LOG#159. Polylogia flashes (I).

In the next series of post, I am going to define (again) and write some cool identities of several objects that mathematicians and physicists know as zeta functions and polylogarithms. I have talked about them already here http://www.thespectrumofriemannium.com/2012/11/07/log051-zeta-zoology/

1. We define the Hurwitz zeta function as:

(1)

2. k-th derivative for Hurwitz zeta function , , , is:

(2)

3. Dirichlet L-series or alternate zeta series:

(3)

with and

4. Dirichlet L-function, with being a periodic sequence of length q or a Dirichlet character:

(4)

More interestingly, there are some interesting constants worth studying. They are called Stieltjes constants. (Stieltjes constants) are the n-th coefficients in the Laurent series expansion of the Riemann zeta function at (the pole!). That is:

(5)

and where

Indeed, the 0-th Stieltjes constant is usually called the Euler-Mascheroni constant . More generally, we can even define the Stieltjes constants for the Hurwitz zeta function, and we will denote it by . Thus, we have:

(6)

or

(7)

(8)

and

(9)

They satisfy as well

(10)

for and

5. Siegel zeta function. It is defined as follows

(11)

where is the Riemann-Siegel theta function. It is defined as

(12)

The Riemann-Stieltjes theta function gives the phase factor for Z(t).

6. Lerch trascendent.

(13)

(14)

(15)

The Lerch trascendent generalizes the Hurwitz zeta function and the polylogarithm. What is a polylogarithm? You will see it now…

7. The polylogarithm.

(16)

8. Clausen functions. They are defined as

(17)

Clausen functions are classically defined as , but is much more general. It is defined for general s and z, both complex numbers and even if the series itself does not converge! The Clausen sine functions (Clausen cosine functions can also be defined) are related to the polylogarithm:

(18)

so

or by analytic continuation.  Moreover, we have the Kummer’s relation

and

where G is the Catalan’s constant. The Clausen sine function (clsin(z)) and the Clausen cosine function (clcos(z)) satisfy a complementarity relationship:

(19)

(20)

and

For Clausen cosine functions we have

(21)

(22)

9. Polyexponential.

(23)

where and , are the Bell polymials and is the circular sine function.

10. Prime zeta and second zeta.

(24)

(25)

where are the imaginary part of the non-trivial zeroes for the Riemann zeta function.

11. Polygamma function (or log-derivative of the gamma function )

(26)

If m=1, we have the digamma function. For m>1, we have polygammas.

(27)

12. Gamma function-Legendre function.

Legendre gives

(28)

and Gauss

(29)

Finally, in connection with all this stuff, there are some additional polynomials. The Bernoulli polynomials are defined as

and the Euler polynomials are given by

for k=1,2,… and

for k=2,4,…

I will see you in my next blog post!!!

LOG#158. Ramanujan’s equation.

Hi, everyone! I am back, again! And I have some new toys in order to post faster (new powerful plugin). Topic today: Ramanujan!

Why Ramanujan liked the next equation?

(1)

This equation can be rewritten as follows

(2)

The full set of solutions in terms of the pair (p,n) can be found to be:

(3)

If we put

the solutions are obvious and, the substitution allows to write

as the transformed equation

Therefore, the Ramanujan’s “seven equation” is deeply related with the following problem:

“(…)Find and calculate the values of the pair (p,n) such as a Mersenne prime number is a triangular number(…)”

Currently, we do know that the above equation has the previously quoted 5 solutions. Curiously, Ramanujan also loved the number 5 (not only the number 7) since he liked the golden ration, continuous fractions and related topics. In fact, the Rogers-Ramanujan identities are important in Mathematics and Physics.
In particular, we also observe:

(4)

This equation has the next set of solutions:

1st.

2nd.

3rd.

4th.

5th.

These relationships are connected (secretly) with the Clifford algebras and orthogonal groups in the following way.

1st. Trivial and nullity

2nd. Trivial as well

3rd. First non trivial result (known to mathematical and physicists)

4th. Non trivial case (a little bit unknown case, linking rotations in six dimensions with the Clifford algebra of spacetime):

5th. Mysterious case (not easily found in the literature), highly non trivial and VERY unknown to physicists and mathematicians (to my knowledge)

Two recent news I liked very much:

1) Pentagraphene: a new graphene-like structure and material.

2) A new type of bond beyond those you know from high school (ionic, covalent and metallic): THE VIBRATIONAL BOND (found in bromine-muonium systems Br-Mu-Br).

See you later, I wish!

PS:

LOG#157. Superstatistics (II).

Following the previous article, now I will discuss some interesting points in the paper:

Generalized information entropies depending only on the probability distribution, by O.Obregón and A.Gil-Villegas.

You can find the paper here http://arxiv.org/abs/1206.3353 The Boltzmann factor for a generalized superstatistics reads

(1) $\boxed{B (E)=\displaystyle{\int_0^\infty d\beta f(\beta) e^{-\beta E}}}$

for a distribution function

(2) $\boxed{f(p_l;\beta)=\dfrac{1}{\beta_0p_l\Gamma\left(\dfrac{1}{p_l}\right)}\left(\dfrac{\beta}{\beta_0}\dfrac{1}{p_l}\right)^{\frac{1-p_l}{p_l}}e^{-\frac{\beta}{\beta_0p_l}}}$

In the case of a gamma distribution, the Boltzmann factor can be evaluated in closed form

(3) $\boxed{B(p_l;E)=\left(1+p_l\beta_0E\right)^{-1/p_l}}$

but, as the paper above remembers, the Boltzmann factor can not generally obtained for general distribution functions in analytical closed form. For instance, for log-normal distribution function

(4) $f(p_l;\beta)=\dfrac{1}{\sqrt{2\pi}\beta (\ln (p_l+1))^{1/2}}\exp{\left(-\dfrac{\left[\ln \frac{\beta (p_l+1)^{1/2}}{\beta_0}\right]^2}{2\ln(p_l+1)}\right)}$

and the F-distribution

(5) $f(p_l;\beta)=\dfrac{\Gamma \left(\frac{8p_l-1}{2p_l-1}\right)}{\Gamma \left(\frac{4p_l+1}{2p_l-1}\right)}\dfrac{1}{\beta_0^2}\left(\dfrac{2p_l-1}{p_l+1}\right)\dfrac{\beta}{\left(1+\frac{\beta}{\beta_0}\frac{2p_l-1}{p_l+1}\right)^{\frac{8p_l-1}{2p_l-1}}}$

have no a closed form Boltzmann factor but the paper provides an approximated form for small variance. Beck and Cohen showed that every superstatistics depending on a constant parameter q are the same for small enough variance of fluctuations! That is a powerful result.

As we saw in the previous blog post, the supestatistics with gamma distribution function provides the entropic function

(6) $\boxed{-\dfrac{S}{k}=\displaystyle{\sum_{l=1}^\Omega p_l\ln p_l+\dfrac{(p_l\ln p_l)^2}{2!}+\dfrac{(p_l\ln p_l)^3}{3!}+\ldots}}$

and the first term is just the Shannon entropy. The functional

(7) $\displaystyle{\Phi=\dfrac{S}{k}-\gamma \sum_{l=1}^\Omega-\beta \sum_{l=1}^\Omega p_l^{p_l+1}E_l}$

is chosen to have $\gamma, \beta$ as Lagrange multipliers. The condition

(8) $\dfrac{\partial \Phi}{\partial p_l}=0$

allows us to calculate $p_l$ in terms of S and the Boltzmann factor

(9) $\boxed{B_\Omega (E)=\left(1+\dfrac{\beta}{\Omega}\right)^{-\Omega}}$

is obtained after the equipartition of energy is applied. From this Boltzmann factor, the entropy formula reads off easily

(19) $\boxed{S=k\Omega\left(1-\Omega^{-1/\Omega}\right)}$

That entropy expands into

(20) $\dfrac{S}{k}=\dfrac{S_B}{k}-\dfrac{1}{2!}e^{-S_B/k}\left(\dfrac{S_B}{k}\right)^2+\dfrac{1}{3!}e^{-2S_B/k}\left(\dfrac{S_B}{k}\right)^3+\ldots$

Superstatistic-like entropy and Shannon entropy differ in the range of small number of microstates $\Omega$, or large probabilities $p_l$.

Next, the paper considers the Kaniadakis entropy, defined by

(21) $\boxed{\displaystyle{S_\kappa=-k\sum_{l=1}^\Omega \dfrac{p_l^{1+\kappa}-p_l^{1-\kappa}}{2\kappa}}}$

Inspired by this entropy function, the authors propose the generalized entropy

(22) $\boxed{\displaystyle{S_\kappa=-k\sum_{l=1}^\Omega \dfrac{p_l^{p_l}-p_l^{-p_l}}{2}}}$

whose Taylor expansion for small $p_l$ reads

(23) $-\dfrac{S}{k}=\displaystyle{\sum_{l=1}^\Omega p_l\ln p_l+\dfrac{\left(p_l\ln p_l\right)^3}{3!}+\ldots}$

The first term is again the Shannon entropy like that of superstatistics, but it has missing the even terms in the expansion, keeping only the odd terms in comparison with the previously studied generalized entropy. A similar procedure can be applied to Sharma-Mittal-Taneja biparametric entropies

(24) $\boxed{S_{\kappa, r}=-k\displaystyle{\sum_{l=1}^\Omega p_l^r\left(\dfrac{p_l^\kappa-p_l^{-\kappa}}{2\kappa}\right)}}$

and where now, $\kappa, r$ are not just constants but functions depending on $p_l$.

The final part of the paper discusses the axiomatic foundation of information theory (Khinchin axioms) and how they are modified in the case of the non-extensive entropies. The axiom that gets modified is related to the joint probability for independent systems. These issues of non-additivity and non-extensive features are very relevant in complex systems (even those having fractal and multifractal structures). I will discuss the Khinchin axioms and the informational ground of physics in the future here on this blog, but I will stop the discussion of this subject today.

See you in my third and last introductory superstatistics blog post!

LOG#156. Superstatistics (I).

This post is the first of three dedicated to some of my followers. Those readers from Mexico (a nice country, despite the issues and particularities it has, as the one I live in…), ;).

Why? Well, …Firstly, they have proved to be unbiased followers of this blog. And I am in debt to them. Secondly,  I will review and explain her some concepts and ideas I am interested in…That is one of the goals of my blog TSOR. Don’t forget, it is all about ideas, information, knowledge and, of course, my own particular obsessions. Physchematics, Physmatics, Physics, Chemistry, Mathematics!!!!!

This first post will discuss the paper by Octavio Obregón titled Superstatistics and Gravitation, published in Entropy 2010, 12, 2067-2076. Link here http://www.mdpi.com/1099-4300/12/9/2067

What is superstatistics about? Well, the super in this case has nothing to do, at least in the current state-of-art of this subject, to supersymmetry and supergravity. Superstatistics is a portmanteau meaning “superposition of statistics”. It was introduced by Beck and Cohen with the aim of describing nonequilibrim systems with a long-term stationary state, such as it posses some kind of fluctuating intensive quantity. Indeed, they showed that “averaged” superstatistics over those fluctuations allows us to recover non-extensive statistics and classical statistics as well, but, as any other generalization, it includes something else. It includes an infinite set of more general statistics they called superstatistics. In their original work, Beck and Cohen selected as fluctuating quantity the thermodynamical temperature among other possible intensive quantities (e.g., the chemical potential or the energy dissipation). For a general distribution $f(\beta)$ of “fluctuating temperature”, we can get an effective Boltzmann factor $B (E)$:

(1) $\boxed{B (E)=\displaystyle{\int_0^\infty d\beta f(\beta) e^{-\beta E}}}$

There, E is the energy of some microstate associated with the cell. You can easily check that choosing $f(\beta)=\delta (\beta-\beta_0)$, you will recover the classical Boltzmann factor. For non-extensive statistical mechanics, the distribution function $f(\beta)$ must be a Gamma or $\chi^2$ distribution, depending on the inverse temperature $\beta$ and the non-extensive parameter q. There are some distributions $f(\beta)$ that don’t provide a “closed form” B(E), since the integral is not analytically solvable. For those cases, the Boltzmann factor B(E) must be calculated numerically.

