# LOG#108. Basic Cosmology (III).

**Posted:**2013/06/09

**Filed under:**Cosmology, Physmatics |

**Tags:**annihilation rate, Boltzmann equation, Bose-Einstein distribution, Cosmology, cross-section, decoupling, equilibrium, equilibrium rate, Fermi-Dirac distribution, freeze out, Maxwell-Boltzmann distribution, out-of-equilibrium processes in cosmology, Saha equation, thermal equilibrium 1 Comment

The current Universe has evolved since its early phase of thermal equilibrium until the present state. The departure from thermal equilibrium in the early Universe made a fossil record we can observe at current time!

There are some easy rules for thermal equilibrium. The easiest one, is that coming from the “interaction rate” . It can be expressed in the following way:

and then, at a given temperature T, we get

and where

**Remark:** If , then

and it implies that a particle interacts less than once after the time .

Moreover, we can understand roughly the so-called decoupling era:

**1st. Any interaction mediated by a massless gauge boson provides**

with and

and this implies that

and

so the equilibrium temperature is found whenever !

**2nd. Interactions mediated by any massive gauge boson provides**

with

and this implies that

and

and then

Moreover,

As a consequence, we can realize that the out-of-equilibrium phenomena in the early and current Universe are very important processes! In particular:

1) They provide the formation of light elements during the * Big Bang Nucleosynthesis (BBN)*, also known as

**, i.e., the formation of the first light elements after the Big Bang (circa 300000 years after the Universe “birth”).**

*primordial nucleosynthesis*2) They provide the path of * recombination* of electrons and protons into hydrogen atoms.

3) They imply the (likely) production of dark matter (or equivalently the presence of some kind of “modified gravity” or/and modified newtonian dynamics).

**Boltzmann’s equation for annihilation of particles in equilibrium**

There is a beautiful equation that condenses the previous physical process of equilibrium at a given temperature and the particle production it yields. Conceptually speaking, we have

Consider a process like

and where the particle 1 is the one we are interested in. Then, we deduce that

where A is certain complicated facter involving “delta functions” of the energies and momenta of the particles 1,2,3,4 and an additional term depending on the statistics of the particle. Explicitly, it takes the form

with

and where

is the Fermi-Dirac (FD, -)/Bose-Einstein (BE,+) distribution. In fact, the above FD/BE factors provide the so-called Pauli blocking/”Bose-Einstein” enhancement effects for the particle production in the processes and . Indeed, particle physics enter into the game here (see above formulae again) and we assume

Do you recognize the *principle* of *detailed balance *in this equation?

We can simplify the assumptions a little bit:

1st. The kinetic equilibrium is taken to be a rapid elastic scattering and we input the FD/BE statistics without loss of generality.

2nd. The annihilation in thermal equilibrium will be calculated from the sum of the chemical potential in any balanced equation.

3rd. Low temperature approximation. Suppose that

then we obtain the Maxwell-Boltzmann approximation to the FD/BE statistics

and , so, since , we get

What is ? After a change of variable

is the “equilibrium number density” deducen from the expression

and it yields

and then we finally get that

equals

Now, we can define the thermally averaged cross section

The Boltzmann equation becomes with these conventions

Remark (I): LHS is similar to and the RHS is similar to

Remark (II): If the reaction rate is , then it provides the chemical equilibrium condition well known in the nuclear statistical equilibrium as the Saha equation, i.e.,

# LOG#106. Basic Cosmology (I).

**Posted:**2013/05/26

**Filed under:**Cosmology, General Relativity, Physmatics |

**Tags:**Big Bang, Bose-Einstein distribution, cosmic microwave background, Cosmological principle, Cosmology, curvature parameter, curved Universe, dark energy, degrees of freedom, dust, early Universe, Einstein tensor, Einstein-Hilbert action, energy density, energy-momentum tensor, equivalence principle, Fermi-Dirac distribution, General Relativity, geodesic equation, geodesics, hot ideal gas, ideal gas, Killing equation, Killing vector, maximally symmetric space, natural units, neutrinos, number density, parsec, particle physics, perfect cosmological principle, perfect fluid, plane Universe, pressure, redshift, relativistic matter, Standard Cosmological Model, thermal equilibrium, yield Leave a comment

The next thread is devoted to Cosmology. I will intend to be clear and simple about equations and principles of current Cosmology with a General Relativity background.

First of all…I will review the basic concepts of natural units I am going to use here. We will be using the following natural units:

We will take the Planck mass to be given by

The solar mass is and the parsec is given by the value

Well, current Cosmology is based on General Relativity. Even if I have not reviewed this theory with detail in this blog, the nice thing is that most of Cosmology can be learned with only a very little knowledge of this fenomenal theory. The most important ideas are: metric field, geodesics, Einstein equations and no much more…

In fact, newtonian gravity is a good approximation in some particular cases! And we do know that even in this pre-relativistic theory

via the Poisson’s equation

This idea, due to the equivalence principle, is generalized a little bit in the general relativistic framework

The spacetime geometry is determined by the metric tensor . The matter content is given by the stress-energy-momentum tensor . As we know one of these two elements, we can know, via Eisntein’s field equations the another. That is, given a metric tensor, we can tell how energy-momentum “moves” in space-time. Given the energy-momentum tensor, we can know what is the metric tensor in spacetime and we can guess how the spacetime bends… This is the origin of the famous motto: “Spacetime says matter how to move, energy-momentum says spacetime how to curve”! Remember that we have “deduced” the Einstein’s field equations in the previous post. Without a cosmological constant term, we get

Given a spacetime metric , we can calculate the (affine/Levi-Civita) connection

The Riemann tensor that measures the spacetime curvature is provided by the equation

The Ricci tensor is defined to be the following “trace” of the Riemann tensor

The Einstein tensor is related to the above tensors in the well-known manner

The Einstein’s equations can be derived from the Einstein-Hilbert action we learned in the previous post, using the action principle and the integral

The geodesic equation is the path of a freely falling particle. It gives a “condensation” of the Einstein’s equivalence principle too and it is also a generalization of Newton’s law of “no force”. That is, the geodesic equation is the feynmanity

Finally, an important concept in General Relativity is that of isometry. The symmetry of the “spacetime manifold” is provided by a Killing vector that preserves transformations (isometries) of that manifold. Mathematically speaking, the Killing vector fields satisfy certain equation called the Killing equation

Maximally symmetric spaces have Killing vectors in n-dimensional (nD) spacetime. There are 3 main classes or types of 2D maximally symmetric that can be generalized to higher dimensions:

1. The euclidean plane .

2. The pseudo-sphere . This is a certain “hyperbolic” space.

3. The spehre . This is a certain “elliptic” space.

**The Friedmann-Robertson-Walker Cosmology**

Current cosmological models are based in General Relativity AND a simplification of the possible metrics due to the so-called Copernican (or cosmological) principle: the Universe is pretty much the same “everywhere” you are in the whole Universe! Remarkbly, the old “perfect” Copernican (cosmological) principle that states that the Universe is the same “everywhere” and “every time” is wrong. Phenomenologically, we have found that the Universe has evolved and it evolves, so the Universe was “different” when it was “young”. Therefore, the perfect cosmological principle is flawed. In fact, this experimental fact allows us to neglect some old theories like the “stationary state” and many other “crazy theories”.

What are the observational facts to keep the Copernican principle? It seems that:

1st. The distribution of matter (mainly galaxies, clusters,…) and radiation (the cosmic microwave background/CMB) in the observable Universe is **homogenous and isotropic.**

2nd. The Universe is NOT static. From Hubble’s pioneer works/observations, we do know that galaxies are receeding from us!

Therefore, these observations imply that our “local” Hubble volume during the Hubble time is similar to some spacetime with homogenous and isotropic spatial sections, i.e., it is a spacetime manifold . Here, denotes the time “slice” and represents a 3D maximally symmetric space.

The geometry of a locally isotropic and homogeneous Universe is represented by the so-called Friedmann-Robertson-Walker metric

Here, is the called the **scale factor. **The parameter determines the geometry type (plane, hyperbolic or elliptical/spherical):

1) If , then the Universe is “flat”. The manifold is .

2) If , then the Universe is “open”/hyperbolic. The manifold would be .

3) If , then the Universe is “closed”/spherical or elliptical. The manifold is then .

**Remark:** The ansatz of local homogeneity and istoropy only implies that the spatial metric is locally one of the above three spaces, i.e., . It could be possible that these 3 spaces had different global (likely topological) properties beyond these two properties.

**Kinematical features of a FRW Universe**

The first property we are interested in Cosmology/Astrophysics is “distance”. Measuring distance in a expanding Universe like a FRW metric is “tricky”. There are several notions of “useful” distances. They can be measured by different methods/approaches and they provide something called sometimes “the cosmologidal distance ladder”:

1st. **Comoving distance.** It is a measure in which the distance is “taken” by a fixed coordinate.

2nd. **Physical distance.** It is essentially the comoving distance times the scale factor.

3rd.** Luminosity distance.** It uses the light emitted by some object to calculate its distance (provided the speed of light is taken constant, i.e., special relativity holds and we have a constant speed of light)

4th. **Angular diameter distance. **Another measure of distance using the notion of parallax and the “extension” of the physical object we measure somehow.

