LOG#229. Mars organics.

Hi, there. Short post today ignites a new category post. Life and Chemistry. The search of life outside Earth, and beyond, is a goal for the current and forthcoming centuries (provided we are not extincted). First targets for life searches in the Solar System include: solar system planets, some of their moons and maybe comets.

Between all the solar system planets (neglecting Moon by naive assumptions, perhaps we should reconsider that if there is ice and water enough on our satellite), Mars is an ideal place to search for life. Other targets like Titan, Ganymede, Ceres, Pluto and others will not be covered today. What do we know about Mars? Mars has a thin atmosphere made of carbon dioxide. And, after decades, we also know some chemical compounds on Mars (with more or less uncertainty):

  • Sulfur-like compounds. A list includes
  1. Thiophen C_4H_4S.
  2. Methyl thiophenes C_5H_6S.
  3. Methanethiol CH_4S.
  4. Dimethyl sulfide C_2H_6.
  5. Benzothiophene C_8H6S.
  • Non sulfur-like compounds.
  1. Benzene C_6H_6.
  2. Toluene (or tropylium ion C_7H_7^+).
  3. Alkylbenzenes (C_8H_9 or benzoate ion C_7H_5O^-).
  4. Chlorobenzene C6H_5Cl.
  5. Nophtahlene C_{10}H_8.

Smaller molecules seen on Mars (of course, beyond CO_2) are:

  1. Carbonyl sulfide COS.
  2. Oxygen O_2.
  3. Carbon disulfide CS_2.
  4. Carbon monoxide CO.
  5. Hydrogen sulfide H_2S.
  6. Sulfur dioxide SO_2.
  7. Methane CH_4. The origin of martian methane is yet a mystery. Proof of life, interior geology or exotics or complex mechanims? We do not know.

There are lots of carbon-chain molecules, with about 1 up to 5 carbon atoms likely on Mars soil. Would we find out azobenzene C_{12}H_{10}N_2 molecules on Mars? Likely not. Of course, we will not found superconductors or MoS_2 in principle, but now we do know there is water below the Mars soils. Ice. And likely salty subterran water cycles. Are there bacteria and other life beings, even microscopic or bigger, on Mars right no? Impossible to say yet! New rovers will try to uncover the biggest mysteries on the red planet and finally to decide if there is some kind of microorganisms or even life beings hidden on Mars! We will know in this century for sure! If not on Mars, Titan and other places of the Solar System like Enceladus have prospective high odds to sustain some king of life. Europe as well. But do not try to land there ;).

See you in another blog post!

LOG#228. The scientific method.

What is the scientific method? There are many definitions out there, but I am providing mine, the one I explain to my students, in this short post.

SCIENTIFIC METHOD (Definition, not unique)

A (cyclic) method/procedure to gather/organize, check (verify or refute) and test, conserve/preserve and transmit/communicate knowledge (both in form of data or organized abstract data/axioms/propositions) or more generally information, based on:

  • Experience. By experience we understand observation of natural phenomena, original thoughts, common sense perceptions and observed data from instruments or data. You can also gather data with emulation or simulation of known data, in a virtual environtment.
  • Intuition and imagination. Sometimes scientific ideas come from experience, sometimes from intuitions and abstractions from real world and/or structures. You can also use imagination to test something via gedanken or thought experiments tied to the previous experiences or new experiences, or use computer/AI/machines to creatively check or do inferences.
  • Logic and mathematical language. Logic, both inductive and deductive, is necessary for mathematical or scientific proofs. Since Galileo, we already know that Mathematics is the language in which Nature is better described with. We can also say that this includes reasoning or reason as a consequence.
  • Curiosity. The will to know is basic for scientists. No curiosity, no new experiments, observations, theories or ideas.

The scientific method has some powerful tools:

  • Computers and numerical simulations. This is new from the 20th century. Now, we can be aided by computer calculations and simulations to check scientific hypothesis or theories. Machine learning is also included here as subtool.
  • Statistics and data analysis. Today, in the era of Big Data and the Rise of AI, this branch and tool from the scientific method gains new importance.
  • Experimental devices to measure quantities predicted or expected from observations and or hypotheses, theories or models.
  • Rigor. Very important for scientists, and mathematicians even more, is the rigor of the method and analysis.
  • Scientific communication, both specialized and plain for everyone. Scientists must communicate their results and findings for testing. Furthermore, they must try to make accessible the uses of their findings or why they are going to be useful or not in the future.

Scientific method can begin from data, or from theories and models. Key ideas are:

  • (Scientific) Hypothesis. Idea, proposition, argument or observation that can be tested in any experiment. By experiment, here, we understand also computer simulations, numerical analysis, observation with telescope or data analysis instruments, machine/robotic testing, automatic check and/or formal proof by mathematical induction or deduction.
  • An axiom is a statement that is assumed to be true without any proof, based on logical arguments or experience.
  • A theory is a set of tested hypotheses subject to be proven before it is considered to be true or false. A theory is also a set of statements that is developed through a process of continued abstractions and experiments. A theory is aimed at a generalized statement or also aimed at explaining a phenomenon.
  • A model is a purposeful representation of reality.
  • A conjecture is proposition based on inconclusive grounds, and sometimes can not be fully tested.
  • A paradigm (Kuhn) is a distinct set of concepts or thought patterns, including theories, research methods, postulates, and standards for what constitutes legitimate contributions to a field.

What properties allow us to say something is scientific and something is not? Philosophy of science is old and some people thought about this question. Some partial answers are known:

  • Falsifiability. Any scientific idea or hypothesis or proposition can be refuted and tested. Otherwise is not science. It is a belief. Scientific stuff can be refutable and argued against with. Experiments or proof can be done to check them. Kuhn defended the addition of additional ad hoc hypotheses to sustain a paradigm, Popper gave up this approach.
  • Verification of data or hypotheses/theories/arguments. Even when you can refute and prove a  theory is wrong, verification of current theories or hypotheses is an important part of scientific instruments.
  • Algorithmic truths and/or logical procedures. Science proceeds with algorithms and/or logic to test things.  Unordered checking looses credibility. Trial and error is other basic procedure of Science.
  • Heuristics arguments based on logic and/or observations. Intuition and imagination can provide access to scientific truths before testing.
  • Reproducibility. Any experiment or observation, in order to be scientific, should be reproducible.
  • Testable predictions. Usually, theories or hypotheses provide new predictions, not observed before.

The scientific method is an iterative, cyclical process through which information is continually revised. Thus, it can be thought as a set of 4 ingredients as well:

  • Characterizations (observations, definitions, and measurements of the subject of inquiry).
  • Hypotheses (theoretical, hypothetical explanations of observations and measurements of the subject).
  • Predictions (inductive and deductive reasoning from the hypothesis or theory).
  • Experiments (tests of all of the above).

Pierce distinguished between three types of procedures:

  • Abduction. It is a mere “guess”, intuitive and not too formal.
  • Deduction. It includes premises, explanations and demonstrations.
  • Induction. A set of classification, probations and sentient reasoning.