Classical gravity and semiclassical gravitational theory are well known theories. From the classical Einstein field equations, passing through the pioneer works of Hawking and Bekenstein about the thermodynamics of black holes, we approach the edge of known theories. We do know that quantum gravity is out there, and we miss some pieces from the whole puzzle. The analogy of thermodynamics AND black hole mechanics has been shown to be robust and it was even more deeper and mysterious with the seminal work of Hawking about quantum Hawking radiation by black holes. In fact, in the end of the 20th century, Jacobson was able to derive the Einstein Field Equations from the proportionality of entropy and the black hole horizon area, together with the thermodynamical equation $\delta Q=TdS$. Padmanabhan has gone further, showing how the variation of the usually “neglected” boundary term in the gravitational theory provides the black hole entropy. He has proved that this rule can be extended to certain (higher derivative) extensions of classical gravity (the so called Lanczos-Lovelock theories). From this picture, the Einstein Field Equations (EFE) are just equations of state. However, Jacobson procedure assumes local equilibrium (quasistatic processes), in such a way that $\delta Q=TdS$ applies between near states of local thermodynamical equilibrium. Jacobson and Padmanabhan ideas can be also tied and related to Verlinde’s entropic gravity approach. However, these ideas have only touched the idea of equilibrium thermodynamics, and have not been applied to nonequilibrium systems. In nonequilibrium thermodynamics, the Clausius equation gets an additional term

(2) $\boxed{dS=\delta Q/T+d_iS}$

where $d_iS$ is some bulk viscosity entropy production term that is determined by imposing energy-momentum conservation. In fact, as the paper by Obregon quotes, General Relativity can still be derived in this context, and the internal entropy production term is related to the tidal heating term of Hartle-Hawking, and the entropy density can be made proportional to the Ricci scalar. The price to be paid is the breakdown of local therymodynamical equilibrium and the emerging theory is a f(R) gravitational theory! Did you know f(R) are “in the mood of theoretical physicists and cosmologists right now?

Spacetime atoms are controversial entities of quantum gravity. Taking into account the analogy between black hole thermodynamics and spacetime theories with classical thermodynamics and quantum solid state theory, we suspect the existence of those “quanta of space and time” objects. Elasticity and other common macroscopic features are just derived from the fundamental quantum theory of matter. The bridge is the thermodynamical/statistical mechanics description. Going from matter to spacetime, at the classical level we do know the spacetime is described by a metric tensor (a two index object). We also know the equations goberning the metric tensor, the EFE. At the microscopical level, we know much less…We expect a quantum description of spacetime in terms of “atoms of spacetime”(or atoms of space and time, separately). However, we do know that we are loosing some key degrees of freedom. In fact, that is likely (but some people think otherwise) one of the origins of the black hole information paradox. We do not know what are the fundalmental gravitational states of the QUANTUM ( not classical) gravitational field. However, Boltzmann guides us yet:  heat (or more precisely thermodynamical temperature) is “motion” (atomic or molecular motion). The solid state theory can not be understood without the knowledge of the atomic/molecular nature of matter. Solids would not been “heatable” if they had not internal microstructure. Motto: if you can heat it (it has temperature), then it has microstructure!

Surprisingly, black hole horizons have a temperature. Horizons would have not any thermal behaviour if spacetime were structureless. And here is where superstatistics enter in the mentioned paper. Superstatistics is a way to generate different entropies and their corresponding statistics. For instante, non-extensive statistics and beyond. Despite the fact we do NOT know yet what are the microstates of the gravitational field since we don’t understand what a spacetime atom is, the paper hints a possible treatment beyond classical thermodynamics. It explores what would happen if spacetime were described by something like a superstatistics! Moreover, it asks and answers the question: what kind of MODIFIED Newton’s law of gravity will we obtain if we departure from generalized entropies or superstatistics? It also tries to answer the question of how gravity would look like with that generalized statistics. It begins assuming a gamma distribution for $\beta$ in the following way:

(3) $\boxed{f(p_l;\beta)=\dfrac{1}{\beta_0p_l\Gamma\left(\dfrac{1}{p_l}\right)}\left(\dfrac{\beta}{\beta_0}\dfrac{1}{p_l}\right)^{\frac{1-p_l}{p_l}}e^{-\frac{\beta}{\beta_0p_l}}}$

Integrating over the inverse temperature (fluctuating!It changes…) we get the generalized Boltzmann factor

(4) $\boxed{B(p_l;E)=\left(1+p_l\beta_0E\right)^{-1/p_l}}$

We make a Taylor expansion of that factor in power series

(5) $B(p_l;E)=e^{-\beta_0 E}\left[1+\dfrac{1}{2}p_l\beta_0^2E^2-\dfrac{1}{3}p_l^2\beta_0^3E^3+\ldots\right]$

Now, we can fix the probability as

(6) $p_l=q-1$

and it yields the known expressions for non-extensive statistical mechanics. Indeed, the gamma function, the log-normal and the F-distributions, depending on $p_l$, provide the smae second term in the above expansion in the 3 cases (but they differ in higher order terms!).

The paper continues explaining how we can derive the entropy functional for superstatistics! Following its calculations, we get

(7) $\boxed{S=k\displaystyle{\sum_{l=1}^\Omega \left(1-p_l^{p_l}\right)}}$

There, $\Omega$ is the number of microstates, and it is related to the probabilities $p_l$ through the equation (normalization condition)

(8) $\boxed{\displaystyle{\sum_{l=1}^\Omega p_l=1}}$

For small $p_l$, we can expand (7) as follows

(9) $\boxed{-\dfrac{S}{k}=\displaystyle{\sum_{l=1}^\Omega p_l\ln p_l+\dfrac{(p_l\ln p_l)^2}{2!}+\dfrac{(p_l\ln p_l)^3}{3!}+\ldots}}$

or equivalently

(10) $\boxed{-\dfrac{S}{k}=\displaystyle{\sum_{l=1}^\Omega\sum_{j=1}^\infty\dfrac{(p_l\ln p_l)^j}{j!}}}$

Remark: The first term in the series correspond to the classical Boltzmann factor.

Remark (II): $p_l$ results in a function of the product $\beta E$ but $p_l$ can not be written explicitly as a fucntion of this product but as an implicit function. It has, however, the usual asymptotic behaviour $p_l\sim \exp(-\beta E)$.

Remark (III): In the case of non-extensive statistical mechanics, such as Tsallis statistics and others, a more elaborated expression relates $p_l$ and the quantity $\beta E$. In fact, it does not coincide with the corresponding $B(E)$ Boltzmann factor in general.

Finally, the paper moves into the entropic approach of gravity, but using superstatistics! As I have explained previously here, here and here, the entropic approach uses that

$\Delta S=2\pi k\dfrac{mc}{\hbar}\Delta x$

$F\Delta x=T\Delta S$

and

$kT=\dfrac{\hbar a}{2\pi c}$

to get the standard theory of gravity

$F=G\dfrac{Mm}{R^2}$

up to a sign and possible quantum corrections. If spacetime is described by some kind of ensemble featured by superstatistics, taking into account that Verlinde states that nonrelativistic gravity is a force caused by the entropy gradient (similarly to polymers in a heat bath), we could vary the deductions of the whole thing and find new corrections. Remember that Verlinde’s idea and claim is that any surface is identified somehow with the relativistic rest mass of the source $E=Mc^2$, and that the whole energy is equipartitioned among N bytes, $E=1/2NkT$. The bytes are proportional to the surface area $A=QN$. Using the entropy functional (7), supposing equiprobability, i.e., $p_l=1/\Omega$, we obtain

(11) $S=k\Omega\left(1-\Omega^{-1/\Omega}\right)$

To study the effect of this on classical non-relativistic gravity `a la Verlinde’, we write it in terms of Boltzmann entropy $S_B=k\ln\Omega$ the above equation (11):

(12) $S=e^{S_B}\left[1-e^{S_Be^{-S_B}}\right]$

where the entropies have been redefined and the factor 1/k has been erased for convenience. In standard General Relativity, we identify

$S_B=\dfrac{1}{4}\dfrac{A}{L_p^2}$

for non-rotating Schwarzschild-like black holes, and $A=4\pi R_S^2$ is the black hole horizon area, $L_p^2=G\hbar/c^3$ is the Planck length. We note that for large entropies $S_B$, it gives $S\sim S_B$, but expanding $S_B$, it yields

(13) $\boxed{S=S_B-\dfrac{S_B^2}{2!}e^{-S_B}+\dfrac{S_B^3}{3!}e^{-2S_B}-\dfrac{S_B^4}{4!}e^{-3S_B}+\ldots}$

or

(14) $\boxed{S=\displaystyle{\sum_{j=1}^\infty\dfrac{S_B^je^{-(j-1)S_B}}{j!}}}$

The modified gravity theory from this modified entropy functional can be read off using the following formula

(15) $\boxed{F=-G\dfrac{Mm}{R^2}\left[1+4L_p^2\dfrac{\partial s}{\partial A}\right]_{A=4\pi R^2}}$

where for $S_B=A/4L_p^2$ we have

(16) $S=\dfrac{A}{4L_p^2}+s$

Calculating the derivative, for superstatistics distribution function ($f(p_l;E)$) and its equipartitioned entropy S above, we get

(17) $\boxed{4L_p^2\left(\dfrac{\partial s}{\partial A}\right)=\dfrac{\partial s}{\partial S_B}=-\dfrac{A}{4L_p^2}e^{-A/4L_p^2}\left(1-\dfrac{A}{2\cdot 4L_p^2}\right)+\left(\dfrac{A}{4L_p^2}\right)^2 e^{-2A/4L_p^2}\left(\dfrac{1}{2}-\dfrac{A}{3\cdot 4L_p^2}\right)-\dfrac{1}{6}\left(\dfrac{A}{4L_p^2}\right)^3e^{-3A/4L_p^2}+...}$

Using that $A=4\pi R^2$, $S_B=A/4L_p^2=\pi R^2/L_p^2$, the entropic corrections due to superstatistics and its equipartitioned entropy read

(18) $\boxed{F=-\dfrac{GMm}{R^2}\left(1-\dfrac{\pi R^2}{L_p^2}+\dfrac{2\pi^2R^4}{L_p^4}-\dfrac{5}{2}\dfrac{\pi^3R^6}{L_p^6}+\ldots\right)}$

The nature and power-like correction terms are different from those coming in other formalisms of entropic gravity (such as extra dimensions, loop quantum gravity,  ungravity, …), where the terms that modify Newont’s gravity are generally proportional to the INVERSE powers of R and they are relevant for radius close to the Planck scale. Here, we obtain the contrary case. Corrections to the Newton’s gravity law by superstatics are important at scales LARGER (not closer) to the Planck scale but they are also important at Planck scale. This result is amazing and surprising. That Newton’s gravitational law do change is changed not only at scales closer to the Planck scale but also to LARGER scalers is a notable result. Specially in the terms of MOND/MOG theories and their competitor theory: dark matter! As the paper clearly states. This result has NOTHING to do with the quantization of gravity, but the type of emerging/entropic gravity theory we do get if spacetime is a NONEQUILIBRIUM system with a long-term stationary state that owns a spatio-temporal fluctuating quantiy (inverse temperature) and that averaging a superstatistics we get a different entropic functional than that of Boltzmann statistics. Remarkably, the generalized entropy has a differential

(19) $\boxed{dS=dS_Be^S_B+dS_Be^{-S_Be^{-S_B}}\left(1-S_B-e^{S_B}\right)}$

To find a simple field model providing these relationships is an open problem. And its generalized relativistic gravitational model, even as the paper says, is yet to be found and its corresponding equations to be unveiled. That is, we don’t know what kind of gravitational theory would provide the same type of corrections that this superstatistics approach via the Verlinde entropic method. In fact, the entropy-area corrections to black holes dominated by superstatistics, as we have seen, are different to those generally seen in extra dimensions, loop quantum gravity or even in the Wheeler-deWitt minisuperspace approach of quantum cosmology. I urge you to read the paper and the references therein.

What do you think? Do you like the superstatistics idea or not?

See you in the next superstatistics post!

LOG#155. Maximal tension, Λ and quantum gravity.

Hi, folks! Before the upcoming series of thematic blog posts (they are on their way, eager readers), I found a very interesting paper to discuss. The paper is this http://arxiv.org/abs/1408.1820 and the pdf file can be found here http://arxiv.org/pdf/1408.1820v1.pdf .

