LOG#178. Divergent sums: The Number awakens!

summary-divergencesAre you divergent?

Divergent1maxresdefaultDivergentdivergent-quoteDivergences are usually sums or results you would consider “infinite” or “ill-defined” (unexistent) in normal terms. But don’t be afraid. You can learn to “regularize” a divergent series or sum. Really? Oh, yes!divFactionsBeyond faction of a well-known book (and movie) series, we can do it. We can sum divergent sums!

Let me consider the sum S = \sum_ {k = 0} ^ \infty (-1) ^k = 1-1 + 1-1 + 1 -\cdots

You think perhaps in adds to zero like

(1)   \begin{equation*} S = (1-1) + (1-1) + (1-1) + \cdots= 0 \end{equation*}

Or, maybe, you even add it to obtain 1 in the following way:

(2)   \begin{equation*} S = 1 + (- 1 + 1) + (- 1 + 1) + \cdots = 1 \end{equation*}

However the mathematician (and some crazy theoretical physicts) have sinister ways of adding this sum you would qualify as “ divergent ”. `What create and do these crazy people of physics and mathematics? I will tell you. But I warn you. They are Dark Arts. Taking common factor:

(3)   \begin{equation*} S = 1- (1-1 + 1-1 + \cdots) = 1-S \end{equation*}

(4)   \begin{equation*} S = 1-S \rightarrow 2S = 1 \rightarrow S = \dfrac{1}{2} \end{equation*}

This seems (black) magic and delirious, because an infinite sum of alternating numbers providing  a fractional number seems to be really from another dimension or parallel universe. But under certain conditions it can be done … And worse, it serves to “ test” results yet more ​​disturbing. The following sum:

(5)   \begin{equation*} A = 1 + 1 + 1 + 1 + 1 + \cdots = \sum_ {k = 1}^ \infty 1 = \sum_ {k = 1} ^\infty (-1) ^{2k} \end{equation*}

you can add using the above sum S = 1-1 + 1-1 + 1-1 + \cdots = 1/2, because if:

(6)   \begin{equation*} A = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + \cdots \end{equation*}

(7)   \begin{equation*} 2A =  2 + 2 + 2 +2+ \cdots \end{equation*}

(8)   \begin{equation*} A-2A = -A = 1-1 + 1-1 + 1-1 + \cdots = S \end{equation*}

and therefore, as we had ominously calculated that S = 1/2, we have the amazing result:

(9)   \begin{equation*} A = 1 + 1 + 1 + 1 + 1 + \cdots = - \dfrac{1}{2} \end{equation*}

Awesome! But there is a “one more thing” … We will now calculate a sum to think it really gives infinite:

(10)   \begin{equation*} B = \sum_ {n = 1} ^\infty n = 1 + 2 + 3 + 4 + 5 + \cdots \end{equation*}

To do this, let’s do another trick of mathematical magician:

(11)   \begin{equation*} B = 1 + 2 + 3 + 4 + \cdots \end{equation*}

(12)   \begin{equation*} 4B = 4 + 8+ \cdots \end{equation*}

(13)   \begin{equation*} -3B=B-4B = 1-2 + 3-4 + \cdots = \sum_ {n = 1}^\infty (-1) ^{n + 1} n = C \end{equation*}

Now calculate, using past computations:

(14)   \begin{equation*} C = 1-2 + 3-4 + \cdots = \sum_ {n = 1} ^\infty (-1) ^{n + 1} n \end{equation*}

Our first concrete result, squared, can be rewritten as follows:

(15)   \begin{equation*} S ^2 = (- 1 + 1-1 + 1-1 + \cdots) ^2 \end{equation*}

Or well

(16)   \begin{equation*} S^2 = S \cdot S \end{equation*}

(17)   \begin{eqnarray*} (-1 + 1-1 + 1-1 + \cdots)\\ \underline{\times (-1 + 1-1 + 1-1 + \cdots)} \end{eqnarray*}

(18)   \begin{eqnarray*} =1-1 + 1-1 + 1-1 + 1-1 + \cdots \\ -1 + 1-1 + 1-1 + 1-1 + \cdots\\ + 1-1 + 1-1 + 1-1 + \cdots\\ \cdots \end{eqnarray*}

(19)   \begin{equation*} 1-2 + 3-4 + 5-6 + \cdots = C \end{equation*}

Therefore: S ^ 2 = C. As S = 1/2, then S ^2 = C = (1/2) ^ 2 = 1/4. Therefore, B = C/ 3 = -1/12. That is (to freak out a last time), we have shown that

(20)   \begin{equation*} \boxed{B = 1 + 2 + 3 + 4 + 5 + \cdots = - \dfrac{1}{12}} \end{equation*}

Astonishing! However, all this stuff is not new. It was known for many people before me, and it was Ramanujan and later Hardy in a book titled Divergent Series where you can find fine theorems about this crazy subject.

Other divergences:

  1. DC comics divergence series…


2. Divergence theorem!

div-div-thmA3. Futurama divergent series…

FuturamaDivergenceSee you in my next blog post!

P.S.: Moral…




Joël Scherk (not Schrek, the ogre) was a French theoretical physicist with a tragic death. Perhaps his contributions were short, but they were important in the fields of supersymmetry (SUSY), supergravity (SUGRA) and superstrings. Moreover, he also introduced some crazy ideas related to these things I am going to discuss. This blog post is essentially a refurbished reedition of his famous talk giving an overview about SUGRA, SUSY and antigravity. I have added some extra stuff by my own, and I plan to update this post in the future with more topics involving the phenomenon of antigravity that Scherk discovered long ago. Enjoy it!

First of all, even when I have not (not yet) dedicated a thread to SUSY or SUGRA, I will be discussing some of his ideas with respect to them.

What is SUSY? SUSY is just a particular type of symmetry or transformation group. Generally speaking, SUSY is defined in terms of transformations which leave invariant certain action AND transforms fermions (or fermionic fields to be more precise) into bosons and viceversa. I have discussed group theory in this blog (elementary group theory) and Lie algebras too. There is a generalization of Lie algebras called Lie superalgebras, also known as graded Lie algebras. Graded Lie algebras (GLA or superalgebras) are the elementary mathematical structures and tools used in SUSY/SUGRA. GLAs used in SUSY and SUGRA models, and their mathematical representations analogue to those in Lie algebras, will be reviewed in this post, based on a famous Scherk’s talk…I am going to explain you some bits and glimpses of the (extended) Poincaré GLA, the (extended) Poincaré de Sitter GLA, and the conformal GLA. I will be also introducing you the N=8 SUGRA and its relevance to physics.


The mere existence of fermions can be traced back to the early 20th century. Around 1925, physicists found the strange and weird character of the electron and similar particles. “Why fermions?” Louis de Broglie asked that question, even when the Pauli exclusion principle was formulated. So we are in the situation where Nature chose the electron to be a fermion (proton an neutrons are fermions too). From the experimental aside, we can not deny but accept the existence of fermions: spin 1/2 particles! What about spin s/2 particles, with odd s greater than 1? We will learn about this just a little…Focusing on the 1/2 case…Quarks are also fermions! Fundamental forces, from the quantum viewpoint, are mediated by integer spin particles (gluons, photons, W and Z bosons and now, it seems, Higgs bosons too). We have also the hypothetical gravitons from the gravitational theories like General Relativity and extensions. Thus,  integer spin particles and its existence is out from any reasonable doubt. However, fermions are just a bit more mysterious. Poor mathematicians and mathematical physicists use to ask and wonder why fermions are relevant.

Simple answer: “There are fermions in Nature because we observe them”

Uhlenbeck and Gouldsmith told us in 1925 that the spin of the electron is 1/2, no matter how weird it is. It is an experimental fact! Furthermore, the spin-statistics theorem tells us that if we have spin 1/2 particles or fields (Dirac fields for simplicity at the moment), these should be quantized through Fermi-Dirac distributions/statistics, while if one deals with integer spin fields, their corresponding distribution is that of Bose-Einstein type. In summary, there are boson and fermions in Nature. The Standard Model (or Standard Theory) says that force carriers are boson fields, and that matter fields are fermion fields. This reasoning is, perhaps, not simpler that saying “A rose is a rose is a rose” (Gertrude Stein).

Elaborated answer: There is a deeper reason of spin 1/2 existence. From a mathematical viewpoint, it is ultimately related to the rotation group SO(3), and the Poincaré group. The SO(3) group is NOT simply connected and its universal covering group is SU(2). SO(3) admits integer representations only, but SU(2) admits 1/2 integer as well as integer representations. So from the SU(2) viewpoint, Nature has realized both integer and 1/2 integer representations. SO(3) is, if you want, more boring than SU(2). This second answer, however, leaves one with an unpleasant feeling, as one may wonder if mathematical physicists would have thought about SU(2) if Nature had chosen a world wher only gluons, photons, W and Z bosons, Higgses and gravitons existed, or if it had played the trick to provide us with fundamental (composite?) spin 0 scalar particles (higgses) and quarks (which is another option!).

Nature said, indeed, that there are Fermions, quarks and leptons, with J=1/2; there are also Bosons, gravity (hypothetical gravitons) with J=2 particles, electroweak particles (massless photons, and massive W,Z bosons, massive Higgses) with spin J=0,1, and strong interactions having gluons with J=1. Mathematical physicists (M\Lambda \Phi physicists) know that J(electron)=1/2, that electrons from spin-statistics imply the existence of Fermions while interactions imply the existence of (gauge/gauge like) compensating fields with J=0,1,2. Fermions (F) do exist. Why F? The response involves representations of the Poincaré group: SO(3) covering group is SU(2), which admits 1/2 spin fields (and other exotica I am not going to discuss here today). Math is OK, but why is realized by Nature? Not simple answer to this does exist. Would you search for some 1/2 piece of art? It does NOT exist spin 1/2 “art”. Metaphysical answers? The western way offers no simple explanation! The M\Lambda \Phi physicist turns towards Western rationalistic enlightenment without concrete solutions. You would only find unsatisfactory theological reasons and you will turn towards Eastern religions. The East gives you a more cool option, namely that a universal principle of harmony and dynamics balances the whole Universe (cf. the Greek idea of arche/arkhe). Chinese people call this Taiji principle the Tao. Don’t worry! I am not going to convert this article into the Tao of Physics ;). Tao refers to the interplay and inerfusion of pairs of opposite “elements” of fundamental character, dubbed Yin and Yang. The Yin refers to the female, dark , cold , soft, negative principle while the Yang (please, I am not the one who said all this even if you find it machist!) refers to the male, bright, warm, hard and positive principle/element (the Universe is mainly dark/female thought ;)).

Scherk tried to temp us with this analogy: one can associate the Yin principle with the Bose principle, that allow interpenetrability, and the Yang principle with the Fermi principle, which doesn’t. It is hard to be dogmatic on this scheme since the darkness/brightness would associate the Bose principle with the Yang and the Fermi principle with the Yin. Since the the light is known to be light/bright and not dark (unless dark photons exist), and the (SM) photon is a boson. More stupid stuff to waste time with Yin/Yang analogies in particle physics: little circles of black within white and vice versa signify that Yin bound states (stable or metastable) of the Yang principle exist and vice versa. This is known to be true as bound states of fermions (e.g. quarks) produce bosons, while the less obvious converse result is true as we know or learn about the study of classical magnetic monopole solutions. This complementarity extends in taoist thinking no only to bosons and fermions, but to many other subjects such as the World as an Object and the Ego as a Subject, which are in this way of seeing not only complementary but also interchangeable. Object/Subject complementarity leads to the cosmologist’s viewpoint (anthropocentrism, if you wish): the Universe or World is as it is in particular state because it is populated by conscious human observers (conscious E.T. beings or even conscious A.I. will also apply). The existence of life and consciousness puts some limitations on various purely physical constants such as gravitational/electromagnetic/weak/strong coupling constants and so on. This class of anthropic idea is not very popular or “well seen” between scientists. Applied to the Fermi/Bose problem, if Nature had decided for instance to build up us with J=0 leptons and J=1/2 quarks only, the nuclei would exist, and so would the atom in a modified way though. However, every atom would collapse under the gravitational strength into one giant molecule at the Earth’s center.

This last answer is also unpleasant. We turn into a more modern view of this problem. It is dominated by SUSY, SUGRA and superstring/M-theory. Local SUSY implies SUGRA. And it necessarily implies a complementarity between Bose (J=0,1,2) particles/quanta and Fermi (J=1/2,3/2) particles/states and their transmutability into each other through SUSY transformations, at least from the pure mathematical viewpoint! SUSY is equal to Fermi-Bose symmetry, and thus, is less “super”. “Super” things are cool, and thus, we prefer SUSY instead that boring Fermi-Bose symmetry naming! The final enlightenment, through some king of French connections thanks to Scherk, allows us to discover SUSY and SUGRA (part of superstring/M-theory models) as the answer to the existence of fermionic fields in Nature. If you feel disappointing at this stage you should give up and not to read any more. However, mathematics is the best tool we have to find the Nature deeper secrets. Are you ready for them? I hope so…


Making a transition phase is not easy. If your path is seriousness, let us consider a small and incomplete set of definitions (dictionary) of the word super as many people ask: “Why do you call it supersymmetry/SUSY?” If you also read my previous list http://www.thespectrumofriemannium.com/2013/05/08/log102-superstuff-the-list/ you will find other alternative and complementary superword set. Many “superwords” are scientific, some others being of common knowledge origin. For instance:

“Super”: word of american (?) origin. Opposite of “regular”. In Switzerland, “super” is a small additional invest compared to “regular” (~1.08 SF/liter vs. ~1.04 SF/liter).

“Scientific superwords”: supernova, superstar, superhelix (DNA), superconductivity, superfluidity, superaerodynamics, superphosphate, superpolyamide, supersonic, superstructure, superoxide, superactinide, superbrane, super p-brane, supermembrane, supercontinent, supermaterial, superspace, superstring, superparticle, superheterodyne, superhuman, superatom, supermolecule, supergroup, superalgebra, superextendon, supermatrix, supersymmetry, supergravity,superfield, supervielbein, super-Higgs mechanism,…

“Political superwords”: superpowers (in 1979 language: Monaco, Lietchtenstein), superphoenix (mythological bird unknown to the Antiquity, of Gallic origins).

“Touristic superwords”: superTignes (where is the CERN staff today?).

“American colloquial superwords”: “Gee, it’s super”, “Super-Duper”.

“Pop music superwords”: supertramp.

“Daily life superwords”: supermarket (Migros, Coop), superman (comic stripo by F. Nietzsche).

“Poetic touch”: superlove; the French poet Jules SUPERvielle (Montevideo 1884-Paris 1960) wrote in 1925 a collection of poems called “Gravitations”.

