Particles Are Vibrations

The music analogy is much to a greater extent than accurate than most people desire to believe

Tetragraviton is a postdoc at the Perimeter Institute who has written several papers on multiloop amplitudes inwards guess theory. Even though none of these papers depends on string theory inwards whatsoever tangible way, I've thought that he's a guy closed plenty to string theory who could potentially operate on it which is why I was surprised past times his weblog post service a calendar week ago,

Particles Aren’t Vibrations (at Least, Not the Ones You Think)

which indicates that I was wrong. The start judgement tells yous what sort of popularizers are supposed to live a target:

You’ve in all likelihood heard this story before, probable from Brian Greene.

I was imagining that in that location was something subtle. People may dislike the overabundant comments most "music as well as string theory" etc. But I didn't give away anything also subtle inwards the weblog post. While there's ever some room for interpretations what a somewhat vague judgement addressed to the laymen could receive got meant, I recollect it's right to conclude that Tetragraviton is just flatly wrong.




Needless to say, the claim that (in weakly coupled string theory) dissimilar particle species are vibration modes of a string isn't just some fairy-tale used past times Brian Greene. It's a translation of an actual defining fact of string theory into evidently English. Brian Greene has inwards no way a monopoly over such a thing. Pretty much everyone else who has talked most string theory agrees that this is the right summary of string theory's ingenious description of the multifariousness of particle species.




Clearly, yous may add together people similar Michio Kaku:

In string theory, all particles are racket on a tiny prophylactic band; physics is the harmonies on the string; chemical scientific discipline is the melodies nosotros play on vibrating strings; the universe is a symphony of strings, as well as the 'Mind of God' is cosmic music resonating inwards 11-dimensional hyperspace.

Kaku as well as fifty-fifty Greene may sometimes live presented as "just some popularizers". But they receive got done highly nontrivial contributions to the field, too. And almost all other string theorists who speak most string theory role real similar formulations. I could give yous dozens of examples. But because of his widely respected technical credentials, permit me pick Edward Witten:

String theory is an essay at a deeper description of nature past times thinking of an simple particle non as a niggling dot but as a niggling loop of vibrating string. One of the basic things most a string is that it tin vibrate inwards many dissimilar shapes or forms, which gives music its beauty. If nosotros take heed to a tuning fork, it sounds harsh to the human ear. And that's because yous hear a pure tone rather than the higher overtones that yous acquire from a pianoforte or violin that give music its richness as well as beauty.

So inwards the instance of 1 of these strings it tin oscillate inwards many dissimilar forms—analogously to the overtones of a pianoforte string. And those dissimilar forms of vibration are interpreted as dissimilar simple particles: quarks, electrons, photons. All are dissimilar forms of vibration of the same basic string. Unity of the dissimilar forces as well as particles is achieved because they all come upward from dissimilar kinds of racket of the same basic string. In the instance of string theory, alongside our acquaint understanding, in that location would live cipher to a greater extent than basic than the string.

The fact that particle species are types of racket isn't just a truth. It's pretty much "the defining truth", the real argue why string theory is unifying forces as well as matter. If yous allow me to quote Barton Zwiebach's undergraduate textbook, A First Course inwards String Theory:

Why is string theory a really unified theory? The argue is simple as well as goes to the centre of the theory. In string theory, each particle is identified as a particular vibrational trend of an simple microscopic string. Influenza A virus subtype H5N1 musical analogy is real apt. Just as a violin string tin vibrate inwards dissimilar modes as well as each trend corresponds to a dissimilar sound, the modes of vibration of a key string tin live recognized as the dissimilar particles nosotros know. One of the vibrational states of strings is the graviton, the quantum of the gravitational field. Since in that location is just 1 type of string, as well as all particles arise from string vibrations, all particles are naturally incorporated into a unmarried theory. When nosotros recollect inwards string theory of a decay process...

Everyone who understands string theory agrees alongside the essence of the disputation that string theory explains particles as vibrations.