The author of the above paper are two old known theoretical physicists, Gary Gibbons (G.W.Gibbons) and John D. Barrow. The paper is titled quite gorgeously as

Maximum Tension: with and without a cosmological constant

Maximum force/tension is not new even to normal world (let me joke a little bit). You can easily find maximum force out there

Beyond marketing strategies in Theoretical Physics, why did that paper recall my attention? There are several reasons. Firstly, I am involved (as free researcher yet) in the study of physical principles for New Physics and Unification. Indeed, the principle of maximal force (tension) is known for me in different approaches I found these years. Maximal force can be linked to maximal acceleration principles, and they have been studied in Finsler geometries, C-space relativity, Born reciprocal relativity, and even from the Quantum Mechanics realm by Papini and Caianiello. Secondly, I am studying higher dimensional physics and mathematics at the moment, as well as dimensional analysis due to some MOOC I am currently doing. Thirdly, I had also read about the maximal force principle in Schiller’s free online book Motion Mountain (a great free e-book despite some of its bias towards non-standard approaches in the speculative final part of the mammoth reading). Fourthly, It discusses (more interestingly) the effect of a non null and positive cosmological constant (and de Sitter scale/units!) over the principle of maximal force. Finally, it is fun when you can complete/correct the paper and realize the typos AND when you disagree about some of the claims.

What about the content of the paper? It is interesting (but it has some typos and some ideas I don’t agree with, and I will tell you why right soon).

Using the Planck systems of units (I have discussed this and other interesting systems of units previously in this blog, here, here, here, herehere, and here, and no more “here” until now). The maximal force is connected to the Planck system of units, according to the authors, in a classical world only, through the equation

(1)    $\boxed{F_m=\dfrac{c^4}{G}}$

in a 4D spacetime (3d space) world. I don’t like that claim too much. Gravity is a fundamental interaction, and we suspect it should have a purely quantum origin (unless of course you find some classical theory reproducing all the Standard Model, Quantum Mechanics rules). Furthermore, you can rewrite the above maximal force in the following way

(2)    $\boxed{F_m=\dfrac{c^4}{G}=\dfrac{\hbar c}{4L_p^2}}$

where $L_p$ is the Planck length

(3)    $\boxed{L_p^2=\dfrac{G\hbar}{c^3}}$

so

(4)    $\boxed{\dfrac{1}{G}=\dfrac{\hbar}{L_p^2c^3}}$

There, $\hbar, c$ are the reduced Planck constant and the speed of light in vacuum, respectively. Remember that the Planck length is a really tiny distance, about $10^{-35}$ meters!

Thus, the claim that the maximal force is “classical” should be taken with care. If you take the gravitational constant as non-fundamental, ant you choose a fundamental microscopic length such as the Planck length as “truly fundamental”, you see that the maximal force is “quantum” in origin from the beginning, non classical at all! It results from the disguise as G that the maximal force seems to be classical in Nature. In my opinion, that a maximal/minimal magnitude be classical or quantum is depends on how you think about it. Of course, that is not the plan as I understand of the whole paper, but I think it should be said that some magnitudes are macroscopic (classical) or microscopic (quantum) depending on how you select your “fundamental physical dimensions”. Moreover, the existence of a maximal force implies a maximal power if you believe in special relativity as it is commonly explained, as $v_m=c$ for known physical objects (matter fields, being massive, travel to less than that maximal speed, and massless objects travel at the speed of light, whenever you accept the postulates and tests of special relativity; no escape of that restriction is possible in the normal special relativity). Therefore,

(5) $\boxed{P_m=F_mv_m=F_mc=\dfrac{c^4c}{4G}=\dfrac{c^5}{4G}=\dfrac{\hbar c^2}{4L_p^2}}$

How is the maximal force principle justified in the introduction of the mentioned paper? It uses 2 arguments (well it uses 3, but as Stoney units are just a rescaled Planck units, they are 2 only):

1st. Maximal force (a conjecture in physics yet) is connected to the structure of static cosmic strings. From a static source, you get

$\rho+\displaystyle{\sum_{i=1}^3p_i}=0$

where $\rho$ is the density and $p_i$ is the pressure in the ith direction. Defining the mass per length as

$\dfrac{M}{L}=\mu$

the tension of the string, and since there is no newtonian source for the cosmic string so

$\nabla^2\phi_N=0$

and as the cosmic string supports a conical metric flat spacetime, it provides an angular deviation

$\Delta \theta =8\pi G\mu c^{-2}>2\pi$

only if $F>F_m$

2nd. Natural units. Natural Planck units (length, time and mass) are introduced according to

$L_p^2=\dfrac{G\hbar}{c^3}$

$T_p^2=\dfrac{G\hbar}{c^5}=\dfrac{L_p^2}{c^2}$

$M_p^2=\dfrac{\hbar c}{G}=\dfrac{\hbar^2}{c^2L_p^2}$

Remark: keeping the idea that G is fundamental, the squared Planck length, time and mass are LINEAR in $\hbar$. However, if you give up G as fundamental, the squared Planck length is LINEAR in $\hbar$ but remarkably the Planck time is quantum due to $L_p$ and the aquared Planck mass is QUADRATIC in $\hbar$. Then, it is claimed that maximal force/tension and power are classical, but that it can only be true if we keep the idea of the gravitational constant as a fundamental “constant”, something we suspect from different arguments that can not be completely true. Maximal force/tension and power are “classical” in the sense they do NOT contain the Planck constant in their expressions. But as we have remarked here, that is only due to the fact we conserve G as a fundamental non-varying constant. Several approaches to quantum gravity strongly suggest that G is not ultimately fundamental, but it would be similar to the Fermi constant $G_F$ of nuclear weak interactions, secretly linked to the mass of the intermediate W gauge boson. Furthermore, without entering into the details you can find in the paper or reading the posts in my blog about different systems of “natural” units, Barrow and Gibbons remember us the first system of natural units, the Stoney units. I will write the equivalence with Planck units for clarification purposes here:

$L_S^2=\dfrac{Ge^2}{c^4}=\dfrac{L_p^2e^2}{\hbar c}=L_p^2\alpha$

$T_S^2=\dfrac{L_p^2e^2}{\hbar c^3}=\dfrac{L_p^2\alpha}{c^2}=T_p^2\alpha$

$M_S^2=\dfrac{e^2}{G}=\dfrac{e^2\hbar}{L_p^2c^3}=\alpha\dfrac{\hbar^2}{L_p^2c^2}=\alpha M_P^2$

Now we go into higher dimensions of space (or spacetime). In N space dimensions (N=D-1 lorentzian spacetime) we have

$\left[G\right]=M^{-1}L^NT^{-2}$

$\left[e^2\right]=ML^NT^{-2}$

$\left[c\right]=LT^{-1}$

$\left[\hbar\right]=ML^2T^{-1}$

and the equivalent to the electromagnetic fine structure constant in N dimensions read

$\boxed{\alpha(N)=\hbar^{2-N}e^{N-1}G^{\frac{3-N}{2}}c^{N-4}}$

Note that only when N=3 is gravity excluded from the fine structure! In general N space dimensions, we can get (according to the paper) a “non quantum” (I disagree with that terminology) maximal quantity

$\boxed{Q=\mbox{mass}\cdot (\mbox{acceleration})^{N-2}=MA^{N-2}}$

Remark: when N=3, you will recover the maximal force $Q_m=F_m=MA$. The main conjecture of the paper is the following statement:

Conjecture. In general relativity, with N spacetime dimensions (N=D-1 lorentzian spacetime), there exists a maximal upper bound to the magnitude Q, heuristically given by

$\boxed{Q=\dfrac{c^{2(N-1)}}{G}}$

up to (possibly) a multiplicative (dimensionless) constant.

The paper suggest using the N space Schwarzschild metric to guess the dimensionless factor. I will work the details for you here (omitted in the paper in its current version). Firstly, use the N space Schwarzschild horizon radius

$R_s=\left(\dfrac{16\pi GM}{(N-1)\Omega_{N-1}c^2}\right)^{1/N-2}$

with the N sphere area being ($\Omega_2=4\pi$)

$\Omega_{N-1}=\dfrac{2\pi^{N/2}}{\Gamma (N/2)}$

the expression for Q (N>3) is then given by

$\boxed{Q=c^{2(N-1)}\left[\dfrac{(N-2)8\pi G}{(N-1)\Omega_{N-1}}\right]^{N-2}\left[\dfrac{(N-1)\Omega_{N-1}}{16\pi G}\right]^{N-1}=MA^{N-2}}$

or

$\boxed{Q=\dfrac{\Omega_{N-1}(N-1)(N-2)^{N-2}}{2^{N-2}\cdot 16\pi}\left(\dfrac{c^{2(N-1)}}{G}\right)=\dfrac{\Omega_{N-1}(N-1)(N-2)^{N-2}}{2^{N+2}\cdot \pi}\left(\dfrac{c^{2(N-1)}}{G}\right)=K(N)\left(\dfrac{c^{2(N-1)}}{G}\right)}$

Check:

$N=3\rightarrow Q_3=MA=F=\left(\dfrac{4\pi G}{\Omega_2}\right)\left(\dfrac{\Omega_2}{8\pi G}\right)^2c^4=c^4G\dfrac{1}{G^2}=\dfrac{c^4}{4G}=F_m$

Proof (N space dimensions):

From the N space Schwarzschild radius we get the following relationships

A)       $R_s^{N-2}=\dfrac{16\pi G M}{(N-1)\Omega_{N-1}c^2}$

B)       $\dfrac{c^2R_s^{N-2}}{2M}=\dfrac{8\pi G}{(N-1)\Omega_{N-1}}$

C)       $\dfrac{(N-2)R_s^{N-2}c^2}{2M}=\left(\dfrac{N-2}{N-1}\right)\dfrac{8\pi G}{\Omega_{N-1}}$

Introducing B) and C) into the formula

$Q=c^{2(N-1)}\left[\dfrac{(N-2)c^2R_s^{N-2}}{(N-1)\Omega_{N-1}}\right]^{N-2}\left[\dfrac{(N-1)\Omega_{N-1}}{16\pi G}\right]^{N-1}$

we obtain

$Q=c^{2(N-1)}\left[\dfrac{(N-2)8\pi G}{2M}\right]^{N-2}\left[\dfrac{M}{R_s^{N-2}c^2}\right]^{N-1}$

Simple algebra provides the result we are searching for

$\boxed{Q=\dfrac{(N-2)^{N-2}}{2^{N-2}}M\dfrac{c^{2(N-2)}}{R_s^{(N-2)}}=\kappa_N\dfrac{Mc^{2(N-2)}}{R_s^{-(2-N)}}}$

where

$\kappa_N=\dfrac{(N-2)^{N-2}}{2^{N-2}}$

is the dimensionless constant we were seeking before. Moreover, dimensional analysis prove that the last N space dimensionful formula has the right dimensions (mass times the acceleration to the N-2 power) since

$\left[R_s^{-N+2}c^{2(N-2)}\right]=L^{-(N-2)}L^{2(N-2)}T^{-2(N-2)}=L^{N-2}T^{-2(N-2)}=\left(LT^{-2}\right)^{N-2}$

and thus

$\left[R_s^{-N+2}c^{2(N-2)}\right]=A^{N-2}$

so

$\boxed{Q=\kappa_NMA^{N-2}}$ q.e.d.

Remark: In Cosmology, the scale factor $a(t)$ is proportional to $t^n$, the cosmic acceleration $\ddot{a}$ is proportional to $t^{n-2}$, and thus, the force acting on the Universe grows as

$F=F_p\left(\dfrac{t}{t_p}\right)^{n-2}$

for $t>t_p$ if $n>2$. By the other hand, the power is

$P=P_p\left(\dfrac{t}{t_p}\right)^{2n-3}$

for times $t>t_p$ if $n<3/2$. Then, force and power decouple after Planck time (likely before that time).

The second part of the paper tries to generalize the above considerations when we have a positive non zero cosmological constant. This is important in physics because our Universe is currently expanding under dark energy on very large scales. What happens when we add a cosmological constant to these arguments? We will study only de N=3 case as the paper does, but similar arguments can be done for general N. Firstly, adding a $\Lambda$ term into the Einstein Field Equations

$R_{\mu\nu}- \dfrac{1}{2}g^{\alpha\beta}R_{\alpha\beta}g_{\mu\nu}=G_{\mu\nu}=\dfrac{8\pi G}{c^4}T_{\mu\nu}-\Lambda g_{\mu\nu}$

or

$G_{\mu\nu}=R_{\mu\nu}-\dfrac{1}{2}Rg_{\mu\nu}=2\pi\dfrac{T_{\mu\nu}}{F_m}-\Lambda g_{\mu\nu}$

where $F_m$ is the maximal force and dimensionally

$\left[x^\mu\right]=L$

$\left[R_{\mu\nu}\right]=L^{-2}$

$\left[T_{\mu\nu}\right]=\dfrac{\mbox{Force}}{\mbox{Area}}=\mbox{Pressure}=ML^{-1}T^{-2}$

$\left[\Lambda\right]=L^{-2}$

$\left[g_{\mu\nu}\right]=1$

The cosmological constant is certain kind of Hooke’s law. A positive cosmological constant implies a cosmic universal force at very large scales of magnitude

$\mathbf{F}_\Lambda=M\dfrac{\Lambda c^2}{3}\mathbf{r}=M\dfrac{\Lambda c^2}{3}R\mathbf{u}_R$

It is obvious than for a very tiny cosmological constant, the effect of that repulsive force is negligible for short distances ($R<<1$). Only when $R>>1$ or similar to the length scale of the cosmological constant this force will be detected (as it does).

