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LOG#249. Basic twistor theory.

 

This my last normal post. Welcome to those who read me. TSOR is ending and from its ashes will arise another project. That is inevitable. I want to use new TeX packages, and that is not easy here, to simplify things, and to write better things I wish to tell the world.

1. Twistor: the introduction

Roger Penrose formulated twistor theory in the hope of making complex geometry and not the real geometry the fundamental arena of geometric theoretical physics, and a better way to understand quantum mechanics. Quantum Mechanics(QM) is baed on complex structure of Hilbert space of physical states (both of finite or infinite dimensions!). The probability amplitudes are the complex numbers. That is, complex numbers describe oscillations interpreted as probability waves! By the other hand, relativity implies that spacetime points is a real four dimensional (in general D-dimensional) vectors. But this is again a restricted option determined by experiments. That coordinates of spacetime are real numbers is just an hypothesis of our mathematical models, despite the fact it is well supported by experiments! The main difficulty is a consistent formulation of special relativistic quantum theory. Even when possible, that it is in the form of quantum field theory, many questions arise: entanglement, the measurement problem, the collapse of wave functions/state vector reduction and the quantum gravity issue.

Twistor theory was created with the idea of treating real coordinates of spacetime points as composed quantities of more general complex objects called twistors. Therefore, in twistor theory, the most fundamental object are twistors instead of spacetime points. Twistor theory is pointless (in the real sense) geometry.

Mathematically, a first approximation to what a twistor is comes from the conformal O(4,2) spinors. Complex 4-vectors in the fundamental representation of a covering conformal group called SU(2,2) is isomorphic to \overline{O(4,2)}. A correspondence between twistors and the spacetime points is given by the so-called incidence equation or Penrose relation.

The twistor formalism originally introduced by Penrose for 4D spacetime can be extended in two or three ways:

  • Extending the Penrose-relation in a supersymmetric way one obtains a correspondence between the supertwistors and the points of D=4 superspaces.

  • Replacing the complex numbers by quaternions (or octonions, Clifford-numbers) in the Penrose relation, one can bring the quaternionic (octonionic, cliffordonic) twistors into connection with the points of the D=6 (D=10, D=2^n) spacetime (superspace, C-(super)space). Even more, one can extend this quaternionic twistor formalism in a SUSY fashion introducting quaternionic fermionic degrees of freedom.

  • Introducing hypertwistors, and likely hypersuperspace or C-hypersuperspace.

2. 4D spacetime as twistor composite

There is a very pedagogical introduction to 4D twistor theory and the fundamental incidence equation and the Penrose relation. It is well known that ANY spacetime point can be described by the 4D (ND) vector X^\mu=(X^0,X^1,X^2,X^3), or X^\mu=(X^0,\ldots,X^{D-1}). This can be connected and linked to a 2\times 2 hermitean matrix, using the Pauli matrics as follows:

(1)   \begin{equation*}X\rightarrow X=\begin{pmatrix} X^0+X^3 & X^1-iX^2\\ X^1+iX^2 & X^0-X^3\end{pmatrix}=X^\mu \sigma_\mu\end{equation*}

This map is one-to-one. We can also consider the complex 4d vector

(2)   \begin{equation*}Z=(Z^0,Z^1,Z^2,Z^3)\end{equation*}

instead of real 4d vectors. The complex 4-vector Z describes a point of the complexified Minkovski spacetime C\mathbb{R}^{4}. A similar relation to the previous equation gives us the correspondence between the points in this complexified Minkovski spacetime and the 2d complex matrices via Z=Z^\mu \sigma_\mu. You can get the real Minkovski spacetime R\mathbb{R}^{4} by putting the reality condition onto the complex matrix Z as Z=Z^+. A point in the twistor construction or model is the use of the isomorphism between complex 2d matrices Z and the Z-plane in 4d complex vector \mathbb{C}: this is the twistor space \mathbb{T}=\mathbb{C}^4. This isomorphism is given by the following correspondence, called Penrose relation:

(3)   \begin{equation*}Z:\mbox{Subspace spanned by columns of $4\times 2$ matrix} \begin{bmatrix}iZ\\ I_2\end{bmatrix}\end{equation*}

More explicitly, the 4\times 2 matrix are identified with a bitwistor, a couple of twistors (T_1,T_2)\in\mathbb{T}:

(4)   \begin{equation*}\begin{bmatrix}iZ\\ I_2\end{bmatrix}=\begin{bmatrix} iZ^0+iZ^3 & Z^2+iZ^1\\ iZ^1-Z^2 & iZ^0-iZ^3\\ 1 & 0\\ 0 & 1\end{bmatrix}\end{equation*}

From a mathematical viewpoint, this gives us an affine system of coordinates for the Z-plane in the twistor space \mathbb{T}. This subspace is a complex Grassmann manifold G_{2,4}(\mathbb{C}). In other words, the Z-plane is given by the two linearly independent twistors (T_1,T_2), the bitwistor in twistor space! This is also a correspondence between the complexified spacetime point Z\in \mathbb{C}^4 and a complex Z-plane in the twistor space. By the other hand, there is NOT a unique relation between the pair of twistors (T_1,T_2) and the Z-plane generated by this pair. It is clear, that every pair of twistors or bitwistor (T_1',T_2') is related to nonsingular matrices 2\times 2 by (T_1',T_2')=(T_1,T_2)M, and that that gives the same Z-plane in the twistor space \mathbb{T}.

Let the pair (T_1', T_2') has the form of previous matrix, then any equivalent pair of twistors satisfy

(5)   \begin{equation*}\begin{bmatrix}iZ\\ I_2\end{bmatrix}=(T_1,T_2)M=\begin{bmatrix}\Omega & M\\ \Pi & M\end{bmatrix}\leftrightarrow iZ=\Omega M, \;\; I_2=\Pi M\end{equation*}

and where the 2\times 2 complex matrices \Omega, \Pi are constructed of the coordinates of the twistors (T_1,T_2). Thus, we have

    \[\tcboxmath{iZ=\Omega\Pi^{-1}\leftrightarrow \Omega=iZ\Pi}\]

