LOG#211. Understanding gravity (I).

Hi, everywhere! Have you missed me?

A little short thread begins. This thread is based on some Nima’s talk on the understanding on gravity. Do we understand gravity? Classically, yes. Quantumly, it is not so easy. String theory is the most successful approach, but canonical quantum gravity or loop quantum gravity is interesting as well, despite some hatred and pessimist reactions to this conservative alternative to strings. Even more recently, two papers appear suggestion a deeper relation between these two approaches, sometimes considered far away from each other…

Nima Arkani-Hamed is a powerful theoretical physicist. In the last years, he has fueled some insights on string theory and quantum field theory pondering about the nature of the ultimate physical theory, the theory of everything, and quantum gravity. What is he working out? A summary:

  • General relativity (GR) and Yang-Mills theory are inevitable and, in the end, they must “merge” or “emerge” somehow from the microworld. Frank Wilczek told me once on twitter, that GR is strikingly similar to some non-linear sigma models known in quantum field theory but not just “the same thing”.
  • The cosmological constant problem and the epic fail of quantum field theory to guess a reasonable value of it (compared with the observed value), and the most certain fail of modified IR gravity theories (till now), as solution. Even MOND (MOdified Newtonian Dynamics is unclear on this stuff) is not accepted as plain solution despite the absence of positive finding of dark matter, so its surprisingly precise fit to galactic scales (not so good at bigger scales) remains as a mystery and issue.
  • Massive gravity theories and related issues with higher derivative theories (Fierz-Pauli, DGP, galileon theories, TeVeS, ghost-free higher derivative theories,…). Experimentally, they are not supported and yet they have been evolving in this experimental era. Likely, the rediscovery of certain ghost-free higher derivative theories, and new BH solutions with scalar hair have rebooted the interest in these theories, again, despite the lack of experimental support, what is not bad since we are limiting the space of available theories to fit the data.
  • Horizon thermodynamics and violent crashes with non-locality. The black hole information problem remains after more than 40 years of the discovery of Hawking radiation (R.I.P., in memoriam). Are black hole horizons apparent as recent ideas claim or are they real? Reality is a powerful but likely relative word.
  • Inflation expectations from the CMB observations. In the next years, the hope to observe the B-mode of the inflationary phase of the Universe in the Cosmic Microwave Background has grown. Living with eternal inflation would push us to adopt the Multiverse scenario, something many people seem to see as cumbersome, but others see it as natural as the solution to the Quantum Mechanics interpretation problem.
  • The end of space-time. Particle physicists are familiar with the idea of unstable or metastable particles. Even the proton would be unstable in many Grand Unified Theories (GUT) and the TOE. Hawking radiation is important because it also would imply that not only black holes, the space-time itself could end or evaporate into “something” or into nothing! Of course, this seems crazy. Accepted known Quantum Mechanics only allows for unitary evolution, allowing something to disappear, would introduce the idea that space-time itself will decay or disappear in the far future. We want to know in such a case why or not the spacetime disappear, and what is the final destiny of the Universe and its spacetime. We don’t know that, as we do not know what happened in the Planck era or before. That is important,…Not for us, but the destiny of our species depends on it. It is a thing that will be important for our descendants unless you think we will not survive the forthcoming crisis.
  • Physics with no space-time. Can we formulate physical theories without any reference to space-time at all. How could such theories look like?
  • Surprising new mathematical structures arising in quantum field theories and general relativity, specially those arising from algebraic geometry and/or combinatorial geometry. Physics has always been boosted by advances in mathematics. It is very likely the next big revolution in physics will happen when new mathematical structures and methods emerge from some sides. In QFT, the polyzeta and polylogarithms have appeared as wonderful recipes to some closed formulae problems. In general relativity and supergravity or superstrings, the duality and the brane revolution are pushing towards a categorical and algebraic differential (likely differentigral in near future) structures yet to be fully understood, and likely appreciated, by the academical community.

Where are we now? It is hard to say. The Heisenberg uncertainty principle, as you know, surely implies that there is something like an “ultimate microscope”. \Delta E\sim 1/\Delta t means that, eventually, as you put more and more energy into a single “point”, you would create a black hole. No distance or time, based on usual or commonly accepted riemannian (and likely differential!), can be have any sense when you reach the fantastic ultrashort size of 10^{-35}m, or the time scale of 10^{-43}s. No operational definition of space or time is available, in terms of conventional geometries, at those scales of distance and time. Volovich suggested, almost 30 years ago, in 1988, that the ultimate physical theory would be number theory. However, this statement has not been developed further, beyond some exotic researches on p-adic and adelic geometries that, however, are being growing in the last years. The point is, of course, where space-time notions end. There are two main places where physicists are for sure convinced they need something else to current theories: the Big Bang and the black hole singularities (and maybe, tangentially, at the black hole horizon and or those dark matter and dark energy stuff nobody understands, due to the fact we do not understand GRAVITY?).

Let me be more technical here. Quantum Field Theory (QFT for short from now), says that vacuum is not static, is also dynamical. It polarizes. So, every old grade course of electromagnetism is not completly fair when telling you vacuum does not polarize. Classically not! Quantumly, YES! Mathematically, QFT provides a recipe to calculate the effect of vacuum polarization through loop integrals in Feynman graphs that are mathematically evaluated into logarithmic integrals, WHENEVER you plug a regulator. That is, a high energy scale \Lambda_{UV} where the usually divergent integral gets regularized to provide a finite value. For one loop, momentum P, and supposing M_P^2 finite, you get something like this:

(1)   \begin{equation*}\dfrac{P^4}{M_P^2}\log \left(\dfrac{\Lambda_{UV}^2}{P^2}\right)\end{equation*}

The Planck mass squared comes from the two vertices from the loop, the 4th power comes from the edges and the logarithmic regulator of the squared cut-off appears due to finite expectations on general grounds. Actually, Higgs physics is important because Higgs mass is sensitive for such logarithmic corrections as there is NOTHING, absolutely nothing, in the Standard Model allowing the Higgs remains so light (125 GeV/c²) as we measure it! Note the similarity between this Higgs physics and the formula above. Any reasonable force, indeed, can be expanded in terms of energy scales in QFT. You would get:

(2)   \begin{equation*}f=F_0+\dfrac{1}{p^2}+\dfrac{1}{M_P^2}\log P^2+\dfrac{1}{M_P^2}\delta^3(r)+\ldots\end{equation*}

and where the divergent parts are usually neglected hoping that some further theoretical approach will teach us why they are certainly negligible under the floor…

Now, enter into the gravity realm. Some times is said that quantum gravity is “hard” or impossible. That is not exactly true or accurate. There are in fact some quantum gravity calculations available. For instance, the leading quantum correction to the newtonian force is provided by the following formula (up to some conventions with the numerical prefactors):

    \[F_q=\dfrac{GM_1M_2}{r^2}\left(1-\dfrac{27}{2\pi^2}\dfrac{G\hbar}{r^2c^2}+\cdots\right)\]

and you can see the classical newtonian force plus a leading order correction. I recommend you the papers by Bjerrum-Bohr on this subject. Then, you may ask, what is the problem? Well, the problem is…You can not do the above calculation for any LOOP order. Only certain theories, like maximal supergravity (due to its hidden and exotic Chern-Simons terms), and superstring theory (I am not sure with que Loop Quantum Gravity approach here), can, in principle provide a recipe to calculate a finite quantum gravity interaction between gravitons and gravitons and matter at any loop order. Why? The increasing number of Feynman diagrams and the mathematically complexity of non-linearity of GR makes the problem formidable. Likely, only a supercomputer or trained AI could manage to add all these diagrams in the most advanced theories and tell us if they are correct when contrasted with experimental data. But that is a future today.

Helicities of particles like gravitons or photons enter into the difficult calculations of quantum gravity or QFT. Locality, imposed by our preconceptions on space, time and field formulations due to Lorentz symmetry and causality, are solid in local QFT based on relativity and Quantum Mechanics (QM). For instance, a photon field can be coded into a vector field with polarization in form of plane waves:

    \[A_\mu=\varepsilon_\mu e^{ipx}\]

Transversality of the photon field, i.e.,

    \[\varepsilon_\mu\cdot p^\mu=0\]

plus gauge redundancy

    \[\varepsilon_ \mu\sim\varepsilon_\mu+\alpha p_\mu\]

    \[A_\mu\sim A_\mu+\partial_\mu\Lambda\]

implies a constraint

    \[\sum_i k_iP_i^\mu=0\]

for all equal k_i. That is, the equivalence principle somehow is telling us that, the whole structure of interactions, based on QM and special relativity, leads long distance interactions for spin 1 or spin 2 forces (electromagnetism and gravity!). Thus, it yields that, whatever the TOE is, relativity (special and general somehow) and QM (the SM somehow as well), they must emerge from it. In fact, the general structure of any Yang-Mills (YM) theory is pretty simple, it has an Y diagram form and three labels for any spin, s=0,1/2,1,3/2,2. We have found fundamental particles of spin 0, 1/2, 1 and 2 (0 with the Higgs, 2 if you count gravitational waves as gravitons). The only lacking fundamental particle is that with spin 3/2, a general prediction of supersymmetry (SUSY). Gravity is unique, at minimal sense, thanks to Einstein discovery of gravity as curved spacetime geometry. Of course, you could extend GR to include extra fields or gravitational degrees of freedom (massive gravitons, dilatons, graviphotons,…), but all of these have failed till now. SUSY has not been found, but it will be found in the future for sure. Black holes, indeed, in GR have some exotic supersymmetries, even in the simple Kerr case. That is not very known but it is true.

