Home 1 2 3 24 25

LOG#247. Seesawlogy.

Introduction 

One of the big issues of Standard Model (SM) is the origin of mass (OM). Usually, the electroweak
sector implements mass in the gauge and matter sector through the well known Higgs mechanism. However, the Higgs mechanism is not free of its own problems. It is quite hard to assume that the same mechanism can provide the precise mass and couplings to every quark and lepton. Neutrinos, originally massless in the old-fashioned SM, have been proved to be massive. The phenomenon of neutrino mixing , a hint of beyond the SM physics(BSM), has been confirmed and established it, through the design and performing of different nice neutrino oscillation experiments in the last 20 years( firstly from solar neutrinos). The nature of the tiny neutrino masses in comparison with the remaining SM particles is obscure. Never a small piece of matter has been so puzzling, important and surprising, even mysterious. The little hierarchy problem in the SM is simply why neutrinos are lighter than the rest of subatomic particles. The SM can not answer that in a self-consistent way. If one applies the same Higgs mechanism to neutrinos than the one that is applied to quarks and massive gauge bosons, one obtains that their Yukawa couplings would be surprisingly small, many orders of magnitudes than the others. Thus, the SM with massive neutrinos is unnatural (In the sense of ‘t Hooft’s naturalness,i.e., at any energy scale \mu, a setof parameters, \alpha _i(\mu) describing a system can be small (natural), iff, in the limit \alpha _i(\mu)\rightarrow 0 for each of these parameters, the system exhibits an enhanced symmetry.).

The common, somewhat minimal, solution is to postulate that the origin of neutrino mass is different and some new mechanism has to be added to complete the global view. This new mechanism is usually argued to come from new physics (NP). This paper is devoted to the review of the most popular (and somewhat natural) neutrino mass generation mechanism the seesaw, and the physics behind of it, the seesawlogy[1](SEE). It is organized as follows: in section~??, we review the main concepts and formulae of basic seesaws; next, in section~?? we study other kind of no so simple seesaws, usually with a more complex structure; in section~??, we discuss the some generalized seesaws called multiple seesaws; then, in section~??, we study how some kind of seesaw arises in theories with extra dimensions, and finally, we summarize and comment the some important key points relative to the the seesaws and their associated phenomenology in the conclusion.

Basic seesawlogy

The elementary idea behind the seesaw technology (seesawlogy) is to generate Weinberg’s dimension-5 operator \mathcal{O}_5=gL \Phi L\Phi, where L represent a lepton doublet, using some tree-level heavy-state exchange particle that varies in the particular kind of the seesaw gadget implementation. Generally, then:

  • Seesaw generates some Weinberg’s dimension-5 operator \mathcal{O}_5, like the one above.
  • The strength g is usually small. This is due to lepton number violation at certain high energy scale.
  • The high energy scale, say \Lambda _s, can be lowered, though, assuming Dirac Yukawa couplings are small.
  • The most general seesaw gadget \textit{is} is through a set of n lefthanded (LH) neutrinos \nu _L plus any number m of righthanded (RH) neutrinos \nu _R written as Majorana particles in such a way that \nu _R=\nu _L^c.
  • Using a basis (\nu _L,\nu_L^c) we obtain what we call the general (n+m)\times(n+m) SEE matrix (SEX):

    (1)   \begin{equation*} M_\nu =\begin{pmatrix} M_L & M_D \\ M_D^T & M_R \end{pmatrix} \end{equation*}

    Here, M_L is a SU(2) triplet, M_D is a SU(2) doublet and M_R a SU(2) singlet. Every basic seesaw has a realization in terms of some kind of seesalogy matrix.

We have now several important particular cases to study, depending on the values that block matrices we select.

Type I Seesaw 

This realization correspond to the following matrix pieces:

  • M_L=0.
  • M_D is a (n\times m) Dirac mass matrix.
  • M_N is a (m\times m) Majorana mass matrix.
  • Type I SEE lagrangian is given by ( up to numerical prefactors)

    (2)   \begin{equation*} \mathcal{L}_S^{I}=\mathcal{Y}_{ij}^{Dirac}\bar{l}_{L_{i}}\tilde{\phi} \nu _{R_{i}}+M_{N_{ij}}\bar{\nu} _{R_{i}}\nu _{R_{j}}^{c} \end{equation*}

    with \phi=(\phi ^+,\phi ^0)^T being the SM scalar doublet, and \tilde{\phi}=\sigma _2 \phi \sigma _2. Moreover,
    \left\langle \phi ^0 \right\rangle =v_2 is the vacuum expectation value (vev) and we write M_D=\mathcal{Y}_Dv_2.

Now, the SEX M_\nu is, generally, symmetric and complex. It can be diagonalized by a unitary transformation matrix (n+m)\times (n+m) so U^TMU=diag(m_i,M_j), providing us n light mass eigenstates (eigenvalues m_i,i=1,...,n) and m heavy eigenstates (eigenvalues M_j,j=1,...,m). The effective light n \times n neutrino mass submatrix will be after diagonalization:

(3)   \begin{equation*} m_\nu = -M_DM_N^{-1}M_D^T \end{equation*}

This is the basic matrix structure relationship for type I seesaw. Usually one gets commonly, if M_D\sim 100\GeV,and M_N=M_R\sim10^{16}\sim M_{GUT} , i.e., plugging these values in the previous formula we obtain a tipically small LH neutrino mass about m_\nu\sim\meV. The main lecture is that in order to get a small neutrino mass, we need either a very small Yukawa coupling or a very large isosinglet RH neutrino mass.

The general phenomenology of this seesaw can substancially vary. In order to get, for instance, a \TeV RH neutrino, one is forced to tune the Yukawa coupling to an astonishing tiny value, typically \mathcal{Y}_D\sim 10^{-5}-10^{-6}.
The result is that neutrino CS would be unobservable ( at least in LHC or similar colliders). However, some more elaborated type I models prevent this to happen including new particles, mainly through extra intermediate gauge bosons w',Z'. This type I modified models are usually common in left-right (LR) symmetric models or some Gran Unified Theories (GUT) with SO(10) or E_6 gauge symmetries, motivated due to the fact we \textit{can not} identify the seesaw fundamental scale with Planck scale. Supposing the SM holds up to Planck scale with this kind of seesaw would mean a microelectronvolt neutrino mass, but we do know from neutrino oscillation experiments that the difference mass squared are well above the microelectronvolt scale. Therefore, with additional gauge bosons, RH neutrinos would be created by reactions q\bar{q}'\rightarrow W'^{\pm}\rightarrow l^{\pm}N or q\bar{q}\rightarrow Z'^{0}\rightarrow NN(or\;\;\nu N). Thus, searching for heavy neutrino decay modes is the usual technique that has to be accomplished in the collider. Note, that the phenomenology of the model depends on the concrete form gauge symmetry is implemented. In summary, we can say that in order to observe type I seesaw at collider we need the RH neutrino mass scale to be around the TeV scale or below and a strong enough Yukawa coupling. Some heavy neutrino signals would hint in a clean way, e.g., in double W’ production and lepton number violating processes like pp\rightarrow W'^{\pm}W'^{\pm}\rightarrow l^{\pm}l^{\pm}jj or the resonant channel pp\rightarrow W'^{\pm}\rightarrow l^{\pm}N^*\rightarrow l^{\pm}l^{\pm}jj.

Type II Seesaw

The model building of this alternative seesaw is different. One invokes the following elements:

  • A complex SU(2) triplet of (heavy) Higgs scalar bosons, usually represented as \Delta =(H^{++},H^+,H^0).
  • Effective lagrangian SEE type II

    (4)   \begin{equation*} \mathcal{L}_S^{II}=\mathcal{Y}_{L_{ij}}l_i^ T\Delta C^{-1}l_j \end{equation*}

    where C stands for the charge conjugation operator and the SU(2) structure has been omitted. Indeed, the mass terms for this seesaw can be read from the full lagrangian terms with the flavor SU(2) structure present:

    (5)   \begin{equation*} \mathcal{L}_S^{II}=-Y_\nu l^T_LCi\sigma _2\Delta l_L+\mu _DH^Ti\sigma _2\Delta ^+ H+ h.c. \end{equation*}

    Moreover, we have also the minimal type II seesawlogy matrix made of a scalar triplet:

    (6)   \begin{equation*} \Delta =\begin{pmatrix} \Delta ^+ /\sqrt{2} & \Delta ^{++} \\ \Delta ^0 & -\Delta ^+/\sqrt{2} \end{pmatrix} \end{equation*}

  • M_L=\mathcal{Y}_Lv_3, with v_3=\left\langle H^0 \right\rangle the vev rising the neutral Higgs a mass. Remarkably, one should remember that non-zero vev of SU(2) scalar triplet has an effect on the \rho parameter in the SM, so we get a bound v_3 \aplt 1 \GeV.
  • In this class of seesaw, the role of seesawlogy matrix is played by the Yukawa matrix \mathcal{Y}_\nu, a 3\times3 complex and symmetric matrix, we also get the total leptonic number broken by two units(\Delta L=2) like the previous seesaw and we have an interesting coupling constant \mu _D in the effective scalar potential. Minimization produces the vev value for \Delta v_3=\mu_Dv_2^2/\sqrt{2}M^2_\Delta and v_2 is give as before.
Then, diagonalization of Yukawa coupling produces:

(7)   \begin{equation*} M_\nu = \sqrt{2}\mathcal{Y}_\nu v_3=\dfrac{\mathcal{Y}_\nu \mu _D v_2^2}{M_\Delta ^2} \end{equation*}

This seesawlogy matrix scenario is induced, then, by electroweak symmetry breaking and its small scale is associated with a large mass M_\Delta. Again, a juidicious choice of Yukawa matrix elements can accomodate the present neutrino mass phenomenology. From the experimental viewpoint, the most promising signature of this kind of seesawlogy matrix is, therefore, the doubly charged Higgs. This is interesting, since this kind of models naturally give rise to M_\Delta=M_{H^{++}}, and with suitable mass, reactions like H^{\pm\pm}\rightarrow l^{\pm}l^{\pm},H^{\pm\pm}\rightarrow W^{\pm}W^{\pm},H^{\pm}\rightarrow W^{\pm}Z or H^{+}\rightarrow l^{+}\bar{\nu}.

Type III Seesaw

This last basic seesaw tool is similar to the type I. Type II model building seesaw is given by the following recipe:

  • We replace the RH neutrinos in type I seesaw by the neutral component of an SU(2)_L fermionic triplet called \sigma, with zero hypercharge ( Y_\Sigma=0), given by the matrix

    (8)   \begin{equation*} \Sigma = \begin{pmatrix} \Sigma ^0 /\sqrt{2} & \Sigma ^{+} \\ \Sigma ^- & -\Sigma ^0/\sqrt{2} \end{pmatrix} \end{equation*}

  • Picking out m different fermion triplets, the minimal elements of seesaw type III are coded into an effective lagrangian:

    (9)   \begin{equation*} \mathcal{L}_S^{III}=\mathcal{Y}_{ij}^{Dirac}\phi ^T\bar{\Sigma}_i^cL_j-\dfrac{1}{2}M_{\Sigma_{ij}}\mbox{Tr}(\bar{\Sigma}_{i}\Sigma _j^c)+h.c. \end{equation*}

  • Effective seesawlogy matrix, size (n+m)\times (n+m), for type III seesaw is given by:

    (10)   \begin{equation*} M_\nu =\begin{pmatrix} 0 & M_D\\ M_D ^T & M_\Sigma \end{pmatrix} \end{equation*}

Diagonalization of seesawlogy matrix gives

(11)   \begin{equation*} m_\nu =-M_D^TM_\Sigma ^{-1}M_D \end{equation*}

As before, we also get M_D=\mathcal{Y}_Dv_2 and similar estimates for the small neutrino masses, changing the RH neutrino by the fermion triplet. Neutrino masses are explained, thus, by either a large isotriplet fermion mass M_\Sigma or a tiny Yukawa \mathcal{Y}_D. The phenomenology of this seesawlogy matrix scheme is based on the observation of the fermion triplet, generically referred as E^{\pm}\equiv\Sigma^\pm,N\equiv\Sigma^0, and their couplings to the SM fields. Some GUT arguments can make this observation plausible in the TeV scale (specially some coming from SU(5) or larger groups whose symmetry is broken into it). Interesting searches can use the reactions q\bar{q}\rightarrow Z^*/\gamma ^* \rightarrow E^+E^-, q\bar{q}'\rightarrow W^* \rightarrow E^\pm N. The kinematical and branching ratios are very different from type II.


The 3 basic seesaw mechanisms are in the figure above. a) Type I. On the left. Heavy Majorana neutrino exchange. b) Type II. In the center. Heavy SU(2) scalar triplet exchange. c) Type III. Heavy SU(2) fermion triplet exchange.}

Combined seesaws

Different seesaw can be combined or the concept extended. This section explains how to get bigger SEE schemes.

a) Type I+II Seesaw

The lagrangian for this seesaw reads:

(12)   \begin{equation*} -\mathcal{L}_m=\dfrac{1}{2}\overline{\left( \nu _L\; N_R^c\right) } \begin{pmatrix} M_L & M_D \\ M_D^T & M_R \end{pmatrix} \begin{pmatrix} \nu _L ^c\\ N_R \end{pmatrix}+h.c. \end{equation*}

where M_D=\mathcal{Y}_\nu v/\sqrt{2}, M_L=\mathcal{Y}_\Delta v_\Delta and <H>=v/\sqrt{2}. Standard diagonalization procedure gives:

(13)   \begin{equation*} M_\nu =\begin{pmatrix} \hat{M}_\nu & 0 \\ 0 & \hat{M}_N \end{pmatrix} \end{equation*}

If we consider a general 3+3 flavor example, \hat{M}_\nu=diag(m_1,m_2,m_3) and also \hat{M}_N=diag(M_1,M_2,M_3).
In the so-called leading order approximation, the leading order seesaw mass formula for I+II seesawlogy matrix type is:

(14)   \begin{equation*} m_\nu = M_L - M_DM_R^{-1}M^T_D \end{equation*}

Type I and type II seesaw matrix formulae can be obtained as limit cases of this combined case. Some further remarks:

  • Both terms in the I+II formulae can be comparable in magnitude.
  • If both terms are small, their values to the seesawlogy matrix may experiment significant interference effects and make them impossible to distinguish between a II type and I+II type.
  • If both terms are large, interference can be destructive. It is unnatural since we obtain a small quantity from two big numbers. However, from phenomenology this is interesting since it could provide some observable signatures for the heavy Majorana neutrinos.

b) Double Seesaw 

A somewhat different seesaw structure in order to understand the small neutrino masses is got adding additional fermionic singlets to the SM. This is also interesting in the context of GUT or left-right models. Consider the simple case with one extra singlet( left-right or scalar under the gauge group, unlike the RH neutrino!). Then we obtain a 9\times9 seesaw matrix structure as follows:

(15)   \begin{equation*} M_\nu =\begin{pmatrix} 0 & M_D & 0 \\ M_D^T & 0 & M_S \\ 0 & M_S^T & \mu \end{pmatrix} \end{equation*}

The lagrangian, after adding 3 RH neutrinos, 3 singlets S_R and one Higgs singlet \Phi follows:

(16)   \begin{equation*} \mathcal{L}_{double}=\bar{l}_L\mathcal{Y}_lHE_R+\bar{l}_L \mathcal{Y}_\nu \bar{H}N_R+ \bar{N}^c_R\mathcal{Y}_S\Phi S_R+\dfrac{1}{2}\bar{S}_L^c M_\mu S_R+h.c. \end{equation*}

The mass matrix term can be read from

(17)   \begin{equation*} -\mathcal{L}_m=\dfrac{1}{2}\overline{\left( \nu _L \; N_R^c \; S^c_R\right) }\begin{pmatrix} 0 & M_D & 0 \\ M_D^T & 0 & M_S \\ 0 & M_S^T & \mu \end{pmatrix} \begin{pmatrix} \nu ^c_L \\ N_R \\ S_R \end{pmatrix} \end{equation*}

and where M_D=\mathcal{Y}_\nu<H>, and M_S=\mathcal{Y}_S<\Phi>. The zero/null entries can be justified in some models (like strings or GUTs) and, taking M_S>>M_D the effective mass, after diagonalization, provides a light spectrum

(18)   \begin{equation*} m_\nu = M_DM^{T^{-1}}_S\mu M_S^{-1}M_D^T \end{equation*}

When \mu >>M_S the extra singlet decouples and show a mass structure m_S=M_S\mu ^{-1}M^T_S, and it can be seen as an effective RH neutrino mass ruling a type I seesaw in the \nu_L -\nu ^c_L sector. Then, this singlet can be used as a “phenomenological bridge” between the GUT scale and the B-L usual scale ( 3 orders below the GUT scale in general). This double structure of the spectrum in the sense it is doubly suppressed by singlet masses and its double interesting limits justifies the name “double” seesaw.
The \textit{inverse type I} is a usual name for the double seesaw too in some special parameter values. Setting \mu=0, the global lepton number U(1)_L is conserved and the neutrino are massless. Neutrino masses go to zero values, reflecting the restoration of global lepton number conservation. The heavy sector would be 3 pairs of pseudo-Dirac neutrinos, with CP-conjugated Majorana components and tiny mass splittings aroung \mu scale. This particular model is very interesting since it satisfies the naturalness in the sense of ‘t Hooft.

c) Inverse type III Seesaw

 

It is a inverse plus type III seesawlogy matrix combination. We use a (\nu _L, \Sigma, S) basis, and we find the matrix

(19)   \begin{equation*} M_\nu =\begin{pmatrix} 0 & M_D & 0 \\ M_D^T & M_\Sigma & M_S \\ 0 & M_S^T & \mu \end{pmatrix} \end{equation*}

Like the previous inverse seesaw, in the limit \mu \rightarrow 0, the neutrino mass is small and suppressed. The Dirac Yukuwa coupling strength may be adjusted to order one, in contrast to the normal type III seesawlogy matrix. This mechanism has some curious additional properties:

  • The charged lepton mass read off from the lagrangian is:

    (20)   \begin{equation*} M_{lep}=\begin{pmatrix} M_l & M_D \\ 0 & M_\Sigma \end{pmatrix}\end{equation*}

  • After diagonalization of M_{lep}, size (n+m)\times (n+m), the n\times n coupling matrix provide a neutral current (NC) lagrangian, and since the matrix shows to be nonunitary, this violates the Glashow-Iliopoulos-Maiani (GIM) mechanism and sizeable tree level flavor-changing neutral currents appear in the charged lepton sector.

d) Linear Seesaw 

Other well known low-scale SEE variant is the so-called linear seesaw. It uses to arise from SO(10) GUT and similar models. In the common (\nu , \nu ^c, S) basis, the seesawlogy matrix can be written as follows:

(21)   \begin{equation*} M_\nu =\begin{pmatrix} 0 & M_D & M_L\\ M_D^T & 0 & M_S \\ M_L^T & M_S^T & 0 \end{pmatrix} \end{equation*}

The lepton number conservation is broken by the term M_L\nu S, and the effective light neutrino mass, after diagonalization, can be read from the next expression

(22)   \begin{equation*} M_\nu= M_D(M_LM^{-1})^T+(M_LM^{-1})M_D^T \end{equation*}

This model also suffers the same effect than the one in the inverse seesaw. That is, in the limit M_L\rightarrow 0, neutrino mass goes to zero and the theory exhibit naturalness. The name linear is due to the fact that the mass dependence on M_D is linear and not quadratic, like other seesaw.