Thus. the existence of the cosmological constant allows us to introduce a new set of “natural” units of length, time and mass related to it. They are called de Sitter units/dS units or “cosmological units” sometimes and they can be found in 2 standard forms:

1st. De Sitter units

$L_{dS}=\sqrt{\dfrac{1}{\Lambda}}$

$T_{dS}=\dfrac{1}{c}\sqrt{\dfrac{1}{\Lambda}}=\dfrac{L_{dS}}{c}$

$M_{dS}=\dfrac{\hbar}{c}\sqrt{\Lambda}$

2nd. Cosmological units (rescaled dS units)

$L_{\Lambda}=\sqrt{\dfrac{3}{\Lambda}}=\sqrt{3}L_{dS}$

$T_{dS}=\dfrac{1}{c}\sqrt{\dfrac{3}{\Lambda}}=\sqrt{3}\dfrac{L_{dS}}{c}=\sqrt{3}T_{dS}$

$M_{dS}=\dfrac{\hbar}{c}\sqrt{\dfrac{\Lambda}{3}}=\dfrac{M_{dS}}{\sqrt{3}}=\dfrac{\sqrt{3}}{3}M_{dS}$

Next, the paper says that we can NOT get CLASSICAL quantities from dS units that are INDEPENDENT of the Planck constant if they include the dS mass (energy). That claim is correct. Indeed, there are two arguments providing mass/force bounds

Argument 1. From Heisenberg Uncertainty Principle (HUP), we get

$MR>\dfrac{\hbar}{c}$

Argument 2. Black hole (BH) solutions, their existence, provide the horizon bound

$R>R_s\rightarrow \dfrac{R}{M}>\dfrac{2G}{c^2}$

In fact, for isotropic fluid spheres, the known Buchdahl’s bound holds

$R>\dfrac{9}{4}\dfrac{GM}{c^2}$

This bound can be generalized to the case $\Lambda\neq 0$ as follows

$R>\dfrac{2GM}{c^2}\left(1-\Lambda R^2-\dfrac{1}{9}\left(1-\dfrac{c^2\Lambda R^3}{GM}\right)^2\right)^{-1}$

Cosmic repulsion with $\Lambda>0$ will blow anything apart unless distance is below the cosmic $R_\Lambda$. Now, HUP provides

$M>\dfrac{\hbar}{c}R_\Lambda=M_\Lambda=\dfrac{\sqrt{3}}{3}M_{dS}$

Cosmic repulsion with $\Lambda>0$ plus the BH horizon condition $R>R_S$ provides an upper bound

$M<\dfrac{c^2}{G}R_\Lambda=\dfrac{\hbar c^2}{G\hbar}R_\Lambda=\dfrac{\hbar c}{G}\dfrac{1}{M_\Lambda}=\dfrac{M_p^2}{M_\Lambda}$

Therefore, the existence of a cosmological constant implies mass itself is confined to the interval

$M_\Lambda\leq M\leq \dfrac{M_p^2}{M_\Lambda}$

due to the HUP and the BH horizon, i.e., due to Quantum Mechanics and General Relativity. In fact, we can be more precise using the so called Kottler-Schwarzschild-de Sitter black hole solution (sometimes referred as Schwarzschild-de Sitter solution only)

$ds^2=-c^2\Delta (r)dt^2+\dfrac{dr^2}{\Delta (r)}+r^2(d\theta^2+\sin^2\theta d\phi^2)$

with

$\Delta (r)=1-\dfrac{2GM}{c^2 r}-\dfrac{1}{3}\Lambda r^2$

The static condition for the metric implies that there are 2 real roots for $\Delta (r)$, say $r=r_B$ and $r=r_C$. If there are 2 roots, we have

$3M\sqrt{\Lambda}<\dfrac{c^2}{G}$

and this implies a maximal force bound, since simple algebra form it yields

$\dfrac{1}{3}Mc^2\sqrt{\Lambda}<\dfrac{c^4}{9G}=F_{m,static}=F_{ms}$

A critical case happens when the 2 roots are the same number, i.e., when $r_B=r_C$, something that is usually referred as Nariai case

$r_B=r_C=\dfrac{1}{\sqrt{\Lambda}}$

and it results in the solution known as $dS_2\times S^2$, i.e., the product of the 2-dimensional dS space with the 2-sphere. Otherwise, when we have two different positive and real roots, we have

$R_s\leq r_B\leq L_{dS}$

or equivalently

$\dfrac{2GM}{c^2}\leq r_B\leq \dfrac{1}{\sqrt{\Lambda}}$

and

$\dfrac{1}{\sqrt{\Lambda}}\leq r_C\leq \sqrt{\dfrac{3}{\Lambda}}$

or equivalently

$L_{dS}\leq r_C\leq L_\Lambda$

In terms of force, that length bounds imply the relationships we were seeking

(6)        $\boxed{F_\Lambda(r=r_B)\leq F_\Lambda (r=r_C=r_{dS})\leq F_{\Lambda}(r=r_D=L_\Lambda)\leq F_{ms}}$

or

(7)        $\boxed{\dfrac{2GM^2}{3}\Lambda\leq \dfrac{Mc^2}{3}\sqrt{\Lambda}\leq \dfrac{\sqrt{3}Mc^2}{3}\sqrt{\Lambda}\leq \dfrac{c^4}{9G}}$

Remark: The addition of a cosmological constant does NOT alter the existence of a maximal force, it only changes the dimensionless coefficient. As you can easily check

$F_{ms}=\dfrac{c^4}{9G}=\dfrac{4}{9}\left(\dfrac{c^4}{4G}\right)=\left(\dfrac{2}{3}\right)^2F_m$

This result can be easily extended to extra dimensions, and you can also work out the influence on it of higher dimensional extended objects (p-branes)…But that, it is another story.

Let me add the references I liked the most from the paper (in addition to Schiller’s e-book, Motion Mountain):

1) Mod. Phys, Lett. A 15, 2153 (2000).  Mak M. K., Dobson Jr. P. N., Harko T.

2) Nariai H., 1951. Sci. Rep. Tohoku Univ. 35, 62. (I love Japanese people and papers like this from his national science).

3) Buchdahl H.A., 1959. Phys. Rev. 116, 1027.

4) Cvetic M., Gibbons G.W., Pope C.N.(2011). Class. Quant. Grav. 28, 195001.

Additional references can be found in the paper if you read it.

In summary:

1st. There seems to be a mysterious maximal force (maximal Q-magnitude) operating in Nature. If it is classical or quantum in origin, I believe, it is an open question. I am more inclined to think the foundation is quantum than classical, but due to the reboot of emergent approach even in Quantum Mechanics, we should be careful about what we are trying to understand. Emergent gravity and Quantum Mechanics are studied in recent times.

2nd. The inclusion of a cosmological constant does not alter the existence of the maximal force (power). It only changes a dimensionless factor and the physical magnitude being maximized. This result holds in any spacetime dimensions.

3rd. The existence of the cosmological constant (de Sitter radius) is connected to a new set of “fundamental” units: the dS/cosmological units. In fact, it is also tied to a MINIMAL force (quantum?), not only to the existence of a maximal force. This result is well known to geeks out there studying de Sitter (extended) relativity and its generalizations. In fact, we do know that Lie algebra stability principles imply that the maximal extension of the Poincare group is the dS (AdS) group under very general conditions. This topic of extended kinematical groups is known since the 20th century (specially, due to the pioneer works by Bacry and Levy-Leblond) and I will discuss some day here.

See you in my next blog post!

May the Maximal Force (Q) be with you!

LOG#154. Moonshine and 42: THE PAPER.

“(…) The answer to the Great Question…is…Forty-two(…)” said Deep Thought with infinity majesty and calm (frequently found quote from The Hitchhiker’s Guide to the Galaxy, Douglas Adams, London 1979)

Dear readers, yesterday, while I was editing, re-editing and writing my next blog posts found an extraordinary short paper by John Mc McKay and Yang-Hui He. You can found it here: http://arxiv.org/abs/1408.2083 Its title said it all: Moonshine and the Meaning of Life.

In that incredible and uncommon short paper (one of the shortest and most wonderful papers I have ever read) the authors hit with the connection and link between the Moonshine conjecture/theorem and the number 42. The number that, according to the cult geek book above, answers the Great Question, i.e., the Meaning of Life.

What is the Moonshine? Essentially, it is a striking link between two very different branches of mathematics: finite group theory and modular forms. In particular, the original moonshine conjecture applied to the biggest sporadic finite group, the Monster group M and the so called $j(q)$ modular (invariant) form. That is the reason why sometimes it was dubbed as monstrous moonshine. The Monster group is the biggest of the sporadic finite groups, and it has about

$N=2^{46}\cdot 3^{20}\cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13^3 \cdot 17 \cdot 19 \cdot 23\cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71$

elements, or equivalently

$N=808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000$

order, and that is about $N\approx 8\cdot 10^{53}$ group elements!!!! Remarkably, note that the Avogadro constant is about $N\approx 6\cdot 10^{23}$, so the number of elements of the Monster is more or less a million of times the Avogadro constant SQUARED! Furthermore, note that the estimated number of protons in the observable Universe is around $10^{80}-10^{83}$, therefore, the Monster group elements is really a big number…Not far away from that cosmological and fundamental number!

What about the quoted paper? It is truly simple in its hypotheses! In fact, the whole plan of the paper is to communicate the following two number theoretic identities (mathematical congruences):

(1)      $\boxed{\displaystyle{\left(\sum_{m=1}^{24} c_m^2\right)\mbox{mod} 70\equiv 42}}$      by John McKay

(2)      $\boxed{\displaystyle{\left(\sum_{m=1}^{24} \tau_m^2\right)\mbox{mod} 70\equiv 42}}$       by Yang-Hui He

What are those things? Well, firstly, let me introduce the elliptic modular form (sometimes called j-invariant) $j(q)$. This $j(q)$ thing is some kind of mathematical object invariant under the group $PSL(2,\mathbb{Z})$, the group of projective special linear transformations of 2×2 matrices with entries in the integers ($\mathbb{Z}$). This object has a Fourier expansion given by the expression

$j(q)=\dfrac{E_4(q)^3}{\Delta (q)}=\displaystyle{\sum_{m=-1}^\infty c_mq^m=\dfrac{1}{q}+744+196884q+21493760q^2+\ldots}$

and where, as the paper remembers, as $z\rightarrow i\infty$, then $q=e^{2\pi i z}$ is the nome for z. Furthermore, we have the theta series for the $E_8$ lattice

$E_4(z)=1+240\displaystyle{\sum_{n=1}^\infty\sigma_3(n)q^n}$

with

$\sigma_3(n)=\displaystyle{\sum_{d\vert n}d^3}$

and

$\Delta (q)=q\displaystyle{\prod_{n=1}^\infty\left(1-q^n\right)^{24}=\sum_{m=-1}^\infty\tau_m q^m=q-24q^2+252q^3-1472q^4+4830q^5-\ldots}$

$\Delta (q)$ is the modular discriminant. The new congruences discovered by McKay and He are the Answer to the Great Question…42…The Meaning of Life… For a geek mathematician (physmatician) ! 🙂

Recall that the lattice vector

$\omega=(0,1,2,\ldots,24:70)$

satisfying

$\displaystyle{\sum_{n=0}^{24}n^2=70^2}$

is truly unique, and it “lives” in the Lorentzian lattice known as $II_{25,1}$ in…26D spacetime dimensions (25 space-like, 1 time-like)! Indeed, it is an “isotropic Weyl vector” that allows us to construct (and build) the Leech lattice as $\omega_\perp/\omega$. The remarkable Watson’s result, that the unique non-trivial solution to the equation

$\displaystyle{\sum_{p=1}^np^2=m^2}$

is

$(n,m)=(24,70)$

is a well-known mathematical result from the 20th century. The Moonshine conjecture is the connection between the Fourier expansion coefficients of the j(q)-invariant and the dimensions of the irreducible representations of the Monster group M. It arrived after this simple sum

$196884=196883+1$

35 years ago, in 1979, as the monstrous moonshine conjecture (now a theorem). That sparked idea revealed deep links and striking connections into two worlds: finite group theory (discrete in elements and origin) and modular forms (complex, continuous and holomorphic/automorphic forms/functions). The moonshine connection was finally proved by Richard Borcherds using tools from string theory, and his invention of the Vertex Operator Algebras (VOA) !

