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

Dualities and equivalences we have met till now (a biased short list):

1st. Energy=Mass.

2nd. Energy=Momentum.

3rd. Energy= $\sqrt{\mbox{mass}^2+\mbox{momentum}^2}$ .

4th. Space-Time merger.

5th. Equivalence principle of mass: inertial mass=gravitational mass.

6th. Energy=frequency (quantum).

7th. Energy=1/Length (de Broglie), or Momentum=1/Length (de Broglie part II).

8th. Probability= $\vert\mbox{Amplitude}\vert^2$ . Born’s rule.

9th. Energy=1/Time (Heisenberg).

10th. EM duality. Electric field=Magnetic field.

11th. S,T,U string dualities.

12th. ER=EPR.

13th. AdS/CFT correspondence.

14th. Fluid/gravity correspondence.

15th. GR=QM correspondence. Holographic principle.

16th. Kerr/CFT correspondende.

17th. Energy (internal)=Heat+Work.

18th. Free energy=enthalpy-TS.

19th. Energy=temperature (kinetic theory).

20th. Temperature=1/Mass (black holes).

21st. Temperature=acceleration (Unruh).

22nd. Acceleration=Mass (Caianiello).

23rd. Gravity=Yang-Mills squared.

24th. Gauge/gravity correspondence.

25th. $g_s$ equals mass. $E_s$ equals mass squared divided by charge.

What about the notion of distance in metric spaces? Well, distance is a metric concept admitting several definitions. The first one I want to show you is the taxi-cab or Manhattan distance:

(11) $\begin{equation*} d_M=\sum_i\vert p_i-q_q\vert \end{equation*}$

There is also a Chebyshev distance

(12) $\begin{equation*} d_C(p,q)=\mbox{max}_i\left(\vert p_i-q_i\vert\right)=\lim_{\k\rightarrow\infty}\left(\sum_{i=1}^n\vert p_i-q_i\vert^k\right)^{1/k} \end{equation*}$

In cryptography, you also have the Hamming distance: it is the minimal number of substitutions for a string length L to be the same after motion. Should we care about distance in gravity? Of course. Distances are measured with the metric in GR and metric theories of gravity. We could guess non-metric theories or even purely topological theories like Chern-Simons theories, BF theories and other theories like conformal field theories. The challenge and frontier is to guess a different theory of gravity explaining dark energy and dark matter in natural way, and to predict something new. Could lagrangians or hamiltonians be substituted by other concepts? Maybe. A 100 TeV collider will explore the nature of the mass generation via Higgs coupling testing. Bonus: to find out terms that induce proton decays, exotic phenomena and to understand better quantum field theory and quantum mechanics if possible. The emergence of both, gravity/spacetime and quantum mechanics/quantum field theory, could be possible in a more general setting. Maybe locality is not required in that setting. That idea comes from the amplituhedron approach, pioneered by Arkani-Hamed. After all, if we can derive stuff like phagraphene (penta-hexa-hepta-graphene) we can find out more things.

Hints:

1st. Black hole physics. Black holes are independent of their method of formation and have lots of internal degrees of freedom we can not understand yet. But we can use thermodynamics and describe them in terms of a few parameters: position, mass, angular momentum, charges, cosmological constant, acceleration, and other scalar hairs…The scalar hairs and other charges are a modern controversial addition. We do not know if black holes evolve unitarily yet. Even Hawking himself, admitting the possibility of breaking down quantum mechanics, introduced the superscattering operator to allow for it. QM could break down in black holes but nobody consider it a plausible option. The option of BH evaporation leaving a planckian remmant is yet considered. However, that is unsatisfactory since small BHs should have an arbitrary large amount of information. That is not the case if you consider the holographic principle. Take it seriously, and then you have to consider that BH are different. And maybe usual fields can not describe them because normal fields contain an infinite amount of information. There is a loophole in all this. QBH could have extra degrees of freedom and quantum and quantum numbers beyond the known one. This approach was launched by Robinson and Hawking, and it has been reconsidered due to the BMS symmetry. Simply, you would expect that GR reduces to SR in the low field approximation. Well, even when that happens often…The general case does not satisfy it! I mean: it is not generally true that GR reduces exactly to SR in low curvature/weak gravitational fields. The Poincaré group is not a simple reduction of the diffeomorphism group in weak fields. There are extra residual infinite-dimensional symmetries. They are called: supertraslations, superrtotations and superboosts. They have a conformal character. The idea of complementarity, saying that the whole information is conserved but can not be accessible for all the observers is a hard one. However, the firewall paradox showed that complementarity was not enough to save the information. We need something else. The current hope is that BMS charges will help us to recover the full information from the black hole. Perhaps, we will have to modify quantum mechanics including non-commutativity in space-time and a generalized Heisenberg principle. Perhaps, BH do not conserve baryon number. Perhaps, there are yet subatomic entities of space-time and matter we do not know.