There is an important (complementary) idea in FRW Cosmology: the** particle horizon**. Consider a light-like particle with . Then,

or

The total comoving distance that light have traveled since a time is equal to

It shows that NO information could have propagated further and thus, there is a “comoving horizon” with every light-like particle! Here, this time is generally used as a “conformal time” as a convenient tiem variable for the particle. The physical distance to the particle horizon can be calculated

There are some important kinematical equations to be known

A) **For the geodesic equation, the free falling particle,** we have

for the FRW metric and, moreover, the energy-momentum vector is defined by the usual invariant equation

This definition defines, in fact, the proper “time” implicitely, since

and the 0th component of the geodesic equation becomes

Therefore we have deduced that . This is, in fact, the socalled “redshift”. The cosmological redshift parameter is more generally defined through the equation

B) **The Hubble’s law.**

The luminosity distance measures the flux of light from a distant object of known luminosity (if it is not expanding). The flux and luminosity distance are bound into a single equation

If we use the comoving distance between a distant emitter and us, we get

for a expanding Universe! That is, we have used the fact that luminosity itself goes through a comoving spherical shell of radius . Moreover, it shows that

The luminosity distance in the expanding shell is

and this is what we MEASURE in Astrophysics/Cosmology. Knowing , we can express the luminosity distance in terms of the redshift. Taylor expansion provides something like this:

where higher order terms are sometimes referred as “statefinder parameters/variables”. In particular, we have

and

C) **Angular diameter distance.**

If we know that some object has a known length , and it gives some angular “aperture” or separation , the angular diameter distance is given by

The comoving size is defined as , and the coming distance is again . For “flat” space, we obtain that

that is

In the case of “curved” spaces, we get

**FRW dynamics**

Gravity in General Relativity, a misnomer for the (locally) relativistic theory of gravitation, is described by a metric field, i.e., by a second range tensor (covariant tensor if we are purist with the nature of components). The metric field is related to the matter-energy-momentum content through the Einstein’s equations

The left-handed side can be calculated for a FRW Universe as follows

The right-handed side is the energy-momentum of the Universe. In order to be fully consistent with the symmetries of the metric, the energy-momentum tensor MUST be diagonal and . In fact, this type of tensor describes a perfect fluid with

Here, are functions of (cosmological time) only. They are “state variables” somehow. Moreover, we have

for the fluid at rest in the comoving frame. The Friedmann equations are indeed the EFE for a FRW metric Universe

for the 00th compoent as “constraint equation.

for the iith components.

Moreover, we also have

and this conservation law implies that

Therefore, we have got two independent equations for three unknowns . We need an additional equation. In fact, the equation of state for provides such an additional equation. It gives the “dynamics of matter”!

In summary, the basic equations for Cosmology in a FRW metric, via EFE, are the Friedmann’s equations (they are secretly the EFE for the FRW metric) supplemented with the energy-momentum conservations law and the equation of state for the pressure :

1)

2)

3)

There are many kinds of “matter-energy” content of our interest in Cosmology. Some of them can be described by a simple equation of state:

Energy-momentum conservation implies that . 3 special cases are used often:

1st. **Radiation (relativistic “matter”).** and thus, and

2nd. **Dust (non-relativistic matter).** . Then, and

3rd.** Vacuum energy (cosmological constant).** . Then, and

**Remark (I):** Particle physics enters Cosmology here! Matter dynamics or matter fields ARE the matter content of the Universe.

**Remark (II):** Existence of a Big Bang (and a spacetime singularity). Using the Friedmann’s equation

if we have that , the so-called weak energy condition, then should have been reached at some finite time in the past! That is the “Big Bang” and EFE are “singular” there. There is no scape in the framework of GR. Thus, we need a quantum theory of gravity to solve this problem OR give up the FRW metric at the very early Universe by some other type of metric or structure.

**Particles and the chemical equilibrium of the early Universe**

Today, we have DIRECT evidence for the existence of a “thermal” equilibrium in the early Universe: the cosmic microwave background (CMB). The CMB is an isotropic, accurate and non-homogeneous (over certain scales) blackbody spectrum about !

Then, we know that the early Universe was filled with a hot dieal gas in thermal equilibrium (a temperature can be defined there) such as the energy density and pressure can be written in terms of this temperature. This temperature generates a distribution . The number of phase space elements in is

and where the RHS is due to the uncertainty principle. Using homogeneity, we get that, indeed, , and where we can write the volume . The energy density and the pressure are given by (natural units are used)

When we are in the thermal equilibrium at temperature T, we have the Bose-Einstein/Fermi-Dirac distribution

and where the is for the Fermi-Dirac distribution (particles) and the is for the Bose-Einstein distribution (particles). The number density, the energy density and the pressure are the following integrals

And now, we find some special cases of matter-energy for the above variables:

1st. **Relativistic, non-degenerate matte**r (e.g. the known neutrino species). It means that and . Thus,

2nd. **Non-relativistic matter** with only. Then,

, and

The total energy density is a very important quantity.** In the thermal equilibrium,** the energy density of non-relativistic species is exponentially smaller (suppressed) than that of the relativistic particles! In fact,

for radiation with

and the effective degrees of freedom are

**Remark:** The factor in the DOF and the variables above is due to the relation between the Bose-Einstein and the Fermi-Dirac integral in d=3 space dimensions. In general d, the factor would be

**Entropy conservation and the early Universe**

The entropy in a comoving volume IS a conserved quantity IN THE THERMAL EQUILIBRIUM. Therefore, we have that

and then

or

Now, since

then

if we multiply by and use the chain rule for , we obtain

but it means that , where is the entropy density defined by

Well, the fact is that we know that the entropy or more precisely the entropy density is the early Universe is dominated by relativistic particles ( this is “common knowledge” in the Stantard Cosmological Model, also called ). Thus,

It implies the evolution of temperature with the redshift in the following way:

Indeed, since we have that , , the **yield** variable

is a convenient quantity that represents the “abundance” of decoupled particles.

See you in my next cosmological post!

# LOG#056. Gravitational alpha(s).

**Posted:**2012/11/29

**Filed under:**Cosmology, Physmatics, Quantum Gravity, Relativity |

**Tags:**alpha, alpha strong, asymptotic freedom, atomic physics, confinement, cosmological constant, cosmological constant problem, cosmological gravitational alpha, cosmological parameter fitting, cosmological parameters, Cosmology, coupling constant, de Sitter radius, Einstein's field equations, energy density, energy density ratios, energy ratios, fine structure constant, gravitational alpha, gravitational constant, gravitational fine structure constant, Hubble parameter, Hubble's length, length ratios, naturalness problem, Planck's energy, Planck's length, QCD, QFT, quantum field theory, quantum theory, ratios, Relativity Leave a comment

The topic today is to review a beautiful paper and to discuss its relevance for theoretical physics. The paper is: **Comment on the cosmological constant and a gravitational alpha **by R.J.Adler. You can read it here: http://arxiv.org/abs/1110.3358

One of the most intriguing and mysterious numbers in Physics is the electromagnetic fine structure constant . Its value is given by

or equivalenty

Of course, I am assuming that the coupling constant is measured at ordinary energies, since we know that the coupling constants are not really constant but they vary slowly with energy. However, I am not going to talk about the renormalization (semi)group in this post.

Why is the fine structure constant important? Well, we can undertand it if we insert the values of the constants that made the electromagnetic alpha constant:

with being the electron elemental charge, the Planck’s constant divided by two pi, c is the speed of light and where we are using units with . Here is the Coulomb constant, generally with a value , but we rescale units in order it has a value equal to the unit. We will discuss more about frequently used system of units soon.

As the electromagnetic alpha constant depends on the electric charge, the Coulomb’s electromagnetic constant ( rescaled to one in some “clever” units), the Planck’s constant ( rationalized by since ) and the speed of light, it codes some deep information of the Universe inside of it. The electromagnetic alpha is quantum and relativistic itself, and it also is related to elemental charges. Why alpha has the value it has is a complete mystery. Many people has tried to elucidate why it has the value it has today, but there is no reason of why it should have the value it has. Of course, it happens as well with some other constants but this one is particularly important since it is involved in some important numbers in atomic physics and the most elemental atom, the hydrogen atom.

In atomic physics, there are two common and “natural” scales of length. The first scale of length is given by the Compton’s wavelength of electrons. Usint the de Broglie equation, we get that the Compton’s wavelength is the wavelength of a photon whose energy is the same as the rest mass of the particle, or mathematically speaking:

Usually, physicists employ the “reduced” or “rationalized” Compton’s wavelength. Plugging the electron mass, we get the electron reduced Compton’s wavelength:

The second natural scale of length in atomic physics is the so-called Böhr radius. It is given by the formula:

Therefore, there is a natural mass ratio between those two length scales, and it shows that it is precisely the electromagnetic fine structure constant alpha :

Furthermore, we can show that the electromagnetic alpha also is related to the mass ration between the electron energy in the fundamental orbit of the hydrogen atom and the electron rest energy. These two scales of energy are given by:

1)** Rydberg’s energy** ( electron ground minimal energy in the fundamental orbit/orbital for the hydrogen atom):

2) **Electron rest energy**:

Then, the ratio of those two “natural” energies in atomic physics reads:

or equivalently

R.J.Adler’s paper remarks that there is a cosmological/microscopic analogue of the above two ratios, and they involve the infamous Einstein’s cosmological constant. In Cosmology, we have two natural (ultimate?) length scales:

1st. The (ultra)microscopic and ultrahigh energy (“ultraviolet” UV regulator) relevant **Planck’s length** , or equivalently the squared value . Its value is given by:

This natural length can NOT be related to any “classical” theory of gravity since it involves and uses the Planck’s constant .

