LOG#217. The 2 pillars.

Time of Cosmic Voyages https://www.youtube.com/watch?v=xEdpSgz8KU4

This is a long post, despite not being a special post (remember I make one of those every 50 posts). What are we going to see here? A five part post!

Part(I). A descriptive prelude.

  • Limitations of our electromagnetic observations? Reductionist vs. holistic visions? We were long ago limited by electromagnetic visible observations. No more today. The end of the reductionism with atomic or quantum physics seems to have reached a limit. Quantum entanglement and contextuality change the global view. Complementary holds.
  • Artifacts. Particle physics uses wonderful detectors and colliders. From cyclotrons, particle accelerators, synchrotrons, others. We will see why this whole stuff matters for your daily life and health.
  • Game rules. Relativity plus quantum mechanics. The two pillars of physics (central topic of this post!). Matter and energy interplay via E=mc^2, wavelength versus momentum interplay via duality \lambda=h/p.Precision and accuracy from these two big pillars make difficult a futher unification.
  • Uncertainty principle. Concentrated energy as a microscope up to scales \Delta X\sim \hbar c/\Delta E, L\sim 1/p. There are some generalized uncertainty principles out there.
  • From the 150 years old Periodic Table to the cosmic roulette of particles. Is a 1/7 reduction of the numbers of elements good enough to be kept forever and ever?
  • Action is quantized! That is the hidden mantra of Quantum Physics. Energy quantization or momentum quantization are secondary. The key magnitude is action (actergy!). Forget what you think about quantization a few seconds. Quantization main object is action (actergy). Other quantizations are truly derived from the action.
  • What is the relative force of the 4 fundamental forces? Compare them to different distance scales. See it below these lines.
  • QCD mass versus Higgs derived masses. You will learn (or remember, if you already knew it) that the mass of your body is essentially a QCD effects. Spoiling the Higgs, not make you massless. Higgs particles via the Higgs field and spontaneous symmetry breaking only give masses to elementary known particles. Of course, Higgs field is important since it manages atoms to become stable and bound as well,…Otherwise electrons would be massless. BUT, the proton mass is, as many other hadron masses, a QCD effect. Why nature chose to keep this accident? Well, it is fortunately for us, since atoms and complex objects depend on proton stability (or long-lifetime metastability), but I am not a big fan of the anthropic principle or why the laws of physics are so well tuned to life to emerge.
  • Protons are complex objects. Textbooks, circa 2020!, seems to be obsolete like the Terminator! I mean, how many of you keep thinking protons or neutrons like solid big balls instead of wibbly wobbly timey wimey stuff? Just joking, whovians. I know you know what the time vortex equation truly is. Are elementary particles balls?
  • Fine tuning of parameters, stars and the origin of the elements from the primordial nucleosynthesis at the early Universe.

The SM is formed by the following set of elementary particles (up to more complex counting systems including charges and polarization modes!):

17 particles only! Usual atoms are made only of the first generation as good approximation, so you passed from 118 (or more) periodic table elements to 17 particle types! That is a factor 7 reduction! A cosmic wheel way of representing these particles also exists:

Moreover, extremely short timelifes are for the Higgs and the top quark. Using the relationship between energy(mass) and lifetime, you get that Higgs bosons or top quarks life about 10^{-25}s or 10^{-26}s, that is about 0.1 or 0.01 yoctoseconds! However, we are far away from Planck time physics (about 10^{-43}s). The above circle seems a pokeball. Anyway, it is a tryumph of reductionism. Everything reduced to combinations of those particles. By the way, the Higgs particle determines at what distances particles interact. Particles that are Higgs-transparent are massless and they act on infinitely large distances. That is the case of electromagnetism and gravity. Subtle point, gluons, despite being massless, are confined into hadrons due to non-abelian features and to confinement. Top quarks interact strongly with the Higgs. strangly very strongly. That is why tops are heavy. Similarly, electroweak bosons W,Z also interact strongly with the Higgs, but a few lesser than the top, and get about 100GeV masses. Particles interacting more with the Higgs field and the higgs particles, are thus more massive. The Universe of particles is being explored right now at the LHC with 14TeV smashes. We see particles everywhere. From subnuclear distances to scales about the zeptometer at the LHC: Any particle is tagged with some particular properties called quantum numbers. Mass, electric charge, angular momentum and spin, parity, weak charges, hypercharges,…Note that you are made of a big number of particles. Being about 70kg, supposing protons are what make you massive, you are about 10^{29} proton units composite.

Fields in the continuum and particles in the discrete are not contradictory views. Particles are just the excitations of the fields. The usual picture of continuum versus discrete view of the Nature is turned into a much more complementary unified view today. Matter-energy and spacetime are made of FIELDS. There are only a few fields in the Universe, maybe manifestations from different mirrors of a single field (this is the unification dream, the final theory treating everything as a single force and field). Kant, Einstein, Faraday, Maxwell, Newton, Leibniz and many others have taught us a lot about these visions. During the 19th century, a new formulation of classical physics was built. It is a non-deterministic approach. It is based on lagrangian and hamiltonian dynamics. Systems are encoded into gadgets called lagrangians (or lagrangian densities in field theory). They overcome the limited and sometimes hard to apply newtonian dynamics \sum F=Ma. In newtonian dynamics, the problem to understand the Universe is reduced to understand what mass is (unanswered in newtonian physics!), and to know what are the forces of the Universe. In the rational mechanics lagrangian and hamiltonian methods, you are reduced to find out the symmetries of the problem, and to compute the minimal action such the equations follow from a minimal (or more generally a critical point) of the action functional. For first order lagrangians and classical hamiltonian dynamics, the equations read

    \[\dfrac{\partial L}{\partial q}-\dfrac{d}{dt}\dfrac{\partial L}{\partial \dot{q}}=0\]

    \[\dfrac{\partial \mathcal{L}}{\partial \phi}-\dfrac{d}{dX^\mu}\dfrac{\partial \mathcal{L}}{\partial \left(\partial_\mu \phi\right)}=0\]

for lagrangians of particles and fields respectively, and

    \[\dfrac{\partial H}{\partial p}=\dot q\]

    \[\dfrac{\partial H}{\partial q}=-\dot p\]

for hamiltonians of particles or fields. The first set of equations are named as Euler-Lagrange (EL) equations, and the latter are the Hamiltonian Equations (HE). The problem to compute the forces are turned into the problem of finding or guessing the lagrangian or hamiltonian, and the dynamics reduced to the understanding of the symmetries via the action principle and the Noether theorem. What is the problem with all this? The quantum mystery is just a mystery of the vacuum. Vacuum mysteries around you. Aristotle had a notion of purpose in his view of Nature. The act. With Galileo, we learned that there is act-less motions! Galileo refuted Aristotle view of the motion. That was further mathematically developed by Newton, Leibniz and others. What Galileo did experimentally and very naively matemathically, Newton and posterior scientists would do it precisely. However, it turns that classical physics, your own perception of the Universe is biased. Classical physics is only an approximation. That was indeed anticipated by the EL and HE approaches, and the action principle, where all the possible configurations in principle are there, but only one (the classical one by reasons we will see later), are tested by Nature. That Nature tests everything is indeed the main argument of quantum physics! The usual non-deterministic view of Nature in quantum physics is surely a bit deeper than the eventual EL or HE approaches to classical mechanics, and have stunned everyone since then. But it is true, at least, with the precision of our current experiments. Nobody knows the future for sure, but quantum mechanics and non-deterministic probabilistic statements are here to stay a long time (if not forever).