SUSY and SUGRA, and their relatives/offsprings, are just a few modern superwords added to this list. SUSY in flat spacetime is described by some elements:

  • Supersymmetric transformation laws (SSTL/S.S.T.L.):
    a set of continuous transformations changing classical, commuting, Bose real (or complex)  integer spin fields into classical anticommuting (sometimes dubbed classical anticommuting  c-numbers), Grassmann Fermi variables with half-integer spin fields and vice versa. Roughly, you can think about this as

    (1)   \begin{equation*} \delta \vert\mbox{Boson}\rangle\sim \vert\mbox{Fermion}\rangle,\;\;\; \delta \vert\mbox{Fermion}\rangle\sim \vert\mbox{Boson}\rangle\end{equation*}

  • SUSY model/theory.  A classical lagrangian/action field theory, in flat spacetime, whose action is invariant under SUSY transformations. Generally speaking, you hear about N=1, N=2, N=4 or even N=8 theories often in the superworld, but you can also build theories with any arbitrary number of supersymmetry generators. Mathematically speaking, your freedom is your (consistent, free of contradictions) imagination!
  • Extended SUSY. Popular expression for SUSY theories with N>1.
  • SUSY (a.k.a. supersymmetry). Any field of M\Lambda \Phi which studies local and rigid supersymmetric transformation laws (SSTL), supersymmetric theories/models and extended SUSY. Theoretical physicists are crazy for (finding) SUSY and adding SUSY to their theories. It is likely an obsession. Experimentally, SUSY is hard to kill, but LHC and other future colliders, also other cool experiments, are pushing forward into finding SUSY. SUSY is a beast. If SUSY does exist, it has to be broken in Nature. How and where is the puzzle. From the mathematical side, SUSY is known to be the only (almost unique) way to relate bosons and fermions through symmetry, evading the celebrated Coleman-Mandula theorem including fermionic generators in the algebra. That is, SUSY (superPoincaré group to be more precise) is (up to some uncommon exceptions) the only non-trivial extension of the Poincaré group containing the internal symmetries of the Standard Model. That is why some people is resistent and resilient to give up SUSY in these times. Other alternatives, like N-graded Lie algebras, quantum groups are seen just as odd or just as certain special representations of (extended/generalized) SUSY.
  • Superspace. A set of generalized coordinates Z^A, containing both, ordinary spacetime coordinates X^\mu, Bose classically commuting coordinates, and \theta^i_\alpha, Grassmann Fermi classically anticommuting coordinates. Here, \mu=1,2,\ldots,D, i=1,2,\ldots,N.         \alpha can also runs from 1 to D, but there are other alternatives and it is a free index right now. Thus, we get                                                                                     

    (2)   \begin{equation*} \boxed{Z^A=\left(X^\mu,\theta^i_\alpha\right)}\end{equation*}

  • Superfield. Any function \Phi(Z^A) with or without indices.
  • Supermultiplet. Any irreducible representation of SUSY (simple N=1, or extended N>1), expressed in terms of the (super)fields.
  • Matter supermultiplet. Any supermultiplet with J_{max}=1,1/2. You can also generalize this definition to other “matter” (dark?), like J_{max}=3/2 or higher, but it is not usual. J_{max}=1 are called the vector supermultiplets, J_{max}=1/2 are called scalar supermultiplets.
  • Supersymmetric Yang-Mills theories/Super Yang-Mills theories for short (SYM). The self-interacting (g\neq 0) field theory based on a vector supermultiplet, or extended YM theory having SUSY symmetries.
  • Goldstino. The m=0, J=1/2 fermions associated with the spontaneous symmetry breaking of SUSY.
  • Gluinos. The SU(3)_c octets with J=1/2 in SYM theories.
  • Superalgebras. Graded Lie algebras (GLAs), dubbed into a cool superword friendly synonim by crazy M\Lambda \Phi addict physicists. These addict physicists could be named superphysicists…

Example: N=1 SYM theory in D-dimensions. It contains the next ingredients,

1st. Spectrum: A_\mu^i, J=1, real fields; \chi^i, J=1/2 Majorana fields.

2nd. Coupling constant: g\neq 0.

3rd. Infinitesimal SUSY parameter:  \varepsilon. J=1/2 constant Majorana field.

4th. SUSY transformations:

(3)   \begin{equation*} \delta A_\mu^j=i\overline{\varepsilon}\gamma_\mu \chi^j \end{equation*}

(4)   \begin{equation*} \delta \chi^j=\sigma^{\mu\nu}F_{\mu\nu}^j\varepsilon \end{equation*}

5th. Action and lagrangian:

    \[S_{SYM}=\dfrac{1}{g^2}\int d^Dx \mathcal{L}_{SYM}\]

(5)   \begin{equation*} \mathcal{L}_{SYM}=-\dfrac{1}{4}F_{\mu\nu}^jF^{\mu\nu j}+\dfrac{i}{2}\overline{\chi}^j\gamma^\mu \mathcal{D}_{\mu j k}\chi^k \end{equation*}

and where

    \[\boxed{F_{\mu\nu}^j=\partial_\mu A_\nu^j-\partial_\nu A_\mu^j+gf^{j}_{mn}A_\mu^m A_\nu^n}\]

    \[\boxed{\mathcal{D}_{\mu r s}=\partial_\mu \delta_{rs}+g\delta_{rs}^l A_\mu^l}\]

If now, we upgrade SUSY by changing our flat (super)spacetime into curved (super)spacetime, we enter into the SUGRA realm. Firstly, let me introduce the main SUGRA superwords:

  • SUGRA models/theories. Any SUSY model/theory built in curved spacetime invariant under N=1 SUSY (however the latter case is sometimes named extended supergravity, but terminology is just a choice in some cases; nevertheless, pure SUGRA generally refers to N=1).
  • Extended SUGRA models/theories. As suggested in the previous definition, any theory/model built in curved spacetime invariant under N>1 SUSY. The issue of allowing or not curvature in the fermionic sector is open. Curved supertheories are usually more subtle but can be handled, in principle, with suitable tools.
  • SUGRA. The field of M\Lambda \Phi physicists (superphysicists?). It studies local SUSY transformation laws, superfields, supergravity and extended SUGRA theories.
  • Pure SUGRA. SUGRA or extended SUGRA uncoupled to any matter supermultiplets, with a self coupling \kappa, dimensionally M^{-1}.
  • Gravitino. The Rarita-Schwinger field quanta. Spin 3/2 particle associated with local simple (or extended!) SUSY.
  • Super Higgs effect/mechanism. The analogue of the Higgs effect/mechanism for SUSY. Wherever SUSY is spontaneously broken, a gravitino (or several gravitini) eats up a goldstino (or several goldstinos) becoming massive fields.

Example (II): N=1 SUGRA. The ingredients of this theory/model are

1st. Spectrum. Vielbein, J=2 real field V_\mu^a. Gravitino, J=3/2 Majorana field \Psi_\mu^a.

2nd. Coupling constant. \kappa. Related to G_N(D). It has several normalizations.

3rd. SUSY parameter. varepsilon; J=1/2 X-dependent. Majorana field.

4th. SUSY transformations. We have

    \[\boxed{\delta V_\mu^a=-i\kappa \overline{\varepsilon}\gamma^a\Psi_\mu}\]

    \[\boxed{\delta \Psi_\mu=\kappa^{-1}\mathcal{D}_\mu\varepsilon}\]

5th. Action and lagrangian.

(6)   \begin{equation*} S_{SG}=\int d^Dx\mathcal{L}_{SG} \end{equation*}

(7)   \begin{equation*} \mathcal{L}_{SG}=-\dfrac{1}{4\kappa^2}VV^{\mu a}V^{\nu b}\mathcal{R}_{\mu\nu ab}-\dfrac{1}{2}\epsilon^{\mu\nu\rho\sigma}\overline{\Psi}_\mu \gamma_5\gamma_\nu\mathcal{D}_\rho\Psi_\sigma \end{equation*}

and where we define

    \[\boxed{\mathcal{R}_{\mu\nu ab}=\partial_\mu\omega_{\nu ab}+\omega_{\mu a}^c\omega_{\nu cb}-(\mu\leftrightarrow \nu)}\]

    \[\boxed{\mathcal{D}_\nu=\partial_\nu+\dfrac{1}{2}\omega_{\nu ab}\sigma^{ab}}\]

Technical details: there are some technicalities in all these things…SUSY/SUGRA and generally field theory involves different types of spinors. Giving up the weirdest types, there are 3 main types of spinors. These spinors are named Weyl, Dirac and Majorana fields (spinors). The esential features come from the Clifford algebra of spacetime

(8)   \begin{equation*} \{\gamma^\mu,\gamma^\nu\}=2\eta^{\mu\nu} \end{equation*}

where \eta^{\mu\nu}=\mbox{diag}(\underbrace{+,\ldots,+}_{t-times},\underbrace{-,\ldots,-}_{s-times}), D=t+s, and the irreducible representation of the \gamma matices are 2^{\left[D/2\right]} dimensional.  The Majorana representation (MR) is any representation of \gamma matrices in which they are all i times real matrices. A celebrated theorem states that a MR does exist in even D if and only if s-t=0,2\mbox{mod} \;\;8. The charge conguration matrix, C, verifies C\gamma^\mu C^{-1}=-\gamma^{\mu T}, where T means transpose matrix. Majorana spinors are any spinor such that \chi=C\gamma^0\chi^*. If t=1, Majorana spinors exist only if a MR does exist. In the MR, we have C=\gamma^0 and \chi=\chi^*. Weyl spinors are any spinor set for even D, such that \gamma^{D+1}=\eta \gamma^0\gamma^1\cdots\gamma^{D-1}, with \eta such as (\gamma^{0\mu})^2=+1, so Weyl spinors satisfy the conditions \gamma^{D+1}\chi=\pm\chi, and they are “two-component like”. Finally, you can combine these conditions, and build a Majorana-Weyl spinor, i.e., \chi spinors that are both, Majorana and Weyl spinors. They do exist in D even if and only if s-t=0\mbox{mod}\;\;8. Either of these two restrictions will cut the number of independent spinor components in half. And thus, we have seen that in some dimensions it is possible to have both Weyl and Majorana spinors simultaneously. This reduces the number of independent spinor components to a quarter of the original size. Spinors without any such restrictions are called Dirac spinors. Which restrictions are possible in which dimensions comes in a pattern which repeats itself for dimensions D modulo 8. Secretly, this phenomenon is connected to Bott’s periodicity, but I will not discuss it here today. The dimensionality of a Dirac spinor as a solution to the Dirac equation in D spacetime dimensions is given by the dimension of the Dirac matrices. In the familiar example of four dimensions, Dirac spinors belong to reducible representations of the Lorentz group. For arbitrary spacetime dimensions one might wonder what dimensionality an irreducible spinor has got.

Remark: The most general free spinor (from Clifford bilinears up to some exotics) action reads

(9)   \begin{equation*} \boxed{S(\Psi)=\int d^Dx \left(\alpha_1\overline{\Psi}\gamma^\mu\partial_\mu\Psi+\alpha_2m\overline{\Psi}\Psi+\alpha_3m_{5}\overline{\Psi}\gamma^{D+1}\Psi\right)} \end{equation*}


Superalgebras are the heart of SUSY. There are 3 main types of SUSY algebras used in SUSY: superconformal algebras (SCA; 15+N^2 Bose generators, and 8N Fermi generators), super-dS (or super-AdS) algebra (SdSA has 10+N(N-1)/2 Bose generators and 4N Fermi generators); and finally the super-Poincaré algebra (SPA has the same number or generators than the SdSA, except in the presence of the so-called central charges). Some relatives from these superalgebras are:

  • SCA: in flat space, you can get super-\lambda\phi^4 theories, and SYM theories (for N=1,2,4). In curved space, you can form SC SUGRA.
  • SdSA: in curved space (there is no flat space cases here) you can get extended (N>1) or simple N=1 dS SUGRA/AdS SUGRA with SO(N) gauge group.
  • SPA: in flat space, you can obtain virtually all supersymmetric models with N=1, and SYM with D=6,10. In curved space, D=4 SUGRA and extended SUGRA theories with SO(N) gauging. D=11, N=1 is the so-called maximal SUGRA (if you give up higher spin excitations with J>2). The most interesting models are the extreme cases, such as that with N=8 SUGRA in D=4 and N=1 with D=11. The N=4 SYM theory (GSO model) is also very interesting. This model has remarkable properties: it is renormalizable, and the Fermi-Feynman gauge is fully 1-loop finite (no renormalization is needed). The 1-loop corrections to the propagator vanish identically (finite and infinite parts), and finally the Gell-Mann-Low \beta (g) function vanishes identically for 1 and 2 loops. This fact makes the GSO model to exhibit no spontaneous symmetry breaking of SUSY, while coupled to SUGRA it does. Another advantage is that this model compared to the SO(8) theory is that the gauge group is arbitrary and can be fitted for Nature. SCA has additional applications. The superconformal SUGRA, whose bosonic sector contains the Weyl theory of gravity (the tensor R_{\mu\nu}^2-R^2/3 correction is present) and has four derivatives in the action, which leads to dipole ghosts. This is generally used to reject this theory on these principles, well as on the classical ground of one word. The SCA has no dimensional constant entering it and thus, the field theories based on it have dimensionless coupling constants, ensuring renormalizability (SC gravity and SYM). The SdSA contains one dimensional coupling constant m, with dimensions of mass/energy. Translations do not commute and give a rotation times m^2. The Universe described by the SdSA is thus not a flat Universe, but a de Sitter one (SO(3,2) rather than SO(4,1)). The radius of the dS Universe is roughly R_0\sim m^{-1} in natural units. The constant m plays the role of a mass term via the term m\overline\Psi_\mu\sigma^{\mu\nu}\Psi_\nu for the gravitino field, and a cosmological constant m^2/\kappa^2 occurs in the lagrangian, where \kappa is related to G_N, the Newton constant of gravity. In extended theories, with N\geq 2, the vector fields such as A_\mu couple with the dimensionless minimal coupling constant g, and \kappa m\sim g. The SO(3,2) AdS Universe is the simple, maximally supersymmetric solution of the field equations, and the actual size of the Universe introduces a stringent bound on g, such as g\leq 10^{-120}! Actually, there is also another viewpoint of this deduction. We can plug g\sim 1 and g^2/\kappa^4\sim (10^{19})GeV^4 and this fits observations even when we have no idea of why this is so. This picture is related to the spacetime foam by Wheeler, Hawking ad Townsend. As they use to point out, the dS theory has solutions where spacetime is flat or nearly flat at large distances (greater than 1 cm) but very strongly curved at distances of the order of Planck’s length (about 10^{-35}m or similar). The physical spacetime may well be a statistical (emergent!) ensemble of fluctuating spacetime foams (with unknown quantum degrees of freedom) of arbitrary sizes and topologies. The issue of the change of topologies in this fluctuating spacetime is not solved in any current candidate of quantum gravity. And finally, SPA, with or without central charges, have the dimension of a mass, and representations of the SPA with central charges occur for massive supermultiplets only (e.g., N=2, J_{max}=1/2) or for the classical solutions of the N=2,4 SYM theories. In these theories, we have respectively 2 and 6 central charges. In the N=2 model, the 2 charges are the electric and the magnetic charges, satisfying the Montonen-Olive relationship


    Olive remarkable idea was to identify mQ and mG with the momenta P_5, P_6 in certain Minkovskian spacetime (s=5, t=1; D=6) where M_6^2=0. Similarly, the N=4 model suggests to define 6 extradimensional central charges Q_i, and these can be identified with 6 extradimensional momenta with P_i=mQ_i and the mass relationship M_{10}^2=0. These facts are not surprising, since the N=2,4 D=4 theories can be obtained by dimensional reduction from the N=1, D=6,10 SYM theories!


We are going to discuss only the representations of the SPA in D=4 dimensions (s=3, t=1), in the massless case, in terms of fields. One can show that only one of the supercharges Q^i_\alpha is relevant (e.g. Q^i_1), and that it decreases the helicity by 1/2. The particle content of a supermultiplet can be easily found. In the next table, we observe the particle contents of N=1,\ldots,10 supersymmetric free field theories with J_{max}\leq 5/2. Scalar multiplets exist up to N=2, vector multiplets up to N=4, SUGRA multiplets up to N=8 and hypergravity with J_{max}=5/2 multiplets up to N=10.

Table 1. Representation contents of N=1,…10 SUSY with 0 mass. Ogievetsky multiplets (having J_{max}=3/2) are shown up to N=4, but exist up to N=6. Hypergravity multiplets (J_{max}=5/2) are shown to exist from N=1 to N=10 but are shown only for N=9 and N=10.


Explicit constructions are necessary to show that that interacting field theories based on these multiplets do exist. This is tru up to J_{max}=2, but no interacting field theory (hypergravity) based on J_{max}=5/2 exists. Hyper seems to suggest that these theories may not exist, but who knows? Hyperspace was a fantasy before the 19th and 20th century. Recently, hypergravity theories, higher spin theories and hypersuperspace theories have been considered by those M\Lambda \Phi crazy physicists you all know…Remarkably, hyperbole itself comes from certain Greek demagog of Samos, he threatened the comic poets Alcibiade and Nicias of ostracism,but he was turned into ridicule by them an was himself ostracized in 417 B.C.


The super prefix associated with SUSY and SUGRA is due first to superunification of fundamental forces somehow. SUSY and SUGRA own good features and they are specially recognized, in some models, by being renormalizable (cf. classical general relativity, a non-renormalizable theory).

1st. Unification goodness.

-Fields of different spins are unified in the same representation, overcoming previous no-go theorems (e.g., the Coleman-Mandula theorem).

-Internal symmetries and spacetime symmetries are unified. SUSY is (almost) unique but with a lot of models/theories to explore. What is the right model/theory? That is an experimental problem.

-Fermions and bosons play symmetrical roles.

-The dichotomy between fields and sources is also solved.

2nd. Renormalization goodness.

-If the bosonic sector of certain supersymmetric field theory renormalizable, so is the whole theory. Further in that case, the associated supersymmetric field theory is more convergent than its bosonic sector. E.g.: the N=4 SYM theory.

-Non supersymmetric theories of gravity with matter interactions (J=0,1/2;1) and their associated matter fields are one loop non-renormalizable, while pure extended SUGRA theories based on the SPA (N=1,…,8) are 1 and 2 (at least, current knowledge improves these arguments) loop renormalizable and finite. Superconformal gravity is also renormalizable, but so is conformal gravity.