From a pure mathematical and theorist way, there are only knowing and understanding facts, analysis, synthesis and reviews or extensions of information/knowledge. From the physical or experimentalist viewpoint, however, we have more:

  • Characterization of experiences and observations.
  • Proposals of hypotheses.
  • Deductions and predictions from hypotheses.
  • Realization of tests and experiments (gathering data).

Note that, from a simple viewpoint, the scientific method and/or main task of Science is to study:

  • Regularities, patterns and relationships between objects and magnitudes.
  • Anomalies or oddities, generally hinting something new beyond standard theories.
  • Reality as something we measure and the link between observers and that reality. What is reality after all? Hard question from the quantum realm side…

By the other hand, a purely bayesianist approach to Science is also possible. In a Bayesian setting, Science is only a set up to test the degree of belief of any proposition/idea/set of hypotheses/model/theory. Theories provide measurable observables and quantities, and scientific predictions are only valid up to certain confidence level with respect some probabilistic distributions. This probabilistic approach to Science does not exclude the existence of purely true or false hypotheses, a frequentist approach to data and error analysis (it complements that tool), and it only focuses on a framework to estimate the probability of propositions, data vectors and experimental parameters fitting certain probability distributions “a prior”.

How to elucidate the degree of (scientific) belief of something? W. K. Clifford discussed this topic with Jaynes in order to give a list. In the Ethics of Belief was argued that: rules or standards that properly govern responsible belief-formation and the pursuit of intellectual excellence are what philosophers call epistemic (or “doxastic”) norms. Widely accepted epistemic norms include:

  • Don’t believe on insufficient evidence.
  • Proportion your beliefs to the strength of the evidence.
  • Don’t ignore or dismiss relevant evidence.
  • Be willing to revise your beliefs in light of new evidence.
  • Avoid wishful thinking.
  • Be open-minded and fair-minded.
  • Be wary of beliefs that align with your self-interest.
  • Admit how little you know.
  • Be alert to egocentrism, prejudice, and other mental biases.
  • Be careful to draw logical conclusions.
  • Base your beliefs on credible, well-substantiated evidence.
  • Be consistent.
  • Be curious and passionate in the pursuit of knowledge.
  • Think clearly and precisely.
  • Carefully investigate claims that concern you.
  • Actively seek out views that differ from your own.
  • Be grateful for constructive criticisms.
  • Question your assumptions.
  • Think about the implications of your beliefs.
  • Persevere through boring or difficult intellectual tasks.
  • Be thorough in your intellectual work.
  • Stick up for your beliefs, even in the face of peer pressure, ridicule, or intolerance.

Unanswered questions by Science are yet to be provided:

  1. Why mathematics is so accurate and precise to describe Nature?
  2. Why is the Universe comprehensible and non-chaotic but regular and structured in general? It could have been very different!
  3. Why numbers and structures are so efficient?
  4. Is Science affected by the Gödel theorems or does it go beyond its applicability?
  5. Can Science explain everything?
  6. Are chaos and other mathematical universes possible and physically realizable or ideally are only unfeasible?

Usually, the scientific method contained theory and experiment only. Now, it also include: computation, big data, machine learning and AI tools!

See you in another blog post!

LOG#227. Cosmic energy.

Short post number two! Surprise!

Have you ever wondered what is the cosmic energy of the Universe? Well, giving up certain General Relativity issues related to the notion of energy in local sense, there is indeed a global notion of energy for the Universe as a whole. I am not considering the Multiverse as an option today. Let me begin with the High School notion of mass and density, particularized for the Universe:


We are considering a closed spherical Universe with 3-d geometry, and then its volume reads

    \[V_U=\dfrac{4}{3}\pi R_U^3\]

What is the radius of the Universe? Well, we could take it as the Hubble radius of the observable Universe, i.e.,


where H\approx 70km/s/Mpc. The density of the Universe can be written as the cosmological value of the vacuum/Hubble scale

    \[\rho_U=\dfrac{\Lambda c^4}{8\pi G}=\dfrac{3c^2H^2}{8\pi G}\]

so \Lambda c^4=3c^2H^2. Therefore, the formula for the mass of the Universe in terms of fundamental constants is


and the expression for the cosmic energy follows up from special relativity greatest formula E_U=M_Uc^2 as


Also, defining L_\Lambda=\sqrt{3/\Lambda} and L_P ^2=G\hbar/c^3, you get


Remark: the cosmological constant fixes not only the biggest mass as the Universal mass of the Universe (I am sorry for pedantic expression), but also fixes the smallest possible mass (the so-called Garidi mass in de Sitter group or de Sitter relativity):


And you can thus prove that

    \[\dfrac{M_U}{M_\Lambda}=\dfrac{1}{2}\dfrac{c^3}{G_N\hbar}\dfrac{3}{\Lambda}\sim \dfrac{L_\Lambda^2}{L_P^2}\sim 10^{122}\]

Note, that




    \[V_P\sim L_P^3\]

    \[V_\Lambda=V_U\sim L_\Lambda^3 \]


    \[\dfrac{M_U}{M_P}\sim\dfrac{ L_\Lambda}{L_P}\]

    \[\dfrac{M_U}{M_\Lambda}\sim\dfrac{ L_\Lambda^2}{L_P^2}\]

and thus




See you in other blog post!!!!!

LOG#226. Higgs vs. lambda.

Hi, there. Short post today!

The dark energy mystery, a.k.a., the cosmological constant problem or why the observed vacuum energy is to tiny is a big problem with no consensus solution yet.

The Standard Model Higgs potential can be written as follows


Here, m_H is the Higgs mass m_H=125GeV, \lambda=0.13 is the Higgs self-coupling from the given mass and \phi_0 is the vacuum expectation value of the Higgs field, namely

    \[\phi_0=\langle 0\vert \phi\vert 0\rangle=246GeV\]

From High School mathematics, you can easily find out the minimum of the above potential

    \[\dfrac{d V_H}{d\phi}=-2m_H^2\phi+4\lambda\phi^3=2\phi\left(-m_H^2+2\lambda\phi^2\right)=0\]

so the non null solution is given by


and the Higgs potential value at that value becomes

    \[V(\phi(min))=-\dfrac{m_H^2\phi^2}{2\lambda}+\phi_0\approx -\dfrac{m_H^2\phi^2}{4\lambda}\]

The Higgs potential provides a natural source for vacuum energy density. Using the values mentioned before we get

    \[\vert V_H(min)\vert\approx 5\cdot 10^8GeV^4\]

The value we get from Cosmology, using the LCDM model is however very tiny

    \[V(\Lambda)=\dfrac{\Lambda c^4}{8\pi G}\approx 3.1\cdot 10^{-47}GeV^4\]

thus, we have a mismatch between theory and experiment about

    \[\dfrac{\rho(Higgs)}{\rho(\Lambda)}\sim 2\cdot 10^{55}\]