What is the point with quantum physics? Well, take for instance the neutron decay process

    \[n\rightarrow p+e+\overline{n}\]

It can happen, if free, in about 15 minutes (888 seconds more or less). Quantum physics tell you that you CAN NOT predict when the decay is going to happen. You are only allowed to ask by the probability of neutron to decay in certain time period. Particle decays are essentially quantum phenomena, and statistically poissonian. You can not predict when something will decay, but knowing some distribution of probabilities, and statistics, you can predict probabilities for events to happen. Thus, quantum physics is just a framework that tell you “probabilities of events”, instead of telling you what is going to happen, you can only ask what is the probability of something to happen? Of course, the caveat here is that even quantum mechanics has tricks…Under certain circumstances, you can find out imposible events or even sure events. The sun is surely wasting its hydrogen fuel. Quantum photons arrive to us thanks to quantum physics (the tunnel effect, essentially). NO matter if you hate the philosophy of quantum mechanics. It allows us to exist and live.

Well, can we do average? Yes. QM let you guess quantum averages from classical and quantum observables. Vacuum polarizes, and you can study particle production processes, even with antiparticles. Accelerator and colliders show that quantum fluctuations are not just bubbles. It is just like a mesh of beams spreading out and interacting with everything in the sorroundings. Quantum physics, then, say that every possible trajectory between A and B happens for quantum particles. Between A and B the states are undetermined (unless you measure, of course) and likely entangled. Quantum physics, in terms of M. Gell-Mann, is the supreme totalitarian theory. Gell-Mann’s totalitarian principle stated originally that everything that is not forbidden is mandatory, today it has been reformulated and upgraded.

Totalitarian QM principle: everything that is not forbidden can happen or will happen.

This principle is essential to have a broader view of what QM is (even if not completely understood for you or experts!). Moreover, any fundamental interaction in the SM has a vertex Y structure (giving up loop corrections) in its simpler terms.

How could we understand the totalitarian principle and the action formulation of classical AND quantum mechanics better? Let me begin stating that Feynman graphs, even if complex, are a useful part of modern physics. Quantum fluctuations and force fundamental interactions are usually represented by Feynman graphs. They represent events in spacetime. Just as SR represents relativistic events as light-cones or space-time diagrams, particle physicists use Feynman graps to model fundamental physics and interactions. Some examples from the SM:

That particles follow A SINGLE history is past. They follow up all the possible histories at once. Thus, Laplace’s dream of a machine predicting the ultimate future of the Universe can not accomplished totally in a QM world. We can only say what are the odds of our future…In fact, QM has to use approximations and statistics since mouses, men and women, elephants, or big things are composite from many particles. Predictions would become IMPOSSIBLE without an statistical and probabilistic approach. Certainly, it is also possible you will know the bayesian approach to probability and science. Well, QM is the ultimate expression of bayesianism in the scientific world. Of course, you can check that some statements are TRUE and FALSE. But only, with the precision of your experimental set-up and current theories or hypothesis. In summary, record this strongly: QM says only probabilities or decays, not when decays will happen…

What about the action principle? Action principle gets enforced in quantum mechanics. In fact, the reason why we “see” the world as classical is twofold: Planck constant is small, and classical trajectories, those who are minimal, provide the main weight to the quantum action. The classical action is a magnitude equal to mass times the proper time, or energy times the proper time (with c=1), modulo a minus sign:

    \[\mbox{Action}=-M\tau=-(\mbox{Mass}\cdot\mbox{Propertime})\]

The quantum amplitde is something like the sum

    \[A(i\rightarrow f)=\exp(iS_1/\hbar)+\exp(iS_2/\hbar)+\cdots+\exp(iS_n/\hbar)\]

This sum is a complex number! The big thing is that non-dominant non-classical paths “cancel” (or almost), and you are left with the minimal action principle of quantum mechanics. It is not that the other trajectories do not happen, it could happen. They are orthogonal and interfere destructively, the classical path is reforced or enhanced by quantum interference!  There is a simple gedanken experiment (sometimes even simulated via applets in some websites). Take a light beam projector with low intensity I. Take a device that counts the number of clicks when photons arrive. Of course, this depends on \lambda and \nu, the wavelength and frequency f of the light. You can count the clicks as they hit a sensor in the screen. Sometimes the light pushes 36 clicks, other times it pushes 16 clicks. In deed, you can write down a formula giving you a discrete relationship between the numbers of clicks and the path the photon followed to the detector, something like this

    \[N=\sum_{paths}\vert\pm 1\vert^2\]

The indeterminism is just result of the interference of the paths! But in this simple example, every trajectory has the same absolute value. Going further, the plus or minus one is due to the phase of a complex number. the action is A_1=-E_1t_1, A_2=-E_2t_2,…A_n=-E_nt_n in general. Divide the action by \hbar to get a pure dimensionless number, multiply by i and then get it exponentiated, so you calculate \exp(iS_j/ \hbar). The probability of any event is given by the sum over all possible path

    \[P=\vert \mathcal{A}(i\rightarrow f)\vert^2=\vert\sum_{\gamma} c_\gamma e^{iS_\gamma/\hbar}\vert^2\]

It shows that this probability is maximized with the critical action, that is the minimal action principle of classical physics. Equivalently, think about paths vectorially! The sum is optimized for classical trajectories, maximal action, minimum time or the shortest paths. Have you ever imagined to be quantumly teleported or abducted by an alien civilization to another galaxy? It could happen spontantenously too, but with very very low probability, despite that event is allowed by the laws of quantum physics. Unfortunately, there is no alien close to me to show me the Universe beyond our galaxy more directly.

Part(II). QFT: a tale of where we stand in high energy physics.

  • Vacuum is the main fundamental object in quantum field theory. It is indeed related to the Fourier expansion of field operarors

    \[\phi=\int\dfrac{d^3x}{(2\pi)^3}\left(a(p)^+e^{-ipx}+a(p)e^{ipx}\right)\]

The quantum realm is the world of the quantum vacuum or the quantum void. Call it voidium if you want. Quantum fluctuations allow you to even surpass the conservation of energy by tiny time shifts (commit a robe, nobody would notice if you return the money before the shop and the register machine open again; that is the essence of the uncertainty principle, everything that can happens will happen, everything? Well, not quite,but this is again another story).

  • What are the symmetries behind the known possible fields and interactions?

There are no forces in quantum physics, indeed, there are only interactions betwen particles. The formal distinction of interactions is just, likely, a misnomer to different classes of phenomena, but it is useful yet at these times. Why? Because if matter and energy are, well, related with E=mc^2, distinctions between mass or matter and energy are quite a question not of mass or energy, but other features, indeed related to the quantum world. Magic word: spin! Particles have angular moment or, internal rotatory properties we call spin. Fundamental forces are transmitted by entire-spin particles, matter fields are spin one-half particles and spin zero or one particles.

  • Bosons and fermions as glue and LEGO pieces making up everything. You and me are not so different after all.

Spreaded out from the origin of time at t\sim 10^{-43}s til the current 14Gyr\sim 10^{18}s, or about ten to the 61 Planck times, the Universe is big, as far as we know 10^{26}m bit, compared to the 10^{-35}m at its birth, ten to the 61 Planck lengths big. However, it contains atoms, planets, comets, stars, galaxies,…Different scales of masses…You are only some tens of kg, planets or moons are 10^{20} times that thing more or less. Stars have also different spectra. You can find out stars with hundreds of solar masses yet in the Universe…Compact objects (even if stellar) are weirder but also existent. Take a solar mass and reduce it to the size of a continent. You get essentially a white dwarf star. To the size of a city, you get a neutron star. If you could compriss them further, you would get black holes. Black holes are indeed the most massive objects we can find in this Universe, beyond our Universe! The Universe has a mass of about 10^{53}kg. Its density is very low, about the vacuum energy density, of about 10^{-27}kg/m^3 (Planck density is about 10^{97}kg/m^3) or 1 proton per cubic meter. Outer space is vacuum, basically. In the other side you could have neutrinos, the less massive particle of the standard model (we do not even know their exact masses!), with about 10^{-39}kg. So, masses in the Universe, in hierarchy we do no understand, are distributed over 92 orders of magnitude, even more if you consider that dark energy could be some king of ultralight particle. Observed scales, not going to Planck scales, are separated by say 44 or 47 orders of magnitude in distances. Currently, the universe has a temperature about 3 K, when likely it began with Planck temperature, 10^{32}K. Old limitations of optical and telescopes were overcome. We have now new tools to observe wavelengths the human eye can not see unaided. Likely, machines will be showing us new ways to see the Universe we have already explored and initiated recently. Gamma rays, radioastronomy, neutrino astronomy or gravitational wave astronomy will the new powerful tools of the future for sure. Electromagnetism is also doomed from cosmological observations (at least in a one-time isotropic, homegenous time coordinate). The limit is the CMB. Beyond that, we will have to use neutrinos or gravitation. We can not see the very early universe before primordial recombination with photons.

