LOG#189. Fundamental challenges.

The marriage between gravity and quantum mechanics is “complicated”. The best physicists and brightest minds have tried, but only with partial success. String/superstring theory, now M-theory, is a curious story. The another story is canonical quantum gravity, or loop quantum gravity in its more modern formulation. Let me go back-wards in time.

During the 20th century, we have created the two greatest theories and frameworks of the Human Science. They have names you know very well. Quantum Mechanics (that crazy theory) and relativity (oh, Einstein’s baby! Isn’t it?).

Quantum Mechanics is the microscopic theory of matter (molecules, atoms, nuclei, elementary particles) and energy. Despite what you have heard, there is no problem with making Quantum Mechanics a relativistic theory. It is called Quantum Field Theory (QFT). The Standard Model is a QFT covering every subatomic (till know) force (electroweak and strong), but it does NOT cover gravity.

Relativity, and here I mean GENERAL relativity, is the simplest theory that is consistent locally with both, special relativity and the equivalence principle. The equivalence principle, as Sheldon tries to explain to Penny in certain The Big Bang theory episode, states that there is only a notion of mass. Gravitational masses and inertial masses are the same. Well, indeed, there are 3 notions of mass. 2 gravitational, and one inertial. Anyway, it shows that, to an incredible precision, these masses are the same, with an accuracy of 1 part in 10^{12-14}. So, for all practical purposes, and for the common experiments or experiences we have, they are equal. Let me remember you the 3 masses:

  1. Active gravitational mass. \int g\cdot dS=-4\pi G_N M.
  2. Passive gravitational mass. P=Mg.
  3. Inertial mass. F=Ma.

I have assumed, but the result is completely general, that D=4.  This theory of general relativity is only built on the general covariance principle and the equivalence principle (some sort of Mach principle contextualized in the 20th theory…). General relativity provides Newton theory as an approximation, but it also includes new effects and an infinite set (in principle) of perturbative corrections. It also explains gravity at large or very large scales (well, indeed, excepting the dark matter or dark energy, that are just plugged by hand into the equations, with no explanation, ad hoc). General relativity explains and describes the expansion of the Universe, predicts the existence of black holes, and gravitational waves (note that there is NO gravitational waves in newtonian gravity!).

Hard marriages use to accomplish big challenges. The biggest of the challenges is that the theory predicts itself its own fall-down. The fall-down events are indeed the structure and dynamics of black holes. It makes us to ask: what is gravity? What is gravitational quantum mechanics? What is, at last, quantum mechanics?

Gravity, the first fundamental force we discovered. It was Newton genius (and likely also the hidden role of Hooke, the most hated person by Newton, defenestrated) who discovered gravity, but some people discuss from time to time if it was discovered by Hooke…Anyway, returning to our theme, Newton (and/or Hooke) managed to unify terretrial mechanics with celestial mechanics. And it yields Universal Gravitation. In fact, there were suspicions by Galileo (the first modern scientist), supported by the Kepler and Copernicus data and observations, that the geocentric view of the Universe was unsustainable. Likely, even some Greek people like Hypparcus also knew it a part of all this stuff.

Gravity is an engine! The simplest way to realize it by Galileo: take any inclined plane. An inclined plane is a machine that transforms height H into energy. In fact E\propto v^2\propto H. You can do it yourself. Pick any round object and an inclined plane. Leave the object at certain H and observe what happens. That was done by Galileo hundreds of times! The coolest thing about this observation is that the fuel it uses is…Gravity. Technicaly, as you learn from High School:



Then, you get v=\sqrt{2gH}, and \dfrac{1}{2}mv^2=mgH. Have you ever imagined that inclined planes were engines fueled by gravity? It is something beautiful. These arguments produce a conceptual revolution in philosophy. And a conceptual revolution in physics, due to the introduction of the notion of energy by Leibniz (another Newton’s foe, but this time, protected such as he managed to teach you differentials and calculus from a more intuitive way than that fluxion calculus by Newton). Descartes, trying to make energy more geometrical, introduced the notion of linear momentum (p=mv). In fact, Leibniz’s monads have a secret link with his findings in physics and mathematics (physmatics, yeah). Monads are equivalent to energy. Monads are the dynamical atoms of the Universe in those times. As a consequence of this thing, the same any inclined explains, it allows us to get a clock. As the YM^2=Gravity says, double the bet. Take 2 inclined planes. You can make any object to “oscillate” between 2 arbitrary inclined planes, if you do it properly. Furthermore, it turns out that the own solar system, as the ancient ones taught us (but using religion or myths, not math, not physics or physmatics), any pair of celestial bodies can be used as CLOCKS. Using the universal law of gravity, and the notion of centripetal force, you obtain


And from this equation, you get the Kepler 3rd law, relating period (time, clock, tick tack, tick tack,…) with distance between two astronomical objects:


I am assuming the space-time is 4D, so the space is 3d. Using the same trick, but with atoms instead of gravitating objects, you see that the same is also true in atomic physics. Any pair of bodies also act as clocks in motion. Since Aristotle times, celestial bodies have served us (well, indeed from much more ancient times), eternal clocks. However, as we do know not, they are not eternal, only they life much longer than a human life. Even more, the whole Universe is a machine made by gravity, taking the machine as an equation


From this viewpoint, the force F_N is associated to motion. The gravitational constant G_N is the universal conversion factor, the energy or fuel is mass (energy in relativity) M_1M_2, and the height is just R_{12}, similarly to the inclined plane. Every celestial body falls down. Even the Doctor falls down, despite he regenerates. The fascinating thing with G_N is that it is not a pure number, it has dimensions. It is not like the Reynolds number in fluid theory, or \pi in mathematics. It is similar to other constant, G_F, the Fermi constant for weak interactions. Fermi constant and the weak interaction explains radioactive decay. Due to weak interactions, some particles are unstable and not eternal, e.g., the neutron. Some grand unification theories (GUTs) predice that even protons do decay (but they hay a very long lifetime). Mathematically speaking


and G_N has units of m^3s^{-2}kg^{-1}=Nm^2/kg^2. In general relativity, we have something just a bit better. As we merge space and time into space-time, if we do not distinguish length and time, we must introduce a new scale or conversion factor. It is the speed of light c=299792458m/s. In fact, c=LT^{-1}. Physical dimensions are categories (category theory fans reading me just know?) that matter into physics. They quantities. Extra example: the fine structure constant is a pure number, created in this way

    \[\alpha_{em}=\dfrac{K_Ce^2}{\hbar c}\approx \dfrac{1}{137}\]

With units such as c=\hbar=K_C=1 you obtain \alpha \cdot 137=e^2.

What is a black hole (BH)? Well, the details are NOT fully now. But there is some ways to understand a BH withe the fashion I have explained here. A black hole is something that traps light. A black hole does not let light to escape. BH are interesting because light is interesting. So, well, what is space-time? What is light? Light are waves or quanta of electromagnetic fields. You are reading me through the screen of your computer or e-device. The device is at certain distance d from your eyes. You are seeing me in the past (OK, only after some minutes since I published and a few fractions of a second. If your eyes are about a meter from the screen, the light uses about 3 nanoseconds to arrive to your eyes and some negligible time to be process in your brain. See any portrait of Nature, a nice girl/boy means that the light travels from them towards us. But, since the speed of light IS finite, it requires some time t_L=d/c. That is common knowledge, and it is used by astronomers, astrophysicists, and collider physicists from all over the world. See the stars (the sun or any other) means to see into our past, or the past of the Universe. You have a time machine every night. For visionaries, a question: would you look forward and away because you could travel with v>c violating special relativity in 3+1 space-time? Hint: multi-temporal relativity and other forgotten relativities (I promise, I will post about them in the future) can go beyond the speed of light, with a price.

Telescopes are real time-machines. The night-sky is a wonderful cosmic TV! It is a screen. Not just like a cinema or theater, but also a nice screen. Space-time is a network. It is just like a continuum limit of a lattice made up with light rays. To understand the space-time IS to interchange concepts like length L or time T in a dual way, using the speed of light. In the same way, it shows than mass and energy are the same thing as well, only a conversion factor is necessary. And as you know, E=Mc^2, don’t you? Moreover, as any physical velocity of the universe, excepting massless force carriers, is less than c, force carriers travel to the speed of light if they are massless and gravity alters not only matter, but also to light rays since they carry energy E=\hbar \omega. Now, we have two constants G_N, c to play with. So,

    \[\dfrac{L^3}{T^2}=L\dfrac{L^2}{T^2}=L c^2\]

Therefore, using the definition of G_N and its dimensions, we can form a gravitational length associated to any relativistic object. Up to a pure number, it reads


Essentially, experts might say, it is half of the Schwarzschild radius. Note that if the speed of light were infinite, the gravitational length would be ZERO. In reality, the Schwarzschild radius is about


and the gravitational length defined above is


This characteristic length does exist for ANY object in the Universe. Even for you. Neglect any other interaction, and plug a mass there. For instant, for the sun is 1.5 km, for Earth is about 4 mm (the Schwarzschild radii of these objects are 3 km and 8 mm, aproximately). In real life you get a short “life”. Gravitational lifetime is much shorter. The hidden reason you don’t notice your gravitational length is that real life scales are much bigger than gravity scales. However, if you were ablo to turn-off any interaction excepting gravity, your length would be L_G. Did Marvel know this? Did Ant-Man know? We have arrived to a surprising new result. Black holes are what is left if you turn-off everything excepting gravity. BH have only mass (or some charges as I told you in the previous post). Of course, this is the no-hair theorem, that could be wrong, or softly wrong, with the recent new researches.

Stage two is Quantum Mechanics. Classical Mechanics is a sort of code or tool that allow us the identify between what is possible it happens, and what it do happens. During the 18th and 19th century, analytical mechanics introduces a new object into physics. It is called the action. Action is usually denoted by S (please do not confuse it the entropy S, another S). Action allows derive the Newton laws from other principle, much more general. It is called the minimal action principle. It is an analogue to the maximal entropy (Max-Ent) principle in thermodynamics. Then, what is action? S is not for Superman, hehehe. Geeknerd quote: “It is not an S. In (our) my planet it means hope (quantization!)”. Dimensions of action are:

    \[S=MT\cdot c^2=ML^2T^{-1}\]

Conceptually, action is a simple thing. Any action is the product of the energy you use in some process by the time the process lasts. Using space-time dimensions, action is:


that is, action IS linear momentum times spatial size or a type of angular momentum times a phase (adimensional phases are). The quantum revolution is born. There exists a MINIMAL VALUE for the action. That is named quantum of action. That is \hbar or h. Classical physics is just the theory in which the minimal action is ZERO, or just, you take the limit in which the quantum of action is zero. Classically, there is nothing that avoids you a finite variation of energy in a instant (\Delta t=0). Quantum Mechanics is different. The new axiom or postulate (physical or mathematic) is that you get a minimal quantum of action h or \hbar=h/2\pi\approx 1.1\cdot 10^{-34}J\cdot s. This happened in 1900. Max Planck had to do it in order to explain the blackbody radiation. Thus, energy quantization is not truly fundamental. Action QUANTIZATION is MORE FUNDAMENTAL. To any minimal action, you get a new quantum length scale. It is the Compton or Quantum wavelength


Old atomists believed that atoms were static, and that mass was a sum of invariable atoms. The new atomism, granted by Quantum Mechanics (and its statistical interpretation), states that atoms are dynamical. This was highlighted by L. Boltzmann, but he was bullied with terrible results for his own life.

That action is a sum of quanta of action is something you could remember from school, if you learned about the Bohr-Sommerfeld correction to the Bohr atom. That quanta of action are what matters, drives you into a deep question about the localization of energy. After all, quantum length is L_Q\propto 1/M, and it is independent of G_N. Einstein, in 1905, envisioned how to apply the localization of photons with light quanta to explain the otherwise paradoxical photoelectric effect. Photoelectric effect is quantum energy being localized. Now, we have three scales for any object. Its own normal size L, the quantum size L_Q, and its gravitational size L_G. L>>L_G>>L_Q in general. For instance, any electron has L_Q>>L_G>>L. Therefore, the electron IS quantum. Real life for humans is generally classical L. How many atoms of action you have for these objects? Imagine some object, like a house. It has a mass M. Houses has a size L. The number of quanta of action of houses you get is, for normal houses:


If the house shows to be gravitational, the number of gravitational houses (boxes) are

    \[N_G=\dfrac{ML_G}{\hbar} c\]

and if the house shows to be quantum


The ideal house, or box, with zero size does not exist in our Universe. If you do the same for the electron, the platonic electron has normal size non-null (you have ensembles of electrons in atoms!), gravitational electrons do not exist (to our knowledge, L_G<1), and for quanta of electrons you easily get N_Q=1. Of course, the deep question is why elephants, cars, houses, humans or entities are NOT electrons. Quantum fluctuations are minimal changes of action, i.e., N\rightarrow N+1 or N\rightarrow N-1, but you can get others exciting more quanta. The macroscopic world, for elephants, cars, houses, humans, or entities is a world where N\sim N+1. When N\neq N+1 you get the QM weirdo. Quantum particles, like the electron. Thus, what is Quantum Gravity? There is no a unique answer. Experts differ about it. According to these lines, since gravity introduce L_G and atoms of action N_G depending on mass in a universal way, quantum gravity must be a theory with quanta of action for certain quanta of action, these quanta of action are UNKNOWN yet, but they are provided in an invariant way by the equations

    \[N_G=\dfrac{ML_G}{\hbar}c=\dfrac{ G_NM^2}{\hbar c}=\dfrac{G_NE^2}{\hbar c^5}=\left(\dfrac{E}{E_P}\right)^2\]

where the Planck energy is about 10^{19}GeV=10^{16}TeV. Thus, quantum gravity introduces a fundamental new number into physics. The minimal number of quantu of action for gravity N_G. It is not 1 but N_G(M). It depends on G_N, \hbar, c. When gravity is turned on, we have to push up the energy. For classical physics, this new quanta of action are zero, for quantum physics is about 1, but for quantum gravity you have to lift it up. To get this easily, you can see yourself, than when you put L_Q=L_G you get the Planck mass, or equivalently, you get the Planck length. If you do it, you will convince yourself of all this thing. Additionally, if you introduce the vacuum energy density as being not-null, you introduce a new quantity, the cosmological constant. It introduces you new length scales (how many scales are there after all?). The idea is that even the vacuum has a “minimal mass” or energy. One option is to use the scale:


Note that G_N is not there, so it is not gravitational at all! Also, you can introduce the scale

    \[R_\Lambda=\sqrt{\dfrac{3}{\Lambda}}=R_U\approx 10^{26}m\]

Compare it to the Planck length, 10^{-35}m. The classical radius of the electron is about 10^{-15}m, similar to the nuclear size. From M_W you can get a scale R_W, and take, e.g., R=(R_P^2R_W)^{1/3}\sim 1 fm, where R_P=L_P is the Planck length, and you get just like the nuclear size again. The remaining scale with gravity is


The radii or length scales:


    \[R_W^{'}=\dfrac{\hbar c}{G_N}\sqrt{\dfrac{\Lambda}{3}}=L_P^2/L_\Lambda\]


represents different systems. The quantity g=c^3/G\hbar\Lambda\sim 10^{-120}-10^{-122} is really tiny. Note that M_WM_W^{'}=M_P^2, R_WR_W^{'}=R_P^2 and that there are also a duality of M_W alone with respect to M_P. How many mass scales are there, then? How many quanta of action? The cosmological quanta of action is very mysterious. You could find

    \[n_\Lambda=\dfrac{M_\Lambda}{M_W}=N^{1/4}\sim 10^{30}\]


    \[N_\Lambda=\dfrac{M_W^{'}}{M_\Lambda}=N^{3/4}\sim 10^{90}\]

Remark: R_W^{'}<<L_P. Thus, for elementary particles in quantum mechanics one should have M_\Lambda<M<M_P, for QM holes (?) one could guess

M_P<M<M_\Lambda^{'} one could expect classical black holes. Let me put the five scales of mass into numbers:

1st. M_W=\dfrac{\hbar}{c}\sqrt{\dfrac{\Lambda}{3}}\approx 10^{-67}g\approx 6\cdot 10^{-35} eV/c^2

2nd. M_\Lambda=\sqrt{M_PM_W}\approx 10^{-35} g\approx 6 meV/c^2.

