LOG#113. Bohr’s legacy (I).

Dedicated to Niels Bohr

and his atomic model

(1913-2013)

1st part: A centenary model

atomElement-117-discoverypurity

This is a blog entry devoted to the memory of a great scientist, N. Bohr, one of the greatest master minds during the 20th century, one of the fathers of the current Quantum model of atoms and molecules.

Niels_Bohr

One century ago, Bohr was the pioneer of the introduction of the “quantization” rules into the atomic realm, 8 years after the epic Annus Mirabilis of A. Einstein (1905). Please, don’t forget that Einstein himself was the first physicist to consider Planck hypothesis into “serious” physics problems, explaining the photoelectric effect in a simple way with the aid of “quanta of light” (a.k.a. photons!). Therefore, it is not correct to assest that N.Bohr was the “first” quantum physicist. Indeed, Einstein or Planck were the first. Said, this, Bohr was the first to apply the quantum hypothesis into the atomic domain, changing forever the naive picture of atoms coming from the “classical” physics.  I decided that this year I would be writting something in to honour the centenary of his atomic model (for the hydrogen atom).

I wish you will enjoy the next (short) thread…

Atomic mysteries

When I was young, and I was explained and shown the Periodic Table (the ordered list or catalogue of elements) by the first time, I wondered how many elements could be in Nature. Are they 103? 118?Maybe 212? 1000? 10^{23}? Or 10^{100}? \infty, Infinity?

We must remember what an atom is…Atom is a greek word \alpha\tau o\mu o\sigma meaning “with no parts”. That is, an atom is (at least from its original idea), something than can not be broken into smaller parts. Nice concept, isn’t it?

Greek philosophers thought millenia ago if there is a limit to the divisibility of matter, and if there is an “ultimate principle” or “arche” ruling the whole Universe (remarkably, this is not very different to the questions that theoretical physicists are trying to solve even now or the future!). Different schools and ideas arose. I am not very interested today into discussing Philosophy (even when it is interesting in its own way), so let me simplify the general mainstream ideas several thousands of years ago (!!!!):

1st. There is a well-defined ultimate “element”/”substance” and an ultimate “principle”. Matter is infinitely divisible. There are deep laws that govern the Universe and the physical Universe, in a cosmic harmony.

2nd. There is a well-defined ultimate “element”/”substance” and an ultimate “principle”. Matter is FINITELY divisible. There are deep laws that govern the Universe and the physical Universe, in a cosmic harmony.

3rd. There is no a well-defined ultimate “element”/”substance” or an ultimate principle. Chaos rules the Universe. Matter is infinitely divisible.

4th. There is no a well-defined ultimate “element”/”substance” or an ultimate principle. Chaos rules the Universe. Matter is finitely divisible.

Remark: Please, note the striking “similarity” with some of the current (yet) problems of Physics. The existence of a Theory Of Everything (TOE) is the analogue to the question of the first principle/fundamental element quest of ancient greek philosophers or any other philosophy in all over the world. S.W. Hawking himself provided in his Brief Story of Time the following (3!) alternative approaches

1st. There is not a TOE. There is only a chaotic pattern of regularities we call “physical laws”. But Nature itself is ultimately chaotic and the finite human mind can not understand its ultimate description.

2nd. There is no TOE. There are only an increasing number of theories more and more precise or/and more and more accurate without any limit. As we are finite beings, we can only try to guess better and better approximations to the ultimate reality (out of our imagination) and the TOE can not be reached in our whole lifetime or even in the our whole species/civilization lifetime.

3rd. There is a well defined TOE, with its own principles and consequences. We will find it if we are persistent enough and if we are clever enough. All the physical events could be derived from this theory. If we don’t find the “ultimate theory and its principles” is not because it is non-existent, it is only that we are not smart enough. Try harder (If you can…)!

If I added another (non Greek) philosophies, I could create some other combinations, but, as I told you above, I am not going to tell you Philosophy here, not at least more than necessary.

As you probably know, the atomic idea was mainly defended by Leucippus and Democritus, based on previous ideas by Anaxagoras. It is quite likely that Anaxagoras himself learned them from India (or even from China), but that is quite speculative… Well, the keypoint of the atomic idea is that you can not smash into smaller pieces forever smaller and smaller bits of matter. Somewhere, the process of breaking down the fundamental constituents of matter must end…But where? And mostly, how can we find an atom or “see” what an atom looks like? Obviously, ancient greeks had not idea of how to do that, or even knowing the “ground idea” of what a atom is, they had no experimental device to search for them. Thus, the atomic idea was put into the freezer until the 18th and 19th century, when the advances in experimental (and theoretical) Chemistry revived the concept and the whole theory. But Nature had many surprises ready for us…Let me continue this a bit later…

In the 19th century, with the discovery of the ponderal laws of Chemistry, Dalton and other chemists were stunned. Finally, Dalton  was the man who recovered the atomism into “real” theoretical Science. But their existence was controversial until the 20th century. However, Dalton concluded that there was a unique atom for each element, using Lavoisier’s definition of an element as a substance that could not be analyzed into something simpler. Thus, Dalton arrived to an important conclusion:

“(…)Chemical analysis and synthesis go no farther than to the separation of particles one from another, and to their reunion. No new creation or destruction of matter is within the reach of chemical agency. We might as well attempt to introduce a new planet into the solar system, or to annihilate one already in existence, as to create or destroy a particle of hydrogen. All the changes we can produce, consist in separating particles that are in a state of cohesion or combination, and joining those that were previously at a distance(…)”.

The reality of atoms was a highly debated topic during all the 19th century. It is worthy to remark that was Einstein himself (yes, he…agian) who went further and with his studies about the Brownian motion established their physical existence. It was a brillian contribution to this area, even when, in time, he turned against the (interpretation of) Quantum Mechanics…But that is a different story not to be told today.

Dalton’s atoms or Dalton atomic model was very simple.

A_New_System_of_Chemical_Philosophy_fp

Atoms had no parts and thus, they were truly indivisible particles. However, the electrical studies of matter and the electromagnetic theory put this naive atomic model into doubt. After the discovery of “the cathode” rays (1897) and the electron by J.J.Thomson (no, it is not J.J.Abrahams), it became clear that atoms were NOT indivisible after all! Surprising, isn’t it? It is! Chemical atoms are NOT indivisible. They do have PARTS.