2nd. String theory/M-theory. The most famous and worked out theory of quantum gravity is not understood yet. It has merits, but today (circa 2020, July 21st), it has no notion of fundamental symmetry or even worst, we can not understand what string theory is…Let me be clear: I am not anti-string theory. I only lack a precise definition of the theory. For instance, consider M-theory, the eleven dimensional theory that has fundamental M2-branes as d.o.f., and dual M5-branes as its fundamental objects and no strings! BPS p-branes in M-theory, in $\mathcal{R}^2\times T^{11-d}$ can be listed:

1st. Kaluza-Klein (KK) modes. p=0 branes. Tension equals 1/R in general.

2nd. Wrapped M2 branes. $p\leq 2$ , and tension $M_{11}^3R^{2-p}$ . Also, tension could be

$M_d^{(d-2)/3}R^{(d-5-3p)/3}$ .

3rd. Wrapped M5 branes. $p\leq 5$ , and tension $M_{11}^6R^{5-p}$ . Also, tension could be

$M_d^{2(d-2)/3}R^{(2d-7-3p)/3}$ .

4th. The KK monopole. Dimension p=d-4. Tension $M_{11}^9R^{12-p}$ or $M_d^{d-2}R$ .

Here, $M_{11}$ is the 11d Planck mass related to the 11d Planck length in a canonical fashion by dimensional analysis. The decoupling of KK modes, with R=0 holding in d-dimensional space and with planckian mass, provides

(13) $\begin{equation*} M_d^{d-2}=M_{11}^9R^{11-d} \end{equation*}$

and

(14) $\begin{equation*} T_pT_{4-d-p}=M_d^{d-2} \end{equation*}$

Thus, the full QM N=8 SUGRA is M-theory on a heptadimensional torus. Hommer Simpson would be happy of this doughnut. In d=3, the KK monopole is an instanton, and its mass is $m=m_0\exp (-\alpha \Delta\varphi/M_p)$ . Therefore, quantum fluctuations of fields or the combination of the fluctuations with coupling constants generate arbitrary masses. The string spectrum is more complicated than we expected. For open strings:

(15) $\begin{equation*} M_s^2=\dfrac{1}{\alpha'}(N-1) \end{equation*}$

For closed strings

(16) $\begin{equation*} M_s^2=\dfrac{2}{\alpha'}\left(N+\overline{N}-2a\right)=\dfrac{1}{\alpha'}\left(2(N+\overline{N})-\dfrac{D-2}{8}\right) \end{equation*}$

Finally, the closed string spectrum plus compactified dimensions reaed

(17) $\begin{equation*} M_s^2=\sum_i\left[\left(\dfrac{m_i}{R_i}\right)^2+\dfrac{1}{\alpha'}(n^iR_i)^2\right]+\dfrac{2}{\alpha'}\left(N+\overline{N}-2\right) \end{equation*}$

There, $\sqrt{\alpha'}=L_s$ is the string length. There is a QM-SR-GR-QFT-String Theory-M-theory connection clear. Even supersymmetry seems inevitable. Even when the version of SUSY is unknown, it happens, even in black holes! Branes have 3 known actions