It's ever amazing to come across how many people similar to pick an of import truth, completely negate it, as well as claim that the termination is a real of import truth. It looks similar they desire to essay out Niels Bohr's famous quote

The contrary of a right disputation is a imitation statement. But the contrary of a profound truth may good live some other profound truth.

Well, he entirely says that the contrary of a profound truth may be some other profound truth. It usually isn't.

OK, as well as then how did Tetragraviton fence that particles aren't vibrations?



We were shown the higher harmonics on a string alongside a claim that this is not how string theory produces the listing of particle species. Except that it is a totally valid sketch of how string theory does it.

In a apartment spacetime background, a unmarried string really has possible higher harmonics \(\alpha^\mu_{\pm n}\) along the string – the \(n\)-th Fourier element inwards the expansion of a combination of \(x^{\prime \mu}(\sigma)\) as well as \(p^\mu(\sigma)\) – as well as \(\alpha_n,\alpha_{-n}\) obey the algebra of annihilation as well as creation operators, respectively.

Influenza A virus subtype H5N1 full general excited opened upward string nation is obtained past times the activeness of these harmonics on the basis nation (usually a tachyonic basis state) \(\ket 0\):\[

\dots (\alpha_{-3})^{N_3} (\alpha_{-2})^{N_2} (\alpha_{-1})^{N_1} \ket 0

\] where the exponents \(N_j\) are non-negative integers (only finitely many are nonzero). For each higher harmonic, the string may live excited past times the corresponding vibration – an integer publish of times because the string obeys the laws of quantum mechanics as well as the quantum harmonic oscillator has an as spaced spectrum. Such an excited string behaves as a particle whose volume is proportional to\[

m^2 = m_0^2 (\dots + 3N_3 + 2N_2 + 1N_1 - 1)

\] The to a greater extent than excitations yous include, the heavier particle yous get. The higher harmonics growth the string theory's volume to a greater extent than quickly. The term \(-1\) is a contribution from the zero-point energies of all these oscillators. You may derive this negative shift as a term proportional to the renormalized amount of integers\[

1+2+3+\dots \to -\frac{1}{12}

\] I've replaced \(=\) past times \(\to\) just because I desire to trim the publish of angry clueless critics past times 70% but live sure that \(=\) would live to a greater extent than accurate.

The characteristic scale \(m\) may live closed to the GUT scale if non the Planck scale. But in that location also be low-string-scale models (brane worlds) where \(m\) is comparable to a few \({\rm TeV}\)s, the energies marginally accessible past times the LHC. I was surprised that Tetragraviton didn't receive got a clue most the possibility of a low string scale.

In the formula above, I suppressed the \(\mu\) index as well as then I was entirely adding racket inwards 1 transverse dimension. Influenza A virus subtype H5N1 realistic 10D superstring requires eight copies of such oscillators, all of them may excite the string past times the same amount, as well as in that location may also live similar fermionic oscillators living on the string. Their contributions to the masses are analogous – except that the corresponding operators "mostly anticommute" as well as the employment numbers are hence \(0\) or \(1\).

For closed strings, nosotros receive got 2 sets of oscillators – left-moving as well as right-moving oscillators \(\alpha\) as well as \(\tilde\alpha\). Both of them may live added to excite the string. The total \(m^2\) calculated from the left-movers must handgrip alongside the total \(m^2\) calculated from the right-movers. The requirement is known as the level-matching condition, \(L_0=\tilde L_0\), as well as it is basically equivalent to the disputation that the selection of the \(\sigma=0\) "origin" of a closed string must live unphysical (the total momentum along/around the closed string must vanish).

Note that our formula calculated \(m^2\) as well as non \(m\) as the integer. This is due to a rather simple kinematic technicality that boils downwards to relativity. In relativity, things simplify when the strings are highly boosted or described inwards the "light cone gauge". In that case, the element \(p^-\) of the energy-momentum vector – a light-cone guess edition of "energy" – turns out to contain a term proportional to \(m^2\). (Explanations without the light-cone guess are possible, too.)