3rd.M_P=\sqrt{\hbar c/G_N}\approx 10^{-5}g=6\cdot 10^{27} eV/c^2=6\cdot 10^{6} ZeV/c^2.

4th. M_\Lambda^{'}=M_P^2/M_\Lambda\approx 10^{25}g\approx 6\cdot 10^{57}  eV/c^2.

5th. M_W^{'}=M_P^2/M_W\approx 10^{56}g\approx 6\cdot 10^{88} eV/c^2.

There are two additional mass scales, you can make:

6th. M_T=\left(\dfrac{\hbar^2\sqrt{\Lambda}}{G_N}\right)^{1/3}\approx 10^{-28}kg\approx 60 MeV/c^2. Note that there is no speed of light here!

7th. M_T^{'}=c\left(\dfrac{\hbar}{G_N^2\sqrt{\Lambda}}\right)^{1/3}\approx 10^{12}kg\approx 6\cdot 10^{47} eV/c^2.

As final ending, let me point out something I missed in previous posts. The Beck deduction of \Lambda with quantum information theory uses the formula

    \[\Lambda=\dfrac{m_e^6G_N^2}{\alpha^6 \hbar^4}\]

It has been deduced as well by Harko, using diffent arguments, to provide

    \[\Lambda=\dfrac{L_p^4}{r_e^6}=\dfrac{\hbar^2 G_N^2m_e^2c^6}{e^{12}}\]

and it gives, as I mentioned, the observed value \Lambda\sim 10^{-56}cm^{-2}. Of course, if this number and formula is correct or just a numerological coincidence, must be elucidated yet.

In summary, quantum mechanics and general relativity ARE important because:

  1. Quantum mechanics introduces quanta of action. It is not only that energy is quantized. Everything is, at the end, quantized. Quantum mechanics is the theory to consider if you want to describe something light and small.
  2. General relativity introduces space-time and G_N, c^2 on equal footing. It is the theory you need in order to describe something BIG and MASSIVE.
  3. Quantum gravity is the theory you need to know how to describe particles that are both, small and very massive. No one had complete success. Maybe, quantum mechanics and general relativity as they are formulated have to be modified. Are QM and GR effective theories at the end? Is space-time (or the quantum of action) emergent?

Some (right or not, who knows?) hints:

  1. The measurement problem.
  2. QFT is not (to my knowledge) affected by maximal acceleration/force, A=Mc^3/\hbar=Ec/\hbar. Maximal or critical acceleration plus newtonian gravity gives you Planck length (check it yourself!).
  3. Quantum rest is impossible due to the HUP or GUP.
  4. Quantum fluctuations of vacuum implies a fluctuating non-null vacuum energy density. QFT gives a wrong result…By 122 orders of magnitude. SUSY does not solves it all…
  5. Electrons can not be extremal black holes, or can they? The Schwarzschild radius of any electron is about 10^{-57}m. The RN radius for an electron R_Q^2=K_CG_Ne^2/c^4 is about R_Q\sim 10^{-36}m<L_P.
  6. Einstein idea of elementary particles as singularities can not be correct. Can they? The issue of geodesic motion in general relativity of such a particle is crucial.
  7. Primordial black holes (PMB), i.e., black holes created in the early Universe not as result of the remaining hypermassive stars but as result of the fluctuations in density due to inflation or post-inflation could be the dark matter we are searching for. The PMB window is essential in the range of mass 10^{14}-10^{23}kg.




LOG#188. Scales of (quantum) gravity?

Hello, world! In the paper Notes on several phenomenological laws of quantum gravity, by Jean-Philippe Bruneton, you find a very interesting discussion I am going to enlarge about the scales of (quantum) gravity. With Planck quantities:




and combinations we can obtain almost every physical quantity. I disagree, in part, since we also need k_B and K_C=1/4\pi\varepsilon_0, but it is essentially correct. However, why not instead (G_N,\hbar, c) take (M_P, L_P, T_P) as fundamental? Indeed, a maximal force/tension arises from c, G_N with F_M=c^4/G_N (there is a mysterious 1/4 factor there, tracked back to the Bekenstein-Hawking formula? Not so clear to me, but it is interesting how numbers and variables appear in physics or physmatics. If you take v\leq c as fundamental basis, the Planck constant and G_N can be derived

    \[\hbar =M_PL_P\]


In fact, there is a natural notion of invariant mass, a mass bound, in D=4. Any physical system with size L has a maximal mass about M\leq Lc^2/G_N, modulo a proportionality constant. In covariant way, any 4D space-time free of horizons, and a lightlike surface B, codimension 2, the integrated mass on the light-sheets L(B) satisfies

    \[M(L(B))^1\leq \dfrac{A(B)}{16\pi G^2}\]

This bound is related to Bousso seminal works about the holographic principle. Next, Jean-Philippe uses the area of a Kerr-Newmann black hole to derive a set of rules any physical mass should verify:

  • Any physical system free of horizons lie above the line L=2M, strictly.
  • Any physical system with an event horizon has M\leq L\leq 2M.
  • No know physical systems lie below the critical line L=M.

The existence of the quantum of action is “derived” from the bound

    \[ML\geq \dfrac{\beta \hbar}{c}\]

for some pure number \beta. Heisenberg uncertainty principle (HUP) suggests \beta=1/2. This is independent of the number of spacetime dimensions D. In fact, the quantum of action can also be derived from the area bound

    \[A>\dfrac{4\pi \beta^2\hbar^2}{M^2c^2}\]

but it implies certain class of \hbar depending on the number of spacetime dimensions! This last quantity can be compared to the gravitational analogue bound

    \[A>\dfrac{16\pi G_N^2M^2}{c^4}\]

Note that these expressions can be related through a duality

    \[\dfrac{\hbar}{Mc}\leftrightarrow \dfrac{G_NM}{c^2}\]

In higher dimensions, we can write

    \[A_{D-2}^{D-3}\leq G_DM^{D-2}\]

where pure numbers have been neglected and A_{D-2} is the area of a codimension 2 surface, G_D is the newtonian gravitational constant in D dimensions of spacetime. From this,


and the conjecture

    \[L\geq M^{1/(D-3)}\]

follows naturally, in terms of a length L=A_{D-2}^{1/(D-2)}. So, we have


while the Planck action constant is now D dependent via

    \[\hbar (D)=\dfrac{M_PL_P^{D/2}}{T_P}\]

and thus, only in 4D it has dimension of action! From a dynamical viewpoint, consider the Einstein-Hilbert action in D dimensions:

    \[S_{EH}=\dfrac{1}{G_D}\int d^D x\mathcal{L}\]



and assume S_D has dimensions of \hbar (D). It implies that the gravitational lagrangian has dimensions of L^{D/2-4}, and it is a curvature only when D=4. As this seems quite unlikely, anyone can build lagrangians (or actions) with the curvature scalar or its generalizations, but these actions are not usually of dimension of \hbar (D). Could it explain why hamiltonian canonical methods are hard with usual methods? What are we missing here? In any case, if \hbar is dependent on the space-time dimension D, it would have far reaching consequences. Specially on extradimensional physics. \left[X,P\right]=i\hbar_D would be a crazy thing…

The next plot is worth mentioning:

The generalized uncertainty principles could allow us to infer some features of quantum gravity, if true. The transplanckian problem is a well-know issue of current approaches of quantum gravity. Quanta in a box satisfy \lambda =L/n, M=E=n/L. The minimal Planckian wavelength can be a bluff of our ignorance. Using dualities


or in terms of frequencies


Only when the wave-length approaches L_P^2/L the black box contain the quanta saturate the mass bounds. Indeed, minimal resolution does not seem to imply a minimal wave-length according this paper we are discussing today. Similarly, with transplanckian momenta


Anyone could protest! After all, we all know the IR-UV regulators in Quantum Field Theory. However, they are related to ignorance and details of interactions and degrees of freedom. Even when physical systems can not be smaller than Planck wavelengths, sub-Planck wavelengths could be allowed! Of course, this is weird…But relativity and quantum mechanics are weird! We should expect quantum gravity in its final formulation be even more weirdo. Wild speculations from relatively unknow researches are related to the existence of the alledged maximal force/tension, or maximal acceleration in Nature. Caianiello and others have studied it. You have also

(1)   \begin{equation*} \boxed{a_P=\left(\dfrac{c^7}{\hbar G_N}\right)^{1/2}} \end{equation*}

that is about 10^{51}m/s^2 when put into numbers. This planckian acceleration is not Caianiello’s maximal acceleration. Interestingly, the paper mention the odd result I am going to tell you now. For scales M>1 the maximal acceleration is the Caianiello’s value


but for M<1, you guess a classical bound


Note that acceleration is like M in natural units, in the Caianiello case, and about 1/M (duality?) in the classical case! Caianiello’s maximal acceleration has an apparent advantage: it does NOT depend on G_N. Caianiello trials to generalize or derive quantum mechanics from geometry were interesting. They are linked to Finsler-like geometries (phase space-time!):



in eight-dimensional spaces! With a proper time, this can be rewritten as follows

    \[-d\tau^2=dx_\mu dx^\mu+\dfrac{\hbar^2}{M^2c^2}d\dot{x}_\mu d\dot{x}^\mu\]

Going beyond, you can also include higher derivatives of x, to get

    \[-d\tau^2=dX_\mu dX^\mu+\dfrac{\hbar^2}{M^2c^2}d\dot{x}_\mu d\dot{x}^\mu+\kappa\dfrac{\hbar^4}{M^4c^4}d\ddot{x}_\mu d\ddot{x}^\mu+\ldots\]

for some constant \kappa. It is some kind of non-local geometry, … Higher-order geometries of Finsler-like type have been studied by Kawaguchi. Another source of acceleration is Hubble parameter (via cosmological constant, if you wish), since

    \[a\sim H_0T_Pa_P\sim 10^{-10}m/s^2\]

However, that seems much more a MINIMAL ACCELERATION, of hidden quantum or subquantum nature. From the above previous arguments, up to constant pure numbers, you get maximal force and maximal powers for the two scales M<1, M>1, as



Maximal energy density (and pressure) are also a follow-up:


for M>1, and


if M<1. The maximal action principle the paper suggests is also a variation of the maximal acceleration principle, and it can be related to the currently popular maximal complexity principle:

    \[\dfrac{d\mathcal{C}}{dt}\leq \dfrac{2M}{\pi}\]

The scale of gravity in the IR and the UV has to be proved more and more. After all, the “constant” G_N is the fundamental constant we do know with worse precision (or accuracy). However, the cosmological constant puzzle has introduced new scales, as I mention in previous posts. Of dark energy? New physics? Let me introduce de Beck-Wesson scales, and the Garidi mass:



Caianiello’s acceleration can also be written in terms of energy as well


and it can be also understood as some type of gravitational Schwinger critical acceleration. The Schwinger effect is related to the creation of a pair of particles by any strong field. For the electric field, in Q.E.D., you can use school physics to derive the critical electric (or magnetic, via E=Bc) as follows. Take a electric field and increase it until you have enough energy to pop-out a particle-antiparticle pair. How strong must the electric field be? Supposing you have plates separated by twice the Compton wavelength of the particle, \lambda=\hbar/Mc, the work to be done is:

    \[qE\cdot 2\lambda=qE\cdot 2\dfrac{\hbar}{Mc}=2Mc^2\]

so the field must be at least

(2)   \begin{equation*} \boxed{E_c=\dfrac{M^2c^3}{q\hbar}} \end{equation*}


(3)   \begin{equation*} \boxed{B_c=\dfrac{M^2c^2}{q\hbar}} \end{equation*}

If you do the same for the gravitational field, you get Caianiello’s acceleration! The easiest way is to put q=M in that case, but you can also derive it from

    \[2\Gamma g\lambda=2mc^2\]

in a similar fashion and it can be generalized to any dimension! A Schwinger ensemble of critical photons (or electrons) in D dimensions will satisfy (check it yourself):





    \[L_S=\dfrac{K_C(D) \hbar q^2}{M^2c^3}\]

This kind of quantization has been proposed by Dvali and Gomez in the Quantum N-portrait interpretation of black holes. For gravity, the same dimensional argument produces


And from these relationships, you can even guess the maximal complexity principle of the Bekenstein-Hawking black hole entropy in any spacetime dimension up to a pure numerical factor. There is a deep link between the equivalence principle and the Hawking or Unruh effects. They are equivalent, provided certain class of the equivalence principle holds. The Unruh-Hawking-Schwinger effects are indeed all THE SAME thing. I mean, they have the same origin. Vacuum. Void.

To end this post, something off-topic…Have you ever wondered the cross-section for two photons become two gravitons or vice versa? It can be done. Skobelev and De Witt did it, to my knowledge, and the result has been rederived with modern mathematics in recent times. The integrated cross-section is:

    \[\sigma (\gamma\gamma\leftrightarrow GG)=\dfrac{\kappa^4\omega^2}{160\pi}\]

and the differerential cross-sections are given differently from Skobelev

    \[\dfrac{d\sigma}{d\cos\theta}=\dfrac{\kappa^4}{64\pi}\dfrac{1+\cos^8 (\theta/2)}{\sin^4 (\theta/2)}\]

and De Witt

    \[\dfrac{d\sigma}{d\Omega}=(2G_NE)^2\dfrac{\cos^{12} (\theta/2)}{\sin^4 (\theta/2)}\]

where E=\hbar \omega and \kappa is related to Newton gravitational constant. Can you check they are equivalent? See you in another blog post!

LOG#187. Hypothetical particles: the list.