This is the Penrose relation in matrix form! If we write

(6)   \begin{equation*}(T_1,T_2)=\begin{pmatrix}\omega^{\dot{1}1} &\omega^{\dot{1}2}\\ \omega^{\dot{2}1} & \omega^{\dot{2}2}\\ \pi_{11} &\pi_{12}\\ \pi_{21} &\pi_{22}\end{pmatrix}\end{equation*}

now we get

(7)   \begin{align*}\omega^{\alpha\dot{1}}=iZ^{\dot{\alpha}\beta}\pi_{\beta 1}\\ \omega^{\dot{\alpha}2}=iZ^{\dot{\alpha}\beta}\pi_{\beta 2}\end{align*}

and where \alpha, \beta=1,2. In short hand notation, we get

    \[\tcboxmath{\omega^{\dot{\alpha}}=iZ^{\dot{\alpha}\beta}\pi_{\beta},\;\; T=\begin{pmatrix}\omega^{\dot{\alpha}}\\ \pi_\beta\end{pmatrix}}\]

This is the celebrated incidence equation postulated firstly by R. Penrose, also named Penrose relation in his honor. It has a simple physical (geometrical) meaning: the pint Z\in \mathbb{C}^4 corresponds to the twistor

    \[T\leftrightarrow \omega^{\dot{\alpha}}=iZ^{\dot{\alpha}\beta}\pi_\beta\]

It is evident that all the twistors lying on the Z-plane given in the same Penrose relation corresponds to a given Z\in\mathbb{C}^4 point and for a given twistor T satisfying the incidence equation, only one complex spacetime point Z is assigned! If anyone need to describe the real spacetime point X\in \mathbb{R}^4, one should require the matrix Z to be hermitian, i.e., to satisfy

(8)   \begin{equation*}  Z=Z^+\leftrightarrow Z=-i\Omega\Pi^{-1}=i(\Pi^{-1})^+\Omega^+\end{equation*}

and then

(9)   \begin{equation*}\Pi^+\Omega+\Omega^+\Pi=0\end{equation*}

Using the notation we introduced above, equivalently

(10)   \begin{align*}\overline{\pi}_{\dot{\alpha}1}\omega^{\dot{\alpha}1}+\overline{\omega}^{\alpha 1}\pi_{\alpha 1}=0\\ \overline{\pi}_{\dot{\alpha}2}\omega^{\dot{\alpha}2}+\overline{\omega}^{\alpha 2}\pi_{\alpha 2}=0\\ \overline{\pi}_{\dot{\alpha}1}\omega^{\dot{\alpha} 2}+\overline{\omega}^{\alpha 2}\pi_{\alpha 1}=0\end{align*}

and where

    \[\overline{\pi}_{\dot{\alpha}\beta}=\overline{\left(\pi_{\alpha\beta}\right)}\]

    \[\overline{\omega}^{\alpha\beta}=\overline{\left(\omega^{\dot{\alpha}\beta}\right)}\]

denotes the complex conjugation. In the twistor framework, these equations say that the twistors (T_1,T_2), the bitwistor, are null-twistors with respect to the U(2,2) norm

    \[\langle T_1,T_2\rangle=\langle T_1,T_1\rangle=\langle T_2,T_2\rangle=0\]

where

(11)   \begin{equation*}\langle T, T\rangle=T^+GT=\begin{pmatrix} \overline{\omega}^\alpha & \overline{\pi}_{\dot{\beta}}\end{pmatrix}\begin{pmatrix}0 & I_2\\ I_2 & 0\end{pmatrix}\begin{pmatrix} \omega^{\dot{\alpha}}\\ \pi_\beta\end{pmatrix}\end{equation*}

Therefore, the reality condition is equivalent to the zero condition for twistors, i.e., to the vanishing of the U(2,2) norm of the bitwistor and couple of twistors. The Z-planes generated by the null twistor (or congruence) are called totally null planes or congruence relation. In this way, we obtain a set of correspondences:

  • Complex planes in twistor space are related one-to-one to points in complexified Minkovski spacetime.

  • Complex planes in twistor space are related  to totally null planes in twistor space, not one-to-one.

  • Totally null planes in twistor space are related one-to-one to points in real Minkovski spacetime.

  • Points of complexified spacetime are related to real spacetime, not one to one, to points of real spacetime.

Remark: from the viewpoint of twistor theory, it is more natural to use twistors (couple of twistors indeed, via a bitwistor), for the description of the complex Minkovski spacetime or the null twistor for the description of the real spacetime!

3. SUSY and Penrose relation

The plan of SUSY is to give a unified mathematical description ob bosonic and fermionic fields. Therefore, one can consider bosons and fermions using the same theoretical background. SUSY or supersymmetry allows us to transorm the descriptions of bosonic fields into fermionic fields and vice versa. In order to have a possible description of bosonic and fermionic fields using the twistor theory, we has to extend it using SUSY. What is SUSY? Surprise…

SUSY replaces the notation of any space-time point X=X^\mu=(X^0,X^1,X^2,X^3) by an appropiate superpoint

(12)   \begin{equation*}Y=(X,\Theta)=(X^0,\ldots,X^3;\theta_1,\ldots,\theta_N)\end{equation*}

Here, the superspace point extends spacetime with a new class of numbers, \theta_i, \theta_i^2=0, \theta_i\theta_j=-\theta_j\theta_i, for all i,j=1,\ldots,N. These numbers are called Grassmann numbers. These numbers allow us to handle fermions, since they anticommute themselves. We can define a supervector representing the D=4 N-extended superspace as follows:

(13)   \begin{equation*} Y=(X,\Theta)\end{equation*}

(14)   \begin{equation*} Y=(X^0,\ldots,X^3;\theta_1,\ldots,\theta_N)=(X^\mu;\Theta_A)\end{equation*}

in such a way

(15)   \begin{equation*}\left[X^\mu,X^\nu\right]=X^\mu X^\nu-X^\nu X^\mu=0\end{equation*}

(16)   \begin{equation*}\{\theta_A,\theta_B\}=\theta_A\theta_B+\theta_B\theta_A=0\end{equation*}

(17)   \begin{equation*}\left[X^\mu,\theta_A\right]=X^\mu\theta_A-\theta_A X^\mu=0\end{equation*}

Commuting coordinates of any supervector are called bosonic coordinates, anticommuting coordinates (c-numbers) are called fermionic coordinatese. In the same spirit, we could generalize twistor theory and the twistor approach introducing N-extended bosonic supertwistors

(18)   \begin{equation*} T^{(n)}=\left(\omega^{\dot{\alpha}},\pi_\beta;\xi_1,\cdots,\xi_n\right)\in \mathbb{C}^{4\vert N}\end{equation*}

and the fermionic N-extended supertwistors

(19)   \begin{equation*}\tilde{T}^{(n)}=\left(\eta_1,\ldots,\eta_4;u_1,\ldots,u_N\right)\in \mathbb{C}^{N\vert 4}\end{equation*}

where the \eta_i quantities are fermionic coordinates ant the u_A quantities are the bosonic degrees of freedom. We will discuss the N=1 case, simple supersymmetry, for simplicity. Firstly, two linearly independent supertwistors (T_1^{(1)},T_2^{(1)}) span (2,0)-superplane int he superspace \mathbb{C}^{4\vert 1}. In analogy with the no superspace case, we define and get

