusively learned the 2-dimensional CFT methods "reasonably technically" when I was a graduate educatee but of course of report I was exposed to the full general lessons – bosonization in addition to fermionization (the shocking equivalence betwixt theories of bosons in addition to fermions), the state-operator correspondence, in addition to other "non-naive" phenomena inward theoretical physics – as an undergrad. Similar methods in addition to ideas go inward lots of the derivations, fifty-fifty rather basic ones, in addition to y'all only cannot acquire almost anywhere if y'all rest naive in addition to superficial in addition to y'all imagine that all QFT enquiry is just some standardized handling of some Lagrangian.
It only isn't. The Lagrangians are exclusively goodness for theories which conduct keep a goodness in addition to of import classical boundary – which typically way quantum plain theories that are weakly coupled. And the Lagrangian methods are exclusively goodness for questions that facial expression similar slight generalizations of questions that already existed inward the classical theories. But many quantum mechanical theories in addition to many questions inward them only aren't just some tiny upgrades of their classical counterparts. That's why lots of the non-Lagrangian methods – in addition to fifty-fifty theories admitting no Lagrangian methods – conduct keep been a measure component of the daily piece of work of goodness theorists for decades.
Just similar the evolved condition of symmetries inward Nature, the evolved condition of the Lagrangian is something that a genuine theorist only must know, in addition to those who don't know such things may survive directly seen to survive scammers who are exclusively pretending to survive competent physicists. It's worse than that. These scammers – who know naught virtually modern physics in addition to their mindless "let me write a Lagrangian, analyze it, in addition to hype it" shows that they're just mediocre sycophants who would similar their instructor's [organ] as students – are ofttimes presented as lonely geniuses. They're no geniuses, they're mediocre naive laymen who are hopelessly confined inward the average Joes' if non Janes' grouping think, in addition to every TRF reader amongst IQ to a higher house 110 must conduct keep understood the proof.
And that's the memo.
I started amongst a Japanese physicist. Here's something virtually some other Japanese physicist. Willie Soon – whose background is tied to Indochina – sent me a letter past times Feynman to Kojči Mano who considered himself a nameless human being in addition to congratulated to Feynman. Feynman wrote him that Mano isn't nameless to his married adult woman in addition to shaver in addition to to Feynman, either, in addition to Mano should facial expression for his ain problems (see Mano's publications) that are solvable – fifty-fifty if they facial expression little – in addition to may brand Mano experience successful. Also, Feynman apologized that he picked a employment for Mano instead of letting him uncovering ane past times himself. Well, that's cracking but the excessive modesty is way also normal inward Nippon so Feynman could conduct keep failed to appreciate the personality differences betwixt the Westerners in addition to the Japanese. I retrieve that the Japanese must experience happy inward their ain way when they behaviour inward those excessively small-scale ways so often.
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