Recently, the so called Mock modularity appears in black hole state counting (DMZ) and in the computation of the elliptic genus of non-compact sigma models (Troost, Ashok, Eguchi, Sugawara). There is a simple physical explanation for the tension between holomorphy and modularity in these examples. Simply put, continuous spectrum is not holomorphic but discrete spectrum IS, and both are “mapped” somehow by certain kind of modular transformation that mixes these two types of spectra, more precisely, mock (modular) transformations do exist connecting the states of the discrete and the continuum worlds!

Thus, mathematics and physics have been connected again. Modular forms/automorphic functions (and their recent MOCK siblings) with finite groups, string theory and gravity (Witten gave. long ago, a talk relating certain Black Holes, the Monster group and the moonshine concept through quantum statistics and the partition function), and some generalized moonshine conjectures have arised in pure mathematics, in parallel sites where the old j(q)-invariant and the Leech lattice connections are essential. Current research is being done about the Umbral Moonshine (generalized moonshine), mock modular forms, and their relationships with the modern superstring theory/M-theory, specially with BPS states, branes and their “(quantum) counting procedure”. Some sophisticated mathematical objects called “shadows” are also introduced, but I am not going to discuss them today, after we have unveiled the Meaning of Life and the Answer to the Great Question… It is…

See you in another (hopefully so meaningful) blog post!!!!!!

P.S.: Read THE PAPER and references therein!!!!

LOG#153. Guardians of the Galaxy and Special Relativity.

Dedicated to my followers, to my (cyber)friends around all over the globe, to those patient people who have been waiting for this reboot and also to Marvel for bringing us Guardians of the Galaxy in a superb and wonderful live-action movie. Finally, very specially, dedicated to you, M.I.G. 😉 Oh, yeah! / Dedicado a mis seguidores, a mis (ciber)amigos alrededor de todo el mundo, a esas pacientes personas que han estado esperando por este “reinicio” y también a Marvel por traernos a Los Guardianes de la Galaxia en una magnífica y maravillosa película de acción. Finalmente, muy especialmente, dedicado a ti, M.I.G. 😉 ¡Oh, sí!

Dear followers and friends…Hello, again! I am sorry for this long “spare time”. I was just a little busy due to different issues. Time to reignite my blog. Today I will post something special…Some problems with solutions about Special Relativity inspired by the movie Guardians of the Galaxy. Moreover, it is a bilingual English-Spanish post, so you will be able to read it and spread it in two languages! Enjoy them all!

Estimados seguidores y amigos. ¡Hola, de nuevo! Siento este largo “tiempo de esparcimiento”. Estuve de hecho un poco ocupado debido a diferentes asuntos. Tiempo de reiniciar mi blog. Hoy escribiré osbre algo especial…Algunos problemas con solutions sobre Relatividad Especial inspirados por la película Los Guardianes de la Galaxia. Además, es un post bilingüe español-inglés, ¡así que seréis capaces de leerlo y propagarlo en 2 lenguajes! ¡Que los disfrutéis todos!

GUARDIANS OF THE GALAXY AND SPECIAL RELATIVITY

PROBLEMAS EN ESPAÑOL/HOMEWORK (IN SPANISH)

Star-Lord y Rocket Raccoon (Mapache Cohete) se dirigen en dos naves diferentes, desde diferentes puntos de origen, a la nueva base lunar de los Guardianes de la Galaxia en el lado oscuro de la Luna de la Tierra, donde les esperan Groot, Gamora y Drax El Destructor (que consideraremos en reposo respecto a Star-Lord y Rocket).

a) Si Ship (The Milano en la película) mide 40 m de longitud propia en el sistema en el que Star-Lord está en reposo, encuentra la longitud de The Milano cuando se acerca a la Luna con velocidad $v_S=0\mbox{.}8c$, donde c es la velocidad de la luz, encuentra la longitud de The Milano (Ship) para los observadores en reposo en la Luna (Groot, Gamora y Drax).

b) ¿Cuál será la longitud de la nave espacial de Rocket que observan el resto de los Guardianes, en la Luna, si la longitud propia de la nave de Rocket es de 10 m y se dirige hacia ella con velocidad $v_R=0\mbox{.}6c$?

c) Encuentra la velocidad relativa con la que se mueven las naves de Star-Lord y Rocket en los siguientes casos: 1) se mueven en la misma dirección y sentido, 2) se mueven en la misma dirección pero en sentidos opuestos. (En ambos casos se mueven hacia la Luna).

EXTRA BONUS (1): Star-Lord pone una cassette en su walk-man y envía una señal electromagnética a la Luna en ese momento que es monitoreada por sus amigos en la luna. En su sistema de referencia (según un reloj en la nave) dura 121 minutos la reproducción completa de la cassette, que es el tiempo que tarda en llegar a la Luna ($v_S=0\mbox{.}8c$). ¿Cuánto dura la reproducción de esa cassette observada desde la Luna?¿A qué distancia se encontraba de la Luna Star-Lord? Explica las medidas de espacio y tiempo desde los dos sistemas de referencia.

EXTRA BONUS (2): Rocket se pone a fabricar uno de sus artilugios mientras llega a la Luna. Envía una señal electromagnética que es monitoreada por sus amigos en la luna. En su sistema de referencia (según su reloj en la nave), tarda 180 minutos en acabar su nuevo bazooka láser guay y en llegar a la Luna ($v_R=0\mbox{.}6c$). ¿Cuánto dura el proceso de fabricación del bazooka láser guay según los Guardianes en la Luna?¿A qué distancia se encontraba Rocket de la Luna? Explica las medidas de espacio y tiempo desde los dos sistemas de referencia.

HOMEWORK (IN ENGLISH)/PROBLEMAS (EN INGLÉS)

Star-Lord and Rocket Raccoon move towards the brand new hidden and secret base of the team, settled on the dark side of Earth’s moon. There, Gamora, Drax The Destroyer and Groot are waiting for them (in rest).

a) If Ship (The Milano in the movie) has a proper length of 40m, find the length of Ship when it approaches to the Moon with speed $v_S=0\mbox{.}8c$, and where Groot, Drax and Gamora are in rest with respect to it.

b) What would be the length of Rocket’s vessel as observerd from the Moon from the remaining (in rest) Guardians if its proper length is 10m and it moves with speed $v_R=0\mbox{.}6c$ towards the moonbase.

c) Find the relative speed of Ship and Rocket’s vehicle ($v_S=0\mbox{.}8c$, $v_R=0\mbox{.}6c$) in the following cases: 1) they move with the same directions, 2) they move with opposite directions (both towards the Moon, though).

BONUS QUESTION (1) : Star-Lord plays a cassette and sends an electromagnetic signal in that moment, which is monitored by his friends on the Moon. In his reference frame (Star-Lord’s clock) the casseta plays for 121 minutes, just the time to arrive to the moonbase ($v_S=0\mbox{.}8c$). What is the duration of the cassette as observed from the moonbase? What was the initial distance from Star-Lord Ship to the Moon? Explain the different observations, both in space and time, as seen by the two frames.

BONUS QUESTION (2): Rocket decides to make one cool new gadget during his trip to the Moon. He sends an electromagnetic signal to the moonbase, which is monitored by the Guardians on the moonbase, when he begins to make it. In Rocket’s reference frame (a clock on his ship), he finishes his brand new laser bazooka in 180 minutes, just when he arrives to the hidden moonbase ($v_R=0\mbox{.}6c$). How long was he making the cool laser bazooka according to the observers on the moonbase? What was the initial distance of Rocket to the Moon? Explain the different observation, both in space ant time, as seen by the two frames.

Solución/Solution A). La contracción de longitud implica/The length contraction implies

$L_S'=L_0(Star-Lord)\sqrt{1-v_S^2/c^2}=24m$

Solución/Solution B). La contracción de longitud implica/The length contraction implies

$L_R'=L_0(Rocket)\sqrt{1-v_R^2/c^2}=10m$

c1) $\boxed{V=\dfrac{v_S-v_R}{1-\frac{v_Sv_R}{c^2}}=0\mbox{.}38c\approx 0\mbox{.}4c}$

c2) $\boxed{V=\dfrac{v_S+v_R}{1+\frac{v_Sv_R}{c^2}}=0\mbox{.}54c\approx 0\mbox{.}5c}$

EXTRA BONUS/BONUS QUESTION (1): Proper time $\tau$ in Special Relativity satisfies/El tiempo propio en relatividad especial satisface

$\Delta t'=\gamma \tau$

$\tau=\int \dfrac{1}{\gamma}dt$

and with/y con $\tau=\Delta t$ and constant speed/y velocidad constante $v_S=0\mbox{.}8c$ ($v_R=0\mbox{.}6c$)

$\boxed{\Delta t'=\gamma \Delta t=\dfrac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}}$

we get for the Star-Lord’s cassette (obtenemos para la cassette de Star-Lord)

$\Delta t_S=\tau_S=121min$

$\Delta t'(Moonbase)=\dfrac{121}{0\mbox{.}6}\approx 202 min$

and we get for the Rocket Raccoon’s cool laser bazooka/y obtenemos para el bazooka láser guay de Rocket Raccoon

$\Delta t_R=\tau_R=180min,\;\;\;\Delta t'=\dfrac{180}{0\mbox{.}8}=225min$

Distances are relative to the frame, so we have to calculate both, the distances to the moonbase as observed by Star-Lord and Rocket in their respective frames, and those observed by the Guardians at rest on the Moon / Las distancias son relativas al referencial, así que tenemos que calcular ambas, las distancias a la base lunar observadas por Star-Lord y Rocket en sus referenciales, y aquellas observadas por los Guardianes en reposo sobre la Luna

$\Delta x=v\Delta t$

$\Delta x'=v\Delta t'$

Thus/Así

$\Delta x(Star-Lord)=v_S\Delta t_S=v_S\tau_S=0\mbox{.}8c\times 121min=96\mbox{.}8min\cdot c$

$\Delta x'(Star-Lord/Moonbase)=\Delta x(Star-Lord)\gamma\approx 161\mbox{.}3min\cdot c$

EXTRA BONUS/BONUS QUESTION (2):

$\Delta x(Rocket)=v_R\Delta_R=v_R\tau_R=0\mbox{.}6c\times 180min=108min\cdot c$

$\Delta x'(Rocket/Moonbase)=\Delta x(Rocket)\gamma=135min\cdot c$

See you soon, in my next blog post! / Os veo pronto, ¡en my próximo post!

P.S.: Galería/Gallery (selected images from the Internet / imágenes seleccionadas de Internet)

P.S.(II): A gift/Un regalo (pdf file/archivo pdf)

GOGandSR-Ex+GDLGyRE-Ej

GOGandSRprintableVersionv2

LOG#152. Bohrlogy (II).