Note that this is, despite the mismatch, better than the crude mismatch due to the even bigger discrepancy between Planck energy density and observed vacuum energy density, since Planck energy density reads:

    \[\rho_P=\dfrac{c^7}{G^2_N\hbar}\approx 5\cdot 10^{113}J/m^3\approx  2.2\cdot  10^{76}GeV^4\]

and thus

    \[\dfrac{\rho_P}{\rho_\Lambda}\sim 10^{122}\]

Or,as well,

    \[\dfrac{\rho_P}{\rho_\Lambda}=\dfrac{\dfrac{c^7}{G^2_N\hbar}}{\dfrac{\Lambda c^4}{8\pi G_N}}=\dfrac{8\pi}{3}\dfrac{c^3}{G_N\hbar}\dfrac{3}{\Lambda}=\dfrac{8\pi}{3}\dfrac{L_\Lambda^2}{L_P^2}\sim \dfrac{L_\Lambda^2}{L_P^2}\]

Solutions? Well, many:

  • The Higgs value is wrong, because the Higgs potential from the SM is not right, but only an approximation.
  • The Cosmological Constant is not the Higgs potential source.
  • Both values are OK, we have bad measurements only.
  • Vacuum energy is scale dependent, you can not compare them without some tricky trick.
  • Some non-perturbative effect in under the floor.
  • A new physics reason not quoted above.

I know ther are some other crude estimates from UV-cutoff given a 122-123 orders of magnitude separation. However, this is much closer. Some SUSY (supersymmetry) theories have some arguments to essentially fit some of the solutions I mentioned. But the issue is not at all clear. A very big cosmological constant would be a disaster for life. The very big vacuum energy of Quantum Field Theory grosser estimates clearly are nonsense, but we do not know yet when the calculation is wrong. Maybe a dS (de Sitter) QFT could provide a better solution using \Lambda as fundamental constant?

Note, do not confuse the Higgs self-coupling \lambda with the cosmological constant \Lambda.

What do you think? A simple minimization of the Higgs potential according the SM gives you a 55 order of magnitude split with observed vacuum energy density? Are they the same or are we lacking something more fundamental?

Off-topic news: it seems the quark-gluon plasma behaves as an ideal relativistic fluid, as the holographic hypothesis suggest, giving experimental increasing support to the bound of shear viscosity to entropy ratio predicted by those holographic models

    \[\dfrac{\eta}{s}\geq\dfrac{\hbar}{4\pi k_B}\]

or in natural units

    \[\dfrac{\eta}{s}\geq \dfrac{1}{4\pi }\]


[1] Jonah E. Bernhard, J. Scott Moreland, Steffen A. Bass, «Bayesian estimation of the specific shear and bulk viscosity of quark–gluon plasma,» Nature Physics (12 Aug 2019), doi: 10.1038/s41567-019-0611-8

[2]  Kari J. Eskola, «Nearly perfect quark–gluon fuid,» Nature Physics (12 Aug 2019), doi: 10.1038/s41567-019-0643-0.

See you in another blog post!!!!

LOG#225. QCD: the matrices.

Quantum Chromodynamics (QCD). The theory I learned that explains the nuclei as a teenager, i.e, the theory what explained why nuclei were not ripped off away due to nuclear forces. Strong nuclear forces (weak nuclear forces are the cause of radioactivity). Quarks and gluons. Matter and particle force carriers or messengers. I remember yet how some teens asked me and smiled with bad intentions what asked me what I was doing during the weekends and the week out of class. Sad news is that yet I have to keep calm seeing the same smile and similar comments about what I like to read, what I like to do or talk at my current age. Dark fate, I think…

Quarks are the matter fields feeling the strong nuclear force. Leptons do not feel color force. Color charges are just a name, not related to physical color though. It is just a way to work out a triad of gauge charges unlike electric charges (a diad) or gravitational charge (aka mass, a monad). How many colors are there? Three: red (R), blue (B) and green (G). Plus anticolor charges, antired \overline{R}, antiblue \overline{B} and antigreen \overline{G}. When two quarks interact, can swap charges or color via gluons. The interaction is something like a chemical reaction

    \[RB\leftrightarrow R\overline{B}\leftrightarrow BR\]

You can indeed imagine anticolor as color charges backwards in time. Expected gluons can be classified as follows (we will see that there are, in fact, only 8 possible color gluons independent to each other):










Note that these letters allow you to get colorless combinations with different quantum color numbers, thus being coherent with Pauli exclusion principle. States like uuu are possible ir you understand u_Ru_Bu_G, so quarks have different quantum numbers. Color is related as well to hypercharge and isospen, different combinations of isospin and hypercharge also label quarks and hadronic states. Experimentally, as color particles are not observable, only colorless states are possible. This fact settles a problem for B\overline{B},R\overline{R},G\overline{G} states, but, as surely you know, there are two main species of hadrons: baryons (quarky threesomes), and mesons (quarky couples). However, today hadron spectroscopy is much richer. Lattice field theories and experiments show us that there are also resonances AND, likely, exotics. Exotics are states not being purely couples or threesomes of quarks. You can get purely gluonic states called glueballs or gluonium by experts, and also you can get in principle quarky foursomes, quarky pentets and more. I read some ago a paper about heptaquarks and octoquarky states. Crazy! I wish I were such a quarky state sometimes.

What about gauge symmetries for quarks? It is called SU(3)_c symmetry. The fact protons get masses is due not to the Higgs mechanism, as I mentioned some days ago, but to a dynamical non-perturbative process of chiral symmetry breaking called sometimes dimensional transmutation. What a name! Gluons can be described by some 3\times 3 special unitary matrices or “grids”. These matrices have to be traceless (the sum over the main diagonal of its entries adds up to zero!). Let me represent color and anticolor states as column vectors and row vectors, respectively:

    \[R=\begin{pmatrix}1\\ 0\\0\end{pmatrix}\]

    \[B=\begin{pmatrix}0\\ 1\\0\end{pmatrix}\]

    \[G=\begin{pmatrix}0\\ 0\\1\end{pmatrix}\]

    \[\overline{R}=\begin{pmatrix}1& 0&0\end{pmatrix}\]


    \[\overline{G}=\begin{pmatrix}0& 0&1\end{pmatrix}\]

Then, you can get by tensor product the 3\times 3 matrices

    \[R\overline{R}=R\otimes\overline{R}=\begin{pmatrix} 1&0&0\\0&0&0\\0&0&0\end{pmatrix}\]

    \[R\overline{G}=R\otimes\overline{G}=\begin{pmatrix} 0&1&0\\0&0&0\\0&0&0\end{pmatrix}\]

    \[R\overline{B}=R\otimes\overline{B}=\begin{pmatrix} 0&0&1\\0&0&0\\0&0&0\end{pmatrix}\]

    \[G\overline{R}=G\otimes\overline{R}=\begin{pmatrix} 0&0&0\\1&0&0\\0&0&0\end{pmatrix}\]

    \[G\overline{G}=G\otimes\overline{G}=\begin{pmatrix} 0&0&0\\0&1&0\\0&0&0\end{pmatrix}\]

    \[G\overline{B}=G\otimes\overline{B}=\begin{pmatrix} 0&0&0\\0&0&1\\0&0&0\end{pmatrix}\]

    \[B\overline{R}=B\otimes\overline{R}=\begin{pmatrix} 0&0&0\\0&0&0\\1&0&0\end{pmatrix}\]

    \[B\overline{G}=B\otimes\overline{G}=\begin{pmatrix} 0&0&0\\0&0&0\\0&1&0\end{pmatrix}\]

    \[G\overline{G}=G\otimes\overline{G}=\begin{pmatrix} 0&0&0\\0&0&0\\0&0&1\end{pmatrix}\]