  • QFT=Special Relativity+Quantum Mechanics=SR+QM. The first results of this fusion is the existence of antimatter (however the known Universe contains a very very low quantity of antimatter, fortunately for us!).

Rule games by relativity and quantum mechanics:

    \[E_\gamma=h/p\gamma=hc/\lambda_\gamma\]

    \[E^2=(mc^2)^2+(pc)^2\]

    \[E=\dfrac{h}{p}=\dfrac{h\gamma}{mv}\]

Relativity=Invariance under the Lorentz or Poincaré symmetry groups. You can classify particles with some numbers, just like you classify elements of the periodic table.  When are relativity and quantum important? Look at this plot size (energy) vs. velocity plot:

  • Leptons and quarks. What are their properties? That means to known about quantum numbers. Quantum numbers of elementary particles include  (rest, invariant) mass, angular momentum (spin), parity, electrical charge, hypercharge (weak charge) and sometime chirality or polarization degree (L or R for usual left-handed or right-handed polarizations).

  • We remember the 3 generations of the SM. But what is mass? Firstly, elementary particles get masses from the Higgs field. BUT, protons that make you and hydrogenated atoms and stars, or heavy nuclei, get masses from the strong force! The so called chiral symmetry breaking is the way in which hadrons get masses. Note that a proton is about 1GeV/c² of mass. But constituent u,u,d quarks are 2.3+2.3+4.8=9.4 MeV. So, where is the remaining proton mass? Hint: the proton is much more complicated than this naive 3-ball picture. Plug about 1 fm into the E-x uncertainty principle: E=hc/\lambda=10^{-26}/10^{-15}J, that are about 10^{-11}J of fluctuating quantum stuff. Protons can can not be imagined like a three-ball coconut. A proton is instead a result of the QCD vacuum! So, the proton is (uud)+(gluon kinetic energy)+(particle+antiparticle hadrons)+…Any kind of rubbish object. What is a proton then? Well, something like this will surprise you:

Mass from QCD is a highly non-trivial process. Indeed, some time ago, that process was called (I think it is yet called so, but it is horrible as name) dimensional transmutation or nonperturbative mass. Yet, 1% of the proton mass come from the Higgs field. You can possible compare this to residual electromagnetism in you daily life. Why walls do not drop off on your head? Electromagnetism is strong compared with gravity at common scales. 10^{42} times  stronger than gravity. BUT, electric charges compensate to each other, except, some residual forces, the Van der Waals forces (and some ionic or covalent variations), and you do not killed by walls thanks to electromagnetic residual forces from chemical bonding!

  • Hard part: Q.E.D. as quantum electromagnetism and how to intuitively get a picture of the quantum fundamental rule \mathcal{A}\sim\vert c_\gamma\exp(iS/\hbar)\vert^2.

Action principle comes from free in a lagrangian formulation. The totalitarian principle applied to strong fields (in both curved or flat spacetime) implies other incredible result in field theory (yet to be experimentally tested). Strongs fields CAN create particle-pairs. This is the Schwinger effect and it can easily derived from the action principle. From purely energetic views of SR and QFT, turning on a big enough electric or magnetic field you could create any suitable particle-antiparticle pair. For electrons in QED, you would get a critical field:

    \[E_c=\dfrac{m^2c^3}{e\hbar}\]

    \[B_c=\dfrac{E_c}{c}=\dfrac{m^2c^2}{e\hbar}\]

The values of these fields is very big. E_c\sim 10^{18}V/m and B_c\sim 10^{9}T. QED is wrong at very large energies, but electromagnetism and weak forces are unified at about 100GeV energies. Weak interactions are essential to understand radioactivity and how some particles “change identity” or flavor. This turns to be necessary for stars to exist. The proton-proton process giving rise to the stellar fusion is energetically possible, accidentally, and it is another surprising fine tuning of the SM:

    \[pp \rightarrow D+D\rightarrow ^3He\]

    \[^3He+^3He\rightarrow ^4He+p^+p^+\]

and the CNO cycle are not possible without the nuclear weak and strong forces observed features. However, we do not know why we observe 3 copies of particles with identical properties excepting masses.

  • Gauge symmetries. The fundamental \Psi'=e^{i\alpha}\Psi global versus local gauge symmetry transformations. These transformations do not change the physics and determine the interactions via the gauge field A=A_\mu(x)dx^{\mu}.

Quantum electromagnetism of QED is the result of the complex field and the QM structure of the vacuum. Wave functions are not directly observable. The can be partially observable due to phase shifts (Bohr-Aharanov effect) or via the Born rule. Majestic: you can only calculated the probability of field distributions in space-time. Quantum fields are generally complex-valued objects, and you get probabilities from amplitudes using the rule P\sim\vert\Psi\vert^2. That is. Physics do not change if you multiply, in QED, the wavefunction by a global phase

    \[\Psi'=\exp(i\alpha)\Psi\]

If you now turn the phase local, instead of global, that is, if you allow the phase to be changing in space-time as well, you are forced to introduce a new field if you want to recover invariance. This field is, for th U(1) case above, the electromagnetic field, A_\mu. This gauge symmetry determines the structure of interactions. And it can be generalized for non-abelian fields, like those required by weak and nuclear forces. Gauge symmetry tells you if you can forbid or allow certain interaction terms in the lagrangian device! Gauge symmetry also determine how the interaction arise between photons, electrons, W and Z boson, gluons (gluon and QCD are the exotic beast here, since they contain self-interactions not seen in weak or electromagnetic interactions).

  • The running of the “fundamental constants”. Due to quantum effects, the vacuum itself is not static, it changes. It polarizes. The amazing consequence of this over usual fundamental physics is that fundamental constants are not constant anymore. That is,

    \[F_e=K\dfrac{Qq}{r^2}\]

K is NOT constant, and thus \alpha is not constant. The polarization of the vacuum makes vacuum permittivity variable. Thus, \alpha=\alpha(r) or equivalently \alpha=\alpha(E).At nuclear distances, about 1 fm or femtometer scale (about 10^{-18}m), the usual fine structure constant is not exactly 1/137. At the LHC, in fact, at energies about 7TeV, the fine structure constant is about \alpha\sim 1/100. So, the running of the “constants” is slow. Indeed, it is a logarithmic variation ruled by the so-called renormalization (semi)group equation and the so-called beta function:

    \[\dfrac{dg(\mu)}{d\ln\mu}=\beta(g(\mu))\]

Note the differences and similarities. In a quantum world, charges and masses get dressed or renormalized due to the quantum fluctuations and the Heisenberg principle. Vacuum polarizes, and in the case of QED, there is screening in the coupling constant, increasing its value (the opposite effect happens in QCD, there is antiscreening).