An interesting but relatively unknown variation of the Bohr model is to use a logarithmic potential energy. In that case, we have

(1) $E=T+U(R)=\dfrac{p^2}{2m}+U(R)$

(2) $E=\dfrac{p^2}{2m}+k\ln\left(\dfrac{R}{R_0}\right)$

(3) $F(R)=-\dfrac{dU}{dR}=-\dfrac{k}{R}$

Bohr quantization rules impose

$L=mvR=n\hbar$

and that

$F_C=F(R)$

so

(4) $m\dfrac{v^2}{R}=k\dfrac{1}{R}$

and then

(5) $mv^2=k$

(6) $\boxed{p^2=mk}$

(7) $\boxed{p=\sqrt{mk}}$

The momentum is NOT quantized in the logarithmic potential Bohr model. By the other hand,

$pR=n\hbar$

implies that

$R_n=\dfrac{n\hbar}{p}$

so

(8) $\boxed{R_n=\dfrac{n\hbar}{p}=\dfrac{n\hbar}{\sqrt{mk}}}$

The velocities are not quantized either

(9) $\boxed{v=\dfrac{p}{m}=\sqrt{k}{m}}$

The energies are easily calculated to be

(10) $\boxed{E_n=\left(\dfrac{1}{2}+\ln\left(\dfrac{n\hbar}{R_0p}\right)\right)k}$

or equivalently

(11) $\boxed{\dfrac{E_n}{k}-\dfrac{1}{2}=\ln\left(\dfrac{n\hbar}{R_0p}\right)}$

The forces and accelerations are quantized

(12) $\boxed{F_n=ma_n=m\dfrac{v^2}{R}=k\sqrt{mk}\dfrac{1}{n\hbar}}$

(13) $\boxed{a_n=\dfrac{F_n}{m}=k\sqrt{\dfrac{k}{m}}\left(\dfrac{1}{n\hbar}\right)}$

Areas are also quantized

(14) $\boxed{S_n=\pi R_n^2=\dfrac{\pi n^2\hbar^2}{mk}=\dfrac{\pi n^2\hbar^2}{p^2}}$

The angular frequencies and the periods are quantized as well

(15) $\boxed{\omega_n=\dfrac{k}{n\hbar}}$

(16) $\boxed{T_n=\dfrac{2\pi}{\omega_n}=2\pi\dfrac{n\hbar}{k}=\dfrac{nh}{k}}$

Interestingly, we can modify and enlarge the Bohr quantization rules. The modified or enhanced Bohr rules imply the addition of the quantization of the area/length via an extra condition

(17) $\boxed{S_n=L_0^2n}$

and thus

(18) $\boxed{R_n=L_0\sqrt{n}}$

Now, we proceed as Bohr himself

$L=n\hbar$

$mvR=n\hbar$

$p_n=\dfrac{n\hbar}{R_n}=\dfrac{\hbar}{L_0}\sqrt{n}$

Therefore, momentum and velocity are again quantized (unlike the usual logarithmic potential with the normal Bohr conditions)

(19) $\boxed{p_n=\dfrac{\hbar}{L_0}\sqrt{n}}$

(20) $\boxed{v_n=\dfrac{p_n}{m}=\dfrac{\hbar}{mL_0}\sqrt{n}}$

Forces and accelerations are quantized in a different form

(21a) $\boxed{F_n=ma_n=\dfrac{k}{L_0}\left(\dfrac{1}{\sqrt{n}}\right)}$

(21b) $\boxed{a_n=\dfrac{k}{mL_0}\dfrac{\sqrt{n}}{n}}$

Energies are also quantized but modified too

(22) $\boxed{E_n=\dfrac{\hbar^2n}{2mL_0^2}+k\ln\left(\dfrac{L_0\sqrt{n}}{R_0}\right)}$

This last equation is something more complicated that the first logarithmic potential. We can play with it a bit. Introduce the areas

$a_0=L_0^2$

$A_0=R_0^2$

and suppose that

$k=\dfrac{\lambda_C^2}{L_0^2}mc^2$

where

$\lambda_C=\dfrac{\hbar}{mc}$

The energies are

$E=\dfrac{\hbar^2n}{2mmc^2L_0^2}nmc^2+k\ln\left(\sqrt{\dfrac{na_0}{A_0}}\right)$

and algebra provides

(23) $\boxed{E_n=\dfrac{k}{2}\left(n+\ln\dfrac{na_0}{A_0}\right)=\dfrac{k}{2}\ln\left(\dfrac{e^nna_0}{A_0}\right)}$

Finally, angular frequencies and periods are also quantized

(24) $\boxed{\omega_n=\sqrt{\dfrac{k}{mL_0^2n}}}$

(25) $\boxed{T_n=\dfrac{2\pi}{\omega}=2\pi\sqrt{\dfrac{mL_0^2n}{k}}}$

As you can observe, some magnitudes change as we modify the quantization rules. This logarithmic model is useful in some interesting problems in theoretical physics and mathematics.

See you in my next blog post!

LOG#151. Bohrlogy (I).

The Bohr model of the hydrogen atom is a cool and nice “toy model”. It serves as a prototype in many applications, even if it is not fully “quantum”. It does provide many applications. In this post, and the followings, we will elaborate some of the most unknown uses of this “atomic” model.

Firstly, take the free non-relativistic hamiltonian (energy) for a point particle and add it a potential energy term U(R):

(1) $H=E=T+U=T+U(R)=\dfrac{p^2}{2m}+U(R)$

In order to apply the Bohr model and its quantization conditions in the (almost) most general condition, choose a homogeneous arbitrary potential energy

(2) $U(R)=kR^D$

Homogeneity of the potential energy means, mathematically, that

(3) $U(aR)=a^DU(R)$

The given “UR” potential energy comes from a central force:

(4) $F(R)=-\dfrac{dU}{dR}=-kDR^{D-1}$

Now, we will study the problem of circular orbits and their quantization rule from the Bohr postulates:

i) Angular momentum (or phase space) is quantized: $L=n\hbar=mvR=pR$, with $n\in \mathbb{Z}^+$

ii) The centripetal force matches the central force providing the given $U(R)$, since

$F_C=F(R)\leftrightarrow\dfrac{mv^2}{R}=kDR^{D-1}$

$mv^2=kDR^D$

$m^2v^2=p^2=mkDR^D$

and then

(5) $\boxed{p^2=mkDR^D}$

Using the condition (i), we get

$p^2R^2=n^2\hbar^2$

$mkDR^DR^2=n^2\hbar^2$

Therefore, the radii of the circular orbits are quantized (space quantization!) as follows

(6) $\boxed{R_n(D)=\sqrt[D+2]{\dfrac{n^2\hbar^2}{mkD}}}$

Momentum is related to radius via

(7) $p_n(D)=\dfrac{n\hbar}{R_n(D)}$

and thus

(8) $\boxed{p_n(D)=n\hbar\sqrt[D+2]{\dfrac{mkD}{n^2\hbar^2}}=\sqrt[D+2]{mkD(n\hbar)^D}}$

Now, we can calculate the energies (note that we could also use the virial theorem for the homogeneous potential energy, but since it is a theorem not usually known by some kind of readers, I will use the brute force approach):

$E=T+U=\dfrac{p^2}{2m}+kR^D$

Inserting the values of the quantized radii and momenta into the energy formula (hamiltonian), we obtain, after some algebra,

$E_n(D)=k^{\frac{2}{D+2}}\left(\dfrac{n^2\hbar^2}{m}\right)^{\frac{D}{D+2}}\dfrac{D+2}{2D^{D/D+2}}$

(9) $\boxed{E_n(D)=C(k,D)\left(\dfrac{n^2\hbar^2}{m}\right)^{D/D+2}}$

where

(10) $\boxed{C(k,D)=k^{\frac{2}{D+2}}\dfrac{D+2}{2D^{D/D+2}}=\left(\dfrac{D+2}{2}\right)\dfrac{k^{2/D+2}}{D^{D/D+2}}}$

Velocities and accelerations are also quantized in this model, and it is something that sometimes is not remarked:

(11) $\boxed{v_n(D)=\dfrac{p_n(D)}{m}=\sqrt[D+2]{\dfrac{kD}{m^{D+1}}(n\hbar)^D}=\sqrt[D+2]{\dfrac{kD}{m}\left(\dfrac{n\hbar}{m}\right)^D}}$

(12) $\boxed{a_n(D)=\dfrac{v_n^2}{R_n}=\sqrt[D+2]{\left(\dfrac{kD}{m}\right)^3\left(\dfrac{n^2\hbar^2}{m^2}\right)^{D-1}}=\sqrt[D+2]{\left(\dfrac{kD}{m}\right)^3\left(\dfrac{n\hbar}{m}\right)^{2(D-1)}}}$

Due to the fact that accelerations are quantized, forces and angular frequencies are also quantized. The quantized forces read

(13) $\boxed{F_n(D)=ma_n(D)=\sqrt[D+2]{(kD)^3\left(\dfrac{n^2\hbar^2}{m}\right)^{D-1}}}$

(14) $m\omega_n^2R_n\rightarrow \omega_n^2(D)=\sqrt[D+2]{(kD)^4\dfrac{(n^2\hbar^2)^{D-2}}{m^{2D}}}$

(15) $\omega_n(D)=\sqrt[D+2]{(kD)^2\dfrac{(n\hbar)^{D-2}}{m^D}}$

The angular frequencies are related to the energies through a pure number:

(16) $\dfrac{E_n(D)}{\omega_n(D)}=\Omega_n(D)$

such as

(17) $\boxed{\Omega_n(D)=\left(\dfrac{D+2}{2}\right)\left(\dfrac{n\hbar}{D}\right)=\left(\dfrac{D}{2}+1\right)\left(\dfrac{n\hbar}{D}\right)=\left(\dfrac{D+2}{2}\right)\dfrac{n\hbar}{D}=n\hbar\left(\dfrac{1}{2}+\dfrac{1}{D}\right)}$

Finally, the periods are also (generally) quantized. Firstly, we will derive the generalized Kepler law for our homogenous potential and central power law for the the force, in the classical sense. Circular orbits satisfy

$m\omega^2R=kDR^{D-1}$

$\omega=\dfrac{2\pi}{T}$

and thus

$m\dfrac{4\pi^2}{T^2}R=kDR^{D-1}$

and hence

$\boxed{T^2=\dfrac{4\pi^2m}{kD}R^{2-D}}$

In the case of an inverse dth-force law (as those related to theories with extra dimensions), we identify

$D=-1-d$

so the previous generalized Kepler third law is rewritten

$\boxed{T^2=-\dfrac{4\pi^2m}{k(1+d)}R^{3+d}}$

and the case $d=0$, with $k=-GMm$, correspond to the usual newtonian case. Now, we proceed with the quantization procedure. It yields

(18) $\boxed{T_n^2(D)=\dfrac{4\pi^2m}{kD}\left(\dfrac{n^2\hbar^2}{mkD}\right)^{\frac{2-D}{2+D}}}$

(19) $\boxed{T_n(D)=\sqrt{\dfrac{4\pi^2m}{kD}\left(\dfrac{n^2\hbar^2}{mkD}\right)^{\frac{2-D}{2+D}}}}$

Remark (I): If $k>0$, repulsive force, $D>0$ in order to provide positive periods. Indeed, there is subtle point here, since $k>0$ with negative $D$ provides attractive force. Otherwise, $k<0$ provides attractive force if $D>0$ and repulsive force if $D<0$. Moreover, the potential energy $U(R)=kR^D$ is attractive if $k<0$ and repulsive if $k>0$ for any D.

Remark (II): There are 2 special cases for the potential energy. The case with D=-1, the so-called “keplerian” problem (or newtonian/coulombian case), and the case with D=+2, the so-called “harmonic oscillator” case. Other uncommon but known case of power-law potential is the so-called string potential, where D=+1 (not -1, like the keplerian case). The string potential is useful in the quantum bouncing ball problem and the quark confinement.

Remark (III): The case with D=-1-d correspond to a world with “d” spatial extra dimensions. The case with D=+2d corresponds to the case “hyperharmonic”, and the case D=2d-1 to the “hyperstring” (d-brane) potential (string or 1-brane, 2-brane, 3-brane,…). The list of potential energies and the corresponding forces would be

1st. Harmonic oscillator: $U(R)=k_2R^2$, $F(R)=-2k_2R$

2nd. The hyperharmonic oscillator: $U(R)=k_{2d}R^{2d}$, $F(R)=-2dk_{2d}R^{2d-1}$

3rd. The keplerian/newtonian/coulombian problem: $U(R)=k_{-1}R^{-1}$, $F(R)=k_{-1}R^{-2}$

4th. The hyperkeplerian/hypernewtonian/hypercoulombian problem: $U(R)=k_{-1-d}R^{-1-d}$, $F(R)=(1+d)k_{-1-d}R^{-2-d}$

5th. The string potential: $U(R)=k_1R$, $F(R)=-k_1$

6th. The hyperstring potential: $U(R)=k_{2d-1}R^{2d-1}$, $F(R)=-(2d-1)k_{2d-1}R^{2d-2}=-(2d-1)k_{2d-1}R^{2(d-1)}$

Example 1. The harmonic oscillator. We have D=2, and $U(R)=kR^2$. Please, note that generally the normalization constant is not k but $k/2$ for the potential energy in the harmonic oscillator (D=2). Therefore, we can compute all the above quantities (radius, velocity, momentum, acceleration, force,…) plugging D=2 in the formulae, to provide

$U(R)=kR^2$, $F(R)=-2kR$

a) $L=n\hbar$

b) $p_n^2=2mkR_n^2$

c) $p_n=\sqrt{2mkR_n}=\sqrt[4]{2mk}(n\hbar)^{1/2}$

d) $R_n=\sqrt[4]{1/2mk}(n\hbar)^{1/2}$

e) $v_n=\sqrt[4]{2k/m^3}(n\hbar)^{1/2}$

f) $a_n=\sqrt[4]{8k^3}{m^5}(n\hbar)^{1/2}$

g) $F_n=ma_n=\sqrt[4]{8k^3/m}(n\hbar)^{1/2}$

h) $E_n=\sqrt{2k/m}n\hbar$

i) $\omega_n=E_n/\hbar=\sqrt{2k}{m}n=\omega_0n$

j) $T_n^2=T^2=\dfrac{2\pi^2m}{k}$. Curiously, the periods (or the time) are nor quantized for the harmonic oscillator, the period is “constant” somehow but you can consider multiples of it even then it is constant (not quantized) itself from the above.