There are 9 matrices, but R\overline{R},B\overline{B},G\overline{G} are not traceless. You get 6 matrices giving up these combinations. The SU(3) group has 8 generators (independent!). Where are the other 2? There are different choices, but you can that the following two matrices provide a good choice



There are 8 total color grids/matrices for independent color changes:


Unlikely photons, gluons also interact with theirselves but not with the Higgs boson! Feynman graphs are typically Y shaped of double Y-shaped for gluon self-interactions. Cinfinement is a complex phenomenon. However, as said before here in TSOR, it can be modeled with a relatively simple potential for two quarks (the quarkonium):


where k=\sigma is the string tension. This model is good for “heavy” quarks. This model can be compared to that of QED:

  • In QED, field lines extend up to infinity as \sim1/r^2. In QCD, quark field lines due to color are stretched or compressed into a tiny region between quarks and antiquarks. Breaking a flux tube requires the creation of a quark-antiquark pair.
  • In QED, Electromagnetic flux is conserved to infinity, but in QCD, color flux is trapped between quarks. No strong interations outside the color flux tube! Breaking the string implies a big energy, much larger than that of bounded atoms.
  • Hybrid states are allows, such as \langle q\overline{q}g\rangle, \langle ggg\rangle,\ldots. These exotica are the subject of some simulations of lattice QCD with supercomputers AND, as well, are being tried to mimic some of the resonant states already known. It is complicated to find out a good match, though.

In summary:

  1. QCD is all about quarks, gluons and internal symmetries modeled by matrices, matrices represent transitions between color states.
  2. QCD is the theory of color, the quantum number (not a physical color truly) of strong interactions. Have you ever imagined if we had called them strawberry, blueberry and mint? Or orange, vanilla and chocolate? Maybe, it would taste better…
  3. Gluons interact with theirselves, unlike photons, due to non abelian properties of the gauge group (something they share with weak and electroweak interactions).
  4. Gluons are Higgs transparent, they do not interact with the Higgs field. Why? Nobody knows. But it is fortunate for live to be it so.
  5. Mesons and baryons get masses due to the chiral symmetry breaking and a complex mechanism of dimensional transmutation. However, valence quarks can get a little mass from Higgs particles. So elementary quarks do interact with the Higgs field a little bit, but composite hadrons are getting masses through QCD.
  6. Hybrid states and resonances are expected to arise in the QCD spectrum. However, the Yang-Mills mass-gap problem is yet unsolved. Prove that tha Yang-Mills equations have a mass-gap and you will win a million dollar from Clay Institute!

See you in another blog post!!!!!

LOG#224. Vis tenebris.

What is dark energy? Where does it come from? Is it constant? Is it a quantum field? Is it vacuum energy? Nobody knows what is the 70% of the Universe!

In the beginning of Physics as Science, Aristotle himself introduced the idea of energy (from the Greek \varepsilon\nu\varepsilon\rho\gamma\varepsilon\iota\alpha., meaning capacity of update or act on something). Stones fall down to Earth with the goal of finding their natural place. The capacity to be placed in the natural place requires something called energy. Energy was also implied to a virtuous ending. Today, the physicist definition uses the capability to do some “work”, i.e., the ability to displace something with a force changing the state of the object requires energy. Energy is also a quantity invariant under time translations (first Noether theorem!). Energy spent to create motion is the the same the body acquires to move. That is the origin of the name “vis viva”. Energy is something required to move something.

There are 4 great equations for energy, related to the biggest revolutions in physics:

1st. \dfrac{1}{2}mv^2=E_k. Kinetic energy or true vis viva, in the sense it was the first to be introduced (up to some numerical constant).

2nd. E_t=k_BT. Thermal energy as the result of atomic-molecular hypothesis. The great idea of Boltzmann.

3rd. E=hf=\hbar\omega. Quantum of energy, Planck desperatation made him invent the quanta of radiation to solve the black body problem.

4th. E=mc^2. Special relativity, 20th equation, Einstein greatest contribution to the understanding of the energy concept. Unfortunately, also used in nuclear weapons.

There is a gravitational potential energy formula as well,


and a electric potential energy


but the above 4 are the most conceptual equations for energy. It was Descartes the first one who discovered  that the followin quantity (or a multiple of it) was constant from Galileo works:


and that it was necessary in order to make up a clock! A gravitational clock is a device that requires some vis viva mv^2 to work! If v=\omega L, you get an energy E=M\omega^2L^2 for the clock, and good clocks have fast ticks, so \omega>>1. The clock gravitational frequency can be written as


L can be done as short as we want … Until quantum mechanics is reached. In quantum mechanics, you get p=mv, and quantization of the action pq=N\hbar.There is a quantum length, critical


Quantum mechanics, thus, allows us to build up clocks with certain uncertainty or fuzziness, instead of gravity, that allows us to build up, in principle, clocks without uncertainty. p=\hbar/L_Q, so then




The critical (optimal) clock in QM is the one with


Deep question: can we simulate with classical gravity a quantum clock? How can we understand gravity from a quantum entangled critical system? Hint is that g=\hbar^2/(M^2L^3).

Leibniz formalized the ideas of Descartes with his calculus. Einstein renewed the idea of vis viva and energy with the aid of his special relativity. Time is slower when you move faster. General relativity enhances this fact, showing that the time passes slower closer of a massive body. E=Mc^2 has indeed a classical meaning as “useful energy” or “binding energy” in nuclear or particle physics. Total energy is useful plus unuseful energy. Related to unuseful energy is the idea of entropy. Entropy, from the Greek meaning “transformation”, is a magnitude equal to S=E/T or S=k_B\ln N, where N is the number of possible configurations or microstates with the given energy.. The Max-Ent principle says the total entropy of the universe tends to be maximized. In the end, at lower energies, every possible configuration has the same probability. Then, equipartition of energy holds, as thermodynamics and statistical physics taught us.