(20)   \begin{equation*}\left(T_1^{(1)},T_2^{(1)}\right)=\begin{bmatrix} \omega^{\dot{1}1} & \omega^{\dot{1}2}\\ \omega^{\dot{2}1} &\omega^{\dot{2}2}\\ \xi_1 & \xi_2 \\ \pi_{11} & \pi_{12}\\ \pi_{21} & \pi_{22}\end{bmatrix}=\begin{bmatrix} iZ\\ \theta^1 & \theta^2\\ 1 & 0\\ 0 & 1\end{bmatrix}\Pi\end{equation*}

Here, Z, \Pi are complex matrices 2\times 2 made up of bosonic elements. This can also be expressed using equations

(21)   \begin{align*}\omega^{\alpha\dot{1}}=iZ^{\dot{\alpha}\beta}\pi_{\beta 1}\\ \omega^{\dot{\alpha} 2}=iZ^{\dot{\alpha}\beta}\pi_{\beta 2}\\ \xi_1=\theta^1\pi_{11}+\theta^2\pi_{21}\\ \xi_2=\theta^1\pi_{12}+\theta^2\pi_{22}\end{align*}

Then, the supersymmetric extension of Penrose relation reads off

(22)   \begin{align*}\omega^{\dot{\alpha}}=iZ^{\dot{\alpha}\beta}\pi_{\beta}\\ \xi=\theta^\alpha\pi_\alpha\end{align*}

These equations mean that every T^{i}=(\omega^{\dot{\alpha}},\pi_\beta,\xi) supertwistor corresponds to a Y=(Z,\Theta)=(z^\mu, \theta^\alpha) superspace point or superpoint. However, note that it is not the only option to generalize the Penrose relation!

Apply three linearly independent supertwistors T^{(1)}_1, T^{(1)}_2, \tilde{T}^{(1)}, where the latter is a fermionic twistor, such as \mathbb{C}^{4\vert 1} is our superspace.

(23)   \begin{equation*}\left(T_1^{(1)},T_2^{(1)},\tilde{T}^{(1)}\right)=\begin{bmatrix}\omega^{\dot{1}1} & \omega^{\dot{1}2} & \rho^{\dot{1}}\\ \omega^{\dot{2}1} & \omega^{\dot{2}2} & \rho^{\dot{2}}\\ \pi_{11} & \pi_{12} & \eta_1\\ \pi_{11} & \pi_{12} & \eta_2\\ \xi^1 & \xi^2 & u\end{bmatrix}=\begin{bmatrix}iZ^{\dot{1}1} & iZ^{\dot{1}2} & \theta^1\\ iZ^{\dot{2}1} & iZ^{\dot{2}2} & \theta^2\\ 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}\pi_{11} & \pi_{12} & \eta_1\\ \pi_{21} & \pi_{22} & \eta_2\\ \xi^1 & \xi^2 & u\end{bmatrix}\end{equation*}

where the fermionic supertwistor includes the four fermionic components

    \[\left(\rho^{\dot{1}},\rho^{\dot{2}},\eta_1,\eta_2\right)\]

and also the bosonic u. The (2;1)-superplane is parametrized by a (Z,\Theta) matrix of 2\times 3 type with elements satisfying the following generalized incidence relations:

(24)   \begin{align*}\omega^{\dot{\alpha}}=iZ^{\dot{\alpha}\beta}\pi_\beta+\theta^{\dot{\alpha}}\xi\\ \rho^{\dot{\alpha}}=iZ^{\dot{\alpha}\beta}\eta_\beta+\theta^{\dot{\alpha}}u\end{align*}

and where the first equation is the bosonic incidence relation, and the second one is the fermionic incidence relation. Thus, this is a different generalization of Penrose relation. For N=1 suspersymmetry, there are 2 possible extensions of Penrose’s relation/incidence equations. In case of the N-extended SUSY, one can generalize those equations in N+1 different ways!

4. Quaternionic extension of Penrose incidence in D=6 spacetime

Taking into account the previous section, there are 2 possible approaches to 6D twistors:

  • Extend Penrose relation from D=4 to D=6 as it has been done in bibligraphy or following these lines.

  • Replace the complex 2\times 2 matrices Z by quaternionic ones. Use quaternionic 2\times 2 matrices describing naturally a 6d real Minkovski spacetime. The previous approach is equivalent to this one if the description of 6d spacetime is careful.

Consider the first case at the moment. 6d twistors are objects

(25)   \begin{equation*} T=\left(\omega^\alpha,\pi_\alpha\right)\in \mathbb{C}^8\end{equation*}

whose structure is determined by the norm of the spinors for 8d complex orthogonal group O(8;\mathbb{C}) given by:

(26)   \begin{equation*}\langle t, t'\rangle=\omega^\alpha\pi'_{\alpha}+\pi_a\omega'^a=0\end{equation*}

Points in 6d complex Minkovski spacetime are represented by a 4\times 4 antisymmetric matrix Z^{\alpha\beta}=-Z^{\beta\alpha}. The Penrose relation becomes

(27)   \begin{equation*}\omega^\alpha= Z^{\alpha\beta}\pi_\beta\end{equation*}

with \alpha,\beta=1,2,3,4. This equation has a nontrivial solution if the twistors T are pure, i.e., if

(28)   \begin{equation*}\langle T,T\rangle=2\omega^\alpha\pi_\alpha\end{equation*}

that is, they have vanishing O(8,\mathbb{C}) norm. The points of the real 6d spacetime are representeed by 4\times 4 complex, antisymmetric matrices Z satisfying a reality condition in the form of

    \[Z=-\overline{Z}\]

    \[\overline{Z}=B^{-1}Z^+B,\;\; B=\begin{bmatrix} 0 & 1& 0 & 0\\ -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & -1 & 0\end{bmatrix}\]

and Z^+ denotes the hermitian conjugated matrix. This reality condition for Z is equivalent to the following equations

(29)   \begin{align*}\overline{\omega}^\alpha\pi_\alpha+\overline{\pi}_\alpha\omega^\alpha=0\\ \overline{\omega}^\alpha=\overline{\omega}^\beta(B^{-1})_\beta^{\;\;\alpha}\\ \overline{\pi}_\alpha=\overline{\pi}_\beta(B)^\beta_{\;\;\alpha}\end{align*}