Moreover, move to the cosmological constant problem. Vacuum energy density problem if you prefer the alternative name for it. The fact that

    \[\Lambda_{observed}\sim 10^{-122}\Lambda_{theory}\]

have been known since the times where Einstein introduced the cosmological term. Worst, now we do know it is not zero, it makes the situation more unconfortable for many. Before 1998 you could simply argue that some unknown symmetry would erase the cosmological term of your gravitational theories. Now, you can not do that. Evidence is conclusive in which the cosmological term seems to be positive and non-null. Just having a theory, that allows for fantastically tiny cancellations, is just weird. Weirder if that cancellation is precise in 122 or lets say the falf 61 orders of magnitude. Such a fine tuning is disturbing. Ludicrous! Ridiculous. Anyway, this fact has not stopped theorists to make some guesses of how to life with that. One idea arised after the second string revolution, circa 1995. Brane worlds. Plug some damping curvature into the bulk of spacetime. The tension on a brane could just explain why gravity is weak, and maybe, explain why the cosmological constant is tiny. The problem is, that the mechanism is much more consistent with NEGATIVE cosmological constant. The DGP model gets 4D GR plus massless matter in an AdS (Anti-deSitter, negative cosmological constant Universe). In technical words, a negative tension provides a modified propagator in QFT solving the cosmological constant problem if this is negative (otherwise this blows up!). Another problem with brane worlds is that no causal modification can work on fields of the main brane Universe. Massive gravities, both in brane gravity and independent models, imply, on very general grounds, that long distance interactions should include SCALAR new degrees of freedom. Remember the Higgs mass problem I mentioned before? Well, you loose the control of scalars in theories without symmetries. In fact, the good and the evil of scalar degrees of freedom: at some point they introduce modifications or violations of either the Einstein equivalence problem (or some soft/hard variation), or the Lorentz symmetry behind it. Exciting news for experimentalists: you can seek these violations in designed experiments. Bad news for conservative theorists: nonlinear interactions can introduce conflicts with the usual conceptions and features of locality (even causality), thermodynamics or Lorentz symmetry expected from current well-tested theories. To save locality, the Einstein equivalence principle or usual properties of QM seems to be inevitable at some point (something also triggered by the yet unsolved black hole information paradox). Unless a loophole is found, it seems the combination of the current theories will imply that we must abandon some yet untouched principle. A toy model in DGP massive gravity, the so called galileon gravity lagrangian:

    \[L_s=\partial^2\phi+\dfrac{\partial^2\phi}{\Lambda^3}\square\phi+\cdots\]

It has a shift symmetry (galilean symmetry) given by transformations

    \[\partial_\mu  \phi\rightarrow \partial_\mu\phi+V_\mu\]

In the simplest Higgs phase of gravity, you would have a zero expectation value for the scalar field. However, radiative corrections to this vacuum are expected to arise. And we do NOT know how to handle with it.

20th century physics is made by the triumph of the current two pilar theories: relativity (special and general) and quantum mechanics. The apparent difficulties to get gravity into the quantum game is much a deep question, but it could allow us to explain the Big Bang, the Universe and the future of it. These difficulties, are also triggering doubts about the role and formulation of QM (annoying for many to accept, from philosophical reasons, more than experimental precision). Surely, quantum mechanics could be modified by a further TOE or GUT theory. However, it will remain true just as the Bohr toy model or newtonian mechanics. Why is our Universe big with big things in it? The reason is QM, or more precisely QFT. QFT=SR+QM. And it is true. Particles are classified as entities with mass (energy) and spin (times \hbar). Experimentally, with the exception of spin 3/2, we have found every particle (GW counted as graviton wavepackets) till spin 2 (spin zero is the Higgs particle). Why not spin 5/2 or 3? Well, there are higher spin theories. They have other issues, with locality or their definition as interacting field theories (excepting some special theories as those by Vasiliev). The simple Y form of Feynman graphs in known theories is particularly striking and simple (beyond some technicalities due to the well-defined processes of regularization and renormalization of physical quantities, that some one should study better at these crisis times?). However, the structure of the Y shape interaction of the Higgs field CANNOT be investigated in the LHC very well. It is a nasty hadron collider. We will need a linear or muon collider. Or a circular collider adjusted to the resonance of the Higgs particle to study his self-interactions (in the standar model, the Higgs potential is simply a cubic plus quartic potential). Otherwise, a 100TeV collider would be better for this as well. A 100 TeV collider would probe vacuum fluctuations of the Higgs field itself (or the muon collider or any other special collider tuned to the Higgs).

There are some critics towards the waste of money those machines would be. Or course, there is no guarantee that we will find something new. But the Higgs particle interactions MUST be studied precisely. The universe is surprisingly very close to Higgs field metastability. It is something that well deserve the money, perhaps, I would only complain about not doing this crazily. We need to plan the Higgs potential study further. However, note that the LHC is about 10TeV, the future colliders will be clean lepton (or photon!) colliders tuned to the Higgs resonance or 100TeV/1000TeV (the latter in my lifetime I wish to see it) and those energies are yet much smaller than Planck energy, 10^{16} TeV. Neutrino physics, gamma rays, likely X-rays and radio bursts or gravitational wave astronomy can probe surely strong gravity and extreme processes much better. For free. Of course, you have to be good enough to catch those phenomena and that, again, cost time and money.

String theory news…But firstly, a little bit history. Strings were discovered in the context of S-matrix theory and strong interactions. The Veneziano amplitude was the key to find out that string theory has something to do with the nuclear realm (despite this is surely ignored by quantum stringers right now, or not so appreciated as 30 years ago!). String amplitudes of four point particles has a simple structure:

(3)   \begin{equation*}A_s=\dfrac{<12>^4\left[34\right]^4}{stu}\mathcal{C}\end{equation*}

and the general amplitude takes the form

(4)   \begin{equation*}\mathcal{A}=\dfrac{<12>^4\left[34\right]^4}{stu}\Pi_{i=1}^\infty\dfrac{(s+i)(t+i)(u+i)}{(s-i)(t-i)(u-i)}=\dfrac{<12>^4\left[34\right]^4}{stu}\dfrac{\Gamma(-s)\Gamma(-t)\Gamma(-u)}{\Gamma(1+s)\Gamma(1+t)\Gamma(1+u)}\end{equation*}

Here, the s, t, u are variables encoding the energies of the colliding strings in certain frame, \Gamma is the gamma function of Euler, a generalization of the factorial function for real AND complex values. The numbers between brackets and powers are certain spinor/vector quantities coding helicities. Well, take weak coupled string theory amplitudes, and 4 point interactions at tree level (no loops in the classical sense), independently of compactification you can get a wonderful universal formulae for gravity and YM amplitudes:

1st. For gravity, you get:

(5)   \begin{equation*}\mathcal{A}_G=\dfrac{<12>^4\left[34\right]^4}{stu}\dfrac{\Gamma(-s)\Gamma(-t)\Gamma(-u)}{\Gamma(1+s)\Gamma(1+t)\Gamma(1+u)}\times \mathcal{K}\end{equation*}

where\mathcal{K} is

    \[\mathcal{K}=-\dfrac{1}{stu}\]

for Type (II) strings at the level of three gravitons interaction, and

    \[\mathcal{K}=-\dfrac{1}{stu}+\dfrac{1}{s(1+s)}\]

for heterotic strings at the level of 2 graviton-scalar interaction, and

    \[\mathcal{K}=-\dfrac{1}{stu}+\dfrac{8}{(1+s)s}-\dfrac{tu}{s(1+s)^2}\]

for bosonic strings at the level of 2 graviton-scalar interaction. Note the universal pole structure of the formulae above.

    \[\mathcal{C}=\dfrac{\Gamma(-s)\Gamma(-t)\Gamma(-u)}{\Gamma(1+s)\Gamma(1+t)\Gamma(1+u)}\]

2nd. YM theory in string theories have the following 4 point tree level amplitudes:

(6)   \begin{equation*}\mathcal{A}_G=<12>^2\left[34\right]^2\dfrac{\Gamma(1-s)\Gamma(1-t)}{\Gamma(1+s)\Gamma(1-s-t)}\times \mathcal{K}'\end{equation*}

where\mathcal{K}' is

    \[\mathcal{K}'=-\dfrac{1}{st}\]

for Type (I) strings at the level of three gravitons interaction, and

    \[\mathcal{K}=-\dfrac{1}{st}-\dfrac{u}{s(1+s)}\]

for bosonic strings at the level of 2 graviton-scalar interaction, and

    \[\mathcal{K}=-\dfrac{1}{stu}+\dfrac{8}{(1+s)s}-\dfrac{tu}{s(1+s)^2}\]

for bosonic strings at the level of 2 graviton-scalar interaction. Heterotic theories would have massive pole structures in the similar expression. The pole structure (beyond the massive corrections in the heteorica case) arises from

    \[\mathcal{C}'=\dfrac{\Gamma(-s)\Gamma(-t)}{\Gamma(1-s-t)}\]

These gamma functions are related to the Euler beta function and can be seen as generalizations of the famous Veneziano amplitude who gave birth to string theory in the 70s or the 20th century. In fact, the S-matrix programme can be approached for any mass and spin. The string theory “magic” procedures with only massless states (external) and playing with another interactions or deformations of the above formula give rise to some open problems (some of them, known from the old string theory times).