SPA exists for any s,t  signature and any spacetime dimension (D=s+t). If we keep D=4, and increase N, we meet the limits N=2,4,8 for the existence of multiplets with J_{max}=1/2,1,2. Similarly, if we keep N=1, and we increase D, we meet the limits D=6,10,11 for the existence of the same multiplets.

An example of supersymmetric field theory in D=10 (s=9, t=1) is easy to provide. Let us take the example fo the N=1 SYM theory. If we keep N=1 and plug in D=10 in the notations we have introduced here, the theory is still SUSY invariant, provided that the \chi^j are Majorana-Weyl spinors, which is possible since s-t=8. The vector A_M^i in D=10 gives in D=4 a vector A_\mu^i and the MW spinor \chi^i reduces to 4 Majorana spinors \chi^i_j, j=1,2,3,4. This is precisely the field contents of the N=4, D=4 SYM theory! What we have done is dimensional reduction. Dimensional reduction consists in starting from certain higher dimensional theory, usually certain supersymmetric field theory with D\geq 5. One lets the size of the internal dimensions shrink into zero size (radius), in which case only the excitations of the fields which are constant in the extra dimensions are relevant, or have a simple y dependence (of the type \Psi(x,y)\sim \exp imy \Psi (x)) so only certain extra dimensional excitations do survive. The resulting D=4 theory is still supersymmetric (invariant under SUSY), but SUSY may be spontaneously broken. The mass m appears as the momentum in the 5th (or any other) extra dimension/direction. This process can be generalized. In generalized dimensional reduction, a phase \exp (imy) dependence is introduced, as in the usual Kaluza-Klein theory. The N=8 theory can then be generalized starting from the D=5, N=8, J_{max}=2 theory which has a rank 4 group of invariance, namely Sp(8). Going from 5 to 4 dimensions, 4 mass parameters m_i, i=1,2,3,4 can be introduced and a 4-parameter family of N=8 theories is obtained which contain the CJ (Cremmer, Julia) theory as a special case (m_i=0). In these theories, SUSY is spontaneously broken, the Bose-Fermi mass degeneracy is lifted but the breaking is “soft” as the following mass relations still hold

    \[\boxed{\sum_{J=0}^{J=2}\left(-1\right)^{2J}\left(2J+1\right)\left(M_J^2\right)^k=0,\forall k=0,1,2,3}\]

These mass relations imply that the one loop correction to the cosmological constant is still fully finite, but apperently non-zero (note that from this viewpoint, these theories do predict -or posdict- a non null cosmological constant). The spectrum of the N=8 theory is the content of the next section.

7. N=8.

In the spontaneously broken N=8 theory, every massive state is complex and the theory has a U(1) summetry. This extra symmetry is actually a local one. Its gauge filed is the vector field A_\mu^n, obtained by reducing the metric tensor from 5 to 4 dimesnions and 2\kappa A_\mu^\curlywedge=V_\mu^5. In the next section, we shall see that the coupling of this vector field leads to the phenomenon of “antigravity” (but be aware of what it does mean!), and we shall denote the particle associated with the quantized A_\mu^\curlywedge by a curlywedge symbol. (In the original papers, it was used the egyptian symbol “shen”, but it is pretty similar to curly wedge, and it is adorable too!).

If every m_i is equal to the same value, the only symmetry of the theory will be U(1) (apart from SUSY, coordinate invariance,…). If we set all the m_i=m, the spectrum is SU(4)\times U(1) degenerate and at zero mass, there are 15+1 gauge bosons. As a gauge group SU(4) is too small to include the SM group but if we disregard the weak interactions, we obtain a model which includes SU(3)_c\times U(1)\times U(1), that is strong interaction plus electromagnetism plus antigravity. Actually, the breaking of SU(4) into SU(3)_c\times U(1) is automatic if we set m_i=m with m_i=m for i=1,2,3 and m_4=M. The electric charge is taken to be equal to 1/3 for the triplet of J=3/2 graviquarks of mass m, 0 for the J=3/2 singlet gravitino of mass M. Once this is accomplished, all the masses and charges are derived by taking the product of representations. This model contains only one electron of mass 3m, but no muon or tau particles. Even the neutrino fields are absent. this is not too surprising and one could tentatively attribute the family structure to composite bound states as well as the SU(2) of weak interactions is taken into account. A more humble attitude is to take the model just a model not as a final theory. As it stands, it is not too bad: it is 1 and 2 loop finite (unlike Einstein’s theory of gravity coupled to leptons, quarks, and the SM group bosons and particles). It also includes a massless graviton, 8 massless gluons, a photon, an electron, a d type quark (Q=-1/3), a u type quark (Q=+1/3) and a c type quark (Q=2/3). The exotic particles are: the sexy quarks (the sextet 6, with Q=1/3), the gluinos of Fayet (Q=0), a triplet of d-graviquarks (the triplet 3: Q=-1/3), a set of 2 neutral gravitini (1,0), massless scalar gluons (2 octets),  massless singlet scalar particles (2 of them) and massive sextets and singlet scalars, and massive scalar quarks (that we could name “sarks”).


Let us consider the scattering of two particles of mass M_1,M_2 having also a coupling to a massless vector field A_\mu^\curlywedge (this vector field is NOT the electromagnetic potential of the SM, so it is some kind of dark electromagnetism) with charges g_1,g_2. In the static limit the total potential energy is given by

(10)   \begin{equation*} V(r)=4k^2 M_1M_2r^{-1}\left[M_1M_2-k^{-2}g_1g_2\right] \end{equation*}

Scherk’s antigravity is defined by the cancellation which would happen if one had systematically the relation between g’s and M’s

(11)   \begin{equation*} \boxed{g=\epsilon k m} \end{equation*}

and where \epsilon=+1,-1,0 for particles, antiparticles and neutral particles (like Z bosons, neutrinos and others). In 1977, it was guessed that since in N=2 SUGRA there is a vector A_\mu, it had to couple minimally to fields with a gauge coupling constant g\sim km. The coupling was actually written in lowest order in k. Later, in 1978, K. Zachos coupled N=2 SUGRA to the multiplet (1/2(2), 0(4)), with a mass m and found \vert g\vert=km, as well as antigravity (the above cancellation phenomenon!). In 1979, the spontaneously broken N=8 theory was found and it was discovered that a vector A_\mu coupled to all the model fields with strength \vert g\vert=2km, a relation that holds for all the 256 states of the model. If there is such an antigravitational force in Nature, and this is an inescapable consequence of SUSY if N>1 (that is, extended SUSY contains antigravity), why don’t see it? Perhaps, we have already seen and we don’t realize it! If we look at the spontaneously broken N=8 model, we may find a beginning of an answer. Suppose we consider the static force between 2 protons (uud bound states). As the graviton couples no to the mass but to the energy-momentum tensor T_{\mu\nu} it sees the total energy of the quarks and gluons, i.e., the mass of the proton. It is mostly the kinetic energy of the quarks and gluons as a whole. This force contribution is given roughly by the term k^2r^{-2}M_p^2, where M_p is the proton mass. The antigraviton (sometimes called graviphoton) is coupled to \overline{\chi}\gamma^\mu\chi times km, where m is the mechanical mass of the quark is question, and \overline{\chi}\gamma^\mu\chi is the conserved electromagnetic current. Therefore, the antigraviton/graviphoton sees directly the quark mass and not the proton mass and its contribution is \pm k^2r^{-1}(2m_u+m_d)^2. Furthermore, we observer that the \dfrac{\mbox{antigraviton}}{\mbox{graviton}} relative contribution is of the order (m_u/M_p)^2, and that is small, about 10^{-4} for u,d quarks. Finally, if we compute the relative difference between the acceleration of a proton and a neutron, we find that it is given by the expression 3m_u(m_d-m_u)/M_p^2, which is even smaller, depending on the u, d mass difference. In the limit of exact SU(2) symmetry for strong interactions, this difference vanishes so that antigravity is in good health…However, if one looks closer, one finds that the \curlywedge exchange leads to serious problems with the equivalence principle! Let me explain it better. Two atoms of atomic numbers A_1, A_2, having Z_1, Z_2 protons and of net charge zero fall with DIFFERENT accelerations towards the Earth! The force between these 2 atoms is given by

(12)   \begin{equation*} F=8\pi G r^{-2}\left[ M(Z_1,A_1)M(Z_2,A_2)-M^0(Z_1,A_1)M^0(A_2,Z_2)\right] \end{equation*}

The negative term is due to the antigraviton/graviphoton exchange; one generally has:

(13)   \begin{align*} M(Z,A)=Z(M_p+m_e)+(A-Z)M_n\\ M^0(Z,A)=Z(2m_u+m_d+m_e)+(A-Z)(m_u+2m_d)\\ \gamma (Z,A)=8\pi G r^{-2}M(Z_1,A_1)\left[1-\dfrac{M^0(Z_1, A_1)M^0(Z_2,A_2)}{M(Z_1,A_1)M(Z_2,A_2)}\right] \end{align*}

If A_2,Z_2 represent the Earth, we can safely replace \dfrac{M^0}{M}(Z_2,A_2) by 3m_u/M_p. Therefore, the acceleration of (Z_1,A_1) towards the Earth is given by

(14)   \begin{equation*} \boxed{\gamma (A_1,Z_1)=a(A_1,Z_1)=\dfrac{8\pi G}{r^2}M_E\left(1-\dfrac{M^0(A_1,Z_1)}{M(A_1,Z_1)}\dfrac{3m_u}{M_p}\right)} \end{equation*}

The relative difference in acceleration of the 2 atoms will be

(15)   \begin{equation*} \Delta=\dfrac{\delta a}{a}=\dfrac{\gamma (Z_1,A_1)-\gamma (Z_2,A_2)}{\gamma (Z_1,A_1)} \end{equation*}

(16)   \begin{equation*} \Delta=\xi\left[m_e(M_n-m_u-2m_d)+\dfrac{2}{3}(m_u+m_d)(M_n-M_p)+\dfrac{1}{2}(m_u-m_d)(M_n-M_p)\right]\Xi \end{equation*}

and  where \xi=\left( Z_2A_1-Z_1A_2\right)\dfrac{3m_u}{M_p}, \Xi=M^{-1}(Z_1,A_1)M^{-1}(Z_2,A_2). Int the last bracket, the dominant contribution is given by m_eM_n, so \Delta reads

(17)   \begin{equation*} \Delta\sim \left(Z_2A_1-Z_1A_2\right)\dfrac{3m_u}{M_p}\dfrac{m_eM_n}{M_1M_2} \end{equation*}

Putting into numbers this expression, we find that \Delta\sim 10^{-6}, bigger than the most accurate bound on the violation of the equivalence principle. This is unaceptable. Is antigravity wrong after all? Theorists do NOT give up a general idea so easy…In order to save the antigravity idea/phenomenon (something quite general in some BSM theories), one must assume that the antigraviton/graviphoton acquires a mass, likely through the Higgs mechanism. This is rather what it happens, since it (the graviphoton/antigraviton) is universally coupled to scalar fields through the lagrangian

(18)   \begin{equation*} \mathcal{L}=-\dfrac{1}{4}F_{\mu\nu}^\curlywedge F^{\mu\nu\curlywedge} -\dfrac{1}{2}\vert \left( \partial_\mu-ikm A_\mu^\curlywedge \right)\phi\vert^2+m^2\phi^*\phi-V(\phi) \end{equation*}

At the classical level, the v.e.v. (vacuum expectation value) is \langle\phi\rangle\neq 0 and m_\curlywedge=km_\phi\langle\phi\rangle. If \langle \phi\rangle\neq 0 is due to SU(2)xU(1) breaking, one has typically \langle \phi\rangle\sim 1=100GeV, and with \langle \phi\rangle\sim 1GeV one finds that m_\curlywedge\sim 10^{-19}GeV. This gives to the antigraviton a Compton wavelength of the order 1 km, which seems to be reasonable. In the case where m_\curlywedge\neq 0, the potential between an atom with (Z,A) and the Earth is provided by the following expression

(19)   \begin{equation*} \boxed{V(Z,A)=\dfrac{8 \pi G}{R_E}\left( 1-\dfrac{M^0}{M}(Z,A)\dfrac{3m_u}{M_p}\exp\left(-m_\curlywedge R_E\right)\right)} \end{equation*}

This formula would be correct if the Earth were a point like object. Taking into account of its actual size leads (for an homogeneous sphere) to multiplying the last term in this expression by a form factor f(m_\curlywedge R_E), where

    \[f(x)=3x^{-3}\left[x\cosh x-\sinh x\right]\]

The altitude from the surface now appears rather than the distance from the center and it leads to the upper bound on m_\curlywedge^{-1}, given by m_\curlywedge^{-1}\leq 1km. Usually, masses are thought to be fixed parameters. However, one knows that they depend upon external conditions such as the temperature T. If one could “heat the vacuum” enough, the phase where \langle \phi\rangle=0 and m_\curlywedge=0 would be restored. Antigravity devices of this kind however still belongs to the field of Ufology and Sci-Fi (Science-Fiction), and apparently not to the field of mathematical physics.


It will be short and we leave it to a great American hero…

Super manofsteel1Remark: SUSY haters have also their propaganda

2yLoLSUSY2215SUSYhopeless2215but also SUSY has big fans/addicts/superphysicists that counter it…


10. APPENDIX: the simplest SUSY.

If you have read up to here, I am going to give you another “gift”. The simplest SUSY you can find from supermechanics. I will add some additional final questions too ;). Let me begin with certain lagrangian. For simplicity, take the mass of the particle equal to the unit (i.e., plug in m=1). The symplest SUSY lagrangian is then build in when you add to the free particle lagrangian (m=1) the Grassmann part as follows:

(20)   \begin{equation*} \boxed{L_{BF}=L_B+L_F=\dfrac{\dot{x}_\mu\dot{x}_\mu}{2}+\dfrac{i\Psi_\mu\dot{\Psi}_\mu}{2}} \end{equation*}

The Bose part corresponds to translations and the Fermi part correspond to spins. This lagrangia is, in fact, a special case where both, angular momentum and spin angular momentum, are invariant under INDEPENDENT rotations in the variables x,\Psi. Any interacting extension from this free case involves that this lagrangian generalization will be inivariant only under SIMULTANEOUS rotation of x,\Psi. In particular, this lagrangian is invariant under

(21)   \begin{eqnarray*} \delta x_\mu=\omega_{\mu\nu}x_\nu\\ \delta \Psi_\mu=\omega_{\mu\nu}\Psi_\nu \end{eqnarray*}

The L_{BF}=L_{SUSY} has variables with the following Poisson algebra

(22)   \begin{equation*} \{x_\mu,x_\nu\}_{PB}=\{p_\mu,p_\nu\}_{PB}=0 \end{equation*}

(23)   \begin{equation*} \{x_\mu,p_\nu\}=\{\Psi_\mu,\Psi_\nu\}=\delta_{\mu\nu} \end{equation*}

The Hilbert space on which this objects acts is given by \mathcal{H}=L^2(\mathbb{R}^2)\otimes \mathbb{C}^{2^N}, where N=\left[d/2\right]. Thus, under quantization, you obtain that the hamiltonian is certain laplacian operator on \mathbb{R}^d. Generally, up to a sign, you write

(24)   \begin{equation*} H=\dfrac{\hat{p}^2}{2}=\Delta/2 \end{equation*}

Remark: the sign convention is important in some applications. It is generally better, for convergence issues, choose the laplacian so that the eigenvalues are asymptotically positive.