Multiple seesaws

 

In ( see book 2011) and references therein, a big class of multiple seesaw models were introduced. Here we review the basic concepts and facts, before introduce the general formulae for multiple seesaws(MUSE):

  • Main motivation: MUSEs try to satisfy both naturalness and testability at TeV scale, in contrast with other basic seesaw. Usually, a terrible fine-tuning is required to implement seesaw, so that the ratio M_D/M_R and the Yukawa couplings can be all suitable for experimental observation, such as new particles or symmetries. This fine-tuning between M_D and M_R is aimed to be solved with MUSEs.
  • Assuming a naive electroweak seesaw so that m \sim (\lambda \Lambda_{EW})^{n+1}/\Lambda ^n _S, where \lambda is a Yukawa coupling and n is an arbitrary integer larger than the unit, without any fine-tuning, one easily guesses:

    (23)   \begin{equation*} \Lambda _S\sim \lambda ^{\frac{n+1}{n}}\left[ \dfrac{\Lambda _{EW}}{100\GeV}\right] ^{\frac{n+1}{n}}\left[ \dfrac{0.1\eV}{m_\nu}\right] ^{1/n}10^{\frac{2(n+6)}{n}}\GeV \end{equation*}

    Thus, MUSEs provide a broad clase of parameter ranges in which a TeV scale seesaw could be natural and testable.

  • The most simple MUSE model at TeV scale is to to introduce some singlet of fermions S^i_{nR} and scalars \Phi_n, with i=1,2,3 and n=1,2,\cdots. This field content can be realized with the implementation of global
    U(1)\times Z_{2N} gauge symmetry leads to two large classes of MUSEs with nearest-neighbours interaction matrix pattern. The first class owns an even number of S^i_{nR} and \Phi_n and corresponds to a straightforward extension of the basic seesaw. The second class has an odd number of S^i_{nR} and \Phi_n, and it is indeed a natural extension of the inverse seesaw.
  • The phenomenological lagrangian giving rise to MUSEs is:

    (24)   \begin{eqnarray*} -\mathcal{L}_\nu =\bar{l}_L\mathcal{Y}_\nu \tilde{H}N_R+ \bar{N}^c_R\mathcal{Y}_{S_1}S_{1R}\Phi _1+ \sum _{i=2}^{n}\overline{S^c_{(i-1)R}}\mathcal{Y}_{S_i}S_{iR}\Phi _i+\nonumber \\ +\dfrac{1}{2}\overline{S^c_{nR}}M_\mu S_{nR}+h.c. \end{eqnarray*}

    Here \mathcal{Y}_\nu and \mathcal{Y}_{S_i} are the 3×3 Yukawa coupling matrices, and M_\mu is a symmetric Majorana mass matrix. After spontaneous symmetry breaking(SSB), we get a 3(n+2)\times 3(n+2) neutrino mass matrix \mathcal{M} in the flavor bases (\nu _L,N_R^c,S_{1R}^c,...S_{nR}^c) and their respective charge-conjugated states, being

    (25)   \begin{equation*} \mathcal{M}=\begin{pmatrix} 0 & M_D & 0 & 0 & 0 & \cdots & 0 \\ M_D^T & 0 & M_{S_1} & 0 & 0 & \cdots & 0 \\ 0 & M_{S_1}^T & 0 & M_{S_2} & 0 & \cdots & 0 \\ 0 & 0 & M_{S_2}^T & 0 & \cdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & \cdots & M_{S_{n-1}} & 0 \\ \vdots & \vdots & \vdots & \cdots & M_{S_{n-1}}^T & 0 & M_{S_n} \\ 0 & 0 & 0 & \cdots & 0 & M_{S_{n}}^T & M_\mu \end{pmatrix} \end{equation*}

    where we have defined M_D=\mathcal{Y}_\nu<H> and M_{S_i}=\mathcal{Y}_{S_i}<\Phi _i>, \forall i, i=1,...,n, and they are 3\times 3 matrices each of them. Note that Yukawa terms exist only if \vert i - j\vert =1,\forall i,j=0,1,...,n and that \mathcal{M} can be written in block-form before diagonalization as

    (26)   \begin{equation*} \mathcal{M}=\begin{pmatrix} 0 & \tilde{M}_D \\ \tilde{M}_D^T & \tilde{M}_\mu \end{pmatrix} \end{equation*}

    with \tilde{M}_D=(M_D \; 0) a 3\times 3(n+1) and \tilde{M}_\mu a symmetric 3(n+1)\times 3(n+1) mass matrix.

  • \textbf{General phenomenological features}:\textit{ non-unitary neutrino mixing} ( in the submatrix boxes) and CP violation (novel effects due to non-unitarity or enhanced CP-phases), \textit{collider signatures of heavy Majorana neutrinos} ( class A MUSEs preferred channel pp\rightarrow l_\alpha^\pm l_\beta^\pm X, i.e., the dilepton mode; class B MUSEs, with M_\mu <<M_{EW}, favourite channel is pp\rightarrow l_\alpha^\pm l_\beta^\pm l_\gamma ^\pm X, i.e., the trilepton mode and the mass spectrum of heavy Majorana would consist on pairing phenomenon, showing nearly degenerate masses than can be combined in the so-called pseudo-Dirac particles).
  • \textbf{Dark matter particles}. One or more of the heavy Majorana neutrinos and gauge-singlet scalars in our MUSE could last almost forever, that is, it could have a very long timelife and become a good DM candidate. It could be fitted to some kind of weakly interacting massive particle (WIMP) to build the cold DM we observe.

Class A Seesaws 

This MUSE is a genaralization of canonical SEE. MUSE A composition:

  • Even number of gauge singlet fermion fields S^i_{nR}, n=2k, \;\; k=1,2,...,.
  • Even number of scalar fields \Phi_n, n=2k, \;\; k=1,2,...,.
  • Effective mass matrix of the 3 light Majorana neutrinos in the leading approximation:

    (27)   \begin{equation*} M_\nu =-M_D\left[ \prod_{i=1}^{k}\left( M^T_{S_{2i-1}}\right)^{-1}M_{S_{2i}} \right] M_\mu ^{-1} \left[ \prod_{i=1}^{k}\left( M^T_{S_{2i-1}}\right)^{-1} M_{S_{2i}}\right] ^T M_D^T \end{equation*}

    When k=0, we obviously recover the traditional SEE M_\nu=-M_D ^TM^{-1}_RM_D if we set S_{0R}=N_R and M_\mu = M_R. Note that since the plugging of M_{S_{2i}}\sim M_D \sim \mathcal{O}(\Lambda _{EW}) and M_{S_{2i-1}}\sim M_\mu \sim \mathcal{O}(\Lambda _{SEE}), then M_\nu \sim \Lambda_{EW}^{2(k+1)}/\Lambda _{SEE}^{2k+1}, and hence we can effectively lower the usual SEE scale to the TeV without lacking testability.

 

Class B Seesaws

 

This MUSE is a generalization of inverse seesaw. MUSE B composition:

  • Odd number of gauge singlet fermion fields S^i_{nR}, n=2k+1, \;\; k=1,2,...,.
  • Odd number of scalar fields \Phi_n, n=2k+1, \;\; k=1,2,...,.
  • Effective mass matrix of the 3 light Majorana neutrinos in the leading approximation:

    (28)   \begin{eqnarray*} M_\nu =M_D\left[ \prod_{i=1}^{k}\left( M^T_{S_{2i-1}}\right)^{-1}M_{S_{2i}} \right] \left( M^T_{S_{2k+1}}\right)^{-1} \nonumber \\ \times M_\mu \left( M^T_{S_{2k+1}}\right)^{-1} \left[ \prod_{i=1}^{k}\left( M^T_{S_{2i-1}}\right)^{-1} M_{S_{2i}}\right] ^T M_D^T \end{eqnarray*}

    When k=0, we evidently recover the traditional inverse SEE but with a low mass scale M_\mu:
    M_\nu =M_D ^T(M^T_{S_1})^{-1}M_\mu (M^T_{S_1})^{-1}M_D^T. Remarkably, if M_{S_{2i}}\sim M_D \sim \mathcal{O}(\Lambda _{EW}) and M_{S_{2i-1}}\sim \mathcal{O}(\Lambda _{SEE}) hold \forall i,i=1,2,...k, the mass scale M_\mu is not necessary to be at all as small as the inverse SEE. Taking, for instance, n=3, the double suppressed M_\nu provides the ratios M_D/M_{S_1}\sim\Lambda _{EW}/\Lambda_{SEE} and M_{S_2}/M_{S_3}\sim \Lambda _{EW}/\Lambda_{SEE}, i.e., M_\nu \sim 0.1 \eV results from Y_{\nu}\sim Y_{S_1}\sim Y_{S_2}\sim Y_{S_3}\sim \mathcal{O}(1) and M_\mu \sim 1 \keV at \Lambda_{SEE}\sim 1\TeV.

Extra dimensional relatives: higher dimensional Seesaws 

Several authors have introduced and studied a higher-dimensional cousin of the seesaw and seesaw matrix. We consider a brane world theory with a 5d-bulk (volume), where the SM particles are confined to the brane. We also introduce 3 SM singlet fermions \Psi _i with i=1,2,3. Being singlets, they are not restricted to the brane and can scape into the extra spacetime dimensions(EDs). The action responsible for the neutrino masses is given by

(29)   \begin{equation*} S=S_{bulk,5d}+S_{brane,4d} \end{equation*}

with

(30)   \begin{equation*} S_{bulk,5d}=\int d^4xdy\left[ i\overline{\Psi}\slashed{D}\Psi - \dfrac{1}{2}\left(\overline{\Psi^c}M_R\Psi +h.c. \right) \right] \end{equation*}

and

(31)   \begin{equation*} S_{brane,4d}=\int _{y=0}d^4x \left[-\dfrac{1}{\sqrt{M_S}}\overline{\nu _L} m^c\Psi -\dfrac{1}{\sqrt{M_S}} \overline{\nu _L^c} m^c\Psi +h.c. \right] \end{equation*}

After a KK procedure on a circle with radius R, we get the mass matrix for the n-th KK level

(32)   \begin{equation*} \mathcal{M}_n= \begin{pmatrix} M_R & n/R\\ n/R & M_R \end{pmatrix} \end{equation*}

and a Dirac mass term with m_D=m/\sqrt{(2\pi M_S R)}. The KK tower is truncated at the level N, and we write the mass matrix in the suitable KK basis, to obtain:

(33)   \begin{equation*} \mathcal{M}=\begin{pmatrix} 0 & m_D & m_D & m_D & m_D & \cdots & m_D \\ m_D^T & M_R & 0 & 0 & 0 & \cdots & 0 \\ m_D^T & 0 & M_R-\dfrac{1}{R} & 0 & 0 & \cdots & 0 \\ m_D^T & 0 & 0 & M_R+\dfrac{1}{R} & \cdots & \vdots & \vdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ m_D^T & 0 & 0 & 0 & 0 & M_R-\dfrac{N}{R} & 0 \\ m_D^T & 0 & 0 & \cdots & 0 & 0 & M_R+\dfrac{N}{R} \end{pmatrix} \end{equation*}

Note that M_R is not assumed to be in the electroweak scale and its value is free. We diagonalize the above matrix to get the light neutrino mass matrix:

(34)   \begin{equation*} m_\nu \simeq m_D\left( \sum _{n=-N}^{N}\dfrac{1}{M_R+n/R}\right) m_D^T= m_D\left( M_R^{-1}+\sum _{n=1}^N\dfrac{2M_R}{M_R^2-n^2/R^2}\right) m_D^T \end{equation*}

Already considered by some other references, the limit N\rightarrow \infty produces the spectrum

(35)   \begin{equation*} m_\nu \simeq m_D\dfrac{\pi R}{\tan (\pi R M_R)}m_D^T \end{equation*}

At level of the highest KK state, say N, the light neutrino mass becomes, neglecting the influence of lower states,

(36)   \begin{equation*} m_\nu \simeq m_D\left( \sum _{n=-N}^{N}\dfrac{1}{M_R+N/R}\right) m_D^T \end{equation*}

Then, irrespectively the value of M_R, if M_R<<N/R, the spectrum get masses that are suppressed by N/R, i.e., m_\nu \simeq m_Dm_D^TR/N. Some further variants of this model can be built in a similar fashion to get different mass dependences on m_D (here quadratic).

Conclusion and outlook

The seesaw has a very interesting an remarkable structure and its a remarkable neutrino mass mechanism BSM. It gives a way to obtain small masses from a high energy cut-off scale, yet to find or adjust. Neutrino oscillation experiment hints that the seesaw fundamental scale is just a bit below of GUT scale, although, as this review has shown and remembered, the nature and value of that seesaw energy scale is highly model dependent: the seesawlogy matrix is a mirror of the GUT/higher gauge-symmetry involved in the small neutrino masses, the EW SSB and the particle content of the theory. Moreover, in spite of seesaw is the more natural way to induce light masses on neutrino( or even every particle using some \textit{universal} seesaw), their realization in Nature is to be proved yet. In order to test the way, if any, in which seesaw is present experimental hints on colliders in the line of this article, DM searches and other neutrino experiments(like those in neutrino telescopes, neutrino superbeams or neutrino factories) will be pursued in present and future time. We live indeed in an exciting experimental era and the discovery of sterile neutrino is going to be, according to Mohapatra, a boost and most impactant event than the one a hypothetical Higgs particle finding will provoke. Their time is just running now.

 

Final note: this text have some LaTeX code errata due to WordPress. I will not correct them. I have a pdf version of this article you could buy cheap soon at my shop here. I am not expensive at all…


  1. Please, do not confuse the term with Sexology!

LOG#246. GR attacks, GR effects!

Newtonian gravity is not coherent with special relativity. Einstein was well aware about it and he had to invent General Relativity (GR). Armed with the equivalence principle, a mystery since ancient times of Galileo, the equivalence between inertial and gravitational mass guided him towards a better theory of gravity. He could envision properties of space-time like geometric features. He deduced that gravity was caused by space-time curvature, and idea that was already anticipated in the XIX century by B. Riemann in this habilitation thesis and by W. K. Clifford with his geometric algebra and calculus. Finally, and rivaling D. Hilbert, he arrived to the field equations (already seen in this blog):

(1)   \begin{equation*} \tcboxmath{G_{\mu\nu}+\Lambda g_{\mu\nu}=\dfrac{8\pi G_N}{c^4}T_{\mu\nu}} \end{equation*}

where G_{\mu\nu}=R_{\mu\nu}+\dfrac{1}{2}g_{\mu\nu}R is the Einstein tensor, G_N is the universal constant of gravity, and  \Lambda is the cosmological constant. g_{\mu\nu} is the metric tensor (acting as gravitational potential in GR!), and the Ricci tensor, the Einstein tensor and the curvature scalar depend upon the metric and its derivatives up to second order in the derivatives. T_{\mu\nu} is the momentum-stress-energy tensor. Space-time (curvature!) says matter and energy how to move, matter-energy tells space-time how to curve!
GR has a large number of tested phenomena! A list (non-exhaustive):

Tidal forces: F_M=\dfrac{2GMm\Delta r}{r^3}. Tidal forces are consequence of the space-time curvature, as source of gravity.

 Gravitational time dilation. It also affects GPS systems (so GR is important for technology):

(2)   \begin{equation*} \Delta t'=\dfrac{\Delta t}{\sqrt{1-\dfrac{2G_NM}{c^2r}}} \end{equation*}

and where R_S=2G_NM/c^2 is the Schwarzschild radius. A simpler way to see the metric effect (gravitational potential) is using the expression:

(3)   \begin{equation*} \Delta t=\dfrac{gh}{c^2}t \end{equation*}

\Delta t is the time of the highest clock, at height h, with respect to the deep observer measuring t. The proof of this result can be done using a simple argument. A clock is some type of oscillator with frequency  \nu. In two different points, that oscillator it will have energy

    \[h\nu_1\left(1+\dfrac{U_1}{c^2}\right)=h\nu_2\left(1+\dfrac{U_2}{c^2}\right)\]

As frequency and period are inversely proportional, it gives

(4)   \begin{equation*} \dfrac{\nu_1}{\nu_2}=\dfrac{1+\dfrac{U_2}{c^2}}{1+\dfrac{U_1}{c^2}}=\dfrac{\Delta t_2}{\Delta t_1} \end{equation*}

If  \Delta t=\Delta t_2-\Delta t_1=\Delta t, and \Delta t_1=t, we recover the first formula \Delta t=ght/c^2 if we suppose that the potential is  0 at h_1 and gh en h_2. In the case the height is not negligible with respect to the radius, the GR correction can be generalized to

(5)   \begin{equation*} \delta_{GPS/GR}=\dfrac{\Delta t}{t}=\dfrac{U(r_s)-U(r_\oplus)}{c^2} \end{equation*}

This formula can be comparede to the SR correction due to motion:

(6)   \begin{equation*} \gamma=\dfrac{1}{\sqrt{1-\frac{v^2}{c^2}}}=1+\delta_{SR} \end{equation*}

Exercise: make the figures for r_s= 26 561 km, y r_\oplus=6370km. The net effect is that the orbital clock is faster than the deeper one by a quantity  1+\delta_R, where \delta_T=\delta_{GR}-\delta_{SR}. We can see what clock is the dom. It is the GR clock. Estimate how many times is more powerful the GR than the SR effect. A more careful deduction, having into account the ellipticity and the orbital parameters, can be derived too:

(7)   \begin{equation*} \Delta t_r(GPS)=-2\dfrac{\sqrt{GM_\odot a}}{c^2}e\sin E=-2\dfrac{\sqrt{GM_\odot a}}{c^2}(E-M) \end{equation*}

and where G, M_\odot, a, e, c, E, M are the universal gravitation constant, the Earth mass, the major semiaxis, the eccentricity, the speed of light, and eccentric anomaly and the average anomaly.