Fast post! List of particles not yet confirmed or completely confirmed to exist:

  1. Preons. (“boojums” and “snarks”?)
  2. Pre-preons.
  3. Prequarks.
  4. Rishons.
  5. Subquarks.
  6. Haplons.
  7. Hexads/Hexons (and trons?).
  8. Bradyons (tardyons, ittyons).
  9. Superbradyons.
  10. Elvisebrions.
  11. Erebons.
  12. Darkons.
  13. Z'.
  14. W'.
  15. X-bosons, X-particles.
  16. Y-bosons, Y-particles.
  17. Luxons (beyond the gluon, photon, graviton).
  18. Planckons.
  19. Maximons (and minimons!).
  20. Tachyons.
  21. Magnetic monopoles.
  22. Dyons.
  23. Tetraquarks.
  24. Pentaquarks.
  25. Hexaquarks.
  26. Multiquarks (N>6).
  27. Glueballs (not yet identified).
  28. Chameleon particles.
  29. Dark photon.
  30. Dilaton.
  31. Dual photon.
  32. Magnetic photon.
  33. Yershovian preons.
  34. Gravitons (should we wonder, having detected gravitational waves?).
  35. Non-linear gravitons (Penrose’s proposal).
  36. Pressuron.
  37. Sterile neutrino (seesaws predict a NEW energy/mass scale not being the Planck scale, but M_N, such as m_\nu=M_D^2/M_N; to get a realistic Dirac mass, the Planck scale is not valid, you need M_N\sim 10^5-10^{12} GeV or M_N\sim 10^2-10^9 TeV in order to use seesaws as neutrino mass generating mechanism).
  38. Acceleron.
  39. Bilepton.
  40. Graviphoton.
  41. Graviscalar.
  42. Majorons.
  43. Majorana particles (fermions being their own antiparticle).
  44. Sparticles (supersymmetric particles).
  45. Wino.
  46. Zino.
  47. Sneutrino.
  48. Squark (sup, sdown, sstrange, scharm, sbottom, stop).
  49. Sfermion (sneutrino, selectron, smuon, stau).
  50. Saxion.
  51. Neutralino.
  52. Sboson.
  53. Photino.
  54. Gravitino.
  55. Higgsino.
  56. Gaugino (sbosons).
  57. Gluino.
  58. Higgsino.
  59. Curvaton.
  60. Doubly charged Higgs bosons (predicted by some seesaws).
  61. Exotic hadron.
  62. Exotic baryon.
  63. Exotic meson.
  64. Cosmon.
  65. Black hole electron.
  66. Holeum.
  67. Macroholeum.
  68. Inflaton.
  69. Plekton.
  70. Q-ons.
  71. QP-ons.
  72. Pomeron.
  73. Skyrmion.
  74. Sphaleron.
  75. Unparticles.
  76. LSP (lightest supersymmetric particle).
  77. R-hadrons.
  78. Minicharged particles.
  79. Fractons.
  80. Monopolium.
  81. Mirror/shadow matter.
  82. p-form fields (e.g., Curtright fields). They appear is superstring/M-theory. 2-branes and 5-branes are naturally coupled to 3-forms and 6-forms (gauge potentials), i.e., they are driven by p-form curvatures (4-forms, 7-forms).
  83. Nothoph.
  84. Anyons.
  85. WIMPs (Weakly Interacting Massive Particles).
  86. SIMPs (Strongly Interacting Massive Particles).
  87. KK-particles.
  88. Technifermions (predicted by technicolor theories, now dormant due to the Higgs boson discovery…Soon rebooted?).
  89. BPS-states.
  90. Subquantum particles.
  91. Superstrings.
  92. p-branes/Dp-branes (critical, fundamental).
  93. Virtual black hole (virtual p-branes…).
  94. Infotons.
  95. Chronons.
  96. Choraeons.
  97. Topons/Spin networks.
  98. Chromons.
  99. Flavons.
  100. Geons (particle-like version).

Did you know any other? Let me know…By the way, some hypothetical astrophysical objects (macroparticles):

  1. Preon stars.
  2. Boson stars.
  3. Planck stars.
  4. Fluid or superfluid stars.
  5. Black stars.
  6. Dark stars.
  7. Quark stars.
  8. Strange stars.
  9. Cosmic strings.
  10. Kugelblitz.
  11. Gravastar.
  12. Fuzzballs.
  13. Quantum star.
  14. Exotic Compact Object (ECO).
  15. Ultra-compact object (UC).
  16. Wormholes.
  17. Cosmic strings.
  18. Cosmic defects.
  19. Black holes (what are they, at the end?).
  20. Gamma Ray Bursts (what are they, at the end?).
  21. Fast Radio Bursts (what are they, at the end?).
  22. Cosmic rays (origin?).
  23. White holes (really equal to black holes?).
  24. Magnetospheric eternally collapsing object (MECO).
  25. Extremal black holes.
  26. Superextremal black holes.
  27. Geons (likely, the thing which made Star Wars light sabers possible…).

With respect to black holes (BH), honouring Stephen W. Hawking 75th birthday, let me add some types of black holes (vacuum solutions to Einstein Field equations). By size, BH can be (exotica like Holeum or macroholeum will not been considered):

  1. Quantum (planckian) BH. I will include here the hypothetical subplanckian radii BHs, if any…
  2. BH electron (or particles).
  3. Micro/mini black holes.
  4. Stellar mass black holes, SMB. (about M_\odot-100M_\odot).
  5. Intermediate mass black holes. IMBH. (10^2M_\odot-10^6M_\odot).
  6. Supermassive black holes, SMBH. (about 10^6M_\odot-10^{10}M_\odot, is the upper limit of BH mass justified? Can they grow even more massive?). SMBH are know to the engines of quasars, blazars and active galactic nuclei (AGN).

The stars like white dwarfs or neutrons stars can evolution into BHs. Specially if they are in binary systems. Binary BHs are the target of gravitational wave observatories (gravoscopes). LIGO has found some binary BH mergers. They have cool names. Specially, GW170104. According to the type of charges BH have (astrophysical BHs are expected to be fully characterized by Kerr solutions, but corrections are envisioned if you believe, as me, that general relativity is not the fully story…), they are:

  1. Schwarzschild BHs. Spherically symmetric, static. Only have mass M and position in space-time. (X,M).
  2. Rotating BHs. In addition to mass, they have spin J. Specified by (X, M, J), or equivalenctly, by M, a, where a=J/Mc.
  3. Charged BHs. Also called Reissner-Nordstrom black holes. They have charge and mass, i.e. (X, M,Q). The charge can be usually electric only, but BHs with electric and magnetic charges can be also constructed, so Q is such as Q^2=Q_e^2+Q_m^2.
  4. Cosmological BHs. They have cosmological constant, so they are dS, AdS black holes. X, M,\Lambda.
  5. Rotating, charged BH. It is called Kerr-Newmann black hole. (X, M, J, Q).
  6. Rotating, charged BH with cosmological constant. (X, M, J, Q, \Lambda).
  7. Taub-NUT like solutions. They include the gravimagnetic mass or NUT parameter N, such as m=M+iN, and m^2=M^2+N^2. So, they are (X, M, N).
  8. Petrov type D BH solutions, also named Plebánski-Demiánski black holes. The most general, to my knowledge, BH solution. Here I am using BH solution as a misnomer (likely) of EFE solution (soliton-like!). (M, N, J, Q, \Lambda, \alpha). The next cube shows you the general map of type D solutions:

Virtual, binary of n-ary BH systems are left apart. Some types of BH have severan event horizons, being multihorizon solutions. The event horizon is, like you know, the surface (hypothetical) where light can not scape from gravity. It is one of the classical BH anatomic parts. What parts? Well:

  1. Event horizon (sometimes, several horizons are possible, even cosmological horizons). They are usually characterized by some radius, like Schwarzschild radius.
  2. Photon sphere (a really cool and weird place).
  3. Ergosphere.
  4. Singularity (the place, where known physical laws are not valid; to be erased, some people argues, by a true quantum gravity theory without singularities). Hawking-Penrose theorems on the gravitational singularities and their inevitable consequence of any gravitational theory, so you need a quantum gravitational theory to supersede them. A quantum gravitational theory to rule them all!

As the black hole information paradox is yet unsolved (after more than 4 decades!), some people are beginning to consider radical solutions. Maybe, event horizons do not exist. Then, what? Proposals: firewalls, Planck stars (remnants are great again?), fuzzballs, fluid analogues, Bose-Einsteind condensates (Dvali-Gomez proposal of BH as graviton condensates is interesting),…Pick one, the stranger, the better. Or imagine your own solution. Higher dimensional analogues of BH do also exist. The Myers-Perry black hole solution is the equivalent to Kerr solution in higher dimensions. Even more, have you wondered if BH with other topologies are possible? Not in D=4. So, if you find out a black ring in Astronomy, it could prove that our Universe is not 4D. Anyway, you get in this way (extra dimensions):

  1. Myers-Perry black holes (higher dimensional BH, Kerr-like).
  2. Black rings.
  3. Black tubes/supertubes.
  4. Black saturns.
  5. Black folds (in general).

Black holes are interesting objects in astrophysics because the natural stellar evolution seems to predict them (or something very similar to it, as condensates or compact objects). From white dwarfs (WD), to neutron stars (NS), giving up the exotic compact objects (quark or preon stars), you are left with black holes. The Chandrasekhar and Tolman-Oppenheimer-Volfoff (Landau too!) limits are quite true. And the relation of these objects with explosive events like supernova, kilonova, hypernova events, and likely to GRB or FRB are going to be studied not now only, but also in the future. Black hole thermodynamics is a solid field right now, and the Hawking process, the M-\sigma relationship, the quasi-periodic oscillations of X-ray in SBH systems, the Blanford-Znajek or Bondi accretion are yet to be more and more studied, and directly observed. The Event Horizon combo-radiotelescope wants to pick an image of our SMBH, that is, this 2017 winter we could see what the Milky Way SMBH, about 4 million of solar masses fat, looks like! I wish you will not suffer spaghetification after it. Tidal forces are related to gravity. And you are not a super-saiyajin in order to support them all, but SMBH have strong gravity and (assuming basic physics holds) few density. I mean, bigger BH are not quantum! You need quantum gravity to describe supermassive TINY (or very dense) objects, not the biggest!

Other EFE solutions, not just “BH”, are the pp-spacetime, the FRW (Friedmann-Robertson-Walker) spacetime we use in cosmology, the Godel spacetime and other exotica.

Primordial BH (PBH) could be evaporating right now via Hawking radiation. They should have about the moon to do it. Primordial BH as dark matter has regained popularity recently. PBH could make all the DM under certain conditions. Erebons? Not yet! Planckons? No, but transplanckons. Since naked singularities are know expected to be reasonable (or to exist!) analogue modes including dark energy stars, gravastars, Planck stars, wormholes or white holes have been analyzed and re-analyzed from time to time. Recently, as well, some BH with hair have appeared. The classical no-hair theorem seems to be very naive in current time. The BH information paradox, as Hawking and Strominger have speculated in the last 2 years, requires likely the existence of (at least) some kind of “soft hair”. Soft hair is related to extra fields and that could be connected to dark matter and dark energy. I am not going to treat supertranslations or super-rotations (the BMS group) today, but it is something worth studying with formidable implications in the near future. We are not, of course, going to build the BH starship Louis Crane proposes, with data

but, anyway, I certainly would be interested if safe in order to go out from my planet! I am not going to do a cosmic censorship to my species today, but I am worried…Perhaps, I should become holographic, according to the holographic principe we can be (innerly) described by a field theory in the boundary of our space-time (outerly). And BH complementarity is not enough to save the information. ER=EPR could do it! And yet, that entanglement is the key, as Keanu and Paul Rudd say here https://www.youtube.com/watch?v=Hi0BzqV_b44

Entanglement and quantum computing (likely quantum gravity too) are approaching. Quantum games and protocols will be revolutionary. And people are not ready yet! Are BH non-singular in the inner shells and we don’t understand them yet?

See you in another blog post. (Quantum) Entanglement is the key! Entanglement reduces uncertainty. Entanglement is related to gravity (how? how do entangled states gravitate?)

P.S.: Let me know about hypothetical weird particles I did not include!

This post is dedicated to S. W. Hawking, for his 75th birthday.

LOG#186. Polylogarithmic condensates.

Today, I return to my best friends. The polylogs! Are you ready for polylog wars? The polylogarithm can be represented by the next integral:

(1)   \begin{equation*} Li_s(s)=\dfrac{z}{\Gamma (s)}\int_0^1\left(-\ln u\right)^{s-1}\dfrac{du}{1-uz} \end{equation*}

As you know, if you follow my blog, you have

    \[Li_1 (z)=\ln (1-z)\]

    \[Li_0 (z)=\dfrac{z}{1-z}\]

    \[z\dfrac{d}{dz}Li_{s+1} (z)=\dfrac{d}{\ln z}\left(Li_{s+1}(z)\right)=Li_s (z)\]

Off-topic: I have fallen in love again (with another mathematical function!). The Lambert W-function is the inverse of Xe^X=Y, so it can be inverted W(z)e^{W(z)}=z, and this is important for solving infinite tetrations ^\infty z=c via

(2)   \begin{equation*} \boxed{c=-\dfrac{W(-\ln z)}{\ln z}} \end{equation*}

Even the most simple polylogarith, the Riemann zeta function, serves for solving in closed form lots of things. For instante, the free energy for a photon in d-dimensions can be written as follows

    \[\mathcal{F}=-\dfrac{1}{\beta^{d+1}}\dfrac{\Gamma (d+1) \zeta (d+1)}{2^{d-1}\pi^{d/2}\Gamma (d/2) d}\]

and the Casimir energy for a massless electron (OK, in the approximation the electron can be thought as “massless”), reads (in D dimensions)

    \[E(D)=\dfrac{f(D)}{L^{D-1}}\Gamma \left(\dfrac{1-D}{2}\right)\zeta \left(\dfrac{1-D}{2}, \dfrac{1}{2}\right)\]

where \zeta (s, Q) is the Hurwitz zeta function. Now, consider the density

    \[\rho =\dfrac{\omega_d}{(2\pi \hbar)^d}\int_0^\infty \dfrac{p^{d-1}}{z^{-1}e^{\beta H}\pm 1}\]

and where H is the hamiltonian, being H=p^2/2m and \beta=1/k_BT, in the non-relativistic case, but the expresion is completely general. The plus sign corresponds to the so-called Bose-Einstein distributions (bosons), and the minus sign can be associated to the Fermi-Dirac distribution (fermions). The awesome fact is that the two integrals are separately PROPORTIONAL, and the proportionality constant is related to ratios of polylogarithms! I wish I had known this when studied Thermodynamics as undergraduate student! Also, define the thermal wavelength

(3)   \begin{equation*} \lambda_T=\hbar\sqrt{\dfrac{2\pi}{mk_BT}}=\hbar \sqrt{\dfrac{2\pi \beta}{m}} \end{equation*}

This thermal wave-length can be generalized to n-dimensions and dispersion relationship E=ap^s as follows

(4)   \begin{equation*} \boxed{\lambda_T (n)=2\pi \hbar \left(\dfrac{a}{k_B T}\right)^{1/s}\left[\dfrac{\Gamma \left(\frac{n}{2}+1\right)}{\Gamma \left(\frac{n}{s}+1\right)}\right]^{1/n}} \end{equation*}