The Noether charge for L_{SUSY} under rotations can be easily work out, and it yields the tensor

    \[Q_{\mu\nu}\equiv J_{\mu\nu}=x_\mu p_\nu-x_\nu p_\mu-\dfrac{i}{2}\left(\Psi_\mu\Psi_\nu-\Psi_\nu\Psi_\mu\right)=L_{\mu\nu}+S_{\mu\nu}\]

This is a good thing. We recover the classical result that rotational invariance implies the conservation of angular momentum J=L+S=angular part+Spin part. In particular, for d=3, we obtain


and it confirms known results of angular momentum under quantization! Now, the full simplest SUSY transformations in action action onto our lagrangian L_{SUSY}=L_{BF}. The most general field-coordinate variation of this lagrangian provides

(25)   \begin{equation*} \boxed{\delta L_{SUSY}=\dot{x}_\mu \delta \dot{x}_\mu+\dfrac{i}{2}\left(\delta \Psi_\mu\right)\dot{\Psi}_\mu+\dfrac{i}{2}\Psi_\mu\left(\delta \dot{\Psi}_\mu\right)} \end{equation*}

Introduce elementary SUSY transformations

(26)   \begin{align*} \delta x_\mu=i\epsilon \Psi_\mu\\ \delta \Psi_\mu=-\epsilon\dot{x}_\mu \end{align*}

Plug in these variations into the \delta L_{SUSY} variation, and operate it to obtain

(27)   \begin{align*} \delta L=i\dot{x}_\mu\epsilon\dot{\Psi}_\mu-\dfrac{i}{2}\epsilon\dot{x}_\mu\Psi_\mu-\dfrac{i}{2}\Psi_\mu\epsilon\dot{x}_\mu=\\ =i\dot{x}_\mu\epsilon\dot{\Psi}_\mu-\dfrac{i}{2}\epsilon\dot{x}_\mu\dot{\Psi}_\mu-\dfrac{i}{2}\dfrac{d}{dt}\left(\Psi_\mu\epsilon\dot{x}_\mu\right)+\dfrac{i}{2}\dot{\Psi}_\mu\epsilon\dot{x}_\mu=\\ =-\dfrac{i}{2}\dfrac{d}{dt}\left(\Psi_\mu\epsilon\dot{x}_\mu\right) \end{align*}

The conserved charge (supercharge is used often) is

(28)   \begin{equation*} \epsilon Q=i\epsilon p_\mu\Psi_\mu=i\epsilon \Psi_\mu p_\mu=i\epsilon \Psi_\mu\dot{x}_\mu \end{equation*}

that is, the Noether supercharge, SUSY generator is

(29)   \begin{equation*} \boxed{Q=i\Psi_\mu\dot{x}_\mu=i\Psi_\mu p_\mu=ip_\mu\Psi_\mu} \end{equation*}

In even dimension, d=2n, we usually quantize the Poisson brackets with the aid of the canonical commutators and anticommutators given by

(30)   \begin{equation*} \left[\hat{x}_\mu,\hat{p}_\nu\right]=i\delta_{\mu\nu} \end{equation*}

for bosons and

(31)   \begin{equation*} \{\hat{\Psi}_\mu,\hat{\Psi}_\nu\}=\delta_{\mu\nu} \end{equation*}

for fermions. Defining

(32)   \begin{equation*} \Psi_\mu=\dfrac{\hat{\gamma}_\mu}{\sqrt{2}} \end{equation*}

it turns that the fermion anticommutator is secretly a (rescaled) Clifford algebra in disguise, since Clifford algebras are defined as

(33)   \begin{equation*} \{\gamma_\mu,\gamma_\nu\}=\{\hat{\Psi}_\mu,\hat{\Psi}_\nu\}=2\delta_{\mu\nu} \end{equation*}

The remaining problem is to find and determine the gamma matrices or some good representation of them. The gamma matrices act onto the Hilbert space \mathcal{H}=L^2(\mathbb{R}^{2d}\otimes\mathbb{C}^{N}, N=2^n in even dimensions, but it can be generalized to odd dimensional spaces too (with care!). Generally speaking, this factorization of the Hilbert space says that SUSY acts on a superpace being “translations times spin”. The quantized Noether operator \hat{Q} associated to SUSY transformations reads

(34)   \begin{equation*} \hat{Q}=i\hat{\Psi}_\mu\hat{p}_\mu=i\dfrac{\gamma_\mu}{\sqrt{2}}\left(-i\partial_\mu\right)=\dfrac{1}{\sqrt{2}}\gamma^\mu\partial_\mu \end{equation*}


(35)   \begin{equation*} \boxed{\hat{Q}=\dfrac{1}{\sqrt{2}}\gamma^\mu\partial_\mu} \end{equation*}

This result teaches us something really cool and amazing: the SUSY quantum mechanical Noether supercharge (operator) is nothing but the Dirac operator (here, acting on the manifold \mathbb{R}^{2n} times the spin group). Remember: the SUSY supercharge is generally speaking certain Dirac-like (Clifford) operator, the product of the Clifford gamma matrix and certain derivative. Indeed, there is something really beautiful in addition to this thing. SUSY transformation can be computed for this operator as well, with new amazing results:

(36)   \begin{align*} \delta Q=i\left(\delta \Psi_\mu\right)\dot{x}_\mu+i\Psi_\mu\delta \dot{x}_\mu=\\ =i(-\epsilon\dot{x}_\mu)\dot{x}_\mu+i\Psi_\mu (i\epsilon\dot{\Psi}_\mu)=-i\epsilon x^\mu\dot{x}^\mu+\epsilon\Psi_\mu\dot{x}_\mu=\\ =-2i\epsilon\left[\dfrac{\dot{x}_\mu\dot{x}_\mu}{2}+\dfrac{i\Psi_\mu\dot{\Psi}_\mu}{2}\right]=-2i\epsilon L_{BF}=-2i\epsilon L_{SUSY} \end{align*}


(37)   \begin{equation*} \boxed{\delta Q=-2i\epsilon L_{SUSY}} \end{equation*}

Motto: the variation of the supercharge is proportional to the SUSY lagrangian (times a constant).

Moreover, compute two successive SUSY transformations, with parameters \epsilon_1,\epsilon_2. Then, you can show that the commutor (and its associated Poisson bracket) reads

(38)   \begin{equation*} \delta_{\epsilon_2}\delta_{\epsilon_1}-\delta_{\epsilon_1}\delta{\epsilon_2}=-2i\epsilon_1\epsilon_2\dfrac{\partial}{\partial t} \end{equation*}

But \partial_t is the generator of translations in time associate to the energy or hamiltonian of the system! This can be easily proved

(39)   \begin{align*} \{\epsilon_2ip_\mu\Psi_\mu,\epsilon_1ip_\nu\Psi_\nu\}_{PB}=(i)^2p_\mu p_\nu\epsilon_2\epsilon_1\{\Psi_\mu,\Psi_\nu\}_{PB}=\\ -p_\mu p_\nu\epsilon_2\epsilon_1\left(-i\delta_{\mu\nu}\right)=i\epsilon_2\epsilon_1 p^2=-2\epsilon_1\epsilon_2 H \end{align*}

and where we have used that 2H=p^2. Thus, under quantization,

(40)   \begin{align*} \{Q,Q\}=2\hat{Q}^2=2(i\hat{p}_\mu\hat{\Psi}_\mu)(i\hat{p}_\nu\hat{\Psi}_\nu)=\\ -2\hat{p}_\mu \hat{p}_\nu\hat{\Psi}_\mu\hat{\Psi}_\nu-2\hat{p}_\mu \hat{p}_\nu\dfrac{1}{2}\left(\hat{\Psi}_\mu \hat{\Psi}_\nu+\hat{\Psi}_\nu\hat{\Psi}_\mu\right)=\\ -\hat{p}_\mu\hat{p}_\nu \delta_{\mu\nu}=-p^2=-2\hat{H} \end{align*}

Thus, the SUSY supercharge is, generally speaking, “the square root” operator of the hamiltonian, since

(41)   \begin{equation*} \boxed{\{\hat{Q},\hat{Q}\}=-2\hat{H}} \end{equation*}

or equivalently

    \[\boxed{\hat{Q}^2=-\hat{H}} \longleftrightarrow \boxed{\hat{Q}=\sqrt{-\hat{H}}}\]

In summary, the main formulae from the simplest SUSY lagrangian are given by

(42)   \begin{align*} \boxed{L_{BF}=L_B+L_F=\dfrac{\dot{x}_\mu\dot{x}_\mu}{2}+\dfrac{i\Psi_\mu\dot{\Psi}_\mu}{2}}\\ \boxed{\delta L_{SUSY}=\dot{x}_\mu \delta \dot{x}_\mu+\dfrac{i}{2}\left(\delta \Psi_\mu\right)\dot{\Psi}_\mu+\dfrac{i}{2}\Psi_\mu\left(\delta \dot{\Psi}_\mu\right)}\\ \boxed{\delta x_\mu=i\epsilon \Psi_\mu\;\;\;\delta \Psi_\mu=-\epsilon\dot{x}_\mu}\\ \boxed{\hat{Q}=i\hat{\Psi}_\mu\hat{p}_\mu=i\dfrac{\gamma_\mu}{\sqrt{2}}\left(-i\partial_\mu\right)=\dfrac{1}{\sqrt{2}}\gamma^\mu\partial_\mu}\\ \boxed{\delta Q=-2i\epsilon L_{SUSY}}\\ \boxed{\delta L=\dfrac{\epsilon}{2}\dfrac{dQ}{dt}}\\ \boxed{\{\hat{Q},\hat{Q}\}=-2H}\\ \boxed{\hat{Q}^2=-\hat{H}} \longleftrightarrow \boxed{\hat{Q}=\sqrt{-\hat{H}}} \end{align*}

Finally, some exercises for addict, eagers readers…I do know you do exist and you are OUT there!

1) Generalize this discussion to a simple manifold with a metric. SUSY covariance reads from

    \[\Psi_\mu=\dfrac{\partial x^\mu}{\partial x'^\mu}\Psi '^\nu\]

and the metric is

    \[ds^2=g_{\mu\nu}(x)dx^\mu dx^\nu\]

The invariant object

    \[ Q=i\langle \dot{x},\Psi\rangle=ig_{\mu\nu}\dot{x}^\mu \Psi^\nu\]

is well defined. Note that it depends on the velocities \dot{x}^\lambda. The lagrangian of this exercise is provided by a covariant version of our simple L_{SUSY}:



    \[L=\dfrac{\langle \dot{x},\dot{x}\rangle}{2}+\dfrac{i}{2}\Bigg\langle\Psi,\dfrac{D\Psi}{Dt}\Bigg\rangle\]

Here, we define the Christoffel connection

    \[\Gamma^\nu_{\lambda\mu}=\dfrac{1}{2}g^{\nu\rho}\left(\partial_\lambda g_{\rho\lambda}+\partial_\mu g_{\lambda\rho}-\partial g_{\lambda\nu}\right)\]

The covariant derivative is given by

    \[\dfrac{D}{Dt}=\dfrac{d}{dt}+\dot{x}^\lambda \Gamma_\lambda\]

and from the expansion

    \[\Gamma^\mu_\nu=dx^\lambda \Gamma^\mu _{\lambda \nu}\]

we can get

    \[ \boxed{R^{\mu}_{\;\;\;\nu}=d\Gamma^{\mu}_{\;\;\; \nu}+\Gamma^\mu_{\;\;\;\sigma}\wedge \Gamma^\sigma_{\;\;\;\nu}}\]

and curvature components

    \[ R^{\mu}_{\;\;\;\nu}=\dfrac{1}{2}R^{\mu}_{\;\;\;\nu\rho\sigma} dx^\rho\wedge dx^\sigma\]

2) Find the Euler-Lagrangian equations for this covariant generalization of our simple lagrangian, L_{SUSY}. Is there any conservation law there? Reason your answer.

3) Express the Dirac operator for any general curved manifold M in local coordinates.

Have fun!!!!!!!

 Let SUSY, SUGRA and ANTIGRAVITY be with you!!!!!!!

antigravityFuture antigravityhoverboard

LOG#176. The cosmological integral.

equation-note.tiff DE Darkness-1 2000px-Vacuum_polarization.svg

Hi, there! In this short note I am going to introduce you into a big problem from a simple integral I use to call “the cosmological integral” (it should be the cosmological energy density integral, but it is too long…).

Any quantum field contributes to the energy density of the Universe through a particular integral…From Quantum Field Theory we learn that this cosmological integral reads

(1)   \begin{equation*} \boxed{C.I.(k)=\rho(k)=\dfrac{1}{(2\pi)^3}\dfrac{1}{2}\int d^3k\sqrt{k^2+m^2}} \end{equation*}

and where I use natural units \hbar=c=1.

Plugin in spherical coordinates for the k-momentum volume element we can rewrite it as

(2)   \begin{equation*} \dfrac{1}{4\pi^2}\int dk k^2\sqrt{k^2+m^2} \end{equation*}

You can also write it again as

(3)   \begin{equation*} \dfrac{1}{4\pi^2}\int dk k^3\sqrt{1+\dfrac{m^2}{k^2}} \end{equation*}

Suppose that we want to perform the integral up to some “big number”/energy/momentum \Lambda and that the lower limit is not too relevant at those big scales…Making a Taylor expansion of the square root for m/k<<1, we get

(4)   \begin{equation*} C.I.=\dfrac{1}{4\pi^2}\int^\Lambda dk k^3\left(1+\dfrac{m^2}{2k^2}-\dfrac{m^4}{8k^4}+\ldots\right) \end{equation*}

It yields

(5)   \begin{equation*} C.I.=\left[\dfrac{k^4}{16\pi^2}+\dfrac{m^2k^2}{16\pi^2}-\dfrac{m^4}{32\pi^2}\log k+\dfrac{m^4}{32\pi^2}\log m+\mbox{cte+finite terms}\right]^\Lambda \end{equation*}


(6)   \begin{equation*} C.I.=\dfrac{1}{16\pi^2}\left(\Lambda^4+m^2\Lambda^2-\dfrac{m^4}{2}\log \left(\dfrac{\Lambda}{m}\right)+\mbox{cte'+finite terms}\right) \end{equation*}

The first two terms, \sim\Lambda^4, \sim m^2\Lambda^2 are powers of the cut-off energy or UV regulator. They are generally absent in a sophisticated method like dimensional renormalization. However, the third term is much interesting…It is a logarithmic term that invades the theory at low energy! In fact, such logarithmic terms (or powers of it as you introduces more and more loops into the corresponding vacuum polarization Feynman graphs) are not negligible unless you introduce certain additional counter-terms in order to erase them or minimize them. That is, ANY quantum field provides a low energy contribution to the vacuum energy proportional to

(7)   \begin{equation*} \boxed{\langle \rho\rangle_{C.I.}=-\dfrac{1}{32\pi^2}\sum_{i}^N(-1)^{F_i} m_i^4\log\left(\dfrac{\Lambda}{m_i}\right)} \end{equation*}

for ANY m_i<<\Lambda, with F_i being the fermion number. This is a serious issue! For the electron, you can easily obtain \rho (e^-)\sim m_e^4\sim 10^{33}meV^4. The observerd vacuum energy (sometimes dubbed or referred as dark energy) is about \rho_{DE}\sim\rho_\Lambda\sim H_0^2M_p^2\sim meV^4. We are off 33 orders of magnitudes for the electron! For heavier particles, the mismatch is even worst! This thing is a modern view of the cosmological constant problem from a very simplified approach…

Remark: You can perform the exact integral for comparison with the obtained results. Any good table of integrals says you that

(8)   \begin{equation*} \int dx x^2\sqrt{x^2+a^2}=\dfrac{1}{8}\left[x(a^2+2x^2)\sqrt{a^2+x^2}-a^4\log\left(\sqrt{a^2+x^2}+x\right)\right]+cte \end{equation*}


1) Expand the integral at the infinite (big enough x) and prove that you get up to finite terms and some constants the previous terms.

2) Expand the integral at x=0 or any good enough small value. What are the meaning of the different asymptotic terms you get at first orders in a. log (a) and x?

Remark(II): Introduction of further loops does not solve the issue, only provides you new logarithmic POWERS of it. The only possible cancellation of the vacuum energy is some unknown yet mechanism, or to include new particle species “almost” nullifying the observed dark energy up to some huge number of numbers. It is highly unnatural. However, this is a fact right now, circa 2015…Further solutions? Screening mechanisms, modified dispersion relationships, or some new symmetry changing your notion of energy density and/or some non perturbative trick we do not know could, maybe, work.

LOG#175. BH and Saint Seiya: Infinite Break and beyond!