The fine structure constant gets bigger with distance, or equivalently, plotted against the logarithm of the distance, alpha decreases with increasing distances, or it increases with the decreasing of the distance. The vacuum is a nasty object in QFT. You can visualize vacuum bubbles or loops from particle-antiparticle virtual particules popping out from the fundamental amplitude:

    \[A=\vert-+-O-+-O-O-+-O-O-O-+\cdots\vert^2\]

and possible many weird loops with subloops and topologies can also arise due to the symmetries of given interactions. For our 3 interactions, we get the SM as gauge theory. Quarks are tied or glued by gluons. There are 3 color charges. There are two electrical charges (one hypercharge) and 6 flavors for both leptons and quarks. Baryons are 3-quarks composite objects. Colors R (RED), G (GREEN), B (BLUE) and anticolor \overline{R}, \overline{G},\overline{B} making quarks must be colorless, since isolated quarks are not observable, every particle must be colorless. The weak interaction (or the electroweak interaction at scales of 100GeV) allows the change of flavor and weak charge.

The ideas above, of renormalization, vacuum polarization, undetermined intermediate states (virtual), running coupling constants, Feynman graphs, are iterated for the 3 interactions different from gravity. In QED you get 1 photon, in electroweak three vector massive photons and you get 8 gluons in QCD. There are 8 colorless ways to produce particles without color plus one extra colorless combination, any strongly interacting quark. Every SM particle has no color (excepting quarks), electric charge, flavor (weak charge) and hypercharge plus spin. Every particle, excitation of a single field, can be seen as wave perturbations in the field. The SM imposes:

  1. A simple gauge symmetry for the L part in (e,\nu), (u,d), the first generation.
  2. Gauge invariance and compensating field trick for any fundamental interaction.
  3. Optional (mandatory): mixing of generations is possible.
  4. Photon and gluons are massless, the W,Z, and H are massive by construction.
  5. Massive W,Z are problematic from the gauge theory viewpoint. This is what generated the creation of the Higgs field and the SSB mechanism. W, Z are exponentially supressed, and thus are unstable, decaying in short times.

Every particle has “polarization” modes, generally denoted by L and R. The SM is a theory for the electroweak theory plus the strong force (quantum chromodynamics, QCD) part explaining nuclei and hadrons. Recipe (oversimplification):

  • The SM is the Glashow-Weinberg-Salam model based on the gauge group G_g=SU(3)_c\times SU(2)_L\times U(1)_Y.
  • Electroweak forces are mediated by photons \gamma (massless) and gauge bosons W^{+},W^{-},Z.
  • The SM mixes particles between different generations and particles inside generations. The first generation comprises the main matter of the universe and it is stable (u,d),(\nu_e,e). Other two replicas of the first generation do exist. Why? Nobody knows for sure.
  • The SM does NOT contain gravity, negligible for particle interactions at the subscale for all the main circumstances.
  • That photons or gluons (the glue of QED and QCD) remain massless is due to the particular structure of the SM. The masses of the Z and the W are obtained like the other fundamental particles, via the Higgs field interaction.
  • SSB is just like a process similar to superconductivity and collective orientations of atoms/spins/particles in condensed matter systems. The fact that the Z-boson or the W-bosons are massive made weak and electroweak interactions short-range, unlikely gravity or usual electromagnetism.
  • The Higgs mechanism is a two-part device or gadget: it contains the SSB (spontaneous symmetry breaking) tool, and the dynamical part via a Higgs self-potential.
  • What is the precisions of the SM? About 1 part in 10^{12} in some cases! It rivals GR precision too! The magnetic moment of the electron measurements are such a precise measurements (anomalous for the muon case, a long standing problem in particle physics pointing out a BSM theory just like massive neutrinos). of (g/2)_{th} vs (g/2)_{exp}.
  • Open problems in the SM: nature of the Dark Matter (it can not be standard known SM particles), strong CP problem (why there is no electric moment of the neutron?), the hierarchy problem, the naturalness problem, anomalous magnetic moment of the muon, neutrino oscillation patters are very different from quark mixing patterns (via the different measurements of the CKM and PMNS matrices due to Cabbibo-Kobayahsi-Maskawa and Pontecorvo-Maki-Nakagawa-Sakata), why there are almost no antimatter in the Universe, the flavor problem (why 6?why 3 generations?), the nature of the QCD resonances, the early Universe picture of the particle physics from the SM particles (in particular the EW phase transition), the properties and nature of quark-gluon plasma.

The SM gives self-consistently a solution to the “problem” of how to get Z-W boson masses without spoiling the local gauge invariance of the fields. Mathematical details will be provided later. The Higgs mechanics is a two piece machine: a) breakdown mechanism, b) Higgs field dynamics (at least from a conservative viewpoint) are fully included in the SM. And, being more precise, the SM gets a maximal precision in the magnetic moment of the electron measurements. Compare the theoretical prediction (supercomputers and high calculus needed to compute it!):

    \[(g/2)_{th}=1.00115965218113(86)\]

with the experimental result

    \[(g/2)_{obs}=1.00115965218073(27)\]

Let me point out these results are not the last data or theoretical predictions. There is certain tension betwen theory and experimental resuls but it is not a huge one. 12 decimals of precision is quite a thing. Imagine to know the distance to the sun with such a precision.

By the other hand, there IS a big one point where new physics does arise in the SM. Neutrino sector. We do know from the last years of the 20th century and from current neutrino beams, solar neutrino experiments, reactor experiments and from cosmological data a lesser hint, that neutrinos are special. Neutrinos are not only the smallest chunks of matter you can get in the SM (yet, their concrete masses are not known!). Neutrinos are transgender or travestis! Neutrinos come in 3 flavors or species (at least, the SM neutrinos, theories do exist with more than 3,…Why 3 light neutrinos? Why left-handed?). It showed that neutrinos can transform into the 3 types when travelling long distances! In fact, there is a similar phenomenon inside hadrons. Quarks also mix! Transitions between neutrino types are modeled by a gadget called PMNS matrix, or neutrino oscillation matrix. Formally, there is also a CKM quark mixing matrix too. Mathematically:

    \[\vert\Psi_\nu\rangle=U_{PMNS}\vert\Psi_m\rangle\]

    \[\vert\Psi_q\rangle=U_{CKM}\vert\Psi_{Q}\rangle\]

Experimental data shows you that U_{CKM}\sim \mbox{diag}(1,1,1), while the neutrino mixing matrix is something much more complicated, somethin with entries like this:

    \[U_{PMNS}=\begin{pmatrix}\square & \bullet & \cdot\\ \circ & \bullet & \square\\ \circ & \bullet &\square\end{pmatrix}\]

The CKM is more ore less diagonal, but it also seems to have substructure

    \[U_{CKM}=\begin{pmatrix}1 & \square & \circ\\ \square & 1 & \cdot\\ \circ & \cdot & 1\end{pmatrix}\]

Furthermore, it seems that somehow mixing angles of these two matrices are complementary to each other, approximately it seems that \Theta_{CKM}\sim\theta_{PMNS}+\pi/2. Nobody knows why, and the third mixing angle of the PMNS matrix, \Theta_{13} was recently (a few years ago) measured. Also, there is some hints of CP-violations (naturally expected from the SM) in the PMNS matrix (something that is well-tested in the quark setting). Did you know we have more neutrino unknowns? Neutrinos are the ONLY fermionic fundamental field in the SM. Beyond knowing its mass, we do not know yet if its spectrum is normal (atomic-like) or inverted. We do not know if neutrinos are Dirac or Majorana particles. That is, neutrinos could be the only fermion in the SM that are their own antiparticles (a very bosonic trait!). Why does it matter? Well, if neutrinos are their own antiparticles, we could in principle understand why there is no almost antimatter in the observable Universe. To explain it, we should be able to know how to cancel out matter and antimatter with a difference of 1 part in 10^{10} such as 10^{10}\neq 10{10}+1. Otherwise, the whole Universe would be very different or it would not exist!

Well, time to go with asymptotically freedom, quark confinement and gluons…

Part(III). A short guide of QCD. From quarks to quark-gluon plasmas.