2nd. The (ultra)macroscopic and ultra-low-energy (“infrared” IR regulator) relevant **cosmological constant/deSitter radius. **They are usualy represented/denoted by and respectively, and they are related to each other in a simple way. The dimensions of the cosmological constant are given by

The de Sitter radius and the cosmological constant are related through a simple equation:

The de Sitter radius is obtained from cosmological measurements thanks to the so called Hubble’s parameter ( or Hubble’s “constant”, although we do know that Hubble’s “constant” is not such a “constant”, but sometimes it is heard as a language abuse) H. From cosmological data we obtain ( we use the paper’s value without loss of generality):

This measured value allows us to derive the Hubble’s length paremeter

Moreover, the data also imply some density energy associated to the cosmological “constant”, and it is generally called Dark Energy. This density energy from data is written as:

and from this, it can be also proved that

where we have introduced the experimentally deduced value from the cosmological parameter global fits. In fact, the cosmological constant helps us to define the beautiful and elegant formula that we can call the **gravitational alpha/gravitational cosmological fine structure constant **:

or equivalently, defining the cosmological length associated to the cosmological constant as

If we introduce the numbers of the constants, we easily obtaint the gravitational cosmological alpha value and its inverse:

They are really small and large numbers! Following the the atomic analogy, we can also create a ratio between two cosmologically relevant density energies:

1st. **The Planck’s density energy.**

Planck’s energy is defined as

The Planck energy density is defined as the energy density of Planck’s energy inside a Planck’s cube or side , i.e., it is the energy density of Planck’s energy concentrated inside a cube with volume . Mathematically speaking, it is

It is an huge density energy!

**Remark:** Energy density is equivalent to **pressure** in special relativity hydrodynamics. That is,

wiht Pa denoting pascals () and where represents here matter (not energy) density ( with units in ). Of course, turning matter density into energy density requires a multiplication by . This equivalence between vacuum pressure and energy density is one of the reasons because some astrophysicists, cosmologists and theoretical physicists call “vacuum pressure” to the “dark energy/cosmological constant” term in the study of the cosmic components derived from the total energy density .

2nd. **The cosmological constant density energy.**

Using the Einstein’s field equations, it can be shown that the cosmological constant gives a contribution to the stress-energy-momentum tensor. The component is related to the dark energy ( a.k.a. the cosmological constant) and allow us to define the energy density

Using the previous equations for G as a function of Planck’s length, the Planck’s constant and the speed of light, and the definitions of Planck’s energy and de Sitter radius, we can rewrite the above energy density as follows:

Thus, we can evaluate the ration between these two energy densities! It provides

and the inverse ratio will be

So, we have obtained two additional really tiny and huge values for and its inverse, respectively. Note that the power appearing in the ratios of cosmological lengths and cosmological energy densities match the same scaling property that the atomic case with the electromagnetic alpha! In the electromagnetic case, we obtained and . The gravitational/cosmological analogue ratios follow the same rule and but the surprise comes from the values of the gravitational alpha values and ratios. Some comments are straightforward:

1) Understanding atomic physics involved the discovery of Planck’s constant and the quantities associated to it at fundamental quantum level ( Böhr radius, the Rydberg’s constant,…). Understanding the Cosmological Constant value and the mismatch or stunning ratios between the equivalent relevant quantities, likely, require that can be viewed as a new “fundamental constant” or/and it can play a dynamical role somehow ( e.g., varying in some unknown way with energy or local position).

2) Currently, the cosmological parameters and fits suggest that is “constant”, but we can not be totally sure it has not varied slowly with time. And there is a related idea called quintessence, in which the cosmological “constant” is related to some dynamical field and/or to inflation. However, present data say that the cosmological constant IS truly constant. How can it be so? We are not sure, since our physical theories can hardly explain the cosmological constant, its value, and why it is current density energy is radically different from the vacuum energy estimates coming from Quantum Field Theories.

3) The mysterious value

is an equivalent way to express the biggest issue in theoretical physics. A **naturalness problem** called the** cosmological constant problem**.

In the literature, there have been alternative definitions of “gravitational fine structure constants”, unrelated with the above gravitational (cosmological) fine structure constant or gravitational alpha. Let me write some of these alternative gravitational alphas:

1) **Gravitational alpha prime**. It is defined as the ratio between the electron rest mass and the Planck’s mass squared:

Note that . Since , we can also use the proton rest mass instead of the electron mass to get a new gravitational alpha.

2) **Gravitational alpha double prime.** It is defined as the ratio between the proton rest mass and the Planck’s mass squared:

and the inverse value

Finally, we could guess an intermediate gravitational alpha, mixing the electron and proton mass.

3) **Gravitational alpha triple prime**. It is defined as the ration between the product of the electron and proton rest masses with the Planck’s mass squared:

and the inverse value

We can compare the 4 gravitational alphas and their inverse values, and additionally compare them with . We get

These inequations mean that the electromagnetic fine structure constant is (at ordinary energies) 42 orders of magnitude bigger than , 39 orders of magnitude bigger than , 36 orders of magnitude bigger than and, of course, 58 orders of magnitude bigger than . Indeed, we could extend this analysis to include the “fine structure constant” of Quantum Chromodynamics (QCD) as well. It would be given by:

since generally we define . We note that by 3 orders of magnitude. However, as strong nuclear forces are short range interactions, they only matter in the atomic nuclei, where confinement, and color forces dominate on every other fundamental interaction. Interestingly, at high energies, QCD coupling constant has a property called asymptotic freedom. But it is another story not to be discussed here! If we take the alpha strong coupling into account the full hierarchy of alphas is given by:

**Fascinating!** Isn’t it?** Stay tuned!!!**

*ADDENDUM:* After I finished this post, I discovered a striking (and interesting itself) connection between and . The relation or coincidence is the following relationship

Is this relationship fundamental or accidental? The answer is unknown. However, since the electric charge (via electromagnetic alpha) is not related a priori with the gravitational constant or Planck mass ( or the cosmological constant via the above gravitational alpha) in any known way I find particularly stunning such a coincidence up to 5 significant digits! Any way, there are many unexplained numerical coincidences that are completely accidental and meaningless, and then, it is not clear why this numeral result should be relevant for the connection between electromagnetism and gravity/cosmology, but it is interesting at least as a curiosity and “joke” of Nature.

*ADDENDUM (II):
*

Some quotes about the electromagnetic alpha from wikipedia http://en.wikipedia.org/wiki/Fine-structure_constant

“(…)There is a most profound and beautiful question associated with the observed coupling constant, e – the amplitude for a real electron to emit or absorb a real photon. It is a simple number that has been experimentally determined to be close to 0.08542455. (My physicist friends won’t recognize this number, because they like to remember it as the inverse of its square: about 137.03597 with about an uncertainty of about 2 in the last decimal place. It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it.) Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the “hand of God” wrote that number, and “we don’t know how He pushed his pencil.” We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly! (…)”. R.P.Feynman, *QED: The Strange Theory of Light and Matter*, Princeton University Press, p.129.

“(…) If alpha [the fine-structure constant] were bigger than it really is, we should not be able to distinguish matter from ether [the vacuum, nothingness], and our task to disentangle the natural laws would be hopelessly difficult. The fact however that alpha has just its value 1/137 is certainly no chance but itself a law of nature. It is clear that the explanation of this number must be the central problem of natural philosophy.(…)” Max Born, in A.I. Miller’s book *Deciphering the Cosmic Number: The Strange Friendship of Wolfgang Pauli and Carl Jung. *p. 253. Publisher W.W. Norton & Co.(2009).

“(…)The mystery about *α* is actually a double mystery. The first mystery – the origin of its numerical value *α* ≈ 1/137 has been recognized and discussed for decades. The second mystery – the range of its domain – is generally unrecognized.(…)” Malcolm H. Mac Gregor, M.H. MacGregor (2007). *The Power of Alpha.*

# LOG#050. Why riemannium?

**Posted:**2012/11/07

**Filed under:**Physmatics, Zeta Zoology and polystuff |

**Tags:**adelic identity, adelic ring, Astrophysics, atom, atomic physics, BE statistics, Berry-Keating conjecture, casimir effect, confinement, cosmological constant, Cosmology, Dirichlet eta function, FD statistics, fractals, group entropy, harmonic oscillator, hawking effect, Hilbert-Polya conjecture, logarithmic potentital, MB statistics, music, non-extensive entropy, Non-trivial Riemann zeroes, p-adic numbers, Paraboson, parafermion, Physmatics, prime numbers, QFT, Quantum chaos, quantum field theory, Quantum Mechanics, Quantum Statistics, random matrix theory, Riemann hypothesis, Riemann zeroes, Riemann zeta function, riemannium, schwinger effect, spectrum of riemannium, trivial Riemann zeroes, Tsallis statistics, Tsallisium, Veneziano amplitude, zeta values 7 Comments

# TABLE OF CONTENTS

DEDICATORY

1. THE RIEMANN ZETA FUNCTION ζ(s)

2. THE RIEMANN HYPOTHESIS

3. THE HILBERT-POLYA CONJECTURE

4. RANDOM MATRIX THEORY

5. QUANTUM CHAOS AND RIEMANN DYNAMICS

6. THE SPECTRUM OF RIEMANNIUM

7. ζ(s) AND RENORMALIZATION

8. ζ(s) AND QUANTUM STATISTICS

9. ζ(s) AND GROUP ENTROPIES

10. ζ(s) AND THE PRIMON GAS

11. LOG-OSCILLATORS

12. LOG-POTENTIAL AND CONFINEMENT

13. HARMONIC OSCILLATOR AND TSALLIS GAS

14. TSALLIS ENTROPIES IN A NUTSHELL

15. BEYOND QM/QFT: ADELIC WORLDS

16. STRINGS, FIELDS AND VACUUM

17. SUMMARY AND OUTLOOK

## DEDICATORY

This special 50th log-entry is dedicated to 2 special people and scientists who inspired (and guided) me in the hard task of starting and writing this blog.