 InfiniteBreak Tech-Aioros-InfinityBreak infinite-break SAGInfinityBreakThis post contains some exercises and challenges for you. In fact, crazy me, I used some of them for my classroom usage (just a fun way to survive is to design and propose maniac ultrageek problems to your students, don’t you think so? A mad scientist I am…). Before going into the problems, let me summarize some useful quantities we saw in the last post http://www.thespectrumofriemannium.com/2015/07/19/log174-basic-black-hole-physics/

The next results are completely valid for a black hole of mass M (energy E=Mc^2), non-rotating, uncharged and in 4 space-time dimensions without cosmological constant:

1) Horizon radius/Schwarzschild radius:

(1)   \begin{equation*} \boxed{R_S=\dfrac{2GM}{c^2}} \end{equation*}

 2) Area of the Schwarzschild’s black hole:

(2)   \begin{equation*} \boxed{A_S=4\pi R_S^2=\dfrac{16\pi G^2 M^2}{c^4}} \end{equation*}

3) Surface gravity \kappa:

(3)   \begin{equation*} \boxed{\kappa=\dfrac{c^4}{4GM}} \end{equation*}

4) Surface tidal acceleration:

(4)   \begin{equation*} \boxed{a_T(BH)=a_T(R=d=R_S)=d\kappa=R_S\kappa=\dfrac{c^6}{4G^2M^2}} \end{equation*}

5) Black hole luminosity:

(5)   \begin{equation*} \boxed{L=A\sigma T^4=\dfrac{\hbar c^2}{3840\pi R^2}} \end{equation*}

6) Black hole luminosity evaluated at the horizon radius:

(6)   \begin{equation*} \boxed{L_{BH}=L(R=R_S)=\dfrac{\hbar c^6}{15360\pi GM^2}} \end{equation*}

 7) Evaporation time of the 4D Schwarzschild’s BH:

(7)   \begin{equation*} \boxed{t_{ev}(BH)=\dfrac{5120\pi G M^3}{\hbar c^4}=8.407\times 10^{-17} M(kg)^3\dfrac{s}{kg^3}=\left(\dfrac{M}{M_\odot}\right)^3\cdot 2.099\cdot 10^{67}yrs} \end{equation*}

 8) BH temperature, for even the classical non-rotating Schwarzschild black hole, is given by the Hawking result:

(8)   \begin{equation*} \boxed{T_{BH}=\dfrac{\hbar c^3}{8\pi k_BGM}} \end{equation*}

9) BH entropy for the 4D Schwarzschild:

(9)   \begin{equation*} \boxed{S_{BH}^{S}=\dfrac{k_B c^3}{4G\hbar} A=\dfrac{1}{4} k_B\left(\dfrac{A}{L_P^2}\right)=\dfrac{4\pi k_BG M^2}{\hbar c}=\dfrac{4\pi  k_BM^2}{M_P^2}=4\pi  k_B\left(\dfrac{M}{M_P}\right)^2=\dfrac{\pi k_B c^3 A}{2Gh}} \end{equation*}

 Sagittarius Aiolos (射手座のアイオロス, Sajitariasu no Aiorosu) is one of the Gold Saints from the anime/manga franchise Saint Seiya. The Infinity Break (infiniti bureiku) is his most powerful  technique and it allows Aiolos to shoot an infinite number of light darts that pierce anything upon contact. With this attack, Aiolos destroyed Horus’ army (Ra’s army) in Saint Seiya: Episode G, with a single hit. According to Aiolos himself, at the moment he uses the Infinite Break, his Cosmo reaches a million of (absolute?) degrees. Supposing he refers to absolute (kelvin) degrees, you are asked to respond the following questions:

1) Calculate the average electron velocity of the atoms forming Aiolos when his Cosmo reaches that million of (absolute) degrees. Compare this average velocity with the typical velocity of a cathode tube electron, a fast neutron and the typical atomic unit of velocity in the hydrogen atom. What would the mass of certain hypothetical black hole whose Hawking temperature were 1 million of kelvin degrees? Obtain the radius, black hole entropy, horizon area, surface gravity, surface tidal acceleration, the black hole luminosity at the event horizon, and the evaporation time for the equivalent black hole. Compare the calculated BH mass with the known masses of Earth’s moon, the sun, Phobos, Hydra and Pluto. Some useful data: m(electron)=9.11\cdot 10^{-31}kg, Boltzmann constant k_B=1.38\cdot 10^{-23}JK^{-1}.

2) Some Zodiac warriors can use the opposite technique: cold freezing attacks! Some knights are able to slow down the atomic/subatomic motion using their Cosmo in order to get glacial strokes! Some examples are Camus, Hyoga or Suykyo. Suppose we do believe in classical canon and that bronze, silver, and gold cloths freeze out at -150ºC, -200ºC and -273.15ºC. Calculate the average electron velocity for the atoms with those 3 temperatures. What would the black hole mass associated to those temperatures?

3) At current time, we do know that the absolute zero can not be reached due to quantum effects (if you want to keep yourself a classical field, think about the third law of thermodynamics instead). That is, no matter how Gold Saint you are, it is impossible at the current state-of-art (and likely in the forthcoming future!) to reach EXACTLY the absolute zero. However, we can approach the absolute zero as close as we want in principle and it explains the anime/mange physics somehow…Having the right tools (you need something better than a home freezer to do it)! (Or in the anime/manga having trained hard enough to low your temperature more than your enemy/rival). It means, being geek and classical with the canon, that if we are able to slow down our molecular, atomic and subatomic motion, with our Cosmo, more than our rival, we will beat the rival and win! Then, calculate the average electron velocity in the following cases: T=100 pK (picokelvin), T=1 pK, T=1fK, T=1zK and T=yK. Compare these temperatures with the Hawking temperatures of black holes having 1, 10, 10000, 1000000, 100 million of solar masses. Comment the results. Data: the solar mass is 1.99\cdot 10^{30} kilograms.

 4) (Advanced) Suppose Aiolos is training to be able to increase his cosmo temperature up to even larger  incredible values. We will study two cases: A) 2 TK (terakelvin, typical scale of quark-gluon plasma), and B) 1 YK (yottakelvin)! Firstly, compare the average electron velocity with the speed of light and deduce if it is a relativistic electron (Hint: calculate the relativistic beta and gamma factors for E=1YK). Calculate the black hole mass and black hole remaining parameters associated to this temperature. Compare them with the associated Planck mass and Planck units. Comment the obtained results. Finally, suppose that no new physical laws appear below the Planck scale. What would the electron velocity be at Planck temperature? Calculate the mass, entropy and physical quantities for a BH with Planck temperature. Calculate the temperature, entropy and physical quantities of a BH with Planck energy and compare the three cases.

5) (Advanced) Suppose some known particles, e.g., the electron, the neutron, the proton, the muon, the tau particle, the top quark and the Higgs boson were tiny black holes. Suppose their masses are effectively described by its PDG value. Calculate the physical parameters of the associated microscopic black holes (temperature, entropy, horizon radius, surface gravity,…). Comment the results.

EXTRA BONUS (Medium difficulty-> Black Hole music): find how many octaves below or above middle C are the frequencies associated to all the given above previous BH (Hint: express the BH masses in Hz, then think about Do frequency in Hz and apply the octave scale for comparison).

EXTRA BONUS (Advanced+hard-> Beyond Schwarzschild BHs): Schwarzschild BHs are not the only Black Hole solution to Einstein Field Equations of General Relativity. A Kerr-Newman-de Sitter (KNdS) black hole is a BH having angular momentum, electric (even magnetic!) charge, and a non null cosmological constant. One subtle aspect of this class of BH is that it owns several horizons. For instance, the particle and cosmological generalized BH temperatures are given by the expressions:

(10)   \begin{equation*} \boxed{\dfrac{1}{\beta_H}=T_H(KNdS)=\dfrac{r_H}{4\pi (r_H^2+a^2)}\left[\left(1-\dfrac{a^2}{l^2}\right)-\dfrac{a^2}{r_H^2}-3\dfrac{r_H^2}{l^2}-\dfrac{Q^2}{r_H^2}\right]} \end{equation*}

(11)   \begin{equation*} \boxed{\dfrac{1}{\beta_c}=T_c(KNdS)=\dfrac{r_c}{4\pi (r_c^2+a^2)}\left[-\left(1-\dfrac{a^2}{l^2}\right)+\dfrac{a^2}{r_c^2}+3\dfrac{r_c^2}{l^2}+\dfrac{Q^2}{r_c^2}\right]} \end{equation*}

i) Find what a, r_H, r_c, l are. Plug in the gravitational constant, the speed of light, the Coulomb constant and the cosmological constant in the above expressions to get the complete dimensionally correct temperature (times the Boltzmann constant, of course).

ii) What of the parameters is/are related to the cosmological constant \Lambda? In what asymptotic limit you get the KN solution without cosmological constant?

iii) Check that the temperatures T_H and T_c can be rewritten as follows:

(12)   \begin{equation*} \dfrac{1}{\beta_H}=T_H(KNdS)=\dfrac{1}{2\pi}\dfrac{r_H-2\dfrac{r_H^3}{l^2}-r_H\dfrac{a^2}{l^2}-M}{(r_H^2+a^2)} \end{equation*}

(13)   \begin{equation*} \dfrac{1}{\beta_c}=T_c(KNdS)=\dfrac{1}{2\pi}\dfrac{r_c-2\dfrac{r_c^3}{l^2}-r_c\dfrac{a^2}{l^2}-M}{(r_c^2+a^2)} \end{equation*}

iv) What is the effect of angular momentum, electric (magnetic) charge and the cosmological constant and their associated generalized BH temperature with respect to the usual Schwarzschild solution? If a Saint Seiya character managed to tune not only his molecular, atomic, subatomic motion but also his/her angular momentum, charge and cosmological constant, would it favor or make it harder to approach a given temperature T? Reason your answers.

P.S. (I): Post your answers if you can/want/are able to! 😉

P.S. (II): Beyond Saint Seiya, this exercise can be exported to any other anime/manga facing black hole/temperature powers. For instance, One Piece. The Yami Yami no Mi devil fruit caused this quote

“Darkness is gravity! The power to pull everything in… and spare not even a ray of light!… Infinite gravity!” Marshall D. Teach

From the One Piece wiki, we read: “(…)The first and foremost strength, as demonstrated by BlackBeard (Marshall D. Teach) is that it allows the user to control darkness and its unique property of gravity. The darkness is visually demonstrated by a black smoke-like substance spreading out from the user’s body. The darkness is a void that devours and crushes everything. Due to this, the user can absorb physical matter and attacks of any type by sucking it into the darkness. The user can also use this ability to irresistibly pull the opponent to them, regardless of their current state. Similar to a BH, the user can absorb things inside a vortex. It seems that it does not completely compress and annihilate that which is sucked in as an actual black hole would do, but rather creates some form of space where things are stored. (…)”

Thus, you can rewrite the whole exercises in terms of One Piece stuff if you wish, of course knowing what Paramecia, Zoan and Logia Devil Fruits are ;).

250px-Yami_Yami_no_Mi_Infobox blackbeard_motivational Black_hole_effectSee you in my next (not so geek) blog post!

LOG#174. Basic black hole physics.

hidden_blackhole_lg BH

In this year, 2015, we celebrate the Light Of Year (LOY) and one century of general relativity (GR), a misnomer for something that should be called relativistic theory of gravity.

Let me celebrate the centenary of general relativity with some (not-ordered) posts about different topics, ultimately related to black holes.

What are black holes? Well, simply put, black holes are objects so dense that the own light can not, at least in classical terms, escape from them! In fact, a naive simple calculation in classical physics (that should be taken with care and not as complete or rigorous in GR or even in the framework of quantum theories) allow us to compute the escape velocity of light! The escape velocity is the speed at which the sum of an object’s kinetic energy and its gravitational potential energy is equal to zero. Mathematically you get it as follows

(1)   \begin{equation*} \dfrac{1}{2}mv_e^2-\dfrac{GMm}{R}=0 \end{equation*}

and thus

(2)   \begin{equation*} \boxed{v_e=\sqrt{\dfrac{2GM}{R}}} \end{equation*}

In fact, escape velocity IS the speed something/someone needs to become freedom from the gravitational attraction of a massive body or material system (with mass M), without further propulsion, i.e., without spending more propellant. When you plug the escape velocity equals to the speed of light, i.e., inserting v_e=c in the previous formulae, you get the so-called Schwarzschild radius R_S, the radius of a non-rotating, uncharged black hole without cosmological constant:

(3)   \begin{equation*} \boxed{R_S=\dfrac{2GM}{c^2}} \end{equation*}

This results are completely valid for a black hole of mass M (energy E=Mc^2), non-rotating, uncharged and in 4 space-time dimensions without cosmological constant. Thus, we have a non-rotating uncharged black hole in 4 dimensions with escape velocity v_e, mass M, energy E and the above Schwarzschild radius. Are there any other magnitudes we should note or calculate in black hole physics and thermodynamics! Yes, there are. Assuming spherical symmetry, we can calculate the Schwarzschild area or event horizon/surface area of the Schwarzschild’s black hole…

(4)   \begin{equation*} \boxed{A_S=4\pi R_S^2=\dfrac{16\pi G^2 M^2}{c^4}} \end{equation*}

We can also calculate the surface gravity \kappa, if the gravitational field of the black hole reads


then, at the Schwarzschild radius it becomes the mentioned surface gravity \kappa=g(R=R_S):

(5)   \begin{equation*} \boxed{\kappa=\dfrac{c^4}{4GM}} \end{equation*}

Interestingly, this surface gravity is 1/M the maximal force c^4/4G allowed by natural units…What else? Surface tides, or more precisely, the tidal acceleration at the black hole surface (sometimes called event horizon). The tidal acceleration is calculated with

(6)   \begin{equation*} a_T=\dfrac{2GMd}{R^3} \end{equation*}

If it is evaluated at R_S you get

(7)   \begin{equation*} \boxed{a_T(BH)=a_T(R=d=R_S)=d\kappa=R_S\kappa=\dfrac{c^6}{4G^2M^2}} \end{equation*}

Blackbody physics…A blackbody with temperature T has a luminosity given by the Stefan-Boltzman law

(8)   \begin{equation*} \boxed{L=A\sigma T^4=\dfrac{\hbar c^2}{3840\pi R^2}} \end{equation*}

The radiation flux is

(9)   \begin{equation*} \phi =\dfrac{L}{4\pi R^2}=\dfrac{\hbar c^6}{61440\pi^2 G^2}\left(\dfrac{1}{M^2 R^2}\right) \end{equation*}

and where L is the black body luminosity. This luminosity gives, evaluated at the Schwarzschild radius

(10)   \begin{equation*} \boxed{L_{BH}=L(R=R_S)=\dfrac{\hbar c^6}{15360\pi GM^2}} \end{equation*}

From this equation, you also have

(11)   \begin{equation*} -\dfrac{dE}{dt}=L=\dfrac{\hbar c^6}{15360\pi GM^2} \end{equation*}

and using E=Mc^2

(12)   \begin{equation*} -\dfrac{dM}{dt}=L=\dfrac{\hbar c^4}{15360\pi GM^2} \end{equation*}

and you can integrate this last equation the evaporation time of a black hole with initial mass M and final mass equal to zero, in order to get the evaporation time of this Schwarzschild’s BH t_{ev}

(13)   \begin{equation*} \boxed{t_{ev}(BH)=\dfrac{5120\pi G M^3}{\hbar c^4}=8.407\times 10^{-17} M(kg)^3\dfrac{s}{kg^3}=\left(\dfrac{M}{M_\odot}\right)^3\cdot 2.099\cdot 10^{67}yrs} \end{equation*}

and where M_\odot is the solar mass, M the mass in kilograms and yr indicates years. Finally, the gold medal: Hawking’s discovery! S. W. Hawking discovered that BH has a temperature, despite the fact he originally thought on the contrary! Jacob Bekenstein works on black hole physics produced a powerful set of analogies between thermodynamics and black holes. Bekenstein suggested that black hole entropy was like an area…Much to the disgust of Hawking, applying his knowledge of Quantum Field Theories (QFT) on curved spacetimes, and basic ideas of Quantum Mechanics, he finished probing that black holes are not black at all, they are hot and radiate like a blackbody with a temperature, for even the classical non-rotating Schwarzschild black hole we are studying here, given by

(14)   \begin{equation*} \boxed{T_{BH}=\dfrac{\hbar c^3}{8\pi k_BGM}} \end{equation*}

This formidable discovery allowed to complete and prove the thermodynamical analogy (identity or duality?) the Bekenstein suggested. In fact, Bekenstein did know that he could only state that black hole entropy was proportional to the event horizon area (or surface area) up to a multiplicative number. Assuming the QFT calculation of Hawking is right (and it is, it is solid and checked), and that

    \[S_{BH}=\alpha A_{BH}\]

You differentiate this to get

    \[dS_{BH}=\alpha k_B dA_{BH}\]


    \[A_{BH}=A_S=4\pi R_S^2\]


    \[dA_{BH}=8\pi R_S dR_S\]

Inserting the value of the Schwarzschild radius in terms of mass M, the gravitational constant G and the speed of light c, we get

    \[dA_{BH}=\dfrac{32\pi G^2}{c^4} M dM\]

The energy/mass of the black hole is


but according to our results

    \[T_{BH}dS_{BH}=\dfrac{\hbar c^3}{8\pi k_B GM} dS_{BH}=\dfrac{\hbar c^3}{8\pi k_B GM} \alpha k_B dA_{BH}=\dfrac{\hbar c^3}{8\pi k_B GM} \alpha k_B \dfrac{32\pi G^2}{c^4} M dM\]

or equivalently

    \[dE=dMc^2=TdS=\dfrac{\alpha 4\hbar G}{c} dM\]

and from here we can determine the numerical constant \alpha in order to fully match the two expressions…It yields


and then, the BH entropy of the Schwarzschild solution of Einstein’s Field Equations becomes

(15)   \begin{equation*} \boxed{S_{BH}^{S}=\dfrac{k_B c^3}{4G\hbar} A=\dfrac{1}{4} k_B\left(\dfrac{A}{L_P^2}\right)=\dfrac{4\pi k_BG M^2}{\hbar c}=\dfrac{4\pi  k_BM^2}{M_P^2}=4\pi k_B\left(\dfrac{M}{M_P}\right)^2=\dfrac{\pi k_Bc^3 A}{2Gh}} \end{equation*}

You got it! The equation that Hawking himself has asked to be in his own tombstone…I thought to title this blog entry as Hawking’s tombstone equation, but it was very creepy so I changed my mind. You can easily check that result by yourself…That is, I suggest you to calculate

    \[ dE=dM c^2=T_{BH}dS_{BH}\]

for this particular case. Many other black holes are similar, but only they have more complex formulae for the BH temperature and the BH entropy.