  • Main two features of QCD: asymptotic freedom and confinement. Asymptotic freedom is just the contrary behaviour of the screening in QED. Strong coupling decreases with decreasing distances! That is very antiintuitive. We are familiarized with forces that increase with distances, strong forces are different. At high energies, short distances, you are essentially “free” of strong force. Subtle. Specially since we call strong force, strong force. Confinement is the weird feature of strong forces making free quarks (or color charges) invisible. You would ask then how do we know quarks exist afterall if they do not exist as isolated objects. The main prove, beyond all the QCD evidence, is the jet structure we get from particle collisions. No free quarks sorry, but quark bunches! Wibbly wobbly quarky quark stuff.
  • Hadrons come into groups: baryons and mesons. Multiquark states for N>3 and even quarkless states (e.g., the glueballs or gluonium) are a known hot topic in QCD.
  • The model of quarks and partons. Partons were introduced by Feynman and Bjorken even before the quark theory was finished. Protons have complex structure. At low energies, you can naively imagine hadrons as valence quarks, but at higher energies hadrons are made from valence quarks PLUS other wibbly wobbly timey wimey stuff. Oh, yes! Microscopically zooming a proton is a fantastic journey itself.
  • The strongness of strong interactions. Why are the strong forces the maximal forces? Well, fortunately for your atomic nuclei and protons, it is so.
  • Quark-gluon plasmas (QGP). At billions (american) of kelvin, you get a wild soup of quarks and gluons unbounded. This is the quark-gluon plasma. It behaves as a perfect fluid.

The Manhattan project, I think you do not know this, provided a fund for particle physicists during and after the second World War. In the 1950s, many investigations provided a surprising subatomic subnuclear world. Many types of particles and resonances arised. Just like Mendeleive built it a periodic table for elements, particle physicists had to creat a big frame for the world of hadrons they were discovering. A famous joke or affirmation those years was that the Nobel Prize was associated to the discovery of new hadron states/particles. Murray Gell-Mann (RIP), and Neemann (RIP), independently by Zweig discovered a way to classify hadrons into schemes using group theory and quantum numbers. Zweig “ace theory” was not popular even when it was pretty similar, but it is an interesting example of how the same ideas arise in different people at the same time, and names do matter to sell your research| The eightfold way was the pavement for the establishment of quark model, and the rise of the QCD as a gauge theory beyond the S-matrix formalism reigning in the 60s. For instance:

You can do hadron spectroscopy with particle physics! I wrote about the names of those hadron states here on TSOR, you can search for them. It is quite curious how many curious unstable particles you have. Funny fact, the omega baryon was taken as inspiration for some Star Trek episodes, like the Omega directive related to Omega particles making warp travel impossible. Any hadron is very complex. Protons at very high energies are messy stuff, you can see inside protons other quarks and strange objects. For energies similar to the proton mass, you can yet keep the 3-ball (uud) making the proton (and similarly with other baryons). BUT, the valence quark picture is only an approximation for certain scale. The quark model arising in the 1970s was deliberately precise to understand the color charge (and why states like the sss or the \Delta particle exist. Quark model is in debt to O. Greenberg parastatistics, an exotic (yet today!) topic related to quantum statistics beyond fermions and bosons.

By the other hand, strong force is so weird due to confinement. Try to ionize a proton just like you do with an atom. Well, unlike to atoms, you get a quark-antiquark state very soon. This feature is similar to particle pair creation due to Schwinger effect in strong QED/gauge theory, but you get it from “free” in QCD. Strong force is the most quantum force of the 3 interactions of the SM. The absence of free quarks is very similar to the absence of magnetic monopoles. Just note that:

    \[\dfrac{E(proton)}{(2m_u+m_d)c^2}\sim 100\]

Thus, quarks are very relativistic as well!!!!! Reamarkly, the same operation done over the hydrogen atom gives you E/(M_e+M_p)c^2 about 10^{-5}. So, unless it is excited or heavier nuclei are considered, simple atoms are not generally relativistic at the level of the binding energies. That the strong interaction (SI) is very quantum and very relativistic is a known fact. Beyond the parton model by Feynman, coding some structure functions for hadrons, there are some interesting simplified models in the description of quarks and gluons. Maybe, the most important model is that of strings. Yes, string theory was born as a strong interaction theory. Hadrons are just flux tubes of color, trapped, and wibbling and wobbling wildly. It shows that the flow of gluon lines carries out almost every the energy of the quarks and hadrons. Indeed, the 99% of matter is due to gluonic field interactions or the color flux tubes. That hadrons are just balls tied up by strings is a useful picture but not too fair today, and spin two states are just and oddity in nuclear and particle physics. However, spin two interactions are known to be those interactions caused by gravity and gravitons, so string theory transformed itself into a theory of everything, as it keeps (but strangely uncomplete) today. You can hear some discussions between theories, and some blogs about that gravity is the square of some Yang-Mills theory. Well, it is not quite precise to say it so simple, but it works in some theories and models, so keep an eye on that.

Other interesting QCD model is that of the spring (string) tension. Essentially, confinement is linear with constant tension. The tension of any hadron is essentially about 1GeV/fm=10^5N. However, if you hit a proton at about 10TeV of energy, and at about a distance of an (deci)attometer, the tension would be instead 10^{12}N or 10^{13}N. Taking into account that tension is the Young-modulus times the strain (or the tension is the pressure times the sectional surface of any material), you get that hadron tension is huge, very huge. It can be compared to that of graphene tension in a meter from stiffness, or that of steel in one cm^2, for the greatest experimentally tested values! Thus, it is not surprising that nuclei and hadrons are so stable, aren’t they? And however, the string/spring model is pretty simple explanation of all of this. You get

 

    \[F_q=\sigma=constant\;\; V_{string}=kr\]

and at shorter distances, you would get asymptotically freedom deconfinement via the potential

    \[V_d=-\dfrac{c\alpha_s}{r}+\sigma r\]

where c is some constant (generally written as 4\lambda/3) and units of V are in GeV. I would like to note that non-perturbativeness is essentially a key property of confined QCD. It yields exponential terms giving rise to particle pair creations similarly to Schwinger effect, via

    \[A=\exp\left(\dfrac{-m_\pi^2}{\sigma}\right)\]

up to unit conversion constants!

Well, quak-gluon plasma has a temperature about 10^{12}K. Take the QCD main typical energy, the chiral vaccum of QCD has energy about 100 MeV, this is about 10^{-11} joules. Use the Boltzmann constant to turn this energy into temperature of fundamental quark-qluon melting, and you get that trillion (american, billion european) of degrees. Similar estimates can be done for the normal electric plasma (to get millions of degrees, easy if you consider that typical atomic energies are about 1 eV), or even more, you can guess the ultimate hot temperature, the Planck temperature from this kind of arguments, about 10^{32}K. What are the properties of QGP? A simple list:

  • It behaves like an almost perfect fluid (without friction!).
  • It is not gluon transparent.
  • It has a complicated phase diagram, much more complicated and subtle that was initially expected.
  • QGP was the main composition of the Universe when the Universe was about between one picosecond up to 1 or 10 microseconds,…Then, protons and neutrons and other hadrons were confined.

As an image is much better, let me show you some representations of the QCD phase diagram (similar to the water phases you study at school):

Let me remember you a thing, the strong coupling “constant” is about 0.118 at the LHC energies, and vacuum Feynman graphs should be counted with care as we have 3 colors and a non-abelian (non-commutative) gauge theory! However, beyond some interesting properties, there is a general framework for the full SM and now we will its full power…

Part(IV). The power of the SM.

The SM lagrangian is, formally simplified, a sum:

    \[L_{SM}=L_m+L_g+L_{int}+L_{H}=i\overline{\Psi}\gamma\cdot D\Psi+F^2+G^2+g_Y\overline{\Psi}\Psi\phi+\vert D\phi\vert^2-V_\phi\]

The simples Higgs potential dynamics is encoded via

    \[V_\phi=-m^2\phi^2+\lambda\phi^4\]

Further interaction terms, more complicated, are allowed in BSM theories, but the SM is just a g\phi^3+\lambda\phi^4 from a perturbative viewpoint.