The overline means complex conjugation. Indeed, this equation is in fact the condition of vanishing the V(4,4) norm. Thus, 6d twistors describe the points of the real Minkovski spacetime RM^6 if the following two norms are zero:

(30)   \begin{align*}\omega^\alpha\pi_\alpha=0\\ \overline{\omega}^\alpha\pi_\alpha+\overline{\pi}_\alpha\omega^\alpha=0\end{align*}

The first equation above is the O(8,C) norm, and the second equation is the U(4,4) norm. It means that 6d twistors describing points in RM^6 are, indeed, invariant under the quaternionic orthogonal group O(4,H), covering the six dimensional group O(6,2). The chain:

    \[O(4,H)\equiv U_\alpha (4,H)=O(8,C)\cap U(4,4)=\overline{O(6,2)}\]

is true as group isomorphism. Therefore, one can look for the quaternionic extension of 4D twistor formalism which can describe RM^6 Minkovski spacetime. Quaternions are algebraic objects

    \[Q=q_0e_1+q_1e_1+q_2e_2+q_3e_3\]

with

 

    \[e_ie_j=-\delta_{ij}+\epsilon_{ijk}e_k\]

and i,j,k=1,2,3. Real numbers are naturally embedded in quaternions. We can also define quaternionic conjugation

(31)   \begin{equation*}\overline{Q}=q_0-q_1e_1-q_2e_2-q_3e_3\end{equation*}

and the norm

(32)   \begin{equation*}N(q)^2=\vert Q\vert^2=\overline{Q}Q=q_0^2+q_1^2+q_2^2+q_3^2\end{equation*}

The quaternion algebra has a natural structure of euclidean 4d spacetime. Complex numbers can be seen too as certain subset of quaternion algebras. Identifying complex numbers is easy from quaternions, if you take the couple

(33)   \begin{equation*}Q=z_1+e_2z_2=(q_0+q_3e_3)+e_2(q_2+q_1e_3)\end{equation*}

In analogy to previous arguments, we can associate a Z-plane in 4D quaternionic space \mathbb{H}^4, in quaternionic twistor space as follows: take Z and associate to it the subspace by columns of 4\times 2 quaternionic matrices

(34)   \begin{equation*}\begin{bmatrix} e_2Z\\ I_2\end{bmatrix}\end{equation*}

By a similar procedure, we get quaternionic Penrose relations

(35)   \begin{equation*}\omega^{\dot{\alpha}}=e_2Z^{\dot{\alpha}\beta}\pi_\beta\end{equation*}

and \alpha,\beta=1,2. The quaternionic twistor will be now t=(\omega^{\dot{\alpha}},\pi_\beta). A real 6d Minkovski spacetime point is desecribed by a 6d vector

    \[X^\mu=(X^0,\ldots,X^5)\in RM^6\]

which can be mapped on a quaternionic hermitian 2\times 2 matrix

    \[\mathbb{X}=\begin{pmatrix}X^0+X^5 & X^4+X^ke_k\\ X_4-X^ke_k & X^0-X^5\end{pmatrix}\]

and k=1,2,3, with e_k the imaginary quaternion units. The reality condition X=X^+, with the plus meaning quaternionic conjugation and transposition, is equivalent to the following condition for quaternionic twistors t:

(36)   \begin{equation*}\langle t,t\rangle=\overline{\omega}^\alpha e_2\pi_\alpha+\overline{\pi}_\alpha e_2\omega^\alpha=0\end{equation*}

Thus, twistors t describe a poitn of RM^6 if their norm, on O(4,H)=U_\alpha(4,H), vanishes. Using the decomposition of quaternionic coordinates of twistor in quaternions one can show that Penrose relations are equivalent to the incidence relations, so descriptions of RM^6  by 6d complex twistors and D=6 quaternions are equivalent.

5. Conclusions

We can summarize some simple possible definitions of twistors related to the written stuff:

  • A twistor is a solution of the twistor equation \nabla_{A'}^{\;\; (A}\omega^{B)}=0, that is called twistor space.
  • A spinor of the conformal group (with two elements!) is a twistor.
Features:
  • Point in twistor space=null-line in Minkovski spacetimes.
  • Point in Minkovski spacetime=line in twistor space.
Twistors are generally available in arbitrary complex dimension for conformal groups, BUT, there is a nice emergent relation with commuting spinors in SL(2,\mathbb{K}), where \mathbb{K} is a division algebra, special relativity in D spacetime, supersymmetry and twistors in dimensions D=3,4,6,10 for the Green-Schwarz action for the superstring, twistors in D=4,5,7,11 for the supermembrane and the Lorentz vector of those dimensions. Superspace version of these arguments are available. There is a match between the number of spacetime dimensions of the (super)p-brane embedding, the number of supersymmetries and the dimension of the p-brane.

 

Twistors are a powerful tool for spacetime geometry in complex manifolds or even real manifolds. We note that 2 approaches seen are equivalent only for real spacetime, though. 6D spacetime is special. 6D spacetime can be xtended in two nonequivalent ways: by complexification or quaternionization methods! The quaternionic formulation of twistor theory leads to serious issues in general, and that is why it is not popular. The main issue is the quantization of twistors because of non-commutative of quaternions. However, the description of 6d spacetime with quaternionic procedures allow us to use the same geometry as in case of the complex description of 4d spacetime! In fact, it is natural to extend this to octonions and 10d spacetime. The problem there is that octonions are generally non-associative and matrix multiplications become nasty due to that: octonionic matrices are hardly associative!

There can only be one!!!!!!

LOG#248. Basic string theory.

3 posts to finish the TSOR adventure!

This blog post will introduce you to basic string theory from my own biased viewpoint. My blog, my rules. I think you concede that!