Now, dark energy as vis tenebris. Inventing his celebrated General theory of Relativity, a locally special relativistic theory of gravitation, Einstein realized that his equations predicted the Universe was expanding (contracting). That was obviously nonsense, the Universe was static (before Hubble discovery!). Then, he used a freedom his equations had, and introduced a term, called the cosmological term (or cosmic repulsion) that balanced gravity and made the Universe static. After Hubble’s discovery of the expanding Universe of galaxies, Einstein thought he had committed the biggest epic fail of his life…I wonder what he would have thought if alive when in 1998 we discovered the expansion is accelerating (positively) by something that mimics his cosmological constant at this time. The name dark energy is due to the fact we are not, in principle, be forced to believe the cosmological constant is indeed constant, since it could be varying very slowly on cosmological times, and it could be some king of field. Anyway, dark energy (vis tenebris) is permeating the space and it is responsible to the increasing speed of the expansion. Every time the space expands, more vis tenebris is added. If it is the cosmological constant, its addition is such that the density energy is constant in all over the space! Vis tenebris energy density is (take a universe as a sphere for simplicity)

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

and then the vacuum energy in a sphere of volume 4\pi R^3/3 reads

    \[E_\Lambda=\dfrac{\Lambda c^4R^3}{6G_N}\]

I you know \Lambda and R, you can compute the dark energy in that volume. As the scale factor increases, the density remains constant so the cosmic energy grows up by

    \[\Delta = \hbar\left(\dfrac{1}{a(t)}-\dfrac{1}{a_0(t)}\right)= \hbar\left(\dfrac{1}{d}-\dfrac{1}{D}\right)\]

Indeed, dark energy (vis tenebris) is the darkly main ingredient of the current Universe (about 2/3 or 3/4 as most). We are dominated by a dark energy “force”. What is it? That is the challenge and the puzzle…

See you in another wonderful post!

LOG#223. Pi-logy.

Hi, there.

Today some retarded Pi-day celebration equations (there is a longer version of this, that I wish I could publish next year). Some numbers and estimates for pi-related equations:

1st. Hawking radiation temperature (Schwarzschild’s 4d black hole case).

(1)   \begin{equation*}T_H=\dfrac{\hbar c^3}{8\mathbf{\pi} G_NMk_B}=6.2\cdot 10^{-8}\left(\dfrac{M}{M_\odot}\right)K\end{equation*}

2nd. Schwarzschild black hole surface area (4d).

(2)   \begin{equation*}A_S=4\mathbf{\pi}R_S^2=\dfrac{16\mathbf{\pi}G_N^2M^2}{c^4}=1.1\cdot 10^8\left(\dfrac{M}{M_\odot}\right)^2\; m^2\end{equation*}

3rd. Black hole power/luminosity (4d).

(3)   \begin{equation*}L_{BH}=P_{BH}=\dfrac{\hbar c^6}{15360\mathbf{\pi}G_N^ 2M^2}=9.0\cdot 10^{-29}\left(\dfrac{M_\odot}{M}\right)^2W\end{equation*}

4th. Black hole evaporation time (4d).

(4)   \begin{equation*}t_{e}=\dfrac{5120\mathbf{\pi}G^2_NM_0^3}{\hbar c^4}=8.41\cdot 10^{-17}\left(\dfrac{M}{1kg}\right)^3s=6.6\cdot 10^{74}\left(\dfrac{M}{1kg}\right)^3s=2.1\cdot 10^{67}\left(\dfrac{M}{1kg}\right)^3yrs\end{equation*}

5th. Time to fall off and arrive to the BH singularity with negligible test mass (4d).

(5)   \begin{equation*}t_f(test)=\dfrac{\mathbf{\pi}}{2c}R_S=\dfrac{\mathbf{\pi}G_NM}{c^3}=1.5\cdot 10^{-5}\left(\dfrac{M}{M_\odot}\right)s\end{equation*}

6th. Time to fall off and arrive to the BH singularity with E=m test mass (4d).

(6)   \begin{equation*}t_f(m)=\dfrac{2}{3}\dfrac{R_S}{c}=\dfrac{4\mathbf{\pi}G_NM}{c^3}=6.2\cdot 10^{-5}\left(\dfrac{M}{M_\odot}\right)s\end{equation*}

7th. Black hole entropy (4d) value in SI units.

(7)   \begin{equation*}S=\dfrac{k_B c^3}{G_N\hbar}A_{BH}=\dfrac{k_BA}{4L_p^2}=\dfrac{4\mathbf{\pi} GM^2}{\hbar c}=\dfrac{\mathbf{\pi}k_Bc^3A_{BH}}{2G_Nh}=1.5\cdot 10^{54}\dfrac{M^2}{M_\odot^ 2}J/K\end{equation*}

8th. M2-M5 brane quantization.

(8)   \begin{equation*}T_{M2}T_{M5}=\dfrac{2\mathbf{\pi}N}{2k_{11}^2}=\dfrac{\mathbf{\pi}N}{k_{11}^2}\end{equation*}

9th. Gravitational wave power or GW luminosity.

    \[L_{GW}=-\dfrac{dE}{dt}=\left(\dfrac{32}{5c^5}\right)G^{7/3}\left(M_c\pi f_{GW}\right)^{10/3}\]

where the gravitational wave frequency is


10th. Chirp frequency or frequency rate.

For circular orbits, you have


11th. Coalescence time for GW merger (circular orbits).


12th. ISCO (inner stable circular orbit) frequency for binary mergers.

    \[f_{max,c}=f_{isco}=\dfrac{c^3}{6^{3/2}\pi GM}\approx 4.4\dfrac{M}{M_\odot} kHz\]

13th. S-matrix in D-dimensions.


14th. Gravitational wave fluxes for gravitons and photons (4d).

    \[F_{GW}=\dfrac{c^3h^2\omega^2}{16\pi G_N}=\dfrac{\pi c^3h^2f^2}{4G_N}\]

where h is the GW strain, and for photons, the GW induced electromagnetic  flux reads

    \[F_{em}=\dfrac{c^3\omega^2 h^4}{8\pi G_N}=\dfrac{\pi c^3 f^2 h^4}{2G_N}\]

15th. Kerr-Newmann black hole area and mass spectrum.

Any massive, rotating, charged black hole have an event horizon given by the following formula


This relation can be inverted to obtain the mass spectrum as function of area, charge and angular momentum as follows (exercise!):


Challenge: modify the above expressions to include a cosmological constant factor.

16th. Universal quantum gravity potential at low energies.

Quantum gravity at low energy provides the following potential energy

    \[V_{QG}=-\dfrac{GM_1M_2}{r}\left[1+\dfrac{3G_N\left(M_1+M_2\right)}{rc^2}+\dfrac{41G_N\hbar}{10\pi r^2}\right]\]

independent of the QG approach you use!

17th. Running alpha strong.


For the general QCD the beta factor reads


and the SM gives \beta_0>0 (N_c=3, n_f=6) and slope \beta(\alpha_s)<0 due to asymptotic freedom (antiscreening).

18th. Graviton energy density and single graviton energy density.

The graviton energy density reads off from GR as

    \[\rho_E=\dfrac{c^2\omega^2f^2}{32\pi G_N}\]

and for a single graviton, it reads

    \[\rho_E(single)=\dfrac{\hbar \omega^4}{c^3}=\dfrac{8\pi^3 h f^4}{c^3}\]

where h is the Planck constant, not the strain here.

I have many other pi-logy equations, but let me reserve them for a future longer post!

See you all, very soon!

LOG#222. The New SI.