(18) $\begin{equation*} S_{NG}=-T_p\int d^{p+1}x\sqrt{-\mbox{det}\left(\partial_\alpha X^\mu\partial_\beta X^\nu\eta_{\mu\nu}} \end{equation*}$

(19) $\begin{equation*} S_{Dp}=-T_p\int d^{p+1}x \sqrt{-\mbox{det}\left(\eta_{\mu\nu}+\partial_\mu Y^i\partial_\nu Y^jg_{ij}\right)} \end{equation*}$

(20) $\begin{equation*} S_{DBI}=-T_p\int d^{p+1}x=-T_p\int d^{p+1}x\sqrt{-\mbox{det}\left(\eta_{\mu\nu}+F_{\mu\nu}\right)} \end{equation*}$

We don’t understand yet duality! Suppose a p-brane carries a charge $Q_p$ , that is, its flux is $\phi_e=Q_p$ in certain units. Take

(21) $\begin{equation*} \phi=\int_{S^{D-p-2}}\star F_{p+2}=\int _{S^{D-p-3}}\tilde{A}_{D-p-3} \end{equation*}$

The phase $\varphi=e^{i\phi \tilde{Q_{D-p-4}}}$ has to be well defined, so

(22) $\begin{equation*} \phi\tilde{Q}_{D-p-4}=Q_p\overline{Q}_{D-p-4}=2\pi N \end{equation*}$

This is the generalized Dirac quantization condition for p-branes. $Q_eQ_m=2\pi N$ . Why?

(23) $\begin{equation*} Q_eQ_m=Q_p\overline{Q}_{D-p-4}=2\pi N \end{equation*}$

A p-form is an antisymmetric tensor having (in the abelian case for simplicity):

(24) $\begin{align*} A_p\rightarrow A_p+d\Lambda_{p-1}\;\;\; \mbox{gauge symmetry}\\ A_p=A_{\mu_1\cdots\mu_p}dx^{\mu_1}\wedge\cdots\wedge dx^{\mu_p}\\ F_{p+1}=dA_{p+1}\\ d\star F_{p+1}=0\\ d\tilde{A}_{D-p-3}=\tilde{F}_{D-p-2}=\star F_{p+2}=\star dA_{p+1} \end{align*}$

The dual of $F_{p+1}$ is $\tilde{F}_{D-p-2}$ . The dual of $A_p$ is $\tilde{A}_{D-p-3}$ , the dual charge of $Q_p$ (electric brane) is $Q_m=\tilde{Q}_{D-p-4}$ (magnetic) brane. The field equations are $F_{p+1}=dA_p$ , $\star F_{p+2}=\star dA_{p+1}$ or $\tilde{F}_{D-p-2}=d\tilde{A}_{D-p-3}$ , and $d\star F_{p+1}=0$ with $d\tilde{F}_{D-p-2}=0$ . Furthermore, the relation of gravity and strings follow from the dilaton-metric relation

(25) $\begin{equation*} G_{\mu\nu}^E=e^{-\frac{2\phi}{D-2}}G_{\mu\nu} \end{equation*}$

and the gravitational constant-string coupling/tension relationship

(26) $\begin{equation*} k_g=g_s\alpha'^{\frac{D-2}{2}} \end{equation*}$

3rd. Fluid/Gravity/Gauge connections. Note the similarity between gauge field and velocity field:

(27) $\begin{equation*} u=u_idx^i\rightarrow A=A_\mu dx^\mu \end{equation*}$

Note the analogy between vorticity and gauge field

(28) $\begin{equation*} \omega=du\rightarrow F=dA \end{equation*}$

Note the striking similarity between enstrophy action and Yang-Mills action:

(29) $\begin{equation*} E=\int\omega\wedge \star\omega\rightarrow S(YM)=\int F\wedge\star F \end{equation*}$

So, the world is a dimensional reduction from higher dimensional M-theory. Moreover, the 11d M-theory and 11d SUGRA provides the Dirac quantization condition

(30) $\begin{equation*} T_2T_5=2\pi N \end{equation*}$

up to multiplicative constant in the left-handed size.