Mercury precession and general orbital objects in GR. Before Einstein, even LeVerrier speculated about Vulcan, a further inner planet of the Solar System. Vulcan does not exist: it is only a non-trivial GR effect:

(8)   \begin{equation*} \Delta \phi=\dfrac{6\pi G_NM}{c^2R(1-e^2)}=\dfrac{24\pi^3 R^2}{c^2T^2(1-e^2)} \end{equation*}

Gravitational lensing (checked with success in the A. Eddington expedition in 1919):

(9)   \begin{equation*} \Theta_{L}=\dfrac{4G_NM}{c^2r}=\dfrac{2R_S}{r}=\dfrac{D_S}{r} \end{equation*}

Prediction of gravitational waves and gravitational radiation (newtonian theory is eternal and dos not shrink orbits). Gravitational waves move at speed of light (or is light the wave that moves at the maximal speed allowed by space-time?)  v_g=c. Every body loses energy with gravitational radiation. It was anticipated by the binary pulsar observations in the 20th century. LIGO detected the first GW in 2016. Without sources, the wave equation for gravitational waves is

(10)   \begin{equation*} \square \overline{h}_{\mu\nu}=0 \end{equation*}

 Gravitomagnetic effect (tested by Gravity Probe B), or Lense-Thirring. The first formula reads:

(11)   \begin{equation*} \dot{\Omega}=\dfrac{R_S ac}{r^3+a^2r+R_Sa^2}\left(\dfrac{360}{2\pi}\right),\;\;\; R_S=\dfrac{2G_NM}{c^2},\;\;\; a=\dfrac{2R_\star^2}{5c}\left(\dfrac{2\pi}{T}\right) \end{equation*}

and the second formula is

(12)   \begin{equation*} \dot{\Omega}=\dfrac{2GJ}{c^2a^3(1-e^2)^{3/2}}\left(\dfrac{360}{2\pi}\right)=\dfrac{2G^2M^2\chi}{c^3a^3(1-e^2)^{3/2}}\left(\dfrac{360}{2\pi}\right) \end{equation*}

 Existence of black holes, objects so dense that light is trapped. Anticipated idea as dark stars by other scientists, a more precise definition of black holes is like vacuum solutions to the EFE (Einstein Field Equations)
\item Vacuum energy density, VIA \Lambda, AND it yields a density energy even for vacuum

(13)   \begin{equation*} \rho_\Lambda=-\dfrac{\Lambda c^4}{8\pi G_N} \end{equation*}

 Cosmic expansion, Big Bang theory, as large scale consequences of EFE. A simplification can be done using the cosmological principle: the Universe is homogeneous and isotropic at very large scales. It simplifies solutions to EFE and a special class of metrics, the so called Friedmann-Robertson-Walker metric, allow us to study the expanding universe and its history.

Other GR effects: de Sitter geodesic effect, strong and weak (or other) equivalence principles, Shapiro delay (fourth classical test of GR), no hair theorem, … Shapiro delay formula reads off

(14)   \begin{equation*} \Delta t_S=-\dfrac{2GM}{c^3}\ln \left(1-\mathbf{R}\cdot \mathbf{r}\right) \end{equation*}

GR also gives the possibility of time machines, wormholes, and TARDIS-like space-times. Other solutions: regular black holes, cosmic strings, cosmic deffects, black p-branes,…Recently, it was realized that the reduction of GR to SR in weak fields is non-trivial due to asymptotic symmetries. The BMS group provides the full set of symmetries in GR, plus a new set of symmetries. Thus, supertraslations, superrotations and superboots have generalized the BMS group in the extended BMS group, a sort of conformal or superconformal symmetry. Gravitational memory effects have also been studied recently. GR is a simple gravitational theory, the simplest of a bigger set of theories.

Extensions of GR can be studied, both as alternative to GR or having GR as approximation: multimetric theories, higher derivative extensions (Finsler geometrey, Lanczos-Lovelock, torsion theories like Cartan’s, non-metric theories, tensor-scalar theories, teleparallelism,…) and many others.

At quantum level, gravitation is not completely understood. The best candidate for TOE and GR extension is string theory, a.k.a. as superstring theory or M-theory. Also supergravity theory remains as option. Other alternative theories are Loop Quantum Gravity, twistor theory and extended relativities. GR involves the existence of space-time singularities where usual physical laws do not apply. No one knows how to treat the issue of the beginning of time and space-time.

See you in other blog post!

P.S.: Off-topic, there are only 3 posts left before I will leave this type of blogging. The post 250 will be special and I will try to do it with my new “face” and interface. WordPress has been doing badly and delaying my posting these weeks (beyond other stuff). I can not loose time checking if LaTeX is encoded here OK. I will change that. I am going to post pure pdf blog posts since the number 250 and so on. I will post in pure pdf format from the 250th and beyond. Maybe I will move the site. I am not sure about that, but anyway I think I will keep this site as the previous (free) since it will be reconverted into a shop of my materials too. Consider make a donation to support my writing job here. Also, you will be able to buy my blog posts edited with LaTeX soon. Let me know if you would buy them all or only a few of them. It would help me to decide how to move forward. Furthermore, I will assist sci-fi and movie/TV-series/plot makes with Scientific Consultancy. Even, I could suggest you what kind of equations or theories could support your stories (if any!).

LOG#245. What is fundamental?

 

Some fundamental mottos:

Fundamental spacetime: no more?

Fundamental spacetime falls: no more?

Fundamentalness vs emergence(ness) is an old fight in Physics. Another typical mantra is not Shamballa but the old endless debate between what theory is fundamental (or basic) and what theory is effective (or derived). Dualities in superstring/M-theory changed what we usually meant by fundamental and derived, just as the AdS/CFT correspondence or map changed what we knew about holography and dimensions/forces.

Generally speaking, physics is about observables, laws, principles and theories. These entities (objects) have invariances or symmetries related to dynamics and kinematics. Changes or motions of the entities being the fundamental (derived) degrees of freedom of differente theories and models provide relationships between them, with some units, magnitudes and systems of units being more suitable for calculations. Mathematics is similar (even when is more pure). Objects are related to theories, axioms and relations (functors and so on). Numbers are the key of mathematics, just as they measure changes in forms or functions that serve us to study geometry, calculus and analysis from different abstract-like viewpoints.

The cross-over between Physics and Mathematics is called Physmatics. The merger of physics and mathematics is necessary and maybe inevitable to understand the whole picture. Observers are related to each other through transformations (symmetries) that also holds for force fields. Different frameworks are allowed in such a way that the true ideal world becomes the real world. Different universes are possible in mathematics an physics, and thus in physmatics too. Interactions between Universes are generally avoided in physics, but are a main keypoint for mathematics and the duality revolution (yet unfinished). Is SR/GR relativity fundamental? Is QM/QFT fundamental? Are fields fundamental? Are the fundamental forces fundamental? Is there a unique fundamental force and force field? Is symplectic mechanics fundamental? What about Nambu mechanics? Is the spacetime fundamental? Is momenergy fundamental?

Newtonian physics is based on the law

(1)   \begin{equation*} F^i=ma_i=\dfrac{dp_i}{dt} \end{equation*}

Relativistic mechanics generalize the above equation into a 4d set-up:

(2)   \begin{equation*} \mathcal{F}=\dot{\mathbcal{P}}=\dfrac{d\mathcal{P}}{d\tau} \end{equation*}

and p_i=mv_i and \mathcal{P}=M\mathcal{V}. However, why not to change newtonian law by

(3)   \begin{equation*}F_i=ma_0+ma_i+\varepsilon_{ijk}b^ja^k+\varepsilon_{ijk}c^jv^k+c_iB^{jk}a_jb_k+\cdots\end{equation*}

i.e.

(4)   \begin{equation*}\vec{F}=m\vec{a}_0+m\vec{a}+\vec{b}\times\vec{a}+\vec{c}\times\vec{v}+\vec{c}\left(\vec{a}\cdot\overrightarrow{\overrightarrow{B}} \cdot \vec{b}\right)+\cdots\end{equation*}

Quantum mechanics is yet a mystery after a century of success! The principle of correspondence

(5)   \begin{equation*} p_\mu\rightarrow -i\hbar\partial_\mu \end{equation*}

allow us to arrive to commutation relationships like

(6)   \begin{align*} \left[x,p\right]=i\hbar\varepsilon^j_{\;\; k}\\ \left[L^i,L^j\right]=i\hbar\varepsilon_{k}^{\;\; ij}L^k\\ \left[x_\mu,x_\nu\right]=\Theta_{\mu\nu}=iL_p^2\theta_{\mu\nu}\\ \left[p_\mu,p_\nu\right]=K_{\mu\nu}=iL_{\Lambda}K_{\mu\nu} \end{align*}

and where the last two lines are the controversial space-time uncertainty relationships if you consider space-time is fuzzy at the fundamental level. Many quantum gravity approaches suggest it.

Let me focus now on the case of emergence and effectiveness. Thermodynamics is a macroscopic part of physics, where the state variables internal energy, free energy or entropy (U,H,S,F,G) play a big role into the knowledge of the extrinsinc behaviour of bodies and systems. BUT, statistical mechanics (pioneered by Boltzmann in the 19th century) showed us that those macroscopic quantities are derived from a microscopic formalism based on atoms and molecules. Therefore, black hole thermodynamics point out that there is a statistical physics of spacetime atoms and molecules that bring us the black hole entropy and ultimately the space-time as a fine-grained substance. The statistical physics of quanta (of action) provides the basis for field theory in the continuum. Fields are a fluid-like substance made of stuff (atoms and molecules). Dualities? Well, yet a mystery: they seem to say that forces or fields you need to describe a system are dimension dependent. Also, the fundamental degrees of freedom are entangled or mixed (perhaps we should say mapped) to one theory into another.

I will speak about some analogies:

1st. Special Relativity(SR) involves the invariance of objects under Lorentz (more generally speaking Poincaré) symmetry: X'=\Lambda X. Physical laws, electromagnetism and mechanics, should be invariant under Lorentz (Poincaré) transformations. That will be exported to strong forces and weak forces in QFT.

2nd. General Relativity(GR). Adding the equivalence principle to the picture, Einstein explained gravity as curvature of spacetime itself. His field equations for gravity can be stated into words as the motto Curvature equals Energy-Momentum, in some system of units. Thus, geometry is related to dislocations into matter and viceversa, changes in the matter-energy distribution are due to geometry or gravity. Changing our notion of geometry will change our notion of spacetime and the effect on matter-energy.

3rd. Quantum mechanics (non-relativistic). Based on the correspondence principle and the idea of matter waves, we can build up a theory in which particles and waves are related to each other. Commutation relations arise: \left[x,p\right]=i\hbar, p=h/\lambda, and the Schrödinger equation follows up H\Psi=E\Psi.

4th. Relativistic quantum mechanics, also called Quantum Field Theory(QFT). Under gauge transformations A\rightarrow A+d\varphi, wavefunctions are promoted to field operators, where particles and antiparticles are both created and destroyed, via

    \[\Psi(x)=\sum a^+u+a\overline{u}\]

Fields satisfy wave equations, F(\phi)=f(\square)\Phi=0. Vacuum is the state with no particles and no antiparticles (really this is a bit more subtle, since you can have fluctuations), and the vacuum is better defined as the maximal symmetry state, \ket{\emptyset}=\sum F+F^+.

5th. Thermodynamics. The 4 or 5 thermodynamical laws follow up from state variables like U, H, G, S, F. The absolute zero can NOT be reached. Temperature is defined in the thermodynamical equilibrium. dU=\delta(Q+W), \dot{S}\geq 0. Beyond that, S=k_B\ln\Omega.

6th. Statistical mechanics. Temperature is a measure of kinetic energy of atoms an molecules. Energy is proportional to frequency (Planck). Entropy is a measure of how many different configurations have a microscopic system.

7th. Kepler problem. The two-body problem can be reduce to a single one-body one-centre problem. It has hidden symmetries that turn it integrable. In D dimensions, the Kepler problem has a hidden O(D) (SO(D) after a simplification) symmetry. Beyond energy and angular momentum, you get a vector called Laplace-Runge-Lenz-Hamilton eccentricity vector that is also conserved.

8th. Simple Harmonic Oscillator. For a single HO, you also have a hidden symmetry U(D) in D dimensions. There is an additional symmetric tensor that is conserved.

9th. Superposition and entanglement. Quantum Mechanics taught us about the weird quantum reality: quantum entities CAN exist simultaneously in several space position at the same time (thanks to quantum superposition). Separable states are not entangled. Entangled states are non-separable. Wave functions of composite systems can sometimes be entangled AND non-separated into two subsystems.

Information is related, as I said in my second log post, to the sum of signal and noise. The information flow follows from a pattern and  a dissipative term in general. Classical geometry involves numbers (real), than can be related to matrices(orthogonal transformations or galilean boosts or space traslations). Finally, tensor are inevitable in gravity and riemannian geometry that follows up GR. This realness can be compared to complex geometry neceessary in Quantum Mechanics and QFT. Wavefunctions are generally complex valued functions, and they evolve unitarily in complex quantum mechanics. Quantum d-dimensional systems are qudits (quinfits, or quits for short, is an equivalent name for quantum field, infinite level quantum system):

(7)   \begin{align*} \vert\Psi\rangle=\vert\emptyset\rangle=c\vert\emptyset\rangle=\mbox{Void/Vacuum}\ \langle\Psi\vert\Psi\rangle=\vert c\vert^2=1 \end{align*}

(8)   \begin{align*} \vert\Psi\rangle=c_0\vert 0\rangle+c_1\vert 1\rangle=\mbox{Qubit}\\ \langle\Psi\vert\Psi\rangle=\vert c_0\vert^2+\vert c_1\vert^2=1\\ \vert\Psi\rangle=c_0\vert 0\rangle+c_1\vert 1\rangle+\cdots+c_{d-1}\vert d\rangle=\mbox{Qudit}\\ \sum_{i=0}^{d-1}\vert c_i\vert^2=1 \end{align*}

(9)   \begin{align*} \vert\Psi\rangle=\sum_{n=0}^\infty c_n\vert n\rangle=\mbox{Quits}\\ \langle\Psi\vert\Psi\rangle=\sum_{i=0}^\infty \vert c_i\vert^2=1:\mbox{Quantum fields/quits} \end{align*}

(10)   \begin{align*} \vert\Psi\rangle=\int_{-\infty}^\infty dx f(x)\vert x\rangle:\mbox{conquits/continuum quits}\\ \mbox{Quantum fields}: \int_{-\infty}^\infty \vert f(x)\vert^2 dx = 1\\ \sum_{i=0}^\infty\vert c_i\vert^2=1\\ L^2(\matcal{R}) \end{align*}

0.1. SUSY The Minimal Supersymmetry Standard Model has the following set of particles:

To go beyond the SM, BSM, and try to explain vacuum energy, the cosmological constant, the hierarchy problem, dark matter, dark energy, to unify radiation with matter, and other phenomena, long ago we created the framework of supersymmetry (SUSY). Essentially, SUSY is a mixed symmetry between space-time symmetries and internal symmetries. SUSY generators are spinorial (anticommuting c-numbers or Grassmann numbers). Ultimately, SUSY generators are bivectors or more generally speaking multivectors. The square of a SUSY transformation is a space-time traslation. Why SUSY anyway? There is another way, at least there were before the new cosmological constant problem (lambda is not zero but very close to zero). The alternative explanation of SUSY has to do with the vacuum energy. Indeed, originally, SUSY could explain why lambda was zero. Not anymore and we do neeed to break SUSY somehow. Otherwise, breaking SUSY introduces a vacuum energy into the theories. Any superalgebra (supersymmetric algebra) has generators  P_\mu, M_{\mu\nu}, Q_\alpha. In vacuum, QFT says that fields are a set of harmonic oscillators. For sping j, the vacuum energy becomes

(52)   \begin{equation*} \varepsilon_0^{(j)}=\dfrac{\hbar \omega_j}{2} \end{equation*}

where

(53)   \begin{equation*} \omega_j=\sqft{k^2+m_j^2} \end{equation*}

Vacuum energy associated to any oscillator is

(54)   \begin{equation*} E_0^{(j)}=\sum \varepsilon_0^{(j)}=\dfrac{1}{2}(-1)^{2j}(2j+1)\displaystyle{\sum_k}\hbar\sqrt{k^2+m_j^2} \end{equation*}

Taking the continuum limit, we have the vacuum master integral, the integral of cosmic energy:

(55)   \begin{equation*} E_0(j)=\dfrac{1}{2}(-1)^{2j}(2j+1)\int_0^\Lambda d^3k\sqrt{k^2+m_j^2} \end{equation*}

Develop the square root in terms of m/k up to 4th order, to get

(56)   \begin{equation*} E_0(j)=\dfrac{1}{2}(-1)^{2j}(2j+1)\int_0^\Lambda d^3k k\left[1+\dfrac{m_j^2}{2k^2}-\dfrac{1}{8}\left(\dfrac{m_j^2}{k^2}\right)^2+\cdots\right] \end{equation*}

(57)   \begin{equation*} E_0(j)=A(j)\left[a_4\Lambda^4+a_2\Lambda^2+a_{log}\log(\Lambda)+\cdots\right] \end{equation*}

If we want absence of quadratic divergences, associated to the cosmological constant, and the UV cut-off, we require

(58)   \begin{equation*} \tcboxmath{ \sum_j(-1)^{2j}(2j+1)=0} \end{equation*}

If we want absence of quadratic divergences, due to the masses of particles as quantum fields, we need

(59)   \begin{equation*} \tcboxmath{\sum_j(-1)^{2j}(2j+1)m_j^2=0} \end{equation*}

Finally, if we require that there are no logarithmic divergences, associated to the behavior to long distances and renormalization, we impose that

(60)   \begin{equation*} \tcboxmath{\sum_j(-1)^{2j}(2j+1)m_j^4=0} \end{equation*}

Those 3 sum rules are verified if, simultaneously, we have that

(61)   \begin{equation*} N_B=N_F \end{equation*}

and

(62)   \begin{equation*} M_B=M_F \end{equation*}

That is, equal number of bosons and fermions, and same masses of all the boson and fermion modes. These conditions are satisfied by SUSY, but the big issue is that the SEM is NOT supersymmetric and that the masses of the particles don’t seem to verify all the above sum rules, at least in a trivial fashion. These 3 relations, in fact, do appear in supergravity and maximal SUGRA in eleven dimensions. We do know that 11D supergravity is the low energy limit of M-theory. SUSY must be broken at some energy scale we don’t know where and why. In maximal SUGRA, at the level of 1-loop, we have indeed those 3 sum rules plus another one. In compact form, they read

(63)   \begin{equation*} \tcboxmath{\sum_{J=0}^{2}(-1)^{2J}(2J+1)(M^{2}_J)^k=0,\;\;\; k=0,1,2,3} \end{equation*}

Furthermore, these sum rules imply, according to Scherk, that there is a non zero cosmological constant in maximal SUGRA.