By the time we will find new physics, we have already redefined the SI in terms of base units and fundamental constants.

The definition of the new SI is the next one: the SI is the system in which the following constants are taken to be exact

  • The unperturbed ground state hyperfine splitting (transition) frequency of the caesium-133 atom \Delta f(Cs-133) is exactly

    \[\Delta f=9192631770 Hz\]

Thus, frequency is fundamental, and the time is a base unit from frequency. One second is the time

    \[1s=\dfrac{9192631770}{\Delta f}\]

and 1 Hz is the reciprocal of the above quantity, exactly too.

  • The speed of light in vacuum c is exactly the quantity

    \[c=299792458m\cdot s^{-1}\]

Using the previous and this definition, you can define the meter to be exactly the amount of length

    \[1m=\dfrac{9192631770c}{299792458\Delta f}\]

  • The elementary charge e is exactly the quantity

    \[e=1.602176565\cdot 10^{-19}\]

Thus, the old electric current unit, base unit, the ampère, is the unit in which you can express charge into current, in corresponding units, with the next conversion constants:

    \[1C=\dfrac{e}{1.602176565\cdot 10^{-19}}=6.241509343\cdot 10^8e\]

    \[1A=\dfrac{1C}{1s}=\dfrac{6.241509343\cdot 10^8e\Delta f}{9192631770}=\dfrac{e}{(1.602176565\cdot 10^{-19})(9192631770)}\]

  • The Planck constant is exactly defined to be

    \[h=6.62607015\cdot 10^{-34}J\cdot s\]

and thus the kilogram is defined in terms of fundamental constants as

    \[1kg=\dfrac{(299792458)^2h\Delta f}{(6.62607015\cdot 10^{-34})(9192631770)c^2}\]

  • The Boltzmann constant is exactly

    \[k_B=1.3806488\cdot 10^{-23}J\cdot K^{-1}\]

so (with 1J=1kg\cdot 1m^2\cdot 1s^{-2} and then

    \[1K=\dfrac{1.3806488\cdot 10^{-23}}{k_B}=\dfrac{1.3806488\cdot 10^{-23}h\Delta f}{(6.62607015\cdot 10^{-34})(9192631770)k_B}\]

  • The Avogadro constant is defined exactly to be

    \[N_A=6.02214129\cdot 10^{23}\]

so the mole is

    \[1mole=\dfrac{6.02214129\cdot 10^{23}}{N_A}\]

  • The luminous efficacy K_{cd} of monochromatic radiation of frequency 540 THz is exactly defined to be 683 lumen/(W\cdot sr). Thus, as stereradian is dimensionless, lm=cd\cdot sr and the candela definition holds exactly to be as well

    \[1cd=1\dfrac{lm}{683W\cdot sr}=\dfrac{K_{cd}}{683}kg\cdot m^2\cdot s^{-3}\cdot sr^{-1}=\dfrac{1\cdot \Delta f^2 hK_{cd}}{(9192631770)^2(6.62607015\cdot 10^{-34})683}\]

See you in another blog post!

LOG#221. Kepler+Cosmic speeds.

Johannes Kepler
Kopie eines verlorengegangenen Originals von 1610

Hi, there! Today some Kepler third law stuff plus cosmic speed calculations and formulae. Cosmic speed sounds cool…But first…

From High School, you surely calculated how fast is  A PLANET moving around the sun (or any star, indeed). To simplify things, take units with G_N=4\pi^2 for a moment (nasty trick that works!). Kepler third law reads


where M=M_\star+M_p is the total mass, sum of the star mass and the planet star, a is the major semiaxis of the ellipse and T is the period of the motion. Note that for binary systems like binary stars, you can not neglect the M_p term since it is comparable to the higher mass. Simple calculus, let you obtain


Have you ever asked yourself what is the STAR speed? Generally speaking, the star is NOT static either in gravitation! So, forget that picture on your head telling you that the sun is fixed, it also moves. What is the star speed? It shows that you can compute easily the star speed with the aid of the conservation of linear momentum. Linear momentum p=mv is conserved, due to translational invariance in 3d space, and thus,


Set the constant to 0, and take the modulus, so you can see now that


Then, you get


or equivalently


In our units with G_N=4\pi^2, speeds are indeed measured in AU/yr! Now, you can not only calculate the speed of Earth around the sun, you can indeed calculate the speed of the sun around the Earth. You can extend this argument and calculate the difference between the planet speed, the star speed and the center of mass speed. It is a quite pedagogical exercise! In fact, there are two extra corrections to the abouve formulae in the general setting of celestial mechanics: you must include the effect of eccentricity and the inclination of the system with respect to the observer. The above formulae suppose you look perpendicular to the system, and the eccentricity is small or zero. If you use standard G_N SI units, you would get instead

    \[v_{p}=\left(\dfrac{2\pi GM}{T}\right)^{1/3}\]


and generally

    \[P_{CM}=\dfrac{m_pv_p+M_\star V_\star}{M}\]

and normally you choose P_{CM}=0 for convenience, but it can also be calculated with respect to the planet or star frames!

Angular speed in Kepler law (or velocities/speeds) are related to the space dimension in space-time! Thus, in D=d+1 space-time you would get angular speeds


and periods would scale as R^{d/2}. Moreover, if you are orbiting a star but a GR rotating object, it is described better by a Kerr metric. In Kerr spacetimes, Kepler third law gets generalized into (G_N=c=1)

    \[\Omega=\pm\dfrac{M^{1/2}}{r^{3/2}\pm aM^{1/2}}\]

where a is now the Kepler parameter. Reintroduce units to get instead


Kepler third law can also be extended, for instance,in Finsler-like general relativity. There, you could get


or even stranger formulae in other gravitationally modified theories of gravity are even possible. Therefore, if you modify gravity with extra dimensions or more general theories, you obtain corrections to the Kepler third law. Even simple rotating black holes provide a generalized Kepler third law (the above formula is for ecuatorial orbits only!) for orbitating bodies! Thus, observations on orbital patterns could provide you hints on modified gravity. Unfortunately, no observation is yet giving you a MOG (MOdified Gravity) or extended theory of gravity confirmation. It implies strong bounds on the possible sizes of these corrections or discard them till now!

To end this post, I will review the so-called cosmic speeds:

  • First cosmic speed. Namely, the orbital speed. For usual spacetime dimensions read


  • Second cosmic speed. It is the escape velocity/speed. It yields


  • Third cosmic speed. It is the escape velocity from the solar system. A naive calculation for Earth third cosmic speed gives


but the fact that Earth is also moving, let us reduce this value to a lower number, since V(oS)=V_S-V_o=12.3km/s, where V_o=29.8km/s is the orbital Earth speed, such as




  • Fourth cosmic speed. You need naively V_4'=350km/s for escape from the Milky way, but as the solary system is moving with respect to it, you can easily show up (exercise for you!) that you would need only V_4=130km/s to escape, similarly to the case of the third cosmic speed.