For external interactions of 3 point interactions, at tree level, you get an Y-amplitude

    \[g\varepsilon^{\mu_1\mu_2\cdots \mu_N}(p_1-p_2)_{\mu_1}\cdots (p_1-p_2)_{\mu_N}\]

For 4 point amplitudes Y+Y at tree level, you get

    \[\dfrac{gg'}{s-M^2}G_N^{(d)}\left[\cos\theta\right]\]

where G_N^{(d)}(\cos\theta) are Gegenbauer polynomials, G_0=1, G_1=x, G_2=dx^2-1, that arise in the expansion of the fraction (newtonian like force) in d-dimensions:

    \[\dfrac{1}{\vert z-r\vert^{d-2}}=\sum_N r^NG_N^{(d)})\left[\cos(\theta)\right]\]

These remarkable formulae link with the Veneziano amplitude:

(7)   \begin{equation*}A_V=\dfrac{\Gamma(-1-s)\Gamma(-1-t)}{\Gamma(-2-s-t)}\end{equation*}

It has a striking pole structure, with residua at s=-1, s=0, s=1,…, s=N; or equivalently at 1, t+2, (t+2)(t+3),...,(t+2)\cdots(t+N+2). It yields the residue at s=1 provides

    \[\mbox{Res}(t+2)(t+3)=\dfrac{25}{4}\left(\cos^2\theta-\dfrac{1}{25}\right)=\dfrac{25}{4}\left(\cos^2-\dfrac{1}{d}\right)+\dfrac{25}{4}\left(\dfrac{1}{d}-\dfrac{1}{25}\right)\]

This expresion is POSITIVE as you keep d\leq 25, a fact known from bosonic string theory (living in 25+1 spacetime). Similarly, the open superstring amplitude

    \[A(1^-2^-3^+4^+)=<12>^2<34>^2\dfrac{\Gamma(-s)\Gamma(-t)}{\Gamma(1-s-t)}\]

has an analogue residue at s=3 (levels 1,t+1,(t+1)(t+2),...,(1+t)\cdots(N-1+t) corresponding to s=1,..,s=N) and you get

    \[\mbox{Res}_{s=3}(t+1)(t+2)=\dfrac{9}{4}\left(\cos^2\theta-\dfrac{1}{9}\right)=\dfrac{9}{4}\left(\cos^2-\dfrac{1}{d}\right)+\dfrac{9}{4}\left(\dfrac{1}{d}-\dfrac{1}{9}\right)\]

that is OK iff d\leq 9. You can enforce positivity to every level as well! But a price is to ensure it. You have to pay a d=2 conformal setting (only known theory to do that is string theory!). If you want a ghost-free theory, positive amplitude theory, such as

    \[a=\sum_k c_k\cos (k\theta)\]

remains positive with c_k>0 you have to live in d=2 dimensions (string theory magic do that, in the abstract worldsheet!). However, this is hard statement as c_k become increasingly exponentially small, but adding a \cos \theta-1 factor makes it false! The positivity of these amplitudes and the analysis of the hidden symmetry structure of the string diagrams have revealed a “jewel” or hidden geometric object in quantum mechanics/string theory/quantum field theory. How? The residue structure of gravity amplitudes are correlated to open superstrings:

    \[ \mbox{Res}_N^{Gravity}(\cos\theta)=\left(\mbox{Res}_N^{OpenS}(\cos\theta)\right)^2\]

This is a new hint of the Gravity=YM^2 mantra of these times, but there is more. Departure of positivity seems to indicate non consistent theories. Positivity magic struggles with massive higher spin states, where problems really live too. What higher spin amplitudes should we include and what to exclude? Not easy task. The idea is that we should search for a way to understand higher spin amplitudes without the worldsheet picture as primary entity (that d=2 restriction is hard for practical purposes in the real world). There is another reason, and that is gravity. Gravity is, from certain viewpoint, more positive than open superstrings (note the power in the amplitude coming from the spinors/polarizations). Here it comes, the amplituhedron. What is the amplituhedron? Well, it is a new object encoding the positivity of amplitudes in QFT and string theory. A formal definition is something like this:

    \[\boxed{\mathcal{M}_{n,k,L}\left[Z_a\right]=\mbox{Vol}\left[\mathcal{A}_{n,k,L}\left[Z_a\right]\right]}\]

What is this? Well, roughly speaking, the amplituhedron is certain generalized polytope (higher dimensional polyhedron) in projective geometry (technically, a generalization of the positive grassmannian) such as its volume is the all-loop scattering amplitude of particle physics processes. There, n is the number of vertices (or particles) interaction, k is the plane dimensionally specifying the helicity structure of the particles, and L is the loop order. Certainly, that amplitudes are lower dimensionally projected shadows of higher dimensional, maybe discrete, structures is a powerful language. I will talk about the amplituhedra and related stuff in the next posts.

Finally, to end this long boring post, let me review some of the magic. The amplitude with no negative probability (positive residue!) for gravity and massive particles reads off:

(8)   \begin{equation*}\mathcal{A}=G_N\dfrac{<12>^4<34>^4}{stu}\dfrac{\Gamma(1-\frac{s}{M^2})\Gamma(1-\frac{t}{M^2})\Gamma(1-\frac{u}{M^2})}{\Gamma(1+\frac{s}{M^2})\Gamma(1+\frac{t}{M^2})\Gamma(1+\frac{u}{M^2})}\end{equation*}

However, emphasis is usally done in the spacetime picture of strings, instead of the amplitude structure inherited from S-matrices! The equation of a superstring is usally given by

    \[\partial_\tau^2 X^\mu(\sigma,\tau)-\partial_\sigma^2 X^\mu(\sigma,\tau)\]

The so-called Green-Schwarz theory contains a Super-Yang-Mills (supersymmetric extension of YM) theory with action

(9)   \begin{equation*}S=\int\left(-\dfrac{1}{4}Tr F^2+i\overline{\Psi} \Gamma \cdot D\Psi\right) dvol\end{equation*}

String theory has a problem. It yields to too many consistent vacua for the Universe. Our SM+GR world is only one betwen 10^{500} in general superstring models, or 10^{272000} F-theory (10+2) different universes. These are Universes very similar to ours (or different), differing in coupling constants and vacuum expectactions values no hint of how to selec. This is the string theory nasty trick. There is no adjustable parameter, but you are driven to accept there are plenty of Universes. To my knowledge, no one has even proved that our constants and field theory expectation values can be derived from one of those Universes in clear plain way. However, there are no too many consistent quantum theories of gravity in the market…

For instance, the (quantum) supermembrane allows you get a bosonic equation (free) of motion given by

(10)   \begin{equation*}\partial_i\left(\sqrt{-g}g^{ij}E^\mu_j\right)=0\end{equation*}

and where

    \[g_{ij}(X)=\eta_{\mu\nu}\partial_iX^\mu\partial_jX^\nu=E^\mu_iE^\nu_j\eta_{\mu\nu}\]

Going to superspace Z=(X,\theta) membranes, the supermembrane equation gets modified

(11)   \begin{equation*}\partial_i\left(\sqrt{-g}g^{ij}E^\mu_j\right)=\varepsilon^{ijk}E_i^\nu\partial_j\overline{\theta}\Gamma^\mu_\nu\partial_k\theta\end{equation*}

SUSY and coherence of the theory in minkovskian(lorentzian) signature force you to match bosonic and fermionic degrees of freedom for branes, i.e., N_B=N_F, such as for a p-brane in D-spacetime (generally lorentzian, giving up this allows you to go beyond eleven dimensions), you have:

    \[N_B=D-p-1\]

    \[N_F=\dfrac{MN}{4}\]

where M is the number of fermionic degrees of freedom, and N is the number of supersymmetries on the superspace target. From this simple equation, you can easily derive that

    \[D-p-1=\dfrac{MN}{4}\]

Take any 1-brane, so you get D-2=\dfrac{MN}{4}. If you impose N=1 SUSY, you get superstring theory with M=4 generators (fermionic d.o.f.) in D=3, you get superstring theory with M=8 generators in D=4, M=16 generators in D=6 and M=32 generators in D=10. You can play with N=2, N=4 and N=8 supersymmetries in these dimensions too.

See you in the next amplituhedron post!

LOG#210. Weak gravity conjectures.

 

Hi, everyone! I’m sorry again for the time lapse, but I required to. To secure my current job (as High School teacher), and well, to fight against doubts about my health. Thank you for your patience!

Gravity is weak (at least on our brane!). Force carriers and couplings tell us this:

Are you weak? Well, I know, I know…Weak or strong depends on the context and the regime you are. Maybe…But why quantum gravity matters here? Well, we want to unify gravity with (Yang-Mills) gauge theories. It is hard.

From general perturbative arguments of quantum field theory (QFT), every theory can be expanded around 1/\Lambda. However, the so-called semiclassical computation is expected to break down at some point, due to our ignorance about the true degrees of freedom of gravity and quantum space-time at the microworld. For instance, a classical similar phenomena does happen in the Standard Model (SM). The SM electron is not “very light” or too heavy. The renormalized photon coupling near the electron, and here is where the Weak Gravity Conjecture (WGC) enters, where the scale eM_P, the electron charge times the Planck mass (natural units are selected here), differs up to order ne from the IR low energy theory given by classical QFT. That is, what is the WGC? Simply, it is the statement that in any gauge theory coupled to gravity, you can not get arbitrary mass or coupling but you are confined to have a bound of the following form (here I suppose the above U(1) gauge theory):

(1)   \begin{equation*}\Lambda \leq eM_P\end{equation*}

Here, the e is the U(1) gauge charge, the electric or more generally speaking the abelian gauge coupling (hypercharge).

Why is this important? Well, the keyword in the mood these days is called: emergence. Gauge theories (and even GR) are expected to be emergent or derived from a more fundamental framework (string theory is such a framework, but also supergravity or supersymmetric theories). Then, the WGC takes a more general form. The smallness of the gauge couplings at low energies (IR, infrared) is caused (likely) by heavy particles in the UV (high energy, short distance, regime). This is sometimes referred as IR/UV “entanglement” but it is a sort of duality like others discovered since 1995, the second superstring revolution. Now, there is a conjecture, by Harlow, in the form of WGC:

“The WGC of any emergent gauge fields is mandatory to enforce factorizability of Hilbert space in quantum gravity (QG) with multiple asymptotic symmetries”.

Here,  let me highlight the complementarity between the UV-cutoff scale \Lambda and the IR-regulator expected from field theory. This is the origin of the above bound in mass-charge and its generalizations, that I am going to discuss a little bit.

Generalization one. Extra dimensions and WGC bounds.