In fact, if you introduce the grand partition function Q, for the previous hamiltonian, as:

(5)   \begin{equation*} Q=Q_0+\dfrac{gV_n \Gamma \left(\frac{n}{s}+1\right)}{(2\pi \hbar)^n a^{n/s}\Gamma \left(\frac{n}{2}+1\right)}\left(k_BT\right)^{n/s}\begin{cases}Li_{n/s+1}(z), BE\\-Li_{n/s+1}(-z),FD\end{cases} \end{equation*}

The result is fully valid, and thus you can write (now, plug \hbar=1 units for simplicity):

(6)   \begin{equation*} \rho=\dfrac{N}{V}=\mbox{density}=\dfrac{1}{(2\pi)^d}\int_0^\infty\dfrac{\varepsilon^{d/2-1}d\varepsilon}{\exp \left(\beta (\varepsilon-\mu)\right)\pm 1} \end{equation*}

as function of

(7)   \begin{equation*} \boxed{\rho\lambda_T^d=Li_{d/2}(z)} \end{equation*}

provided you define the auxiliar function

    \[\zeta (z)=\begin{cases}z, BED\\ -z, FDD\end{cases}\]

so you get that

    \[\rho=\lambda_T^{-d} Li_{d/2} (z)\]

    \[\rho=-\lambda_T^{-d} Li_{d/2} (-z)\]

can be rewritten in compact form as

(8)   \begin{equation*} \boxed{\rho=sgn(\zeta)\lambda_T^{-d}Li_{d/2}(\zeta)} \end{equation*}

Therefore, the grand partition function becomes

    \[\dfrac{\lambda^2}{V}\ln Q(\zeta)=sgn(\zeta)Li_{d/2+1}(\zeta)\]

and Q_{BED}Q_{FDD}=1. From the polylogarithmic grand partition function you can derive ALL the thermodynamics you want to know. For instance, as

    \[U=-\partial_\beta \ln Q\]

    \[N=\langle N\rangle=z\partial_z \ln Q\]

you can easily find out that

(9)   \begin{equation*} \boxed{\beta P \rho^{-1}=\dfrac{Li_{d/2-1}(\zeta)}{Li_{d/2}(\zeta)}} \end{equation*}

(10)   \begin{equation*} \boxed{\dfrac{\beta U}{N}=\dfrac{d}{2}\dfrac{Li_{d/2-1}(\zeta)}{Li_{d/2}(\zeta)}} \end{equation*}

(11)   \begin{equation*} \boxed{\dfrac{\beta T S}{N}=\left(\dfrac{d}{2}+1\right)\dfrac{Li_{d/2-1}(\zeta)}{Li_{d/2}(\zeta)}\ln \zeta} \end{equation*}

(12)   \begin{equation*} \boxed{Y=yield=\dfrac{Li_{d/2-1}(\zeta)}{Li_{d/2}(\zeta)}} \end{equation*}

Indeed, the proportionality constant between the BED and the FDD as integral in any number of dimensions (e.g., d) is given by

(13)   \begin{equation*} \boxed{K_d=-\dfrac{Li_{d+1}(-1)}{Li_{d+1}(1)}=\left(1-2^{-d}\right)} \end{equation*}

The Riemann zeta function and the polylogarithms help us to write analytical formulae for the Bose-Einstein condensation (the reason for this post). For a non-relativistic boson gas in d dimensions you obtain

(14)   \begin{equation*} \boxed{k_BT_C (NR)=\dfrac{2\pi \hbar^2}{m}\left(\dfrac{n}{\zeta (d/2)}\right)^{2/d}} \end{equation*}

For the 3d case (d=3), you can easily write the common result

    \[k_BT=\dfrac{2\pi \hbar^2}{m\zeta (3/2)}n^{2/3}\approx 3.31\hbar^2 \dfrac{n^{2/3}}{m}\]

where \zeta (3/2)\approx 2.612. Indeed, there are variations of the formulae above. Firstly, M. Casas studied a massless boson gas weakly interacting with dispersion relationship E=a(d)v_F \hbar K, where K is a constant (not related with previous K_d) and v_F is certain Fermi velocity of the bosonic Cooper pair created by two fermions. The condensation formula reads in this case:

(15)   \begin{equation*} \boxed{k_BT=a(d)\hbar v_F\left(\dfrac{\pi^{(d+1)/2}n}{\Gamma\left(\frac{d+1}{2}\right)\zeta (d)}\right)^{1/d}} \end{equation*}

Note that the fact of having a BE condensation depends on the details of the dimensionality of the system and the dispersion relationship. In the end, is the polylogarithm and the Riemann zeta function the main characters there! For instance, in the same non-relativistic case (where before you find out that BEC is impossible in 2d), if you use a confining potential energy U=Ar^\alpha, you can derive the temperature in d-dimensions:

(16)   \begin{equation*} k_BT_C =\left( \left(\dfrac{2\hbar^2}{m}\right)^{d/2}A^{d/2} \dfrac{\Gamma \left(\frac{d}{2}+1\right)n}{\Gamma \left(\frac{d}{\alpha}+1\right)\zeta\left(\frac{d}{2}+\frac{d}{\alpha}\right)}\right)^{\left(\frac{d}{2}+\frac{d}{\alpha}\right)^{-1}} \end{equation*}

and hence, there is only BEC iff


In the case of the ultra-relativistic case, the condensation formula in d-dimensions reads

(17)   \begin{equation*} \boxed{k_BT_C^{UR}=\left[\dfrac{(\hbar c)^d2^{d-1}\pi^{d/2}\Gamma\left(d/2\right)}{\Gamma (d)\zeta (d)}\right]^{1/d}} \end{equation*}

All these formulae can be even generalized to a fully particle-antiparticle treatment or even take the non-extensive thermostatistics generalization of all of them. More? Yes. Define the biparametric symbols



such as


and the biparametric deformed polylogarithm


Q-ons are the case with symmetrical (q,p)-symbol, so p=1/q. Let me define


and again take E=ap^\sigma. Then, you can derive the magic formula of BEC condensation of q-ons (existing in any dimension!)

(18)   \begin{equation*} \boxed{k_BT_c(q-ons)=\left(\dfrac{\sigma}{2}\right)^{\sigma/d}a(2\pi \hbar)^\sigma 2^\sigma\pi^{\sigma/2}\left(\dfrac{\Gamma (d/2)(q-q^{-1})n}{\Gamma (d/\sigma)H_{d/q+1}(q^2,1,q)}\right)^{\sigma/d}} \end{equation*}

The Stefan-Boltzmann law in D-dimensions (space-like!) can also be related to the zeta function if you write

(19)   \begin{equation*} \boxed{R_T=a(D)T^{D+1}} \end{equation*}


(20)   \begin{equation*} \boxed{a(D)=\left(\dfrac{2}{hc}\right)^D\pi^{(D-1)/2}k_B^{D+1}D(D-1)\Gamma\left(\dfrac{D+1}{2}\right)\zeta (D+1)} \end{equation*}

and where the Wien law in D-dimensions (or ND) is related to the Lambert function via


Absolutely gorgeous! Isn’t it? See you in another blog post!


LOG#185. Geometricobjects.

I have the power! I have a power BETTER than the Marvel’s tesseract. It is called physmatics. Hi, there! We are back to school. This time, I am going to give you a tour with some geometrical objects, or geometricobjects, if you want a full tense portmanteau.

Geometry is found in Nature. It seems Nature likes geometry, and even more, it favors the emergence of geometrical forms. Firstly, I am sure you know the cube. A polyhedron, such as, its volume or content is V=L^3, just in the same way a square is a polygon, with are A=L^2. For a cube, thus:

    \[V_C=L^3\;\;\; A_C=6A=6L^2\;\;\; \]

I am confident you also now the tetrahedron. The volume any tedrahedron and its total area are given by the formulae (A, B, C are the vertices of the tetrahedron, and “a” its edge length)

    \[V_T=\dfrac{V_C (A,B,C)}{6}=\dfrac{Ah}{3}=\dfrac{\sqrt{2}L^3}{12}\approx 0.1179L^3\;\;\;\;\;\;\; A_T=4A=\sqrt{3} L^2\approx 1.732L^2\]

I can not resist myself to show you a relatively unknown formula to obtain the area of any triangle, if you know the sides (lengths) a, b, c, and the semiperimeter:

    \[A_\Delta (Heron)=\sqrt{s(s-a)(s-b)(s-c)}\]



There is a Heron-like formula for the volume of the tetrahedron as well. Brahmagupta area for a cyclic quadrilateral (side lengths a, b, c, d) reads


with semiperimeter


or if you avoid the semiperimeter, you get


This Brahamgupta formula is a special case of the Bertschneider’s formula. Next step, the octahedron:

    \[V_O=\dfrac{\sqrt{2}L^3}{3}\;\;\; A_t=8L^2=2\sqrt{3}L^2\]

The dodecahedron has also

    \[V_D=\dfrac{(15+7\sqrt{5})L^3}{4}\approx 7.66 L^3\;\;\; A_t=3\sqrt{(25+10\sqrt{5})}L^2=12A\approx 20.65L^2\]

The icosahedron volume and surface area read

    \[V_I=\dfrac{5(3+\sqrt{5})L^3}{12}=\dfrac{5\phi^2}{6}\approx 2.18L^3\;\; A_t=20A=5\sqrt{3}L^2\approx 8.66L^2\]

where \phi=(1+\sqrt{5})/2 is the golden ratio.

Even when it is not any polyhedron, the 2-sphere has volumen and area

    \[V_S=\dfrac{4\pi}{3}R^3\approx 4.189R^3\;\; A_S=4\pi R^2\approx 12.57R^2\]

Wow! I am not sure you learned all these formulae at school (I didn’t!), but they are lovely! Mathematics is beautiful, radicals are nice! It is obvious ;). Platonic solids are cool.

Well, now we are going to go BEYOND…After all, polygons or polyhedrons are only special cases of polytopes in any dimension! Are you a beyonder? In order to study hypergeometrical geometricobjects, a bit on terminology. Polytopes is the word to generalize polyhedra and polygons to higher (finite!) dimensions. If the dimension of the “body” is:

    \[Dim=-1,0,1,2,3,\ldots,j,\ldots, (n-3), (n-2),(n-1),n\]

and thus the polytope “section”or j-dimensional element is named null (vacuum), vertex (vertices), edge, face, cell,…, j-face [the j-rank element],…,peak-(n-3)-face, ridge or subfacet-(n-2)-face, facet-(n-1)-face, body or n-polytope itself. The case is that even you could consider the infinite-dimensional case. An n-dimensional polytope is bounded by a number of (n − 1)-dimensional facets. These facets are themselves polytopes, whose facets are (n − 2)-dimensional ridges of the original polytope. Every ridge arises as the intersection of two facets (but the intersection of two facets need not be a ridge). Ridges are once again polytopes whose facets give rise to (n − 3)-dimensional boundaries of the original polytope, and so on. These bounding sub-polytopes may be referred to as faces, or specifically j-dimensional faces or j-faces. A 0-dimensional face is called a vertex, and consists of a single point. A 1-dimensional face is called an edge, and consists of a line segment. A 2-dimensional face consists of a polygon, and a 3-dimensional face, sometimes called a cell, consists of a polyhedron.

Till infinity and beyond…Polygons with infinite edges are called apeirogons! A polyhedron with infinite faces could be named apeirohedron, and an infinite dimensional polytope is called apeirotope. Even more, you have skew apeirohedra. There is also strange polyhedra called pseudopolyhedra. Note that something related, but for fermionic or spinorial geometry has appeared in the analysis of quantum gravity. It is called the amplituhedron or positive grassmannian. I am not going to discuss the amplituhedron today, but it is nice to see we are coming back to Greek jobs on Geometry and Nature. The platonic higher dimensional polyhedra are named regular polytopes. They are fully classified. Regular polytopes are called in general the simplex (n-simplex, simplices in plural; the equilateral triangle and the tetrahedron are special cases), the hypercube (n-cube or cross polytope; the 4-D cube or tesseract, the cube and the square are special cases). There is also convex and star polytopes.

Simplices of dimension d=0,1,2,3,…,n are named the point, line/segment, equilateral triangle, tetrahedron, pentachoron (4-simplex),…, n-simplex. Any simplex has (n+1) vertices. The  n-cube with dimension d=0,1,2,3,4,5,…,n is called point, line, square, cube, tesseract (octachoron), penteract (regular decateron),…,n-cube. Any orthoplex, n-orthoplex has 2n vertices or any n-cube has 2^n vertices. Great? Now some formulae. For the n-simplex with edge length L, you have the following formula for the volume and the total surface area:

(1)   \begin{equation*} \boxed{V_N (simplex)=\dfrac{\sqrt{2^{-N}(N+1)}L^N}{N!}}\;\;\; \boxed{S_N(simplex)=\dfrac{\sqrt{2^{1-N}N}(N+1)L^{N-1}}{(N-1)!}} \end{equation*}

and where the N! is the factorial of N, i.e. N(N-1)(N-2)\cdots 3\cdot 2\cdot 1. For the n-cube (do you love tesseracts? I love even more n-cubes!), you get for n-cubes

(2)   \begin{equation*} \boxed{V_N(cube)=L^N}\;\;\;\; \boxed{S_N(cube)=2NL^{N-1}} \end{equation*}

and for the n-orthoplex

(3)   \begin{equation*} \boxed{V_N(orthoplex)=\dfrac{\sqrt{2^N}L^N}{N!}}\;\;\;\boxed{S_N(orthoplex)=\dfrac{\sqrt{2^{N+1}N}L^{N-1}}{(N-1)!}} \end{equation*}

Remember: n-simplex has (n+1) vertices, and n-orthoplex has 2n vertices, and n-cube has 2^n vertices. Admit yourself you love these formulae!

Naming is also fun with these things. As you have remembered, 3D platonic solids are five. The tetrahedron (3-simplex), the cube (3-cube, hexahedron), the octahedron (3-orthoplex), the dodecahedron and the isahedron. In four (euclidean) dimensions, 4D convex regular polytopes are six: the 5-cell (4-simplex), the 8-cell (4-cube, tesseract), the 16-cell (4-orthoplex), the 24-cell, the 120-cell and the 600-cell. In general dimension, greater or equal to 5D, you only have the n-simplex, the n-cube and the n-orthoplex as regular polytopes. You see there are some exotica in 4D! 3D and 2D are also special for regular polytopes. Any relation with the Poincaré conjecture or why we live in 4D? I am not sure, but you can write me if you want after passing the bot-spam test! Cross polytopes in any dimension are also fun. As you have read above, the point, the line, the square, the octahedron, the hexadecachoron (16-cell, 4-orthoplex), the 5-triacontakaiditeron (5-orthoplex or pentacross), and the n-orthoplex. Any n-orthoplex has n vertices. Apeirotopes are much less studied, to my knowledge, and maybe the recent interest in the amplituhedron or this post will drive the interest for these objects in the mathematical community. Apeirotopes are just interesting for honeycombs structures (like those in graphene!), and they have also variations like skew apeirotopes. In order to classify these objects, the Swiss mathematician Ludwig Schläfli created an array of numbers now known as Schläfli symbols. You can easily find out the Schläfly symbols for the regular polytopes I mentioned here. There is also a very interesting notion of duality (yes! duality!) between some polyhedra and polytopes that are more obvious as permutations of Schläfli symbols.