\textbf{Exercise}. Prove that the photon, gluon or graviton energy density can be written in the following way

    \[U_b=\dfrac{E}{V}=\dfrac{hf^4}{c^3}=\dfrac{(hf)^4}{(hc)^3}\]

In addition to that, prove that the energy density of a fermionic massive m field is given by

    \[U_f=\dfrac{E}{V}=\dfrac{m^4c^5}{h^3}=\dfrac{(mc^2)^4}{(hc)^3}\]

Compare the physical dimensions in both cases.

0.2. Extra dimensions D-dimensional gravity in newtonian form reads:

(64)   \begin{equation*} F_G=G_N(D)\dfrac{Mm}{r^{D-2}} \end{equation*}

Compatifying extra dimensions:

(65)   \begin{equation*} F_G=G_N(D)\dfrac{Mm}{L^Dr^2} \end{equation*}

and then

(66)   \begin{equation*} \tcboxmath{ G_4=\dfrac{G_N(D)}{L^D}} \end{equation*}

or with  M_P^2=\dfrac{\hbar c}{G_N},

(67)   \begin{equation*} \tcboxmath{M_P^2=V(XD)M_\star^2} \end{equation*}

Thus, weakness of gravity is explained due to dimensional dilution.
Similarly, for gauge fields:

(68)   \begin{equation*} \tcboxmath{ g^2(4d)=\dfrac{g^2(XD)}{V_X}} \end{equation*}

 

 

LOG#244. Cartan calculus.

I am going to review the powerful Cartan calculus of differential forms applied to differential geometry. In particular, I will derive the structure equations and the Bianchi identities. Yes!

Firstly, in a 2-dim manifold, we and introduce the Cartan 1-forms

(1)   \begin{align*} d\theta^1+\omega^1_{\;\;2}\wedge \theta^2=0\\ d\theta^2+\omega^2_{\;\;1}\wedge \theta^1=0 \end{align*}

The connection form reads

(2)   \begin{equation*} \omega=\begin{pmatrix} \omega^1_{\;\; 1} & \omega^1_{\;\; 2}\\ \omega^2_{\;\; 1} & \omega^2_{\;\; 2} \end{pmatrix} \end{equation*}

Now, we can introduce the so-called curvature k=\Omega^1_{\;\; 2}(e_1,e_2) and the curvature 2-form, since from d\omega^1_{\;\;2}=k\theta^1\wedge\theta^2, we will get

(3)   \begin{equation*} \Omega=\begin{pmatrix} \Omega^1_{\;\; 1} & \Omega^1_{\;\; 2}\\ \Omega^2_{\;\; 1} & \Omega^2_{\;\; 2}\end{pmatrix} \end{equation*}

The generalization to n-dimensional manifolds is quite straightforward. The torsion 1-forms \Theta are defined through the canonical 1-forms \theta via

(4)   \begin{equation*} \theta=\begin{pmatrix}\theta^1 \\ \vdots \\ \theta^n\end{pmatrix} \end{equation*}

such as

(5)   \begin{equation*} \Theta=\begin{pmatrix} \Theta^1 \\ \vdots \\ \Theta^n\end{pmatrix} \end{equation*}

With matrices \omega=\omega^i_{\;\; j} and \Omega^i_{\;\; j}, being antisymmetric n\times n, we can derive the structure equations:

(6)   \begin{equation*} \tcboxmath{\Theta=d\theta+\omega\wedge \theta} \end{equation*}

(7)   \begin{equation*} \tcboxmath{\Omega=d\omega+\omega\wedge\omega} \end{equation*}

Note that

(8)   \begin{equation*} \Theta^k=T^k_{ij}\theta^i\wedge\theta^j \end{equation*}

The connection forms satisfy

(9)   \begin{align*} \nabla_X e=e\omega(X)\\ \nabla e=e\omega \end{align*}

The gauging of the connection and curvature forms provide

(10)   \begin{align*} \overline{\omega}=a^{-1}\omega a+a^{-1}da\\ \overlin{\Omega}=a^{-1}\Omega a \end{align*}

since \overline{e}=ea, and e=\overline{e}a^{-1}, as matrices. Note, as well, the characteristic classes

(11)   \begin{equation*} \int_M e(M)=\int_M \mbox{Pf}\left(\dfrac{\Omega}{2\pi}\right)=\chi(M) \end{equation*}

is satisfied, with

(12)   \begin{equation*} \mbox{det}\left(I+\dfrac{i\Omega}{2\pi}\right)=1+c_1(E)+\cdots+c_k(E) \end{equation*}

Now, we also have the Bianchi identities

(13)   \begin{equation*} \tcboxmath{d\Theta=\Omega\wedge\theta-\omega\wedge\Theta} \end{equation*}

(14)   \begin{equation*} \tcboxmath{d\Omega=\Omega\wedge\omega-\omega\wedge\Omega} \end{equation*}

Check follows easily:

(15)   \begin{align*} d\theta=\Theta-\omega\wedge\theta\\ d\omega=\Omega-\omega\wedge\omega\\ d\Theta=\Omega\wedge\theta-\omega\wedge\Theta\\ d\Omega=\Omega\wedge\omega-\omega\wedge\Omega\\ d(\Omega^k)=\Omega^k\wedge\omega-\omega\wedge\Omega^k \end{align*}

From these equations:

    \[d\Theta=d(d\theta)+d\omega\wedge\theta-\omega\wedge d\theta\]

    \[d\Theta=(\Omega-\omega\wedge\omega)\wedge\omega-\omega\wedge(\Theta-\omega\wedge\theta)\]

and then

    \[d\Theta=\Omega\wedge\omega-\omega\wedge\Theta\;\;\; Q.E.D.\]

sinde \omega\wedge\omega\wedge\theta=0. By the other hand, we also deduce the 2nd Bianchi identity

    \[d\Omega=d^2\omega+d\omega\wedge\omega-\omega\wedge d\omega\]

Note that d(d\omega)=d^2\omega=0. Then,

    \[d\Omega=d\omega\wedge\omega-\omega\wedge d\omega=(\Omega-\omega\wedge\omega)\wedge \omega-\omega\wedge(\Omega-\omega\wedge\omega)\]

and thus

    \[d\Omega=\Omega\wedge\omega-\omega\wedge\Omega\;\; Q.E.D.\]

Remember: d\theta gives the 1st structure equation, d\omega gives the 2nd structure equation, d\Theta gives the first Bianchi identity, and d\Omega provides the 2nd Bianchi identity.

LOG#243. Elliptic trigonometry.

Jacobi elliptic functions allow to solve many physical problems. Today I will review briefly some features. Let me first highlight that the simple pendulum, Euler asymmetric top, the heavy top, the Duffing oscillator, the Seiffert spiral motion, and the Ginzburg-Landau theory of superconductivity are places where you can find Jacobi functions to arise.

Firstly, you can know there are three special Jacobi functions, named \mbox{sn}, \mbox{cn} and \mbox{dn}. The addition formulae for these 3 functions resembles those from euclidean or hyperbolic geometry:

(1)   \begin{equation*} \tcboxmath{\mbox{sn}(\alpha+\beta)=\dfrac{\mbox{sn}(\alpha)\mbox{cn}(\beta)\mbox{dn}(\beta)+\mbox{sn}(\beta)\mbox{cn}(\alpha)\mbox{dn}(\alpha)}{1-k^2\mbox{sn}^2(\alpha)\mbox{sn}^2(\beta)}} \end{equation*}

(2)   \begin{equation*} \tcboxmath{\mbox{cn}(\alpha+\beta)=\dfrac{\mbox{cn}(\alpha)\mbox{cn}(\beta)-\mbox{sn}(\alpha)\mbox{sn}(\beta)\mbox{dn}(\alpha)\mbox{dn}(\beta)}{1-k^2\mbox{sn}^2(\alpha)\mbox{sn}^2(\beta)}} \end{equation*}

(3)   \begin{equation*} \tcboxmath{\mbox{dn}(\alpha+\beta)=\dfrac{\mbox{dn}(\alpha)\mbox{dn}(\beta)-k^2\mbox{sn}(\alpha)\mbox{sn}(\beta)\mbox{cn}(\alpha)\mbox{cn}(\beta)}{1-k^2\mbox{sn}^2(\alpha)\mbox{sn}^2(\beta)}} \end{equation*}

and where k^2=m is the modulus fo the Jacobi elliptic function. To prove these addition theorems, we can take some hard paths. Let me define the derivatives:

(4)   \begin{equation*} \dfrac{d\mbox{dn(\gamma)}}{d\gamma}=\mbox{cn}(\gamma)\mbox{dn}(\gamma) \end{equation*}

(5)   \begin{equation*} \dfrac{d\mbox{cn(\gamma)}}{d\gamma}=-\mbox{sn}(\gamma)\mbox{dn}(\gamma) \end{equation*}

(6)   \begin{equation*} \dfrac{d\mbox{dn(\gamma)}}{d\gamma}=-k^2\mbox{sn}(\gamma)\mbox{cn}(\gamma) \end{equation*}

and where

(7)   \begin{align*} \mbox{sn}^2(\alpha)+\mbox{cn}^(\gamma)=1\\ k^2\mbox{sn}^2(\gamma)+\mbox{dn}^2(\gamma)=1\\ \mbox{dn}^2(\gamma)-k^2\mbox{cn}^2(\gamma)=1-k^2 \end{align*}

and where the initial conditions \mbox{sn}(0)=0, \mbox{cn}(0)=1, \mbox{dn}(0)=1 are often assumed. A more symmetric form of these equations can be deduced (exercise!):

(8)   \begin{equation*} \tcboxmath{\mbox{sn}(\alpha+\beta)=\dfrac{\mbox{sn}^2(\beta)\mbox{dn}^2(\alpha)-\mbox{sn}^2(\alpha)\mbox{dn}^2(\beta)}{\mbox{sn}(\beta)\mbox{cn}(\alpha)\mbox{dn}(\alpha)-\mbox{sn}(\alpha)\mbox{cn}(\beta)\mbox{dn}(\beta)}} \end{equation*}

(9)   \begin{equation*} \tcboxmath{\mbox{cn}(\alpha+\beta)=\dfrac{\mbox{sn}(\beta)\mbox{cn}(\beta)\mbox{dn}(\alpha)-\mbox{sn}(\alpha)\mbox{cn}(\alpha)\mbox{dn}(\beta)}{\mbox{sn}(\beta)\mbox{cn}(\alpha)\mbox{dn}(\alpha)-\mbox{sn}(\alpha)\mbox{cn}(\beta)\mbox{dn}(\beta)}} \end{equation*}

(10)   \begin{equation*} \tcboxmath{\mbox{dn}(\alpha+\beta)=\dfrac{\mbox{sn}(\beta)\mbox{cn}(\alpha)\mbox{dn}(\beta)-\mbox{sn}(\alpha)\mbox{cn}(\beta)\mbox{dn}(\alpha)}{\mbox{sn}(\beta)\mbox{cn}(\alpha)\mbox{dn}(\alpha)-\mbox{sn}(\alpha)\mbox{cn}(\beta)\mbox{dn}(\beta)}} \end{equation*}

Using that

(11)   \begin{align*} \mbox{dn}^2(\gamma)-\mbox{cn}^2(\gamma)=(1-k^2)\mbox{sn}^2(\gamma)\\ \dfrac{\mbox{dn}(\gamma)+\mbox{cn}(\gamma)}{\mbox{sn}(\gamma)}=(1-k^2)\dfrac{\mbox{sn}(\gamma)}{\mbox{dn}(\gamma)-\mbox{cn}(\gamma)} \end{align*}

you can derive the third form of the addition theorem for Jacobi elliptic functions:

(12)   \begin{equation*} \tcboxmath{\mbox{sn}(\alpha+\beta)=\dfrac{\mbox{sn}(\beta)\mbox{cn}(\alpha)\mbox{dn}(\beta)+\mbox{sn}(\alpha)\mbox{cn}(\beta)\mbox{dn}(\alpha)}{\mbox{dn}(\alpha)\mbox{dn}(\beta)+k^2\mbox{sn}(\alpha)\mbox{cn}(\alpha)\mbox{sn}(\beta)\mbox{cn}(\beta)}} \end{equation*}

(13)   \begin{equation*} \tcboxmath{\mbox{cn}(\alpha+\beta)=\dfrac{\mbox{cn}(\alpha)\mbox{dn}(\alpha)\mbox{cn}(\beta)\mbox{dn}(\beta)-(1-k^2)\mbox{sn}(\alpha)\mbox{sn}(\beta)}{\mbox{dn}(\alpha)\mbox{dn}(\beta)+k^2\mbox{sn}(\alpha)\mbox{cn}(\alpha)\mbox{sn}(\beta)\mbox{cn}(\beta)}} \end{equation*}

(14)   \begin{equation*} \tcboxmath{\mbox{dn}(\alpha+\beta)=\dfrac{(1-k^2)\mbox{sn}^2(\beta)+\mbox{cn}^2(\beta)\mbox{dn}^2(\alpha)}{\mbox{dn}(\alpha)\mbox{dn}(\beta)+k^2\mbox{sn}(\alpha)\mbox{cn}(\alpha)\mbox{sn}(\beta)\mbox{cn}(\beta)}} \end{equation*}

Finally the fourth form of the addition theorem for these functions can be found from algebra, to yield:

(15)   \begin{equation*} \tcboxmath{\mbox{sn}(\alpha+\beta)=\dfrac{\mbox{sn}(\alpha)\mbox{cn}(\alpha)\mbox{dn}(\beta)+\mbox{sn}(\beta)\mbox{cn}(\beta)\mbox{dn}(\alpha)}{\mbox{cn}(\alpha)\mbox{cn}(\beta)+\mbox{sn}(\alpha)\mbox{sn}(\beta)\mbox{dn}(\alpha)\mbox{dn}(\beta)}} \end{equation*}

(16)   \begin{equation*} \tcboxmath{\mbox{cn}(\alpha+\beta)=\dfrac{\mbox{cn}^2(\beta)\mbox{dn}^2(\alpha)-(1-k^2)\mbox{sn}^2(\alpha)}{\mbox{cn}(\alpha)\mbox{cn}(\beta)+\mbox{sn}(\alpha)\mbox{sn}(\beta)\mbox{dn}(\alpha)\mbox{dn}(\beta)}} \end{equation*}

(17)   \begin{equation*} \tcboxmath{\mbox{dn}(\alpha+\beta)=\dfrac{\mbox{cn}(\alpha)\mbox{cn}(\beta)\mbox{dn}(\alpha)\mbox{dn}(\beta)+(1-k^2)\mbox{sn}(\alpha)\mbox{sn}(\beta)}{\mbox{cn}(\alpha)\mbox{cn}(\beta)+\mbox{sn}(\alpha)\mbox{sn}(\beta)\mbox{dn}(\alpha)\mbox{dn}(\beta)}} \end{equation*}

Du Val showed long ago that these 4 forms can be derived from a language of five 4d vectors that are parallel to each other. The vectors are

(18)   \begin{equation*} V_1=\begin{pmatrix} \mbox{sn}(\alpha+\beta)\\ \mbox{cn}(\alpha+\beta)\\ \mbox{dn}(\alpha+\beta)\\ 1 \end{pmatrix} \end{equation*}

(19)   \begin{equation*} V_2=\begin{pmatrix} \mbox{sn}^2(\alpha)-\mbox{sn}^2(\beta)\\ \mbox{sn}(\alpha)\mbox{cn}(\alpha)\mbox{dn}(\beta)-\mbox{sn}(\beta)\mbox{cn}(\beta)\mbox{dn}(\alpha)\\ \mbox{sn}(\alpha)\mbox{cn}(\beta)\mbox{dn}(\alpha)-\mbox{sn}(\beta)\mbox{cn}(\alpha)\mbox{dn}(\beta)\\ \mbox{sn}(\alpha)\mbox{cn}(\beta)\mbox{dn}(\beta)-\mbox{sn}(\beta)\mbox{cn}(\alpha)\mbox{dn}(\alpha) \end{pmatrix} \end{equation*}