The SM does NOT contain gravity. It only codes 3 out 4 fundamental interactions. However, the framework is self-consistent (up to some technical problems we do not know how to solve yet) with the Higgs mechanism (SSB plus the Higgs potential). Interactions?

  1. Strong force by 8 gluons.
  2. Weak force by W and Z bosons.
  3. Electromagnetic force by photons.
  4. Mass giver to elementary particles with Higgs and interaction terms by Yukawa-like interactions not coming from any symmetry but from the own Higgs mechanism and dynamics.

Gravity is a force apart from the SM, even when you can in principle calculate graviton scattering processes. Taking into account loop corrections is a nightmare with gravity. Feynman graphs blow up in number and you can not control the destiny of the nasty infinite terms spoiling renormalizability. Thus, we are hoping a new BSM theory will help us to solve this. QG as superstrings/M-theory or LQG were designed to live with these problems better. Today, they have helped to some theoretical fundamentally mathematical details, but we lack stringy/loopy experimental support. We have, however, three mysterious generations, the 6 quarks and 6 leptons, 4 gauge bosons and the Higgs-like boson at 125 GeV saying us that something else is required, but we do not know how or what is it. This post is already showing you:

  • There are known knowns.
  • There are known unknowns.
  • There are unknown unknowns, likely, out there, hidden in the noise of our current data.

How does the LHC work? Take water, tons of water, and spoils the electrons with electric fields. Electrolysis apart, you need high frequency electric fields. Then inject particles with frequencies between 40MHz to 10kHz. Create tubes of low temperature (1.9K or about 2K) at the LHC main tubes. Get superconductivity with about 8.3T magnetic fields. From ionized hydrogen atoms, got about bunches of protons (10^{11}p per bunch. Take about groups of 3000 packets and insert them into a 26 km long collider by sucessive injections. At 40 MHz, you could show that protons circle the full LHC about 11245 times per second. Then, build up some cool detectors (ATLAS, CMS, ALICE and LHCb are their names, plus some new detectors under construction to test other fantastic theories). Then, build up some cool wonderful informatical system and architecture, such as you can pile up data with detector times about 3 microseconds. Note than you can even “detect” particles like the tau with about 300 femtoseconds of timelife, response times of 25 ns and even more, you can indeed hint the existence of very short-lived resonances like the Higgs, the top quark and other known particles with lifetimes of 0.1 yoctoseconds!!!!! But you see at energies, not time detector responses. Essentially, any particle collider is a time-series for the energy-mass-frequency and the number of events, after good statistical and experimental analysis. That is.

If you think all this is useless, let me talk you…About history and later about medicine. Any particle collider has transversal applications. A complex detector like ATLAS or CMS are a set of wires, electronics and aparatus with high-end applications. Of course, you can think that a 14 G€ machine is expensive. But you must think globally. The LHC has cost every european citizen only a few euros per person. Does it deserve the spending of the money? Why to make 9T magnetic fields and 15 meters wires/detectors in size? The same question arises in different time periods of human history. Nobody, for sure, predicted that GR and the gravitational correction to time measurements would be necessary to get you not lost with the GPS system, but it is true. You can not get GPS devices work properly if GR were not correct. There, at speed of 14000km/s satellites, and times of a few 7-38 microseconds, the gravitational corrections by GR are necessary to get a proper position in the sea, in any remote part of the globe. What about accelerators? First use of accelerators were, in fact, TV monitors. Did you enjoy old TV? Cathodic rays by J.J. Thomson were a basic tool for the first TV designs. Van de Graaff generators are yet common tools for showing the effects of electricity, but you also have other interesting accelerators named pelletrons and the one by Gockcrofth-Walton. There are also linear colliders (LINAC) used for alimentary uses, cyclotrons in nuclear research and medicine, and synchrotons, … Synchroton radiation is important in medicine, but also in material reseach and electron microscopy. You can erase microbes and clean materials with synchroton radiation, and what about those X-rays sources? What about radiation therapy? You have now PET and proton therapy too! You MUST know an important thing. PET arised in the LHC site previous collider, the LEP. And much of current collider technology, based on calorimetry and crystals at the detectors, is being reused for prediagnosing terrible illness. The production of radiodrugs is also made with collider-aided tools. Ionizing radiation detectors are also important pieces of technology natural in colliders that find a refurbished use in medicine. Ionizing particles are naturally charged. Radiation is counted or detected by semiconductors (diods). You can find out beyond nuclear weapons a full set of pacific uses of radiation tools: archeology, vulcanology, nuclear safety from nuclear reactors, fire detectors,…How can we detect radiation?

  1. Ionizing radiation is detected with counters (gas or semiconductors) and dosimeters using photographic thin films.
  2. Exciting radiation is detected with thermoluminiscent materials, sparking counters and other gas detectors.

Who had said to Dirac that antimatter would be useful for Positron-Electron-Tomographies when he derived his famouse equation

    \[\left[i\hbar\gamma^\mu\partial_\mu-e\gamma^\mu A_\mu-mc\right]\Psi=(i\hbar \gamma\cdot D-mc)\Psi=0\]

Medical imaging is a wonderful branch of particle physics. Beyond ecography, NMR (nuclear magnetic resonance) requires high magnetic fields, non ionizing radiation and nuclear relaxing. X-rays are ionizing (that is why you can not X-ray yourself every day), the TAC implies higher dosis but they are not too much used per year per person, and PET is important in oncology with low resolution, and fluoride-18 isotopes essentially. Neurology and cardiology are benefiting from particle physics too. Furthermore, isotopes are necessary for those devices, so we need curiously one of the most fascinating predictions of Mendeleiev table 150 years ago: technetium. Technetium sources are required in many medical nuclear resources, not only in PTAC or PET+TAC, but also in SPECT, using the metastable technetium-99 isotope. Thus, radiation therapy, historically using X-rays have evolved into using a more multiparticle setting. You can use not only limited gamma rays for some processes, you can also use electrons and proton particles for therapy. Proton therapy is new and promising, specially promising with the great precision of 220 MeV proton residuals. Old therapies based on Co-60 are known to have generated patient issues some decades after the treatment. Proton therapy machines are now in top hospitals. In principle you could use any “soft” particle for therapy. Neutrinos? Carbon atoms? The CERN is aware of all of this, and it has some multidisciplinar projects like ENLIGHT and BroLEIR about how to simulate proton radiation and the efects of proton therapy or oxygen-16 to cure your health. Gammagraphies are also a medical tool, dosis are important though. Dosis is defined as radiation energy per mass unit. And radiodrugs specifically desiged for customized treatments are on the way. So, please, neve say that particle or fundamental physics is useless. You have induction kitchens and microwaves at your cooking stations thanks to radiation studies! Nuclear transmutation is currently possible and the alchemy promises of ancient times secure your health if done properly. You can forget what a barn is (a particle physicist unit of area equal to 10^{-24}cm^2, or that the top quark or Higgs mass are about 173 and 125 GeV, but you should know particle physics does matter in your timelife. Even if you don’t know or don’t find new physics because your job is very different, the search for SUSY, Dark Matter, extra dimensions, black holes, surely will affect you collaterally. Particles colliders are not designed to produce, in the high energy physics community, a single concrete particle, but medical applications are different. Now, while reading these lines, you are being crossed over by billions of neutrini, and for some muons.

Part(V). General Relativity and the LCDM model.