Remember the instantaneous speed is:


Let me remark a final two constants (the hidden secret constant will be the topic of a future blog post about the full Kepler problem and its generalizations) formulae for the Kepler reduced 2-body problem:



where E_t, L are the total energy and angular momentum M is the total mass, \mu=(M_1+M_2)/M the reduced mass, and R the orbital major semiaxis, and e is the orbital eccentricity. Are you eccentric today? Problem: What would be the escape velocity from our Universe?

A summary table:

See you in a future new blog post!

P.S.: Some earthling speeds are

i) Rotational speed of Earth on the equator is about 1670km/h or 0.46km/s.

ii) Rotational speed of Earth around the sun is about 107000km/h, or about 30 km/s.

iii) Rotational speed of the solar system (the sun) around the Milky Way is about 220km/s (828000km/h). The Milky Way spins at  about 270 km/s with respect to its center.

iv) Milky Way speed towards the Big Attractor is about 611km/s, or about 2.2 million km/s.

v) Milky Way speed with respect to the CMB is about 2268000km/h or about 630 km/s.

vi) What about the fifth cosmic velocity? Yes, it can be defined. The fifth cosmic speed is the velocity necessary to escape from this Universe. Take R_U, M_U as the radius and mass of this Universe. You can calculate the 5th cosmic speed as follows


LOG#220. Higgs&Symmetries.

    \[V(H)=m_H^2H^2+gH^3+\lambda H^4\]

Why the leaves are green? Why the sky is blue? Why diamonds are hard? The Quantum Mechanics can indeed answer those questions, and even more subtle questions. The Standard Model (SM) is the frontier in our knowledge of microscopic things, that is, the SM represents the superior quantum theory explaining the building blocks of the Universe (well, a 5% of it actually, but it does not matter for the purposes of this new blog post).

  • Six leptons (plus their antiparticles): (e,\nu_e), (\mu,\nu_\mu), (\tau,\nu_\tau).
  • Six quarks (plus their antiparticles times 8, color factor): (u,d), (c,s), (t,b).
  • Four bosons (up to gauge charges and their antiparticles): W^+,Z^0,\gamma, g.
  • The mass giver, the Higgs field, H_0, for elementary particles. Gluons and photons are higgs-transparent, bosons that interact more strongly with the Higgs field are more massive.

Timeline(short review):

  • 1932: there are protons, electrons, the neutron and the positron.
  • 1937: the muon is discovered.
  • 1940s: hadron explosion. Pions, kaons, lambdas, deltas, sigmas and other exotic states are discoverd. Puzzlement.
  • 1970s: quark theory (aces). Previous S-matrix approaches are substituted by the QCD gauge theory.
  • 1980s: gauge bosons discovered.
  • 1987: supernova in Magallanic Clouds. Neutrinos come first than photons at SuperKamiokande.
  • 1995: top quark is found.
  • 1998: neutrino oscillations are confirmed. Neutrinos are massive. First evidence that there is something beyond the SM.  Dark energy is found.
  • 2012: Higgs-like boson discovered at the LHC. No new physics signal is found, circa 2019.

Interactions and forces keep united together everything. Why there are 3 generations with increasing mass? Flavor problem. Why the Higgs mass or the electroweak scale so small compared with the Planck mass/scale? Hierarchy problem. Little hierarchy problem: why are neutrinos lighter than other SM particles? Why relative forces between interactions are the way we measure them? Pions are indeed like Van der Waals forces like the one for atomic nuclei. Pion theory is an effective theory for nuclear forces.

There are 4 forces with messengers or messenger particles: gluons, photons, W and Z bosons, Higgs particles and the graviton. Should we consider the mass giver Higgs like another interaction? Particles are classified by angular momentum:


for any entire number n=0,1,2,3,\ldots\infty . Then, there are entire half-spin particles (following the Pauli exclusion principle), and there are entire-spin particles. These are fermions and bosons, matter particles and force messengers. How did interactions arise from fields? The keyword is symmetry. I am not going to talk about SUSY and why it is inevitable in some form we do not know today, but I will tell you about symmetry and its consequences: conservation laws. Indeed, Emmy Noether two theorems go even beyond simple conservation laws. E. Noether two theorems are about invariance under finite and infinite symmetry groups. Global symmetries give you conserved quantities, local (infinite dimensional) symmetries give you identities between field equations/equations of motion for particles/fields. Likely, Noether’s theorems are the 2 most beautiful theorems in mathematical physics (if you know any other theorem rivaling them, let me know!).

Have you ever wonder why energy is conserved? Let me give you some hints:

Example 1. Free particles. 

Energy is conserved. There are symmetries under translations in time! Suppose there is a particle with mass M. Take the Newton law:


for free particles, it yields that




Name that constant kinetic energy, or total energy, so you get


Simple and beautiful. Energy is something that is conserved when you have something that is invariant under temporal translations, i.e., motion in time.

Example 2. Motion under constant force.

Generalize the above example to the motion of a free particle under constant force F. Then, you get


Then, again this is invariant under shifts in time t'\rightarrow t+\varepsilon, and there is a conserved energy function, namely




Now, use the equation of motion above \ddot{x}=F/m, and you effectively get that \dot{E}=0. Energy is conserved again! This is important in the case of gravitational field near the surface and other examples!

Example 3. Motion of simple harmonic oscillators.

Now, you get the equation of motion (EOM):


Again, under time translations, the EOM is invariant, so there is some energy functional. It yields the classical formula holds:




Example 4. Motion in a gravitational (newtonian) field.


Again, the classical energy function (by time invariance) reads


Check of constancy, using the EOM as above:


Symmetries or invariance are tied deeper into conservation laws by the Noether first theorem:

  • Temporal translation invariance implies energy conservation.
  • Invariance under spatial translations imply linear momentum conservation as well.
  • Rotational invariance means conservation of angular momentum, certain bivector or antisymmetric matrix in higher dimensions.
  • Changes in reference frames are tied to conservation of invariant mass and spin of particles. Boosts or Lorentz transformations have certainly non-trivial conservation laws (a center of mass-like conservation law). Boosts in galilean relativity has motion of center of mass constancy under these transformations.
  • Higher non-trivial symmetries have correspondingly conservation laws. Some non trivial examples are the Kepler problem, string theory or general relativity theories. Also, invariance under scale transformations have conservation laws.

Remark: under boosts (SR), you get the conserved quantities


Remark(II): discrete symmetries, produce multiplicative conservation laws of parity (P), charge conjugation (C) and time reversal (T). Particles are usually classified in the Particle Data Booklet by J^{P}.

Remark(III): hidden anomalous symmetries in the SM are baryon number or lepton number conservation. Beyond the Standard Model, these numbers can be violated and thus, proton or other generally stable particles could decay. Experimentally, fortunately for us, \tau_p\geq 10^{34}yrs. The proton is the lightest particle with baryon number not zero, so it can not decay. Baryon number is usually defined as B=(n_q-n_{\overline{q}})/3. Muons are unstable in microseconds, rho particles are unstable in yoctoseconds.