Remark: Generally, black hole entropy units are in the international system and in base e (or natural), so they are written in terms of nats . There are many other cool entropy units:  hartleys (previously called bans, ban in singular) or shannons, where

    \[ 1 \mbox{nat}=\dfrac{1}{\ln 10} \mbox{ban}=\dfrac{1}{\ln 10}\mbox{hartley}=\dfrac{1}{\ln 2} \mbox{shannon}\]


    \[ 1 \mbox{ban}= 1 \mbox{hartley} = \log_2 10 \mbox{bit}= \ln 10 \mbox{nat}\]

You can calculate all these wonderful black hole magnitudes and many others in the site http://xaonon.dyndns.org/hawking/ but you must be aware that the entropy units are bans/hartleys and not nats!

See you in my next blog post!

LOG#173. Proton decay.



Smaragde HEARTdiamond diamond x-kryptonite aquamarine-gemstone

Are diamonds and other (crystalline) jewels/gemstones forever? Well, if protons are stable they are. If not, you are lost! We are all lost indeed in such a case…(However, I am sure that Superman would be fine to know the fact stable non radioactive kryptonite would not last forever in case protons are unstable, would you?)

The issue of the proton decay is a high-end subject, and it has visited periodically the mind of theoretical physicists since the 70s of the 20th century. In this post, I am going to discuss some very broad ideas about this interesting topic.

Let me begin remembering some basic ideas of unification and scales of energy. Usually the Planck mass is thought to be

\[M_P\sim 10^{19}GeV=10^{16}TeV\]

Compare this energy with our ridiculous LHC energy…About 10TeV (even the enlarged successors will reach about 100 TeV at most!). We are 15 orders of magnitude away from the hypothetical scale where quantum gravity lives…Unless superstring/M-theory or any other new physics and its own energy scale enter the game. For instante, grand unified theories (GUTs for friends) live roughly at $10^{15}GeV$. There were some hopes (indeed we have not lost all of them) that quantum gravity/stringy effects would show at the TeV scale…Modern views of superstring theory and other quantum gravitational theories do not assume that the string or quantum gravity energy scale is just equal to the Planck mass/energy (remember that mass IS energy, according to $E=Mc^2$, $E=M$ if you use units with c=1!). Therefore, $M_s$ or $M_{QG}$ could be LOWER than $M_P$.

Let me now discuss proton decay in 4D spacetime. There are generally two scenarios for this process. I mean, there are usually two mechanisms for proton becoming unstable particles (if they are almost stable, they are due to some hidden symmetry we could find some day…):

Mechanism 1. GUT triggered proton decay. Grand Unified Theories involving the unification of the electroweak and the strong forces imply generally the existence of a new kind of particle species, called leptoquarks, that catalyze proton decays. It can also be caused due to some magnetic monopole or GUT monopole, but I will not enter into these details today…However, I will point out that effective field theory (EFT) approach applied to GUTs imply that a dimension 6 operator

\[\hat{O}_6=\dfrac{\Psi \Psi \Psi \Psi}{\Lambda^2}\]

introduce typically a coupling to four fermions inducing generally proton decay.

Mechanism 2. Quantum Gravity/Spacetime foam triggered proton decay. This is a less know reaction. Due to quantum fluctuations of spacetime at ultrashort distances, inevitable due to the Heisenberg principle, virtual black holes and/or foamy spacetime at distances about the Planck length $L_P\sim 10^{-35}$ m (or when spacetime becomes foamy, typically at $L_s$ or $L_{QG}$), protons can suffer a foamy/quantum gravitational decay…

Let me discuss with more detail the second mechanism. The probability for 2 quarks to be within a distance equal to one Planck length is

\[P\sim\left(\dfrac{m_{p}}{M_P}\right)^3\sim 10^{-57}\]

where $m_p$ is the proton mass. According to Quantum Mechanics, the typical decay time for a particle of mass m is inversely proportional to the mass, more precisely unstable particles (resonances) have lifetimes about


where $\Gamma$ is the energy width of the resonance, proportional to its mass through the Einstein celebrated equation $\Gamma=Mc^2$. Thus, for the proton case, we get, supposing the proton is a high lived resonance…The previous probability per proton, crossing a typical resonance time of

\[\tau_p\sim\dfrac{1}{m_p}\sim 10^{-24}s \sim 10^{-31}yrs\]

if we assume that the particles move close to the speed of light. If we take into account the “nearness” for two protons, this gives an extra factor $m_p/M_P$, and therefore we obtain that typically, virtual black holes or quantum gravity fluctuations trigger proton decays in a time

\[\boxed{\tau_p(BH)\sim\left(\dfrac{1}{m_p}\right)\left(\dfrac{M_P}{m_p}\right)^3\left(\dfrac{M_P}{m_p}\right)=\dfrac{1}{m_p}\left(\dfrac{M_P}{m_p}\right)^4=\dfrac{M_P^4}{m_p^5}\sim 10^{45}yrs}\]

This bound is well above the current experimental bounds (circa 2015), with say that $\tau_p\geq 10^{33} yrs$ (SuperKamiokande bounds are closer to the $10^{34} yrs$ right now). In fact, this result is much more general. As I said above, GUTs already imply the proton decay option…However it has to be highly suppressed in order to fit the observations. Typically, GUTs with energy scale $\Lambda=M_X$ trigger proton decays in a time

\[\boxed{\tau_p(GUT)\sim \left(\dfrac{M_X}{10^{15}GeV}\right)^4\cdot 10^{30}yrs}\]

Thus, we have to be very careful with out GUTs and QG theories in order to avoid too fast proton decays!

Remark (I): Low energy black holes, if they have any experimental role in particle physics, need generally something else than a bigger symmetry to avoid superplanckian/transplanckian corrections to the proton lifetime! One possible solution is to introduce a generalized uncertainty principle (GUP, but please, not confuse it with GUT) in order to modify these estimates. GUPs have been known since the early times of modern quantum physics (Snyder and Heisenberg pioneer some original works in the 2nd third of the 20th century, paving the way to non-commutative geometry and even giving us some spoilers of modern string theory and de Sitter spacetime geometry):

\[\Delta x\geq \dfrac{1}{2}\left(\dfrac{1}{\Delta p}+\alpha L_P^2\Delta p\right)\]

\[\Delta t\geq \dfrac{1}{2}\left(\dfrac{1}{\Delta E}+\beta L_P^2\Delta p\right)\]

coming from Heisenberg uncertainty foamy/stringy corrections

\[\Delta x\geq \alpha L_P^2\]

\[\Delta t\geq \beta L_P^2\]

Remark (II): Kinematically forbidden processes enlarge or increase the lifetime $\tau$, since emission-absorption  of virtual particles suppress them with typical time $t\sim 1/M_X$, with X being the intermediate virtual particle.

Remark (III): Using GUP and virtual BH you can guess the lifetime

\[\boxed{\tau_p(BH+GUP)\sim \dfrac{M_P^8}{m_p^5M_{BH}^4}}\]

Note that, if so, excited states with about $10^3M_P$ induce too quick proton decay! Proton decay is a subtle issue for ANY GUT and BSM theory, even for (quantum) gravity! For instance, suppose that $M_P$ were about 1TeV instead our known $10^{16}TeV$ value…If you compute the proton lifetime in that case you get $10^{-12}$ s. Do you want to save theories with low quantum gravity energy scale or even superstrings/GUT from proton decay? Be aware!

Take now two ideas for proton decay “escape”:

1st. $M_P>M_s\sim M_{QG}=\dfrac{1}{R_S}$, where $R_S$ is the Schwarzschild radius.

2nd. Consider extra dimensions and that quarks inside protons can propagate into $n\geq d$ dimensions.

A similar argument to those we use previouly, allow us to obtain that

\[\boxed{\tau_p(XD)\sim \dfrac{1}{m_p}\left(\dfrac{M_{QG}}{m_p}\right)^{4+d}}\]

Using the experimental bound, and the GUT scale $\Lambda=M_X$, you have


and then

\[\boxed{\Lambda\equiv (m_p^5\cdot 10^{33}yrs)^{1/4}\simeq 1.4\cdot 10^{16}GeV\sim M_X=M_{GUT}}\]

Combining $\tau_p(XD)$ with $\Lambda$, we can obtain the bound

\[\boxed{M_{QG}>\left(m_p^d\Lambda^4\right)^{\frac{1}{4+d}}\approx 10^{\frac{64}{d+4}}GeV}\]

and where we have used that the proton mass is about 1 GeV and that $\Lambda \sim 10^{16}GeV$. Note that if $d=0$, you get $M_{QG}\sim M_{GUT}$. For d=1,2 you obtain that the quantum gravity scale is high but LOWER than GUT/Planck energy. Finally, d=7 offers a weak bound about 700TeV than even with the VLHC or a big collider we could find hard to test. As an exercise, you can guess how many extra dimensions you should have to probe them with the LHC using these arguments (interestingly, what happens with $d=\infty$?). The above energy scales can be associated to some size. Corresponding sizes using the equation

\[L^n=M_P^2 M_{QG}^{-(2+n)}\]

where n is the number of LARGE extra dimensions (we take d=0 for simplicity), provide typical lengths about



\[\boxed{l< 10^{6/n}\cdot 10^{-32}m}\]

Question: if you plug as l the distance we can probe with colliders, that is about 1 zeptometer or 0.1 zeptometer with the LHC, how many extra dimensions should you have in order to feel if with it? Calculate n for 1 zm and 0.1 zm! I would like to remark that these distances would be very hard to probe with experiments of modified gravity. Thus, testing the proton lifetime is an important thing for BSM theories!

Proton decay is an example of something much more general. Processes with violations of the baryon number B. For proton decays, $\Delta B=1$ but you could generally have $\Delta B\geq 1$ in general! Or even, you can have leptonic number violations, for instance, processes like

\[\nu_i\rightarrow \mbox{nothing}\]

\[\nu_i\rightarrow BH\rightarrow \nu_f\]

\[\nu_i\overline{\nu_j}\rightarrow \mbox{nothing}\]

\[\mu\rightarrow BH\rightarrow e\gamma\]

due to fluctuations in spacetime or produced due to loops of virtual black holes. Of course, we have not observed that, but you can guess some results from all you have learned here. Again, for simplicity, take d=0 and a baryonic number violation with $\Delta B=N$. You get

\[\boxed{\tau_p(\Delta B=N)\sim\dfrac{1}{m_p}\dfrac{1}{\alpha^{2N}}\left(\dfrac{M_{QG}}{m_p}\right)^{4N}\sim \left(\dfrac{M_{QG}}{10^{16}GeV}\right)^{4N} 10^{64(N-1)}\cdot 10^{33}yrs}\]

and the corresponding lowered quantum gravity scale

\[\boxed{M_{QG}>m_p\left(\dfrac{\Lambda}{M_P}\right)^{1/N}\approx 10^{16/N}GeV}\]

Therefore, higher order violations of baryonic number B, $\Delta B\geq 2$, put stringent bounds on the scale of quantum gravity!!!!!! You can ignore them if you have to build your own unified theory. The idea of baryon number violations limiting the scale of quantum gravity is quite independent model, so you should study it a little bit not just for pure diversion!

Are there any other tools you could use for testing quantum gravity scale and not so tough like proton decays? Yes, there are…Not too much, and they are not very well known for some people, but they are interesting. First of all, you could study the so-called neutron-antineutron oscillations. The time scale of these oscillations are not as tough as those put by other processes. Today, you have (to my knowledge, I am not expert on that):

\[\tau_{n\overline{n}}>10^8 s\]

and that implies roughly that

\[ M_P> 10^5GeV=10^2TeV\]

Curiously, this result was not very well known for some experts, so superstringers and some people, BEFORE the LHC, tried to dream with 1 TeV quantum gravity. It is not a realistic expectation in my opinion. Indeed, neutrino masses via seesaw type I imply a very high intermediate scale, so living with high energy quantum gravity is inevitable unless something incredible happens…

Secondly, you could search for lepton number violations in the neutrino sector (complementary to the baryon number violations in the quark/QCD sector) or try to search for hints in atomic physics using muonic atoms, muonium, and other exotica. Thirdly, you can seek hints of strange processes using neutrino astronomy/astrophysics/cosmology, a new-born field with a promising future! Cosmic rays are Nature’s collider, and you can not even be more fortunate to have something out there providing PeV, EeV or even ZeV/YeV particles!!!! High energy neutrinos produce scattering in the atmosphere, and you could even probe some of the previous reactions involving virtual BH. Are there virtual BH out there? If the QG energy scale were low, taking into account that a typical BH cross-section is about $\sigma(BH)\approx \pi R_S^2$ and that BH’s mostly radiate OFF the brane, you could deduce hints of this reaction searching for missing energy. Neutrinos hitting atomic nuclei N, to produce BH via $\nu N\rightarrow BH+X$ are possible and, I think, they are the last hope for Hawking to get his Nobel prize (detection of Hawking radiation is a very hard subject, no matter what you have heard; beyond lab experiments, the Hawking radiation can only be detected through some radical events and hypotheses!).

A little bit off-topic: How many degrees of freedom has a graviton in D=N+1 spacetime?


You get, 35 for N=9 (D=10), 44 if N=10 (D=11),  54 if N=11 (D=12), 65 if N=12 (D=13), 77 if N=13 (D=14), 90 if N=14 (D=15) and 104 if N=14 (D=16).

See you in my next blog post!!!!!

LOG#172. Higgscelebration: 3 years of a Higgs-like boson!


In 2012, July 4th…We had the Higgsdependence day! A SM Higgs-like resonance was discovered at the LHC and was announced to the world. Today, we celebrate this discovery of the Higgs-like thing or particle almost 3 years ago. So let me talk you about what were the expectations and especulations (some of these are yet discussed!) about finding or not the Higgs boson!

Facts: there was some SM exclusions from previous experimental data at the Tevatron and theoretical hints that if the Higgs existed, it should be relatively light (this was assumed taking into account the precision measurements in the SM and certain solid theoretical principles). More ore less, the Higgs shoould be excluded around the 160-170 GeV window (the top quark is near that energy, curiously). The vacuum understanding of the SM is, however, too low. In fact, something weird happens with the Higgs…Even when experimental data and some principles hinted that it should be light, no symmetry in the SM avoided that the Higgs mass received radiative corrections to the mass through virtual loops. Roughly, a single calculation with Feynman graphs provides in the SM:

\[\Delta M_H^2\simeq \Lambda^2\left(\dfrac{3(2M_W^2+M_Z^2+m_h^2-4m_t^2)}{32\pi^2 v^2}\right)\]

Plug the numbers and observe what you get! Things get worse if you include more loops with gauge bosons and fermions…The SM predicts its own fall with this phenomenon! Otherwise, we don’t understand QFT and how to make calculations that have provided to be very predictive. Theories beyond the SM include extra loops to cancel these contributions but it is not known if we can solve this issue without some amount of fine tuning…For instance, SUSY models include logarithmic corrections to these terms just as the inclusion of antiparticles solved the problem of the electron charge and mass in Quantum Electrodynamics. Renormalization and renormalizability of any QFT theory ARE important features! However, their origin is yet a mystery…Could we build a finite theory without the need of renormalization and regularization?

In fact, there is some problem with this, since the cosmological constant, roughly the vacuum energy we observe from cosmology, does not fit with the Higgs mass or more precisely with the minimum of the Higgs potential in the context of the SM. To make things worse…We have $\Lambda\sim 10^{-123}M_P^4$ as gravitational vacuum. In the QCD sector of the SM, moreover, another BIG issue is the strong CP problem, or why the CP violating term that is allowed by the theory and parametrized by the so called theta QCD angle is really tiny, about $\theta_{QCD}\leq 10^{-10}$. Back in time to the pre-2012 Higgs discovery era, the electroweak vacuum was known to be around the W, Z masses, roughly 100 GeV, and the Higgs should not be too far from them without dangerous or risky issues for the whole SM picture. Thus, the discovery of the SM Higgs was an important challenge for all the theoretical physics community!