Thomson’s model or “plum pudding” model, came into the rescue…Dalton believed that atoms were solid spheres, but J.J.Thomson was forced (due to the electron existence) to elaborate a “more complex” atomic model. He suggested that atoms were a spherical “fluid” mass with positive charge, and that electrons were placed into that sphere as in a “plum pudding” cake.  I have to admit that I were impressed by this model when I was 14…It seemed too ugly for me to be true, but anyway it has its virtues (it can explain the cathode ray experiment!).cathode-rays-formation

thomsonAndNagaokaModels

The next big step was the Rutherford experiment! Thomson KNEW that electrons were smaller pieces inside the atom, but despite his efforts to find the positive particles (and you see there he had and pursued his own path since he discovered the reason of the canal rays), he could not find it (and they should be there since atoms were electrically neutrial particles). However, clever people were already investigating radioactivity and atomic structure with other ideas…In 1911, E. Rutherford, with the aid of his assistants, Geiger and Marsden, performed the celebrated gold foil experiment.

Rutherford_experiment

To his surprise (Rutherford’s), his assistants and collaborators provided a shocking set of results. To explain all the observations, the main consequences of the Rutherford’s experiment were the next set of hypotheses:

1st. Atoms are mostly vacuum space.

2nd. Atoms have a dense zone of positive charge, much smaller than the whole atom. It is the atomic nucleus!

3rd. Nuclei had positive charge, and electrons negative charge.

He (Rutherford) did not know from the beginning how was the charge arranged and distributed into the atom. He had to improve the analysis and perform additional experiment in order to propose his “Rutherford” solar atomic model and to get an estimate of the nuclei size (about 1fm or 10^{-15}m). In fact, years before him, the japanase Nagaoka had proposed a “saturnian” atomic model with a similar looking. It was unstable, though, due to the electric repulsion of the electronic “rings” (previously there was even a “cubic” model of atom, but it was unsuccessful too to explain every atomic experiment) and it had been abandoned.

And this is the point where theory become “hard” again. Rutherford supposed that the electron orbits around nuclei were circular (or almost circular) and then electrons experimented centripetal forces due to the electrical forces of the nucleus. The classical electromagnetic theory said that any charged particle being accelerated (and you do have acceleration with a centripetal force) should emit electromagnetic waves, losing energy and, then, electrons should fall over the the nuclei (indeed, the time of the fall down was ridiculously small and tiny). We do not observe that, so something is wrong with our “classical” picture of atoms and radiation (it was also hinted with the photoelectric effect or the blackbody physics, so it was not too surprising but challenging to find the rules and “new mechanics” to explain the atomic stability of matter). Moeover, the atomic spectra was known to be discrete (not continuous) since the 19th century as well. To find out the new dynamics and its principles became one of the oustanding issues in the theoretical (and experimental) community. The first scientist to determine a semiclassical but almost “quantum” and realistic atomic spectrum (for the simpler atom, the hydrogen) was Niels Bohr. The Bohr model of the hydrogen atom is yet explained at schools not only due to its historical insterest, but to the no less important fact that it provides right answers (indeed, Quantum Mechanics reproduces its features) for the simplest atom and that its equations are useful and valid from a quantitative viewpotint (as I told you, Quantum Mechanics reproduces Bohr formulae). Of course, Bohr model does not explain the Stark effect, the Zeeman effect, or the hyperfine structure of the hydrogen atom and some other “quantum/relativistic” important effects, but it is a really useful toy model and analytical machine to think about the challenges and limits of Quantum Mechanics of atoms and molecules. Bohr model can not be applied to helium and other elements in the Periodic Table of the elements (its structure is described by Quantum Mechanics), so it can be very boring but, as we will see, it has many secrets and unexpected surprises in its core…

Bohr model for the hydrogen atom

bohr_transitionsBohr_atom_model_EnglishBohr_atombohrAndBalmer

Bohr model hypotheses/postulates:

1st. Electrons describe circular orbits around the proton (in the hydrogen atom). The centripetal force is provided by the electrostatic force of the proton.

2nd. Electrons, while in “stationary” orbits with a fixed energy, do NOT radiate electromagnetic waves ( note that this postulate is againsts the classical theory of electromagnetics as it was known in the 19th century).

3rd. When a single electron passes from one energetic level to another, the energy transitions/energy differences satisfy the Planck law. That is, during level transitions, \Delta E=hf.

In summary, we have:

bohrPostulatesbohrmodelHypotheses

Firstly, we begin with the equality between the electron-proton electrostatic force and the centripetal force in the atom:

\begin{pmatrix}\mbox{Centripetal}\\ \mbox{Force}\end{pmatrix}=\begin{pmatrix}\mbox{Electron-proton}\\ \mbox{electric force}\end{pmatrix}

Mathematically speaking, this first postulate/ansatz requieres that q_1=q_2=e, where e=1\mbox{.}602\cdot 10^{-19}C is the elementary electric charge of the electron (and equal in absolute value to the proton charge) and m_e=9.11\cdot 10^{-31}kg is the electron mass:

F_c=\dfrac{m_ev^2}{R} and F_C=K_C\dfrac{q_1q_2}{R^2}=K_C\dfrac{e^2}{R^2} implies that

(1) \boxed{F_c=F_{el,C}}\leftrightarrow \boxed{\dfrac{m_ev^2}{R}=\dfrac{K_Ce^2}{R^2}}\leftrightarrow \boxed{v^2=\left(\dfrac{K_C}{m_e}\right)\left(\dfrac{e^2}{R}\right)}

Remark: Instead of having the electron mass, it would be more precise to use the “reduced” mass for this two body problem. The reduced mass is, by definition,

\mu=m_{red}=\dfrac{m_1m_2}{m_1+m_2}=\dfrac{m_em_p}{m_e+m_p}

However, it is easy to realize that the reduced mass is essentially the electron mass (since m_p\approx 1836m_e)

\mu=\dfrac{m_e}{1+\left(\dfrac{m_e}{m_p}\right)}\approx m_e(1-\dfrac{m_e}{m_p}+\ldots)=m_e+\mathcal{O} \left(\dfrac{m_e^2}{m_p}\right)

The second Bohr’s great idea was to quantize the angular momentum. Classically, angular momentum can take ANY value, Bohr great’s intuition suggested that it could only take multiple values of some fundamental constant, the Planck’s constant. In fact, assuming orbitar stationary orbits, the quantization rule provides

(2) \boxed{L=m_ev(2\pi R)=nh} or \boxed{L=m_evR=n\dfrac{h}{2\pi}=n\hbar} with \hbar=\dfrac{h}{2\pi} and n=1,2,3,\ldots,\infty a positive integer.

Remark: h=6\mbox{.}63\cdot 10^{-34}Js and \hbar=\dfrac{h}{2\pi}=1\mbox{.}055\cdot 10^{-34}Js are the Planck constant and the reduced Planck constant, respectively.