What happens with internal (non-space-time) symmetries? In any quantum theory and Quantum Field Theory, global internal symmetries are associated to conservation laws and quantum numbers. Even discrete symmetries have conserved quantities (discrete and multiplicative quantities inded). Particles are field excitations. So, how internal symmetry arise in field equations. Let me assume A=A(x) and B=B(x). Suppose a symmetry A\rightarrow A+q and B\rightarrow B-q is certain symmetry. The field equation is:


Under global q=constant symmetry transformations, you can check easily the invariance of the above field equation. Local field symmetries imply the existence of compensating fields and identities called Noether identities between field equations. That local gauge invariance implies the existence of gauge fields is a tryumph of modern mathematical physics, and it is due to Noether in the end. Under LOCAL gauge transformations,

    \[(A-B)\rightarrow (A-B)+2q\]

    \[(A+B)\rightarrow (A+B)\]

And thus, the above EOM is NOT invariant. However, you can RESTORE invariance introducing gauge (compensating) fields. How????? Let me show you a Dirac-like equation example. Write the EOM


The local U(1) invariance under phase transformations of the wave function or field

    \[\Psi\rightarrow \exp\left(i\alpha\right)\Psi\]

    \[\partial_x\left(\exp\left(i\alpha\right)\Psi\right)=\exp\left(i\alpha\right)\partial_x\Psi+i\partial_x\alpha \exp\left(i\alpha\right)\Psi\]

    \[\partial_t\left(\exp\left(i\alpha\right)\Psi\right)=\exp\left(i\alpha\right)\partial_t\Psi+i\partial_t \alpha \exp\left(i\alpha\right)\Psi\]

You can check that the Dirac equation above i(\partial_t-\partial_x)\Psi+m\Psi=0 is NOT locally gauge invariant. You must introduce a new field, the gauge field A, transforming under symmetry as

    \[A\rightarrow A-\partial_x\alpha+\partial_t\alpha\]

Then, a modified field theory (charged) arise, that IS invariant under gauge transformations. Let us do the calculations explictly:



    \[\Psi\rightarrow e^{i\alpha}\Psi\;\;\;A\rightarrow A-\partial_x\alpha+\partial_t\alpha\]

transforms and you get


Local invariance holds now!!!!!!!This U(1) trick is mimicked for SU(2) and SU(3) symmetries (non-abelian) and weak charge (flavor) and color charge are thus related to non-abelian gauge invariances! A problem arises, however, within the SU(2) case (not in the color force case). The symmetries of the W and Z, when local, have a bad behaviour when mass is present. In other words, the mass of the gauge bosons W,Z spoils gauge invariance. That is where Higgs fields and the Higgs mechanics arise. In 1964, Higgs (and indepently other researches, by the particles are due to Higgs himself, with Nobel Prize merits) introduced a new field and new particles to restore weak gauge invariance when mass terms are present. The name spontaneous symmetry breaking or hidden symmetry is also used here. The ideas are powerful:

  • There is a new field, the Higgs field, permeating the space-time, like a fluid. Mass is similar to friction with this relativistic field.
  • Associated to the Higgs field, there are excitations of the field, called Higgs particles. Higgs particles are waves in the Higgs field, and these waves give masses to interacting particles with the Higgs. Transparent particles to the Higgs fields get no masses.
  • To get a Higgs field requires a lot of concentrated energy.

Simplest Higgs set-up:

Take an electron field, \Psi (it can also be a boson field like the W or Z).

Under gauge symmetry, it transforms as \Psi\rightarrow q\Psi and its mass term m\Psi^2\rightarrow mq^2\Psi^2. Here, q=e^{i\alpha}. Thus, mass term is not invariant under gauge local transformations. Define H such as under symmetry

    \[H\rightarrow \dfrac{H}{q^2}\]

Note that we will get gauge invariance plus an interaction of type H\Psi^2. Then, the product H\Psi^2 is invariant under gauge transformations. Expand H around a vacuum as

    \[H=\langle H\rangle_0+h\]


    \[H\Psi^2=\langle H \rangle_0\Psi^2+h\Psi^2=m\Psi^2+h\Psi^2\]

where we can define “mass” as

    \[m\equiv \langle H\rangle_0 \lambda\]

and the Higgs coupling

    \[h=\lambda \overline{h}\]

In general,

    \[y_pH\Psi^2=y_p\langle H\rangle_0\Psi^2+y_ph\Psi^2\]

Thus, you get a generic mass term for ANY fundamental (elementary, not composite) particle

    \[m=y_p\langle H\rangle_0\]


In other words,

    \[\mbox{Mass}=\mbox{Yukawa constant}\times\mbox{Higgs v.e.v.}\]

Experimentally, the Higgs v.e.v is about 246 GeV, and it was know much before the Higgs boson discovery in 2012. However, some issues in the SM can not be solved:

  • The Higgs field introduces introduces new Yukawa interactions not coming from any SM symmetry.
  • The Higgs mass itself is not fixed by any SM symmetry. Indeed, it could have been in principle very heavy, but some theoretical arguments were known before the Higgs discovery. It the Higgs existed, it could not be very heavy without spoiling the properties of the known SM. You should have invented other mechanism if the Higgs field were not been found at the LHC. SSB and Higgs fields could not be too heavy. However, something protects the Higgs to become heavy from loop corrections to mass. What it is? We do know. It could be supersymmetry (SUSY) or any other new kind of symmetry.
  • Higgs field is close to be metastable in about t\sim 10^{500}yrs. New particles can stabilize the Higgs vacuum, but it is a hard problem!
  • The Higgs field can NOT be the dark energy, it is too heavy. However, the Higgs field could be the inflaton. No proof of this is known, it is speculative.
  • The Higgs field is not consistent with the known value of the cosmological constant. However, we can not be eager here, since we do not know what dark energy is and we do not what the Higgs-like partice truly is. GR is not a quantum theory, not a YM theory at least, and we do not know how to make GR consistent with the SM. That is why they are usually considered apart to each other.
  • The searches for any theory beyond the SM, BSM theories, like string theory or loop quantum gravity, is being guided by the same principles that Maxwell used to derive his electromagnetic synthesis in the 19th century. Similarly, Einstein was guided by symmetry to SR and to GR from simple principles. The SM and the GR are believed to be effective field theories, approximate theories at low energies. We need GUTs or TOE for a further final unification.
  • The origin of mass is now turned into a problem of why Yukawa Higgs couplings are the values we observed. Even the Higgs field has its own self-coupling. We need a better theory to understand the origin of mass.

What is the future of fundamental physics? Expensive experiments and cheap experiments. Brilliant minds observing the Universe with new tools. Philosophy is not useful as before. Scientific advances require feedback from theory and experiment. New colliders (CLIC, the muon collider, the Chinese 100TeV collider, the FCC,…) will require complementary projects and dark matter/dark energy extra experiments. Gravitational waves, gamma ray astronomy, neutrino astronomy and multimessenger astronomy is a new exciting field. Theoretical speculations, like those guided by philosophy, are not useful without experimental support. We need data and test hypotheses!