(20)   \begin{equation*} V_3=\begin{pmatrix} \mbox{sn}(\alpha)\mbox{cn}(\alpha)\mbox{dn}(\beta)+\mbox{sn}(\beta)\mbox{cn}(\beta)\mbox{dn}(\alpha)\\ \mbox{cn}^2(\beta)\mbox{dn}^2(\alpha)-(1-k^2)\mbox{sn}^2(\alpha)\\ \mbox{cn}(\alpha)\mbox{cn}(\beta)\mbox{dn}(\alpha)\mbox{dn}(\beta)-(1-k^2)\mbox{sn}(\alpha)\mbox{sn}(\beta)\\ \mbox{cn}(\alpha)\mbox{cn}(\beta)+\mbox{sn}(\alpha)\mbox{sn}(\beta)\mbox{dn}(\alpha)\mbox{dn}(\beta) \end{pmatrix} \end{equation*}

(21)   \begin{equation*} V_4=\begin{pmatrix} \mbox{sn}(\alpha)\mbox{cn}(\beta)\mbox{dn}(\alpha)+\mbox{sn}(\beta)\mbox{cn}(\alpha)\mbox{dn}(\beta)\\ \mbox{cn}(\alpha)\mbox{cn}(\beta)\mbox{dn}(\alpha)\mbox{dn}(\beta)-(1-k^2)\mbox{sn}(\alpha)\mbox{sn}(\beta)\\ (1-k^2)\mbox{sn}^2(\beta)+\mbox{cn}^2(\beta)\mbox{dn}^2(\alpha)\\ \mbox{dn}(\alpha)\mbox{dn}(\beta)+k^2\mbox{sn}(\alpha)\mbox{sn}(\beta)\mbox{cn}(\alpha)\mbox{cn}(\beta9 \end{pmatrix} \end{equation*}

(22)   \begin{equation*} V_5=\begin{pmatrix} \mbox{sn}(\alpha)\mbox{cn}(\beta)\mbox{dn}(\beta)+\mbox{sn}(\beta)\mbox{cn}(\alpha)\mbox{dn}(\alpha)\\ \mbox{cn}(\alpha)\mbox{cn}(\beta)-\mbox{sn}(\alpha)\mbox{sn}(\beta)\mbox{dn}(\alpha)\mbox{dn}(\beta)\\ \mbox{dn}(\alpha)\mbox{dn}(\beta)-k^2\mbox{sn}(\alpha)\mbox{sn}(\beta)\mbox{cn}(\alpha)\mbox{cn}(\beta)\\ 1-k^2\mbox{sn}^2(\alpha)\mbox{sn}^2(\beta) \end{pmatrix} \end{equation*}

Du Val also grouped the vectors V_2 to V_5 in a compact matrix \mathcal{A} invented by Glaisher in 1881:

    \[ \begin{pmatrix} s_1^2-s_2^2 & s_1c_1d_2+s_2c_2d_1 & s_1c_2d_1+s_2c_1d_2 & s_1c_2d_2+s_2c_1d_1\\ s_1c_1d_2-s_2c_2d_1 & c_2^2d_1^2-(1-k^2)s_1^2 & c_1c_2d_1d_2-(1-k^2)s_1s_2 & c_1c_2-s_1s_2d_1d_2\\ s_1c_2d_1-s_2c_1d_2 & c_1c_2d_1d_2+(1-k^2)s_1s_2 & (1-k^2)s_2^2+c_2^2d_1^2 & d_1d_2-k^2s_1s_2c_1c_2\\ s_1c_2d_2-s_2c_1d_1 & c_1c_2+s_1s_2d_1d_2 & d_1d_2+k^2s_1s_2c_1c_2 & 1-k^2s_1^2s_2^2 \end{pmatrix} \]

This matrix has a very interesting symmetry \mathcal{A}^T(\alpha,\beta)=\mathcal{A}(\alpha,-\beta). You can also define the antisymmetric tensor F_{jk}=a_jb_k-a_kb_j from any vector pair a_i, b_j. In fact, you can prove that the tensor

(23)   \begin{equation*} F_{kl}=m\varepsilon_{klpq}\mathcal{A}_{pq} \end{equation*}

where m equals to 1, k^2, 1-k^2, and the \varepsilon tensor is the Levi-Civita tensor, holds as identity between the matrix \mathcal{A}, and the division in two couples the quartets of vectors.  It rocks!

How, a refresher of classical mechanics. The first order hamiltonian Mechanics reads

(24)   \begin{equation*} \begin{pmatrix} \dot{q}\\ \dot{p} \end{pmatrix}=\begin{pmatrix} 0 & +1\\ -1 & 0\end{pmatrix}\begin{pmatrix}\dfrac{\partial H}{\partial q}\\ \dfrac{\partial H}{\partial p}\end{pmatrix} \end{equation*}

From these equations, you get the celebrated Hamilton equations

(25)   \begin{equation*} \dot{p}^i=-\dfrac{\partial H}{\partial q_i} \end{equation*}

(26)   \begin{equation*} \dot{q}^i=+\dfrac{\partial H}{\partial p_i} \end{equation*}

Strikingly similar to F_i=-\nabla_i \varphi, or \dot{p}^i=-\nabla_i U. First order lagrangian theory provides

(27)   \begin{equation*} \dfrac{\partial L}{\partial q}-\dfrac{d}{dt}\left(\dfrac{\partial L}{\partial \dot q}\right)=0 \end{equation*}

Also, it mimics classical newtonian mechanics if you allow

(28)   \begin{equation*} \dfrac{\partial L}{\partial q}=-\dfrac{d}{dt}p \end{equation*}

There is a relation between the lagrangian L and the hamiltonian function H via Legendre transforamations:

(29)   \begin{equation*} H(q,p,t)=\sum_i p_i\dot{q}_i-L \end{equation*}

where the generalized momentum is

(30)   \begin{equation*} p_i=\dfrac{\partial L}{\partial \dot{q}_i} \end{equation*}

There is also routhian mechanics, by Routh, where you have (n+s) degrees of freedom chosen to be n q_i and s p_j, such as

(31)   \begin{align*} R=R(q,\zeta, p,\dot{\zeta},t)=p_i\dot{q}_i-L(q,\zeta,p,\dot{\zeta},t)\\ \dot{q}_i=\dfrac{\partial R}{\partial p_i}\\ \dot{p}_i=-\dfrac{\partial R}{\partial q_i}\\ \dfrac{d}{dt}\left(\dfrac{\partial R}{\partial \dot{\zeta}_j}\right)=\dfrac{\partial R}{\partial \zeta_j} \end{align*}

and where there are 2n routhian-ham-equations, and s routhian-lag-equations. The routhian energy reads off easily

(32)   \begin{equation*} E_R=R-\sum_j^s\dot{\zeta}_j\dfrac{\partial R}{\partial\dot{\zeta}_j} \end{equation*}

and

(33)   \begin{equation*} \dfrac{\partial R}{\partial t}=\dfrac{d}{dt}\left(R-\sum_j^s \dot{\zeta}_j\dfrac{\partial R}{\partial\dot{\zeta}_j}\right) \end{equation*}

Finally, the mysterious Nambu mechanics. Yoichiru Nambu, trying to generalize quantum mechanics and Poisson brackets, introduced the triplet mechanics (and by generalization the N-tuplet) with two hamiltonians H,G as follows. For a single N=3 (triplets):

(34)   \begin{equation*} \dot{f}=\dfrac{\partial(f,G,H)}{\partial(x,y,z)}+\dfrac{\partial f }{\partial t} \end{equation*}

and for several triplets

(35)   \begin{equation*} \dot{f}=\displaystyle{\sum_{a=1}^N\dfrac{\partial(f,G,H)}{\partial(x_a,y_a,z_a)}+\dfrac{\partial f }{\partial t}} \end{equation*}

 and where f=f(r_1,r_2,\cdots,r_N,t). Sometimes it is written as \dot{f}=\nabla G\times \nabla H. In the case of N-n-plets, you have

(36)   \begin{equation*} \tcboxmath{\dfrac{df}{dt}=\dot{f}=\{f,H_1,H_2,\ldots,H_{N-1}\}} \end{equation*}

and also you get an invariant form for the triplet Nambu mechanics

(37)   \begin{equation*} \omega_3=dx_1^1\wedge dx_1^2\wedge dx_1^3+\cdots+dx_N^1\wedge dx_N^2\wedge x_N^3 \end{equation*}

This 3-form is the 3-plet analogue of the symplectic 2-form

(38)   \begin{equation*} \omega_2=\displaystyle{\sum_i dq_i\wedge dp_i} \end{equation*}

The analogue for N-n-plets can be easily derived:

(39)   \begin{equation*} \tcboxmath{\omega_n=\displaystyle{\sum_j dx^1_j\wedge dx^2_j\wedge\cdots\wedge dx^n_j}} \end{equation*}

 

The quantization of Nambu mechanics is a mystery, not to say what is its meaning or main applications. However, Nambu dynamics provides useful ways to solve some hard problems, turning them into superintegrable systems.

See you in other blog post!

LOG#242. Hyperbolic magic.

Beta-gamma fusion! Live dimension!

Do you like magic? Mathemagic and hyperbolic magic today. Master of magic creates an “illusion”. In special relativity, you can simplify calculations using hyperbolic trigonometry!

 

(1)   \begin{equation*} E=Mc^2=m\gamma c^2=\dfrac{mc^2}{\sqrt{1-\beta^2}} \end{equation*}

(2)   \begin{equation*} p=Mv=m\gamma v \end{equation*}

are common relativistc equation. Introduce now:

(3)   \begin{equation*} \tcboxmath{\beta=\dfrac{v}{c}=\tanh\varphi}\;\; 0\leq \beta<1,  -\infty<\varphi<\infty \end{equation*}

as the rapidity. Then:

(4)   \begin{equation*}  \gamma=\dfrac{E}{mc^2}=\dfrac{1}{\sqrt{1-\beta^2}}=\dfrac{1}{1-\tanh^2\varphi}}=\sqrt\dfrac{\cosh^2}{\cosh^2-\sinh^2}}=\cosh\varphi \end{equation*}

so

(5)   \begin{equation*} \tcboxmath{\gamma=\cosh\varphi}\;\;\; \gamma\geq 1, -\infty<\varphi<\infty \end{equation*}

Similarly, you get that

(6)   \begin{equation*} p=m\gamma v=mc\beta\gamma=mc\tanh\vaphi\cosh\varphi=mc\sinh\varphi \end{equation*}

and thus

(7)   \begin{equation*} \tcboxmath{p=mc\sinh\varphi}\;\;\; -\infty<\varphi<\infty \end{equation*}

Also, you can write

(8)   \begin{equation*} \tcboxmath{\tanh\varphi=\dfrac{pc}{E}}\;\; 0\leq \beta<1,  -\infty<\varphi<\infty \end{equation*}

(9)   \begin{equation*} \tcboxmath{\dfrac{vE}{mc^3}=\beta\gamma}\;\;\; 0\leq \beta<1,  -\infty<\varphi<\infty, \gamma\geq 1 \end{equation*}

and

(10)   \begin{equation*} \tcboxmath{\beta=\dfrac{pc}{E}} \;\; \;\; 0\leq \beta<1,  -\infty<\varphi<\infty \end{equation*}

The above equations can be inverted, and it yields

(11)   \begin{equation*} \tcboxmath{\beta=\dfrac{v}{c}=\tanh\varphi=\tanh\left(\sinh^{-1}\left(\dfrac{p}{mc}\right)\right)} \end{equation*}

(12)   \begin{equation*} \tcboxmath{\beta=\dfrac{v}{c}=\tanh\left(\cosh^{-1}\left(\gamma\right)\right)=\tanh\left(\cosh^{-1}\left(\dfrac{E}{mc^2}\right)\right)=\sqrt{1-\left(\dfrac{mc^2}{E}\right)^2}} \end{equation*}

(13)   \begin{equation*} \tcboxmath{\gamma=\dfrac{1}{\sqrt{1-\beta^2}}=\cosh\left(\tanh^{-1}\left(\beta\right)\right)} \end{equation*}

(14)   \begin{equation*} \tcboxmath{\gamma=\dfrac{1}{\sqrt{1-\beta^2}}=\cosh\left(\tanh^{-1}\left(\dfrac{pc}{E}\right)\right)} \end{equation*}

(15)   \begin{equation*} \tcboxmath{\varphi=\tanh^{-1}\left(\beta\right)=\tanh^{-1}\left(\dfrac{pc}{E}\right)} \end{equation*}

Hyperbolic functions also simply the Lorentz transformations into a more symmetric form! Consider the spacetime interval:

(16)   \begin{equation*} s^2=x^\mu x_\mu=x^2-(ct)^2=x^2+(ict)^2 \end{equation*}

and a rotation matrix

(17)   \begin{equation*} R(\theta)^T=R^{-1}=\begin{pmatrix}\cos \theta & \sin\theta\\ -\sin\theta & \cos\theta\end{pmatrix} \end{equation*}

Now, make a rotation with imaginary angle \varphi=i\theta and apply it to the vector X=(x,ict):

(18)   \begin{equation*} \begin{pmatrix} x'\\ ict'\end{pmatrix} =\begin{pmatrix} \cos i\theta & \sin i\theta\\ -\sin i\theta & \cos i\theta\end{pmatrix}\begin{pmatrix} x\\ ict\end{pmatrix} \end{equation*}

Then

(19)   \begin{equation*} \begin{pmatrix} x'\\ ict'\end{pmatrix} =\begin{pmatrix} \cosh\theta & i\sinh \theta\\ -i\sinh \theta & \cosh\theta\end{pmatrix}\begin{pmatrix} x\\ ict\end{pmatrix} \end{equation*}

(20)   \begin{equation*} \begin{pmatrix} x'\\ ict'\end{pmatrix}=\begin{pmatrix}\beta & i\beta\gamma\\ -i\beta\gamma & \beta\end{pmatrix}\begin{pmatrix} x\\ ict\end{pmatrix}=\begin{pmatrix} \beta x -\beta\gamma ct\\ i\left(-\beta\gamma x+\beta ct\right)\end{pmatrix} \end{equation*}

and thus

(21)   \begin{equation*} \begin{pmatrix}x'\\ ct'\end{pmatrix}=\begin{pmatrix} \beta & -\beta\gamma\\ -\beta\gamma & \beta\end{pmatrix}\begin{pmatrix}x\\ ct\end{pmatrix} \end{equation*}

That is the Lorentz transformation! A Lorentz transformation is just a rotation matrix of an imaginary angle with imaginary time! But you can give up imaginary numbers using hyperbolic functions! Indeed, L(\varphi)=L^T for Lorentz transformations, while R(\theta)=(R^{-1})^T for rotations.

Finally, something about particle spin and “rotations”, secretly related to Lorentz transformations of spinors. Spin zero particles are the same irrespectively how you see them, so if you turn them 0 degrees (radians), spin zero particles remain invariant. Vector spin one particles like A_\mu are the same if you turn them 360^\circ=2\pi \;rad. Tensor spin two particles like h_{\mu\nu} are the same if you rotate them about 180^\circ=\pi \; rad. Now, the weird stuff. Electrons and spin one-half fermions are the same only if you rotate them…720^\circ=4\pi\;rad!!! They see a largest world than the one we observe! The hypothetical gravitino field remains invariant only when you twist it about 240^\circ=4\pi/3\; rad. You can also iterate the argument for higher spin particles. Even you could consider the case with infinite (continous) spin.

Remark(I): in natural units with c=\hbar=k_B=1 you can prove that

    \[1kg=6.61\cdot 10^{35}GeV\]

    \[1K=8.617\cdot 10^{-14}GeV\]

    \[1m=8.07\cdot 10^{14}GeV^{-1}\]

Remark(II): in natural units with G_N=c=1 you also get

    \[1kg=7.42\cdot 10^{-28}m\]

    \[1kg=1.67 ZeV=8.46\cdot 10^{27}m^{-1}\]

Now, perhaps you have time for a little BIG RIP in de Sitter spacetime with phantom energy \omega<-1

(22)   \begin{equation*} \tcboxmath{T_{BRip}=-\dfrac{2}{3(1+\omega)H_0\sqrt{1-\Omega_{m,0}}}} \end{equation*}

Perhaps, now you face the proton decay crisis of your life due to pandemic, any time?

Challenges for you:

Challenge 1. Some recent reviews of proton decay in higher dimensional models derive the estimate

    \[\tau_{proton}\sim\left(\dfrac{M_P}{M_{proton}}\right)^D\dfrac{1}{M_{proton}}$$ For $D=4$, it yields about $\tau\sim 10^{52}s\sim 10^{45}yrs\]

However, Hawking derived a similar but not identical estimate

    \[\tau_{proton}\sim\left(\dfrac{M_P}{M_{proton}}\right)^8\dfrac{1}{M_{proton}}\sim 10^{120}yrs\]

using processes with virtual black holes and spacetime foam. I want to understand this formulae better, so I need to understand the origin of the powers and the absence (or presence if generalized GUT/TOE arises) of gauge couplings. In short:

1) What is the reason of the D-dependence in the first formula and the 8th power in the second formula?

2) Should the proton decay depend as well and in which conditions of gauge (or GUT,TOE) generalized couplings too?

Challenge 2. 

Derive the formulae

    \[\dot{f}=\dfrac{96}{5}\pi^{8/3}\mathcal{M}^{5/3}f_0^{11/3}\]

    \[A=\dfrac{2\mathcal{M}^{5/3}\pi^{2/3}f_0^{2/3}}{D_L}\]

Finally, string theory…To crush you even more…Gravitational constant is just derived from the string coupling and the dilaton field in superstring theories. The recipe is

(23)   \begin{equation*} \langle e^\phi\rangle =e^{\phi_\infty} \end{equation*}

such as

(24)   \begin{equation*} g_s(d)=\langle e^\phi\rangle_0 \end{equation*}

Define the string tension \alpha'=L_s^2, and the string lenth L_s=\sqrt{\alpha'}. Then, in a 10d Universe

(25)   \begin{equation*} \tcboxmath{G_N(10d)=8\pi^6g_s^2\left(\alpha')^4=8\pi^6 g_s^2 L_s^8} \end{equation*}

Furthermore, with n compactified dimensions, you get

(26)   \begin{equation*} \tcboxmath{G_N(10d)=G_N(n)V_{10-n}} \end{equation*}

and

(27)   \begin{equation*} \tcboxmath{g_s^2(10d)=\dfrac{V_{10-d}}{(2\pi L_s)^{10-n}(g_s^{(n)})^2}} \end{equation*}

In summary, you can obtain

(28)   \begin{equation*} \tcboxmath{\dfrac{g_s^2(2\pi L_s)^{10-n}}{16\pi G_N(10d)}=\dfrac{g_s^2(n)}{16\pi G_N(n)}} \end{equation*}

Have I punched hard?