You may receive got been afraid that inwards relativity, the liberate energy formula would unavoidably contain lots of foursquare roots from \(E=\sqrt{M^2+P^2}\) which would brand all the oscillators unharmonic. But this trap may live avoided past times a selection of coordinates on the basis sheet. In particular, inwards the light-cone guess (really a conformal guess is enough), the internal liberate energy of the string is linked to \(m^2\) of the corresponding particle as well as the formula for \(m^2\) reduces to simple harmonic oscillators without foursquare roots. None of these things may live clear to anyone "without whatsoever calculations" but the students larn as well as verify the reasons earlier the fifth lecture of string theory. The termination is that fifty-fifty though the string is a relativistic object (the vibration equations are Lorentz-covariant), the relevant Hamiltonians may live written past times simple formulae involving harmonic oscillators as well as no foursquare roots.

So inwards some units, the squared masses \(m^2\) of allowed vibrating strings are literally integers inwards sure units.

The squared masses of known particle species are non as spaced inwards this way. It's mostly because

1. the strings by as well as large vibrate inwards a curved spacetime background
2. the particles – racket of strings – interact alongside each other (because strings split as well as join) as well as this has a similar lawsuit on the masses as acre theory phenomena such as the Higgs mechanism; inwards fact, the Higgs machinery as well as all similar things operate inwards string theory "just like" inwards acre theory

Tetragraviton says that the disputation "particles are vibrations" is invalidated inwards some way because string theory also has extra dimensions as well as supersymmetry. But neither extra dimensions nor supersymmetry invalidate the moving painting above. In fact, neither extra dimensions nor supersymmetry imply that fifty-fifty the simple as spaced spectrum based on the higher harmonics has to live generalized.

Extra dimensions may live apartment (torus or its orbifolds) as well as supersymmetry may live expressed inwards damage of gratuitous fields (whose spectrum is precisely gotten past times adding the liberate energy of the higher harmonics).

Moreover, Tetragraviton's extra comments most extra dimensions as well as supersymmetry are absolutely demagogic given the fact that he claimed to present something inaccurate most Brian Greene's statements most string theory. Brian Greene has ever discussed extra dimensions as well as supersymmetry inwards much to a greater extent than item than Tetragraviton. For example, several total chapters are dedicated to these topics inwards The Elegant Universe.

I desire to emphasize that in that location are really semirealistic models of string theory – which basically attain the minimal supersymmetric criterion model or something similar that consistently coupled to quantum gravity – which nevertheless build the spectrum pretty much past times the simple add-on of the higher harmonics that I discussed above. In particular, I hateful the orbifolds of tori as well as the heterotic models inwards the gratuitous fermionic formulation.

Influenza A virus subtype H5N1 novelty of such orbifolds is that some of the states come upward from twisted sectors. Influenza A virus subtype H5N1 twisted sector has some novel boundary conditions. Influenza A virus subtype H5N1 circular trip around the closed string doesn't furnish yous to the same dot inwards the infinite (or configuration space) but 1 related past times a global symmetry (isometry of the compactification manifold or a generalization of an isometry). Consequently, the indices \(n\) of \(\alpha_{n}\) are no longer integers but they are shifted past times a fractional shift such as \(1/2\) or \(1/4\) away from an integer. This isn't changing the story qualitatively. It's nevertheless truthful that the squared masses are integer multiples of a quantum. Also, the negative additive shift inwards the formula for \(m^2\) – the basis nation liberate energy – depends on which (twisted – or untwisted) sector yous consider.

Let me speak over Tetragraviton's claims inwards to a greater extent than detail.

It’s a dainty story. It’s fifty-fifty partly true. But it [the claim that increasingly heavy particle species are obtained from the add-on of higher harmonics etc.] gives a completely incorrect thought of where the particles we’re used to come upward from.

Sorry but it gives a completely right qualitative thought where all particle species inwards string theory come upward from. All of them come upward from string racket as well as it's ever the instance that (as long as 1 ignores subleading corrections to the masses from acre theory effects etc.) the to a greater extent than racket are added to a string, the heavier particle species nosotros obtain.