From this quantization rule (2), we can easily get

vR=\left(\dfrac{n\hbar}{m_e}\right) and then v^2R^2=\left(\dfrac{n\hbar}{m_e}\right)^2

Thus, we have

R^2=\left(\dfrac{n\hbar}{m_e}\right)^2\dfrac{1}{v^2}

Using the result we got in (1) for the squared velocity of the electron in the circular orbit, we deduce the quantization rule for the orbits in the hydrogen atom according to Bohr’s hypotheses:

R^2=\left(\dfrac{n\hbar}{m_e}\right)^2\left(\dfrac{m_eR}{K_Ce^2}\right)

R=\dfrac{n^2\hbar^2}{m_e^2}\dfrac{m_e}{K_Ce^2}

(3) \boxed{R_n=R(n)=\left(\dfrac{\hbar^2}{m_eK_Ce^2}\right)n^2}\leftrightarrow \boxed{R_n=a_Bn^2}

where n=1,2,3,\ldots,\infty again and the Bohr radius a_B is defined to be

(4) \boxed{a_B=\dfrac{\hbar^2}{m_eK_Ce^2}}

Inserting values into (4), we obtain the celebrated value of the Bohr radius

a_B\approx 0\mbox{.}53\AA=53pm=5\mbox{.}3\cdot 10^{-11}m

The third important consequence in the spectrum of energy levels in the hydrogen atom. To obtain the energy spectrum, there is two equivalent paths (in fact, they are the same): use the virial theorem or use (1) into the total energy for the electron-proton system. The total energy of the hydrogen atom can be written

E=\mbox{Kinetic Energy}+\mbox{(electrostatic) Potential Energy}

E=\dfrac{p^2}{2m_e}-\dfrac{K_Ce^2}{R}=\dfrac{m_ev^2}{2}-\dfrac{K_Ce^2}{R}

Substituting (1) into this, we get exactly the expected expression for the virial theorem to a 1/r^2 potential (i.e. E=E_p/2):

E=\dfrac{m_ev^2}{2}-\dfrac{K_Ce^2}{R}=-K_C\dfrac{e^2}{2R}

(5) \boxed{E=-K_C\dfrac{e^2}{2R}}

Inserting into (5) the quantized values of the orbit, we deduce the famous and well-known formula for the spectrum of the hydrogen atom (known to Balmer and the spectroscopists at the end of the 19th century and the beginning of the 20th century):

(6) \boxed{E_n=E(n)=-\dfrac{m_eK_C^2e^4}{2\hbar^2n^2}=-\dfrac{m_e}{2}\left(\dfrac{K_Ce^2}{n\hbar}\right)^2=-\dfrac{\mbox{Ry}}{n^2}} \;\;\forall n=1,2,3,\ldots,\infty

and where we have defined the Rydberg (constant) as

(7) \boxed{\mbox{Ry}=\dfrac{m_e(K_Ce^2)^2}{2\hbar^2}=\dfrac{m_eK_C^2e^4}{2\hbar^2}=\dfrac{1}{2}\alpha^2 m_ec^2}

Its value is Ry=R_H=2.18\cdot 10^{-18}J=13\mbox{.}6eV. Here, the electromagnetic fine structure constant (alpha) is

\alpha=K_C\dfrac{e^2}{\hbar c}

and c is the speed of light. In fact, using the quantum relation

E=\dfrac{hc}{\lambda}

we can deduce that the Rydberg corresponds to a wavenumber

k=1\mbox{.}097\cdot 10^{7}m^{-1}

or a frequency

f=\nu=3\mbox{.}29\cdot 10^{15}Hz

and a wavelength

\lambda =912\AA=91\mbox{.}2nm

Please, check it yourself! :D.

The above results allowed Bohr to explain the spectral series of the hydrogen atom. He won the Nobel Prize due to this wonderful achievement…

Hydrogenic atoms

(and positronium, muonium,…)

In fact, it is easily straightforward to extend all these results to “hydrogenic” (“hydrogenoid”) atoms, i.e., to atoms with only a single electron BUT a nucleus with charge equal to Ze, and Z>1 is an integer (atomic) number greater than one! The easiest way to obtain the results is not to repeat the deduction but to make a rescaling of the proton charge, i.e., you plug q_2=Ze or/and make a rescaling of the electric charge q_2=e\longrightarrow Ze (be aware of making the right scaling in the formulae). The final result for the radius and the energy spectrum is as follows:

A) From R_n=\left(\dfrac{\hbar^2}{m_eK_Ce^2}\right)n^2, with e\longrightarrow Ze, you get

(8) \boxed{\bar{R}_n=\bar{R}(n)=\dfrac{\hbar^2}{m_eK_CZe^2}n^2=\dfrac{a_Bn^2}{Z}}

B) From E_n=-m_e\dfrac{(K_Ce^2)^2}{2\hbar^2n^2}, with the rescaling e\longrightarrow Ze, you get

(9) \boxed{\bar{E}_n=\bar{E}(n)=-m_e\dfrac{Z^2(K_Ce^2)^2}{2\hbar^2n^2}=-\dfrac{Z^2\alpha^2m_ec^2}{2n^2}=-\dfrac{Z^2Ry}{n^2}}

Therefore, the consequence of the rescaling of the nuclear charge is that energy levels are “enlarged” by a factor Z^2 and that the orbits are “squeezed” or “contracted” by a factor 1/Z.

Exercise: Can you obtain the energy levels and the radius for the positronium (an electron and positron system instead an electron a positron). What happens with the muonium (strange substance formed by electron orbiting and antimuon)?And the muonic atom (muon orbiting an proton)? And a muon orbiting an antimuon? And the tau particle orbiting an antitau or the electron orbiting an antitau or a tau orbiting a proton(supposing that it were possible of course, since the tau particle is unstable)? Calculate the “Bohr radius” and the “Rydberg” constant for the positronium, the muonium, the muonic atom (or the muon-antimuon atom) and the tauonium (or the tau-antitau atom). Hint: think about the reduced mass for the positronium and the muonium, then make a good mass/energy or radius rescaling.