Form language is very powerful. Let $\omega$ be a 0-form (function), and let $\eta$ be a 1-form (dual to a 1-vector). Then classical differential operators can be reviewed as follows…

(31) $\begin{equation*} d\omega=\dfrac{\partial f}{\partial x}dx+\dfrac{\partial f}{\partial y}dy+\dfrac{\partial f}{\partial z}dz \end{equation*}$

The dual of this object by the musical isomorphism is the gradient 1-vector $\nabla f=\vec{F}$ . Moreover, this result is also independent of the number of dimensions. Now, take a 1-form $\eta=A_idx^i$ , dual of the 1-vector $\vec{\eta}=A_ie^i$ via musical isomorphism. Then, taking the Hodge dual, applying the exterior differential and taking again the Hodge dual you will get the divergence:

(32) $\begin{equation*} \star d\star \eta=\mbox{div}\vec{\eta}=\sum_i \partial_i A_i \end{equation*}$

Finally, the rotational or curl in any dimension can also be calculated with the aid of differential form in a straightforward fashion:

(33) $\begin{equation*} d\eta=\sum_i\partial_iA_j-\partial_jA_idx^i\wedge dx^j \end{equation*}$

The rotational or curl as vector (not as bivector!) is the Hodge dual of this. Only in D=3 dimensions, the Hodge dual of the 2-form $d\eta$ is a 1-form, and by musical duality and isomorphism, dual to a 1-vector! Also, the laplacian operator in 3d is a 0-form usually (or a 3-form by Hodge duality). However, in $D-2$ space dimensions, the laplacian reads

(34) $\begin{equation*} \Delta\omega=\nabla^2\omega=\star d\star d\omega=\left(\delta d+d\delta\right)\star d\eta \end{equation*}$

and where $\delta$ is the so-called codifferential. $\delta = (-1)^k \star^{-1} d\star$ . En 3d euclidean space $\star^2=1$ , but that is not true in general spacetimes. Form language is also used by physicists in M-theory. In terms of differential forms, the bosonic sector action of M-theory can be written in the following way:

(35) $\begin{equation*} S_b(11d)=\dfrac{1}{2k_{11}^2}\int R_{AB}\star_{11d} e^{AB}-\int\dfrac{1}{2}F\wedge \star_{11d} F-\int A\wedge F\wedge F \end{equation*}$

where $F_4=dA$ is a four-form from a 3-form, and $g_{AB}=(e^{AB}+e^{BA})/2$ . The field equations from this action are:

(36) $\begin{align*} \star_{11}(\iota_{X^B}P_A)=\dfrac{1}{2} \iota_{X_A}F\wedge \star\iota_{X_B} F-\dfrac{1}{6}g_{AB} F\wedge\star_{11} F\\ d\star_{11} F=\dfrac{1}{2}F\wedge F\\ dF=0\\ \nabla_{X^A}\epsilon=-\dfrac{1}{24}\left(e^AF-3F\epsilon^A\right)\epsilon e^Ae^B+e^Be^A=2g^{AB}\\ P_A=\iota_{X^B} R_{BA} \end{align*}$

SUGRA or supergravity solutions in 11d have solitons. In the framework of M-theory, not well-defined, we know that fundamental objects are 2-branes and 5-branes, not strings! A 2-brane is written as follows

(37) $\begin{equation*} ds^2=\left(1+\dfrac{k_2}{r^6}\right)^{-2/3}\eta_{\mu\nu} dx^\mu dx^\nu+\left(1+\dfrac{k_2}{r^6}\right)^{1/3}\delta_{mn}dx^mdx^n \end{equation*}$

(38) $\begin{equation*} ds^2=\left(1+\dfrac{k_5}{r^3}\right)^{-1/3}\eta_{\mu\nu}dx^\mu dx^\nu+\left(1+\dfrac{k_5}{r^3}\right)^{2/3}\delta_{mn}dx^mdx^n \end{equation*}$