Example 2. The keplerian hydrogen atom. We write $D=-1$ and $k=-k_Ce^2$, where $k_C$ is the Coulomb constant and $e$ is the absolute value of the electric charge of a single electron/proton. We define $\alpha=k_Ce^2/\hbar c$ as the fine structure constant, the Bohr radius $a_B=\hbar^2/mk_Ce^2=\lambda_C/\alpha$ and $\lambda_C=\hbar/mc$ as the reduced Compton wavelength of the electron. Thus, we get the quantities

$U(R)=-k_C\dfrac{e^2}{R}$, $F(R)=-k_C\dfrac{e^2}{R^2}$

a) $E_n=-\dfrac{\alpha^2}{2}\dfrac{mc^2}{n^2}$

b) $v_n=\alpha \dfrac{c}{n}$

c) $p_n=mv_n=m\alpha\dfrac{c}{n}$

d) $R_n=a_Bn^2$

e) $L=n\hbar=mv_nR_n$

f) $a_n=\dfrac{\alpha^3}{\hbar}\dfrac{mc^3}{n^4}$

g) $F_n=ma_ n=\dfrac{\alpha^3}{\hbar}\dfrac{m^2c^3}{n^4}$

h) $T^2_n=4\pi^2\dfrac{m}{k_ce^2}R^3_n$

A variation of the hydrogen atom is the so-called “gravitational atom”, where $k=G_NMm$ or $k=G_Nm^2$ if the 2 orbiting masses are identical.

See you in my next blog post!

LOG#150. Bohr and Doctor Who: A=mc³.

The year 2013 is coming to its end…And I have a final gift for you. An impossible post!

This year was the Bohr model 100th anniversary. I have talked about this subject already, herehere and here. The hydrogen spectrum is very important in Astronomy, Astrophysics and even cosmology, chemistry, and quantum mechanics. Quantum mechanics provides indeed the same results for the hydrogen atom than that of the original Bohr model, but it also includes novel effects: the Stark effect, the Zeeman effects- normal and anomalous, and many others like the hyperfine structure of the atom! However, one of the magical things of the hydrogen atom is that it yields the right results if you neglect purely quantum effects as spin, and other subtle effects like the finite mass effect (this effect can be even obtained with the aid of classical mechanics and the notion or reduced mass but it does not matter to the needs of this post!). The energy levels of the hydrogen atom can be summarized in the next pictures:

Is it cool?

Other useful scheme for you:

The Bohr model was revolutionary…

Recently, this year happened the Doctor Who 50th anniversary as well!!!!!And it has been a revolutionary year for whovians too!

Therefore, I thought (some weeks ago): I am a geek,  I am a whovian, I am a physicist, I am a theoretical physicist, I am a physmatician/physchematician as well. Thus, I have to merge all this crazy stuff together…Then, this special 150th post will try to do it! The Physics, Chemistry, Mathematics of a revolutionary theory and a wonderful (inmortal?) TV show…Impossible? Impossible post? Let me know after you read the final result…

Let me begin with a short review of the popular and well known Bohr model (for hydrogenlike atoms) formulae:

1) Quantization of angular momentum (quantization of “rotational features”).

$\boxed{L(n)=L_n=mv_nr_n=\hbar n=\dfrac{hn}{2\pi}\;\;\;\; n=1,2,\ldots,\infty}$

2) Quantization of radius (quantization of allowed orbits, or quantization of length/space!) and area.

$\boxed{R(n)=R_n=a_Bn^2=a_0n^2=\dfrac{1}{Z\alpha}\dfrac{\hbar}{mc}n^2=\dfrac{1}{Z}\dfrac{\hbar^2}{mK_Ce^2}n^2\;\;\;\; n=1,2,\ldots,\infty}$

The length traveled by an electron in the hydrogen atom is also quantized by the rule $l_n=2\pi R_n$ as well.

Area of circular orbits are given by $S=\pi R_n^2$, and thus

$\boxed{S(n)=S_n=\pi (a_Bn^2)^2=\pi a_B^2n^4=\pi\dfrac{1}{Z^2\alpha^2}\dfrac{\hbar^2n^4}{m^2c^2}=\pi\dfrac{1}{Z^2}\dfrac{\hbar^4}{m^2K_C^2e^4}n^4\;\;\;\; n=1,2,3,\ldots,\infty}$

they are also quantized!

3) Quantization of velocities.

The rate of change in position with respect to time is also quantized through the rule

$\boxed{v(n)=v_n=\dfrac{Z\alpha c}{n}=\dfrac{ZK_Ce^2 }{\hbar n}\;\;\;\; n=1,2,\ldots,\infty}$

4) Quantization of linear momentum.

As a consequence of quantization of velocities (or space quantization as well), the linear momentum is also quantized

$\boxed{p(n)=p_n=mv_n=\dfrac{n\hbar}{R_n}=\dfrac{Z\alpha mc}{n}=\dfrac{ZK_Ce^2 m}{\hbar n}\;\;\;\; n=1,2,\ldots,\infty}$

5) Quantization of acceleration. The centripetal acceleration is quantized as well

$\boxed{a(n)=a_n=\dfrac{v_n^2}{R_n}=\dfrac{Z^3\alpha^3mc^3}{\hbar n^4}= \dfrac{Z^3}{\hbar n^4}mc^3\dfrac{K_C^3e^6}{\hbar^3c^3}=\dfrac{Z^3K_C^3e^6}{\hbar^4n^4}m\;\;\;\;n=1,2,3,\ldots,\infty}$

This equation will be very useful and vital in this post. Keep it in your mind! (In fact, whovians KNOW that $mc^3$ in the time vortex but they are wrong in what it means!).

6) Quantization of centripetal force. The last quantization rule implies a quantization rule for the centripetal force

$\boxed{F(n)=F_n=ma_n=\dfrac{Z^3\alpha^3m^2c^3}{\hbar n^4}=\dfrac{Z^3}{\hbar n^4}m^2c^3\dfrac{K_C^3e^6}{\hbar^3c^3}=\dfrac{Z^3K_C^3e^6}{\hbar^4n^4}m^2\;\;\;\;n=1,2,3,\ldots,\infty}$

7) Quantization of energies.

$\boxed{E(n)=E_n=\dfrac{1}{2}ma_nR_n=\dfrac{Ry}{n^2}=\dfrac{Z^2\alpha^2mc^2}{2n^2}=\dfrac{Z^2(K_Ce^2)^2m}{2\hbar^2n^2}\;\;\;\;n=1,2,\ldots,\infty}$

In summary, radius, length, area, velocity, linear momentum, angular momentum, acceleration, force and energy is quantized in the Bohr model. Awesome! Are you Bohred? LoL. Now, the whovian part. According to the trivia (and whovian “culture”), it is said that (wikipedia):

“(…)In the science fiction television series Doctor Who, the Time Vortex is the medium that TARDISes and other time machines travel through. The “howlaround” tunnel in most versions of the series’ title sequence is supposed to be a representation of the Time Vortex, although it is sometimes also shown as nothingness.

The Vortex is outside normal spacetime, and therefore normal rules of physics to not apply. For instance, in the Vortex the equation for the relationship between energy and matter is E=mc³ (The Time Monster).

The Vortex is an extremely hostile environment. In the serial Warriors’ Gate opening the TARDIS in flight exposes the interior to the Time Winds, which age whatever they come into contact with.

Beings that dwell in the Vortex include the Chronovores (The Time Monster), the Vortex Wraiths (the Eighth Doctor Adventures novels The Slow Empire and Timeless) and the vortisaurs(the Big Finish audio play Storm Warning).

In the Eighth Doctor Adventures, Sabbath‘s employers set up their headquarters in the Vortex, casting many of the natives out into the linear universe.(…)”

And we also find the following comments in the world wide web:

“(…)E=MC³ in the extra temporal physics of the time vortex. ‘Being without becoming, an ontological absurdity!’ The Doctor makes a ‘time flow analogue’ from a Moroccan burgundy bottle, spoons, forks, corks, keyrings, tea leaves and a mug. ‘The relationship between the different molecular bonds and the actual shapes form a crystalline structure of ratios.’ [Since the Doctor and the Master made these at school (the Academy) to spoil each other’s time experiments, it is only the shape of the things that are important.] The Master works out his landing coordinates with map and compass. A lot of polarities get reversed.(…)”

The point I want to remark is that of the physical interpretation of $E=mc^3$ is wrong. Indeed, that has no dimensions of energy. Energy has (physical, not numerical) dimensions of $ML^2T^{-2}$ irrespectively the number of spacetime dimensions! That is how Physics works. But, you know, The Doctor lies (rule number One!). In fact, the right interpretacion of $mc^3$ is a quantum interpretation. If you have kept your attention to the Bohr model formulae, you should have realized something that is “almost” $mc^3$. Plug $Z=\alpha=n=1$, then you get that accelerations in the Bohr model are quantized and are less than the maximal acceleration

$\boxed{a_{max}=A=\dfrac{mc^3}{\hbar}}$

And if you take units in which the (rationalized) Planck constant is taken to be equal to the unit ($\hbar=1$), you get the “time vortex” equation for acceleration (or gravitational field if you take into account the equivalence principle, being “naive”):

$\boxed{A=mc^3}$

Here you are!!!!!! The geekiest cool explanation of the right physical meaning of A=mc³ (Not E=mc³!!!!!!!). In fact, there is more interesting cool stuff to discuss. Common special relativity is related to the existence of one maximal speed (velocity): the speed of light. Of course, in quantum theory this becomes a bit fuzzy because of the Heisenberg uncertainty principle and even general relativity and some Beyond Standard Model theories (like the Magueijo-Moffat theories of varying speed of light) can change it, but, in the end, “classically” there seems to exist a maximal speed in 3+1 spacetime. It is the speed of light. Similarly, with the aid of the Bohr model (note that the Bohr model as stated and studied here is NOT a model consistent with special relativity! Exercise: state why!;)) you can guess a new “extended” relativity principle: the principle of maximal acceleration! From the Bohr model you obtain quantized accelerations decreasing as $n^{-4}$. $A=mc^3/\hbar$ is the maximal acceleration. Not surprisingly, there has been some speculations about the existence of a “new extended relativity principle” related to that maximal acceleration principle (MAP). The MAP was pioneered by Born in his reciprocal relativity, Caianiello, Nesterenko, Scarpetta, and others. Nowadays, the principle of maximal acceleration (MAP) is seriously considered by some scientists like Brandt, C. Castro, and some quantum gravity theorists like Rovelli himself! It is not crackpottery. MAP is out there. Even more, we can extended MAP further…Is there a maximal jerk? A maximal length? A maximal n-th derivative of position? A maximal -nth (negative or integral) derivative of position? A maximal fractional n-th derivative? What about minimal bounds? Quantum gravity is generally assumed to provide area (and length) quantization, and it is generally assummed that quantum gravity provides a minimal length on very general assumptions (irrespectively you use superstring theory or not!). Thne next “fact” was one of the main lectures obtained from the physics of Black Holes (whovian comment: Time Lords get its power source from a Black Hole!): the Bekenstein conjecture states that the Black Hole entropy, proportional to the area, is quantized. So, area and length are quantized (remarkably similar to some of the features of the Bohr atom!). Thus, momentum itself will be quantized, also angular momentum, velocity and acceleration as well! What about the derivative of acceleration? And the derivative of derivative of acceleration? We can speculate if any derivative (even integrals of position or fractional differentegrals) is quantized! Position, velocity, acceleration, jerk,… I have discussed the naming and physical interpretations of higher order velocities (n-th derivatives) here.