Now, we can also calculate the velocity of an electron in the quantized orbits for the Bohr atom and the hydrogenic atom. Using (3) and (8),

mvR=n\hbar\leftrightarrow mR=\dfrac{n\hbar}{m_e}\leftrightarrow v^2R^2=\dfrac{n^2\hbar^2}{m_e^2}

or

v^2=\left(\dfrac{n\hbar}{m_e}\right)^2\dfrac{1}{R^2}

and inserting the quantized values of the orbit radius

v_n^2=\dfrac{K_Ce^2}{m_eR_n}=\dfrac{m_e(K_Ce^2)^2}{m_en^2\hbar^2}

so, for the Bohr atom (hydrogen)

(10) \boxed{v_n=v(n)=\dfrac{K_Ce^2}{\hbar n}=\dfrac{\alpha c}{n}}

In the case of hydrogenic atoms, the rescaling of the electric charge yields

(11) \boxed{\bar{v}_n=\bar {v}(n)=\dfrac{ZK_Ce^2}{\hbar n}=\dfrac{Z\alpha c}{n}}

so, the hydrogenic atoms have a “enlarged” electron velocity in the orbits, by a factor of Z.

The feynmanium

This result for velocities is very interesting. Suppose we consider the fundamental level n=1 (or the orbital 1s in Quantum Mechanics, since, magically or not, Quantum Mechanics reproduces the results for the Bohr atom and the hydrogenic atoms we have seen here, plus other effects we will not discuss today relative to spin and some energy splitting for perturbed atoms). Then, the last formula yield, in the hydrogenic case,

v_1=Z\alpha c

Furthermore, suppose now in addition that we have some “superheavy” (hydrogenic) atom with, say, Z>137 (note that \alpha\approx 1/137 at ordinary energies), say Z=138 or greater than it. Then, the electron moves faster than the speed of light!!!!! That is, for hydrogenic atoms, with Z>137 and considering the fundalmental level, the electron would move with v>c. This fact is “surprising”. The element with Z=137 is called untriseptium (Uts) by the IUPAC rules, but it is often called the feynmanium (Fy), since R.P. Feynman often remarked the importance of this result and mystery. Of course, Special Relativity forbids this option. Therefore, something is wrong or Z=137 is the last element allowed by the Quantum Rules (or/and the Bohr atom). Obviously, we could claim that this result is “wrong” since we have not consider the relativistic quantum corrections or we have not made a good relativistic treatment of this system. It is not as simple as you can think or imagine, since using a “naive” relativistic treatment, e.g., using the Dirac equation , we obtain for the fundamental level of the hydrogenic atom the spectrum

(12) \boxed{E_1=E=m_ec^2\sqrt{1-Z^2\alpha^2}}. This result can be obtained from the Dirac equation spectrum for the hydrogen atom (in a Coulomb potential):

(13) \boxed{E_{n,k;Z,\alpha}=E(n,k;Z,\alpha)=mc^2\left[1+\left(\dfrac{Z\alpha}{n-\vert k\vert+\sqrt{k^2-Z^2\alpha^2}}\right)^2\right]^{-1/2}}

where n is a nonnegative integer number n=N+\vert k\vert and k^2=(j+\frac{1}{2})^2. Putting these into numbers, we get

HydrogenAtomSpectrumDiracEquationFirstLevelsor equivalently (I add comments from the slides)

HydrogenicAtomFirstLevelsDiracEq

If you plug Z=138 or more into the above equation from the Dirac spectrum, you obtain an imaginary value of the energy, and thus an oscillating (unbound) system! Therefore, the problem for atoms with high Z even persist taking the relativistic corrections! What is the solution? Nobody is sure. Greiner et al. suggest that taking into account the finite (extended) size of the nuclei, the problem is “solved” until Z\approx 172. Beyond, i.e., with Z>172, you can not be sure that quantum fluctuations of strong fields introduce vacuum pair creation effects such as they make the nuclei and thus atoms to be unstable at those high values of Z. Some people believe that the issues arise even before, around Z=150 or even that strong field effects can make atoms even below of Z=137 to be non-existent. That is why the search for superheavy elements (SHE) is interesting not only from the chemical viewpoint but also to the fundamental physics viewpoint: it challenges our understanding of Quantum Mechanics and Special Relativity (and their combination!!!!).

Is the feynmanium (Z=137) the last element? This hypothetical element and other superheavy elements (SHE) seem to hint the end of the Periodic Table. Is it true? Options:

1st. The feynmanium (Fy) or Untriseptrium (Uts) is the last element of the Periodic Table.

2nd. Greiner et al. limit around Z=172. References:

(i) B Fricke, W Greiner and J T Waber,Theor. Chim. Acta, 1971, 21, 235.

(ii)W Greiner and J Reinhardt, Quantum Electrodynamics, 4th edn (Springer, Berlin, 2009).

3rd. Other predictions of an end to the periodic table include Z = 128 (John Emsley) and Z = 155 (Albert Khazan). Even Seaborg, from his knowledge and prediction of an island of stability around Z,N= 126, 184,\ldots , left this question open to interpretation and experimental search!

4th. There is no end of the Periodic Table. According to Greiner et al. in fact, even when superheavy nuclei can produce a challenge for Quantum Mechanics and Special Relativity, indeed, since there is always electrons in the orbitals (a condition to an element to be a well-defined object), there is no end of The Periodic Table (even when there are probabilities to a positron-electron pair to be produced for a superheavy nuclei, the presence of electrons does not allow for it; but strong field effects are important there, and it should be great to produce these elements and to know their properties, both quantum and relativistic!). Therefore, it would be very, very interesting to test the superheavy element “zone” of the Periodic Table, since it is a place where (strong) quantum effects and (non-negligible) relativistic effects both matter. Then, if both theories are right, superheavy elements are a beautiful and wonderful arena to understand how to combine together the two greatest theories and (unfinished?) revolutions of the 20th century. What awesome role for the “elementary” and “fundamental” superheavy (composite) elements!

Probably, there is no limit to the number of (chemical) elements in our Universe… But we DO NOT KNOW!

In conclusion: what will happen for superheavy elements with >173 (or Z>126, 128, 137, etc.) remains unresolved with our current knowledge. And it is one of the last greatest mysteries in theoretical Chemistry!