Suppose you formulate (D+1) gravity and compactify some extra dimension on a circle. Then, you get the celebrated relationship

(2)   \begin{equation*} M_P^{D-2}(D)=2\pi R M_P^{D-1} (D+1)\end{equation*}

By the other hand, gauge theory on the circle provides a KK charge coupling

(3)   \begin{equation*}\dfrac{1}{e_{KK}}=\dfrac{1}{g^2_{YM}}=\pi R^3M_P^{D-1}(D+1)\end{equation*}

From these, and the hypothesis that the UV scale is less or equal to the Planck scale, i.e., \Lambda\leq M_P(D+1), counting degrees of freedom you obtain something like this

(4)   \begin{equation*} \Lambda_{QG}^{D-2}\leq N_{dof} M_P^{D-2}(D)\end{equation*}

and where the number of degrees of freedom with mass BELOW the quantum gravity scale can not exceed the value m\sim eM_P in the U(1) case, such as

    \[N(\Lambda)\geq \dfrac{\Lambda}{eM_P}\]

implies that

    \[M_P^2\geq N(\Lambda_{QG})\Lambda^2_{QG}\]

and thus

    \[M_P^2\geq \dfrac{\Lambda}{eM_P}\Lambda^2\]

or

    \[M_P^2\geq\dfrac{\Lambda^3_{QG}}{eM_P}\]

and finally

    \[\boxed{\Lambda_{QG}\leq e^{1/3}M_P}\]

In fact, a more precise bound would be

    \[\boxed{\Lambda_{U(1)}\leq (ke)^{1/3}M_P}\]

since our calculation involved a nasty trick of gauge KK into the gravity realm of the bulk. Even more surprisingly, in any dimension, loops and UV cut-offs allow us to generalized that into

(5)   \begin{equation*} \Lambda_{U(1)}\sim \left(eM_P^{3\frac{D-2}{2}}\right)^{1/D-1}\end{equation*}

and the scale of gravity fixes into

(6)   \begin{equation*}\lambda_g(E)=G_NE^{D-2}\sum \mbox{dim}R_i\end{equation*}

and where the sum extends on any dimension of gauge representations for the i-particle into the E-configuration after compactification or the cut-off regularization. Again, M_P^{D-2}\geq N_{dof} \Lambda^{D-2} and now:

-For U(1) G_Nm^2\leq c_{WGC} k^2e^2n^2, where k is the lattice spacing, and n is some large quantum number, such as

    \[n_{max}\leq \dfrac{\Lambda_{QG}}{ekM_P^{(D-2)/2}}\]

and therefore, the scale of quantum gravity is bound in the form

    \[\Lambda_{QG}\leq (ek)^{1/D-1}M_P^{3(D-2)/2(D-1)}\]

In fact. the requirement to avoid the Landau pole in U(1) also imply a similar bound, provided

    \[\lambda_{gauge}(E)\sim e^2E^{D-4}\sum (kn)^3\sim e^2k^2E^{D-4}\left(\dfrac{E}{ekM_P^{(D-2)/2}}\right)^3\]

You can also generalize all this for non abelian gauge theories in any dimension!!!!! For a SU(2) in D-dimensions, you obtain

(7)   \begin{equation*}\Lambda_{QG}\leq k^{1/D}g^{2/D}M_P^{2(D-2)/D}\,\forall D\end{equation*}

You can easily evaluate for our 4D world to get

    \[\Lambda\leq k^{1/4}g^{1/2}M_P\]

For larger groups, as SU(3), the formula is more complicated but after group theory and mathematics you can obtain

    \[\Lambda_{QG}\leq k^{2/(D+3)}g^{5/(D+3)}\left(M_P^{7( D-2)/2(D-3)}\right)\]

and for 4D

    \[\Lambda_{QG}\leq k^{2/7}g^{5/7}M_P\]

The gauge coupling for the SU(2) and SU(3) groups also follow up these calculations

    \[\lambda_g(E,SU(2))\sim\dfrac{E^D}{g^ 2kM_P^{2(D-2)}}\]

and

    \[\lambda_g(E,SU(3))\geq g^2E^{D-4}\dfrac{(E/gM_P^{(D-2)/2})^7}{k^2}\]

For a general non-abelian gauge group G, the two mail formulae for the WGC are the following

(8)   \begin{equation*}\boxed{\Lambda_{QG}\leq k^{\frac{r_G}{n_G+D-2}}g^{\frac{n_G}{n_G+D-2}}M_P^{\frac{n_G+2}{n_G+D-2}\left(\frac{D-2}{2}\right)}}\end{equation*}

(9)   \begin{equation*}\boxed{\lambda_g(E)\geq g^2E^{D-4}\dfrac{}Q_{max}^{r_G+L_G+2}{k^{r_G}}\sim\dfrac{E^{n_G+D-2}}{k^{r_G}g^{n_G}M_P^{(n_G+2)(D-2)/2}}}\end{equation*}

These equations and formulae have the following meaning in D-dimensions: they provide a bound betwen the mass of the lightest particles and the heaviest, and provide a relationship betwen the range of the group r_G, the lattice spacing L_G, as a function of the range plus half of the roots of the groups. Note that for SU(N), you have d_G=N^2-1, r_G=N-1, and that you also get the asymptotic limit for “big numbers”(r_G<<d_G)

    \[\lim_{n\rightarrow\infty}\Lambda_{QG}=gM_P^{(D-2)/2}\]

as usual from compatification scenarios and other brane worlds or BSM theories. Indeed, for instance, in a KK scenario in SU(2)xU(1), you would get something like this:

    \[M^2(j,q)\leq (q^2j^2+e^2q^2)M^{(D-2)}_P\]

and

    \[N(E)\sim\dfrac{1}{eg^2}\left(\dfrac{E}{M_P^{(D-2)/2}}\right)^3\]

with

    \[\Lambda_{QG}\leq e^{1/D+1}g^{2/(D+1)}M_P^{\frac{5}{2}\frac{D-2}{D+1}}\]

The general gauge-gravity unification, supposing n=\sum_i^Dn_i, n_i=r_i+l_i, implies that the lightes mass can not be arbitrary and are related to the higher energy maximal mass in such a way that

(10)   \begin{equation*}\boxed{\Lambda_{QG}\leq (\mbox{det}\tau)^{-1/2}(\Pi_i g_i^{n_i})^{1/(n+D-2)}M_P^{\frac{(n+2)(D-2)}{2(n+D-2)}}}\end{equation*}

(11)   \begin{equation*}\boxed{\lambda_i\geq \dfrac{(det\tau)^{1/2}E^{n+D-2}}{\Pi_i g_i^{n_i}}M_P^{(n+2)\frac{(D-2)}{2}}}\end{equation*}

In summary, gauge-gravity unification IMPLIES under very general grounds the Weak Gravity Conjecture:

(12)   \begin{equation*}\boxed{\Lambda^2_{gauge}\leq e^2<q^2>_\Lambda M_P^{D-2}}\end{equation*}

Generalization two.String theories and WGC bounds.

What about string theory? It yields that string theory as a generalized KK theory also contains a genberalized form of the WGC as well! In the weak coupling limit, any stringer has derived in some moment something like this

    \[\lambda_g(E)\sim g_s^2\dfrac{e^{2\pi(2+sqrt{2})\sqrt{N}}}{N^{3/2}}\]

for N=E^2\alpha'/4 for gravity, and

    \[\lambda_G(E)\sim g_s^2\dfrac{e^{2\pi(2+sqrt{2})\sqrt{N}}}{N^{2}}\]

for N=E^2\alpha'/4 for gauge theory in the background. Also, you probably know that

    \[G_N\sim g_s^27M_s^8\]

in 10d heterotic string, and

    \[g^2_Y\sim g_s^2/M_s^6\]

This implies that

    \[\lambda_g\sim\left(\dfrac{E}{M_s}\right)^8\]

for gravity, and

    \[\lambda_G\sim g_s^2\left(\dfrac{E}{M_s}\right)^6\]

for gauge theory below the string scale up to numerical constants. Moreover, below strings, \lambda(gravity)<<\lambda(gauge), but at the string scale we would have \lambda(gravity)\sim,\lambda(gauge), at least if g_s<<1. For a p-dimensional torus compact manifold:

    \[G_N\sim\dfrac{g_s^2}{M_s^2R_1\cdots R_p}\]

    \[g^2_Y\sim \dfrac{g_s^2}{M_s^6R_1\cdots R_p}\]

    \[g_i^2\sim\dfrac{g_s^2}{M_s^8R_1\cdots R_p}\]

where g_i is the KK photon coupling to the compact R_i circle. Therefore, you can derive the following results:

    \[\lambda_{gravity}\sim\dfrac{g_s^2E^{8-p}}{M_s^8R_1\cdots R_p}\]

    \[\lambda_{gauge}\sim \dfrac{g_s^2E^{6-p}}{M_s^6R_1\cdots R_p}\]

and then

    \[\lambda(gravity)\sim g_s^2\left(\dfrac{E}{M_s}\right)^8\]

    \[\lambda(gauge)\sim g_s^2\left(\dfrac{E}{M_s}\right)^6\]

    \[\lambda(KK,i)\sim g_s^2\left(\dfrac{E}{M_s}\right)^8\]

Similarly, lightly different but analogous, in Type I strings you have

    \[G_N\sim \dfrac{g_s^2}{M_s^8}\]

    \[g^2\sim\dfrac{g_s}{M_s^6}\]

Note that the gauge coupling is the only difference with respect to the heterotic case. Then, you proceed and get

    \[\lambda(gravity)\sim g_s^2\left(\dfrac{E}{M_s}\right)^8\]

    \[\lambda(gauge)\sim g_s^2\left(\dfrac{E}{M_s}\right)^6\]

Therefore, \lambda(gauge)\sim g_s>>\lambda (gravity)\sim g_s^2 in Type I strings, and \lambda (gravity, M_s)\sim g_s^2. Comparing both cases, you will note surely that \lambda (gauge,open strings,M_s)\sim g_s, and that \lambda (gauge, closed string, M_s)\sim g_s^2.

What about D-branes? Well, they are also included. D-branes are “heavy” particles in general. Take IIA 10d superstring theory. And take that

    \[g^2\sim \dfrac{1}{M_s^6}\]

    \[M_P\sim\dfrac{M_s^8}{g_s^2}\]

For a D-brane, you get that

    \[gM_P^4\sim \dfrac{M_s}{g_s}>>M_s\]

For a D0-brane, gM_P^4\sim M_s/g_s there is no change below \Lambda_{QG}\sim M_s, and thus, the relationship

    \[eM_P^{(D-2)/2}\geq \Lambda_{QG}\]

also holds for Dp-branes!!!!!!