These branes carry charges $Q_2$ and $P_5$ electric/magnetic charges:

(39) $\begin{equation*} Q_2=\sqrt{2}k_{11}\int_{\Sigma_8}\star J^{(3)}=\dfrac{1}{\sqrt{2}k_{11}}\int_{\partial \sigma_8}\left( \star F^{(4)}+\dfrac{1}{2}A^{(3)}\wedge F^{(4)}\right) \end{equation*}$

(40) $\begin{equation*} P_5=\dfrac{1}{\sqrt{2}k_{11}}\int_{\partial\Sigma_5}F^{(4)} \end{equation*}$

We algo have

(41) $\begin{align*} F_{mnpq}=3k_5\epsilon_{mnpqr}\dfrac{y^r}{r^5}\\ P_5=\dfrac{2\pi N}{\sqrt{2} k_{11}T_2}\\ k_2=\dfrac{k_{11}^2 T_2}{3\Omega_7}\\ Q_2P_5=2\pi N\\ A_{\mu\nu\sigma}=\epsilon_{\mu\nu\sigma}\left(1+\dfrac{k_2}{r^6}\right)^{-1}\\ R_{MN}-\dfrac{1}{2}g_{MN}R=k_{11}^2 T_{MN}\\ d\left(\star F^{(4)}+\dfrac{1}{2}A^{(3)}\wedge F^{(4)}\right)=-2k_{11}^2 \star J^{(3)} \end{align*}$

Thus, duality, holography and the AdS/CFT have challenged us with what we usually meant by fundamental. Even spacetime, is itself fundamental or made of something else? Lee Smolin and others before have speculated about the issue of atomic space-time. Even more, what is the symmetry of the fundamental ground state of quantum gravity pervades the approach of loop quantum gravity! Global Lorentz symmetry and Poincaré symmetry are NOT symmetries of the classical GR. They are only symmetries if the cosmological constant vanishes! There are some options tried in the literature:

-Broken Lorentz Invariance. It means (at least in principle) a privileged spacetime frame.

-Nothing new, only special QFT or conformal QFT.

-New symmetries: DSR, TSR, asymptotic symmetries (BMS, supertraslations, superrotations and superboosts, Virasoro symmetrey).

-SUSY/supersymmetry.

-Hypersymmetry or any other variant of graded symmetries.

-Quantum groups. E.g.: $SO_q(1,4)$ implies

(42) $\begin{equation*} P_\mu \sim \left(\dfrac{\sqrt{\Lambda} L_p}{\alpha}\right)^r\sqrt{\Lambda} M_{0,\mu} \end{equation*}$

If $r>$ you get a singular momentum, if $r<1$ you get Poincaré symmetry, and if $r=1$ you get a special spacetime known as $\kappa$ (kappa) spacetime. With gravity in 2+1d, the so called Kodama state is a good candidate for quantum state of gravity. Gravitational theories could be constrained topological field theories like BF theories or Chern-Simons theories. Anyway, normal and future colliders are yet far far away to test granular spacetime if you consider them Planck sized. Space atoms (even if spin networks) are about $10^{-99}cm^3$ small. Time atoms would be $10^{-43}s$ short. Far from current technology, yet…Note that thermodynamics in the 19th century hinted the quantum matter revolution, even the black body problem hinted the quantum electromagnetic revolution. Ironically speaking, the quantum gravity revolution would complete the cycle back to the origin and that is what black hole thermodynamics is also hinting. Modern speculations like the ER=EPR or gravity from entanglement are just revivals of quantum ideas for space and time. Quantum entanglement has been proved to reduce uncertainty (separability enlarges it), and relativity is affected by entanglement too. Note that gravity in $d+1$ dimensions is equivalente to $d$ quantum field theory. It from bit, or just gravity from information as Wheeler thought and said long ago. However, it is not clear the way…

About gravity…Strongly symmetric spacetimes have a beautiful metric:

(43) $\begin{equation*} \tcboxmath{ds^2=-f(r)dt^2+f(r)^{-1}dr^2+r^2d\Omega^2} \end{equation*}$

This metric unifies many types of black holes and pseudo-black holes. For instance:

-Minkovski spacetime is obtained if $f(r)=1$ with $c=1$ .