More about the fine structure constant, the Sommerfeld corrections and the Dirac equation+QED (Quantum ElectroDynamics) corrections to the hydrogen spectrum, in slides (think it yourself!):

bohrsommIdea

sod1dirac04onelectronSpectrum

Final remarks (for experts only): Some comments about the self-adjointness of the Dirac equation for high value of Z in Coulombian potentials. It is a well known fact that the Dirac operator for the hydrogen problem is essentially self-adjoint if Z<119. Therefore, it is valid for all the currently known elements (circa 2013, June, every element in the Periodic Table, for the 7th period, has been created and then, we know that chemical elements do exist at least up to Z=118 and we have tried to search for superheavy elements beyond that Z with negative results until now). However, for 119\leq Z\leq 137 any “self-adjoint extension” requires a precise physical meaning. A good idea could be that the expectation value of every component of the Hamilton is finite in the selected basis. Indeed, the solution to the Coulombian potential for the hydrogenic atom using the Dirac equation makes use of hypergeometric functions that are well-posed for any Z\leq 137. If Z is greater than that critical value, we face the oscillating energy problem we discussed above. So, we have to consider the effect of the finite size of the nucleus and/or handle relativistic corrections more carefully. It is important to realize this and that we have to understand the main idea of all this crazy stuff. This means that the s states start to be destroyed above Z = 137, and that the p states begin being destroyed above Z = 274.  Note that this differs from the result of the Klein-Gordon equation, which predicts s states being destroyed above Z = 68 and p states destroyed above Z = 82. In summary, the superheavy elements are interesting because they challenge our knowledge of both Quantum Mechanics and Special Relativity. What a wonderful (final) fate for the chemical elements: the superheavy elements will test if the “marriage” between Quantum Mechanics or Special Relativity is going further or it ends into divorce!

Epilogue: What do you think about the following questions? This is a test for you, eager readers…

1) Is there an ultimate element?

2) Is there a theory of everything (TOE)?

3) Is there an ultimate chemical element?

4) Is there a single “ultimate” principle?

5) How many elements does the Periodic Table have?

6) Is the feynmanium the last element?

7) Are Quantum Mechanics/Special relativity consistent to each other?

8) Is Quantum Mechanics a fundamental and “ultimate” theory for atoms and molecules?

9) Is Special Relativity a fundamental and “ultimate” theory for “quick” particles?

10) Are the atomic shells and atomic structure completely explained by QM and SR?

11) Are the nuclei and their shell structure xompletely explained by QM and SR?

12) Do you think all this stuff is somehow important and relevant for Physics or Chemistry (or even for Mathematics)?

13) Will we find superheavy elements the next decade?

14) Will we find superheavy elements this century?

15) Will we find that there are some superheavy elements stable in the island of stability (Seaborg) with amazing properties and interesting applications?

16) Did you like/enjoy this post?

17) When you was a teenager, how many chemical elements did you know? How many chemical elements were known?

18) Did you learn/memorize the whole Periodic Table? In the case you did not, would you?

19) What is your favourite chemical element?

20) Did you know that every element in the 7th period of the Periodic table has been established to exist but th elements E113, E115,E117 and E118 are not named yet (circa, 2013, 30th June) and they keep their systematic (IUPAC) names ununtrium, ununpentium, ununseptium and ununoctium? By the way, the last named elements were the coperninicium (E112, Cn), the flerovium (Fl, E114) and the livermorium (Lv, E116)…

13502276-green-atom-electron-llustration-on-black-background

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LOG#099. Group theory(XIX).

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racah1

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Final post of this series!

The topics are the composition of different angular momenta and something called irreducible tensor operators (ITO).

Imagine some system with two “components”, e.g., two non identical particles. The corresponding angular momentum operators are:

J_1\cdot J_1, J_2\cdot J_2, J_1^z, J_2^z

The following operators are defined for the whole composite system:

J=J_1+J_2

J_z^T=J_z^1+J_z^2

J^2=(J_1+J_2)^2

These operators are well enough to treat the addition of angular momentum: the sum of two angular momentum operators is always decomposable. A good complete set of vectors can be built with the so-called tensor product:

\vert j_1j_2,m_1m_2\rangle =\vert j_1,m_1\rangle \otimes \vert j_2,m_2\rangle

This basis \vert j_1j_2,m_1m_2\rangle could NOT be an basis of eigenvectors for the total angular momentum operators J^2_T,J_z^T. However, these vector ARE simultaneous eigenvectors for the operators:

J_1\cdot J_1,J_2\cdot J_2,J_z^1, J_z^2

The eigenvalues are, respectively,

\hbar^2 j_1(j_1+1)

\hbar^2 j_2(j_2+1)

\hbar m_1

\hbar m_2

Examples of compositions of angular momentum operators are:

i) An electron in the hydrogen atom. You have J=l+s with l=r\times p. In this case, the invariant hamiltonian under the rotation group for this system must satisfy

\left[H,J\right]=0

ii) N particles without spin. The angular momentum is J=l_1+l_2+\cdots+l_N

iii) Two particles with spin in 3D. The total angular momentum is the sum of the orbital part plus the spin part, as we have already seen:

J=l+s=l_1+l_2+s_2+s_2

iv) Two particles with spin in 0D! The total angular momentum is equal to the spin angular momentum, that is,

J=S=s_1+s_2

In fact, the operators J^2,J_1\cdot J_1,J_2\cdot J_2,J_z commute to each other (they are said to be mutually compatible) and it shows that we can find a common set of eigenstates

\vert j_1j_2,JM\rangle

The eigenstates of J^2, J_z, with eigenvalues \hbar^2 J(J+1) and \hbar M are denoted by

\vert \Omega J,M\rangle

and where \Omega is an additional set of “quantum numbers”.

The space generated by \vert \Omega,JM\rangle, for a fixed number J, and 2J+1 vectors, -J\leq M\leq J, is an invariant subspace and it is also irreducible from the group theory viewpoint. That is, if we find a vector as a linear combination of eigenstates of a single particle, the remaining vectors can be built in the same way.

The vectors \vert j_1j_1,JM\rangle can be written as a linear combination of those \vert j_1j_1,m_1m_2\rangle. But the point is that, due to the fact that the first set of vectors are eigenstates of J_1\cdot J_1,J_2\cdot J_2, then we can restrict the search for linear combinations in the vector space with dimension (2j_1+1)(2j_2+1) formed by the vectors \vert j,m\rangle with fixed j_1,j_2 quantum numbers. The next theorem is fundamental:

Theorem (Addition of angular momentum in Quantum Mechanics).

Define two angular momentum operators J_1,J_2. Define the subspace, with (2j_1+1)(2j_2+1) dimensions and j_1\geq j_2, formed by the vectors

\vert j_1j_2,m_1m_2\rangle=\vert j_1,m_1\rangle \otimes \vert j_2,m_2\rangle

and where the (quantum) numbers j_1,j_2 are fixed, while the (quantum) numbers m_1,m_2 are “variable”. Let us also define the operators J=J_1+J_2 and J^2,J_z with respective eigenvalues J,M. Then:

(1) The only values that J can take in this subspace are

J\in E=\left\{ \vert j_1-j_2\vert, \vert j_1-j_2+1\vert,\ldots,j_1+j_2-1,j_1+j_2\right\}

(2) To every value of the number J corresponds one and only one set or series of 2J+1 common eigenvectors to J_z, and these eigenvector are denoted by \vert JM\rangle.