Conclusion: any reasonable form of gauge-gravity unification implies a WGC and a limit of the number or masses of the lightest species as function of the heaviest ones! GUT, string theory or any other BSM seems to be bound through quantum gravity!!!!!! Equivalently, it seems that quantum gravity limits the number of particle species and degrees of freedom (or masses) of particle through the Planck mass, the number of dimensions and the gauge couplings. Yet, in other form, gravity is weak because unification with gauge theories limits the mass to charge ration between particles.

Remark: the WGC has surely implications in black hole physics. Extremal black holes are likely unstable (or non-existent), and they decay into charged particles. It could have experimental signatures…

Well, more is to come soon. Indeed, I have some news to give you in brief (provided my health doesn’t collapse again), and big news are happening for this blog. Probably a change of format, and well, maybe I will end posting with this style at about log#300. Life is changing and moving, and other change, like the one I did when I acquired this domain years ago are to begin when I prepare the infrastructure I am eager to own. By the way, you can follow me at my instagram @riemannium.

See you in another new blog post!!!!!!!!

 

LOG#209. A cosmic link with a non-trivial zeta zero?

Some weeks ago, the Riemann zeta function and the Riemann hypothesis were again in the news. Sir Michael Atiyah proposed a (wrong) failed proof of the Riemann hypothesis. My blog has an history of being controversial from time to time. So I am going to reveal another of my craziest ideas here. I can be wrong and it is just a curious coincidence I noted long ago but as it is short, I decided to postpone it to a transitional post like this. What is this thing about? Let me explain me better…

The cosmological constant that Einstein introduced like a property of spacetime is related to a possible new length scale (usually tied to the Hubble length in Cosmology), L_\Lambda. It is also related to de Sitter spacetimes and the so-called Garidi mass in de Sitter group representations. Taking \Lambda as fundamental constant, together the triple (h,c,G_N), you get the quadruple set of parameters (h,c,G_N,\Lambda). With this quadruple set, you can build up 8at least) two mass/length scales:

1st. M_W=\dfrac{h}{c}\sqrt{\dfrac{\Lambda}{3}}\sim 10^{-68}kg. This is known from Wesson or Garidi in de Sitter spacetime representation theory.

2nd. M_U=\dfrac{c^2}{G_N}\sqrt{\dfrac{3}{\Lambda}}\sim 10^{53}kg.

We recall that we can define the Planck length l_p^2=\dfrac{Gh}{c^3} and the cosmological length l_\Lambda^2=\dfrac{3}{\Lambda}, and you can easily prove (I have done this before in this blog) that

(1)   \begin{equation*} \dfrac{M_W}{M_U}=\dfrac{l_p^2}{l_\Lambda^2}=\dfrac{Gh\Lambda}{3c^3}=\alpha_C\approx 6.51\cdot 10^{-122} \end{equation*}

is a sort of cosmic fine structure constant \alpha_C^{-1}\approx 1.54\cdot 10^{121}. What has all this to do with my post and the Riemann zeta function? In principle nothing!!!! Just a weird numerical coincidence I noted some years ago and I did not publish because is completely crazy. We have as you probably know two different determinations of \Lambda, we can denote them by \Lambda_{th} and \Lambda_{exp}. They differ about just the above cosmic fine structure constant factor. It is huge. Theoretical physicists are just depressed when they got a prediction that it is about 121 orders of magnitude different from observational data! Now, a curious coincidence (I like to compare it to the moonshine conjecture since it involves large numbers, like those also used in Dirac-Eddington hypothesis):

(2)   \begin{equation*} \boxed{\Lambda_{th}=\sqrt{\exp\left(4\pi^2\cdot  t_1\right)}\Lambda_{exp}\simeq 1.5\cdot 10^{121}} \end{equation*}

or

(3)   \begin{equation*} \boxed{\Lambda_{th}=\sqrt[8]{\exp\left(16\pi^2\cdot  t_1\right)}\Lambda_{exp}\simeq 1.5\cdot 10^{121}} \end{equation*}

and where t_1 is the first non-trivial Riemann zeta function zero

    \[\boxed{14.1347251417346937904572519835624702707842571156992}\]

Of course, I am not pretending to say the above relation is exact or even true. It is just another moonshine-like coincidence I wanted to share with my readers and all over the world…Why the hell  the first Riemann zeta function non-trivial zero has to do with the ratio between the Planck length and the cosmological constant (Hubble) length? Nothing, unless the Riemann zeta function associated system, the riemannium, has something to do with quantum gravity and cosmology. We already know that zeroes behave like the spectrum of some “heavy” system. The nature of that system is presently just unknown. A heavy nuclei? A black hole? A complex system? A quantum quasicrystal? Maybe some n-body classically chaotic system? We don’t know.

See you in a new blog bost!

P.S.: Do you want to win one million dollars? Prove the Riemann hypothesis. Do you want to live forever? Prove the Riemann hypothesis. Fortune and glory? Prove the Riemann hypothesis!

LOG#208. Monsters and LISA.

Black holes and other astrophysical objects are true monsters. Some interesting tools from web pages to learn about these free catalogues: WATCHDOG and BLACKCAT. Also, the blackhole.org page and the sounds of spacetime for gravitational waves are suitable for you. Furthermore, you can enjoy the BH stardate online encyclopedia.

Two of the biggest and coolest (literally) black holes are quasars. Their names: OJ 287 and 3C 273. Persistent BH sources do emit X-rays, and they are powerful sources of X-rays in our galaxy (and beyond!). Transient BH events are also studied. Beyond X-rays, you study BH accretion and merging, likely BH growing if you are “lucky” or a true believer.

A list of “monsters” (not necessarily BH):

  1. Moon’s magnetic field.
  2. Asteroid with rings. 10199 Chariklo its name.
  3. Sixtail asteroids (not steroids!)
  4. Red storm (Jupiter’s famous biggest spot!).
  5. Hot Jupiter messes.
  6. HD 106906b.
  7. Uranus storms.
  8. KIC2856960.
  9. UV underproduction sources.
  10. Shape of dark matter (DM).
  11. Galaxies with age of billions of years (10 billions or more!).
  12. Iciness Saturn rings.
  13. GR bursts and fast radio bursts.
  14. Cataclysmic variable stars.
  15. Universe smoothness.
  16. Strange structures in the Universe (Multiverse?).

The new born gravitational wave astronomy is going to outer space. LISA (Laser Interferometer Space Antenna) will study a completely different set of sources beyond the ground based gravitational wave observatories. What are they?

  1. WD (White dwarf) binaries.
  2. NS (Neutron Star) binaries.
  3. BH+NS/WD systems (binaries).
  4. BH+BH mergers (supermassive or even intermediate; EMRI and IMRI are expected to be observed). EMRI=Extreme Mass-Ratio Inspirals. IMRI=Intermediate Mass-Ratio Inspirals.

LISA science targets in its timelife (2-10 years):

  1. Hubble constant (after 2 years, by binary inspirals, with accuracy of a few per cent).
  2. Equation of state of dark energy.
  3. EMRI/IMRI sources/observations.
  4. Tests of fundamental physics with gravitational waves.
  5. Ultracompact binaries.
  6. Surprises we have not expected or thought about ;).

By the other hand, beyond GW astronomy, some cool experiments are the HAWC water telescope, Cherenkov telescopes (CTA!), and the AMON network.

Primordial BH are interesting objects. In the window 10^{-16}-10^{-7}M_\odot they offer the option to be ALL the dark matter.

Gravitational wave luminosity (power!) is given for a binary (non-eccentric) system by:

    \[L_{GW}=-\dfrac{dE}{dt}=\left(\dfrac{32}{5c^5}\right)G^{7/3}\left(M_c\pi f_{GW}\right)^{10/3}\]

where the gravitational wave frequency is

    \[f_{GW}=2f_{orb}=\dfrac{1}{\mathbf{\pi}}\sqrt{\dfrac{GM}{r}}\]

and M_c=\mu^{3/5}M^{2/5} is the chirp mass, and the reduced mass \mu=M_1M_2/(M_1+M_2), with M=M_1+M_2. Sometimes, the symmetric mass ratio \eta=\mu/M is used. Non-zero binary eccentricity formula do exist but it will not be considered in this post. GW higher harmonics are useful in GW astronomy. However, for circular orbits, you have

    \[\dot{f}_{GW}=\left(\dfrac{96}{5c^5}\right)G^{5/3}\pi^{8/3}M_c^{5/3}f_{GW}^{11/3}\]

For the strain amplitude, averaging, you get

    \[h=1.5\cdot 10^{-21}\left(\dfrac{f_{GW}}{10^{-3}Hz}\right)^{2/3}\left(\dfrac{M_c}{M_\odot}\right)^{5/3}\left(\dfrac{D}{kpc}\right)^{-1}\]

Remember: the gravitational wave frequency evolution for a binary system from t_0 up to coalescence is twice the orbital frequency:

(1)   \begin{equation*} f_{GW}=2f_{orb}=\dfrac{1}{\pi}\left(\dfrac{GM_c}{c^3}\right)^{-5/8}\left(\dfrac{5}{256(t_0-t)}\right)^{3/8} \end{equation*}

where the chirp mass reads off

(2)   \begin{equation*} M_c=\sqrt[5]{\dfrac{(M_1M_2)^3}{(M_1+M_2)}} \end{equation*}

and the power of the gravitational wave is equal to

(3)   \begin{equation*} P_{GW}=\dfrac{32G}{5c^5}\mu^2\omega^6 r^4 \end{equation*}

with \omega=2\pi f_{GW} and \mu=M_1M_2/M, M=M_1+M_2.

By the way, the maximal frequency on Earth that we can detect from binary black hole mergers is related to the innermost  stable circular orbit (ISCO). Roughly, this orbit corresponds to a radius or separation from the center of mass equal to r=3r_s/2=6GM/c^2. For this orbit, the frequency should be:

(4)   \begin{equation*} f_{max,c}=f_{isco}=\dfrac{c^3}{6^{3/2}\pi GM}\approx 4.4\dfrac{M}{M_\odot} kHz \end{equation*}

Thus, ground bases observatories could catch on mergers of intermediate black holes if sensitive enough. However, they will catch more easily mergers of tens or hundreds of solar masses. What are the biggest monsters? The new created black holes species with more than millions of solar masses: the ultramassive black holes (with more than 10^{10} solar masses). Example: IC 1101, TON 618, NGC 4889, NGC 6166, NGC 1270, and others you can read off from the this wiki-list.

