Dear Lubos, thank you lot for this article. Please excuse my ignorance but could you lot delight briefly explicate why nosotros don't respect quantization of higher dimensional objects (such equally branes) discussed inwards diverse measure references? Shouldn't the rules of QM decide the dynamics of branes too? Or may live it is done somewhere but I accept non been able to respect it.Well, the ground why this "quantization" isn't discussed inwards whatsoever of these texts is that it is non possible. The enquiry is analogous to the question: "Why don't textbooks of zoology characteristic photographs of flight elephants that would resemble the flight eagles?" You know, elephants don't fly. In the same sense, at that topographic point is no "brane theory" that would live fully analogous to "string theory".
You tin run across a difference betwixt the 2 situations: most kids tin figure out that different eagles, elephants don't wing afterward some fourth dimension – when they run across no flight elephants. Jee_Jee has seen that no counterpart of string theory is existence constructed amongst other objects – but he even so believes that this non-existent entity exists. It must accept been omitted because string theorists are stupid or they shroud some muddy hush-hush or something similar that. Why does he believe such a thing instead of making the same straightforward determination equally the kid makes nigh the flight elephants?
The curt enquiry truly encompasses numerous laymen's delusions whose invalidity has been shown inwards my texts for decades – amongst really limited results (the bathroom of stupidity inwards the the world is almost infinite). Jee_Jee seems to assume every unmarried one of these 3 fundamentally flawed basic assumptions:
- Some sort of egalitarianism. In this case, if something tin live "done" amongst strings, it may live done amongst whatsoever other objects such equally branes.
- String theory is a collection of inexpensive operations too lots of other analogous things may live constructed. The myth nigh the cheapness or genericity of string theory is encouraged past times self-described critics of string theory i.e. crackpots such equally Sabine Hossenfelder.
- Classical physics is the starting dot patch quantum mechanical theories are derived – too the quantization is just a formality or a cherry on a pie that is ever possible to add.
The encephalon defect known equally egalitarianism
The get-go 1 is the egalitarianism. Jee_Jee clearly assumes that if some mathematical constructions may live done starting amongst \(D-1=1\) strings too \(D=2\) the world sheets (the strings' history inwards space), it must live possible to practise the same thing amongst two-dimensional membranes too higher-dimensional branes, too. Everything must piece of work "analogously", right?
That is just never the case. When 2 objects \(X,Y\) are different, it is ever wrong to assume that things that may live done amongst \(X\) may live done amongst \(Y\) or vice versa. It is wrong to assume that \(f(X)=f(Y)\) – that they accept the same properties. When things are different, they also accept different properties too different operations may live applied to \(X\) too \(Y\).
Needless to say, it is mainly a well-known large laid of the stupid people of the introduce era – the leftists – who endure from this misconception. And they seem to assume the egalitarianism nigh everything. The most of import subclass of such assumptions concerns the groups of people. Men must ever accept the same characteristics equally women, races too nations must accept the same characteristics equally others. Well, they just don't. There is no ground why they should too elementary observations demo that indeed, when they're different, they accept different properties, too.
But it's non just the sexes, races, too nations. People brand egalitarian assumptions nigh mathematical structures too fifty-fifty numbers, too. For example, at that topographic point exists a elementary compact Lie grouping \(E_8\) whose dimension is \(248\). Surely, at that topographic point must also be a elementary compact Lie grouping whose dimension is \(842\) or whatsoever other large number, right? Otherwise nosotros would violate the equality of the numbers \(248,842\). Well, allow me order you lot a secret. The numbers \(248\) too \(842\) are non equal. They differ "quantitatively" but they also differ "qualitatively". Lots of complex operations or structures that be amongst \(248\) appearing at a sure as shooting house just don't be amongst the release \(842\) substituted instead.
(A real-world example: at that topographic point are octonions which are isomorphic to \(\RR^8\) equally a vector space. So according to some elementary people, including people who pretend to live theoretical physicists at top Great Britain universities, at that topographic point must also live \(2^k\)-ions for whatsoever integer \(k\), similar 64-ions, right? Well, it's non the case. The octonions are the largest partitioning algebra constructed out of reals inwards a similar way.)