What is the Universe made of? This really ancient question (both philosophically and scientifically addressed differently from time to time) has not an ultimate answer. Today, we believe there are atoms (elements) that make up the chemistry of life we need. Atoms are not fundamental! Since the 19th century and through the 20th century we discovered lots of particles: electrons, protons, neutrons…An even worst, protons an neutrons are now believed not to be fundamental but made up from quarks. There are 6 quarks (6 flavors or types: up, down, charm, strange, top, bottom). There are 6 leptons (electron, muon, tau, electron neutrino, muon neutrino, tau neutrino). Moreover, there are gauge fields: photon, gluons, W bosons and Z bosons, plus the Higgs bosons found in 2012, 8 years ago. BUT, particles are not fundamental either! They are excitations of quantum fields. Quantum fields are fluid-like stuff permeating the whole Universe. Missing something? Of course: gravity. The Standard Model does NOT contain gravitational fields as gauge fields. Gravity is described not by a Yang-Mills theory but with General Relativity, a minimal theory treating spacetime as the field potential. The metric of spacetime is somehow the field of gravity (more precisely, the metric is the gravitational potential function). Using a torsion-less theory of gravity, you find out that the so called affine connection (the Christoffel symbols) are the equivalent of the classical gravitational field. \Gamma\sim \partial g. However, the nature of the gravitational field is not the affine connection, but the spacetime curvature. The tensor field defined by

(1)   \begin{equation*} G_{\mu\nu}+\Lambda g_{\mu\nu}=f(g,\partial g)\end{equation*}

describes the free source theory of relativistic gravity known as general relativity. Gravitons are not included in principle in the theory, gravitational waves are derived in the weak limit of perturbations of flat spacetime g_{\mu\nu}=\eta_{\mu\nu}+\varepsilon h_{\mu\nu}. Thus, gravitons are hypothetical transmitters of gravity for quantum gravity. Indeed, if you believe in quantum mechanics, gravitons are inevitable quanta behind the gravitational waves. In summary, we get bosons and fermions. We have reduced the ancient periodic table to a new set of fundamental ingredients. Not counting colors, or helicities or antiparticles, we have 6 quarks, 6 leptons, the photon, the gluon, the W, the Z and the H. Thats all, folks. 17 particles. 17 fields. Why to stop there? 17 are too many. What if some of them are again NOT fundamental and made of other stuff? Are they truly point-like?

Purely point-like particles provide some known issues. The first issue is the divergence of electric (or gravitational or alike) potential energy. Suppose the electron were a uniform sphere with radius r_e (density \rho_e). The electric energy of such an sphere is

(2)   \begin{equation*} U_e=m_ec^2=\dfrac{3K_Ce^2}{5r_e}\end{equation*}

If purely point-like, you see there is an infinity electric energy! Giving up the 3/5 factor arising from spherical symmetry, the classical electron radius is

(3)   \begin{equation*} r_e=\dfrac{K_Ce^2}{m_ec^2}\sim 10^{-15}m\end{equation*}

Electrons are known to be fundamental much below this scale. In conclusion: a pure point-particle is meaningless at the end, because of the divergence in classical electrodynamics of the electromagnetic energy. But, as you know since you read me here, electromagnetism is not classical at the current state-of-art of the theory. Quantum electrodynamics (QED) is the theory we should use to answer the above questions. Well, even when it is a disaster as a single theory (it produces a so called Landau Pole at very very high energies), we know QED is an approximation to electroweak theory. However, a mystery is the final destiny of the divergence above even in the quantum regime!!!!!!! Yes, for all practical purposes, the QED is an effective theory that, in the end, neglect the infinities due to refined versions of the above ultraviolet (short scale, high energy) catastrophe. The fact that calculations work fine even neglecting those infinities is puzzling. An unsolved theoretical problem to understand why those infinities can be ignored without (excepting the vacuum energy problem) leaving us a completely nonsense mess of theory. The problem of quantum radiative self-corrections to energy-mass of particles is even worse with gravity. Using similar techniques to those used in the Standard Model do NOT work. Then, what is a particle or graviton? Two main alternative paths are possible:

  • 1) Save gravity and pointlike particles, but change quantization.  This path is the one donde in Loop Quantum Gravity (formerly non-perturbative canonical quantum gravity) or LQQ. Also, this method is followed up by the approach called Asymptotic Safety, and some minor approaches.

  • 2) Save usual quantisation, but give-up a purely point-like nature of particles. That’s string theory, or now p-brane theory (even when a first quantized theory of p-branes is not yet available for p>1).

Adopting the second approach, if not a point, how do we describe strings or stuff? After all, the simplest object beyond structureless points are strings. We need some kinematical and dynamical basics for doing the math. We should describe strings in a natural invariant way, sticking to special relativity and quantum rules, even trying to extend that to general relativity. It is a surprising historical remark that strings do describe not only strong forces, but also have spin two excitations, aka gravitons! The miracle of string theory, beyond point particle theories, is that it allows us to describe a consistent theory of gauge interactions including the gravitational field. Plus a bonus: string theory (modulo some details concerning the critical dimension and the number of quantum fields) is free of UV divergences in perturbation theory. It is a finite theory of quantum gravity from the beginning. There is no free dimensionless parameters though. There is a link between the string coupling g_s, the string tension \alpha' and the string length L_s=\sqrt{\alpha'}, dependent of the type of string theory, the spacetime dimensions and the nature of the fundamental objects (not only string-like!) in the spectrum of the theory.

String theory fundamental object is a single tiny string, generally speaking L_s\sim L_p in old string theory, but massive states can mismatch that. For instance, if  L_s=g_s^2L_p, so you could get objects and string greater than Planck length in non-perturbative fashion with g_s>>1. Or, you can have things below the Planck scale if soft enough. Particles or field are really excitation modes of fundamental strings (in critical string theory). Different modes correspond to different fields or particles.

1. Main mathematics

Classical points are given a a line of world in spacetime, i.e., x^\mu (\tau). Strings should be described by a surface area in spacetime, i.e., X^\mu(\tau,\sigma). Here, 0\leq \sigma\leq L_s and strings can be open or closed strings. Open strings have X^\mu(\tau,0)=f^\mu(\tau,0) and X^\mu(\tau,L). Closed strings have periodicity in the space-like worldsheet, such as X^\mu(\tau,\sigma)=X^\mu(\tau,\sigma+L_x). You see that X^\mu(\sigma,\tau) depends on the target spacetime time, as any string in D-spacetime, D=d+1 minkovskian spacetime is the usual selection, consists really of a set of D (generally scalar) fields. The next question is: what is the equation of motion of a FREE string? Well, without giving more advanced details, and noting the similarity between strings and waves, recall that free point particles have the equation of motion (in newtonian mechanics, but also single special relativistic case):

(4)   \begin{equation*} \dfrac{d^2x^i}{dt^2}=\ddot{x}^i=0\rightarrow \dfrac{d^2x^\mu}{d\tau^2}=\ddot{x}^\mu=0\end{equation*}

or

(5)   \begin{equation*} \dfrac{\partial^2}{\partial \tau^2}X^\mu(\tau)=\partial_{\tau\tau}X^\mu(\tau)=\partial^2 X=0\end{equation*}