See you in another blog post dimension!

LOG#241. Flatland & Fracland.

Flatland is a known popular story and book. I am going to review the Bohr model in Flatland and, then, I am going to strange fractional (or fractal) dimensions, i.e., we are going to travel to Fracland via Bohrlogy today as well.

Case 1. Electric flatland and Bohrlogy.

(1)   \begin{equation*} F_c(2d)=K_c(2d)\dfrac{e^2}{r} \end{equation*}

Suppose that

(2)   \begin{equation*} E_p(2d)=K_c(2d)e^2\ln\left(\dfrac{r}{a_0}\right) \end{equation*}

Then, we have

(3)   \begin{equation*} m\dfrac{v^2}{r}=K_c\dfrac{e^2}{r} \end{equation*}

and thus

(4)   \begin{equation*} v=\sqrt{\dfrac{K_c}{m}}e \end{equation*}

Moreover, imposing Bohr quantization rule L=pr=mvr=n\hbar, then you get

(5)   \begin{equation*} r=\dfrac{n\hbar}{mv} \end{equation*}

(6)   \begin{equation*} \tcboxmath{r_n=na_0=n\dfrac{\hbar}{e\sqrt{mK_c}}} \end{equation*}

Total energy becomes

(7)   \begin{equation*} E=E_c+E_0=E_m=\dfrac{1}{2}mv^2+K_ce^2\ln\left(\dfrac{r_n}{a_0}\right)} \end{equation*}

(8)   \begin{equation*} \tcboxmath{E_m=E_0\left(\dfrac{1}{2}+\ln n\right)-E_0\ln\left(\dfrac{\hbar}{e\sqrt{mK_c}}\right)} \end{equation*}

where E_0=K_ce^2.

Case 2. Gravitational flatland and Borhlogy.

(9)   \begin{equation*} F_N(2d)=G_N(2d)\dfrac{e^2}{r} \end{equation*}

Suppose that

(10)   \begin{equation*} E_p(2d)=G_N(2d)m^2\ln\left(\dfrac{r}{a_0}\right) \end{equation*}

Then, we have

(11)   \begin{equation*} m\dfrac{v^2}{r}=G_N(2d)\dfrac{m^2}{r} \end{equation*}

and thus

(12)   \begin{equation*} v=\sqrt{G_Nm} \end{equation*}

Moreover, imposing Bohr quantization rule L=pr=mvr=n\hbar, then you get

(13)   \begin{equation*} r=\dfrac{n\hbar}{mv} \end{equation*}

(14)   \begin{equation*} \tcboxmath{r_n=na_0=n\dfrac{\hbar}{m\sqrt{mG_N}}} \end{equation*}

Total energy becomes (up to an additive constant)

(15)   \begin{equation*} E=E_c+E_0=E_m=\dfrac{1}{2}mv^2+G_Nm^2\ln\left(\dfrac{r}{a_0}\right) \end{equation*}

(16)   \begin{equation*} \tcboxmath{E_m=E_0\left(\dfrac{1}{2}+\ln n\right)-E_0\ln\left(\dfrac{\hbar}{m\sqrt{G_Nm}}\right)} \end{equation*}

where E_0=G_Nm^2.

Exercise: Gravatoms. 

Suppose a parallel Universe a where electrons were neutral particles and no electric charges existed. In such a Universe, 2 electrons or any electron and a proton would form a gravitational bound state called gravatom (gravitational atom for short). The force potencial would be V_g=Gm^2/r. And we could suppose that electron mass and G_N are the same as those in our Universe.

a) Calculate the ratio between the gravitational potential and the electric potential in our universe. Comment the results. 1 point.

b) Compute the analogue of Bohr radius in the gravatom. Comment the result. 1 point.

c) Compute the analogue of Rydberg constant for the gravatom. Is it large or small compared with the usual Rydberg constant? 1 point.

d) Compute the period of the electron in the lowest energy level. Compare it with the age of our Universe. 1 point.

e) Imagine a parallel Universe B, where the electrons were indeed supermassive. Higher the electron mass is, lower the size of the gravatom. If big enough, the size of the gravatom is smaller than the Compton wavelength of a free electron, measuring the size of the irreducible wave function of the electron. In that limit, there is no free electron but a bound state of a black hole. Compute the critical mass for the cross-over. Compare that scale to a human lifetime. 1 point.

Case 3. Welcome to Fracland, land of fractional Bohrlogy. 

3A. Fractional H-atom.

Consider the potential energy

(17)   \begin{equation*} U(r)=-\dfrac{Ze^2}{r} \end{equation*}

and the hamiltonian

(18)   \begin{equation*} H_\alpha=D_\alpha\left(-\hbar^2\Delta\right)^{\alpha/2}=D_\alpha\left(-\hbar \Delta^{1/2}\right)^\alpha \end{equation*}

where, in principle, we allow only 1<\alpha\leq 2, but a suitable analytic continuation could be feasible somehow. Then, \alpha\overline{E_k}=-\overline{U} and pr_n=n\hbar provide

(19)   \begin{equation*} \omega(n_1\rightarrow n_2)=\dfrac{E_2-E_1}{\hbar} \end{equation*}

such as

(20)   \begin{equation*} \alpha D_\alpha\left(\dfrac{n\hbar}{r_n}\right)^\alpha=\dfrac{Ze^2}{r_n} \end{equation*}

And, finally, you get the radii and energy levels for the fractional H-atom as follows

(21)   \begin{equation*} \tcboxmath{r_n=a_0 n^{\frac{\alpha}{\alpha-1}}\;\;\;  a_0=\left(\dfrac{\alpha D_\alpha \hbar^\alpha}{Ze^2K_C}\right)^{\frac{1}{\alpha-1}}} \end{equation*}

(22)   \begin{equation*} \tcboxmath{E_n= (1-\alpha)\overline{E_k}}\;\;\; \tcboxmath{E_n=\left(1-\alpha\right)E_0 n^{-\frac{\alpha}{\alpha-1}}\right)} \end{equation*}

(23)   \begin{equation*} \tcboxmath{\omega_n(\alpha)=\dfrac{(1-\alpha)E_0}{\hbar}\left(\dfrac{1}{n_1^{\frac{\alpha}{\alpha-1}}}-\dfrac{1}{n_2^{\frac{\alpha}{\alpha-1}}}\right)} \end{equation*}

and where now

(24)   \begin{equation*} \tcboxmath{E_0=\left(\dfrac{Z^{\alpha}(K_Ce^2)^\alpha}{\alpha^\alpha D_\alpha \hbar^\alpha\right)^{\frac{1}{\alpha-1}\right)}}} \end{equation*}

Note that E_k=D_\alpha p^\alpha. Virial theorem implies \overline{E_k}=\overline{U}(n/2) if U=\alpha r^n.

3B. Fractional harmonic oscillator (in 3d).

Consider now

(25)   \begin{equation*} H(\alpha,\beta)=D_\alpha(-\hbar^2\Delta)^{\alpha/2}+q^2 r^\beta \end{equation*}

In the case \alpha=\beta you get

(26)   \begin{equation*} H_\alpha=D_\alpha(-\hbar^2\Delta)^{\alpha/2}+q^2r^\alpha \end{equation*}

For a single d.o.f., i.e., if D=1, you can write

(27)   \begin{equation*} E=D_\alpha p^\alpha+q^2x^\beta \end{equation*}

The energy levels can be calculated

(28)   \begin{equation*} \tcboxmath{E_n=\dfrac{\pi \hbar \beta D_\alpha^{1/\alpha} q^{2/\beta}}{2B\left(\frac{1}{\beta},\frac{1}{\alpha}+1\right)}\left(n+\dfrac{1}{2}\right)^{\frac{\alpha\beta}{\alpha+\beta}}} \end{equation*}

and where B(x,y) is the beta function. Remarkably, only the standard QM simple HO has equidistant energy spectrum!

3C. Fractional infinite potential well.

Let the potential be

(29)   \begin{equation*} V=\begin{cases}V(x)=+\infty, x<-a\\ 0,-a\leq x\leq a\\ V(x)=+\infty, x>a\end{cases} \end{equation*}

Then, the energy spectrum becomes

(30)   \begin{equation*} \tcboxmath{E_n=D_\alpha\left(\dfrac{\pi\hbar}{\alpha}\right)^\alpha n^\alpha} \end{equation*}

The ground state energy is

(31)   \begin{equation*} \tcboxmath{E_0=D_\alpha\left(\dfrac{\pi\hbar}{2\alpha}\right)^\alpha} \end{equation*}

3D. Delta potential well.

Consider 1<\alpha\leq 2, and the \delta-function potential V(x)=-\gamma\delta(x), with \gamma>0. The energy spectrum is, for the bound state,

(32)   \begin{equation*} \tcboxmath{E=-\left[\dfrac{\gamma B\left(\frac{1}{\alpha},1-\frac{1}{\alpha}\right)}{\pi \hbar \alpha  D_\alpha^{1/\alpha}}\right]^{\frac{\alpha}{\alpha-1}}} \end{equation*}

3E. Fractional linear potential.

Consider the potential

(33)   \begin{equation*} V(x)=\begin{cases} Fx, x\geq 0, F>0\\ \infty, x<0\end{cases} \end{equation*}

The energy spectrum will be

(34)   \begin{equation*} \tcboxmath{E_n=\lambda_n F\hbar \left(\dfrac{ D_\alpha}{\left(\alpha + 1\right) F\hbar}\right)^{-\frac{1}{\alpha + 1}}} \end{equation*}

and where \lambda_n are solutions to certain trascendental equation, with 1<\alpha\leq 2.

Hidden connection with the riemannium. Some time ago, I posted in physics stack exchange this question https://physics.stackexchange.com/questions/60991/mysterious-spectra

Thus, fractional H-atoms and oscillators, with care enough, can also be seen as riemannium-like.

See you in another blog post dimension!

LOG#240. (Super)Dimensions.

 

Hi, everyone! Sorry for the delay! I have returned. Even in this weird pandemic world…I have to survive. Before the blog post today, some news:

  1. Changes are coming in this blog. Whenever I post the special 250th post, the format and maybe the framework will change. I am planning to post directly in .pdf format, much like a true research paper.
  2. I survive, even if you don’t know it, as High School teacher. Not my higher dream, but it pays the bills. If you want to help me, consider a donation.
  3. Beyond the donation, I am aiming to offer some extra stuff in this blog: free notes (and links) to my students or readers, PLUS, a customized version of them that of course you should pay me for the effort to do. It would me help me to sustain the posting or even managing independence of my other job that carries from me time to post more often.
  4. I will offer a service of science consulting to writers, movie makers, and other artists who wish for a more detailed scientific oriented guide.
  5. I will send the full bunch of my 250 blog posts as soon as possible, with customized versions if paid. 1 euro/dollar per blog post will be my price. The customized election of blog posts will be negotiated later, but maybe I will offer 25 blog posts packs as well, plus the edition cost. Expensive? Well, note that I had to put lot of effort to build this site alone. I need to increase income in these crisis times. You will be able to find the posts here for free anyway, but if you want them edited, you can help me further. If I could I would leave my current job since I am unhappy with it and with COVID19 is a risk to be a teacher (if presence is required into class of course).

Topic today are dimensions. Dimension is a curious concept. Fractal geometry has changed what we used to consider about dimensions, since fractals can have non-integer dimensions. Even, from certain viewpoint, you can also consider negative dimensions, complex dimensions and higher versions of it. With fractals, you have several generalized dimensions:

(1)   \begin{equation*} \tcboxmath{D_{box}=D_0=-\lim_{\varepsilon\rightarrow 0}\left(\dfrac{\log N(\varepsilon)}{\log\dfrac{1}{\varepsilon}}\right)} \end{equation*}

Next, information dimension

(2)   \begin{equation*} \tcboxmath{ D_1=\lim_{\varepsilon\rightarrow 0}\left[-\dfrac{\log p_\varepsilon}{\log\dfrac{1}{\varepsilon}}\right]} \end{equation*}

Generalized Renyi dimensions are next

(3)   \begin{equation*} \tcboxmath{D_\alpha=\lim_{\varepsilon\rightarrow 0}\dfrac{\dfrac{1}{\alpha-1}\log \sum p_i^\alpha}{\log\varepsilon}} \end{equation*}

Now, we can also define the Higuchi dimension:

(4)   \begin{equation*} \tcboxmath{ D_h=\dfrac{d \log (L(X))}{d\log (k)}} \end{equation*}

Of course, you also have the celebrated Hausdorff dimension

(5)   \begin{equation*} \tcboxmath{\mbox{dim}_h(X)=\mbox{inf}\left{d\geq 0: C_H^d(X)=0\right}} \end{equation*}

In manifold theories, you can also define the codimension:

(6)   \begin{equation*} \tcboxmath{\mbox{codim}(W)=\mbox{dim}(V)-\mbox{dim}(W)=\mbox{dim}\left(\dfrac{V}{W}\right)} \end{equation*}

if W is a submanifold W\subseteq V. Also, if N is a submanifold in M, you also have

(7)   \begin{equation*} \mbox{codim}(N)=\mbox{dim}(M)-\mbox{dim}(N) \end{equation*}

such as

(8)   \begin{equation*} \tcboxmath{\mbox{codim}(W)=\mbox{codim}\left(\dfrac{V}{W}\right)=\mbox{dim}\left(\mbox{coker}(W\rightarrow V)\right)\right)} \end{equation*}

Finally, superdimensions! In superspace (I will not go into superhyperspaces today!), you have local coordinates

(9)   \begin{equation*} \tcboxmath{X=(x,\Xi)=(x^\mu, \Xi^\alpha)=(x^\mu, \theta,\overline{\theta})} \end{equation*}

where \mu=0,1,2,\ldots, n-1 and \alpha=1,2,\ldots,\nu. Generally, \nu=2m, so the superdimension is the pair (n,\nu)=(n,2m) in general. In C-spaces (Clifford spaces) you have the expansion in local coordinates:

(10)   \begin{equation*} \tcboxmath{X=X^A\gamma_A=\left(\tau, X^\mu,X^{\mu_1\mu_2},\ldots,X^{\mu_1\dots\mu_D}\right)} \end{equation*}

and if you go into C-superspaces, you will also get

(11)   \begin{equation*} \tcboxmath{Z=Z^W\Gamma_W=(X^A; \Xi^\Omega)=\left(\tau,X^\mu,X^{\mu_1\mu_2},\ldots,X^{\mu_1\dots\mu_D}; \theta, \theta^\alpha,\theta^{\alpha_1\alpha_2},\ldots,\theta^{\alpha_1\ldots\alpha_m}\right)} \end{equation*}

With superdimensions, you can also have superdimensional gauge fields and supermetric fields, at least in principle (in practice, it is hard to build up interacting field theories with higher spins at current time). For supergauge fields, you get

(12)   \begin{equation*} \tcboxmath{A=A^W\Gamma_W=(A^Z; \Xi^\Omega)=\left(\tau,A^\mu,A^{\mu_1\mu_2},\ldots,A^{\mu_1\dots\mu_D}; \Theta, \Theta^\alpha,\Theta^{\alpha_1\alpha_2},\ldots,\Theta^{\alpha_1\ldots\alpha_m}\right)} \end{equation*}

The C-space metric reads

(13)   \begin{equation*} \tcboxmath{ds^2=dX_AdX^A=d\tau^2+dx^\mu dx_\mu+dx^{\mu_1\mu_2}dx_{\mu_1\mu_2}+\cdots dx^{\mu_1\cdots \mu_D}dx_{\mu_1\cdots\mu_D}} \end{equation*}

and more elaborated formula for C-supermetrics and C-superhypermetric could be done (I am not done with them yet…). The mixed type of gauge fields in C-superspaces (even C-superhyperspaces) is yet hard to even myself. Work for another day!

Definition 1 (UR or eTHOR conjecture).

There is an unknown extended theory of relativity (eTHOR), ultimate relativity (UR), and it provides transformation rules between any type of field (scalar, spinorial,vector, tensor, vector spinor, tensor spinor, and general multitensor/multiform multispinor) and their full set of symmetries. Consequences of the conjecture:

  • UR involves coherent theories of higher spins AND higher derivatives, such as there is a full set of limits/bound on the values of the n-th derivatives, even those being negatives (integrals!).

  • UR involves a generalized and extended version of relativity, quantum theory and the equivalence principle.

  • UR provides the limits of the ultimate knowledge in the (Multi)(Uni)verse, even beyond the Planck scale.

  • UR will clarify the origin of space-time, fields, quantum mechanics, QFT and the wave function collapse.

  • UR will produce an explanation of M-theory and superstring theory, the theory of (D)-p-branes and the final fate of the space-time singularities, black hole information and black hole evaporation, and the whole Universe.

See you in other blog post!

P.S.: Please, if you want to help me, I wish you can either donate or buy my stuff in the near future. My shop will be launching soon,…In September I wish…

LOG#239. Higgspecial.

The 2012 found Higgs particle is special. The next generations of physicists and scientists, will likely build larger machines or colliders to study it precisely. The question is, of course, where is new physics, i.e., where is the new energy physics scale? Is it the Planck scale? Is it lower than 10^{28}eV?

What is a particle?

Particles are field excitations. Fields satisfy wave equations. Thus particles, as representations of fields, also verify field or wave equations. Fields and particles have also symmetries. They are invariant under certain group transformations. There are several types of symmetry transformations:

  1. Spacetime symmetries or spacetime invariance. They include: translations, rotations, boosts (pure Lorentz transformations) and the full three type combination. The homogeneous Lorentz group does not include translations. The inhomogeneous Lorentz group includes translations and it is called Poincaré group. Generally speaking, spacetime symmetries are local spacetime transformations only.
  2. Internal (gauge) symmetries. These transformations are transformations of the fields up to a phase factor at the same spacetime-point. They can be global and local.
  3. Supersymmetry. Transformations relating different statistics particles, i.e., relating bosons and fermions. It can be extended to higher spin under the names of hypersymmetry and hypersupersymmetry. It can also be extended to N-graded supermanifolds.