Experimentally, nosotros receive got entirely observed a few dozens of particle species. But they come upward from the string tower of vibrating strings, too. In some approximation, they're usually coming from states alongside \(m^2=0\). But that does not hateful that the counting of the higher harmonics as well as their contributions to \(m^2\) may live avoided.

Why?

It's because of the negative shift inwards the \(m^2\). The basis nation of a (closed or open) string is ordinarily a tachyon alongside \(m^2\lt 0\). This nation is projected out past times the so-called GSO projection. At the end, the spacetime supersymmetry is a sufficient (but non necessary!) status to acquire rid of all the tachyons. But in that location are ever numerous massless states – inwards the approximation of gratuitous strings. And these states are massless because the negative basis nation contribution to \(m^2\) is cancelled past times the positive contributions from the oscillators. This cancellation may accept a dissimilar numerical score for dissimilar states – as well as particularly for states inwards dissimilar twisted sectors.

But again, the counting of the basic frequency's as well as higher harmonics' liberate energy is unavoidable fifty-fifty if yous desire to empathise the source of the massless states. If all the non-constant modes along the string could live completely ignored as well as omitted, the whole added value of string theory would live "redundant garbage" as well as nosotros could just operate alongside the as consistent massless truncation of string theory.

However, nosotros just can't. In particular, 1 of the massless i.e. \(m^2=0\) states of the vibrating string is the graviton, the quantum of the gravitational moving ridge (or field), the messenger of the gravitational force. Even inwards the simplest \(D=26\) bosonic string theory, the spin-two graviton states are obtained from a closed string past times the activeness of 2 oscillators:\[

\alpha^\mu_{-1} \tilde \alpha^\nu_{-1} \ket{0}

\] Similarly, inwards the RNS \(D=10\) Definition of superstring theory, it is\[

\alpha^\mu_{-1/2} \tilde \alpha^\nu_{-1/2} \ket{0}_{NS,NS}

\] The basis nation is a tachyon (which survives inwards bosonic string theory, a source of infrared inconsistencies equivalent to an instability, but is removed past times the GSO projections inwards superstring theory). But its negative \(m^2\) is precisely cancelled past times 1 left-moving excitation of the "basic frequency wave" on the string, as well as 1 right-moving 1 (note that the level-matching status holds). There is no way to acquire the same results without the non-constant "sinusoidal waves" on the string.

The dot that 4gravitons is missing is that massless states (in the free-string approximation) coming from the quantized strings are massless "by accident". Most states receive got positive masses, some states hap to receive got null masses when the damage are added. But the latter aren't separated from the balance inwards whatsoever a priori way. The massive excitations are inwards no way added artificially to some massless starting point. There is no massless starting dot inwards string theory. String theory unavoidably generates the massless as well as massive states simultaneously, alongside no consistent way to separate them. It is non quite trivial to derive the massless spectrum inwards a full general string compactification. It's most as difficult as to derive the states at whatsoever massive level.

OK, dorsum to the graviton nation that had 2 (minimal nonzero frequency) moving ridge excitations around the closed string.

Once yous allow the "basic frequency" of the moving ridge on the string, yous automatically allow all of them because splitting as well as joining strings are capable of producing truncated sines on a shorter interval which may entirely live Fourier-expanded on the shorter string if yous allow all the higher harmonics as well. And a consistent theory of quantum gravity may entirely live obtained if yous contain all of them. There's just no way to consistently truncate the higher harmonics because fifty-fifty the "simple" graviton depends on the non-constant modes along the string.

Again, fifty-fifty for massless states, the careful counting of the liberate energy from nontrivial sinusoidal excitations of the string is essential to acquire the right mass.

Disappointingly, the entirely interpretation of Tetragraviton's words that "the vibrating string moving painting alongside harmonics is a completely incorrect explanation of the well-known particle species" is that he just doesn't receive got a clue how string theory explains the massless as well as low-cal states.