Some examples:

i) Two spin 1/2 particles. J=s_1+s_2. Then j=0,1 (in units of \hbar=1). Moreover, as a subspaces/total space:

E(1/2)\otimes E(1/2)=E(0)\oplus E(1)

ii) Orbital plus spin angular momentum of spin 1/2 particles. In this case, j=l+s. As subspaces/total space decomposition we have

E(l)\otimes E(1/2)=E(l+1/2)\oplus E(l-1/2) if l\neq 0

E(l)\otimes E(1/2)=E(1/2) if l=0

iii) Orbital plus two spin parts. j=l+s_1+s_2. Then, we have

E(l)\otimes E(1/2)\otimes E(1/2)=E(l)\otimes (E(0)+E(1))=E(l)\otimes E(0)\oplus E(l)\otimes E(1)

This last subspace sum is equal to E(l)\oplus E(l+1)\oplus E(l)\oplus E(l-1) if l\neq 0 and it is equal to E(0)\oplus E(1) if l=0.

In the case we have to add several (more than two) angular momentum operators, we habe the following general rule…

E=E(j_1)\otimes E(j_2)\otimes E(j_3)\otimes \ldots \otimes E(j_n)

We should perform the composition or addition taking invariant subspaces two by two and using the previous theorem. However, the theory of the addition of angular momentum in the case of more than 2 terms is more complicated. In fact, the number of times that a particular subspace appears could not be ONE. A simple example is provided by 2 non identical particles (2 nucleons, a proton and a neutron), and in this case the total angular momentum with respect to the center of masses and the spin angular momentum add to form j=l+s_1+s_2. Then

E(l)\otimes E(1/2)\otimes E(1/2)=E(l)\otimes (E(0)\oplus E(1))=E(l)\otimes E(0)\oplus E(l)\otimes E(1)

This subspace sum is equal to E(l)\oplus E(l+1)\oplus E(l)\oplus E(l-1) if l\neq 0 and E(0)\oplus E(1) if l=0.

Clebsch-Gordan coefficients.

We have studied two different set of vectors and bases of eigentstates

(1) \vert j_1j_2,m_1m_1\rangle, the common set of eigenstates to J_1^2,J_2^2,J_{z1},J_{z2}.

(2) \vert j_1j_1,JM\rangle, the common set of eigenstates to J_1^2,J_2^2,J^2,J_z.

We can relate both sets! The procedure is conceptually (but not analytically, sometimes) simple:

\displaystyle{\vert j_1j_2,JM\rangle=\sum_{m_1=-j_1}^{j_1}\sum_{m_2=-j_2;m_1+m_2=M}^{j_2}\vert j_1j_2,m_1m_2\rangle\langle j_1j_2,m_1m_2\vert JM\rangle}

The coefficients:

\boxed{\langle j_1j_2,m_1m_2\vert JM\rangle}

are called Clebsch-Gordan coefficients. Moreover, we can also expand the above vectors as follows

\displaystyle{\vert j_1j_2,m_1m_1\rangle=\sum_{J=\vert j_1-j_2\vert}^{J=j_1+j_2}\sum_{M=-J}^{M=J}\vert J M\rangle \langle J M\vert j_1j_2,m_1m_2\rangle}

and here the coefficients

\boxed{\langle J M\vert j_1j_2,m_1m_2\rangle}

are the inverse Clebsch-Gordan coefficients.

The Clebsch-Gordan coefficients have some beautiful features:

(1) The relative phases are not determined due to the phases in \vert j_1j_2,JM\rangle. They do depend on some coefficients c_m. For any value of J, the phase is determined by recurrence! It shows that

\langle j_1j_2,j_1 J-j_1\vert J,J\rangle \in \mathbb{R}^+

This convention implies that the Clebsch-Gordan (CG) coefficients are real numbers and they form an orthogonal matrix!

(2) Selection rules. The CG coefficients \langle j_1j_2,m_1m_2\vert J,M\rangle are necessarily null IF the following conditions are NOT satisfied:

i) M=m_1+m_2.

ii) \vert j_1-j_2\vert \leq J\leq j_1+j_2

iii) j_1+j_2+J\in \mathbb{Z}

The conditions i) and ii) are trivial. The condition iii) can be obtained from a 2\pi rotation to the previous conditions. The two factors that arise are:

R(2\pi)\vert j,m\rangle=(-1)^{2j}\vert j,m\rangle \leftrightarrow (-1)^{2J}=(-1)^{( j_1+j_2)}

(3) Orthogonality.

\displaystyle{\sum_{m_1=-j_1}^{j_1}\sum_{m_2=-j_2}^{j_2}\langle j_1j_2,m_1m_2\vert J,M\rangle\langle j_1j_2,m_1m_2\vert J' M'\rangle=\delta_{JJ'}\delta_{MM'}}

\displaystyle{\sum_{J=\vert j_1-j_2\vert}^{j_1+j_2}\sum_{M=-J}^{J}\langle j_1j_2,m_1m_2\vert J,M\rangle\langle j_1j_2,m'_1m'_2\vert J M\rangle=\delta_{m_1m'_1}\delta_{m_2m'_2}}

(4) Minimal/Maximal CG coefficients.

In the case J,M take their minimal/maximal values, the CG are equal to ONE. Check:

\vert j_1j_2,J=j_1+j_2, J=M\rangle=\vert j_1j_2,m_1=j_1,m_2=j_2\rangle

(5) Recurrence relations.

5A) First recurrence:

C_J=\sqrt{J(J+1)-M(M-1)}\langle m_1m_2\vert J,M-1\rangle=

=\sqrt{j_1(j_1+1)-m_1(m_1+1)}\langle m_1+1,m_2\vert J,M\rangle+

+\sqrt{j_2(j_2+1)-m_2(m_2+1)}\langle m_1,m_2+1\vert J,M\rangle

5B) Second recurrence:

C'_J=\sqrt{J(J+1)-M(M+1)}\langle m_1,m_2\vert J,M+1\rangle=

=\sqrt{j_1(j_1+1)-m_1(m_1-1)}\langle m_1-1,m_2\vert J,M\rangle+

+\sqrt{j_2(j_2+1)-m_2(m_2-1)}\langle m_1,m_2-1\vert J,M\rangle

These relations 5A) and 5B) are obtained if we apply the ladder operators J_\pm in both sides of the equation defining the CG coefficients and using that

J_\pm \vert JM\rangle=(J_{1\pm}+J_{2\pm})\vert JM\rangle

J_\pm \vert JM\rangle=\hbar \sqrt{J(J+1)-M(M\pm1)}\vert J,M\pm 1\rangle

Irreducible tensor operators. Wigner-Eckart theorem.