Well, the hypersphere  or n-ball case is worth mentioning in order to continue this post.

(4)   \begin{equation*} \boxed{V_N (ball)=\dfrac{\pi^{N/2}R^N}{\Gamma\left(\frac{N}{2}+1\right)}=\dfrac{2\Gamma \left(\frac{1}{2}\right)^N R^N}{N\Gamma \left(\frac{N}{2}\right)}} \end{equation*}

(5)   \begin{equation*} \boxed{S_N (ball)=\dfrac{dV_N(R)}{dR}=\dfrac{N\pi^{N/2}R^{N-1}}{\Gamma\left(\frac{N}{2}+1\right)}=\dfrac{2\Gamma \left(\frac{1}{2}\right)^N R^{N-1}}{\Gamma \left(\frac{N}{2}\right)}} \end{equation*}

and where \Gamma (z) is the gamma function that generalizes the factorial to any number. Note, as well, a very strange thing about the factorial. It can be thought as the square of the contraction of a Levi-Civita symbol, since

    \[\boxed{\varepsilon_{i_1i_2\ldots i_n}\varepsilon^{i_1i_2\ldots i_n}=n!}\leftrightarrow \varepsilon^2=n!\]

Some string theorists usually prefer to compactify extra dimensions on balls or the n-torus to reduce calculations, but in principle, extra dimensions could have any topology or be discrete like a lattice or a polytope. However, dimensional analysis says that it only varies a number, not the scale L_X of the eXtra dimension. How to do sphere packings? Or hypersphere packings? This is an old mathematical problem. Recently, the young female mathematician Maryna Viazovska solved an old open (not now!) in mathematics of hypersphere packings. In her papers https://arxiv.org/abs/1603.04246 and https://arxiv.org/abs/1603.06518 , The sphere packing in dimension 8 and The sphere packing in dimension 24, she solved the old conjecture linking the hypersphere packing with theta functions, and the Leech lattice. In fact, I can not resist myself to write her theorem in D=24 (bosonic string fanatics are here?):

“The Leech lattice achieves the optimal sphere packing density in \mathbb{R}^{24} , and it is the only periodic packing in \mathbb{R}^{24} with that density, up to scaling and isometries. In particular, the optimal sphere packing density in \mathbb{R}^{24} is that of the Leech lattice, namely \Delta_{24}=\dfrac{\pi^{12}}{12!}\approx 0.00192.”

For the 8D (D=8) case (non-assiociative octonionic fans and friends here?) the result is the (hyper)sphere packing  constant \Delta_8=\pi^4/384=\frac{\pi^4}{2^4\cdot 4!}\approx 0.25367. I am sure John C. Baez will love this thing (his talks about numbers 8 and 24 are legendary! http://math.ucr.edu/home/baez/numbers/).

The sphere packing problem in euclidean dimensions is, thus, “solved”. The kissing number problem is also related. But, even when you think about the maximal density you can package n-balls, it is also interesting. The sphere packing problem is the three-dimensional version of a class of ball-packing problems in arbitrary dimensions. In two dimensions, the equivalent problem is packing circles on a plane. In one dimension it is packing line segments into a linear universe.

Wikipedia says: “(…)In dimensions higher than three, the densest regular packings of hyperspheres are known up to 8 dimensions. Very little is known about irregular hypersphere packings; it is possible that in some dimensions the densest packing may be irregular. Some support for this conjecture comes from the fact that in certain dimensions (e.g. 10) the densest known irregular packing is denser than the densest known regular packing. In 2016, Maryna Viazovska announced a proof that the E8 lattice provides the optimal packing (regardless of regularity) in eight-dimensional space, and soon afterwards she and a group of collaborators announced a similar proof that the Leech lattice is optimal in 24 dimensions.(…)”

The sphere packing constant measures which portion of d-dimensional Euclidean space can be covered by non-overlapping unit balls. Let \Delta_d be the sphere (hypersphere) packing constant. In dimension 1, D=1, you have \Delta_1=1 trivially. It has long been known that a best packing in dimension 2 is the familiar hexagonal lattice packing, in which each disk is touching six others. The first proof of this result was given by A. Thue at the beginning ot twentieth century. The density of the hexagonal lattice packing is \Delta_2=\frac{\pi}{\sqrt{12}}=\frac{\pi}{2\sqrt{3}}\approx 0.90690, that is something greater than 90 per cent!


    \[\boxed{\Delta_2=\dfrac{\pi}{\sqrt{12}}=\dfrac{\pi}{2\sqrt{3}}=\dfrac{\pi\sqrt{12}}{12}=\dfrac{\pi\sqrt{3}}{6}=\dfrac{\pi\sqrt{3}}{3!}\approx 0.90690}\]

For 3D (D=3) you have the FCC/HCP maximal density (Kepler 1611!) given by

    \[\boxed{\Delta_3=\dfrac{\pi}{\sqrt{18}}=\dfrac{\pi}{3\sqrt{2}}=\dfrac{\pi\sqrt{2}}{6}=\dfrac{\pi\sqrt{2}}{3!}\approx 0.74048}\]

In fact, the values of \Delta_n are known also for n=4,5,6,7 (8 was rediscovered by Maryna, see http://mathworld.wolfram.com/HyperspherePacking.html ):

    \[\boxed{\Delta_4=\dfrac{\pi^2}{16}\approx 0.61685}\]

    \[\boxed{\Delta_5=\dfrac{\pi^2\sqrt{2}}{30}\approx 0.46526}\]

    \[\boxed{\Delta_6=\dfrac{\pi^3\sqrt{3}}{144}\approx 0.37295}\]

    \[\boxed{\Delta_7=\dfrac{\pi^3}{105}\approx 0.29530}\]

    \[\boxed{\Delta_8=\pi^4/384=\frac{\pi^4}{2^4\cdot 4!}\approx 0.25367}\]

    \[\boxed{\Delta_{24}=\dfrac{\pi^{12}}{12!}=\dfrac{\pi^{12}}{\Gamma (13)}=\dfrac{\pi^{12}}{\Pi (12)}\approx 0.00192}\]

Thus, the Leech lattice density is the best of the best. High dimensions are everywhere! Low dimensions are anomalous. All data can be described by numbers, so any large collection of data is high dimensional. Why is this important? This is sphere packing! The error spheres of any message should form a sphere packing. This is called an error-correcting code. For rapid communication, want as large a vocabulary as possible. I.e., to use space efficiently, want to maximize the packing density. Rapid, error-free communication requires a dense sphere packing. Real-world channels correspond to high dimensions. Of course some channels require more elaborate noise models, but sphere packing is the most fundamental case. So, therefore, if you allow me this, communication is…An issue about n-balls! Have you got n-balls?Perhaps you should also explore hyperbolic spheres, named as horospheres/paraspheres, and hypercycles.

On a scale from one to infinity, a million is small, but we know almost nothing about sphere packing in a million dimensions. A naive n-grid argument says that the sphere packing in \mathbb{R}^n is at most

    \[\Delta_n\leq \dfrac{R^n \pi^{n/2}}{ (n/2)!2^n}\]

Why is the sphere packing problem difficult? Many local maxima for density. Lots of space. Complicated geometrical configurations. No good way to rule out implausible configurations rigorously. High dimensions are weird. However, it is a very important issue for communication theory, coding theory, quantum computing and information theory. Horocycles!!!! (No, they are not horrocruxes!) We really don’t know what the best sphere packing in a million dimensions looks like. Our estimate of its density might be off by an exponential factor. We don’t even know whether it is ordered (like a crystal, quasicrystal, polycrystal, quasipolycrystal) or disordered (like a lump of dirt, a foam or turbulent fluids). But these problems DO really matter. Every time you are using a cell phone or a computer, you can thank Shannon and his relationship between information theory and high-dimensional sphere packing!

Another piece, not totally unrelated to this geometric theme, is the common issue of folding a piece of paper. In 2002, a female high school student, Britney Gallivan, became famous and she is best known for determining the maximum number of times that paper or other materials can be folded in half. She worked out two beautiful theorems that, in my opinion, have to be known by mathematicians, teachers and people. The reason is that they are completely general and they use arithmetical and geometrical series in a wonderful way.

1st Gallivan theorem. For single-direction folding (using a long strip of paper), the exact required strip length L is

(6)   \begin{equation*} \boxed{L=\dfrac{\pi}{6} t\left(2^{n}+4\right)\left(2^{n}-1\right)} \end{equation*}

where t represents the thickness of the material to be folded, L is the length of a paper piece to be folded in only one direction, and n represents the number of folds desired.

Proof. Let me consider a paper with length L and thickness t. In the first fold, via a semicircle, the lost length is \pi t. After the second fold, the lost length is

    \[\pi t+\pi t+2\pi t\]

The third fold lost length becomes

    \[\pi t+(\pi t +2\pi t)+(3\pi t)+(4\pi t)\]

And after n-folds

    \[\pi t+(\pi t +2\pi t)+(3\pi t)+(4\pi t)+\ldots +(\pi t+\ldots +2^{n-1}\pi t)\]

Adding the lost length after the n-folds, you have

    \[\pi  \left(1+(1+2)+\ldots+(1+2+\ldots+2^{n-1})\right)\]

Using the result about how to sum any arithmetic sum (Gauss, the king!):

    \[\dfrac{\pi t}{2}\left(1\cdot 2+2\cdot 3+4\cdot 5+8\cdot 9+\ldots+ 2^{n-1}\cdot (2^{n-1}+1)\right)\]

Now, you can split this last sum as two, as follows

    \[\dfrac{\pi t}{2}\left( (2^0+2^2+\ldots+2^{2(n-1)})+(2^0+2^1+\ldots+2^{n-1})\right)\]

Finally, use the result for getting the sum of geometric series, to obtain

    \[L=\dfrac{\pi t}{2}\left(\dfrac{ (2^{2n}-1)}{3}+(2^n-1)\right)\]

or equivalently, after basic algebra (hint, use x=2^n),

    \[L=\dfrac{\pi t}{6}\left(2^n+4\right)\left(2^n-1\right)\]

as we wanted to demonstrate. Q.E.D.

2nd Gallivan theorem. An upper bound and a close approximation of the actual paper width needed for alternate-direction folding is

(7)   \begin{equation*} \boxed{W=\pi t 2^{3(n-1)/2}} \end{equation*}

where W is the width of a square piece of paper with a thickness of t, and n is the desired number of folds to be carried out in alternate directions. For paper that is not square, e.g., having a 2:1 ratio, the above equation still gives an accurate limit.

These theorems highlight two facts:

1st. In order to fold anything in a half, it must be \pi-times longer than its thickness. Did you love \pi? \pi is now better! \pi is also related not only to spheres but also to folding something in a half (of course, the process is obviously related to the circle or semicircle).

2nd. Depending of how you fold something, the amounts its length decreases with each fold DIFFERS.

I found myself reading an application of Gallivan 2th theorem to Cosmology. Find it here https://physics.le.ac.uk/journals/index.php/pst/article/viewFile/858/631

I am going to summarize it anyway…Take the Hubble sphere with v(t)=c/H(t) and from the picture

plug W_0=\sqrt{2} d_H. Using the Hubble law with

    \[H(t)=\dfrac{1}{a(t)}\dfrac{d a(t)}{dt}\]

and taking the current time t_0, the current values of the cosmological constant energy density \Omega_{\Lambda,0}, then if space itself has folded from cosmic past to current time, the number of folds will be

    \[n(H)=\dfrac{2}{3\ln 2}\ln \left(\dfrac{\sqrt{2} c}{\pi H(t) t_0}\right)+1\]

Interestingly, today we have dn/dt\sim 0 (Why? Who knows? Who nose?). Then, rewriting the Hubble parameter in terms on the number of folds, you get

    \[H(t)=\sqrt{\Omega_{\Lambda,0} }H_0 \coth \left(\dfrac{3}{2}\sqrt{\Omega_{\Lambda,0} }H_0 t\right)\]

For the accepted values (today) of H_0 and \Omega_{\Lambda, 0}, you get that the maximal number of folds in our Universe is about 67. The following picture show it better

You could also ask if elementary particles are “stacked” or “folded” branes (p-branes). Gallivan’s theorems provide a very general way to estimate the number of times you have something folded or stacked to get something of ANY “size”.

Exercise 1. Provide a proof of Gallivan’s 2nd theorem.

Exercise 2. How many times or folds should a superstring have been curled up or wrapped around an extra dimension of size L in order to get something like the classical electron radius or the asummed right proton radius. Plug L as the Planck length as initial guess. Plug another “reasonable” values of L. What do you learn? I believe this exercise should be done by ANY stringer/M-branist/Dp-branist or fan of extra dimensions. Compactified space as folded space has curious trickery related with the worst understood mathematical scaling: exponential scaling. It is very counterintuitive. Do it yourself. Some numbers used in known compactifications can be compared with your results.


LOG#184. Absement, Fourier and transforms.

In this short post, I am going to relate the relatively unknown notion of absement, the quantity

    \[A=\int X(t)dt\]

with some common transforms, in particular, with Fourier series and transforms more naturally.

The Fourier series for a periodic signal is

    \[x(t)=a_0+\sum_{n=1}^\infty a_n \cos (n\omega_0 t)+ b_n \sin (n\omega_0 t)\]

The complex forms reads

    \[x(t)=\sum_{n=-\infty}^\infty c_n\exp (j n \omega_0 t)\]

The coefficients for the real case read

    \[a_0=\int x(t)dt/T_0\]

    \[a_n=\dfrac{2}{T_0}\int x(t) \cos (n\omega_0 t)dt\]

    \[b_n=\dfrac{2}{T_0}\int x(t) \sin (n\omega_0 t) dt\]

and in the complex case

    \[c_n=\dfrac{1}{T_0}\int x(t) \exp (-jn\omega_0t) dt\]

Here, the Fourier transforms reads

    \[X(f(t))=\int_{-\infty}^\infty x(t) \exp (-2\pi j f t) dt\]

Key remark: the a_0 coefficient in the Fourier transforms is the average absement of x(t).Indeed, the remaining coefficients are also the absement weighted with sinusoidal functions.

Exercise: given the Laplace transform

    \[F(s)=L(f(t))=\int_0^\infty f(t) \exp (-st) dt \]

the discrete Fourier transformation

    \[X(\exp (j\omega))=\sum_{n=-\infty}^\infty x(n) \exp (-nj\omega)\]

and the zeta transform

    \[X(z)=\sum_{n=-\infty}^\infty x(n)z^{-n}\]

What is the relationship with the absement of the transformation coefficients? Interpret your results.

LOG#183. Bohrlogy: some pocket formulae.

This post continues the Bohrlogy thread I did some time ago. It aims to parallel electric Bohr (atom), the gravitational Bohr atom and the black hole atom (as new!). Note that \alpha= K_Ce^2/\hbar c, \lambda_C=\hbar/mc and L_C=\hbar/Mc, R_S=2GM/c^2.