These two people are

1st. John C. Baez, a mathematical physicist. Author of the old but always fresh/brand new *This Week Finds in Mathematical Physics,* and now involved in the Azimuth blog. You can visit him here

http://johncarlosbaez.wordpress.com/

and here

http://math.ucr.edu/home/**baez**/

I was a mere undergraduate in the early years of the internet in my country when I began to read his TWF. If you have never done it, I urge to do it. Read him. He is a wonderful teacher and an excellent lecturer. John is now worried about global warming and related stuff, but he keeps his mathematical interests and pedagogical gifts untouched. I miss some topics about he used to discuss often before in his hew blog, but his insights about virtually everything he is involved into are really impressive. He also manages to share his entusiastic vision of Mathematics and Science. From pure mathematics to physics. He is a great blogger and scientist!

2nd. The professor Francis Villatoro. I am really grateful to him. He tries to divulge Science in Spain with his excellent blog ( written in Spanish language)

http://francisthemulenews.wordpress.com/

He is a very active person in the world of Spanish Science (and its divulgation). In his blog, he also tries to explain to the general public the latest news on HEP and other topics related with other branches of Physics, Mathematics or general Science. It is not an easy task! Some months ago, after some time reading and following his blog (as I do now yet, like with Baez’s stuff), I realized that I could not remain as a passive and simple reader or spectator in the web, so I wrote him and I asked him some questions about his experience with blogging and for advice. His comments and remarks were incredibly useful for me, specially during my first logs. I have followed several blogs the last years (like those by Baez or Villatoro), and I had no idea about what kind of style/scheme I should addopt here. I had only some fuzzy ideas about what to do, what to write and, of course, I had no idea if I could explain stuff in a simple way while keeping the physical intuition and the mathematical background I wanted to include. His early criticism was very helpful, so this post is a tribute for him as well. *After all, he suggested me the topic of this post! I encourage you to read him and his blog (as long as you know Spanish or you can use a good translator).
*

Finally, **let me express and show my deepest gratitude to John and Francis**. Two great and extraordinary people and professionals in their respective fields who inspired (and yet they do) me in spirit and insight in my early and difficult steps of writing this blog. I am just convinced that Science is made of little, ordinary and small contributions like mine, and not only the greatest contributions like those making John and Francis to the whole world. I wish they continue making their contributions in the future for many, many years yet to come.

Now, let me answer the question Francis asked me to explain here with further details. My special post/log-entry number 50…It will be devoted to tell you why this blog is called The Spectrum of Riemannium, and what is behind the greatest unsolved problem in Number Theory, Mathematics and likely Physics/Physmatics as well…Enjoy it!

## 1. THE RIEMANN ZETA FUNCTION ζ(s)

The Riemann zeta function is a device/object/function related to prime numbers.

In general, it is a function of complex variable defined by the next equation:

or

Generally speaking, the Riemann zeta function extended by analytical continuation to the whole complex plane is “more” than the classical Riemann zeta function that Euler found much before the work of Riemann in the XIX century. The Riemann zeta function for real and entire positive values is a very well known (and admired) series by the mathematicians. due to the divergence of the harmonic series. Zeta values at even positive numbers are related to the Bernoulli numbers, and it is still lacking an analytic expression for the zeta values at odd positive numbers.

The Riemann zeta function over the whole complex plane satisfy the following functional equation:

Equivalently, it can be also written in a very simple way:

where we have defined

Riemann zeta values are an example of beautiful Mathematics. From , then we have:

1) .

2) . The harmonic series is divergent.

3) . The famous Euler result.

4) . And odd zeta value called Apery’s constant that we do not know yet how to express in terms of irrational numbers.

5) .

6) . Trivial zeroes of zeta.

7) , where are the Bernoulli numbers. The first 13 Bernoulli numbers are:

8) We note that .

9) .

For instance, , , and . Indeed, arises in string theory trying to renormalize the vacuum energy of an infinite number of harmonic oscillators. The result in the bosonic string is . In order to match with Riemann zeta function regularization of the above series, the bosonic string is asked to live in an ambient spacetime of D=26 dimensions. We also have that

10) . The Riemann zeta value at the infinity is equal to the unit.

11) The derivative of the zeta function is . Particularly important of this derivative are:

or

This allow us to define the factorial of the infinity as

and the renormalized infinite dimensional determinant of certain operator A as:

, with

12) . This is a result used by theoretical physicists in dimensional renormalization/regularization. is the so-called Euler-Mascheroni constant.

The alternating zeta function, called Dirichlet eta function, provides interesting values as well. Dirichlet eta function is defined and related to the Riemann zeta fucntion as follows:

This can be thought as “bosons made of fermions” or “fermions made of bosons” somehow. Special values of Dirichlet eta function are given by:

**Remark(I)**: is important in the physics realm, since the spectrum of the hydrogen atom has the following aspect

and the Balmer formula is, as every physicist knows

**Remark (II)**: The fact that is finite implies that the energy level separation of the hydrogen atom in the Böhr level tends to zero AND that the sum of ALL the possible energy levels in the hydrogen atom is finite since is finite.

**Remark(III)**: What about an “atom”/system with spectrum ? If , we do know that is the case of the Kepler problem. Moreover, it is easy to observe that corresponds to tha harmonic oscillator, i.e., . We also know that is the infinite potential well. So the question is, what about a spectrum and so on?

In summary, does the following spectrum

with energy separation/splitting

exist in Nature for some physical system beyond the infinite potential well, the harmonic oscillator or the hydrogen atom, where , and respectively?

It is amazing how Riemann zeta function gets involved with a common origin of such a different systems and spectra like the Kepler problem, the harmonic oscillator and the infinite potential well!

## 2. THE RIEMANN HYPOTHESIS

The Riemann Hypothesis (RH) is the greatest unsolved problem in pure Mathematics, and likely, in Physics too. It is the statement that the only non-trivial zeroes of the Riemann zeta function, beyond the trivial zeroes at have real part equal to 1/2. In other words, the equation or feynmanity has only the next solutions:

I generally prefer the following projective-like version of the RH (PRH):

The Riemann zeta function can be sketched on the whole complex plane, in order to obtain a radiography about the RH and what it means. The mathematicians have studied the critical strip with ingenious tools an frameworks. The now terminated ZetaGrid project proved that there are billions of zeroes IN the critical line. No counterexample has been found of a non-trivial zeta zero outside the critical line (and there are some arguments that make it very unlikely). The RH says that primes “have music/order/pattern” in their interior, but nobody has managed to prove the RH. The next picture shows you what the RH “say” graphically:

If you want to know how the Riemann zeroes sound, M. Watkins has done a nice audio file to see their music.

You can learn how to make “music” from Riemann zeroes here http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/munafo-zetasound.htm

And you can listen their sound here

http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/zeta.mp3

Riemann zeroes are connected with prime numbers through a complicated formula called **“the explicit formula”**. The next equation holds integer numbers, and non-trivial Riemann zeroes in the complex (upper) half-plane with :

and where is the celebrated Gauss prime number counting function, i.e., represents the prime numbers that are equal than x or below. This explicit formula was proved by Hadamard. The explicit formula follows from both product representations of , the Euler product on one side and the Hadamard product on the other side.

The function , sometimes written as , is the logarithmic integral

The explicit formula comes in some cool variants too. For instance, we can write

where

and

For large values of x, we have the asymptotics

and

**Remark:** Please, don’t confuse the logarithmic integral with the polylogarithm function .

Gauss also conjectured that

## 3. THE HILBERT-POLYA CONJECTURE

**Date:** January 3, 1982. Andrew Odlyzko wrote a letter to George Pólya about the physical ground/basis of the Riemann Hypothesis and the conjecture associated to Polya himself and David Hilbert. Polya answered and told Odlyzko that while he was in Göttingen around 1912 to 1914 he was asked by Edmund Landau for a physical reason that the Riemann Hypothesis should be true, and suggested that this would be the case if the imaginary parts, say of the non-trivial zeros

of the Riemann zeta function corresponded to eigenvalues of an unbounded and unknown self adjoint operator . That statement was never published formally, but it was remembered after all, and it was transmitted from one generation to another. At the time of Pólya’s conversation with Landau, there was little basis for such speculation. However, Selberg, in the early 1950s, proved a duality between the length spectrum of a Riemann surface and the eigenvalues of its Laplacian. This so-called Selberg trace formula shared a striking resemblance to the explicit formula of certain L-function, which gave credibility to the speculation of Hilbert and Pólya.