-De Sitter spacetime is got if $f(r)=1-\dfrac{r^2}{\Lambda^2}$ .

-Anti de Sitter spacetime is obtained with $f(r)=1+\dfrac{r^2}{\Lambda^2}$ .

-Schwarzschild spacetime (black hole) is got if $f(r)=1-\dfrac{2m}{r}$ .

-Kerr-Newman black holes are obtained with $f(r)=1-\dfrac{2m}{r}+\dfrac{Q^2}{r^2}$ .

-Hayward spacetimes (regular black holes) are got from $f(r)=1-\left(\dfrac{2mr^2}{r^3+2l^2}\right)$ .

Gravity is a quantum mystery…In part, …There is the Buchdahl limits for ECOs, you get the Killing-Yano tensor equation for hidden symmetris (like those in Kerr or Kerr-like blackholes):

(44) $\begin{equation*} \nabla_{\mu} \mathcal{Y}_{\nu_1\nu_2\cdots\nu_p}+\nabla_{\nu_1}\mathcal{Y}_{\mu\nu_2\nu_3\cdots\nu_p}=0 \end{equation*}$

For instance, Myers-Perry black holes have Killing tensors

(45) $\begin{equation*} \mathcal{Y}^{2(n-k)}=\star (\wedge h^k) \end{equation*}$

Not long ago, there are some inequalities called Higuchi bounds appearing in gravity and string theory. For higher spin theories in dS spacetime

(46) $\begin{equation*} M^2\geq s(s+1)H^2 \end{equation*}$

Instead, if you take a field of spin $j$ in $d-dim$ dS you have a mass-spin and mass helicity bounds:

(47) $\begin{equation*} \tcboxmath{M_j^2\geq H^2(j-1)(d+j-4)} \end{equation*}$

(48) $\begin{equation*} \tcboxmath{M_{j,h}^2\geq H^2(j-h-1)(d+j+h-4)} \end{equation*}$

String theories, by the other hand, have these bounds:

(49) $\begin{equation*} M_2^2=2M_s^2 \end{equation*}$

and

(50) $\begin{equation*} M_s^2\geq \dfrac{d-2}{2} H^2 \end{equation*}$

Should quantum gravity protect baryon/lepton number conservation or it will be violating them generally? Protons are stable because they are the lighter states with baryon number/charge different from zero…

(51) $\begin{equation*} B=\dfrac{n_q-n_{\overline{q}}}{3} \end{equation*}$

From neutrino physics, we suspect that baryon number symmetry is not fundamental. Also GUTs hint that. Baryogenesis from leptonogenesis is a common idea to create the Universe in such a way there will be more matter than antimatter, thus explaining why we exist after all. If not, note that the universe would be a complete equilibrium of photons and radiation coming from the total matter-antimatter annihilation.

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.

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

Challenge problem today: how many protons have human being? Easy one.

Challenge problem (II): how many dark energy do you have at a 5x5x4 cubic meters room?

“Nadie sabe realmente qué sentido tiene la vida, y realmente no importa. Explora el mundo. Todo lo que te rodea es interesante si se indaga lo suficiente.” R.P. Feynman.

“Elijan un destino que les satisfaga, pese a felicidades o infelicidades” JF.

(69) $\begin{equation*} A_\mu=\varepsilon_\mu e^{iXP} \end{equation*}$

(70) $\begin{equation*} h_{\mu\nu}=\varepsilon_{\mu\nu}e^{iXP} \end{equation*}$

May the quantum absement $\mathcal{A}\sim 10^{-79}ms$ be with you!!!!!

May the Chronons be with you!

$\theta_0=\dfrac{e^2}{6\pi\varepsilon_0 mc^3}=\dfrac{2K_Ce^2}{3mc^3}=\dfrac{2\alpha_e\hbar}{3mc^2}$

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