Then, it would be natural for free strings to have the following equation of motion

(6)   \begin{equation*}\left[\dfrac{\partial^2}{\partial \tau^2}-\dfrac{\partial^2}{\partial\sigma^2}\right]X^\mu(\tau,\sigma)=\left(\partial_{\tau\tau}-\partial_{\sigma\sigma}\right)X^\mu(\tau,\sigma)=\overleftrightarrow{\partial}X=0\end{equation*}

You can see that D-dimensional free strings are only a set of D-dimensional 2d wave equations. Strings carry energy and momentum, but also spin degrees of freeedom. Generally speaking, we have that the string tension and the string length are related via L_s=2\pi\sqrt{\alpha'} (some people usually prefers L_s=\sqrt{\alpha'} as normalized string length \overline{L_s}=L_s/2\pi. The string also owns a typical energy scale:

(7)   \begin{equation*}M_s=L_s^{-1}\end{equation*}

Experimentally, circa 2020, we know that the string mass scale is in a range

    \[3.5TeV\leq M_s\leq M_P\leq 10^{15}TeV\]

You could try to generalize the above for classical p-branes (p is the number of space-like dimensions) in D=p+1 spacetime, and classical (p,q)-branes (p is the number of space-like dimensions, q is the number of time-like dimensions) in D=p+q spacetime as follows:

(8)   \begin{equation*}\displaystyle{\left[\sum_{i=1}^q\dfrac{\partial^2}{\partial \tau^2_i}-\sum_{j=1}^p\dfrac{\partial^2}{\partial\sigma^2_j}\right]X^\mu(\vec{\tau},\vec{\sigma})=\left(\vec{\partial}_{\tau\tau}-\vec{\partial}_{\sigma\sigma}\right)X^\mu(\vec{\tau},\vec{\sigma})}\end{equation*}

and where

(9)   \begin{equation*}X^\mu(\vec{\tau},\vec{\sigma})=X^\mu(\tau^a,\sigma^b)=X^\mu(\tau^1,\cdots,\tau^q,\sigma^1,\cdots,\sigma^p)\end{equation*}

Thus, general p-branes /(p,q)-branes or extended objects are described by hyperbolic/ultrahyperbolic wave equations in D-dimensional target space-time. Only the first quantized version of strings is known at current time.

2. Dynamics of strings

General solution for 2D dimensional wave equations are available:

(10)   \begin{equation*}X^\mu(\tau,\sigma)=X^\mu_R(\tau-\sigma)+X^\mu_L(\tau+\sigma)\end{equation*}

This describes a traveler string wave as the sum of a stringy right-wave/oscillation plus a stringy left-wave/oscillation. Assuming periodic boundary conditions, the most general solution is a Fourier expansion

(11)   \begin{equation*}X^\mu(\tau,\sigma)_{R,L}=\dfrac{x^\mu}{2}+\dfrac{\pi \alpha' p^\mu_{R,L}(\tau\pm\sigma)}{L_s}+i\sqrt{\dfrac{\alpha'}{2}}\displaystyle{\sum_{k\in\mathbb{Z}\neq 0}\dfrac{\alpha^\mu_{k(L,R)}}{k}e^{-i\frac{2\pi k}{L_s}(\tau\pm\sigma)}\end{equation*}

Here, x^\mu, p^\mu are the center of mass and center of momentum of the string. The first quantization of strings is simple and well understood (unlike general p-branes!). Quantization of every wave oscillation in a single string is a mode. Every mode is quantum harmonic oscillator. A string is secretly a field or infinite number of harmonic oscillators.

Every excitation mode \alpha^\mu(L)_k, \alpha^\mu(R)_k represents a harmonic oscillator. States in vacuum are labeled by center of mass momentum \vert 0,p_i\rangle. Excitations of L/R type gives a frequency 2\pi k/L_s. And finally, the quantum string state is a ket:

(12)   \begin{equation*}\vert Q_s\rangle=\displaystyle{\prod_{k>0,\mu}\left(\alpha^\mu_{-k}(R)\right)^{n_{k,\mu}(L)}\prod_{k>0,\mu}\left(\alpha^\mu_{-k}(R)\right)^{n_{k,\mu}(R)}\vert 0,p\rangle}\end{equation*}

Remark: consider equal number of left/right moving stringy quanta. Then, there is a Tower of String EXcitations  characterized by oscillation number N_L=N_R. The first levels of this quantization provides:

(13)   \begin{align*}N_L=N_R=0,\mbox{vacuum state},\;\;\vert Q_s\rangle=\vert 0,p\rangle\\ N_L=N_R=1,\mbox{first excited level},\;\;\vert Q_s\rangle=\varepsilon_{\mu\nu}\alpha^\mu_{-1}(L)\alpha^\nu_{-1}(R)\vert 0,p\rangle\\ \vdots\end{align*}

For bosonic strings, this represents a spectrum spin-dependent

(14)   \begin{equation*}M^2=4M_s^2\cdot (N-a)\end{equation*}

with a=1, N_L=N_R=N. The tachyon mode N=0=N_L=N_R is erased with supersymmetry (SUSY) in superstring theory (in 10d spacetime!). M_s sets the string scale, and N_L=N_R=1 represents massless tensorial states (gravitons!). In the higher energy regime, we expect resonances depending of mass and spin, roughly M^2\simeq NM_s^2, with N\simeq J. Just a further comment: closed strings give sense with the above argument to graviton-like excitations, where do photon-like degrees of freedom come from? From open strings! Copy-cat the same program of classical solutions and quantisation with the right suitable boundary conditions. The result is that free endpoints can move freely along an object called a Dp-brane, or (p+1)-dimensional hypersurface of spacetime (cf. Polchinski 1996). With boundary conditions assumed, the massless gauge U(1)-like fields will be

(15)   \begin{equation*}\vert Q_s\rangle=\varepsilon_{\mu}\alpha^\mu_{-1}(L,R)\vert 0,p\rangle\end{equation*}

String excitations along 1 Dp-brane: U(1) gauge fields  A^i, i = 0,\cdots,p, with N-coincident Dp-branes is promoted to U ( N ) gauge symmetry N\times N gauge bosons! Moreover,  Dp-branes at intersection provides  matter fields (chiral fermions) in bifundamental reprentations  (\overline{N_a}, N_b). Thus, we have a stringy/Dp-brane machine to generate \Prod_i U(N_i) gauge field theory and derive the Standard Model. The problem is: there are too many ways to do it! Even if with string theory, gauge theory implies gravity, or gravity and gauge theory are included in the same set-up, we do not know how to generate uniquely the Standard Model, and the vacuum we call our Universe… However:

  • Strings interact by joining and splitting. Open string endpoints can join to form a stable closed string. (The converse is not always true).