String theory is unique
The 2nd misconception that I listed is truly a special representative of the myth of egalitarianism – but a special representative that is really of import too prominent too that lives its ain life. Laymen typically assume that "string theory" is just a phrase constructed past times adding a random substantive representing an object, "string", to the discussion "theory". Surely nosotros tin supervene upon the "string" amongst whatsoever other analogous "object" too nosotros volition create some other theory that is on par amongst string theory.
Well, nosotros can't. There is alone 1 string theory. You know, sometimes it's existence said that a whole is to a greater extent than than the amount of the pieces. Well, "string theory" is much to a greater extent than than the amount of "string" too "theory". Only when "strings" play the corresponding role inwards the construction of a "theory", you lot destination upwardly amongst 1 of the most precious laid of ideas that are known to the mankind.
The get-go mathematical appear that was identified to arise from string theory was establish past times Gabriele Veneziano precisely 50 years ago. Half a century is a pretty long fourth dimension but it's even so truthful that nosotros don't know whatsoever other "competitor" or "counterpart" of the special mathematical construction where strings play a key role. The construction has the properties it has. If you lot supervene upon 1 slice past times a different one, you lot are almost guaranteed to destination upwardly amongst an inconsistent laid of ideas – unless you're lucky, you lot supervene upon many things properly, too you lot destination upwardly amongst a new, equivalent Definition of string theory.
Mr Tau, a mute gentle wizard from Czech fairy-tales, has a chapeau on his head: \(\hat\tau\). The chapeau is what truly allows all the magic. Germans clearly didn't telephone telephone him "Herr Tau" because "Herr" would signal he was a Nazi. ;-)
Quantum mechanics is primary, classical physics is derived from it (if the quantum mechanical starting dot exists at all)
Finally, Jee_Jee manifestly manipulates amongst the discussion "quantization" equally if it were a straightforward physical care for that is always possible. One starts amongst a classical theory or fifty-fifty a classical motion-picture demo of an object (such equally a string or a brane) too so he just adds hats (like Mr Tau's chapeau on the motion-picture demo above) which brand the theory "quantum mechanical". In his the world view, "quantization" is just a tiresome bureaucratic physical care for that alone requires patience. Unfortunately, that's an thought nigh the human relationship betwixt a classical theory too a quantum mechanical theory that most students believe when they perish through some basic courses.
But this isn't how Nature plant at all.
The "quantization" should live understood equally a heuristic physical care for that attempts to respect a quantum mechanical theory whose "classical limit" is a particular classical theory. For this reason, "quantization" should live understood equally a method to solve an "inverse problem". But inwards reality, the quantum mechanical theory is the generic, full-blown theory amongst \(\hbar \gt 0\) too the classical theory (like a theory of an oscillating classical string) is just the \(\hbar \to 0\) bound of the quantum mechanical theory.
The quantization is possible – a quantum mechanical theory may live establish – if you lot start amongst a classical bound from a elementary plenty class, e.g. amongst the Hamiltonian of the cast \(H=p^2/2m+V(x)\). But the thought that at that topographic point is a one-to-one correspondence betwixt classical theories too quantum mechanical theories is completely wrong.
Some quantum mechanical theories don't accept whatsoever classical limits – the proper quantity written equally \(\hbar\) may live impossible to define and/or the bound may spend upwardly to be (limits sometimes don't exist).
On the reverse too to a greater extent than importantly, for some classical theories, at that topographic point just exists no quantum mechanical theory whose \(\hbar \to 0\) bound would live equivalent to the classical theory. (The non-existence is completely analogous to the non-existence of a "relativistic counterpart" of a randomly picked non-relativistic theory.) It agency that the "addition of the hats" just isn't guaranteed to exist. These obstructions are analogous to Cumrun Vafa's "Swampland" but at a simpler level. Vafa's "Swampland" includes effective quantum champaign theories that can't live completed to a string theory vacuum (that also describes quantum gravity consistently). At a to a greater extent than primitive level, at that topographic point exists a "Classical Swampland" of classical theories that can't live completed to a full-blown quantum mechanical theory.