  • General relativity=Equivalence principle+SR. GR=EP+SR.
  • Curvature=Energy-Momentum.
  • Gravity=Pseudoforce=Geometry.
  • There are gravitational waves.
  • Gravity is weak generally, but it can also be strong at big masses or high densities.
  • GR needs quantum gravity when the Schwarzschild radius (or gravitational size) equals to the quantum size (Compton’s wavelength). That is reached (naively) at the Planck length, 10^{-35}m. Check:

    \[2\Lambda_Q=R_s=L_P\leftrightarrow\; L_P^2=\dfrac{G\hbar}{c^3}\]

  • Spacetime tells matter-energy how to move, matter-energy tells spacetime how to curve (warp). There is no torsion in classical GR, but you can include it to get Einstein-Cartan theory with nonsymmetrical energy-momentum or Einstein tensors.
  • The large structure of the spacetime and the LCDM standard cosmological model. This is the current analogue of the SM for the largest cosmic structures we have today. It predicts lot of things, and explain current data as fair as the SM.
  • We need dark matter (or MOND/MOG) to explain galactic rotation curves and elliptical galaxies dispersion speed:

    \[V^4=G^2M\Sigma=G^2M^2R^2\]

or

    \[V^4=2GMa_0\]

with \Sigma=M/R^2, a_0=G\Sigma/2=GM/2R^2.

  • The Universe is expanding with H=0\sim 70km/s/Mpc at large structures. There are some divergences between the value of the Hubble parameter H_0 at current time. But average, it is about 70 in conventional units (km/s/Mpc).
  • The Big Bang model: the cosmic microwave background and its anisotropies confirm the LCDM previsions.
  • Current CMB temperature is about 2.73K up to some anisotropies in the sky. It is also expected a cosmic neutrino background at about 1.945K or less (depending on the extra non-SM neutrini and other BSM physics). The relic graviton background is also expected at about 0.9 K or lesser. T_{CgB}\sim\sqrt{2/N}T_{CMB}. You get the 0.9 K counting the particle species degrees of freedom of the SM. If there were additional particles, the cosmic graviton background would be lesser than 0.9K.
  • The Universe density is close to the critical density, about 1proton per cubic centimeter, or about 1 electron per litre. Primordial fluctuations were the seeds of current irregularities given by galaxies and other cosmic structures.
  • The universe is flat (euclidean) at cosmic scales (despite being intrisically curved spacetime). This recalls for inflation.
  • Dark matter, if real, should create a wind or flux with speed of 300km/s.
  • The main evidence of cold dark matter comes from flat rotation curves in spiral galaxies and the dispersion of speed in elliptic galaxies. Zwicky, however, found evidence of this in 1933 (more mass was required to explain galactic motions), and Vera Rubin confirmed it in 1975. There are evidence of DM due to gravitational lensing observations or the Bullet Cluster.
  • Galaxies are “flat” due to interactions with matter and gravity.  The role of DM in galaxy formation and evolution is yet a hot topic of research.
  • Simulations of the Universe with DM and or Dark Energy are consistent with the LCMB paradigm. However, it does not imply that models without dark matter can not exist. However, DM evidence is from different sources, not only a single one. Simulations have the power to discriminate some models.
  • Dark matter particles is collisionless and do not cluster, but it forms haloes. Halo dynamics is however poorly understood. It is believed to be spherically symmetric, but we do not know for sure.
  • Supernovae type Ia measure the Hubble constant and hint that the Universe is accelerating, not decelerating.
  • Dark energy or the cosmological constant remains a big puzzle even today. What is it? Quintessence? Phantom energy? The mere vacuum energy? 70% dark energy, 25% dark matter, 5% normal matter. We are a mere 5% in the Universe. DM are likely some neutral particles not in the SM (one or several types!). We have some ideas of what DM is but no prove of their existence by direct production has been managed till now.
  • GW plan to measure the Hubble parameter with precision in order to spoil some tensions in the current data. H_0=\dfrac{\dot{a}}{a} at current time t=0.
  • Friedmann equations. For homegeneous and isotropic Universes, like ours, the GR field equations can be recasted into two simple equations, called Friedmann equations:

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

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

  • Known issues: singularities, nature of dark matter and dark energy, the finding of the cosmic neutrino background and the relic cosmic graviton background, the finding of the stochastic gravitational wave background, the finding of the inflation signatures of the universe, the test of the multiverse ideas (possible?)., the final fate of the observable universe (will protons decay? will the space-time disappear?what will black holes leave after they evaporate?).

After the two modern revolutions of the 20th century, physicists are yet stunned by their precision. Relativity (both special and general) and Quantum Physics (in its ultimate current form, the Standard Modern) rule with surprisingly accuracy and precision in the realm of experimental physics.

2 pillars, ying and yang of physics, relativity (general relativity) reigns into the macroworld, while the quantum mechanics (the Standard Model) remain unbreakable at the microworld. Where do we stand in our search for the ultimate theory? Let me trip with you into an overview of what we do we know (more or less) that is true at both extreme theories and scales.

What is time? What is space? What is mass or energy? What are the fundamental forces? These questions, even when translated into the quantum are classical as well. In fact, it was the genius of Einstein and many other great scientists what operationally say what they are (more or less, since our scientific knowledge is provisional).

Special relativity (SR) was build in order to unify the laws of mechanics and the (galilean) principle of relativity with those laws of electromagnetism (specially, the symmetry of Maxwell equations). What is motion? The change of position in time with respect to some reference frame. What is time? Just a parameter or coordinate in a four-dimensional set-up. It shows up the the marriage of classical mechanics and electromagnetism (in 4d) can be done, saving the relativity principle up to a cost. In unidimensional time relativity, 3-speed (4-speed) are limited to signals lesser or equal to the speed of light. In any space-time diagram, you get a cone when propagating at the speed of light. Light signals relate space with time, and light also relates mass with energy:

    \[X=ct\]

    \[E_0=mc^2\]

However, this fact also implies that Newton’s gravity can not be right. Reason? Gravity propagates instantaneously in newtonian gravity! That is forbidden in special relativity. Einstein realized this, and he had to struggle with superior mathematics to find out a theory locally consistent with special relativity containing gravity. This theory, a locally special relativistic theory of graviy is what general relativity (GR) is. However, the theory proved itself to be greater than his own creator and inventor could ever imagined at his lifetime. If you plug G_N=0 or c=\infty in GR, basically (there are some nasty stuff like we saw in come recent log), you recover SR from GR. GR is a theory that models spacetime with a metric field. A metric field is a matrix, usually symmetric (I will be only considering as is usual gravity without torsion here, so the Einstein tensor remains symmetric). Flat spacetime is just the normal Minkovski metric (a diagonal matrix!):

    \[ds^2=c^2dt^2-dx^2-dy^2-dz^2\]

up to a global sign convention. This GR theory is fascinating. It explains gravity as a curvature of spacetime. In fact, the deviation of the circle usual length by curvature is due to gravity itself:

    \[\dfrac{Perimeter}{Diameter}-\pi\sim \dfrac{G_NM}{c^2r}\]

Gravity, being a force, it is really a pseudoforce. Why do we want quantum gravity? It is not only why we want something else that calculate the cross-section of 2 gravitons transforming into 2 photons (Skovelev):

(1)   \begin{equation*} \sigma(GG\rightarrow\gamma\gamma)=\dfrac{k^4\omega^2}{160\pi}=\dfrac{\pi d_S^2}{10}\end{equation*}

For the electron at rest, it would be very tiny, \sigma\approx 10^{-110}cm^2.