We will learm more about the sounds of spacetime in forthcoming entries! See you in other wonderful blog post…

P.S.: From Orosz et al. and other sources from the arxiv, you get something like this list (several versions included)

LOG#207. Beck, Zeldovich and maximal acceleration: the vacuum.

Some time ago, Zeldovich derived the following expression for the vacuum energy density:

(1)   \begin{equation*} \rho_{V}=\dfrac{Gm^6c^2}{\hbar^4} \end{equation*}

or equivalently, with a link to Caianiello’s maximal acceleration formula,

(2)   \begin{equation*} \rho_V=G_N\left(\dfrac{mc^3}{\hbar}\right)^6\left(\dfrac{\hbar}{c^8}\right)^2 \end{equation*}

Remark: the above vacuum energy density is related to the cosmological constant via the mathematical formula

(3)   \begin{equation*} \rho_V=\dfrac{\Lambda c^4}{8\pi G}=\rho_{CC} \end{equation*}

Now to far away, Christian Beck also proposed a formula for the measured cosmological constant and he derived it from pure informational axioms. It reads:

(4)   \begin{equation*} \rho_{CC}^{Beck}=\left(\dfrac{c}{\hbar}\right)^4\left(\dfrac{G_N}{8\pi}\right)\left(\dfrac{m_e}{\alpha_{em}}\right)^6 \end{equation*}

and where \alpha_{em}=K_Ce^2/\hbar c is the known fine structure constant. Why the cosmological constant is so small when our current theories based on standard Quantum Field Theories predice it should be HUGE is a mystery. We have some ideas based on supersymmetry and non-perturbative damping due to Schwinger effects that could work, but no one has managed a clear explanation. Some people believe we need a better theory. I agree partly, we need also experiments. Are the Beck formula or Zeldovich proposal right? Can we test them somehow? It is the work of future physicists to enlighten the dark issue of the vacuum energy and its radical mismatch between microscales and macroscales. What is vacuum or vacuum energy after all?

 

LOG#206. Multitemporal theories.

Newton’s gravity reads:

    \[F_N=G_N\dfrac{Mm}{R^2}\]

In extra dimensions, D=d+1, d-1=2+n, it reads

    \[F_{D}=G_D\dfrac{Mm}{R^{D-2}}=G_D\dfrac{Mm}{R^{d-1}}\]

For extra dimensions, if their size are much smaller than considered distances, R>>R_{XD}, then by matching

    \[F_N=F_D\]

    \[\boxed{G_N=\dfrac{G_D}{R^n}}\]

So, gravitational is weak in our scale because it gets diluted through extra dimensions. Real Planck scale gravity is much stronger. In terms of mass scales (large ADD scenario):

    \[\boxed{M_P^2(4D)=M_D^2R^n}\]

More generally, we can substitute R^n by a volumen V_n:

    \[\boxed{G_N=\dfrac{G_D}{V_n}}\]

    \[\boxed{M^2_D V_n=M^2_P}\]

What if you get extra time-like dimensions. Let N=n+1+d the number of dimensions. Then,

    \[\boxed{F(XT)=G^{xt}_{(ij)}\dfrac{M^iM^j}{R^d}\cos^2\theta}\]

The proof is also straightforward:

    \[F^{xt}=G_D\dfrac{\varepsilon_i M^{ij}\varepsilon_j}{R^d}\]

with \varepsilon_i the time vectors, such that

    \[\cos\theta=\varepsilon_i\cdot \varepsilon_j/\vert\varepsilon_i\vert\vert\varepsilon_j\vert\]

    \[F^{xt}=G^{(ij)}\dfrac{M^i_{\; k}M^{\; k j}}{R^d}\]

so

    \[\boxed{F^{xt}_{ij}=G_{(ij)}\dfrac{M^i_{\; k}M^{\; k j}}{R^d}\cos^2\theta}\]

where, with d=2+s, N=n+1+d=n+1+2+s=n+2+s. Therefore,

    \[\boxed{G_{4D,eff}=G_N\cos^2\theta}\]

and gravity is “small” due to the almost orthogonality of time vectors. Equivalently,

    \[\boxed{M_{D,eff}^2=M_{4D}^2\cos^2\theta}\]

We can indeed combine the extradimensional space-like case with the time-like case:

    \[\boxed{G_{4D,eff}^{(ij)}=G_N^{(ij)}\cos^2\theta}\]

    \[\boxed{M_P^{2 (ij)}=M_D^{2 (ij)} V^s \cos^2\theta}\]

Questions:

  1. What are more interesting, extra time-like or extra space-like dimensions?
  2. Why extra time-like dimensions are IMPORTANT despite being generally neglected by theorists, excepting a few excepcional cases?

LOG#205. Ether wind, SR and light clocks.

From the Michelson-Morley experiment, some clever experimentalist tried to derive the light speed through the “ether wind”. It is very similar to a river being rowing by sailors in a boat. The times for journey “upstream” and “downstream” are:

    \[cT_\parallel (1)=L+vT_\parallel (1)\]

    \[cT_\parallel (2)=L-vT_\parallel (2)\]

Journey across the ether wind uses the following pythagorean theorem:

    \[c^2\left(\dfrac{T_\perp}{2}\right)^2=L^2+v^2\left(\dfrac{T_\perp}{2}\right)^2\]

From

    \[T_\parallel=T_\parallel (1)+T_\parallel (2)=\dfrac{L}{c-v}+\dfrac{L}{c+v}\]

    \[\boxed{T_\parallel=\dfrac{2Lc}{c^2-v^2}}\]

and

    \[\boxed{T_\perp=\dfrac{2L}{\sqrt{c^2-v^2}}}\]

you get the time difference

    \[\Delta T=T_\parallel-T_\perp=\dfrac{2Lc}{c^2-v^2}-\dfrac{2L}{\sqrt{c^2-v^2}}\approx \dfrac{L}{c}\left(\dfrac{V}{c}\right)^2\]

The Michelson-Morley experiment had enough resolution to detect fringes caused by the above time difference. However, there were no time difference. There is no shift in light ways through the heavens. However, the ether hypothesis was kept until Einstein times…

Einstein derived the relativity of time with the tool of light clocks. Suppose a rest frame and a moving rocket with constant speed v.. Inside the rocket a light beam perpendicularly, to measure time in its frame. Supposing it travels d upside and d upside down, the time it yields is

    \[t_{LC}=\dfrac{2d}{c}\]

That is the lab light clock time. From the outside, the rocket light clock time is different, since it follows an oblique trajectory. The distance of one side is D, so the time in this frame will be

    \[t_{RC}=\dfrac{2D}{c}\]

By trigonometry, the base the rocket travels is x=vt_{RC}, so the semibase reads x/2=vt_{RC}/2. Let \Delta t the time interval between events in lab frame, and \Delta t' the time interval the lab sees or the rocket clock measures between some events. Again, pythagorean theorem rocks:

    \[D^2=d^2+\dfrac{v^2t_{CR}^2}{4}=d^2+\dfrac{v^2D^2}{c^2}\]

so

    \[D=\dfrac{d}{\sqrt{1-v^2/c^2}}\]

and then

    \[\Delta t'=\Delta t\sqrt{1-v^2/c^2}\]

Time moves “slower” for rocket clocks seen from outside, and measured by the lab outside. Similar arguments work out for lenght and we have length contraction! If L_0 is the length of a rod measured with a light beam in the rocket frame, and L is the length of the rod as measured in the LAB frame OUTSIDE. Front of rod crosses a point P at time t'_1 in the rocket and t_1 in the lab. The back of rod crosses a the point P at time t'_2 in the rocket and t_2 in the lab. Since

    \[L=v(t_2-t_1)\]

    \[L'_0=v(t'_2-t'_1)\]

then

    \[L=v\Delta t=v\Delta t'\sqrt{1-v^2/c^2}=L'_0\sqrt{1-v^2/c^2}\]

so

    \[L=L'_0\sqrt{1-v^2/c^2}\]

Therefore, the movin rod in the LAB frame outside appears (to the lab observers) length contracted L<<L_0. The rod would be normal from the rocket inside observers. There is an invariant interval of spacetime, as it was shown in my notes on special relativity here:

    \[\Delta\tau^2=c^2\Delta t^2-\Delta x^2\]

That number is the same in all frames moving at constant speed with respect to each other. Simultaneity is also relative, as space and time measurements as well.

What happens with energy and momentum? In the lab frame, particle has at time t position x. In the particle frame (rocket frame), we have t', x'=0. Thus,

    \[\dfrac{dt'}{dt'}=1\;\;\;\dfrac{dx'}{dt'}=0\]

Then, we form the invariant

    \[m^2c^2\left(\dfrac{dt'}{dt'}\right)^2-m^2\left(\dfrac{dx'}{dt'}\right)^2=m^2c^2\]

provided the transverse momentum

    \[p_T=mc^2\left(\dfrac{dt}{dt'}\right)\]

and the canonical momentum

    \[p=m\dfrac{dx}{dt'}=mv\]

satisfy

    \[m^2c^2\left(\dfrac{dt}{dt'}\right)^2-m^2\left(\dfrac{dx}{dt'}\right)^2=\left(p_T/c^2\right)^2-p^2=m^2c^2\]

Thus,

    \[p_T=\sqrt{p^2c^2+m^2c^4}\approx mc^2+\dfrac{p^2}{2m}\]

Note that p_T=E_{total}=Mc^2. However, p=0 yields E_0=mc^2 are rest energy.