## 4. RANDOM MATRIX THEORY

*Dialogue(circa 1970). “(…)Dyson*: So tell me, Montgomery, what have you been up to? *Montgomery*: Well, lately I’ve been looking into the distribution of the zeros of the Riemann zeta function. *Dyson*: Yes? And? *Montgomery*: It seems the two-point correlations go as….(…) *Dyson*: Extraordinary! Do you realize that’s the pair-correlation function for the eigenvalues of a random Hermitian matrix? It’s also a model of the energy levels in a heavy nucleus, say U-238.(…)”

A step further was given in the 1970s, by the mathematician Hugh Montgomery. He investigated and found that the statistical distribution of the zeros on the critical line has a certain property, now called **Montgomery’s pair correlation conjecture**. *The Riemann zeros tend not to cluster too closely together, but to repel*. During a visit to the Institute for Advanced Study (IAS) in 1972, he showed this result to Freeman Dyson, one of the founders of the theory of random matrices. Dyson realized that the statistical distribution found by Montgomery appeared to be the same as the pair correlation distribution for the eigenvalues of a random and “very big/large” Hermitian matrix with size NxN. These distributions are of importance in physics and mathematics. Why? It is simple. The eigenstates of a Hamiltonian, for example the energy levels of an atomic nucleus, satisfy such statistics. Subsequent work has strongly borne out the connection between the distribution of the zeros of the Riemann zeta function and the eigenvalues of a random Hermitian matrix drawn from the theoyr of the so-calle Gaussian unitary ensemble, and both are now believed to obey the same statistics. Thus the conjecture of Pólya and Hilbert now has a more solid fundamental link to QM, though it has not yet led to a proof of the Riemann hypothesis. The pair-correlation function of the zeros is given by the function:

In a posterior development that has given substantive force to this approach to the Riemann hypothesis through functional analysis and operator theory, the mathematician Alain Connes has formulated a “trace formula” using his non-commutative geometry framework that is actually equivalent to certain generalized Riemann hypothesis. This fact has therefore strengthened the analogy with the Selberg trace formula to the point where it gives precise statements. However, the mysterious operator believed to provide the Riemann zeta zeroes remain hidden yet. Even worst, we don’t even know on which space the Riemann operator is acting on.

However, some trials to guess the Riemann operator has been given from a semiclassical physical environtment as well. Michael Berry and Jon Keating have speculated that the Hamiltonian/Riemann operator is actually some kind of quantization of the classical Hamiltonian where is the canonical momentum associated with the position operator . If that Berry-Keating conjecture is true. The simplest Hermitian operator corresponding to is

At current time, it is still quite inconcrete, as it is not clear on which space this operator should act in order to get the correct dynamics, nor how to regularize it in order to get the expected logarithmic corrections. Berry and Germán Sierra, the latter in collaboration with P.K.Townsed, have conjectured that since this operator is invariant under dilatations perhaps the boundary condition for integer may help to get the correct asymptotic results valid for big . That it, in the large we should obtain

## 5. QUANTUM CHAOS AND RIEMANN DYNAMICS

Indeed, the Berry-Keating conjecture opened another striking attack to prove the RH. A topic that was popular in the 80’s and 90’s in the 20th century. The weird subject of “quantum chaos”. Quantum chaos is the subject devoted to the study of quantum systems corresponding to classically chaotic systems. The Berry-Keating conjecture shed light further into the Riemann dynamics, sketching some of the properties of the dynamical system behind the Riemann Hypothesis.

In summary, the dynamics of the Riemann operator should provide:

1st. The quantum hamiltonian operator behind the Riemann zeroes, in addition to the classical counterpart, the classical hamiltonian , has a dynamics containing the scaling symmetry. As a consequence, the trajectories are the same at all energy scale.

2nd. The classical system corresponding to the Riemann dynamics is chaotic and unstable.

3rd. The dynamics lacks time-reversal symmetry.

4th. The dynamics is quasi one-dimensional.

A full dictionary translating the whole correspondence between the chaotic system corresponding to the Riemann zeta function and its main features is presented in the next table:

## 6. THE SPECTRUM OF RIEMANNIUM

In 2001, the following paper emerged, http://arxiv.org/abs/nlin/0101014. The Riemannium arxiv paper was published later (here: Reg. Chaot. Dyn. 6 (2001) 205-210). After that, Brian Hayes wrote a really beautiful, wonderful and short paper titled The Spectrum of Riemannium in 2003 (*American Scientist, Volume 91, Number 4 July–August, 2003,pages 296–300*). I remember myself reading the manuscript and being totally surprised. I was shocked during several weeks. I decided that I would try to understand that stuff better and better, and, maybe, make some contribution to it. The Spectrum of Riemannium was an amazing name, an incredible concept. So, I have been studying related stuff during all these years. And I have my own suspitions about what the riemannium and the zeta function are, but this is not a good place to explain all of them!

The riemannium is the mysterious physical system behind the RH. Its spectrum, the spectrum of riemannium, are given by the RH and its generalizations.

Moreover, the following sketch from Hayes’ paper is also very illustrative:

What do you think? Isn’t it suggestive? Is it amazing?

## 7. ζ(s) AND RENORMALIZATION

Riemann zeta function also arises in the renormalization of the Standard Model and the regularization of determinants with “infinite size” (i.e., determinants of differential operators and/or pseudodifferential operators). For instance, the -dimensional regularized determinant is defined through the Riemann zeta function as follows:

The dimensional renormalization/regularization of the SM makes use of the Riemann zeta function as well. It is ubiquitous in that approach, but, as far as I know, nobody has asked why is that issue important, as I have suspected from long time ago.

## 8. ζ(s) AND QUANTUM STATISTICS

Riemann zeta function is also used in the theory of Quantum Statistics. Quantum Statistics are important in Cosmology and Condensed Matter, so it is really striking that Riemann zeta values are related to phenomena like Bose-Einstein condensation or the Cosmic Microwave Background and also the yet to be found Cosmic Neutrino Background!

Let me begin with the easiest quantum (indeed classical) statistics, the Maxwell-Boltzmann (MB) statistics. In 3 spatial dimensions (3d) the MB distribution arises ( we will work with units in which ):

Usually, there are 3 thermodynamical quantities that physicists wish to compute with statistical distributions: 1) the number density of particles , 2) the energy density and 3) the pressure . In the case of a MB distribution, we have the following definitions:

We can introduce the dimensionless variables $late z=\dfrac{mc^2}{k_BT}$, . In this way,

With these definitions, the particle density becomes

This integral can be calculated in closed form with the aid of modified Bessel functions of the 2th kind:

or equivalently

And thus, we have the next results (setting for simplicity):

Even entropy density is easiy to compute:

These results can be simplified in some limit cases. For instance, in the massless limit . Moreover, we also know that . In such a case, we obtain:

We note that in this massless limit.

**Remark (I):** In the massless limit, and whenever there is no degeneracy, holds.

**Remark (II):** If there is a quantum degeneracy in the energy levels, i.e., if , we must include an extra factor of for *massive* particles of spin j. For massless photons with helicity, there is a degeneracy.

**Remark (III):** In the D-dimensional (D=d+1) Bose gas with dispersion relationship , it can be shown that the pressure is related with the energy density in the following way

**Remark (IV): **Let us define as the number of ways an integer number can be expressed as a sum of the sth powers of integers. For instance,

because

because

If with and , then and the partition function is

We will see later that

with is nothing but the generatin function of the partitions

The Hardy-Ramanujan inversion formula reads (for the case s=1 only):

**Remark (V):** There are some useful integrals in quantum statistics. They are the so-called Bose-Einstein/Fermi-Dirac integrals

The BE-FD quantum distributions in 3d are defined as follows:

where the minus sign corresponds to FD and the plus sign to BE.

We will firstly study the BE distribution in 3d. We have:

Introducing a scaled temperature , we get

Again, we can study a particularly simple case: the massless limit with . In this case, we get:

The FD distribution in 3d can be studied in a similar way. Following the same approach as the BE distribution, we deduce that:

and again the massless limit and provide

**Remark (I):** For photons with degeneracy we obtain

**Remark (II):** In Cosmology, Astrophysics and also in High Energy Physics, the following units are used

The Cosmic Microwave Background is the relic photon radiation of the Big Bang, and thus it has a temperature due to photons in the microwave band of the electromagnetic spectrum. Its value is:

Indeed, it also implies that the relic photon density is about

It is also speculated that there has to be a Cosmic Neutrino Background relic from the Big Bang. From theoretical Cosmology, it is related to the photon CMB temperature in the following way:

or equivalently

This temperature implies a relic neutrino density (per species, i.e., with ) about

The cosmological density entropy due to these particles is

and then we get

**Remark (III): **In Cosmology, for fermions in 3d ( note that BE implies , and that we must drop the factors in the next numerical values) we can compute

**Remark (IV):** An example of the computation of degeneracy factor is the quark-gluon plasma degeneracy . Firstly we compute the gluon and quark degeneracies

Then, the QG plasma degeneracy factor is

In general, for charged leptons and nucleons , for neutrinos (per species, of course), and for gluons and photons. Remember that massive particles with spin j will have .

**Remark (V):** For the Planck distribution, we also get the known result for the thermal distribution of the blackbody radiation

**Remark (VI):** Sometimes the following nomenclature is used

i) Extremely degenerated gas if

ii) Non-degenerated gas if

iii) Extremely relativistic gas ( or ultra-relativistic gas) if

iv) Non-relativistic gas if

** **

## 9. ζ(s) AND GROUP ENTROPIES

Let us define the following shift operator :

where . Moreover, there is certain isomorphism between the shift operator space and the space of functions through the map .

We define the **generalized logarithm **as the image under the previous map of . That is:

where , with , and . Furthermore, the next contraints are also given for every generalized logarithm:

1st. .