But I believe that this is non the entirely employment alongside his views most string theory. Another paragraph says:

String theory’s strings are nether a lot of tension, as well as then it takes a lot of liberate energy to brand them vibrate. From our perspective, that liberate energy looks similar mass, as well as then the to a greater extent than complicated harmonics on a string represent to extremely massive particles, closed to the Planck mass!

I've discussed that, it's non necessarily the case. I practise believe that the string scale is closed to the Planck volume but in that location practise be low-string models where it's as depression as a few \({\rm TeV}\)s. This is just a technical difference. The heavier excited string racket are as existent inwards both scenarios.

But it's primarily the next paragraph that I believe to live seriously flawed:

Those aren’t the particles you’re used to. They’re non electrons, they’re non nighttime matter. They’re particles nosotros haven’t observed, as well as may never observe. They’re non how string theory explains the key particles of nature.

Electrons as well as particles of nighttime thing (if the latter is composed of particles) are excited strings as well, as well as fifty-fifty for those, the add-on of energies from racket on the string is needed to acquire the right volume (despite its existence null inwards the free-string approximation). There just doesn't be whatsoever feel inwards which the disputation that "the states inwards the infinite tower of arbitrarily excited strings don't depict the electron or nighttime matter" could live correct. It's just wrong, wrong, wrong.

But the generalization of this statement, "they [strings excited past times the harmonics] are non how string theory explains the key particles of nature" is sure enough the contrary of a deep truth. That is exactly how string theory explains the key particles of Nature. Barton Zwiebach's quote higher upward may live used as the best explanation inwards this weblog post service why this observation is both right as well as essential.

There may also live some confusion most "what counts as a key particle of Nature". Tetragraviton seems to count the electron but non some heavy states close the string scale. But both of them are key particles of Nature. Moreover, both of them receive got masses whose essential contribution comes from the liberate energy of racket added to the string. There is no qualitative divergence betwixt the electron as well as the graviton; as well as heavier string states. We may receive got detected some particles as well as non others but all of them are as existent as well as as fundamental.

So how does string theory acquire from 1 key type of string to all of the particles inwards the universe, if non through these vibrations? As it turns out, in that location are several dissimilar ways it tin happen. I’ll depict a few.

The start as well as most of import flim-flam hither is supersymmetry. ...

Again, it's just non truthful that supersymmetry replaces or invalidates the fact that the head contribution to the volume of particles inwards string theory comes from the racket of a string. Supersymmetry is a special characteristic of a subset of the string vacua (and similarly quantum acre theories). But the elements of this subset are constructed inwards the same way as elements exterior this subset. In string theory, they are constructed past times counting the liberate energy that racket on a quantum relativistic string may carry. Supersymmetry almost ever requires some fermionic degrees of liberty but they may live viewed as extra coordinates of the (super)space as well as they add together racket as well as liberate energy through (fermionic) harmonic oscillators just similar their bosonic friends (well, they're non just Platonic friends, they're superpartners). They also receive got higher harmonics alongside \(n=2,3\) etc., entirely the employment numbers are \(N_a=0,1\).

Supersymmetry relates dissimilar types of particles to each other. In string theory, it agency that along alongside racket that acquire higher as well as higher, in that location are also low-energy racket that demeanor similar dissimilar sorts of particles.

Supersymmetry makes it to a greater extent than probable that in that location volition live massless or low-cal particles but it is non a necessary condition. There be non-supersymmetric (yet tachyon-free) string vacua alongside the analogous massless part of the spectrum (massless is meant at the floor of the gratuitous string, the string scale). Despite the absence of supersymmetry, the publish of massless bosonic as well as fermionic particle species – massless states of a vibrating string – is basically the same as inwards the similar supersymmetric models (I am talking most the tachyon-free non-SUSY heterotic strings). The states are just different, non less numerous or "worse".

Even alongside supersymmetry, string theory doesn’t give ascension to all of the right sorts of particles. You postulate something else, similar compactifications or branes.