Particles of light, from the classical side, are radiation. Wave light phenomena are classical electromagnetic waves. Usually, accelerated point-like particles of matter emit electromagnetic waves. Waves are also associated to the Maxwell field described by the pair E, B. In the quantum world, things are a little different. However, we see (yet!) phenomena like interference at the classical level!

    \[\vert A\vert^2=I=\vert A_1+A_2\vert^2\neq I_1+I_2\]

The Heisenberg uncertainty principle provides \Delta x\Delta p\geq \hbar/2. Quantum physics says that probability is related to \vert A_1+A_2\vert^2. Hydrogen atoms are quantized by Bohr rules, via L=n\hbar =hh/2\pi. The interaction of light with matters surprised people again when we found out that wave physics could NOT explain the photoelectric effect! A linear relation between kinetic energy and the frequency of light is NOT expected from wave light theory! Exercise: use what you know from the harmonic oscillator or waves to prove this fact. However, quantum light theory, as Einstein taught us, solves the issue of the photoelectric effect giving us the right theory with

    \[K=hf-W\]

Photons are quanta of light, with E=hf=\hbar\omega. Classically, beyond a different dependency of kinetic energy and frequency of light, we would obtain f=f'. However, interaction with atoms or matter quanta changes this naive idea. The total momentum and energy of light and atoms are conserved. Take p=h/\lambda for light, and E=pc. You invest some of the light momentum for make electrons free of the bounding forces at the matter surface. p=hf/\lambda changes, but the total momentum and energy is conserved before and after the photon hitting the electron and metal surface in the photoelectric effect! Interactions of light and matter are quantum in nature. Quantum interactions are more complicated due to fluctuations. However, in general, energy, momentum and angular momentum are conserved. Left-handed and right-handed electrons interact in the same way. Compton scattering is another interesting phenomenon. It can be seen as a consequence of gauge U(1) invariance associated to charge conservation. Antimatter interacts in a parallel way, only changing the sign of charge and we have also the CPT theorem in any local special relativistic framework. Annihilation of matter and antimatter becomes possible at quantum level. Radiation arises from high energy physics. Particle colliders use these facts to create particles. Quantum Field Theory (QFT) is a misnomer for a quantum mechanical special relativistic theory that allows to the particle number to vary! Number of particles changes in any QFT. Particle creation/destruction phenomena is the ABC of QFT. For instante, in Q.E.D., the QFT theory for light and matter, any gauge (electromagnetic) compesating field is A_\mu(x,t), it has a potential \varphi(x,t), and matter fields are \Psi(x,t). The Heisenberg principle applies, to yield:

    \[\Delta p\Delta x\sim h\]

    \[\Delta E\geq \dfrac{\hbar c}{L}\]

    \[\Delta p\geq \dfrac{\hbar}{L}\]

    \[\Delta E>E\]

    \[L<\dfrac{\hbar}{mc}=\lambda_C\]

Beyond light, beyond photons…What happens to quanta of MATTER? The question is complex. A complete theory for quanta of matter required time, 15-20 years, in the first third of the 20th century. Using the de Broglie relation, p=h/\lambda, just as we have a wave-like equation for “light”

    \[\partial_\mu\partial^\mu\varphi(x,t)=0\]

given suitable \varphi, the wave-like theory for electrons is much more complex because it implies the square root of the wave equation to understand that. Dirac derived the next equation in 1928:

    \[\left(\partial_x-1/c\partial_t\right)\Psi_+(x,t)=\dfrac{mc}{\hbar}\Psi_-\]

    \[\left(\partial_x+1/c\partial_t\right)\Psi_-(x,t)=\dfrac{mc}{\hbar}\Psi_+\]

so

    \[\left(\partial_x-1/c\partial_t\right)\left(\partial_x+1/c\partial_t\right)\Psi_{\pm}=\left(\dfrac{mc}{\hbar}\right)^2\Psi_{\pm}\]

Matter fields follow Pauli exclusion principle (PEP), they have negative energy states and they imply the existence of antimatter. Light is its own antiparticle and photons are bosons. Electrons and other matter field are FERMIONS. Under rotations, fermions are described by spinors, they need 4\pi radians or twists to become the same object. If not, their wavefunction changes by a minus sign! Dirac equation predicts antimatter. Positrons were discovered a years after Dirac wrote its equation (a Clifford algebra structure is behind it, to be discussed here in the near future!).

See you in another blog post!

LOG#204. Mechanics and light.

Gravitational or electrical forces are inverse squared laws:

    \[F_N=G_N\dfrac{Mm}{R^2}\]

    \[F_C=K_C\dfrac{Qq}{R^2}\]

Strikingly similar, they are both also conservative forces. For gravity:

    \[U_g=-G_N\dfrac{Mm}{R}\]

and

    \[U_e=K_C\dfrac{Qq}{R}\]

for the electrical force. Defining the potentials V_g=U_g/m, and V_e=U_e/q, you get the gravitational and electrical potentials

    \[V_g(r)=-G_N\dfrac{M}{R}\]

    \[V_e(r)=K_C\dfrac{Q}{R}\]

Conservative fields are defined from these potentials

    \[E=-\nabla V_e\]

    \[g=-\nabla V_g\]

In general, for any field \Psi, if conservative, \Psi=-\nabla V. The gravitational field reads, from newtonian gravity (module a sign)

    \[g=\dfrac{F}{m}=G_N\dfrac{M}{R^2}\]

and you would get E=-K_CQ/R^2 in the coulombian field case! Focusing on the gravitational case (a similar field could be done with the electrical field)…The momentum

    \[p=mv=m\dfrac{dx}{dt}\]

is conserved under any vertical (radial) gravitational field. Imagine you do a traslation

    \[x'=x-\alpha\]

The momentum in the x component reads p_x=m\dfrac{dx}{dt}=m\dfrac{dx'}{dt}! Note the momentum in the y or vertical component would not be conserved due to F_g! Thus, symmetry is important. Imagine a spring holding from the upper horizontal surface. Then

    \[x(t)=A\sin(\omega t)\]

where A=constant and

    \[\dot{x}=A\omega \cos (\omega t)\]

with m\omega^2=k, then

    \[m\ddot{x}=-kx\]

and

    \[E=T+U=\dfrac{m}{2}\dot{x}^2+\dfrac{k}{2}x^2=A^2\dfrac{k}{2}=\dfrac{m\omega^2 A^2}{2}=constant\]

Since the spring force is conservative, F=-kx=-kd(x^2/2)/dx, the total energy is conserved. Note the symmetry that says E does not depend on the time and it is constant!

Going 3D is important here. We will use components to avoid vector arrows for convenience. Newton’s second law is

    \[\sum_i F_i=ma_i\]

    \[v_i=\dfrac{d}{dt}x_i\]

    \[a_i=\dfrac{d}{dt}v_i=\dot{v}_i=\dfrac{d^2}{dt^2}x_i=\ddot{x}_i\]

    \[F_i=\dfrac{d}{dt}p_i\]

If F_i=0, then p_i=constant!

Kinetic energy for non-relativistic particles read

    \[T=\dfrac{1}{2}mv^2=\dfrac{1}{2}m\dot{q_i}^2=\dfrac{p_i^2}{2m}\]

If p_i is conserved, then the kinetic energy is also conserved. This is valid for the free particle. In the case of conservative forces, the potential energy reads

    \[a_i=\dfrac{d^2}{dt^2}x_i=\dfrac{f_i}{m}\]

and it yields a uniform motion with solution

    \[v_i(t)=v_{0i}+\dfrac{f_i}{m}t\]

    \[x_i(t)=x_{0i}+v_{0i}t+\dfrac{f_i}{2m}t^2\]

and

    \[f(x_i-x_{0i})=\dfrac{1}{2}m(v^2_i-v_{0i})\]

The first term is precisely:

    \[W(0\rightarrow f)=\Delta T=\dfrac{1}{2}\Delta v^2\]

And thus,

    \[W(i\rightarrow f)=-\Delta E_p=-\Delta U\]

with

    \[U=-F\cdot \Delta x\]

or

    \[f_i=-\dfrac{dU}{dx^i}\]

i. e., f=-\nabla U, Q.E.D. for any conservative force. E=T+U holds for conservative forces with certain properties in the potential energy (depending on coordinates in a homogeneus way). For the harmonic oscillator:

    \[a=\ddot{x}\]

    \[\ddot{x}+\omega^2x=0\]

and the solution

    \[x(t)=A\sin (\omega t)+B\cos (\omega t)\]

with x(0)=x_0 at t=t_0. t=t_0=0 in general, so

    \[x(t)=x_0\cos (\omega t)\]

or any other sinusoidal waveform as well. The velocity

    \[v(t)=\dot{x}=-x_0 \omega \sin (\omega t)\]

    \[a(t)=-x_0\omega^2\cos^2(\omega t)\]

and then

    \[E=\dfrac{1}{2}v(t)^2+\dfrac{1}{2}x(t)^2=\dfrac{1}{2}mx_0^2\omega^2=constant\]

as before!

Light can NOT be described with classical NEWTONIAN mechanics. It took several decades an roughly speaking several centuries to code electromagnetic laws into a single set of equations. Maxwell wrote the synthesis of our current electromagnetic knowledge of light:

  1. Gauss law for the electric field: \nabla \cdot E=div E=\rho/\varepsilon_0=4\pi K_C\rho. Equivalently, \phi=\oint_S E\cdot dS=4\pi K_CQ=Q/\varepsilon_0. For point particles, this law provides E_i=K_CQx_i/r_i^3=Qu_i/4\pi\varepsilon_0 r_i^2. Moreover, F_i=qE_i, and the gauge field E_i=-\nabla_i\varphi. \varphi=V=Q/4\pi\varepsilon_0 r_i is the potential.
  2. Faraday’s law: \nabla\times E=-\partial_t B, or equivalently \oint_\Gamma E\cdot dl=-\partial_t\oint_C B\cdot dS.
  3. Gauss law for the magnetic field (no magnetic monopoles in standard electromagnetism): \nabla\cdot B=0, or \oint_SB\cdot dS=\phi_B=0.
  4. Ampere’s law: \nabla\times B=j/\varepsilon_0c^2. This original Ampere’s law does not conserve electric charge, so Maxwell added an extra term, the displacement current, yielding

        \[\nabla\times B=j/\varepsilon_0c^2+(\partial_t E)/c^2\]

    .