Exercise: derive the above cross-section (be aware of identical particles effects) from

    \[\dfrac{d\sigma}{d\cos\theta}=\dfrac{k^4\omega^2}{64\pi}\left(\cos^8(\theta/2)+\sin^8(\theta/2)\right)\]

Exercise (II): check the graviton-photon to graviton-photon formula and tell what is the main problem it has compared to the previous formula.

    \[\sigma(G\gamma\rightarrow G\gamma)=\dfrac{k^4\omega^2}{64\pi}\dfrac{1+\cos^4(\theta/2)}{sin^4(\theta/2)}\]

It is not only that strong fields allow particle pair creation rates, e.g., via the Schwinger effect

    \[\Gamma_S=\dfrac{e^2E^2}{(2\pi)^3}\sum_{n=1}^\infty e^{-\frac{\pi m^2c^4}{e\hbar E}}\]

GR is uncomplete. However, GR is a wonderful theory. As SR. SR says:

    \[\mbox{Proper time}=(\mbox{Time})^2-(\mbox{distance})^2\]

using units in which c=1. The time measured by a rest clock is equal to certain combination of the time of the clock in motion minus the distance, using something similar to euclidean triangles. They are indeed hyperbolic triangles. GR says space-time is elastic and dynamical. And the equivalence principle, in any of its forms, say in the end that gravity is just curvature or geometry. The shape of space in time implies that space-time grows somehow. It expands (despite the static preconception of the theory when Einstein created it, it soon proved to predict the Universe as something moving itself!). The Einstein Field theory equations are pretty simple:

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

The vacuum density energy is yet a mystery, so that is one of the reasons of why to go beyond GR:

    \[\rho_\Lambda=\dfrac{\Lambda c^4}{8\pi G_N}\]

Essentially, its value today is about a proton per cubic meter (do the numbers yourself!), and it is very similar (coincidence problem!) to matter energy density or dark matter energy density today. But it was not so in the past! The cosmological constant can be interpreted as a Lagrange multiplier, a volume pressure term or a negative contraterm in the EFE. Nobody knows why it has the value it has today. Curvature of spacetime has a temporal component. The purely temporal components of the curvature trigger the newtonian potential at weak gravitational fields. It also implies the existence of gravitational fields/waves traveling at the speed of light…Or, is the light the one who travels at the maximal possible speed in local spacetime? The stiffness of spacetime is inversely proportional to G_N. It is pretty big, so you need big masses or densities in order to make curved spacetime effects to appear clearly. GW were so weak, we had to search for them into the noise on Earth (in space the story is completely different, but this is the subject of a future blog post!). Just like the LHC probes zeptometric scales, GR probes bigger scales. The SM has a precision of 1 part into 10^{10} or so. GR has similar values of precision in some observables. Gravity bends spacetime and triangles are, in principle, curved (it yields that the largest spacetime structures of the Universe are euclidean though! This flatness can be solved hardly using inflationary ideas). What else? Einstein theory of gravity, GR, predicts as well:

  • Precession of orbits. Abandon your keplerian world. Ellipses precess.  The critical case in the solar system is the famous Mercury orbit. Was the genius of Einstein realize that his theory could explain Mercury without dark matter, Vulcan.
  • Gravitational time delay. Closer you are to any heavy mass, slower is your time flow.
  • Gravitational lensing of light. Eddington checked this a century ago, in 1919. Einstein was already a celebrity, but this confirmation of his GR theory elevated him to the level of God of Physics.
  • The existence of vacuum solutions we call black holes. Even when other people had speculated about black stars before in newtonian gravity, GR contains naturally solutions with features or darkness. We do not understand their ultimate destiny. That is another reason why GR is not complete.
  • Deviations from euclidean geometry is measured with angular measurements:

    \[\alpha+\beta+\gamma-\pi\sim \dfrac{GM}{c^2r}\]

  • The universe is expanding, and it likely had a beginning in time.
  • The universe has a vacuum energy that is not null (it is a 20 years old rediscovery of Einstein’s cosmological constant).
  • Gravitational waves exist (radically new astronomy by LIGO is going on at these moments). The era of multimessenger astronomy is just beginning.
  • Black holes are real things. Recently, we picked up the photo of M87 and SgA*, our galactic BH, is being analyzed right now.

Let me show you some GW formulae:

  • Gravitational luminosity formula reads

    \[L_{GW}=-\left(\dfrac{dE}{dt}\right)_{GW}=-\dfrac{G}{5c^5}\langle\dfrac{\partial^3Q}{\partial t^3}\rangle\]

such as, for a binary circular system

    \[L_{GW}=\dfrac{32G}{5c^5}\mu^2a^4\Omega^6=\dfrac{32G^4\mu^2M^3}{5c^5}\]

for M=M_1+M_2 and \mu=M_1M_2/M.

  • Temporal radius reduction due to GW emission and the coalescence time are given by:

    \[\dot{a}=-\dfrac{64G^3}{5c^5}\dfrac{\mu M^2}{a^3}\]

    \[\tau_c=\dfrac{5}{256}\dfrac{c^5a_0^4}{G^3\mu M^4}\]

Is the LCDM model fair? Yes:

  • It predicts all the GR classical tests, as it is basend on it.
  • It predicts the observed expansion of the Universe.
  • The Universe is spatially homogeneous and isotropic. The perfect cosmological principle is wrong. Universe is not aethernal. A restricted cosmological principle holds, though, today. We are not special. At large scales, the Universe look the same everywhere.
  • Scale ractor measures expansion via R(t)=a(t)R. GREFE for LCDM reduce to the Friedmann equations.
  • Critical density is close to the cosmic density, \rho_c=3H_0^2/8\pi G. Knowing H_0 allows you to measure the age and size (up to scale factor) of the Universe. Knowing H_0 and G allows you to compute the critical density for the Universe to collapse. We are close to that value, but accelerating Universe is diluting galaxies into vacuum. To calculate the critical density, take the Hubble law v=HR and equal kinetic energy to potential gravitational energy:

    \[\dfrac{1}{2}M_UH^2_0R^2=\dfrac{4}{3}\pi \rho G R^2\]

and then \rho_c follows straightforward from elementary algebra.

  • Some parts of the cosmic history are yet sorrounded by mystery and unknowns. We have to live with ignorance there. Planck era or inflation era are  yet hard to test. We believe we understand the QCD era, more or less, but not with absolute safety.
  • Light element abundance is another great prediction of LCDM. Indeed, it fits nicely to observations. We have to check some early Universe young stars here, the famous population III. The James Webb Space Telescope (JWST) will be looking at Pop III stars and it will show us wonderful things for sure. Current universe is old, stars are second or third generation stars, like our sun (3rd generation star).
  • GW are there. We will see the Universe BH and other fantastic GW sources invisible for light with these tools. GW evidence before LIGO discovery 3 years ago was found in pulsars (1993 Nobel Prize).

The Big Bang happened not in a single place, but everywhere, in a single moment of time, 13.7Gyr ago. There is no centre of the Universe. We see the past of the Universe during the sky night. During the first seconds, the Universe created radiation, then arised the first particles via QCD decoupling, later it created nuclei and elements (primordial nucleosynthesis), and, finally, we got astronomically/astrophysically bound objects like galaxies, clusters,…Evidences from the Big Bang are the elements that form everything, radiation from the CMB and its anisotropies, structure formation…The nuclei areised after the 3 first minutes, and in about 380000 years, recombination was possible and the light creating the elements and the CMB were produced. Inflation is required to explain some puzzles (flatness and anisotropies the main two, but there are other problems difficult without inflation). Inflation naturally requires scalar fields (or similar) exponentially inflating the universe. The matter-antimatter asymmetry is responsible of being us here. Neutrinos can keep likely part of that dark mystery, or even the dark matter partial solution. BH naturally can violate the conservation of baryon number and thus trigger proton decays. Planck mass naturally gives a lower theoretical bound of 10^{45} years (or even longer with care, about 10^{140} yrs) for proton lifetime from BH virtual fluctuations and/or spacetime foam models.

That is all…folks. The end? Choose a final death for our Universe:

  1. Big Freeze (No Freezer will save you).
  2. Big Crunch (No piston will save you).
  3. Big Rip (No gravity will save you).
  4. Little Rip (No soft gravity will save you).
  5. Big Decay (Higgs field unstability/metastability: no force will save you).

See you in other blog post soon!

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