What was to be found? The “pole mass” of the Higgs, since it was known to be a resonant or unstable particle/state…Masses are generally running, and the best behaviour of the mass is for pole masses: more “convergent” in some sense, easy to calculate renormalon effects and to show the finiteness of the theory, but as it happens with the W or the Z masses, where the Higgs mass was in energy was completely unknown. The SM alone does NOT fix the value of the Higgs mass and its coupling $\lambda$ to the SM particles, much unlike the Z mass, related to the W mass through some mathematical relationships and the Weinberg angle $\theta_W$. How to find the Higgs signal? Well, it is not like find a bat-signal but it is pretty similar in some points. Firstly we searched for a peak in the energy, showing the resonant state production (pole mass signal) plus another quantum numbers (like spin, helicity, parity, …some of which are not completely established now in 2015!). Finally, we should check if it is REALLY a SM Higgs particle or some subtle impostor that mimics him very, very well…That is another unsolved question but, at least with some great confidence, people believe the discovered $\sim 125GeV$ particle is some class of Higgs-like particle,…If it is composite or another cool impostor is yet to be analyzed! Assuming it is a SM Higgs, the 125 GeV particle should be a $0^+$ state or something very close to it in order to fit all the current data. Particles and resonances are usually classified by angular moment and spin, and it is denoted by $J^P$ in the particle data group and its reviews.

Now, suppose it is the SM or a subtle impostor…How do we go beyond the SM? How to go BSM? Well, as we all know, if you read this blog or similar sources, there are some pretty solid hints of BSM physics:

1st. Neutrino masses and oscillations. Neutrino masses are too tiny with respect the rest of the SM particles. Indeed, the SM prediction for neutrino masses is ZERO provided there is no flavor oscillations…However, neutrino change its flavor through big distances…What kind of neutral spinor is a neutrino if it is not a Weyl neutrino? Even if you can fit them with new vacuum expectation values through the Higgs field and the spontaneous symmetry breaking, supposing that the neutrino is a Dirac field, some physicists work out the idea that the neutrino mass hierarchy and its origin is different from the SM particles. If neutrino were Majorana particles (equal to their antiparticles) the neutrino field would be even more special… The little hierarchy problem, or why the neutrino masses are lower than other SM fields could be solved with the aid of the mechanism called the seesaw. However, there are other ways to generate neutrino masses without seesaws. Even you can be conservative and take Dirac neutrinos and the SM way via the Higgs and vev to generate neutrino masses, if you give away some naturalness…

2nd. The SM fine tuning of the masses and couplings is another well known problem that has no explanation with the tools of the SM. Beyond the little hierarchy problem stated above, we have the hierarchy problem (why the electroweak scale is so different from the Planck scale). Is there a desert between the electroweak and the Planck scale? Can we test it?

3rd. The flavor problem. Why 3 generations and 6 flavors? Nobody knows…

The hierarchy problem has been one of the guiding principles to build BSM theories. In the past 40 years we have thought or study loopholes and ways to solve that gap: supersymmetry (SUSY) and supergravity (SUGRA), superstrings/M-theory, TeV-scale gravity, preonic models, extra dimensions of space (and time, but it is not mainstream), little Higgses (or theories with extra gauge bosons in which the Higgs particle is the Nambu-Goldstone pseudoboson particle), and lots of model building with the framework of the braneworlds during the last 2 decades. The advent of the LHC has changed some too optimistic expectations (fiction-science?) and I believe some big classes of the models in the last list has gone to the trash…Literally.

Sadly, there is another big problem not too known for the public…The flavor problem, that has not received the required attention those years…Or at least, it has not made big advances ultimately. Without progress, there is no science…So the LHC is going to shed light to this one much more than, perhaps, the hierarchy problem…B physics (physics related to the B quarks) is putting strong bounds on new physics BSM and, indirectly, paving the way to…A desert? The explanation of the neutrino masses? We have no idea yet, but understanding the pattern in the mixing of the quarks and neutral leptons, neutrinos, is VERY important. In fact, the mixing of the quarks is minimal to some extent, since the CKM matrix is almost diagonal, while the mixing of the neutrinos is close to be maximal, since the PMNS matrix has the matrix elements “very mixed” or “rotated”.

The attack to the flavor methods with SUSY has provided to be awful and terrible. In fact, the breaking of SUSY produces lots of flavors, much more than we observe. In fact, this is used to put stringent bounds on SUSY models (Oh, yeah! But you should NOT talk too much about this with SUSY specialists or stringers if you don’t want to be into trouble with them!). Roughly, soft terms in the MSSM and the constrained MSSM (or MSUGRA) drive us into the question: is SUGRA flavor blind? Moreover, in this to equilibrate the discussion with stringers and SUSY fans, some regions in the parameter space of these models DO contain light higgs masses as the one we have discovered…Indeed, some bayesian and mMSSM tools applied to them produce the following result (also expected from the pure SM view and precision data): light Higgs masses are favored over heavy scalar masses.

The flavor problem contains another subtle concept: the flavor changing neutral currents (FCNC). These are very dangerous “beasts”. However, they can also be friendly: FCNC can help to reduce the 100 free parameter space in some of the monstrous models above. Flavor changing neutral currents in the SM are very suppressed and hints of such an event could shed light to what kind of theory stands beyond the SM. These currents can modify the “order zero” CKM expectation of quasi-diagonal matrix, i.e., $m_q^2\sim m^2\delta_{ij}$. Of course, the relevant part of this is “almost”.

4th. Dark matter (DM). Galactic rotation curves does not fit the Kepler laws/Universal gravitation/GR expectation. Some galactic cluster observations do support the idea of having more mass in galaxies that the mass we can see with light. However, where is this matter? How is it distributed? And more…What particle/s are dark matter? DM can not be any SM particle. That is a fact. However, it introduces the possibility of finding new particles and interactions at the LHC, the ILC and their successors. Many physicists think that the whole dark matter is likely not a single particle but a completely new set of new particles (of course, you could be minimal and conservative and set the dark energy density of the universe with a single new field as well!) and their interactions. These new interactions should be restricted to gravity, maybe the weak force and a really weakly or feeble coupling to the SM (if any!). Thus, you have lots of models with dark matter out there. For instance, gauge mediated models introduce “messenger particles” that connect the hidden SUSY breaking scale (or the new physics scale) with the SM or some subsector of the MSSM that reproduces the SM. However, problems arise again: we need loop suppressions in the calculations of the masses and problems with the so called couplings of the superpotential. Perhaps some models sith tree level gauge mediation are possible in order to recover the SM fields through interactions of Z’ and Z with heavy neutrinos and other mediators. The SM is thought to be an effective theory valid up to some energy scale. If you put the energy-cut off at the Planck scale, you find a wonderful fine tuning of the parameters or some class of symmetry should protect the Higgs (and other extra scalars that effective SUSY models produce, for instance) to get heavier and make your brain boils like water…Effective field theories (even superstring theory and M-theory are effective despite the fact their fans don’t use to know it) require the expansion around some energy scale (with the aid of perturbation theory) and higher dimensional (quantum) operators. These operators are tools used to parametrize our ignorance about New Physics and the true “final theory”. New Physics involves some new degrees of freedom in general. New fields, new interactions. However, many of these models of new physics does not fix our Higgs potential ambiguity or they do not impose positivity in the (scalar/super) potential. A class of these models is the NMSSM. Please, note that the usually adored MSSM plus effective approaches produces problems. So, what about giving up SUSY after all? How to solve the hierarchy problem without SUSY/SUSY breaking? Compositeness theories like technicolor theories or preonic models are known examples of all this…But the recipe is very similar to that with SUSY…Use additional symmetries to protect the Higgs mass (and other likely scalar particles/new particles). This compositeness theories involve generally new ultrastrong dynamics. It is tied to certain class of models called little Higgses: Higgs as pseudo-Nambu-Goldstone particle of broken symmetry. Compare this situation with the breaking of chiral symmetry in QCD. A new superstrong energy scale is usually introduced. The absence of FCNC plus electroweak corrections provide a rough idea of this superQCD dynamics energy scale. It should be around 3 TeV. Early ideas that the SM is effective up to a few TeVs is managed with these models. However, some fine tuning of the parameters is required…

Before the LHC, the flavor problem made us to wonder about new generations of heavy particles (quarks and leptons). Even technicolor and other ideas would try to explain what would happen were the Higgs not been detected:


What about a fourth generation with weird quantum numbers? What about magnetic monopoles or dyons -particles with both electric and magnetic charges? Electric and magnetic supercharges have been known in the physics literature. However, again, there are some problems. For instance, in the presence of magnetic monopoles (or magnetic charges), baryon number (and even angular moment) can change and catalyze baryon decay (e.g. proton decays!). Protons are very stable, since current bounds say that $\tau(p)>10^{33-34}yrs$, so any BSM theory including magnetic monopoles can get into trouble quickly if you don’t avoid this event. The Callan-Rubakov effect and the Witten effect are also related to this issue. You generally require high mass quark effects because of QCD. Skyrmions (certain QCD configurations) can decay due to magnetic monopoles at low energy but certain topological baryon number is conserved instead. Neutrinos are also neglected in this picture due to the fact they are almost massless. However, it is believed that in order to get magnetic monopoles interact with neutrinos, some enlarged or generalized mechanism should be invented. Thus, the question of why there are not other heavy quarks beyond the top quark -or the quarks of the second/third generations is easily solved by compositeness. There is no Higgs but a new superstrong dynamics, but there should be a magnetic charge! Magnetic monopoles produce striking signals at some detectors (fireballs or “rainbow” cascades, even they produce entire gap leaps in superconducting devices-recall the unreproduced Cabrera monopole in 1982, 14th February). In summary, technicolor plus some magnetic charges in a hidden U(1) theory would not need a (fundalmental) Higgs at all.

Extra dimensions have been very popular in Physics, not only in Science-Fiction in the last centuries! Modern superstring theory/M-theory relates the Newton constant, the string scale and the Kaluza-Klein scale in clever ways. The second superstring revolution and the new braneworlds models have shifted the attention to some usually assumed as true facts. In old Kaluza-Klein theories (KK theories for short), the string scale was about the KK scale. That is not necessarily true anymore. Up to a numerical constant, we have:

\[\dfrac{1}{G_N}=\dfrac{M_s^8}{M^2_{KK}}\approx 10^{38}GeV\]

Old KK arguments implied that $M_s\sim M_{KK}$ and then $M_s\sim M_p\sim 10^{19}$ GeV. Current theories put some fields in the “TeV”-scale and other in the bulk (extradimensional space). The bulk is connected with the SM 3-brane with some fields depending on some fine tuning and clever model building. The radion, the dilaton and other extra fields propagate under certain subspaces. Even you have other braneworld models like that by Randall and Sundrum where you get weak gravity in the SM 3-brane but strong gravity at the parallel brane due to a cool and clever non factorizable metric! Anyway, standard KK methods work: you get the light plus heavier higgses in general plus whole sets of KK resonances. You can also have higher spin particles with care (specially if you require SUSY!).

Another unsolved question by the SM is its own geometrical origin. Some smart people (Schucker, Connes,…) tried in the last decades to formulate the SM and its messy lagrangian from solid and strong geometrical foundations. Indeed, they managed it…However, they were not successful with the Higgs mass prediction. However, non-commutative geometry has provided its power to derive the SM lagrangian from the spectral action principle (see the works by Connes et alii, like Ali Chamseddine, Matilde Marcolli, …). One good thing about the NC construction of the SM is that it predicts a single Higgs. Gauge groups are restricted and the fermions are in the fundamental representation. However, it was a pity they failed in the Higgs mass prediction…That M. Shaposhnikov guessed around 2010 using asymptotically safety gravity+SM! His only addition were 3 right-handed heavy/very heavey new neutrinos…I am not going to discuss this idea in this post, but I should say that predicting a Higgs mass about 128 GeV without SUSY is quite an statement!!!!

What if…no Higgs or Higgslike partice had been found? Then, we would have been into trouble…But troubles are fun sometimes. We have fun with this Higgs-like state and the questions we have know are yet pretty similar to those we would have had in the case of no Higgs. That is: is the Higgs gaugephobic for some of the SM fields? Is there some unHiggs particle or some conformal hidden sector BSM? What is the state of art with extra dimensions and NC with the current data? In any case, we should watch all the data with care…To solve the flavor problem, the minimal solution is to introduce the Yukawa as the only sources of FCNC through some dimension 6 operators. Essentially, the effective lagrangian should contain two pieces: the SM piece and some piece involving dimension 6 operators or higher. The scale induced is larger that roughly 11 TeV (maybe more with the last data but I am not updated in this subject)! Thus, flavor physics, as I stated above, constraints harder the scale of new physics! With neutrinos, Ice Cube will be studying high energy neutrinos, neutrinos with energies much much larger than ANY accelerator can produce in a human lifetime and likely in the next centuries. We are lucky that Nature produces naturally such amount of highly energy neutrinos that we could not even produce with current technologies! Maybe, some FCNC hints could be discovered in 5-10 years of data taking in some neutrino detectors. Who knows?

Dark matter is yet mysterious…Dark energy is yet even more mysterious…And are they related to the electroweak SSBthrough some hidden portal? There is no conclusive answer yet. In principle, interactions between the DM or hidden sector with the SM are possible, but highly suppressed…Otherwise, we would have noticed it!!!

What lies beyond the SM?

See you in my next blog post!!!!

PS: From the SM and beyond in a few pics…

Standard-ModelThe SUSY scape…

BSMSusy-particles BSMMindOfSUSY

Shaposhnikov’s model: the “almost desert/nightmare” of theoretical physics (asymptotically safe gravity is also suggested)


From the light to the dark side of the Universe using braneworlds…

braneworld1and the Multiverse/Polyverse…

multiverse8 multiverse7 multiverse6

Original caption: Parallel universes, conceptual computer artwork. --- Image by © Mehau Kulyk/Science Photo Library/Corbis

multiverse3 Multiverse2 multiverse1Desperate solutions?

LOG#171. From Bohrlogy to dualities.


Old (old fashioned!) Quantum Mechanics is understood as the quantum theory before its final formulation around 1927-1931…It includes the Bohr model, the Wilson-Bohr-Sommerfeld quantization and some other tricks like the one to take into account the finite nuclear size. For finite nuclear mass (not infinite), considering it as fixed, the hypothesis implies that (M is the nuclear mass and m is the electron mass around the nucle-we take the hydrogenic one single electron atom for simplicity) thre reduced mass will be:

(1)   \begin{equation*} \mu=\dfrac{Mm}{M+m}=\left(\dfrac{1}{M}+\dfrac{1}{m}\right)^{-1} \end{equation*}


(2)   \begin{equation*} \dfrac{1}{\mu}=\dfrac{1}{M}+\dfrac{1}{m} \end{equation*}

Compare this equation with the association of two electrical resistances (or inductances) in parallel (exactly with the same formal expression) or two capacitors in series! Simple and neat/clear analogy!

Following the Bohr model of the hydrogen atom and its success (today we know it was only partial) to explain the hydrogen spectrum, people began to study if it could be generalized to atoms with higher atomic number. Even Bohr himself tried to solve the issue…Bohr’s formula and model had worked well to give the previously known Rydberg for the hydrogen atom, but it was not known then if it would also give spectra for other elements with higher Z numbers, or even precisely what the Z numbers (in terms of charge) for heavier elements were.

It was known that he ordering of atoms in the periodic table did tend to be according to atomic weights or mass, but there were a few famous “reversed” cases where the periodic table demanded that an element with a higher atomic weight (such as cobalt at weight 58.9) nevertheless be placed at a lower position (Z=27), before an element like nickel (with a lower atomic weight of 58.7), which the table demanded take the higher position at Z=28. Moseley inquired if Bohr thought that the electromagnetic emission spectra of cobalt and nickel would follow their ordering by weight, or by their periodic table position (atomic number, Z), and Bohr said it would certainly be by Z. Moseley’s reply was “We shall see!”

Since the spectral emissions for high Z elements would be in the soft X-ray range (easily absorbed in air), Moseley was required to use vacuum tube techniques to measure them. Using X-ray diffracton techniques in 1913-1914, Moseley found that the most intense short-wavelenght line in the x-ray spectrum of a particular element was indeed related to the element’s periodic table atomic number, Z.