The combination of the 4 equations above produces wave-like equations for E, B:

    \[\dfrac{1}{c^2}\partial_t^2 E_i-\nabla^2 E_i=\left(\dfrac{1}{c^2}\partial_t^2-\nabla^2\right) E_i=0\]

    \[\dfrac{1}{c^2}\partial_t^2 B_i-\nabla^2 B_i=\left(\dfrac{1}{c^2}\partial_t^2-\nabla^2\right) B_i\]

Plane wave solutions are allowed, with \sin(\omega t-k_i x^i) or generally

    \[E_i\sim c B_i\sim e^{\omega t-k_i x^i}\]

Wave speed is given by

    \[\dfrac{1}{c^2}=\varepsilon_0\mu_0\]

so Maxwell cleverly pointed out that light should be an electromagnetic wave! Furthermore, E\perp B\perp v in general. Light can also be polarized. Polarization or fluctuations in the directions of (E, B) is due to the transverse character of the electromagnetic waves. Maxwell’s equations unify E,B into a single framework. All the electromagnetic phenomena from a common dynamics. Special relativity allows to condense Maxwell equations into \partial_\mu F^{\mu\nu}=j^\nu and \varepsilon_{\mu\nu\sigma\tau}\partial^\nu F^{\sigma\tau}=0. Clifford algebra simplify these equations into a single \partial F=j. Differential forms also allows for such a simplification. Maxwell equations have a new invariance beyond galilean invariance: Lorentz invariance. Essentially, Maxwell equations imply Special Relativity.

In presence of matter, Maxwell equation must be completed with constitutive relations for the electromagnetic fields, plus

    \[\nabla \cdot D=\rho_{free}\]

    \[\nabla \cdot B=0\]

    \[\nabla \times E=-\partial_t B\]

    \[\nabla \times H=j_{free}+\partial_t D\]

and the boundary conditions for

    \[D=\varepsilon_0 E+P\;\;\; P=P(E)\]

    \[H=\dfrac{B}{\mu_0}+M\;\;\; M=M(E)\]

where P, M are the polarization and the magnetization for the (D, H) pair.

Remark: Beyond mechanics and light, today we care about entropic forces,

    \[F_i^{ent}=T\dfrac{\partial S}{\partial q^i}\]

Entropic forces and conservative forces can be added

    \[F_t=F_i^{ent}+F_i^{cons}=T\dfrac{\partial S}{\partial q^i}-\dfrac{\partial U}{\partial q^i}\]

Only in the zero temperature limit, we get the usual conservative terms. The above force can be obtained from

    \[A=U-TS\]

i.e., the Helmholtz free energy.

LOG#203. Action gym.

A challenge post! 😉

Quantum physics becomes important when the magnitude called action is of order of Planck constant (an action itself!). Action is quantized. It is a much more essential quantization than that of energy or angular momentum (action itself the latter!). Whenever S\sim 10^{-34}J\cdot s you have quantum effects.

Exercise 1. Compute the action from an antenna with power 1kW and frequency f=1MHz. Is it quantum?

Exercise 2. A pocket clock. It uses a device of size 10^{-4}m and mass 10^{-4}kg to get times with precision of 1s. Compute its action. Is it quantum?

Exercise 3. Compute the action for a single atomic nucleus. Typical energies are about 1 MeV and distances about 1 fm (10^{-15}m). Binding energy per nucleon is about 10^{-12}J, mass of the nucleus can be taken as the proton mass. Compute the action. Is it quantum?

Exercise 4. Compute the action for an electron in the 1s shell of the hydrogen atom. Is it quantum?

Exercise 5. Compute the so-called Fermi energy in the case of potassium (metal). The atomic weight is about 39 g/mol, and its atom density 0.86 g per cubic centimeter. Suppose a single electron per atom. Compute its action. Is it quantum?

Remark:

    \[\left[\hbar\right]=ML^2T^{-1}=\mbox{Energy}\cdot \mbox{Time}=\mbox{Momentum x Length}=\mbox{angular momentum x angle}\]

LOG#202. Harmonic oscillator.

Some clever physicists say that everything is an harmonic oscillator, and that every hard problem is just solvable in terms of a suitable set of harmonic oscillators (even true with string theory!):

In classical mechanics (CM) you have a the following standard harmonic oscillator lagrangian:

    \[L_{HO}=\dfrac{1}{2}m\dot{q}^2-\dfrac{1}{2}kq^2=T-U=\mbox{Kinetic Energy-Potential Energy}\]

The first order lagrangian given above depends upon the generalized velocities in the kinetic energy part. It provides the following Euler-Lagrange equations (EL):

    \[E(L)\equiv \dfrac{\partial L}{\partial q}-\dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{q}}\right)=0\]

For the L_{HO} given above, you obtain

    \[ \dfrac{\partial L}{\partial q}=-kq\]

    \[\dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{q}}\right)=\dfrac{dp}{dt}=m\ddot{q}\]

where p=m\dot{q} is the generalized momentum. Putting the two terms together, you get

    \[-m\ddot{q}-kq=0\leftrightarrow m\ddot{q}+kq=0\leftrightarrow \ddot{q}+\dfrac{k}{m}=0\leftrightarrow \ddot{q}+\omega^2 q=0\]

Indeed, you recognize this equation as the classical harmonic oscillator solution, that of course you can also get from Newton’s second for a Hooke’s law F=-kq. Moreover, you can also be general, and from the prescription:

    \[L(q,\dot{q};t)=T(\dot{q})-U(q)\]

derive the Newton’s law from this energetic approach, since EL applied to it implies

    \[\dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{q}}\right)=\dfrac{dp}{dt}\]

if you define the generalized momentum as

    \[p\equiv \dfrac{\partial L}{\partial \dot{q}}\]

and, by the other hand, for the potential energy depending ONLY on the generalized coordinates you get

    \[\dfrac{\partial L}{\partial q}=-\dfrac{\partial U}{\partial q}\]

Note that the last term is only the prescription for a conservative force F=-\nabla U.

Questions:

  1. What if U=U(q,\dot{q}) or U(q,\dot{q},t). Nothing changes, unless the potential energy depends explicitly on time, what renders issues to the problem. Energy could be not conserved. And generally it is not conserved, unless time is not present explicitly in the lagrangian.
  2. What about non-conservative forces? Well, there are some issues too. Several methods have been developed to account for in into the lagrangian method. E.g.: Rayleigh dissipative function, fractional calculus techniques and others.
  3. What if lagrangians act on fractional derivatives?
  4. Riewe’s mechanism using dynamics is remarkable.
  5. What about the field theory extension of the harmonic oscillator?

Furthermore, consider the canonical HO lagrangian

    \[L_{HO}\equiv L_1=\dfrac{1}{2}m\dot{q}^2-\dfrac{1}{2}kq^2\]

Next, consider the change of the lagrangian by a piece (extra langrangian)

    \[L_2=\dfrac{d}{dt}\left(-\dfrac{mq\dot{q}}{2}\right)\]

A change or variation in the lagrangian of the form

    \[L\rightarrow L+\dfrac{d}{dt}f(q,\dot{q})\]

is generally called gauge invariance for L. The addition of the total time derivative to the lagrangian does not change the equations of motion (EOM). In a field theory, the addition of a divergence (total derivative with respect all the spacetime indices) does not change the EOM. Then,

    \[L_3=L_1+L_2\]

This trick, however, has a caveat here, since I used a function f(q,\dot{q}). The lagrangian L_3 depends on the generalized accelerations and you will have to use a second order EL equations to do the job. The proof is simple:

    \[L_3=\dfrac{1}{2}m\dot{q}^2-\dfrac{1}{2}kq^2+\dfrac{d}{dt}\left(-\dfrac{mq\dot{q}}{2}\right)=\dfrac{1}{2}m\dot{q}^2-\dfrac{1}{2}kq^2-\dfrac{1}{2}m\dot{q}^2-\dfrac{mq\ddot{q}}{2}\]

so

    \[\boxed{L_3\equiv L_{HO}^{HO}=-\dfrac{mq\ddot{q}}{2}-\dfrac{1}{2}kq^2}\]

is the higher order harmonic oscillator (HOHO) lagragian! If you dislike the minus signs, you can even define L_4=-L_3 and to continue the next steps below. The second order (higher order) EL equations read as follows:

    \[E(L)\equiv \dfrac{\partial L}{\partial q}-\dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{q}}\right)+\dfrac{d^2}{dt^2}\left(\dfrac{\partial L}{\partial \ddot{q}}\right)=0\]

Now, you can recover the HO equation from this higher (second) order lagrangian. Proof:

    \[\dfrac{\partial L}{\partial q}=-\dfrac{m\ddot{q}}{2}-kq\]

    \[\dfrac{\partial L}{\partial \dot{q}}=0\rightarrow \dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot{q}}\right)=0\]

    \[\dfrac{\partial L}{\partial \ddot{q}}=-\dfrac{mq}{2}\rightarrow \dfrac{d^2}{dt^2}\left(\dfrac{\partial L}{\partial \ddot{q}}\right)=-\dfrac{m\ddot{q}}{2}\]

And so, from E(L)=0, adding the above last equations, you also get in the second order formalism

    \[-m\ddot{q}-kq=0\]

    \[m\ddot{q}+kq=0\]

    \[\ddot{q}+\dfrac{k}{m}q=0\]

    \[\ddot{q}+\omega^2q=0\]

Thus, the theories with L_1 and L_3, equivalently, L_{HO} and L_{HO}^{HO}, are completely equivalent at the level of the EOM, since they are related by a gauge transformation, or they differ by total time derivative. It is also related to (non) canonical transformations in phase space. But this will be treated in a future classical mechanics thread…

In conclusion:

    \[L_3=L_1+\dfrac{d}{dt}f(q,\dot{q})=L_1+\dfrac{d}{dt}\left(-\dfrac{mq\dot{q}}{2}\right)\]

is a higher (second) order lagrangian giving us the EOM of a single harmonic oscillator, and it is related to the canonical standard HO lagrangian via a gauge transformation (a time derivative of a function f(q,\dot{q})).

Remark (I): You can generalize this to field theory as well.

Remark (II): The theory of equivalent lagrangians and higher mechanics is subtle, but it exists.