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, here, here 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”).

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

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

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

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

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

they are also quantized!

3) Quantization of velocities.

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

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

4) Quantization of linear momentum.

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

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

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

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

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

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

7) Quantization of energies.

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

8) Quantization of periods (or time quantization!).

Since $E_n=\hbar\omega_n$ and $T_n=\dfrac{2\pi}{\omega_n}$ , periods in orbital circles are also quantized accordingly to the rule:

(9) $\begin{equation*} \boxed{T(n)=T_n=\dfrac{2\pi\hbar }{E_n}=\dfrac{Ry}{n^2}=\dfrac{4\pi\hbar^3n^2}{Z^2(K_Ce^2)^2m}\;\;\;\;n=1,2,\ldots,\infty}\end{equation*}$

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

(10) $\begin{equation*} \boxed{a_{max}=A=\dfrac{mc^3}{\hbar}}\end{equation*}$

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”):

(11) $\begin{equation*}\boxed{A=mc^3}\end{equation*}$

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

(12) $\begin{equation*}\boxed{E=\left(D^{-n}_t (x)\right)\left( D_t^{n+2} (x)\right)(\mbox{Mass})}\end{equation*}$

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

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

In fact, you can symmetrize the above equations as follows:

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

(15) $\begin{equation*}\boxed{E=ML^2\left(D^{-\alpha+1}_t X\right)\left( D_t^{\alpha+1} X\right),\;\;\forall\alpha\in\mathbb{Z}}\end{equation*}$

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

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

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

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

A completely symmetric form is also available, from previous results:

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

or, at formal level of “fractional derivatives”

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

From any of these last equations, you can derive a complete set of kinematical and dynamical relations for integer values of $n, \alpha$ . You can recover classical relationships like

(20) $\begin{equation*} E=xD_t^2(x)M=Fx=(\mbox{Force})(\mbox{Displacement})=(\mbox{Mass})(\mbox{Acceleration})(\mbox{Displacement})\end{equation*}$

(21) $\begin{equation*} E=\mbox{Energy}=MaX=(\mbox{Mass})(\mbox{Velocity})^2\end{equation*}$

and you can also build up new ones, like

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

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:

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

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

(24) $\begin{equation*} \displaystyle{S(q,\dot{q};t)=\int L(q,\partial_t q;t)dt}\end{equation*}$

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

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

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

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

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

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!):

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

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

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

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

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!

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The Spectrum Of Riemannium

The Art Of Science and Physmatics. This is the geeknerd way!

LOG#150. Bohr and Doctor Who: A=mc³.

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