First of all, the common Bohr atom (electric case):

1. Quantized energy (frequency) via E=n\hbar \omega, f=E/nh.
2. Quantized action/angular momentum: L=n\hbar.
3. Quantized orbital radii:

    \[r_n=n^2 a_0=n^2 \dfrac{\hbar}{Z\alpha mc}=\dfrac{n^2\lambda_C}{Z\alpha}=\dfrac{n^2\hbar^2}{ZK_Cme^2} \]

4. Quantized velocity: v_n=Z\alpha c/n
5. Quantized spectrum:

    \[E_n=-\dfrac{Z^2\alpha^2 mc^2}{2n^2}\]

6. Quantized acceleration:

    \[a_n=\dfrac{Z^3\alpha^3 mc^3}{\hbar n^4}\]

7. Quantized density (\rho=m/V):


Now, the gravitational Bohr atom:

1. Quantized energy (frequency) via E=n\hbar \omega, f=E/nh.
2. Quantized action/angular momentum: L=n\hbar.
3. Quantized orbital radii:

    \[r_n(G)=n^2 a_0^G=n^2 \dfrac{\hbar^2}{NG_NMm^2}=\dfrac{2n^2\lambda_C^2}{NR_S} \]

4. Quantized velocity:


5. Quantized spectrum:

    \[E_n(G)=-\dfrac{N^2G_N^2 M^2m^3}{2n^2\hbar^2}=-\dfrac{N^2}{8}\dfrac{R_S^2}{\lambda_C^2}mc^2\]

6. Quantized acceleration:

    \[a_n(G)=\dfrac{N^3R^3_S}{\lambda_C^3} \dfrac{mc^3}{8 \hbar n^4}=\dfrac{N^3R^3_S}{(2\lambda_C)^3} \dfrac{mc^3}{\hbar n^4}\]

7. Quantized density (\rho=m/V):


Finally, the Bohr black hole atom (yeah!). It is obtained from the previous one, plugging \lambda_C\sim R_S. It happens when \lambda_C\sim R_S\sim L_P, where L_P is the Planck length. You get:

1. Quantized energy (frequency) via E=n\hbar \omega, f=E/nh.
2. Quantized action/angular momentum: L=n\hbar.
3. Quantized orbital radii:

    \[r_n(BHA)=2n^2 a_0(BH)/N=2n^2R_S/N=2n^2L_P/N \]

4. Quantized velocity:


5. Quantized spectrum:


6. Quantized acceleration:


7. Quantized density (\rho=m/V):

    \[\rho_n(BHA)=\dfrac{3}{32\pi}\dfrac{N^3m}{R_S^3 n^6}=\dfrac{3}{32\pi}\left(\dfrac{N}{ n^2}\right)^3\dfrac{m}{L_P^3}\]

Black holes have temperature T_{BH}=\hbar c^3/(8\pi G_N M k_B), and frequency, then

    \[f_{BH}=\dfrac{ c^3}{16\pi^2 G_NM k_B}\]

in S.I. units. Note that frequencies do not contain \hbar but energy does via Planck E=hf=\hbar \omega.

Interestingly, electrons can not be black holes. The Schwarzschild radius R_S=2G_NM/c^2 and a Planck length electron has nothing to do with this. There is A CLASSICAL way to have BH electrons, via the Reissner-Nordström solution in General Relativity. You can introduce a charge radius about

    \[R_{Q}=\sqrt{\dfrac{G_NQ^2}{4\pi \varepsilon_0 c^4}}=\sqrt{\dfrac{K_C G_N}{c^4}}Q\]

For the electron, it yields R_Q\approx 10^{-57}m< L_P. Taking a charged rotating BH electron is different. The Kerr parameter for any elementary electron with J=\hbar/2 is a=J/mc\approx 10^{-13}m. Some time ago, Wheeler talked about geons, gravitationally self-sustained electromagnetic fields, very non-linear and intense fields. A similar current concept is the so-called “kugelblitz”. No geons or kugelblitz have been observed in Nature, but they could exist via laser sabers (Jedi, you are?), or they could arise from numerical solutions in the field equations of certain gravitational or extended gravitational theories.

Finally, a similar concept to Bohr BH atoms is known in the literature. They are called holeums. For holeums

    \[E_n=-\dfrac{mc^2 \alpha_g^2}{4n^2}\]


    \[\alpha_g=\dfrac{G_Nm^2}{\hbar c}=\dfrac{m^2}{m_P^2}\]


and m=2m(MBH), 2 microblackholes. Even more, if you generalized this to a set of madroholeums. Macroholeums have been considered even as the internal states of BH. Black holeums would be BH with internal structure, indeed quantum black holes. They would be stable. This is very speculative.

As curious comment, I will compare some numbers from the Bohr atom and the gravitational Bohr atom. Bohr radius is about 1 angstrom, 10^{-10} meters. The gravitational analogue is big, very big. It is about 10^{34}m. The Rydberg for the hydrogen atom is 13.6 eV, about 10^{-18} joules. In the case of the gravitational atom is about 10^{-105} joules.

Final exercise, and think about it. Imagine a world where electrons and protons are (electrically) uncharged and they are bound by gravitational forces. What is the minimal radius? What is the minimal energy in the fundamental level? In order to fit the radii to the known values, assuming, e.g., that you have equal proton and electron mass, what should the electron mass be in order to get the usual Bohr atom result? Compare all the numbers and scales you get with your every day experience.

LOG#182. Duality and gravity: extra thoughts.

In the previous post, I reviewed very naively and shallowly some of the relationships between duality and gravity, and that from electromagnetic duality. There are many other interesting points. This post is aimed to highlight some issues and works that have not (to my knowledge) been exposed in the way I like.

Magnetic monopoles are that crazy but wonderful idea that make theoretical physicists to burn into their heads. As it is well known, from Dirac, the existence of a single magnetic monopole could provide an explanation of WHY electric charge is quantized (irrespectively the unobservable fractional quark charge and other fractional charges observed in condensed matter, I might say). There is a deep link between magnetic flux and mass, and therefore, one is tempted (not without troubles!) to identify mass as certain kind of “trapped flux”. It was an heritage that the second superstring revolution has left unended, to my opinion, because it points out to certain ambiguity of the choice of the fundamental degrees of freedom of any dual-theory that has remained until current times. Before to enter the game of the Dirac quantization condition and its brane analogue, let me expose some other basic stuff, a very well know fact that is sometimes left unexplained (or in the air).

Imagine you pick 2 electrons of mass M_e and charge e. What is the mass of that electron in order to have the electric ofrce it yields equal to the gravitational force it causes? To answer this, you only need school physics. What it is more interesting, is that due to the specific power laws of the Coulomb and Newton laws, the result is independent of the distance you put the electrons. You get:


where G_N is the newton gravitational constant and K_C is the Coulomb constant. This results even holds in any space-time dimension provided the electric force and the gravitational force “see” the same number of the dimensions, i.e., the follow the SAME power law. From this, the electron mass reads


Compare this to the Planck mass

    \[M_P=\dfrac{\hbar c}{G_N}\]

to get

    \[\dfrac{M_e}{M_P}=\sqrt{\dfrac{K_C e^2}{\hbar c}}=\sqrt{\alpha}\]

The mistic and mythic electromagnetic fine structure constant. A pure number that has attracted everyone. Do you dislike \alpha. Take instead Planck units, Stoney units where the Stoney mass reads

    \[M_S=\sqrt{\dfrac{K_C e^2}{G_N}}\]

and no Planck constant. The same argument leads to M_e/M_S=1 or M_e=M_S. That is indeed a deeper result, since you can change from Stoney units to Planck units inserting factors of \sqrt{\alpha} in the appropiate way. Of course, asking why the limit mass is the Planck mass or the Stoney mass to balance electrostatic and gravitational forces is other way to ask why the gravitational constant G_N and the K_C number have the values they have. No one knows why, but it is an interesting question to imagine what would happen if their values were different of if they were variable in time (some old speculations are about this) or energy (this is more or less accepted, at least in Quantum Field Theories, QFT). Note, however, that if they do vary in energy, they had to vary in cosmic time (by duality, T\sim \hbar /E, in quantum theories due to the Heisenberg uncertainty principle, but it is the way in which they should be the real problem). Therefore, the Coulomb constant and the gravitational constant combine with the electric charge to produce a mass scale, independently of details and the number space-time dimensions! However, the number of space-time dimensions can vary the proportion in which this is attained. The ration between the electric mass and the electron mass IS (and this is IMPORTANT), in Stoney units,

(1)   \begin{equation*} \boxed{\dfrac{e}{M_e}=\sqrt{\dfrac{K_C}{G_N}}} \end{equation*}


(2)   \begin{equation*} \boxed{\dfrac{e}{M_e}=\sqrt{\dfrac{\alpha K_C}{G_N}}} \end{equation*}

Conclusion: the trio (\alpha, G_N, K_C) allow us to define the charge to mass electron ratio (and other electric charge to mass by construction) in certain invariant way. Let me remember that Barrow et al. have probed (and I wrote about this in this blog) that for N space (space-time) dimensions, the next adimensional quantity can be made

(3)   \begin{equation*} A=e^{N-1}\hbar^{2-N}c^{N-4}G^{(3-N)/2} \end{equation*}

Note the absence of K_C (due to the system of units they use). N=3 give you \alpha. Note what happens in N=2 (no \hbar!), N=1 (no electric charge) and N=0. The last case \hbar^2 G^{3/2}/(ec^4) is important for multitemporal theories. Also, D=5 (assuming space-time dimensions in a positive or null number), is the first number for which you need the quartet (e, \hbar, c, G_N), and likely K_C in hidden format, to get the adimensional quantity.

Duality is a relationship, as far as you imply the original T-duality or S-duality, involving interchangeable roles of magnetic and electric roles. Surprisingly, it also implies a dual role of G_N and K_C. It is not strange. The speed of light is related to K_C via K_C=1/4\pi\varepsilon_0 and c^2=1/\sqrt{\varepsilon_0\mu_0}=. Now, if you want to have a single valued electron wave-function, certain phase quantity (a flux) must be quantized. In particular, using gaussian units, SI units (weber convention) or SI unites (ampere times meter convention), you get respectively

    \[Q_eQ_m=\dfrac{\hbar c}{2}N\]

    \[Q_eQ_m=2\pi \hbar N\]

    \[Q_eQ_m=\dfrac{\hbar c^2}{2K_C} N\]

where N is an integer number. Taking the first one, with N=1, Q_e=e and Q_m=g, it allows you to define the scale of magnetic monopole mass as

    \[\boxed{M_m=M(g)=\dfrac{\hbar c}{e}\dfrac{1}{\sqrt{G_NK_C}}}\]

and the magnetic charge to mass ratio reads (compare with the electric case!):


Surprised? A duality in K_C! If you write units with K_C=1 (a common choice), you get the disturbing result that the square root of the gravitational constant is related to the ratio between the magnetic mass and a monopole mass! I don’t remember if this has been pointing out before, but it is a striking result! Have you ever heard that Gravity=(Yang-Mills)²? This could be tracked from this result. Is gravity caused my hidden and virtual “magnetic monopoles”? I know, it is a crazy idea. In summary

    \[\boxed{\dfrac{g}{M(g)}=\sqrt{G_NK_C}}\leftrightarrow \boxed{\dfrac{e}{M_e}=\sqrt{\dfrac{\alpha K_C}{G_N}}}\]

What else? Well, additional mysteries. Christian Beck has introduced the so-called Planck-Einstein scale with the aid of the cosmological constant. In particular, the Planck-Einstein length reads:

    \[L_{PE}=\sqrt{L_PL_E}=\sqrt{\dfrac{G\hbar}{c^3\Lambda}}\sim 0.037 mm\]

By the other way, there is another mass scale in stellar physics worth mentioning. The Chandrasekhar mass. Being M_P the Planck mass and m_p the proton mass

    \[M_C=\left(\dfrac{\hbar c}{G_Nm_p^2}\right)^{3/2}M_P=\dfrac{M_P^3}{m_p^2}\approx 1.5M_\odot\]

Scaling these numbers, a la Dirac, you can get that the number of baryons in the Universe is about (M_P/m_p)^4, the photon to baryon ratio is about \sqrt{M_P/m_p}\sim 10^{10}, and the Hubble time to Planck time ratio is about M_U/M_P\sim 10^{60}. Note that the BH entropy also scales as the ratio (M_{BH}/M_P)^2.

In searches for quantum gravity, sometimes is highlighted the resemblance of G_N with the Fermi constant G_F and the weak interaction before the rise of the electroweak theory that mades Fermi interaction an effective field from massive gauge particles. Using the Fermi constant, you can indeed define a length and mass scale. The Fermi length is

    \[L_W=\hbar c \sqrt{G_F}=\lambda_W\]

and the Fermi mass is, using L_W as a Compton wavelength

    \[M_W=\dfrac{\hbar}{L_W c}\]

This number is about 290 GeV and it is closely related to the Higgs vacuum expectation value, about 246GeV, since the latter is v_H=1/\sqrt{\sqrt{2}G_F}. L_W is a measure of the decay length of beta decays. Fermi constant allows us to introduce in a natural way a magnetic moment for neutrinos (yet unveiled!) as follows

    \[\mu_\nu\approx \dfrac{ eG_F m_\nu}{\hbar^2}\]

Even much more interesting is that the following equation gives an approximate (close to expectations from neutrino oscillations measurements) value for neutrino mass

    \[M=\dfrac{\hbar^3\sqrt{G_N}}{eG_F c}\sim m_\nu\]

Or even you can guess something like

    \[m_\nu=\dfrac{c\sqrt{\Lambda}}{\hbar}\dfrac{G_F}{G_N}\dfrac{\Omega_{DM}}{\Omega_{DE}}\approx 0.7 eV\]

and where the omegas are the energy densities of dark matter and dark energy, and \Lambda\sim 10^{-56}cm^{-2}.