  • Behaviour consistent with universality of gravity: photons provide gravity, and somehow, gravity is the square of a gauge theory. That is the motto gravity=YM^2 popular theses days.

  • In string theory, gauge interactions and gravity are not independent. They are linked by the internal consistency of the theory. String theory is the only known theory with this property. Even more, consistency implies critical dimensions: 26d the original bosonic string theory, 10d the superstring, and 11d M-theory (12d F-theory, 13d S-theory,\ldots).

  • UV finiteness and the end of divergences. The general picture is that string theory has an intrinsic UV regulator (the string length). High energy scattering probes that lenght and non-local behaviour is obtained. Point-like interaction vertices are smothened/erased. Quantitatively precise, loop diagrams in perturbative string theory can be checked to be FINITE. No more UV divergences.

  • Strings are special? Can a particle have even higher-dimensional substructure? Model particle as a membrane (Dirac pioneered this with the electron-membrane model): 2 spatial dimensions .Tubes of length L and radius R have spatial V=LR. Quantum fluctuations of p-branes: Long, thin tubes can form without energy cost and that is an issue. Membranes automatically describe multi-particle states. No first quantisation of higher-branes à la strings possible. Quantum membranes have continuous spectrum no one know how to discretize like strings.

However, string theory puzzles further. It implies:

  • Internal consistency conditions make further predictions: spacetime is not 4-dimensional, but 10-dimensional (26d in bosonic string theory with no supersymmetry).

  •   In 10 dimensions there is only one unique type of string theory. It has many equivalent formulations which are dual to each other.

  • Witten 1995 showed that there is a single theory, dubbed M-theory, with six duality-related limites: 11d SUGRA, Heterotic SO(32), Heterotic E_8\times E_8, Type IIA,  Type IIB, and Type I.  These 6 theories are related with a web of T-dualities and S-dualities, and a general U-duality group much more general.

  • The 10-dim. theory/11d maximal SUGRA/11d M-theory is supersymmetric, and every boson has a fermionic superpartner. This does NOT imply that supersymmetry must be found at LHC. SUSY energy scale can be in any point between tested energies and Planck energy.

  • Superstring theory is well-defined and unique (up to dualities) in 10d/11d and lower/higher dimensions(higher dimensions are usually neglected due to higher spins or extra time-like dimensions).

  • The low energy regime E<<M_s  os superstring predicts indeed Einstein general relativity plus stringy corrections with several gauge fields.

  • Within the full 10d bulk a graviton propagates, and along lower dimensional D-branes a gauge boson propagates.

  • Within the high energy regime E\geq M_s, characteristic tower of massive string excitations provide  measurable (in principle) as resonances (Kaluza-Klein states)! Energy dependence of interactions differs from field theory.

  • The scattering amplitudes are ultra-violet finite without the need for renormalisation. It is believed that string theory interactions represent the fundamental (as opposed to effective) theory, but heavy Dp-brane states also arised in the second string revolution.

3. Issues with contemporary string theory

Our world/Universe is apparently 4d (3+1, minkovskian metric). We need to compactify extra unobserved string theory dimensions, with or without brane world metrics, to derive our Universe (SM plus gravity in the form of General Relativity). The problem is, as I told you before, there is no a unique way to do it. Thus, the model building to get the Universe as a single solution is doomed in current string theory! The set of every possible string theory compatification providing a Universe like ours has a name: the string landscape.

 Superstring theory is well-defined only if spacetime is 10 d/11 d as M-theory. It is thus an example of a theory of extra dimensions. You can build up string theories having point particles 0-branes, strings 1-branes, 2-branes (M-theory, 11d SUGRA), and so on. Extra dimensions are compact and very small. For instance, pick a 5d world with coordinates X^M=(x^\mu, x^4)=(x^0,x^1,x^2,x^3;x^4). The extra invisible dimension is folded/wrapped a tiny circle S^1 with radius R_4. If this radius becomes very tiny, the world will appear to be 4d, but, with enough energy, you could reach that dimension. To arrive at 4 large extra dimensions we need to compactify 6 dimensions (or 7 in M-theory). The simplest solution (of course, not the only solution), every dimension is a circle , i.e. internal space is a six-dimensional torus :

(16)   \begin{equation*}T^6=S^1\times \cdots\times S^1\end{equation*}

More general 6-dimensional/7-dimensional spaces allowed (Calabi-Yau manifolds were popular in the past). Every consistent compactification yields a solution to string equation of motions with specific physics in 4D. This gives you a landscape of theories.  Configuration of multiple branes are related to  gauge groups. The intersection pattern is related to charged matter and specifics of geometry is related to interactions (computable!).The field of model building or String phenomenology is to explore interplay of string geometry and physics in 4 dimensions.

The landscape of string vacua biggest puzzle: every consistent compactification is a solution to string equations of motion. Every 4d solution is called a 4d string vacuum. In 10d: All interactions uniquely determined. In 4d: Plethora of consistent solutions exists – the landscape of string vacua Existence of many solutions is typical in physics: Einstein gravity is one theory with many solutions! Pressing question: Consequences for physics in 4D physics? Solution to fine-tuning problems (Higgs, Cosmological Constant)? Harder in the Landscape!

Even worse is the more recent ideas of Swampland versus Landscape: Which EFT(effective field theories) can be coupled to a fundamental theory of QG? There is also a Swampland of inconsistent EFTs related to the Landscape of consistent quantum gravitational theories. Swampland conjectures of general scope, but not sharply proven. The Weak Gravity Conjecture is also analyzed from the viewpoint of the string theory landscape these times.

Is string theory as a framework for QG allows to test explicit conjectures? More ideas (for quantitative check of swampland conjectures and sharper formulation):

  • Study manifestations of swampland conjectures in string geometry.

  • String Geometry Geometry of compactification space involves Physics in 4d (or higher). Holographic principle is tested as well.

  • Strings as extended objects probe geometry differently than points. it opens door for fascinating interplay between mathematics and physics: new physics ways to think about geometry by translating into physics. For instance: classification of singularities in geometry, singularities occur when submanifolds shrink to zero size and branes can wrap these vanishing cycles and give rise to massless particles in effective theory.