2nd. , , and .

3rd. , and where .

With these definitions we also have that

A)

B)

Examples of generalized logarithms are:

**1) The Tsallis logarithm. **

**2) The Kaniadakis logarithm.**

**3) The Abe logarithm.**

**4) The biparametric logarithm.**

with and in the case of the Abe logarithm.

**Group entropies** are defined through the use of generalized logarithms. Define some discrete probability distribution with normalization . Therefore, the group entropy is the following functional sum:

where we have used the previous definition of generalized logarithm and the Boltzmann’s constant is a real number. It is called group entropy due to the fact that is connected to some universal formal group. This formal group will determine some correlations for the class of physical systems under study and its invariant properties. In fact, the Tsallis logarithm itself is related to the Riemann zeta function through a beautiful equation! Under the Tsallis group exponential, the isomorphism is defined to be , and thus we easily get:

such as and .

## 10. ζ(s) AND THE PRIMON GAS

The **primon gas/free Riemann gas** is a statistical mechanics toy model illustrating in a simple way some correspondences between number theory and concepts in statistical physics, quantum mechanics, quantum field theory and dynamical systems.

*primon gas IS a quantum field theory*. It is also named a gas or a free model because the particles are non-interacting. There is no potential. The idea of the primon gas was independently discovered by Donald Spector (

**(QFT)**of a set of non-interacting particles, called the “primons”**D. Spector, Supersymmetry and the Möbius Inversion Function, Communications in Mathemtical Physics 127 (1990) pp. 239-252**) and Bernard Julia (

**Bernard L. Julia, Statistical theory of numbers, in Number Theory and Physics, eds. J. M. Luck, P. Moussa, and M. Waldschmidt, Springer Proceedings in Physics, Vol. 47, Springer-Verlag, Berlin, 1990, pp. 276-293**).

*There have been later works by Bakas and Bowick*(

**I. Bakas and M.J. Bowick, Curiosities of Arithmetic Gases, J. Math. Phys. 32 (1991) p. 1881**) and Spector (

**D. Spector, Duality, Partial Supersymmetry, and Arithmetic Number Theory, J. Math. Phys. 39 (1998) pp.1919-1927**) in which it was explored the connection of such systems to string theory.

This model is based on some simple hypothesis:

1st. Consider a simple quantum Hamiltonian, , having eigenstates labelled by the prime numbers “p”.

2nd. The eigenenergies or spectrum are given by and they have energies proportional to . Mathematically speaking,

with

Please, note the natural emergence of a “free” scale of energy . What is this scale of energy? We do not know!

3rd. The second quantization/second-quantized version of this Hamiltonian converts states into particles, the “primons”. Multi-particle states are defined in terms of the numbers of primons in the single-particle states :

This corresponds to the factorization of into primes:

The labelling by the integer “N” is unique, since every number has a unique factorization into primes.

The energy of such a multi-particle state is clearly

4th. The statistical mechanics partition function IS, for the (bosonic) primon gas, the Riemann zeta function!

with , and where is the Boltzmann’s constant and T is the absolute temperature. The divergence of the zeta function at the value (corresponding to the harmonic sum) is due to the divergence of the partition function at certain temperature, usually called Hagedorn temperature. The Hagedorn temperature is defined by:

This temperature represents a limit beyond the system of (bosonic) primons can not be heated up. To understand why, we can calculate the energy

A similar treatment can be built up for fermions rather than bosons, but here the Pauli exclusion principle has to be taken into account, i.e. two primons cannot occupy the same single particle state. Therefore can be 0 or 1 for all single particle state. As a consequence, the many-body states are labeled not by the natural numbers, but by the square-free numbers. These numbers are sieved from the natural numbers by the Möbius function. The calculation is a bit more complex, but the partition function for a non-interacting fermion primon gas reduces to the relatively simple form

The canonical ensemble is of course not the only ensemble used in statistical physics. Julia extended the Riemann gas approach to the grand canonical ensemble by introducing a chemical potential (*Julia, B. L., 1994, Physica A 203(3-4), 425*), and thus, he replaced the primes p with new primes . This generalisation of the Riemann gas is called the **Beurling gas**, after the Swedish mathematician Beurling who had generalised the notion of prime numbers. Examining a boson primon gas with fugacity , it shows that its partition function becomes

**Remarkable interpretation:** pick a system, formed by two sub-systems not interacting with each other, the overall partition function is simply the product of the individual partition functions of the subsystems. From the previous equation of the free fermionic riemann gas we get exactly this structure, and so there are two decoupled systems. Firstly, a fermionic “ghost” Riemann gas at zero chemical potential and, secondly, a boson Riemann gas with energy-levels given by . Julia also calculated the appropriate Hagedorn temperatures and analysed how the partition functions of two different number theoretical gases, the Riemann gas and the “log-gas” behave around the Hagedorn temperature. Although the divergence of the partition function hints the breakdown of the canonical ensemble, Julia also claims that the continuation across or around this critical temperature can help understand certain phase transitions in string theory or in the study of quark confinement. The Riemann gas, as a mathematically tractable model, has been followed with much attention because the asymptotic density of states grows exponentially, , just as in string theory. Moreover, using arithmetic functions it is not extremely hard to define a transition between bosons and fermions by introducing an extra parameter, called kappa , which defines an imaginary particle, the non-interacting parafermions of order . This order parameter counts how many parafermions can occupy the same state, i.e. the occupation number of any state falls into the interval , and therefore belongs to normal fermions, while are the usual bosons. Furthermore, the partition function of a free, non-interacting κ-parafermion gas can be defined to be (Bakas and Bowick,1991, in the paper *Bakas, I., and M. J. Bowick, 1991, J. Math. Phys. 32(7), 1881*):

Indeed, Bakas et al. proved, using the Dirichlet convolution , how one can introduce free mixing of parafermions with different orders which do not interact with each other

where the symbol means d is a divisor of n. This operation preserves the multiplicative property of the classically defined partition functions, i.e., . It is even more intriguing how interaction can be incorporated into the mixing by modifying the Dirichlet convolution with a kernel function or twisting factor

Using the unitary convolution Bakas establishes a pedagogically illuminating case, the mixing of two identical boson Riemann gases. He shows that

This result has an amazing meaning. Two identical boson Riemann gases interacting with each other through the unitary twisting, are equivalent to mixing a fermion Riemann gas with a boson Riemann gas which do not interact with each other. Therefore, one of the original boson components suffers a transmutation/mutation into a fermion gas!

**Remark (I)**: the Möbius function, which is the identity function with respect to the operation (i.e. free mixing), reappears in supersymmetric quantum field theories as a possible representation of the operator, where F is the fermion number operator! In this context, the fact that for square-free numbers is the manifestation of the Pauli exclusion principle itself! In any QFT with fermions, is a unitary, hermitian, involutive operator where is the fermion number operator and is equal to the sum of the lepton number plus the baryon number, i.e., , for all particles in the Standard Model and some (most of) SUSY QFT. The action of this operator is to multiply bosonic states by 1 and fermionic states by -1. This is always a global internal symmetry of any QFT with fermions and corresponds to a rotation by an angle . This splits the Hilbert space into two superselection sectors. Bosonic operators commute with whereas fermionic operators anticommute with it. This operator really is, therefore, more useful in supersymmetric field theories.

**Remark (II)**: potential attacks on the Riemann Hypothesis may lead to advances in physics and/or mathematics, i.e., progress in Physmatics!

**Remark (III)**: the energy of the ground state is taken to be zero and the energy spectrum of the excited state is , where , , runs over the prime numbers. Let N and E denote now the number of particles in the ground state and the total energy of the system, respectively. The fundamental theorem of arithmetic allows only one excited state configuration for a given energy

where n is an integer. It immediately means that this gas preserves its quantum nature at any temperature, since only one quantum state is permitted to be occupied. The number fluctuation of any state (even the ground state) is therefore zero. In contrast, the changes in the number of particles in the ground state predicted by the canonical ensemble is a smooth non-vanishing function of the temperature, while the grand-canonical ensemble still exhibits a divergence. This discrepancy between the microcanonical (combinatorial) and the other two ensembles remains even in the thermodynamic limit.

One could argue that the Riemann gas is fictitious/unreal and its spectrum is unrealisable/unphysical. However, we, physicists, think otherwise, since the spectrum does not increase with N more rapidly than , therefore the existence of a quantum mechanical potential supporting this spectrum is possible (e.g., via inverse scattering transform or supplementary tools). And of course the question is: what kind of system has such an spectrum?

Some temptative ideas for the potential based on elementary Quantum Mechanics will be given in the next section.

## 11. LOG-OSCILLATORS

Instead of considering the free Riemann gas, we could ask to Quantum Mechanics if there is some potential providing the logarithmic spectrum of the previous section. Indeed, there exists such a potential. Let us factorize any natural number in terms of its prime “atoms”:

Take the logarithm

where are prime numbers (note that if we include “1” as a prime number it gives a zero contribution to the sum).

Now, suppose a logarithmic oscillator spectrum, i.e.,

with

with . In order to have a “riemann gas”/riemannium, we impose an spectrum labelled in the following fashion

Equivalently, we could also define the spectrum of interacting riemannium gas as

In addition to this, suppose the next quantum postulates:

**1st. Logarithmic potential:**

with positive constants

From the physical viewpoint, the positive constant means repulsive interaction (force).