Beck, Wesson and others using de Sitter relativity have pointed out that the cosmological constant introduces other mass scale into physics:


Put it into numbers to get something like 10^{-66}g. You can even wonder if the number of neutrinos, and the neutrino density, in the universe could be respectively

    \[N_\nu=\dfrac{\hbar c}{G_F\Lambda}\approx 10^{88}\]

    \[n_\nu=\dfrac{\hbar c \sqrt{\Lambda}}{2\pi^2 G_F}\]

In fact, using purely formal arguments from information theory, Beck arrive to a striking proposal to get the electron mass from first principles. Writing \Lambda_P=1/L_P^2, Beck writes

    \[\boxed{m_e=\dfrac{\alpha \hbar}{c}\Lambda^{1/3}\Lambda^{1/6}}\]

or, inverting the relationship, the scale of \Lambda is

    \[\boxed{\Lambda=\dfrac{G_N^2}{\hbar^4}\left(\dfrac{m_e}{\alpha_{em}}\right)^6=\left(\dfrac{G_Nm_e^3}{\hbar^2 \alpha^3}\right)^2}\]

It is similar, but not equal, to some previous results by Nottale using scale relativity since

    \[\alpha \dfrac{M_P}{m_e}=\Lambda^{-1/6}L_P^{-1/3}\]

is not exactly the Nottale proposal, but certainly something similar to it. Beck proposes that the vacuum energy density is related to electrons

    \[\rho_{Beck}=\dfrac{\Lambda c^4}{8\pi G_N}=\dfrac{G_N}{8\pi}\left(\dfrac{c}{\hbar}\right)^4\left(\dfrac{m_e}{\alpha}\right)^6\]

Beck thoughts through the Planck-Einstein units is much more explicit

Other place where duality arises naturally is in Heisenberg Uncertainty Principle, or Generalized Uncertainty Principle, popular effective general results in some proposals of quantum gravity, but also popular in extensions of relativity like Born reciprocal relativity or dS relativity. Born conjectured that you should have to promote the duality of phase space to space-time. It can be done to a price. Several consequences arise, like maximal force/acceleration (to be discussed elsewhere) but, from the GUP perspective, the X\leftrightarrow P symmetry implieds a GUP of the following type

(4)   \begin{equation*} \boxed{\Delta X\Delta P\geq \dfrac{\hbar}{2}\left(1+\alpha+\beta (\Delta P)^2+\gamma (\Delta X)^2\right)} \end{equation*}

The question IS, what are \alpha, \beta,\gamma? For \beta the natural choice (known from non-commutative geometry geometry, string theory or simply mixing general relativity with quantum mechanics is \beta=L_P^2/\hbar. For \gamma, Bambi or Arraut provide the natural Hubble length (or inverse cosmological constant length) quantity \gamma=1/L^2_\Lambda. Since, L_p<<1 and L_\Lambda>>1 (about 10^{-35}m and 10^{26}m respectively) is normal the correction is unnoticed. However, as far as I know, the \alpha correction is less known. It is a pure number, it could be also very small (or zero) and related (or not) to \alpha, \beta. Thus, we know the more general GUP in phase spacetime reads

(5)   \begin{equation*} \boxed{\Delta X\Delta P\geq \dfrac{1}{2}\left(1+\alpha+\dfrac{L_p^2}{\hbar}(\Delta P)^2+\left(\dfrac{\Delta X}{L_\Lambda}\right)^2\right)} \end{equation*}

Going to space-time only, non-commutative geometry argues that space-time coordinates theirselves are non-commuting. The natural regulator could be the Planck length

    \[\Delta X_i \Delta X_j\geq L_p^2 \delta_{ij}\]

Likely, it could exist a maximal length regular too, as a consequence of GUP! Inspired by the previous result, one searches for it in the form

    \[\Delta X^2\geq L_p^2+\left(\dfrac{\Delta X}{L_\Lambda}\right)^n\]

In order to get a maximal length NC-geometry HUP+GUP you need at least n=3. But, you can also get it with n=2 iff (as it happens) L_\Lambda>>1. One is tempted to ask why is NC necessary, as far as we know space-time is very smooth on large scales. We know that quantum fluctuations of large tiny (dense) stuff should introduce these modifications in some way. The precise nature of these modifications (or even non-associative modifications) will be also a phenomenological issue to discuss in quantum gravity phenomenology. \left[x,x\right]=iL_p^2 and \left[x,y\right]=iL_p^2 formulae are known since the early times of quantum Mechanics, with Snyder, Born, and other less known characters. That L_S^2\Lambda\sim \exp (-1/\alpha), where L_S is the string length or scale, it is also comprehensible from a non-perturbative way through the Schwinger effect.

Off-topic: if you are afraid of tiny numbers, you should remember that quantities like 1 shed (10^{-24}barns), 1 barn=10^{-24}cm^2 are today common in nuclear or particle physics.

Some cool exercises:

Exercise 1. In order to get N=10^{78}, 10^{80} baryons in the Universe, what is the value of M you should plug into (\alpha_S is the strong coupling constant):

    \[\dfrac{\hbar^3}{e}\dfrac{\sqrt{G_N}}{G_F c}\dfrac{ \sqrt{N}}{\alpha_S}=M\]

Exercise 2. Could you fit the known hadronic and leptonic spectra from the following two-parameter formula for suitable values of k, n (integer numbers, fractional numbers:

    \[\boxed{M(n,k)=\dfrac{\alpha \hbar \Lambda_P^{1/3}\Lambda^{1/6}}{c}\left(\dfrac{\sqrt{G_N} M_P}{2 e}\right)^nn\cdot k}\]

or using the Beck formula

    \[\boxed{M(n,k)=n\cdot k \left[\dfrac{M_P c}{2e\sqrt{T_G}}\right]^nm_e}\]

and where \Lambda\approx 10^{-56}cm^2, \Lambda_P=1/L_P, L_P is the Planck length, c, \hbar, G_N, \alpha are the speed of light, reduced Planck constant, Newton gravitational constant and the electromagnetic fine structure constant. m_e is the electron mass and T_G=c^4/G_N is related to the maximal tension principle via F_M\leq c^4/4G_N. Note the resemblance of this equation to the string mth octave pth note formula given by f(m,p)=f_0\cdot 2^m(\sqrt{12})^p.

For any space-time dimension D you can have a p+1 gauge field that naturally couples to a p-brane, via a p+2 field strength. Its dual is a D-p-2 form with D-p-3 gauge field. Dirac quantization reads Q_eQ_m=N\hbar c/2, with Q_e=\int_\Sigma ^\star F and Q_m=\int_{\Sigma_d}F over suitable manifolds \Sigma, \Sigma_d.

Bunster, Henneaux et alii, proposed long time ago a strange (but wonderful) extension of duality for higher spin bosonic fields, with spin s. It is written as follows:

    \[\boxed{\dfrac{1}{2\pi\hbar}Q_{\mu_1\cdots \mu_{s-1}}(v)P^{\mu_1\cdots \mu_{s-1}}(u)=\dfrac{MN}{2\pi\hbar}f_{\mu_1\cdots \mu_{s-1}}(v)f^{\mu_1\cdots \mu_{s-1}}(u)\in \mathbb{Z}}\]

For s=1, you get a phase transformation (module an integer number), and the Dirac quantization condition naturally follows. For s=2, you obtain

    \[\dfrac{4G}{\hbar}P_\mu Q^\mu\in \mathbb{Z}\]

where P_\mu=Mv_\mu is the 4-momentum (generalization to arbitrary D is obvious), and Q^\mu=Nu^\mu, N being the magnetic mass, and where v^\mu is the magnetic 4-momentum. Thus, this higher spin quantization is not a quantization condition for electric-like or magnetic-like masses, but a quantization on 4-momentum. In the rest frame, this s=2 quantization reads

    \[\dfrac{4GMN}{\hbar}\in \mathbb{Z}\]

There are also the option to get gravitational dyons. Dyons are particles with both, electric and magnetic charges is given by (Zwanziger–Schwinger quantisation)

    \[e_1g_2-e_2g_1\in \mathbb{Z}\]

For the gravitational case, s=2, you get the analogue formula:

    \[\dfrac{4G_N}{\hbar}\left(P_\mu \overline{Q}^\mu-\overline{P}_\mu Q^\mu\right)\equiv \dfrac{4G_N\varepsilon_{ab}Q_\mu^a\overline{Q}^b}{\hbar}\in \mathbb{Z}\]

Remark: the higher symmetric (spin) tensors Q_{\mu_1\cdots\mu_{s-1}} and P^{\mu_1\cdots\mu_{s-1}} are the magnetic-like and electric-like conserved charges associated to the asymptotic symmetries of any spin s tensor field. For s=1, as I mentioned, asymptotic symmetries ARE internal gauge symmetries, indeed, constant phase transformations, the P would be the electric charge and the Q the magnetic charge. For s=2, we have a space-time index, so conserved masses have dimensions of mass, and the P, Q tensors are associated to space-time traslations. In fact, in principle, they are associated only to linearized spacetime diffeomorphisms. The back-reactions of these charges in the case s>1 imply some technical difficulties, as we can not neglect non-linear (self)-interactions. Interestingly, Einstein theory does support the existence of gravitational dyons! I mean, general relativity can sustain the existence of both, electric and magnetic masses!

See you in another wonderful post!

LOG#181. Duality and gravity: a short note.

The 2nd superstring revolution circled/circles about (electromagnetic) duality. In D=4 (or general D space-time), duality transformations for the electromagnetic field strength read

F^{\mu\nu}\rightarrow \cos \alpha F^{\mu\nu}-\sin \alpha ^{\star}F^{\mu\nu}

and for the dual field 2-curvature

^\star F^{\mu\nu}\rightarrow \sin \alpha F^{\mu\nu}+\cos \alpha ^{\star}F^{\mu\nu}

They are true symmetry for the Maxwell equations in vacuum, dF=d^\star F=0, and for the whole inhomogeneous Maxwell equations if you add magnetic charges (e, g) and current (J_e, J_m) if you interchange both, electric and magnetic degrees of freedom. For the electric and magnetic part:

E\rightarrow \cos \alpha E+\sin \alpha B

B\rightarrow -\sin \alpha B+\cos \alpha B

Indeed, this symmetry can be thought as some kind of phase-space or phase space-time symmetry. Note the resemblance of the integrals

S=\dfrac{1}{2}\int dt \left(\dot{q}^2-q^2\right)


S=\dfrac{1}{2}\int d\mu \left( B^2-E^2\right)


S=S(q,p)=\int dt \left(p\dot{q}-H\right)

where H=(p^2+q^2)/2. In fact, the last action has symmetry under

q\rightarrow \cos \alpha q - \sin \alpha p

p\rightarrow \sin \alpha q+ \cos \alpha p

Some people argue against because it is not Lorenz invariant in the (q-\dot{q}) plane, but a deeper analysis suggests this criticism as naive and wrong. Well, the point is…What happens with gravity? Gravitational “charge” is mass/energy. Thus, one would ask if there is a kind of duality in the gravitational sector of our gravitational theories. It is subtle. Effectively, you can dualize the theory. You can take the dual of the scalar curvature

R\rightarrow \cos \alpha R -\sin \alpha ^\star R

^\star R\rightarrow \sin \alpha R+\cos \alpha ^\star R

But it comes to certain price. You have to introduce the “gravitational magnetic charge”, also called NUT parameter N into the theory. The Schwarzschild mass is complexified so, M_G=M_S+iN, donde M_S is the Schwarzschild mass, N the NUT charge and M_G the fully dualized gravitational mass (cf. the electromagnetic full dualized charge, Q=Q_e+iQ_m=e+ig. Moreover, EM duality is often an off-shell symmetry (i.e. a symmetry of the action, not just of equations of motion, or on-shell symmety). It has certain caveats, and people got disoriented about how to define a theory where the degrees of freedom are so large and with such a big symmetry.

After the 1995 revolution, many people thought in the duality principle: all fields and dual fields must be treated “democratically”. Have a look to this table and the branes on it:

Therefore, under duality invariance, any p-form gauge field and its dual D-p-2-forms should appear equally in the action or equations of motion in a dual formulation. To use the electromagnetic duality in the gravity sector has provided to be a formidable task, since promoted to a symmetry of the theory, it implies certain infinite-dimensional group content that has not been fully solved. Peter West has conjectured that the Kac-Moody groups E_{10} or E_{11} are behind M-theory, maximal SUGRA in D=11 and the final theory of everything (TOE), but it is yet to be probed. Are you a SUGRA fan? Read Peter West papers on this subject. It is hard but englightened to see the good and bad points of the proposal.

Nowadays, duality is used for much more. After the 1995 revolution, it was also found that theories defined on anti-de Sitter space-times (AdS spacetime for short) could be described by a QFT on the boundary thanks you to a clever application of the holographic principle by Maldacena. This equivalence (duality) is now dubbed as AdS/CFT correspondence but it is also a “duality” in generalized sense. That is what the duality revolution left us. Theories can be described very differently according to the degrees of freedom or “coordinates” you use. But, then, the issue is…Why to choose some over others? And even more…Are they “real”? Does it even matter? After all, no one has seen yet a magnetic monopole, a dyon (particles with both electric and magnetic charges) or a NUT gravimagnetic mass…What do you think?

See you in another blog post! May the duality and the Dp-branes be with you!

P.S.: I like TOEs, but I dislike the preference by certain classes of branes in the M-theory current formulation. I believe some day, we will formulate M-theory with any p-brane dimension in a self-consistent way in any space-time dimension enlarging our idea of space-time. Beyond time, beyon space, beyond space-time and phase space-time. If SUSY matters, in any form, is yet ahead of us.

LOG#180. Hadronic mysteries.

Hi, everyone!

Today is not today. Elementary particles are known to fall in two species, taking for granted the Standard Model (SM): bosons and fermions. The SM bosons are photons, gluons, W and Z bosons and the recently found Higgs boson. Bosons are “force” carriers (May the bosons be with you!), so photons carry electromagnetic forces, gluons carry chromodynamic forces, and W, Z carry weak forces (namely, they are responsible of flavor changing and ultimately of radioactive decay!). Moreover, at high energy, about or around hundreds of GeV, electromagnetic and weak forces are “unified” into electroweak forces. Fermions or leptons come into 3 families (only differing into mass, i.e., they are just one generation at common energies, but there are two more “families” and higher energies). Fermions can be leptons or hadrons. Leptons are the electron, the muon, the tau particle and their associated (massive, at least one state or flavor, due to neutrino oscillations). Leptons do NOT feel the strong nuclear force. However, hadrons do FEEL strong nuclear forces. Hadrons and their types are “complicated”. Hadrons are made of quarks (quarks and leptons make the fundamental fermions in the SM framework). But quarks, due to the asymptotic freedom of QCD (Quantum Chromodynamics) have the weird feature that can not be seen single! Quarks are never and never isolated. We deduce their existence due to phenomenological jets at colliders. Even worse, hadrons classification is puzzling. With the recent announcement of the \Xi_{cc}^{++} particle, I feel the need to make a little bit of quark chemistry, and to summarize some issues and enigmas not yet understood in the SM.