Yup, except that Brian Greene as well as many others receive got explained all these things alongside extra dimensions etc. far to a greater extent than accurately as well as pedagogically than Tetragraviton. Incidentally, a compactification is always needed to obtain at to the lowest degree a semi-realistic string vacuum.

In string theory, the particles we’re used to aren’t just higher harmonics, or racket alongside to a greater extent than as well as to a greater extent than energy. They come upward from supersymmetry, from compactifications as well as from branes.

Again, in that location is absolutely no "contradiction" betwixt racket on 1 side as well as compactifications, SUSY, or branes on the other side. They're independent concepts. Strings vibrate fifty-fifty when they're placed inwards a compactified spacetime manifold, fifty-fifty when they're supersymmetric, fifty-fifty when in that location are D-branes around, as well as fifty-fifty if the strings are opened upward strings attached to these D-branes. It's similar to the bass strings: they also vibrate fifty-fifty when they are surrounded past times an musical instrument whose shape resembles Meghan Trainor, no treble. The shape of the instrument, similar the shape of the compactification manifold, influences the audio (and spectrum) of the racket but it inwards no way invalidates or removes the vibrations.

But a dot that he repeats all the fourth dimension as well as that annoys me is 1 most the "particle you're used to". String theory isn't primarily a theory hateful to speak over "just the particles yous used to". String theory is a theory of everything – which includes all particles, including those that no 1 is used to because nosotros haven't observed them (yet).

If someone isn't interested inwards the full list of particles inwards Nature, I give away it obvious that he has no argue to live interested inwards string theory, either – because string theory is almost past times Definition a theory going good beyond the technical limitations of electrical flow experiments. If someone isn't interested what is hiding beneath the surface, an effective acre theory is an easier mental attitude for those whose involvement is this narrow-minded. The effective acre theories are really defined to live the response to the questions that never essay to acquire beyond a sure authorities defined past times practical limitations. But inwards that case, if someone is interested inwards these low-energy things only, I don't come across why he would live reading weblog posts or books most string theory at all.

It just makes no feel whatsoever. The somebody isn't interested inwards these questions as well as then he in all likelihood doesn't report them as well as hasn't studied them. He almost certainly knows cipher most the things he could receive got learned (about string theory) but he hasn't as well as he should improve closed his mouth.

The higher harmonics are nevertheless important: in that location are theorems that yous can’t laid quantum gravity alongside a finite publish of extra particles, as well as then the infinite tower of racket allows string theory to exploit a key loophole.

Right. All the excited string modes are totally needed for the consistency of the quantum gravity, as I said. Also, as I discussed inwards a comment on the 4gravitons blog, when yous gradually growth the value of the string coupling constant, the excited string states are gradually turning to dark hole microstates. The exponential growth of the publish of excited string states is a precursor or an approximation to the quasi-exponential growth of the publish of dark hole microstates nosotros postulate inwards a consistent quantum theory of gravity.

They just don’t hap to live how string theory gets the particles of the Standard Model.

If the basis is described past times a weakly coupled string theory, string theory does derive all the particles of the Standard Model precisely past times the same algorithm that 4gravitons irrationally denounces.

The thought that every particle is just a higher vibration is a mutual misconception, as well as I promise I’ve given yous a improve thought of how string theory really works.

Is it not a misconception as well as Tetragraviton has entirely brought confusion as well as falsehoods to this topic.

Quite generally, a popularizer of scientific discipline ever runs the gamble of existence separated from the large shots who practise the best research, as well as and then on. People realize this (true) full general fact as well as that's also why popularizers are sometimes attacked alongside similar words. However, inwards the instance of string theory, almost all these attacks are just evidently rubbish.

In particular, Brian Greene has been extremely careful what he was maxim most string theory. His explanations of these topics represent pretty much to the most accurate sketch that is accessible to a large plenty subset of the lay public. And people who are criticizing some basic claims such as the deep insight that "in string theory, particles are vibrations" are just total of šit. The identification of the particle species (all of them) as well as the vibration states of a string is a profound truth (of weakly coupled string theory).