**2nd. Bohr-Sommerfeld quantization rule:**

a)

or equivalently we could also get

b)

**3rd. Turning point condition**:

In the case of 2a) we would deduce that

so

and then

Then, using the turning point condition in this equation, we finally obtain

In the case of 2b) we would obtain

In summary, the logarithmic potential provides a model for the interacting Riemann gas!

## 12. LOG-POTENTIAL AND CONFINEMENT

Massive elementary particles (with mass m) can be understood as composite particles made of confined particles moving with some energy inside a sphere of radius R. We note that we do not define futher properties of the constituent particles (e.g., if they are rotating strings, particles, extended objects like branes, or some other exotic structure moving in circular orbits or any other pattern as trajectory inside the composite particle).

Let us make the hypothesis that there is some force needed to counteract the centrifugal force . The centrifugal force is equal to , i.e., the balancing force F is . Then, assuming the two forces are equal in magnitude, we get

where is some constant, and that equation holds regardless the origin of the interaction. The potentail energy necessary to confine a constituent particle will be, in that case,

with some integration constant to be determined later. The center of mass of the “elementary particle”, truly a composite particle, from the external observer and the mass assinged to the composited system is:

The logarithmic potential energy is postulated to be proportional to , and it provides

with is another constant. In fact, are parameters that don’t depend, a priori, on the radius R but on the constitutent particle properties and coupling constants, respectively. Indeed, for instance, we could set and fix the ratio to the constant , where is the gravitational constant. However, such a constraint is not required from first principles or from a clear physical reason. From the following equations:

and

we get

Quantum Mechanics implies that the angular momentum should be quantized, so we can make the following generalization

so

Using the previous integral and this last result, we obtain

This is due to the fact that and

Combining these equations, we deduce the value of as a function of the parameters

The ratio can be calculated from the above equations as well, since

for the case n=0 implies that

, and after exponentiation, it yields

Introducing the variable we have to solve the equation

The solution is from which the relationship between and can be easily obtained. Indeed, we can make more deductions from this result. From , then

If we take , with , then

so

with and

Equivalently, the masses would be dynamically generated from the above equations, since

and

so we would deduce a particle spectrum given by a logarithmic spiral, through the equation

**Remark**: The shift implies that the spiral would begin with as the lowest mass and not the biggest mass, turning the spiral from inside to the outside region and vice versa.

In summary, the logarithmic oscillator is also related to some kind of confined particles and it provides a toy model of confinement!

## 13. HARMONIC OSCILLATOR AND TSALLIS GAS

Is the link between classical statistical mechanics and Riemann zeta function unique or is it something more general? C. Tsallis explained long ago the connection of non-extensive Tsallis entropies an the Riemann zeta function, given supplementary arguments to support the idea of a physical link between Physics, Statistical Mechanics and the Riemann hypothesis. His idea is the following.

A) **Consider the harmonic oscillator with spectrum**

, are the H.O. eigenenergies.

B) **Consider the** **Tsallis partition function**

where and the deformed q-exponential is defined as

and

and the inverse of the deformed exponential is the q-logarithm

It implies that

Now, defining the Hurwitz zeta function as:

the last equation can be rewritten in a simple and elegant way:

This system can be called the** Tsallis gas** or **the Tsallisium**. It is a q-deformed version (non-extensive) of the free Riemann gas. And **it is related to the harmonic oscillator!** The issue, of course, is the problematic limit .

In the limit we get the Riemann zeta function from the Hurwitz zeta function:

or

The above equation, the partition function of the Tsallis gas/Tsallisium, connects directly the Riemann zeta function with Physics and non-extensive Statistical Mechanics. Indeed, C.Tsallis himself dedicated a nice slide with this theme to M.Berry:

**Remark (I):** The link between Riemann zeta function and the free Riemann gas/the interacting Riemann gas goes beyond classical statistical mechanics and it also appears in non-extensive statistical mechanics!

**Remark (II):** In general, the Riemann hypothesis is entangled to the theory of harmonic oscillators with non-extensive statistical mechanics!

## 14. TSALLIS ENTROPIES IN A NUTSHELL

For readers not familiarized with Tsallis generalized entropies, I would like to expose you the main definitions of such a generalization of classical statistical entropy (Boltzmann-Gibbs-Shannon), in a nutshell! I have to discuss more about this kind of statistical mechanics in the future, but today, I will only anticipate some bits of it.

Tsallis entropy (and its Statistical Mechanics/Thermodynamics) is based on the following entropy functionals:

**1st. Discrete case.**

plus the normalization condition

**2nd. Continuous case.**

plus the normalization condition

**3rd. Quantum case. Tsallis matrix density.**

plus the normatlization condition

In all the three cases above, we have defined the q-logarithm as , , and the 3 Tsallis entropies satisfy the non-additive property:

## 15. BEYOND QM/QFT: ADELIC WORLDS

Theoretical physicsts suspect that Physics of the spacetime at the Planck scale or beyond will change or will be meaningless. There, the spacetime notion we are familiarized to loose its meaning. Even more, we could find those changes in the fundamental structure of the Polyverse to occur a higher scales of length. Really, we don’t know yet where the spacetime “emerges” as an effective theory of something deeper, but it is a natural consequence from our current limited knowledge of fundamental physics. Indeed, it is thought that the experimental device making measurements and the experimenter can not be distinguished at Planck scale. At Planck scale, we can not know at this moment how the framework of cosmology and the Hilbert space tool of Quantum Mechanics could be obtained with some unified formalism. It is one of the challenges of Quantum Gravity.

Many people and scientists think that geometry and topology of sub-Planckian lengths should not have any relation with our current geometry or topology. We say and believe that geometry, topology, fields and the main features of macroscopic bodies “emerge” from the ultra-Planckian and “subquantum” realm. It is an analogue to the colours of the rainbow emerging from the atoms or how Thermodynamics emerge from Statistical Mechanics.

There are many proposed frameworks to go beyond the usual notions of space and time, but the p-adic analysis approach is a quite remarkable candidate, having several achievements in its favor.

Motivations for a p-adic and adelic approaches as the ultimate substructure of the microscopic world arise from:

1) Divergences of QFT are believed to be absent with such number structures. Renormalization can be found to be unnecessary.

2) In an adelic approach, where there is no prime with special status in p-adic analysis, it might be more natural and instructive to work with adeles instead a pure p-adic approach.

3) There are two paths for a p-adic/adelic QM/QFT theory. The first path considers particles in a p-adic potential well, and the goal is to find solutions with smoothly varying complex-valued wavefunctions. There, the solutions share certain kind of familiarity from ordinary life and ordinary QM. The second path allows particles in p-adic potential wells, and the goal is to find p-adic valued wavefunctions. In this case, the physical interpretation is harder. Yet the math often exhibits surprising features and properties, and some people are trying to explores those novel and striking aspects.

Ordinary real (or even complex as well) numbers are familiar to everyone. Ostroswski’s theorem states that there are essentially only two possible completions of the rational numbers ( “fractions” you do know very well). The two options depend on the metric we consider:

**1) The real numbers.** One completes the rationals by adding the limit of all Cauchy sequences to the set. Cauchy sequences are series of numbers whose elements can be arbitrarily close to each other as the sequence of numbers progresses. Mathematically speaking, given any small positive distance, all but a finite number of elements of the sequence are less than that given distance from each other. Real numbers satisfy the triangle inequality .

**2) The p-adic numbers.** The completions are different because of the two different ways of measuring distance. P-adic numbers satisfy an stronger version of the triangle inequality, called ultrametricity. For any p-adic number is shows

Spaces where the above enhanced triangle inequality/ultrametricity arises are called **ultrametric spaces**.

In summary, there exist two different types of algebraic number systems. There is no other posible norm beyond the real (absolute) norm or the p-adic norm. It is the power of Mathematics in action.

Then, a question follows inmediately. How can we unify such two different notions of norm, distance and type of numbers. After all, they behave in a very different way. Tryingo to answer this questions is how the concept adele emerges. The ring of adeles is a framework where we consider all those different patterns to happen at equal footing, in a same mathematical language. In fact, it is analogue to the way in which we unify space and time in relativistic theories!

**Adele numbers are an array consisting of both real (complex) and p-adic numbers!** That is,

where is a real number and the are p-adic numbers living in the p-adic field . Indeed, the infinity symbol is just a consequence of the fact that real numbers can be thought as “the prime at infinity”. Moreover, it is required that all but finitely many of the p-adic numbers lie in the entire p-adic set . The adele ring is therefore a restricted direct (cartesian) product. The idele group is defined as the essentially invertible elements of the adelic ring:

We can define the calculus over the adelic ring in a very similar way to the real or complex case. For instance, we define trigonometric functions, , logarithms and special functions like the Riemann zeta function. We can also perform integral transforms like the Mellin of the Fourier transformation over this ring. However, this ring has many interesting properties. For example, quadratic polynomials obey the Hasse local-global principle: a rational number is the solution of a quadratic polynomial equation if and only if it has a solution in and for all primes p. Furthermore, the real and p-adic norms are related to each other by the remarkable adelic product formula/identity:

and where is a nonzero rational number.

Beyond complex QM, where we can study the particle in a box or in a ring array of atoms, p-adic QM can be used to handle fractal potential wells as well. Indeed, the analogue Schrödinger equation can be solved and it has been useful, for instance, in the design of microchips and self-similar structures. It has been conjectured by Wu and Sprung, Hutchinson and van Zyl,here http://arXiv.org/abs/nlin/0304038v1 , that the potential constructed from the non-trivial Riemann zeroes and prime number sequences has fractal properties. They have suggested that for the Riemann zeroes and for the prime numbers. Therefore, p-adic numbers are an excellent method for constructing fractal potential wells.