Let me begin this post naming the two more known baryons. You know them if you have basic education. Atoms are made of electrons AND the nucleus. The nuclei contain or are formed mainly (at least, the normal matter you know, you will know about other exotics in this post) by two particles or states. These states are baryons (baryons are made of 3 quarks; or even more odd number of quarks, read below…). Other states in the nuclei are mesons (quark + antiquark particles). Protons and neutrons are made of quarks as follows

(1)   \begin{equation*} p/p^+=N^+\equiv (u u d)=\mbox{proton} \end{equation*}

(2)   \begin{equation*} n=n^0/N^0\equiv (u d d)=\mbox{neutron} \end{equation*}

Sometimes, nuclear physicists call protons and neutrons as nucleons, and think about them as different STATES of the same entity, the NUCLEON. As quantum states, protons have a mass about 938 MeV/c², while neutrons a little bigger, 939.6 MeV/c². Both of them, are just about 1 GeV. The tiny difference of neutron and proton masses is far reaching. It is due to symmetry breaking and it is necessary in order life to exist (otherwise, atoms as we know would not exist!). You could imagine a world with neutrons lighter than protons, and it would be a weird universe. Baryons and mesons have to be colorless. So, the color charge, generally red (Red), green (G) and B (blue) has to be “white” inside any hadron (that is why mesons do form; you can glue red-antired, and so on, bound states). You can know the whole quantum numbers of quarks from the next table:

What else? Well, that is the question! We discovered that protons and neutrons ARE NOT alone. There are many others. Perhaps too many? Perhaps too few? You will think about this. Beyond mass, the “elementary particles” like hadrons are classified with two numbers (yep, do you like numbers?). The first one is the angular momentum in units of \hbar. It is denoted by J. The other number is called “parity”, and it is symbolized by P. Thus, a complete particle is listed within a J^P order. Of course, this is not even enough, and new “quantum” numbers must be introduced. But I am not going to go that far today. I am going to be simple. Using the J^P numerology, I am going to show you every hadron (baryon and meson) that the SM says it exist. Do you remember the quarks above? There are three generations, but, as leptons come in “pairs”, so there are 6 “flavored leptons” (do you like hexapods?), and so, you find experimentally 6 quark species (why? That is a good question! But again, not to be covered today…). The six quarks are named up (u), down (d), charm (c), strange, top or truth (t), bottom or beauty (b). You surely noticed neutrons and protons are made of up and down. Yes! And it was the discovery of new hadrons what hinted about the existence of quarks (at least, valence quarks are useful; fundamental quarks too).

Angular momentum for the simplest baryons is J=1/2. And parity comes in two types: plus (+) and minus (-). Plus parity is usually watched as “vector”-like, while minus parity is pseudovector-like for baryons (it would be scalar or pseudoscalar for mesons). We begin the baryonology zoo with the lambda particle:

\Lambda^0=(u d s) \Lambda^+_c=(u d c) \Lambda^0_b=(u d b)

These particles are three, and they are essentially udx states.

Surprised? You should not, not yet! There are many other baryons (net yet observed all of them!). Let me introduce you to the sigma particles, 9 states:

\Sigma^+=(uus) \Sigma^0=(uds) \Sigma^-=(dds)

\Sigma_{c}^{++}=(uuc) \Sigma_c^+=(udc) \Sigma^0_c=(ddc)

\Sigma_{b}^{+}=(uub) \Sigma_b^0=(udb) \Sigma_b^-=(ddb)

9 sigmas. Do you like the nonet, 3×3 array? Did you see it? Yes, certain sigma particles have the same quark composition that some lambda particles. How do we know they are different? Mass and other quantum numbers…Puzzled? Well, let me continue the tour.  Now, the awesome xi particle (that who triggered this post is listed here), sometimes dubbed cascade B or cascade particles:

\Xi^{0}=(uss) \Xi^-=(dss) \Xi_{c}^{+}=(usc) \Xi_{c}^{0}=(dsc) \Xi^{'+}_c=(usc) \Xi^{'0}_{c}=(dsc)

\Xi_{cc}^{++}=(ucc) \Xi_{cc}^{+}=(dcc) \Xi^{0}_{b}=(usb) \Xi_{b}^{-}=(dsb) \Xi^{'0}_{b}=(usb) \Xi_{b}^{'-}=(dsb)

\Xi^{0}_{bb}=(ubb) \Xi^{-}_{bb}=(dbb) \Xi^{+}_{cb}=(ucb) \Xi^{0}_{cb}=(dcb) \Xi^{'+}_{cb}=(ucb)  \Xi^{'0}_{cb}=(dcb)

So, you have 18 wonderful xi particles. Note that they are all made up from (uxx) or (dxx), with “x” being a quark of the second/third generation. Differences with the sigmas are quantum. Finally, the last hadron with J=1/2. The Omega baryon/particle.  These baryons come in 8 types:

\Omega_c^0=(ssc) \Omega_b^-=(ssb) \Omega_{cc}^+=(scc) \Omega^0_{cb}=(scb)

\Omega_{cbb}^0=(cbb) \Omega_{ccb}^+=(ccb) \Omega^{-}_{bb}=(sbb) \Omega^{'0}_{cb}=(scb)

8 omega particles! Do you have the \Omega? Do you watched the Saint Seiya Omega cartoon? Even, did you know Star Trek and the Borg veneration of the Omega particle? Do you know the omega Directive? 😉

Let’s count the above J=1/2^+ states: nucleons (2), deltas (3), sigmas (9), xis (18), omegas (8). Total: 40. Twice if you count the antiparticles. 80 particles plus antiparticles. Now, we proceed with the J=3/2^+ particles…Again, let me introduce the higher angular cousins of deltas, delta “resonances”:

\Delta^{++}=(uuu) \Delta^+=(uud) \Delta^0=(udd) \Delta^-=(ddd)

Curiously, we have here 4 (not 3!) delta particles. Again, we get the sigmas, now dubbed starred sigmas \Sigma^\star, to distinguish them from the previous sigmas.

\Sigma^{\star +}=(uus) \Sigma^{\star 0}=(uds) \Sigma^{\star -}=(dds)

\Sigma_{c}^{\star ++}=(uuc) \Sigma_c^{ \star +}=(udc) \Sigma^{\star 0}_c=(ddc)

\Sigma_{b}^{\star +}=(uub) \Sigma_b^{\star 0}=(udb) \Sigma_b^{\star -}=(ddb)

Now, we find out 12 starred xi particles (the primed versions are not known, but you can wondered if they should be allowed to exist as exercise):

\Xi^{\star 0}=(uss) \Xi^{\star -}=(dss) \Xi_{c}^{\star +}=(usc) \Xi_{c}^{\star 0}=(dsc)

\Xi_{cc}^{\star ++}=(ucc) \Xi_{cc}^{\star +}=(dcc) \Xi^{\star 0}_{b}=(usb) \Xi^{\star 0}_{bb}=(ubb)

\Xi^{\star +}_{cb}=(ucb) \Xi^{\star -}_{b}=(dsb) \Xi^{\star -}_{bb}=(dbb) \Xi^{\star 0}_{cb}=(dcb)

And now, the last \Omega particles. Now a decuplet (not an octet)! The fascinating “excited” (starred) omega particles are (2 unstarred as it is conventional):

\Omega^-=(sss) \Omega^{\star 0}_{c}=(ssc) \Omega_b^{\star -}=(ssb) \Omega^{\star +}_{cc}=(scc) \Omega_{cb}^{\star 0}=(scb)

\Omega_{bb}^{\star -}=(sbb) \Omega^{\star ++}_{ccc}=(ccc) \Omega_{ccb}^{\star +}=(ccb) \Omega_{cbb}^{\star 0}=(cbb) \Omega_{bbb}^{-}=(bbb)

Let’s count again the states: 4+9+12+10=35. Up to 70 if you count the antiparticles as well! Thus, we have no less than 40+35=75 particles, or 150 particles and antiparticles of baryonic nature. Note the unexisting (by the moment) of higher angular moment xi particles, and that there is no hadron with top quarks! That is simple. The top quark is so heavy, that it can not hadronize. Top quarks are not able to bound and form hadrons because they are too massive. A riddle: why is the top quark so massive with respect to the remaining quarks? Nobody knows for sure. Write me if you guess a good reason.


Are you ready for more? Let’s begin with meson listing. The first big group are the pseudoscalars with J^P=0^-. These are the pions (Yukawa intermediate effective strong force mediators!):

\pi^+=(u\overline{d}) with antiparticle \pi^-=(\overline{u}d)


You have 2 pions (3 if you consider the pion minus antiparticle). Then you have the 4 eta particles (their own antiparticles):





Now, 4 kaons. The kaons are “special”. Due to CP violations, some of the following states are not really quite “accurate”. I mean, the K_S, K_L states are not exactly as I will write them, but only “approximately”. Yeah, it has become stranger. Stranger things happen in particle physics! States that are composite and a bit “fuzzy” due to CP-violations! Strange are kaons, thus:

K^+=(u\overline{s}) with antiparticle K^+=(\overline{u}s)

K^0=(d\overline{s}) with antiparticle K^0=(\overline{d}s)

K_S^0=\left(\dfrac{d\overline{s}+s\overline{d}}{\sqrt{2}}\right) equal to its antiparticle

K_L^0=\left(\dfrac{d\overline{s}-s\overline{d}}{\sqrt{2}}\right) equal to its antiparticle

Therefore, we get now 4 (6) kaons (kaons and antikaons). Next, the D-mesons and the B-mesons:

D^+=(c\overline{d}) with antiparticle D^-=(\overline{c}d)

D^0=(\overline{u}c) with antiparticle \overline{D}^0=(u\overline{c})

D^+_S=(c\overline{s}) with antiparticle D^-_S=(\overline{c}s)

B^+=(u\overline{b}) with antiparticle B^-=(\overline{u}b)

B^0=(d\overline{b}) with antiparticle \overline{B}^0=(\overline{d}b)

B^0_s=(s\overline{b}) with antiparticle \overline{B}^0=(\overline{s}b)

B^+_c=(c\overline{b}) with antiparticle B^-_c=(\overline{c}b)

So, 7 mesons (14 counting their antiparticles). In total, we have 7 (14)+ 4 (6) + 4 (4)+2 (3)=17 (27).

Finally, the vector mesons with J^P=1^-. Let me begin in the inverse way this time. The excited D-mesons and B-mesons are again 7 (14):

D^{\star +}=(c\overline{d}) with antiparticle D^{\star -}=(\overline{c}d)

D^{\star 0}=(\overline{u}c) with antiparticle \overline{D}^{\star 0}=(u\overline{c})

D^{\star +}_S=(c\overline{s}) with antiparticle D^{\star -}_S=(\overline{c}s)

B^{\star +}=(u\overline{b}) with antiparticle B^{\star -}=(\overline{u}b)

B^{\star 0}=(d\overline{b}) with antiparticle \overline{B}^{\star 0}=(\overline{d}b)

B^{\star 0}_s=(s\overline{b}) with antiparticle \overline{B}^{\star 0}=(\overline{s}b)

B^{\star +}_c=(c\overline{b}) with antiparticle B^{\star -}_c=(\overline{c}b)

You then get the rho particles 2 (2+1 antiparticle=3):

\rho^+=(u\overline{d}) with antiparticle \rho^-=(d\overline{u})


Then, you obtain the four self-particles (particles = antiparticles):





And finally, 2 excited kaons (plus their antiparticles, if you wish…2+2=4)

K^{\star +}=(u\overline{s}) with antiparticle K^{\star -}=(s\overline{u})

K^{\star 0}=(d\overline{s}) with antiparticle K^{\star 0}=(s\overline{d})

We sum now again for excited meson states: 2 (4) + 4 (4) + 2 (3) + 7 (14) = 13 (25). Together with previous results above 17 (27)=30 (52) particles. WoW. Anyway, for comparison, have a look the the following light meson table (it includes some resonances and states I gave up for simplicity):


You must be saying that it is over…Well, not so easy. I have only scheduled the main predicted (sometimes already confirmed as with the recent \Xi_{cc}^+ particle) particles. The hadron phenomenology is very complex and rich due to the subtle Yang-Mills theory it provides. Indeed, the existence of some of the above states came under surprise. Who ordered that? Said Rabi about the muon…But also with all these strange replicas…And more. Even even even worse. QCD calculations can indeed fit the hadron spectrum. The full match between experimental spectrum and the theoretical expectation coming from lattice QCD in supercomputers is far from clear. For instance, see this:

Furthermore, did you know that the mass-gap conjecture in the pure Yang-Mills theory is a Millenium Clay problem not yet solved? Did you know that experimentalists have found mischievous resonant particles of the listed particles I gave you? By resonance I mean that they are short-lived higher mass higher angular states of the “parent” particle (you can guess it with the prime particles and some baryons/mesons I listed). No one understand the inner degrees of freedoms of resonances or why they even exist for sure. In fact, lattice QCD suggests many other states, like the so-called glueball (hadron states made of purely gluon states), the exotics (HYBRIDS, hadrons states combining quarks with GLUONS in more complex manners), or even multi-quark species (tetra-quarks, penta-quarks, …),…So, the vacuum of QCD is not yet understood (just like its phenomenology, complex and hard, as you see). I mean, some people think that any baryon state could be written as follows:

(3)   \begin{equation*} \Psi=\vert \mbox{hadron}\rangle=\alpha_{3}\vert qqq\rangle +\beta_{5}\vert qqqq\overline{q}\rangle+\gamma\vert qqqg\rangle+\delta\vert ggg\rangle+\ldots \end{equation*}

What is an hadron (baryon, meson)? What are those resonances we see and their degrees of freedom? Where are the known missing “lost resonances” we expected and we did not found? What is the true nature of the QCD dynamics? One issue is how to understand some states/particles or resonances. Some people suggest that we could be seen the rise of quark chemistry. I have additional suggestive captures and pictures for you about this subject. Firstly, the debate

Secondly, the big molecular models of quarks that are yet to be decided

The issue of constituent vs. valence quarks, an old phenomenological discussion along the parton model by Feynman and others, and how to “image” not only the new multi-quark states (molecules?) but also the traditional and common nucleon states (what is the right “molecular geometry of the uud/proton and the udd/neutron quark states inside the hadron molecule? New studies suggest they are arranged like an “Y” instead a triangle…).

Perhaps quark-gluon chemistry is a better name, since indeed a big part of nucleons is due not the Higgs but the vacuum of the QCD! That is why the YM Clay problem matters. By the way, neutrons decay in about 888 seconds, but protons have been found to be very very stable. Current limitis are about 10^{33} years for proton lifetime! Hyper-Kamiokande will probe up to proton lifetimes about 10^{34-35}yrs. Lucky of us, of course, proton life-time is so big! But some GUTs and theories predict proton decay (bad news for future cosmic life-forms based on protons). Thus, the decay of formerly stable particles like the proton could be FINITE. We are protonically doomed. We or our descendants. Provided the Human Kind survives other cosmic catastrophes, we will have to face the proton decay rate in the far away cosmic future.

May the (strong) Force be with you! May the hadrons (QCD) and hyperons be with you!

P.S.: Some exercises for you!

Assignment 1. Any other insteresting hadron (baryon/meson) I did not include? Let me know!

Assignment 2. Search for the missing resonances I mentioned and what were/are the SM predictions not yet fully fullfilled in the quark/hadron espectrum. How many of them I did not include? Any others you found interesting?

Assignment 3. Why do particles DECAY after all? In addition, imagine a world or Universe where protons, neutrons or hadrons (or even any other SM particle, like heavy leptons) do NOT decay even being different entities/particles. What would it happen? Could it have been possible?

Assignment 4. Search for theories with quarks being non-fundamental. Pre-quarks? Pre-pre-quarks? Preons? Rishons? Why are they “complicated” and hard?

Assignment 5. Combinatorics with quarks is fun. Imagine we study quark chemistry from a purely mathematical viewpoint, and you gave up linear combinations for both mesons and baryons. You know there are 6 quark species or flavors. How many mesons can you make up from these quarks (consider a meson is a quark-antiquark couple)? How many baryons can you make up from these quarks? Do NOT count the corresponding antiparticles.