This line was known as the so-called K-alpha line. Following Bohr’s lead, Moseley found that this relationship could be expressed by a simple formula, later called Moseley’s Law. Mathematically:

(3)   \begin{equation*} \sqrt{f}=k_1(Z-k_2) \end{equation*}

or equivalently

(4)   \begin{equation*} f=k_1^2(Z-k_2)^2 \end{equation*}

Moseley’s two given formulae for K-alpha and L-alpha lines, in his original semi-Rydberg style notion were derived to be:

(5)   \begin{equation*} f(K_\alpha)=\dfrac{3}{4}\cdot (3.29\cdot 10^{15})(Z-1)^2 Hz \end{equation*}

(6)   \begin{equation*} f(L_\alpha)=\dfrac{5}{36}\cdot (3.29\cdot 10^{15})(Z-7.4)^2 Hz \end{equation*}

 The energy of photons that a hydrogen atom can emit in the Bohr deduction of the Rydberg formula  is given by the difference of any two hydrogen energy levels:

    \[E_i-E_f=-\Delta E_{i,f}=-\dfrac{m_e^4Q_e^2Q_Z^2}{8h^2\varepsilon^2_0}\left(\dfrac{1}{n_i^2}-\dfrac{1}{n_f^2}\right)\]

For the hydrogen atom, the quantity of charge reads


because Z (the nuclear positive charge, in fundamental units of the electron charge is essentially Q=Ne) is equal to 1. That is, the hydrogen nucleus contains a single charge. However, for the hydrogenic atoms (those in which the electron acts as though it circles a single structure with effective charge Z), Bohr realized from his derivation that an extra quantity would need to be added to the conventional charge to the fourth power, in order to account for the extra pull on the electron, and thus the extra energy between levels, as a result of the increased nuclear charge. In 1914 it was realized that Moseley’s formula could be adapted from Bohr’s, if two assumptions were made:

1st. The electron responsible for the brightest spectral line (K-alpha) which Moseley was investigating from each element, results from a transition by a single electron between the K and L shells of the atom (i.e., from the nearest to the nucleus and the one next farthest out), with energy quantum numbers corresponding to 1 and 2.

2nd. The Z in Bohr’s formula, though still squared, required a reduction by 1 to calculate K-alpha. This effect arises because the initial and final states of the atom have different amounts of electron-electron repulsion. A widespread oversimplification is the idea that the effective charge of the nucleus decreases by 1 when it is being screened by an unpaired electron. In any case, Bohr’s formula for Moseley’s K-alpha X-ray transitions became:

    \[E_i-E_f=-\Delta E_{i,f}=-\dfrac{m_e^4Q_e^4(Z-1)^2}{8h^2\varepsilon^2_0}\left(\dfrac{1}{n_i^2}-\dfrac{1}{n_f^2}\right)\]

In the case with the transition with initial n=1 and final n=2, dividing both sides by h to convert energy to frequency, we get:

    \[f=\nu=\dfrac{m_eQ_e^4}{8h^3\varepsilon^2}\left(\dfrac{3}{4}\right)(Z-1)^2=(2.48\cdot 10^{15}(Z-1)^2 Hz\]

The final value of the theoretical frequency

    \[f_0=2.47\cdot 10^{15} Hz\]

is in good agreement with Moseley’s empirically-derived value. This fundamental frequency is the same as that of the hydrogen Lyman-alpha line, because the 1s to 2p transition in hydrogen is responsible for both Lyman-alpha lines in the hydrogen atom, and also the K-alpha lines in X-ray spectroscopy for elements beyond hydrogen, which are described by Moseley’s law. Moseley was indeed fully aware that his fundamental frequency was Lyman-alpha, the fundamental Rydberg frequency resulting from two fundamental atomic energies, and for this reason differing by the Rydberg-Bohr factor of exactly 3/4, and he explicitly showed it clearly in his original papers.

As regards Moseley’s L-alpha transitions in relation with current modern Quantum Mechanics, the modern view associates electron shells with principle quantum numbers n, with each shell containing two times n squared electrons, giving the n=1 shell of atoms 2 electrons, and the n=2 shell 8 provides electrons. The empirical value of 7.4 for Moseley’s second K is thus associated with n=2 to n=3, then called L-alpha transitions (not to be confused with Lyman-alpha transitions), and occurring from the “M to L” shells in Bohr’s later notation. This value of 7.4 is now known to represent an electron screening effect for a fraction (specifically 0.74) of the total of 10 electrons contained in what we now know to be the n=1 and n=2 (or K and L) “shells”.

What else? The theory of special relativity!!!!! Considerations from the special theory of relativity implied that the circular orbits of the electron in the atom should be considered an approximation. Just as it happens with the elliptical orbits in the solar system. Circles are a particle type of ellipse. That is how Wilson, Bohr and Sommerfeld, with the aid of some ideas of classical mechanics and the special theory of relativity, arrived to a more general quantization rule. Mathematically speaking, it reads

(7)   \begin{equation*} \oint_C pdq=nh \end{equation*}

 for some “periodic” curve C. Classical Bohr quantization can be read off from this rule. Take a periodic orbit be a circle with angular moment L. The Wilson-Bohr-Sommerfeld above is a quantization of the action-angle variable (p,q) to be

    \[\oint Ld\phi=nh\]


    \[ L\oint d\phi=2\pi L=nh\]

and thus the angular moment is quantized as Bohr’s model hypotheses/rules argued


However, the power of Wilson-Bohr-Sommerfeld ideas is that we can go beyond Bohr’s model. For simple harmonic motion, we obtain, from classical mechanics


and thus

    \[x=x(t)=A\sin (\omega t)\]

We know that

    \[\omega=2\pi f=\sqrt{k/m}\]


    \[dx=\omega A \cos\omega t dt\]


    \[ E_c=\dfrac{1}{2}mv^2=\dfrac{p^2}{2m}\]




    \[p=mv=m\omega A\cos\omega t\]

Wilson-Bohr-Sommerfeld rule provides

    \[\oint pdx=\oint m\omega^2A^2\cos^2\omega t dt=nh\]


    \[2E\oint \cos^2\omega t dt=nh\]


    \[m\omega^2 A^2=2E\]


    \[\theta=\omega t\]


    \[2E\oint \cos^2\omega t dt=\dfrac{2E}{\omega}\int_0^{2\pi}\cos^2\theta d\theta=\dfrac{2E\pi}{\omega}=nh\]

Thus, we have arrived to

    \[E=\dfrac{nh\omega}{2\pi}=nhf=n\hbar \omega\]

but this is the Einstein’s and Planck’s quantization rule for harmonic oscillators/quanta of “light” (or more generally bosons).

 Finally, our third example of Wilson-Bohr-Sommerfeld quantization rule. A free particle in a box. Firstly we consider the unidimensional (D=1) box with length equal to L. The action-angle variable is

    \[\oint pdx=\int_0^L(+mv)dx+\int_{L}^0(-mv)dx=2mvL=nh\]

Thus, we have that


Now, going further, we have three cases:

1st. Massive non-relativistic particle. We have


Energy is quantized, so linear momentum is also quantized

    \[p=\dfrac{n\hbar \pi}{L}\]

Remarkably, this quantized momentum also appear in Kaluza-Klein theories when you use dimensional reduction of a periodic extra space-like dimension of size pi times L!

2nd. Massless (m=0) relativistic particle (ultrarelativistic particle).

    \[E=pc=\dfrac{nh}{2L}=\dfrac{n\pi \hbar}{L}\]

The momentum is also quantized too

    \[p=\dfrac{n\pi \hbar}{Lc}\]

Remark: in natural units (c=1), up to a numerical 2 (1/2) factor, this is the same momentum as the previous case! The energy spectrum differs from the presence of mass, a number AND the power in the Planck constant and n. Of course, it coincides with the KK quantization for a periodic space-like extra dimension!

3rd. Relativistic massive particle.


 From this, we get a rest mass shifted quantized relativistic momentum

    \[p=\sqrt{\left(\dfrac{n\hbar \pi}{Lc}\right)^2-m^2c^2}\]

These formulae can be easily generalized to a particle in a multidimensional box of size


or more generally

    \[V=\prod_{i=1}^DL_i=L_1L_2\cdots L_D\]

You only have to take D different angle-action variables and mimic the procedure. For a non-relativistic massive particle you will have




For the massless relativistic D-dimensional case you have


and the massive relativistic D-dimensional case




Tired of boring point particle spectra? Go to superstring theory!!!!!! We have already seen that a single particle in a box (Am I a mad man with a box????? LoL) of size L has the spectrum


where I am using natural units right now. Note that when L goes to zero the energy diverges! Thus, finite energy states is n=0 only and that does NOT depend on the box size L at all! Going into D spacetime dimensions ambient, if you reduce dimensionally the theory and compactify k extra space-like dimensions, you can only get D-k non compact dimensions AND possible zero modes. That is, fields can only, at least in principle from this framework, propagate along D-k non compact dimensions (today, current modern theories can do this better and this feature depends on the particular model building of your theory, i.e., you can guess theories with fields propagating not only in the non compact space, the brane, but also into the compact space-that can be LARGE- or in the bulk; these are commonly referred as brane-worlds). In summary, small dimensions probe energies greater than 1/L, not lesser. Moreover, closed strings (loops!) have an intersting spectra:


where the last two terms are purely stringy from the winding modes and the harmonic oscillator excitations of the string (they can possible include a zero point contributions). By the other hand, open strings have the spectrum


There are no winding modes in the open string spectrum. Indeed, a quite mysterious remark was that even when open strings DO contain closed strings as excitations, in a dual picture there is no empty space. It was discovered that a hidden extended object, with p=D-k-1 spatial components, is linving in the edges of the open string. These objects are D-branes (from Dirichlet boundary conditions) or Dp-branes (D-k-1=p-membrane or p-brane)! The role of these new objects in the theory formerly known as superstring theory is very striking and surprising. Indeed, there are cool mathematical objects involving their dualities. We have seen above that T-duality exchange the roles of winding modes and KK modes, and much more interestingly, relates a compactified space with size L with another of 1/L is suitable units. So, somehow, large and small quantities are not differnt from the viewpoint of T-duality and the related theories! That is a very strange result and indeed has nothing to do with common experience. But it is true and a solid result in superstring theory, now completely established. Very small and very big things are related with T-duality. What else? S-duality!!!!! But before talking about S-duality, we will go back in time to the early times of quantum physics, and we will met the thoughts of a guy called Paul A. M. Dirac…

See you in my next blog post!

LOG#170. The shortest papers ever: the list.


Hi, everyone.

In this short post I am going to discuss some of the shortest papers ever written all over the world!!!!! This idea came to my mind due to a post by @seanmcarroll on twitter and other social networks…

Shortest papers on mathematics…? I know the following two papers (one quoted by Sean, the next one I have discussed it here):

1. Counterexample to Euler’s Conjecture of sums of like powers.

2. Moonshine and the meaning of life.

The texts are given below. Firstly:

shortest-math-paperThe second one is this one: http://arxiv.org/abs/1408.2083 Its title said it all: Moonshine and the Meaning of Life. I talked about it in this post http://www.thespectrumofriemannium.com/2014/08/13/log154-moonshine-and-42-the-paper/

In pictures, it can be shown here:

moonshineShortPaper1AmoonshineShortPaper1B Finally, my next example is a old example from physics. And it is also related to some of my old posts here http://www.thespectrumofriemannium.com/2012/12/28/log069-cpn-spheres-1836/

Here you are the shortest paper on Physics I do know:

3. The ratio of proton and electron masses.

Lenz-mpme-valueAnother one about Mathematics appeared in THE MATHEMATICAL ASSOCIATION OF AMERICA, Monthly 121…

4. A refinement of a Theorem of J.E. Littlewood

littlewoodPaperThe shortests abstract? Maybe this one:

5. Can apparent superluminal neutrino speeds be explained as a quantum weak measurement?

URL: http://arxiv.org/ftp/arxiv/papers/1110/1110.2832.pdf

Title: Can apparent superluminal neutrino speeds be explained as a quantum weak measurement?

Authors: M V Berry, N Brunner, S Popescu & P Shukla.

Published: November 11 2011, J.Phys.A 44 492001

Abstract: Probably not.

Caption: probablyNot

I also found here this:

“(…)In 1974, clinical psychologist Dennis Upper found himself stricken with writer’s block. Though pen was to paper, no words would flow. He decided to solve his problem with a scientific experiment. Yet, as is frequently the case in science, his experiment didn’t work as intended, and that’s putting it euphemistically. Despite the failure, his work, “The unsuccessful self-treatment of a case of “writer’s block,” was published in the prestigious Journal of Applied Behavioral Analysis. It is reproduced in its entirety below:


Despite the paper’s glaring brevity, Upper’s reviewer hailed its brilliance:

I have studied this manuscript very carefully with lemon juice and X-rays and have not detected a single flaw in either design or writing style. I suggest it be published without revision. Clearly it is the most concise manuscript I have ever seen-yet it contains sufficient detail to allow other investigators to replicate Dr. Upper’s failure. In comparison with the other manuscripts I get from you containing all that complicated detail, this one was a pleasure to examine. Surely we can find a place for this paper in the Journal-perhaps on the edge of a blank page.


Do you know any other short paper about Physics, Chemistry, Medicine or Mathematics that deserve to be in this list?

Let me know, if you pass my spam-filter, Trivial Pursuit alike, questionnaire… XD…


 P.S.: A shortest abstract was communicated to me after writing this post, you can see it here: http://inspirehep.net/record/110443

Some screenshots

4Quarks1A4Quarks1BIt seems that the referees made Montonen and Roos to add some words to the abstract, so the final published version has something else that the above Ulysses famous word…

LOG#169. A two level problem.

lasersword Laser_beam

This post is the solution of the following problem: pick an atom with two energy levels. They have a transition wavelength of 580nm. At room temperature 5\cdot 10^{20}atoms are in the lower state.

1) How many atoms are in the upper state, if we have thermal equilibrium?

2) Suppose instead we had 9\cdot 10^{20} atoms pumped into the upper state, with 5\cdot 10^{20} atoms in the lower state. This is a non-equilibrium state. How much energy (in Joules) could be released in a single pulse of light if we restored the equilibrium?

Data: Boltzmann factor is \exp\left(-E/k_BT \right) and k_B=1\mbox{.}38\cdot 10^{-23}J/K

Solution to 1). If N_b is the higher energy state and N_a is the lower energy state, then the energy difference between these two atomic states can be calculated using the transition wavelength (580nm), i.e.,

    \[E_b-E_a=hf=\dfrac{hc}{\lambda}=\Delta E\]

    \[\Delta E=3\mbox{.}43\cdot 10^{-19}J\]


    \[k_BT=1\mbox{.}38\cdot 10^{-23}J/K\cdot 300K=4\mbox{.}10^{-21}J\]

Now, according to the Boltzmann distribution, the population in any state is given by


Therefore, the ration of N_a and N_b in the thermal equilibrium should be

    \[\dfrac{N_b(eq)}{N_a(eq)}=e^{-\frac{\Delta E}{k_BT}}=1\mbox{.}1\cdot 10^{-36}\]

and thus

    \[N_b(eq)=N_a(eq)\cdot 1\mbox{.}1\cdot 10^{-36}\]

Using the giving value of N_a, we get

    \[N_b(eq)=5\mbox{.}5\cdot 10^{-16}\]

and that is a very tiny number of atoms. This is showing to us that the energy gap between the given atomic states at room temperature is so large that almost all the electrons “choose” to stay in the lower state and hardly we will find electrons in the upper state.

Solution to 2). When we pump atoms into the upper state, we create a non-equilibrium atomic state. Energy will be released up until the equilibrium is restored (in fact, this is the working principle of the laser, in a simplified fashion!). We obtain now

    \[N_a(non-eq)=5\cdot 10^{20}\]

    \[N_b(non-eq)=9\cdot 10^{20}\]

    \[N(non-eq)=N_a(non-eq)+N_b(non-eq)=14\cdot 10^{20}\]

and we are interested in the number of atoms which restores the thermal equilibrium. Since the ratio should be

    \[\dfrac{N_a}{N_b}=1\mbox{.}1\cdot 10^{-36}\]

remains the same (the gap width should not change), we find

    \[N_a(non-eq)+N_b(non-eq)=14\cdot 10^{20}\]

    \[N_b(non_eq)(1+9\mbox{.}09\cdot 10^{35})=9\cdot 10^{20}\]

and then

    \[N_b(non-eq)=1\mbox{.}54\cdot 10^{-15}\]

The number of atoms that contribute to restore the thermal equilibrium is

    \[\Delta N=N(non-eq)-N_b(non-eq)\approx 9\cdot 10^{20}\]

The energy released in a single monochromatic pulse would be

    \[E=\Delta N\cdot \dfrac{hc}{\lambda}=9\cdot 10^{20}\cdot 3\mbox{.}43\cdot 10^{-19}\approx 309J\]