There are 4 important previous definitions for this topic:

1st. Irreducible Tensor Operator (ITO).

We define (2k+1) operators T^{(k)}_q, with q\in \left[-k,k\right] the standard components of an irreducible tensor operator (ITO) of order k, T^{(k)}, if these components transform according to the following rules

\displaystyle{U(\alpha,\beta,\gamma)T^{(k)}_qU^{-1}(\alpha,\beta,\gamma)=\sum_{q=-k}^{k}D^{(k)}_{qq'}(\alpha,\beta,\gamma)T^{(k)}_{q'}}

2nd. Irreducible Tensor Operator (II): commutators.

The (2k+1) operators T^{(k)}_q, q\in \left[-k,k\right], are the components of an irreducible tensor operator (ITO) of order k, T^{(k)}, if these components satisfy the commutation rules

\left[J_{\pm},T^{(k)}_q\right]=\hbar \sqrt{k(k+1)-q(q\pm 1)}T^{(k)}_{q\pm 1}

\left[ J_z,T^ {(k)}_q\right]=q\hbar T^{(k)}_q

The 1st and the 2nd definitions are completely equivalent, since the 2nd is the “infinitesimal” version of the 1st. The proof is trivial, by expansion of the operators in series and identification of the involved terms.

3rd. Scalar Operator (SO).

We say that S=T^0_0 is an scalar operator, if it is an ITO with order k=0. Equivalently,

U(\alpha,\beta,\gamma)SU^{-1}(\alpha,\beta,\gamma)=S

One simple way to express this result is the obvious and natural statement that scalar operators are rotationally invariant!

4th. Vector Operator (VO).

We say that V is a vector operator if

\displaystyle{U(\alpha,\beta,\gamma)V^{(1)}_qU^{-1}(\alpha,\beta,\gamma)=\sum_{q=-1}^1D^{(1)}_{qq'}(\alpha,\beta,\gamma) V^{(1)}_{q'}}

Equivalently, a vector operator is an ITO of order k=1.

The relation between the “standard components” (or “spherical”) and the “cartesian” (i.e.”rectangular”) components is defined by the equations:

V_1=-\dfrac{1}{2}(V_x+iV_y)

V_0=V_z

V_{-1}=\dfrac{1}{2}(V_x-iV_y)

In particular, for the position operator R=(r_1,r_0,r_{-1}), this yields

r_1=-\dfrac{1}{\sqrt{2}}(x+iy)

r_=z

r_{-1}=\dfrac{1}{\sqrt{2}}(x-iy)

Similarly, we can define the components for the momentum operator

p=(p_1,p_0,p_{-1}) or the angular momentum

L=(L_+,L_-,L_z)\equiv (L_1,L_0,L_{-1})

Now, two questions arise naturally:

1) Consider a set of (2k+1)(2k'+1) operators, built from ITO T^{(k)}_qT^{(k')}_{q'}. Are they ITO too? If not, can they be decomposed into ITO?

2) Consider a set of (2k+1)(2J+1) vectors, built from certain ITO, and a given base of eigenvalues for the angular momentum. Are these vectors an invariant set? Are these vectors an irreducible invariant set? If not, can these vectors be decomposed into irreducible, invariant sets for certain angular momentum operators?

Some theorems help to answer these important questions:

Theorem 1. Consider T^{(k_1)}_{q_1}, T^{(k_2)}_{q_2}, two irreducible tensor operators with q_1\in \left[-k_1,k_1\right] and q_2\in \left[-k_2,k_2\right]. Take k and q\in \left[-k,k\right] arbitrary. Define the quantity

\displaystyle{S^{(k)}_q\equiv \sum_{q_1=-k_1}^{k_1}\sum_{q_2=-k_2}^{k_2}T^{(k_1)}_{q_1}T^{(k_2)}_{q_2}\langle k_1 k_2,q_1 q_2\vert k q\rangle}

Then, the operators S^{(k)}_q are the “standard” components of certain ITO with order k. Moreover, we have, using the CG coefficientes:

\displaystyle{T^{(k_1)}_{q_1}T^{(k_2)}_{q_2}=\sum_{q_1=-k}^{k}\sum_{q_2=\vert k_1-k_2\vert}^{k_1+k_2}S^{(k)}_q\langle k q\vert k_1 k_2, q_1 q_2\rangle}

Theorem 2.  Let T^{(k)}_{q_1} be certain ITO and \vert j_2 m_1\rangle a set of (2j_2+1) eigenvectors of angular momentum. Let us define

\displaystyle{\vert \omega_{JM }\rangle =\sum_{q_1=-k_1}^{k_1}\sum_{m_2=-j_2}^{j_2}\left(T^{(k_1)}_{q_1}\vert j_2 m_2\rangle\right)\langle k_1 j_2, q_1 m_2\vert J M\rangle}

These vectors are eigenvectors of the TOTAL angular momentum:

J^2\vert \omega_{JM}\rangle =J(J+1)\hbar^2\vert \omega_{JM}\rangle

J_z\vert \omega_{JM}\rangle=M\hbar \vert \omega_{JM}\rangle

Note that, generally, these eigenstates are NOT normalized to the unit, but their moduli do not depend on M. Moreover, using the CG coefficients, we algo get

\displaystyle{T^{(k)}_{q_1}\vert j_2 m_2\rangle =\sum_{M=-J}^J\sum_{J=\vert k_1-j_2\vert}^{k_1+j_2}\vert J M\rangle \langle J M\vert k_1 j_2, q_1 m_2\rangle}

Theorem 3 (Wigner-Eckart theorem).

If T^{(k)}_q is an ITO and some bases for angular momentum are provided with \vert j_1 m_1\rangle and \vert j_2 m_2\rangle, then

\boxed{\langle j_2 m_2\vert T^{(k)}_{q}\vert j_1 m_1\rangle = \langle j_1 j_2, m_1 m_2\vert k q\rangle \dfrac{1}{2k+1}\langle j_2\vert \vert \mathbb{T}^{(k)}_q\vert\vert j_1\rangle}

and where the quantity

\boxed{\langle j_2\vert \vert \mathbb{T}^{(k)}_q\vert\vert j_1\rangle}

is called the reduced matrix element.  The proof of this theorem is based on (4) main steps:

1st. Use the (2k+1)(2j+1) vectors (varying q, m),  T^{(k)}_q\vert j m\rangle.