Again, equally implicitly or explicitly stated inwards all weblog posts nigh the foundations of quantum mechanics, the primary work is that almost all the laymen, beginners, too eternal beginners maintain on thinking classically. Classical physics too classical pictures are even so at the center of their thinking. They sometimes psyche to a judgement or two, write an equation or 2 that are "quantum mechanical" but those are ever assumed to live just some cherries on a pie that truly don't play whatsoever of import role too that may alone modify some details.
All the of import questions, e.g. all the qualitative ones, are even so existence answered past times classical physics past times these folks. But this is just completely wrong. Everything inwards Nature is quantum mechanical too when classical physics is plenty to reply some qualitative question, it's just an representative of a coincidence or expert luck. If your thinking isn't fundamentally quantum mechanical, it is just wrong according to modern physics.
Some mathematics: why strings are unique
Fine. What goes wrong when nosotros assay to replicate the basic constructions from rudimentary perturbative string theory amongst branes substituted instead of strings?
Early chapters of string theory textbooks hash out the basic dynamics of the string the world sheet, the two-dimensional submanifold of the spacetime that describes the history of a one-dimensional string. The coordinates along the the world canvas are frequently called \(\sigma,\tau\) – the get-go 1 is a spatial coordinate along the string too the 2nd 1 is a time, or closer to a time.
At each dot \((\sigma,\tau)\) of the two-dimensional the world sheet, nosotros accept \(X^\mu(\sigma,\tau)\), the numbers remembering the seat of the corresponding dot of the the world canvas (a dot on the string at some minute of time) inwards the spacetime. The index \(\mu\) is 1 arising from the spacetime vectors. We ultimately desire the spacetime to accept some metric tensor\[
g_{\mu\nu}(X^\kappa)
\] at each spacetime point. When this metric tensor champaign inwards the spacetime is known, it is possible to calculate the induced metric on the the world canvas – nosotros switch to \(\tau,\sigma\to \sigma^\alpha,\alpha=0,1\):\[
h_{\alpha\beta} = \partial_\alpha X^\mu \partial_\beta X^\nu g_{\mu\nu}(\sigma^\gamma).
\] My primary dot is that at that topographic point is some metric tensor on the the world canvas that is inherited from the spacetime. So the theory on the the world canvas is unavoidably a theory of gravity. When nosotros search for a quantum mechanical version of the theory, it is a theory of quantum gravity. And quantum gravity is difficult – too when done inwards the straightforward way, it is inconsistent.
Why? Because nosotros larn all the divergences inwards the quantum multi-loop Feynman diagrams for the scattering of the gravitons, the physical excitations of the metric tensor. The alone instance when these inconsistencies are avoided is when at that topographic point are truly no physical excitations of the metric tensor at all.
Now, the the world canvas has \(D=2\) dimensions. The metric tensor is a symmetric 1 too it has \[
\frac{D(D+1)}{2}
\] components. However, the theory also has a diffeomorphism symmetry. The coordinates \(\sigma^\alpha\) may live replaced amongst novel ones – past times picking \(D\) functions of the coordinates \(\sigma^\alpha\). This liberty (a estimate symmetry of a general-relativity-like theory, directly used for the the world canvas or the the world volume) may live used to eliminate \(D\) components out of \(D(D+1)/2\) components of the metric tensor. So the release of (off-shell) components of the metric tensor that are left is\[
\frac{D(D+1)}{2} - D = \frac{D(D-1)}{2}.
\] It is just similar the release of components of an antisymmetric tensor. You tin cheque that for \(D\geq 2\), the release higher upwardly is even so positive so at that topographic point would even so live some physical modes of the gravitational field, fifty-fifty afterward the gauge-fixing (a special selection of the coordinates \(\sigma^\alpha\)). So no consistent theory would exist. All such theories would live on par amongst a naively quantized Einsteinian gravity which is non-renormalizable.