Indeed, a simple and naive (physical) dimensional analysis allow us to recover the main relations of Quantum Mechanics and Special Relativity, and to build new interesting relationships I have discussed in the blog post mentioned above. Let me put this statement into equation a bit further. We define negative derivatives (integrals!) of position as the kinematical variables:

$\displaystyle{\boxed{\mathcal{A}=\int xdt=\mbox{Absement}}}$

$\displaystyle{\boxed{\tilde{\mathcal{A}}=\int x d^2t=\mbox{Absity}}}$

$\displaystyle{\boxed{\tilde{\mathcal{A'}}=\int xd^3t=\mbox{Abseleration}}}$

$\ldots$

Physical dimensions of some classical magnitudes (giving up purely numerical prefactors):

$E=\mbox{Energy}=(\mbox{Mass})(\mbox{Velocity})^2=(\mbox{Force})(\mbox{Space})=(\mbox{Force})(\mbox{Displacement})$

$\mbox{Linear momentum}=(\mbox{Mass})(\mbox{Velocity})$

$\mbox{Angular momentum}=(\mbox{Mass})(\mbox{Velocity})(\mbox{Displacement})$

Some “new” kinematical and dynamical variables:

$\mbox{Yank}=\left(\dfrac{\mbox{Force}}{\mbox{Time}}\right)=(\mbox{Mass})(\mbox{Jerk})$

$(\mbox{Absement})=(\mbox{Displacement})(\mbox{Time})$

And there are many others I will not rewrite here again. In terms of physical dimensions, we also know that:

$E=ML^2T^{-2}=(\mbox{Absity})(\mbox{Tug})=(\mbox{Abseleration})(\mbox{Snatch})$

Can it be generalized further? Yes, it can! Note that:

$E=(LT^2)(MLT^{-4})=(LT^3)(MLT^{-5})$

so we can write

$(1)\boxed{E=\left(D^{-n}_t (x)\right)\left( D_t^{n+2} (x)\right)(\mbox{Mass})}$

for $n=0,1,\ldots,...$. Indeed,  we can also write (writing x=LX, X adimensional)

$(2)\boxed{E=ML^2\left(D^{-\alpha}_t X\right)\left( D_t^{\alpha+4} X\right),\;\;\forall\alpha\in\mathbb{Z}}$

In fact, you can symmetrize (1) as follows:

$(3)\boxed{E=(\mbox{Mass})\left(D^{-n+1}_t (x)\right)\left( D_t^{n+1} (x)\right)}\;\;\;\forall n\in\mathbb{Z}$

and where we define the negative differentiation (integration!) as

$I=\left(\dfrac{d}{dt}\right)^{-n}=D_t^{-n}=\left[T^n\right]$

and where the last term being the physical dimensions. The positive derivatives are defined as usual

$\mathcal{D}=D_t^n=\left(\dfrac{d}{dt}\right)^n=\left[T^{-n}\right]$

A completely symmetric form is also available, from (3):

$(4)\boxed{E=(\mbox{M})\left(D^{\frac{2(-n+1)}{2}}_t (x)\right)\left( D_t^{\frac{2(n+1)}{2}} (x)\right)}\;\;\forall n\in\mathbb{Z}$

or, at formal level of “fractional derivatives”

$(5)\boxed{E=(\mbox{M})\left(\sqrt{D_t^{2(-n+1)}}(x)\right)\left(\sqrt{D_t^{2(n+1)}} (x)\right)}\;\;\forall n\in\mathbb{Z}$

From any of these last equations (3)-(4)-(5) (or (1), (2)), you can derive a complete set of kinematical and dynamical relations for integer values of $n, \alpha$. You can recover classical relationships like

$E=xD_t^2(x)M=Fx=(\mbox{Force})(\mbox{Displacement})=(\mbox{Mass})(\mbox{Acceleration})(\mbox{Displacement})$

or

$E=\mbox{Energy}=MaX=(\mbox{Mass})(\mbox{Velocity})^2$

and you can also build up new ones, like

$E=\mathcal{A}\mathcal{Y}=\mbox{(Absement)}\mbox{(Yank)}=(\mbox{Absement})(\mbox{Mass})(\mbox{Jerk})=(\mbox{Mass})(\mbox{Presement})^{-2}$

This (fractional) operator (differentegral calculus!) formalism is very useful to determine and quickly remember some old results and produce new interesting variables. Furthermore, if you implement a generalized principle of maximal velocity, and maximal (linear) momentum, you recover the usual relations of special relativity! I mean, take the formal expressions

$E=pc$

$E=mc^2$

which are obtained from the relativistic dispersion relationship

$E^2=p^2c^2+m^2c^4\leftrightarrow E=\sqrt{p^2c^2+m^2c^4}$

in the limits of $m=0, p\neq 0$ and $m\neq 0, p=0$. These formal relationships can be obtained from our general expressions above:

$(\mbox{Energy})=(\mbox{Linear momentum})(\mbox{Velocity})=pD_t(X)$

$(\mbox{Energy})=(\mbox{Mass})(\mbox{Velocity})^2=MD_t^2(X)$

from maximal velocity $D_t(x)\leq c$! One should be wonder if similar things do exist for higher order derivatives (likely yes, but nobody knows for sure today, and it is speculative). In fact, from Newton’s 2nd law:

$\displaystyle{F=\mbox{Force}=D_tp=D_t(\mbox{Linear Momentum})\leftrightarrow p=D_t^{-1}F=\int F dt=D_t^{-1}\mbox{(Force)}}$

Is force (acceleration) bounded from maximal force (accelerations)? What about mass, length or absement? What about any n-th order derivative? And fractional derivatives? Minimal action is common in the variational approach of mechanics (lagrangian and hamiltonian dynamics), but we lack a Maximal Action Principle as we lack a Maximal Acceleration Principle in phycics at current time. Physics are based on critical points of action functionals

$\displaystyle{S(q,\dot{q};t)=\int L(q,\partial_t q;t)dt}$

or its higher order generalizations (sometimes referred as higher order lagrangian theory)

$\displaystyle{S(q,\dot{q},\ddot{q},\ldots;t)=\int L(q,\partial_t,\partial_{tt}q,\ldots;t)}dt$

and, for this particle actions, their field theory analogues are, respectively

$\displaystyle{S(\phi,\partial\phi;x)=\int \mathcal{L}(\phi,\partial\phi;x)d^Dx}$

$\displaystyle{S(\phi,\partial \phi,\partial^2\phi,\ldots;x)=\int \mathcal{L}(\phi,\partial\phi,\partial^2 \phi,\ldots;x)d^Dx}$

in D spacetime dimensions, with

$x=(x^\mu)$

$\partial=D_\mu=\partial_{\mu}$

$\partial^2=\partial_{\mu_1\mu_2}$

and so on! Why am I commenting this? Well, recently, the so-called “double field theory” became popular in several branches of theoretical physics: string theory/M-theory, QFT, dualities and related stuff. Well, it seems pretty interesting to me that the doubling of fields can be suggested by this operator formalism (both in the particle and field pictures) by the following map:

$q\rightarrow 1/q$

$\partial_t\rightarrow \partial_t^{-1}=\dfrac{1}{\partial_t}$

$\phi\rightarrow 1/\phi$

$\partial_\mu\rightarrow\partial_\mu^{-1}$

$\partial\rightarrow 1/\partial=\partial^{-1}$

This T-duality (S-duality if you go to the realm of coupling constants as well) is very suggestive…It extends the classical actions and (at least) enlarge (duplicates,multiplies) the degrees of freedom (double field theory is contained in this operational approach, but it gets generalized!):

$\displaystyle{S=\int\mathcal{L}(q,\partial_tq;t)dt\rightarrow S=\int\mathcal{L}\left(q,\dfrac{1}{q}=q^{-1},\partial_tq,\partial^{-1}_tq,\partial^{-1}_tq^{-1};t\right)dt}$

$\displaystyle{S=\int\mathcal{L}(q,\partial_tq,\partial^2_tq,\ldots;t)dt\rightarrow S=\int\mathcal{L}\left(q,\dfrac{1}{q}=q^{-1},\partial_tq,\partial^{-1}_tq,\partial^{-1}_tq^{-1},\partial^2_tq,\partial^2_tq^{-1},\partial^{-2}_tq,\partial^{-2}_tq^{-1},\ldots;t\right)dt}$

$\displaystyle{S=\int\mathcal{L}(\phi,\partial\phi;x)d^Dx\rightarrow S=\int\mathcal{L}\left(\phi,\phi^{-1},\partial \phi,\partial^{-1}\phi,\partial^{-1}\phi^{-1};x\right)d^Dx}$

$\displaystyle{S=\int\mathcal{L}(\phi,\partial\phi,\partial^2\phi,\ldots;x)d^Dx\rightarrow S=\int\mathcal{L}\left(\phi,\phi^{-1},\partial\phi,\partial^{-1}\phi,\partial^{-1}\phi^{-1},\partial^2\phi,\partial^2\phi^{-1},\partial^{-2}\phi,\partial^{-2}\phi^{-1},\ldots;x\right)d^Dx}$

Does such a generalized MAP/generalized special relativity/Maximal Length Principle exist? I think so. Even more, minimal action, minimal length, minimal velocity, minimal acceleration,…principles should exist as well to explain the quantum realm in some generalized enlarged “pseudoclassical” extended theory of relativity. I can tell you more: I work on it! The existence of such a principle, I think, is the true door to quantum gravity and the true Theory Of Everything (TOE). Indeed, classical mechanics has a principle of minimal action (minimal velocity?) and it also holds in Quantum Mechanics (due to the Heisenberg Uncertainty Principle, HUP). Generalized Uncertainty Principles (GUP) are also discussed in the phenomenology of Quantum Gravity. Of course, all this stuff speculative. I have also asked this question in physics stack exchange:

http://physics.stackexchange.com/questions/61522/extended-born-relativity-nambu-3-form-and-ternary-n-ary-symmetry

Indeed, I wrote a message to a young graduate student who wrote a preprint time AFTER I posted the above question. The preprint is this one:

http://arxiv.org/abs/1308.4044

Its title (and I urge you to read the whole paper) says it all: Notes on several phenomenological laws of quantum gravity.

It is a great paper! Elementary but fascinating! It serves as a good starting point for newbies into the fascinating topic of extended theories of relativity! A subject in which I am actively involved, you have really no idea of how! :).

Only to say a little bit, quantum physics is (currently) believed to quantize (or be based on the quantization of) angular momentum and (specially) energy (neglecting continuous spectra obtained in the large n limit). Quanta of energy are sometimes called quanta, but you could call them “energons” (I am sorry, I am a fan of the transformers as well, so let me borrow the term at the moment and call them quanta in your daily life if you want). Quanta of angular momentum is related to spin, so we could call “spinons” to those elementary quanta. Quanta of charge could be called “chargons”. And you could guess similar quanta of physical magnitudes, even for those new (generally unknown) quantities we call absement, presement, absity, abseleration, tug,…But what about a quantum of space or time? You can call them choraons and chronons (chronons have been discussed in the past by Finkelstein, Caldirola and some other physicists), and these quanta are related to the quanta of linear momentum or energy in a simple way. Loop Quantum Gravity using spin networks derive a similar concept. Indeed, mass itself is related to the reation of quanta of energy, time and space by simple physical dimensions! Remember: $E=ML^2T^{-2}$. If you quantize energy, length and time, mass itself MUST be quantized (to be more precise, it should be “fractionalized” or it will be a rational multiple of some integer/rational number! It is strongly similar to the phenomenon of fractional charge quantization -fractionalization of charge- now popular in condensed matter physics!).

In summary: There is no choice there! Everything is a quantum/some quanta of something! Every magnitude should be quantized in physics. There should be a minimal/maximal or MIN/MAX X-principle acting in fundamental physics over any fundamental system of magnitudes. Position or length, velocity, acceleration, energy, mass, angular momentum or action, linear momentum, even absement, jerk and any n-th derivative (positive, integer or fractional) should reach some minimal/maximal values. Surprised? You should not! It is pure (atomic like) logic!

By the other hand, the role of generalized Nambu mechanics in physics (and its quantization) is yet unclear. There has been some recent advances I follow closely, and some cool new applications of Nambu Mechanics in hydrodynamics, atmospheric dynamics (essentially, the Lorenz system can be written as a Nambu system) or pure electromagnetism and electrodynamics. However, this Nambu stuff will be told in the near future, in a forthcoming TSOR post!

And all these lines were my impossible 150th post!!!!!! I hope you liked it! See you in a new amazing TSOR post! See you in 2014! Stay tuned!

PS: Happy new year 2014 everyone and everywhere out there! Best wishes for all of you from my inner soul, heart and (crazy) mind! Hahahahahaha… Take care and enjoy your mortal existence!