We say a transformation is global when the group parameter does not depend on the base space (generally spacetime). We say a transformation is local when it depends of functions defined on the base space.

Quantum mechanics is just a theory relating “numbers” to each other. Particles or fields are defined as functions on the spacetime momentum (continuum in general) and certain discrete set of numbers (quantum numbers all of them)

    \[\ket{p^\mu,\sigma}\]

and thus

    \[U(\Lambda)\ket{p,\sigma}=D_{\sigma\sigma'}\ket{\Lambda p,\sigma'}\]

represents quantum particles/waves as certain unitary representations of the Poincaré group (spacetime)! Superfields generalize this thing. Diffferent particles or fields are certain unitary representations of the superPoincaré group (superspacetime)! Equivalently, particles are invariant states under group or supergroup transformations. Particle physics is the study of fundamental laws of Nature governed by (yet hidden and mysterious) the fusion of quantum mechanics rules and spacetime rules.

From the 17th century to 20th century: we had a march of reductionism and symmetries. Whatever the ultimate theory is, relativity plus QM (Quantum Mechanics) are got as approximations at low or relatively low (100GeV) energies. Reductionism works: massless particles interact as an Greek Y (upsilon) vertices.

    \[\chemfig{A-B(-[:-30]C)-[:30]D}\]

Massless particles can be easily described by twistors, certain bispinors (couples of spinors):

    \[p_{\alpha\dot{\alpha}}=\begin{pmatrix} p_0+p_3 & p_1-ip_2\\ p_1+ip_2 & p_0-p_3\end{pmatrix}=\lambda_{\alpha}\overline{\lambda}_{\dot{\alpha}}\]

Indeed, interactions are believed to be effectively described by parallel twistor-like variables \lambda_A\propto \lambda_B\propto \lambda_C and \overline{\lambda_A}\propto\overline{ \lambda_B}\propto \overline{\lambda_C}. The Poincaré group completely fixes the way in which particles interact between each other. For instante, the 4-particle scattering constraints

    \[(\langle 1 2 \rangle \left[3 4 \right])^{2S}F(s,t,u)\]

where s is the spin of the particle. Be aware of not confusing the spin with the Mandelstam variable s. Locality implies the factorization of the 4-particle amplitude into two Y pieces, such as

    \[F(s,t,u)=\begin{cases}\dfrac{g^2}{s t^S}\\ \dfrac{g^2}{t u^S}\\ \dfrac{ g^2}{u s^S}\end{cases}\]

Two special cases are S=0 (the Higss!) and S=2 (the graviton!):

    \[F(s,t,u)=g^2\left(\dfrac{1}{s}+\dfrac{1}{t}+\dfrac{1}{u}\right)\]

    \[F(s,t,u)=g^2\dfrac{1}{stu}\]

where the latter represents the 2×2 graviton scattering. For spin S=1 you have

    \[Y\propto gf^{abc}\dfrac{\langle 1 2 \rangle^3}{\langle{13}\rangle\langle 23\rangle}\]

Interactions between both, massive and massless spin one particles must contain spin zero particles in order to unitarize the scattering amplitudes! Scalar bosons are Higgs bosons. Of course, at very high energies, the Higgs and the chiral components of the massive gauge bosons (spin one) are all unified into a single electroweak interaction. A belief in these principles has a paid-off: particles have only spin 0,1/2,1,3/2,2,… The 21st century revelations must include some additional pieces of information about this stuff:

  • The doom or end of spacetime. Is the end of reductionism at sight?
  • Why the Universe is big?
  • New ideas required beyond spacetime and internal symmetries. The missing link is usually called supersymmetry (SUSY), certain odd symmetry transformations relating boson and fermions. New dogma.
  • UV/IR entanglement/link/connection. At energies bigger than Planck energy, it seems physics classicalize. We have black holes with large sizes, and thus energies (in rest) larger than Planck energy. High energy is short distance UV physics. Low energy is large distance IR physics.
  • Reductionism plus wilsonian effective field theory approaches plus paradigmatic model is false. Fundamental theories or laws of Nature nothing like condensed matter physics (even when condensed matter systems are useful analogues!). Far deeper and more radical ideas are necessary. Only at Planck scale?

Photons must stay massless for consistent Quantum Electrodynamics, so they are Higgs transparent. 2\neq 3 is the Nima statement on this thing. massless helicities are not the same of massive helicities. This fact is essential to gauge fields and chiral fermions. So they can be easily engineered in condensed matter physics. However, Higgs fields are strange to condensed matter systems. Higgs is special because it does NOT naturally arise in superconductor physics and other condensed matter fields. Why the Higgs mass is low compared to the Planck mass? That is the riddle. The enigma. Higgs particles naturally receive quantum corrections to mass from boson and fermion particles. The cosmological constant problem is beyond a Higgs-like explanation because the Higgs field energy is too-large to handle it. Of course, there are some ideas about how to fix it, but they are cumbersome and very complicated. We need to go beyond standard symmetries. And even so, puzzles appear. Flat spacetimes, de Sitter (dS) spacetimes or Anti de Sitter (AdS) spacetimes? They have some amount of extra symmetries: SO(5,1)\rightarrow \mbox{Poincaré}\rightarrow SO(4,2). The cases of \Lambda>0 (dS), \Lambda=0 (flat spacetime), \Lambda<0 (AdS). Recently, we found (not easily) dS vacua in string and superstring theories. But CFT/AdS correspondences are better understood yet. We are missing something huge about QM of the relativistic vacuum in order to understand the macroscopic Universe we observe/live in.

Why is the Higgs discovery so important? Our relativistic vacuum is qualitatively different than anything we are seen (dark matter, dark energy,…) in ordinary physics. Not just at Planck scale! Already at GeV and TeV scale we face problems! The Higgs plus nothing else at low energies means that something is wrong or at least not completely correct. The Higgs is the most important character in this dramatic story of dark stuff. We can put it under the most incisive and precise experimental testing! So, we need either better colliders, or better dark matter/dark energy observations. The Higgs is new physics from this viewpoint:

  1. We have never seen scalar (fundamental and structureless?) fields before.
  2. Harbinger of deep and hidden new principles/sectors at work at the quantum realm.
  3. We must study it closely.

It could arise that Higgs particles are composite particles. How pointlike are Higgs particles? Higgs particles could be really composite of some superstrong pion-like stuff. But also, they could be truly fundamental. A Higgs factory working at 125 GeV (pole mass) of the Higgs should serve to see if the Higgs is point-like (fundamental). Furthermore, we have never seen self-interacting scalar fields before. A 100 TeV collider or larger could measure Higgs self-coupling up to 5%. The Higgs is similar to gravity there: the Higgs self-interacts much like gravitons!

Yang-Mills fields (YM) plus gravity changes helicity of particles AND color. 100 TeV colliders blast interactions and push High Energy Physics. New particles masses up to 10 times the masses accessible to the LHC would ve available. They would probe vacuum fluctuations with 100 times the LHC power. The challenge is hard for experimentalists. Meanwhile, the theorists must go far from known theories and go into theory of the mysterious cosmological constant and the Higgs scalar field. The macrouniverse versus the microuniverse is at hand. On-shell lorentzian couplings rival off-shell euclidean couplings of the Higgs? Standard local QFT in euclidean spacetimes are related to lorentzian fields. UV/IR changes this view! Something must be changed or challenged!

Toy example

Suppose F=1/t. By analytic continuation,

    \[\dfrac{1}{2\pi i}\oint dt F(t)=1\]

Is Effective Field Theory implying 0 Higgs and that unnatural value? Wrong! Take for instance

    \[F(t)=\dfrac{1-10^{-120t/t_{uv}}}{t}\]

Then

    \[\oint dt F(t)=0!\]

This mechanism for removing bulk signs works in AdS/CFT correspondences and alike theories. For \Lambda=0 we need something similar to remove singularities and test it! For instance, UV-IR tuning could provide sensitivities to loop processes arising in the EFT of the Higgs potential

    \[V(1-loop)=\lambda^4h^4\log( \lambda^2+k^2)+(M\pm h)^4\log (M\pm k)^2=\sum M^4\log M^2(k)\]

However, why should we tree cancel the 1-loop correction? It contains UV states and \lambda^2 M^2 h^2 terms. Tree amplitudes are rational amplitudes. Loop amplitudes are trascendental amplitudes! But long known funny things in QFT computations do happen. For instance,

    \[\Gamma(positronium)=\mbox{something}\times (M^2-9)\]

Well, this not happens. There is a hidden mechanism here, known from Feynman’s books. Rational approximations to trascendental numbers are aslo known from old mathematics! A High School student knows

    \[\ln 2=\int_0^1 \dfrac{dx}{1+x}\]

This is trascendental because of the single pole at x=-1. If you take instead

    \[I(x)=\int \dfrac{dx P(x)}{1+x}=P(-1)\log 2+\mbox{Rational part}\]

you get an apparen tuning of rational to trascendental numbers

    \[\int_0^1\dfrac{dx}{1+x}\left(\dfrac{x(1-x)}{2}\right)^N=\pm \log 2+\mbox{Rational part}\]

and thus, e.g., if N=5, you get a tiny difference to \log 2 by a factor of 10^{-5} (the difference up to this precision is 2329/3360). The same idea works if you take

    \[4\int_0^1\dfrac{dx}{1+x^2}\left(\dfrac{x(1-x)}{4})\right)^{4N}=\pi +\mbox{Rational number}\]

You get for N=1 \pi-22/7\sim 10^{-3} and \pi-47171/15015\sim 10^{-6} if N=2. Thus, we could conjecture a fantasy: there is a dual formulation of standard physics that represents physical amplitudes and observeables in terms of integral over abstract geometries (motives? schemes? a generalized amplituhedron seen in super YM?). In this formulation, the discrepancy between the cosmological constant scale and the Higgs mass is solved and it is obviously small. But it can not be obviously local physics. Another formulation separates different number theoretical parts plus it looks like local physics though! However, it will be fine-tuned like the integrals above! In the end, something could look like

    \[V(h)=\int dk^2 F(k^2)=\mbox{Logs+rational}=\mbox{exponentially small}\]

Fine-tuning could, thus, be only an apparent artifact of local field theory!

A final concrete example:

    \[F(h)=\int_k^1 \dfrac{dx (x-h)^4}{1+x}\left(\dfrac{x(1-x)}{2}\right)^N\]

Take V(h)=F(h)+F(-h). Then,

    \[V(h)=\sum_{\pm}(1\pm h)^4\log (1\pm h)+\mbox{Rational parts}\]

And it guarantees to be fine-tuning! This should have critical tests in a Higgs factory or very large LHC and/or 100TeV colliders of above. In the example above, if N=5

    \[V(h)=\ldots+10^{-6}+10^{-5}h^2+\ldots+h^{10}\]

with no sixth power or eight power terms. Precision circular electron-positron colliders could handle with this physics. Signals from tunning mechanisms could be searched. It is not just m_h^2 terms only. High dimensional operators and corrections to the Higgs potential (the vacuum structure itself!) could be studied. But also, we could search for new fields or tiny effects we can not do within the LHC.

Summary: scientific issues today are deeper than those of 1930s or even 1900s. Questions raised by the accelerated universe and Higgs discovery go at the heart of the Nature of the spacetime and the vacuum structure of our Universe!

What about symmetries? In the lagrangian (action) approach, you get a symmetry variation (quasiinvariance) of the lagrangian as follows

    \[\delta_s L=\partial_\mu \left[\left(\dfrac{\partial L}{\partial(\partial_\mu\phi)}\right)\delta_s \phi\right]+\left[\dfrac{\partial L}{\partial\phi}-\partial_\mu\left(\dfrac{\partial L}{\partial(\partial_\mu\phi)}\right)\right]\delta_s\phi\]

Then, by the first Noether theorem, imposing that the action is extremal (generally minimal), and the Euler-Lagrange equations (equations of motion), you would get

    \[E(L)=0, \mbox{Plus quasiinvariance}\;\; \delta_s L=\partial_\mu K^{\mu}\]

a conserved current (and a charged after integration on the suitable measure):

    \[\partial_\mu J^\mu=0=\partial_\mu\left[\dfrac{\partial L}{\partial(\partial_\mu \phi)}\delta_s \phi -K^\mu\right]=\varepsilon \partial_\mu J^\mu\]

such as

    \[J^\mu=\dfrac{\partial L}{\partial( \partial_\mu \phi)}\Delta_s\phi-\dfrac{K^\mu}{\varepsilon}\]

where \Delta_s\phi= \delta_s\phi/\varepsilon.

This theorem can be generealized for higher order lagrangians, in any number of dimension, and even for fractional and differentigral operators. Furthermore, a second Noether theorem handles ambiguities in this theorem, stating that gauge local transformations imply certain relations or identities between the field equations (sometimes referred as Bianchi identities but go further these classical identities). You can go further with differential forms, exterior calculus or even with Clifford geometric calculus. A p-form

    \[A_p=\dfrac{1}{p!}A_{\mu_1\cdots\mu_p} dx^{\mu_1}\wedge\cdots \wedge dx^{\mu_p}\equiv A_{\mu_1\cdots\mu_p} dx^{\mu_1}\wedge\cdots \wedge dx^{\mu_p}\]

defines p-dimensional objects that can be naturally integrated out. For a p-tube in D-dimensions

    \[\tau_{\mu_{p+1}\cdots \mu_D}=\dfrac{1}{p!}\int_C\varepsilon_{\mu_1\cdots\mu_p\mu_{p+1}\cdots\mu_D}\delta(x-y) dy^{\mu_1}\wedge\cdots dy^{\mu_p}\]

On p-forms, the Hodge star operator for a p-form A_p in D-dimensions turn it into a (D-p)-form

    \[\star A=\dfrac{\sqrt{\vert g\vert}}{p!(D-p)!}A_{\mu_1\cdots\mu_p}\varepsilon^{\mu_1\cdots\mu_p} dx^{\nu_{p+1}}\wedge\cdots\wedge dx^{\nu_D}\]

As D=Dim(X), then we have \star^2=\star\star=(-1)^{p(D-p)+q}, where q=1 if the metric is Lorentzian, q=0 for euclidean metrics and q=T, the number of time-like dimensions, if the metric is ultrahyperbolic. Moreover,

    \[vol=\star 1=\dfrac{\sqrt{\vert g\vert}}{D!}\varepsilon_{\mu_1\cdots \mu_{D}}dx^{\mu_1}\wedge\cdots\wedge dx^{\mu_D}=\sqrt{\vert g\vert} dx^{\mu_1}\wedge\cdots \wedge dx^{\mu_D}\]

For \star:\Omega^p\rightarrow \Omega^{D-p} maps, you can also write

    \[\langle A,B\rangle=\int A\wedge \star B=\int B\wedge \star A \]

    \[\int A\wedge B=\int \star A\wedge \star B\]

where the latter is generally valid up to a sign. The Hodge laplacian reads

    \[\Delta=(d^++d)^2\]

and you also gets

    \[\langle A, dB\rangle=\langle d^+A,B\rangle\]

If \partial X is not zero (the boundary is not null), then it implies essentially Dirichlet or Neumann boundary conditions for d, d^+. When you apply the adjoint operator d^+ on p-forms you get

    \[d^{+}=(-1)^{Dp+D+1}\star d\star\]

in general but you pick up an extra -sign in euclidean signatures.

To end this eclectic post, first a new twist on Weak Gravity Conjectures(WGC). Why the electron charge is so small in certain units? That is, e<m_e. Take Coulomb and Newton laws

    \[F_C=K_C\dfrac{Qq}{r^2}=\dfrac{1}{4\pi\varepsilon_0}\dfrac{Qq}{r^2}\]

    \[F_N=G_N\dfrac{Mm}{r^2}\]

Then,

    \[\dfrac{F_N}{F_C}\sim 10^{-42}\]

Planck mass is

    \[M_P=\sqrt{\dfrac{\hbar b}{G_N}}\]

and then

    \[Q_P=\sqrt{4\pi\varepsilon_0\hbar c}=\left(\dfrac{\hbar c}{K_C}\right)^{1/2}\]

Planckian entities satisfy instead F_G/F_C=1! then, the enigma is why

    \[\dfrac{m_e}{M_p}<10^{-22}<<\dfrac{q_e}{q_P}\sim 0.1=10^{-1}\]

In other words, q_e/m_e\approx 10^{21} in relativistic natural units with c=\hbar=4\pi\varepsilon_0=1 with \hbar\neq 1. The WGC states that the lightest charge particle with m,q in ANY U(1) (abelian gauge) theory admits UV embedding into a consistent quantum gravity theory only and only if

    \[\dfrac{qg}{\sqrt{\hbar}}\geq \dfrac{m}{M_p}\]

where g is the gauge coupling and QED satisfies WGC since ge/\sqrt{\hbar}\sim 10^{-3}>>10^{-22}. WGC ensures that extremal black holes are unstable and decay into non-extremal black holes rapidly (if even formed!) via processes (and avoid to be extremal too) that are QG consistent. Furhtermore, WGC could imply the 3rd thermodynamical law easily. For Reissner-Nordström black holes

    \[T=\dfrac{\hbar \sqrt{M^2-Q^2}}{2\pi\left(M+\sqrt{M^2-Q^2}\right)^2}\]

and a grey body correction to black hole arises from this too

    \[\langle N_{j\omega l p}\rangle=\dfrac{\Gamma(j\omega l p)}{e^{(\omega-e\phi)/T}\pm 1}\]

Generalised Uncertainty principles (GUP) plus Chandrasekhar limits enjoy similarities with the WGC:

    \[M_C\sim \dfrac{1}{m_e^2}\left(\dfrac{\hbar c}{G}\right)^{3/2}\simeq 1.4M_\odots\]

The S-matrix

    \[\braket{\Psi(+\infty) |\Psi(+\infty)}=1=\bra{\Psi(- \infty)} S^+S\ket{\Psi(-\infty)}=\braket{\Psi(-\infty) |\Psi(-\infty)}\]

and by time reversal, the principle of deteailed balance holds so

    \[\braket{\Psi(-\infty) |\Psi(+\infty)}=1=\bra{\Psi(+ \infty)} S^+S\ket{\Psi(-\infty)}=\braket{\Psi(+\infty) |\Psi(-\infty)}\]

Quantum determinism implies via unitarity \Psi'=U\Psi. However, Nature could surprise us. And that would affect Chandrasekhar masses or TOV limits. Stellar evolucion implies luminosities L=4\pi R^2 \sigma T_e^4, where T_e is the effective temperature for black bodies (Planck law), and \sigma is the Stefan-Boltzmann constant

    \[\sigma=5.67\cdot 10^{-5}\dfrac{erg}{cm^2 K^4 s}\]

Maximal energy for a set of baryons under gravity is

    \[E_G=-\dfrac{GMm_B}{R}=-G\dfrac{Nm_B^2}{R}=\hbar c\dfrac{N^{1/3}}{R}-G\dfrac{Nm_B^2}{R}\]

as the baryon number for a star is

    \[N_B=\left(\dfrac{\hbar c}{Gm_B}\right)^{3/2}\simeq 2\cdot 10^{57}\]

Using the Wien law

    \[\lambda_{max} T_e\simeq 2.9\cdot 10^6 mm\cdot K\]

the stars locally have an equation for baryon density about

    \[\left(\dfrac{V}{N}\right)^{1/3}=n^{1/3}=\dfrac{n^{1/3}}{M_P^{1/3}}\sim \left(\dfrac{4}{3} \pi R_\odot^3\dfrac{m_P}{M_{ \odot}}\right)^{1/3}\sim 10^{-8}\]

Stars are sustained by gas and radiation against gravitational collapse. The star pressure

    \[P_\star=P(gas)+P(radiation)=\dfrac{K}{\mu}\rho T+\dfrac{1}{3}n T^4\]

Thus, the maximal mass for a white dwarf star made by barions is about 1.5M_\odot. Ideal gas law implies the HR diagram!!!! Luminosity scales as the cube of mass. The Eddington limit maximal luminosity for any star reads off as

    \[L_E=\dfrac{AMc Gm(r)}{K}\]

and a Buchdahl limit arises from this and the TOV limit as follows

    \[TOV\rightarrow \dfrac{GM}{c^2R}<\dfrac{4}{9}\]

and then

    \[M>\dfrac{4}{9}\dfrac{Rc^2}{G}\]

implies a BH as inevitable consequence iff M>3M_\odot approximately!