However, nosotros may respect 1 additional symmetry on the the world canvas or the the world volume, the scaling symmetry whose transformations (Weyl transformations) are determined past times a scalar champaign \(\varphi\) equally a parameter. The metric tensor is just existence multiplied past times \(\exp(\varphi(\sigma^\alpha))\) at each point. If nosotros larn by to rewrite the theory inwards such a way, the release of components of the the world canvas or the the world majority metric tensor that are left undetermined volition be\[
\frac{D(D-1)}{2} - 1 = \frac{(D+1)(D-2)}{2}.
\] And this release of components may truly vanish for a positive \(D\) but alone if \(D=2\). It agency that when \(D=2\), nosotros may gauge-fix the diffeomorphism symmetry on the the world canvas too (locally on the the world sheet) eliminate all the "problematic" components of the metric tensor on the the world sheet. We destination upwardly amongst a theory of a gratuitous string that oscillates nicely, fifty-fifty inwards quantum mechanics – because the quantum mechanical oscillation is equivalent to some Klein-Gordon theories on the the world canvas (if the spacetime is flat).
We may also allow these strings to bring together too separate – too that results inwards the interactions of the resulting particles inwards the spacetime. It is a non-trivial fact that the theory continues to live consistent fifty-fifty afterward the interactions are allowed. The residuum of string theory may live uncovered past times analyzing the strings, their joining too splitting, too strings amongst different boundary conditions. You also respect branes inwards the theory but their properties depend on the "more fundamental" strings – they're non fundamental inwards the same thing equally the strings too you lot can't utilization them equally the starting dot of a construction of the theory.
If you lot tried to practise the same "straightforward derivation" but the \(D=2\) the world sheets were replaced amongst \(D\gt 2\) the world volumes, you lot would destination upwardly amongst a theory on the the world majority that has some components of the metric tensor that cannot live eliminated. It would live a theory of gravity similar to Einstein's full general relativity – but 1 on the the world volume, non inwards the spacetime – too at that topographic point is no known consistent way to "quantize" such a theory. The problems (non-renormalizable loop divergences) is the ground why nosotros considered a theory of extended objects inwards the get-go place.
So it is alone the strings whose dimensionality is high plenty to brand the theory smoother at curt distances inwards the spacetime; but depression plenty to avoid a replication of the spacetime-like problems of full general relativity inwards the the world volume.
Obstructions preventing you lot from finding a quantum mechanical theory amongst a classical limit
In the explanation above, I in conclusion argued that the "brane theory" of fundamental branes, constructed inwards analogy amongst strings, was impossible because of the same problems amongst "divergent multi-loop Feynman diagrams" that spoil the straightforward quantization of Einstein's equations inwards the spacetime. You could accept written "analogous" things amongst a higher release of the world canvas dimensions upwardly to a sure as shooting dot but at some minute of an attempted calculation, the expressions would larn divergent, meaningless, too incurable.
When nosotros destination upwardly amongst a non-renormalizable theory afterward nosotros start amongst a classical champaign theory equally a starting point, nosotros may say that the corresponding quantum mechanical theory doesn't exist. Well, a theory amongst a given classical bound (such equally Fermi's theory of weak interactions or Einstein's equations) may ultimately be but it must comprise "a greater collection of degrees of freedom" than just those assumed inwards the classical theory that nosotros wanted to quantize (e.g. W-bosons, Z-bosons, too the Higgs bosons inwards the instance of the weak interactions; too excited string states inwards the instance of gravity).
Non-renormalizability isn't the alone obstacle that prevents us from "quantizing" sure as shooting classical theories. An fifty-fifty nicer too "more fatal" work that nosotros may frequently human face upwardly are anomalies. They are quantum mechanical corrections to symmetries or conservation laws. Typically, a conservation police describe for a electrical flow \(\partial_\mu J^\mu = 0\) gets some loop Feynman diagrams on the right mitt side. By some choices of the renormalization procedures, such damage on the right mitt side may frequently live laid equal to nix for a single current or a single symmetry, but non for all the symmetries of the classical theory simultaneously.