  • String theories and the Landscape/Swampland give interpretation for classification of singularities in mathematics and guidelines for new situations unknown to mathematicians.

  • String theory is a maximally economic quantum theory of gravity, gauge interactions and matter.

  • Assumption of stringlike nature of particles leads to calculable theory without UV divergences.

  • Challenge for String Phenomenology: understanding the vacuum of this theory

  • String Theory as modern mathematical physics: deep interplay with sophisticated mathematics(e.g.: Mirror symmetry, D-brane categories,. . . ).

  • String Theory as a tool: Holographic principle like the AdS/CFT, or Kerr dS/dS correspondences. String Theory is a framework for modern physics.

4. Moduli and fluxes issues

String theories have extra field-theoretic degrees of freedom. Consider firstly the next four dimensional action

(17)   \begin{equation*}S=\int d^4x\sqrt{-g}\left(\dfrac{1}{2\kappa^2}R-\Lambda_{bare}-\dfrac{Z}{48}F_4^2\right)\end{equation*}

where F_4 is a four-form with solutions to the EOM(equations of motion)

(18)   \begin{equation*}F^{\mu\nu\rho\sigma}=c\epsilon^{\mu\nu\rho\sigma}\end{equation*}

It is easily probed that it gives a contribution to the cosmological constant/vacuum energy

(19)   \begin{equation*}\Lambda=\Lambda_{bare}+\dfrac{1}{2}\dfrac{Zc^2}{2}\end{equation*}

This gives rise to the moduli (flux compatifications) problem in string theory. In string theory, c is quantized, but you are provided many of such four-form (and even other grade forms!) contributions:

    \[\Lambda=\Lambda_{bare}+\dfrac{1}{2}\displaystyle{\sum_{i=1}^{N_{flux}}n_i^2y_i^2\]

If you have MANY y_i and N_{flux} is arbitrary, \Lambda can be tuned to a very somall value under VERY special conditions, but not all clear! You can try to get how many values you need of these vacua, to get N_{V}=N_{values}^{N_{flux}}, and see how many string theory solutions give you the SM plus Gravity Universe we live in! Terrible result: usual string theory gives you 10^{500} possible Universes in 10d/11d, or even worst, using F-theory technology, you guess an upper bound about 10^{272000} (Vafa)…

4.1. The string spectrum and M-atrix models

As strings need extra dimensions, they also have different quantum numbers in addition to common particle quantum numbers. The dimensional compactification provides the level n of Kaluza-Klein resonance. BUT, the winding number w around the extra dimension/s is also a purely stringy quantum number. For KK-modes:

(20)   \begin{equation*} m_{KK}c^2=E_{KK}=\dfrac{\hbar c}{R} \end{equation*}

and for a the winding w-modes

(21)   \begin{equation*} m_{W}c^2=\dfrac{\hbar c w R}{L_s^2} \end{equation*}

so finally, including the R/L excitation modes and the continous part of the quantized string, we get with c=\hbar=1 units:

(22)   \begin{equation*} \tcboxmath{E^2=m_{s,0}+p^2+\dfrac{n^2}{R^2}+\dfrac{w^2R^2}{L_s^4}+\dfrac{2}{L_s^2}\left(N_L+N_R-2\right)} \end{equation*}

Note the symmetry under n\leftrightarrow w and R\leftrightarrow L_s^2/R, known as T-duality. In non-pertubative settings, we also get a symmetry between the strong and weak coupling g_s\leftrightarrow 1/g_s (S-duality). It anticipated the Dp-brane revolution with monopoles in a famous Montonen-Olive conjecture.

Even when a general formulation of what M-theory is, one proposal was made called M-atrix theory. This M-atrix theory reveals that M-theory is a emergent model from the dynamics of a matrix model of D0-branes. Pick up a very large set of N\times N matrices X^a. These matrices (one for each space dimension, a=1,2,\ldots,D_1) represent the position of N-pointlike D0-branes. The energy is a hamiltonian object, formally

(23)   \begin{equation*}\displaystyle{H=\sum_{a=1}^{D-1}\sum_{i,j=1}^N\left(P^a_{ij}\right)^2+\sum_{a,b=1}^{D-1}\sum_{i,j=1}^{N}\left(\left[X^a,X^b\right]_{ij}\right)^2+\cdots}\end{equation*}

At low energies, these matrices all commute, their eigenvalues behave like normal spatial coordinates. Thus, ordinary spacetime is emergent from the M-atrix. But, in the regime where quantum fluctuations become large or strong, the full M-atrix structure, non-commutative and (sometimes non-associative in strings!) highly non-linear must be considered. M-theory is a highly non-local theory seen in this way. If it can be simulated with quantum computing is something to be tested in the future! Maybe, even M-theory tools will be more powerful than usual quantum computational tools.

 

5. Epilogue: a Multiverse of Madness nightmare?

Even when string theory or SUSY are very powerful, the duality revolution has touched two angular pieces they have left unanswered:

  • The selection of our vacuum or Universe. There are too many possible solutions, and that leaves us with the option of the Multiverse or that our vacuum could be not stable but metastable.

  • What is the fundamental theory/degrees of freeedom of superstring/M-theory. Duality maps change and challenge what is the fundamental entity in a dual theory. Holographic maps included, you can have a field theory with no gravity and change into a higher-dimensional gravitational theory and vice versa. You can calculate with magnetic branes instead electric branes. What is string theory? After all, we have no first quantized theory of membranes like those in M-theory yet.

Parallel to all this, Nima-Arkami Hamed discovered the amplituhedron: a new tool to simplify Feynman diagram computations based on higher-dimensional entities of polytopal class. They are intrinsically non-local. He has envisioned a future in which locality, SR, GR, and QFT are derived from a new set of structures. Long ago, when the 26d four-point function for the scattering of four tachyons is the Shapiro-Virasoro amplitude was derived to be:

    \[A_4 \propto (2\pi)^{26} \delta^{26}(k) \dfrac{\Gamma(-1-s/2) \Gamma(-1-t/2) \Gamma(-1-u/2)}{\Gamma(2+s/2) \Gamma(2+t/2) \Gamma(2+u/2)}\]

people wondered what was behind that. It showed to be the string…What is the p-brane generalization of this amplitude, if it exists?

In the end, the problem with String Theory is that we are not sure of what the symmetry of the whole theory is. We lack an invariant notion/relativity+equivalence principle for string/p-branes! There are only a few ideas circulating about what these new relativity principle/new equivalence principle are! But that is the subject of my final TSOR blog post, after a twistor interlude!

See you in my next blog post!

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…

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