Epilogue: heterodynes or superheterodynes? Jansky? dB scales? Photoelectric effect is compatible with multiphoton processes and special relativity too. SR has formulae for Compton effect, inverse Compton effect, pair creations, pair annihilations, and strong field effects!

 

 

LOG#238. Cosmostuff.

Cosmology is facing again some troubles. Some estimations of the Hubble parameter H_0 differ up to four standard deviations from the accepted value. Even when a few km/s/Mpc are not quite a huge difference, it turns than they can provide anomalies if measured with enough precision.

I am going to review Cosmology today, as far as we know it. It is a quite dynamical field, so changes are expected in the next years with new telescopes, the JWST (James Webb Space Telescope), and other tools or ideas that make this subject so fascinating. After all, isn’t  the Universe as a whole the biggest place where we live? I am neglecting the Multiverse idea as untestable at this moment.

Cosmology can use the so-called standard candles and standard rulers to measure how fast the Universe is expanding. Two measures of distance are

(1)   \begin{equation*} D_L\rightarrow F=\dfrac{L}{4\pi D_L^2}\end{equation*}

(2)   \begin{equation*} D_A=\dfrac{R}{\theta}\end{equation*}

and they are called the luminosity distance and the angular distance. They use, respectively, the flux by some strong sources (e.g., SNIa) and the ability to measure the size of an object at long distances (of course, far, far away objects look like point sources, so angular measurement is only possible when having good resolution and/or close enough objects with respect to our instruments).

Considering luminosities, you can measure the bolometric magnitudes:

(3)   \begin{equation*} m-M=5\log \left(\dfrac{d}{10pc}\right)\end{equation*}

and

(4)   \begin{equation*} m_A-m_B=-2.5\log \left(\dfrac{f_A}{f_B}\right)\end{equation*}

Known from the beginning of the second third of the 20th century (Zwicky), we do know that galaxies are not bright enough to explain the galactic motions. The Tully-Fisher relation

(5)   \begin{equation*}L\propto v^4\end{equation*}

and the Faber-Jackson relation

(6)   \begin{equation*} L\propto \sigma (v)^4\end{equation*}

are the celebrated luminosities of spiral galaxies and elliptical galaxies, respectively. They hint there are dark matter out there (or equivalently, a component of gravity we do not see with the electromagnetic fluxes).

There are two faint objects we are not sure yet of how many of them are in the galaxies. White dwarfs and neutron stars (I give up the option of black holes and other exotic compact objects, ECOs, at this moment). There are quite strong and robust limitations to the white dwarf and neutron star masses due to nuclear physics. For the former we have the Chandrasekhar limit at about 1.44M_\odot, for the latter we have the Tolman-Oppenheimer-Volkoff limit, at about 2-3M_\odot. The uncertainties in the knowledge of the nuclear equation of state for neutron star are the responsible of the relative big mass spreading (1 solar mass). Gravitational waves are going to be the tool to measure the nuclear equation of state, very likely.

The Dark Energy Survey (DES) is a 15000 times 70, i.e., 1050000 dollar project of astronomical Cosmology. PTOLEMY is another future experiment. If there is “no” privileged distance, then we could measure some interesting distances. For instance, a recently new method is the baryonic acoustic oscillations (BAO). What the hell are the BAO? BAO are fluctuations or rapid changes in the density of the visible baryonic matter (normal matter) of the universe, caused by acoustic density waves (changes in the density of baryons, i.e., the proton oscillations at the distances where galaxies are correlated in distance in-and-out) in the primordial plasma of the early universe. The critical acoustic distance is about 150 Mpc or about 500 Mlyr. Then, the correlation function is a device that gives you the probability of two galaxies have to be separated certain distance d_S. Cosmology, also, measures the rate of expansion of the Universe via a modified Hubble law. Original Hubble law is

    \[v=Hd\]

but the REAL and more precise Hubble law depends on the redshift z

(7)   \begin{equation*}v=H(z)d\end{equation*}

In the 21st century, we have more tools to measure what the Universe looks like beyond the electromagnetic visible spectrum. We can handle X-rays, gamma rays, radio waves, neutrinos (extragalactic from 2006 and SN 1987A are known) and gravitational waves (since 2015). Software analysis is diverse: tools like Sextractor or DESI are known in astronomy. General relativity is tested and established as comoslogical theory at the disguise of LCDM. Perfect cosmological principle has been abandonated. We solve the Olbers paradox giving a lifetime of 13.8 Gyrs to our Universe (e.g., see the LSST data and observations, or PLANCK probe ultimate data).  The non-perfect cosmological principle, saying that the Universe is isotropic and homogeneous at distances above 100 Mpc has been tested. GR equations hold

    \[G_{\mu\nu}=\dfrac{8\pi G}{c^4}T_{\mu\nu}\]

with

    \[G_{\mu\nu}=R_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu} R+\Lambda g_{\mu\nu}\]

Pressure is a source of gravity, through the universal equation of state

    \[P=\omega \rho\]

There is a critical density to collapse (gravitationally) the Universe. It is given by

(8)   \begin{equation*}\rho_c=\dfrac{3H^2}{8\pi G}\end{equation*}

or

(9)   \begin{equation*}\rho_c(E)=\rho_c(m)c^2\dfrac{3H^2c^2}{8\pi G}\end{equation*}

Plugging the measured values, it has a value about 8\cdot 10^{-27}kg/m^3, or 10 hydrogen atoms (protons) per cubic meter. Also, you can rewrite that value as

    \[\rho_c(measured)\sim \dfrac{1.5\cdot 10^{11}M_\odot}{Mpc^3}\]

Then, knowing \rho_c, you can define the omega densities as the density of species X over the critical density (dimensionless):

    \[\Omega_X=\dfrac{\rho_X}{\rho_c}=\dfrac{8\pi G\rho_X}{3H^2}\]

The measured value of the cosmological constant energy (or mass) density, is close to the Zeldovich estimate of vacuum energy:

    \[\rho_\Lambda=\dfrac{Gm^6c^2}{\hbar^4}=\dfrac{\Lambda c^4}{8\pi G}\]

This relation has been derived by C. Beck from purely information theory arguments

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

being m=m_e the electron mass and \alpha the electromagnetic fine structure constant. The Higgs vacuum energy density is much bigger than the measured vacuum energy density, it is about v=(\sqrt{2})^{-1/2}\approx 246 GeV, and that is a problem we can not completely understand yet. If so, the Universe should have collapsed. And that is not the case.

In cosmology, theoretical cosmology, we have to measure the redshift in order to know the scale factor of the expansion. They are linked with the equation

    \[a(t_e)=\dfrac{1}{1+z}\]

The Friedmann equations are only the Einstein Field Equations for a homogeneous and isotropic Universe (described by a metric and a perfect fluid):

(10)   \begin{equation*}\left(\dfrac{\ddot{a}}{a}\right)=-\dfrac{4\pi G}{3}\left(\rho +\dfrac{3P}{c^2}\right)+\dfrac{\Lambda c^2}{3}\end{equation*}

(11)   \begin{equation*}\left(\dfrac{\dot{a}}{a}\right)^2=\dfrac{8\pi G\rho}{3}-\dfrac{\kappa c^2}{a^2}+\dfrac{\Lambda c^2}{3}\end{equation*}

(12)   \begin{equation*}T_{\mu\nu}=\left(\rho+\dfrac{P}{c^2}\right)u_\mu u_\nu+Pg_{\mu\nu}\end{equation*}

Conservation of energy at cosmological scales provides

    \[\dot{\rho}+3\left(\rho+\dfrac{P}{c^2}\right)\dfrac{\dot{a}}{a}=0\]

For matter, dust, we have \omega=0 and pressureless composition P=0. For radiation, we have P=\rho/3 and \omega=1/3. For dark energy, the cosmological constant Einsteind thought was his biggest mistake, we have \omega=-1 and P=-\rho. If -1<\omega<1/3 we have the so-called quintessence field. A dynamical field that resembles scalar field but it can also be other classes of field. Quintessence and scalar fields are generally speaking a prediction of superstring theories and other unification models. If \omega<-1 we have the so-called phantom energy, a field that could destroy literally the Universe and provide a future singularity. It is the Big Rip (or little Rip in some lite models). There is a relationship between the \omega parameter, the scale factor and the density:

(13)   \begin{equation*}\rho\propto a^{-3(1+\omega)}\end{equation*}

Thus, for dust, \rho\propto a^{-3}, for radiation \rho\propto a^{-4} and \rho\propto constant for dark energy (cosmological constant is constant in density, due to its name of course). Cosmology, at the observational level, tries to fit data to the theory of LCDM via a global fit

    \[\sum\dfrac{\left(\mbox{(Data)}-\mbox{(Theory)}\right)^2}{\sigma}\]

of some parameters like \Omega_X, H_0, and other more specific parameters. We can measure the distance

    \[r(z)=\dfrac{c}{H_0}\int_0^{z}\dfrac{dz'}{\sqrt{\Omega_\Lambda+\Omega_k(1+z^{'})^2+\Omega_M(1+z^{'})^3+\Omega_r(1+z^{'})^4}}\]

with

    \[d_L(z)=r(z)(1+z)\]

    \[d_A=\dfrac{r(z)}{1+z}\]

The effect of the cosmic expansion is not a Doppler shift. It just enlarges wavelengths. This is the cosmological redshift. It is not a Doppler shift as one usually thinks. Indeed, we have observed objects with redshift above 1, so the relationship v\sim cz=HD=\dfrac{\dot{a}}{a}d implies that \dot{a}=a H=cz>1 if z>1. Of course, this not implies relativity is wrong. It is only the apparent effect of expansion.

That the Universe suffered a hot dense primordial phase, the Big Bang, is already proved beyond doubts. The COBE satellite and further CMB (cosmic microwave background) probes have even probed that the young Universe was very close to a perfect blackbody body. Indeed, COBE itself proved that at the level of 400\sigma. We can now measure the fluctuations of the blackbody CMB temperature in the sky according to some direction. It fits a perfect blackbody with some anisotropies. The power spectrum is the name of the Fourier transform of the temperatura in the sky map. It is measured at the level of some \mu K, microkelvins. The first peak in that power spectrum fits that our Universe is, surprisingly, flat, since our Universe is close to the critical energy density. \rho_U\sim\rho_c implies, thus, \kappa\sim 0. Please, do not tell that to flatlanders! The second peak gives us that there is dark matter out there. There are other hints about that. For instance, the peculiar velocity of galaxies (measured with real Doppler shifts) or the rotational curves of spiral galaxies and the dispersion speed of elliptical galaxies. We expect a cosmic neutrino background at the level of

    \[T_\nu=\left(\dfrac{4}{11}\right)^{1/3}T_{CMB}\simeq 1.945 K\]

and a relic graviton background at the level of

    \[T_{RGB}=\left(\dfrac{g_s(t_0)}{g_s(t_p)}\right)^{1/3}T_\gamma\]

If we plug g_s=3.91 as the number of relativistic particle species today, and g_s=106.75 as the same number evaluated at the Planck time decoupling (that can be only estimated with a GUT/TOE exactly, but we can extrapolate roughly the Standard Model up to that energy to get a bound), you would get a 1K relic graviton background as upper bound (it can be lower if the number of particle species grows at higher energies), so the 1 kelvin degree is a good robust bound. Other theories would lower that value. Extra dimensional theories would also reduce the RGB by modifying the power law of the ratio of the number of particle species.

What else? Dark matter. WIMPs are being testing as candidates since decades ago. They have failed. The naive cross section of Dark Matter would be typically \sigma_{DM}\sim G_FM_W\sim 10^{-36} cm^2. After 25 years of experimental searches, we do know that \sigma_{DM}\leq 10^{-46}cm^2, provided it exists of course. WIMPS would have freezed at

    \[\rho_c=\rho_f\left(\dfrac{T}{T_0}\right)^{-3}\]

and annihilation happens with rates \Gamma=n\sigma v. Relativistic dark matter are constrained by the number of known light neutrino species, not being radiation! No experiment has detected dark matter particles yet. Anyway, it could be anything between about 80 magnitude orders.

Finally, have you ever wondered how many eras in the cosmic evolution have the Universe faced? Let me list them:

  1. Definition 1 (Luminosity distance). F=\dfrac{L}{4\pi D_L^2}.
  1. The Planck era: t\leq 10^{-43}s. Quantumg gravity is necessary here, and likely the next two or three eras.
  2. The inflation era: 10^{-43}s\leq t\leq 10^{-35}s. Inflation, exponential expanding Universe, was suffered her. Otherwise, the flatness problem and the observed structure could be impossible. B-modes in the CMB would be a proof of inflation. After BICEP2 blunder, we have to wait for further evidences.
  3. Leptogenesis and baryogenesis. Where do quark and leptons come from? Why and how were they formed? We do not know, but this era is a future target of some experiments and theoretical studies. If neutrinos were Majorana particles or Majorana states for the neutrino species (right-handed!) wher found, we could explain, in principle, the unbalance between matter and antimatter we observe today.
  4. Electroweak and QCD phase transitions. In the edge of our current knowledge thanks to ALICE and quark-gluon plasma studies and the discovery of the Higgs field permeating the Universe. Further studies of this era will require gravitational waves and neutrinos, since no photon can show us information of this a previous dark eras.
  5. Neutrino decoupling. At about 1 second from the Big Bang, neutrino decouples from the primordial plasma.
  6. Electron-positron annihilation. After neutrino decoupled, electron and positros begin annihilating.
  7. Big Bang nucleosynthesis. The Universe is cooling and at some point, first nuclei (hydrogen, helium and lithium) confine. The point where matter and radiation pressures equalize is found. It happens at about 3 minutes. The cosmic temperature is about 3000 K, and it is lower than the 158000 K of the 13.6 eV ground state of the hydrogen atom.
  8. Recombination. The Universe expands and cools further. After 380000 years, electrons bound to the first atoms (H, He and Li). Big Bang theory predict the relative abundances of the main isotopes. The first stars (Population III) will form in the next millions of years, breeding the Universe with black hole seeds and galaxies after billions of years (Gyrs).
  9. At some point, a few Gyrs ago, dark energy begins dominate the cosmic expansion, previously dominated by matter. Thus, from radiation dominated expansion, we passed a several Gyr era of matter dominated expansion until we end in this dark energy era. Is emergent life related to dark energy dominating the cosmic expansion? Probably not, but it is a striking coincidence.

See you in the next blog post!

Definition 2 (Angular distance). D_A=\dfrac{R}{\theta}.

Definition 3 (Friedmann equations).

    \[\left(\dfrac{\ddot{a}}{a}\right)=-\dfrac{4\pi G}{3}\left(\rho +\dfrac{3P}{c^2}\right)+\dfrac{\Lambda c^2}{3}\]

    \[\left(\dfrac{\dot{a}}{a}\right)^2=\dfrac{8\pi G\rho}{3}-\dfrac{\kappa c^2}{a^2}+\dfrac{\Lambda c^2}{3} \]

Definition 4 (Perfect fluid energy-momentum tensor). T_{\mu\nu}=\left(\rho+\dfrac{P}{c^2}\right)u_\mu u_\nu+Pg_{\mu\nu}.

Definition 5 (Cosmic neutrino and graviton backgrounds).

    \[T_\nu=\left(\dfrac{4}{11}\right)^{1/3}T_{CMB}\simeq 1.945 K\]

    \[T_{RGB}=\left(\dfrac{g_s(t_0)}{g_s(t_p)}\right)^{1/3}T_\gamma\]

1 visitors online now
1 guests, 0 members
Max visitors today: 5 at 02:54 am UTC
This month: 9 at 08-05-2020 09:50 am UTC
This year: 54 at 01-21-2020 01:53 am UTC
All time: 177 at 11-13-2019 10:44 am UTC