When the anomalous symmetry is a estimate symmetry, the theory is rendered completely inconsistent because the estimate symmetry has to live on inwards social club to "kill" (well, "decouple") the time-like fashion of the photon or some other estimate boson (whose existence would Pb to predictions of negative probabilities for every other process). Anomalies ordinarily require left-right-asymmetric interactions.
Even the ordinary Standard Model – whose left-right asymmetry is concentrated inwards the sector of the weak interactions – would live anomalous too so inconsistent if you lot omitted all leptons or if you lot omitted all quarks. You powerfulness think that the leptons (electrons etc.) too the quarks (\(u,c,t;d,s,b\)) are existence added equally "matter fields" to the Standard Model inwards the cast of 2 independent packages, the leptons too the quarks, too you lot could add together these packages separately. Classically, this controversy looks obvious. When the Standard Model is consistent amongst both leptons too quarks, it should live consistent if you lot take 1 of them, you lot could think.
But this is non the case. Leptons truly require quarks too vice versa. Only when leptons too quarks are added simultaneously, all the anomalies cancel too the classically expected estimate symmetries may live reproduced at the quantum level.
This is an representative of a concept that is omnipresent inwards quantum champaign theories – including the quantum champaign theories that define the the world canvas dynamics of strings (or, to a greater extent than problematically, the the world majority dynamics of branes). Also, aside from anomalies too other inconsistencies, the quantum mechanical effects may qualitatively modify the deportment of the classical theory that you lot assumed equally an inspiration to create the quantum mechanical theories. An representative of that are the monodromies inwards the Seiberg-Witten \(\NNN=2\) estimate theory inwards 4 dimensions. Without quantum mechanics, you lot wouldn't fifty-fifty think of such monodromies, allow lonely nigh their existence unavoidable.
The anomalies, monodromies, too other effects are proportional to \(\hbar\) and/or its higher power. They are invisible classically because \(\hbar=0\) inwards the classical theory but they are of import inwards the full-blown, quantum mechanical theory too may homecoming many proposed theories inconsistent. Note that if nosotros focus our enquiry on the characteristic properties of a quantum mechanical theory, nosotros frequently laid \(\hbar=1\) so the quantum effects are non "small" (or some zero) inwards whatsoever feel – they're at to the lowest degree equally of import equally everything else. It may live a expert thought for me to give you lot the next explanation why quantum mechanics is capable of rendering many theories inconsistent fifty-fifty though they would live consistent classically. H5N1 outcome inwards the quantum mechanical theory may live computed inwards 2 different ways. The expressions could live \(C+2\hbar\) too \(C-3\hbar\). In the \(\hbar\to 0\) limit, both expressions are equal to \(C\) too everything is fine. But inwards quantum mechanics, \(\hbar\neq 0\) so the 2 expressions are different which is a take inconsistency – an inconsistency that is just invisible inwards the classical limit.
So objects that aren't equal must live assumed to accept different, occasionally qualitatively different characteristics – too they are frequently capable or incapable of different treatments. String theory is a really special too unique mathematical structure. There is alone 1 theory inwards theoretical physics that has this marking of depth too relevance to the explanations of the most fundamental features of the Universe – too it's string theory. The uniqueness is analogous to the uniqueness of \(E_8\) amid the special compact elementary Lie groups or the uniqueness of the monster grouping amid the sporadic groups. The uniqueness of \(E_8\) too the monster grouping amid the isolated groups of the 2 kinds has been rigorously proven. The uniqueness of string theory hasn't been proven quite rigorously but the evidence supporting the uniqueness is extremely potent and, inside "patches" where extra assumptions are made nigh the cast of the theory or description, the restricted proofs are rigorous, too.
And quantum mechanics isn't just a cherry on a pie. It's the primary theory, classical physics is derived from quantum mechanics equally a bound (not the other way around), too if there's no quantum mechanical theory amongst a sure as shooting classical limit, it agency that this classical bound is prohibited inwards Nature. Fundamental, weakly coupled branes are an representative of that ban.