By the other hand, following Feynman, we do know that path integrals for quantum particles/entities manifest fractal properties. Indeed we can use path integrals in the absence of a p-adic Schrödinger equation. Thus, defining the adelic version of Feynman’s path integral is a necessary a fundamental object for a general quantum theory beyond the common textbook version. However, we need to be very precise with certain details. In particular, we have to be careful with the definition of derivatives and differentials in order to do proper calculations. Indeed we can do it since both, the adelic and idelic rings have a well defined translation-invariant Haar measure

and

These measures provide a way to compute Feynman path integrals over adelic/idelic spaces. It turns out that Gaussian integrals satisfy a generalization of the adelic product formula introduced before, namely:

where is an additive character from the adeles to complex numbers given by the map:

and is the fractional part of in the ordinary p-adic expression for x. This can be thought of as a strong generalization of the homomorphism .Then, the adelic path integral, with input parameters in the adelic ring and generating complex-valued wavefunctions follows up:

The eigenvalue problem over the adelic ring is given by:

where U is the time-development operator, are adelic eigenfunctions, and is the adelic energy. Here the notation has been simplified by using the subscript , which stands for all primes including the prime at infinity. One notices the additive character which allows these to be complex-valued integrals. The path integral can be generalized to p-adic time as well, i.e., to paths with fractal behaviour!

How is this p-adic/adelic stuff connected to the Riemannium an the Riemann zeta function? It can be shown that ground state of adelic quantum harmonic oscillator is

where is 1 if is a p-adic integer and 0 otherwise. This result is strikingly similar to the ordinary complex-valued ground state. Applying the adelic Mellin transform, we can deduce that

where are, respectively, the gamma function and the Riemann zeta function. Due to the Tate formula, we get that

.

and from this the functional equation for the Riemann zeta function naturally emerges.

**In conclusion:** it is fascinating that such simple physical system as the (adelic) harmonic oscillator is related to so significant mathematical object as the Riemann zeta function.

## 16. STRINGS, FIELDS AND VACUUM

The Veneziano amplitude is also related to the Riemann zeta function and string theory. A nice application of the previous adelic formalism involves the adelic product formula in a different way. In string theory, one computes crossing symmetric Veneziano amplitudes describing the scattering of four tachyons in the 26d open bosonic string. Indeed, the Veneziano amplitude can be written in terms of Riemann zeta function in this way:

These amplitudes are not easy to calculate. However, in 1987, an amazingly simple adelic product formula for this tachyonic scattering was found to be:

Using this formula, we can compute and calculate the four-point amplitudes/interacting vertices at the tree level exactly, as the inverse of the much simpler p-adic amplitudes. This discovery has generated a quite a bit of activity in string theory, somewhat unknown, although it is not very popular as far as I know. Moreover, the whole landscape of the p-adic/adelic framework is not as easy for the closed bosonic string as the open bosonic strings (note that in a p-adic world, there is no “closure” but “clopen” segments instead of naive closed intervals). It has also been a source of controversy what is the role of the p-adic/adelic stuff at the level of the string worldsheet. However, there is some reasearch along these lines at current time.

Another nice topic is the vacuum energy and its physical manifestations. There are some very interesting physical effects involving the vacuum energy in both classical and quantum physics. The most important effects are **the Casimir effect** (vacuum repulsion between “plates”) ,** the Schwinger effect** ( particle creation in strong fields) ,** the Unruh effect** ( thermal effects seen by an uniformly accelerated observer/frame) , **the Hawking effect** (particle creation by Black Holes, due to Black Hole Thermodynamcis in the corresponding gravitational/accelerated environtment) , and **the cosmological constant effect** (or vacuum energy expanding the Universe at increasing rate on large scales. Itself, does it gravitate?). Riemann zeta function and its generalizations do appear in these 4 effects. It is not a mere coincidence. It is telling us something deeper we can not understand yet. As an example of why zeta function matters in, e.g., the Casimir effect, let me say that zeta function regularizes the following general sum:

**Remark:** I do know that I should have likely said “the cosmological constant problem”. But as it should be solved in the future, we can see the cosmological constant we observe ( very, very smaller than our current QFT calculations say) as “an effect” or “anomaly” to be explained. We know that the cosmological constant drives the current positive acceleration of the Universe, but it is really tiny. What makes it so small? We don’ t know for sure.

**Remark(II):** What are the p-adic strings/branes? I. Arefeva, I. Volovich and B. Dravogich, between other physicists from Russia and Eastern Europe, have worked about non-local field theories and cosmologies using the Riemann zeta function as a model. It is a relatively unknown approach but it is remarkable, very interesting and uncommon. I have to tell you about these works but not here, not today. I went too far, far away in this log. I apologize…

## 17. SUMMARY AND OUTLOOK

I have explained why I chose The Spectrum of Riemannium as my blog name here and I used the (partial) answer to explain you some of the multiple connections and links of the Riemann zeta function (and its generalizations) with Mathematics and Physics. I am sure that solving the Riemann Hypothesis will require to answer the question of what is the vibrating system behind the spectral properties of Riemann zeroes. It is important for Physmatics! I would say more, it is capital to theoretical physics as well.

Let me review what and where are the main links of the Riemann zeta function and zeroes to Physmatics:

1) Riemann zeta values appear in atomic Physics and Statistical Physics.

2) The Riemannium has spectral properties similar to those of Random Matrix Theory.

3) The Hilbert-Polya conjecture states that there is some mysterious hamiltonian providing the zeroes. The Berry-Keating conjecture states that the “quantum” hamiltonian corresponding to the Riemann hypothesis is the corresponding or dual hamiltonian to a (semi)classical hamiltonian providing a classically chaotic dynamics.

4) The logarithmic potential provides a realization of certain kind of spectrum asymptotically similar to that of the free Riemann gas. It is also related to the issue of confinement of “fundamental” constituents inside “elementary” particles.

5) The primon gas is the Riemann gas associated to the prime numbers in a (Quantum) Statistical Mechanics approach. There are bosonic, fermionic and parafermionic/parabosonic versions of the free Riemann gas and some other generalizations using the Beurling gas and other tools from number theory.

6) The non-extensive Statistical Mechanics studied by C. Tsallis (and other people) provides a link between the harmonic oscillator and the Riemann hypothesis as well. The Tsallisium is the physical system obtained when we study the harmonic oscillator with a non-extensive Tsallis approach.

7) An adelic approach to QM and the harmonic oscillator produces the Riemann’s zeta function functional equation via the Tate formula. The link with p-adic numbers and p-adic zeta functions reveals certain fractal patterns in the Riemann zeroes, the prime numbers and the theory behind it. The periodicity or quasiperiodicity also relates it with some kind of (quasi)crystal and maybe it could be used to explain some behaviour or the prime numbers, such as the one behind the Goldbach’s conjecture.

8) A link between entropy, information theory and Riemann zeta function is done through the use of the notion of group entropy. Connections between the Veneziano amplitudes, tachyons, p-adic numbers and string theory arise after the Veneziano amplitude in a natural way.

9) Riemann zeta function also is used in the regularization/definition of infinite determinants arising in the theory of differential operators and similar maps. Even the generalization of this framework is important in number theory through the uses of generalizations of the Riemann zeta function and other arithmetical functions similar to it. Riemann zeta function is, thus, one of the simplest examples of arithmetical functions.

10) There are further links of the Riemann zeta function and “vacuum effects” like the Schwinger effect ( pair creating in strong fields) or the Casimir effect ( repulsive/atractive forces between close objects with “nothing” between them). Riemann zeta function is also related to SUSY somehow, either by the striking similarity between the Dirichlet eta function used in Fermi-Dirac statistics or directly with the explicit relationship between the Möbius function and the operator appearing in supersymmetric field theories.

In summary, Riemann zeta function is ubiquitious and it appears alone or with its generalizations in very different fields: number theory, quantum physics, (semi)classical physics/dynamics, (quantum) chaos theory, information theory, QFT, string theory, statistical physics, fractals, quasicrystals, operator theory, renormalization and many other places. Is it an accident or is it telling us something more important? I think so. Zeta functions are fundamental objects for the future of Physmatics and the solution of Riemann Hypothesis, perhaps, would provide such a guide into the ultimate quest of both Physics and Mathematics (Physmatics) likely providing a complete and consistent description of the whole Polyverse.

Then, the main unanswered questions to be answered are yet:

**A) What is the Riemann zeta function? What is the riemannium/tsallisium and what kind of physical system do they represent really? What is the physical system behind the Riemann non-trivial zeroes? What does it mean for the Riemann zeroes arising from the Riemann zeta function generalizations in form of L-functions?**

**B) What is the Riemann-Hilbert-Polya operator? What is the space over the Riemann operator is acting?**

**C) Are Riemann zeta function and its generalization everywhere as they seem to be inside the deepest structures of the microscopic/macroscopic entities of the Polyverse?**

I suppose you will now understand better why I decided to name my blog as The Spectrum of Riemannium…And there are many other reasons I will not write you here since I could reveal my current research.

However, stay tuned!

*Physmatics is out there and everywhere, like fractals, zeta functions and it is full of lots of wonderful mathematical structures and simple principles!*