2nd. Form the linear combination/superposition

\displaystyle{\vert \omega_{JM}\rangle=\sum_{m,q}\left( T^{(k)}_q\vert j m\rangle\right)\langle k j, q m\vert J M\rangle}

and use the theorem (2) above to obtain

\langle J' M'\vert J M\rangle=\delta_{JJ}\delta_{MM}F(J)

3rd. Use the CG coefficients and their properties to rewrite the vectors in the base with J and M. Then, irrespectively the form of the ITO, we obtain

\displaystyle{T^{(k)}_q\vert j m\rangle=\sum_{J,M}\langle J M\vert k j, q m\rangle \vert \omega_{JM}\rangle}

4th. Project onto some other different state, we get the desired result

\displaystyle{\langle \omega_{J'M'}\vert T_q^{(k)}\vert j m\rangle=\sum_{J,M}\langle \omega_{J' M'}\vert \langle J M\vert k j, q m\rangle \vert \omega_{JM}\rangle}

or equivalently

\displaystyle{\langle \omega_{J'M'}\vert T_q^{(k)}\vert j m\rangle=\sum_{J,M}\langle \delta_{J' M'}\delta_{J'M'}F(J)\langle J M\vert k j,q m\rangle}

i.e.,

\displaystyle{\langle \omega_{J'M'}\vert T_q^{(k)}\vert j m\rangle=F(J)\langle J M\vert k j, q m\rangle}

Q.E.D.

The Wigner-Eckart theorem allows us to determine the so-called selection rules. If you have certain ITO and two bases \vert j_1,m_1\rangle and \vert j_2, m_2\rangle, then we can easily prove from the Wigner-Eckart theorem that

(1) If m_1-m_1\neq q, then \langle j_1 m_1\vert T^{(k)}_q\vert j_2 m_2\rangle=0.

(2) If \vert j_1-j_2\vert < k < j_1+j_2 does NOT hold, then \langle j_1 m_1\vert T^{(k)}_q\vert j_2 m_2\rangle=0.

These (selection) rules must be satisfied if some transitions are going to “occur”. There are some “superselection” rules in Quantum Mechanics, an important topic indeed, related to issues like this and symmetry, but this is not the thread where I am going to discuss it! So, stay tuned!

I wish you have enjoyed my basic lectures on group theory!!! Some day I will include more advanced topics, I promise, but you will have to wait with patience, a quality that every scientist should own! 🙂

See you in my next (special) blog post ( number 100!!!!!!!!).


LOG#098. Group theory(XVIII).

AngularMomentum

This and my next blog post are going to be the final posts in this group theory series. I will be covering some applications of group theory in Quantum Mechanics. More advanced applications of group theory, extra group theory stuff will be added in another series in the near future.

Angular momentum in Quantum Mechanics

Take a triplet of linear operators, J=(J_x,J_y,J_z). We say that these operators are angular momentum operators if they are “observable” or observable operators (i.e.,they are hermitian operators) and if they satisfy

\boxed{\displaystyle{\left[J_i,J_j\right]=i\hbar\sum_k \varepsilon_{ijk}J_k}}

that is

\left[J_x,J_y\right]=i\hbar J_z

\left[J_y,J_z\right]=i\hbar J_x

\left[J_z,J_x\right]=i\hbar J_y

The presence of an imaginary factor i makes compatible hermiticity and commutators for angular momentum. Note that if we choose antihermitian generators, the imaginary unit is absorbed in the above commutators.

We can determine the full angular momentum spectrum and many useful relations with only the above commutators, and that is why those relations are very important. Some interesting properties of angular momentum can be listed here:

1) If J_1,J_2 are two angular momentum operators, and they sastisfy the above commutators, and if in addition to it, we also have that \left[J_1,J_2\right]=0, then J_3=J_1+J_2 also satisfies the angular momentum commutators. That is, two independen angular momentum operators, if they commute to each other, imply that their sum also satisfy the angular momentum commutators.

2) In general, for any arbitrary and unitary vector \vec{n}=(n_x,n_y,n_z), we define the angular momentum in the direction of such a vector as

J_{\vec{n}}=n\cdot J=n_xJ_x+n_yJ_y+n_zJ_z

and for any 3 unitary and arbitrary vectos \vec{u},\vec{v},\vec{w} such as \vec{w}=\vec{u}\times\vec{v}, we have

\left[J_{\vec{u}},J_{\vec{u}}\right]=i\hbar J_{\vec{w}}

3) To every two vectors \vec{a},\vec{b} we also have

\left[\vec{a}\cdot\vec{J},\vec{b}\cdot\vec{J}\right]=i\hbar (\vec{a}\times \vec{b})\cdot \vec{J}

4) We define the so-called “ladder operators” J_+,J_- as follows. Take the angular momentum operator J and write

J_+=J_x+iJ_y

J_-=J_x-iJ_y

These operators are NOT hermitian, i.e, ladder operators are non-hermitian operators and they satisfy

J_+^+=J_-

J_-^+=J_+

5) Ladder operators verify some interesting commutators:

\left[J_x,J_+\right]=J_+

\left[J_x,J_-\right]=-J_-

\left[J_+,J_-\right]=2J_z

6) Commutators for the square of the angular momentum operator J^2=J_x^2+J_y^2+J_z^2

\left[J^2,J_k\right]=0,\forall k=x,y,z

\left[J^2,J_+\right]=\left[J^2,J_-\right]=0

7) Additional useful relations are:

J^2=\dfrac{1}{2}\left(J_+J_-+J_-J_+\right)+J_z^2

J_-J_+=J^2-J_z(J_z+I)

J_+J_-=J^2-J_z(J_z-I)

8) Positivity: the operators J_i^2,J_\pm,J_{+}J_{.},J_-J_+,J^2 are indefinite positive operators. It means that all their respective eigenvalues are positive numbers or zero. The proof is very simple

\langle \Psi \vert J_i^2\vert \Psi\rangle =\langle \Psi\vert J_iJ_i\vert\Psi\rangle =\langle \Psi\vert J^+_iJ_i\vert\Psi\rangle =\parallel J_i\vert\Psi\rangle\parallel\geq 0

In fact this also implies the positivity of J^2. For the remaining operators, it is trivial to derive that

\langle \Psi\vert J_-J_+\vert\Psi\rangle\geq 0

\langle \Psi\vert J_+J_-\vert \Psi\rangle\geq 0

since

\langle\Psi\vert J_-J_+\vert\Psi\rangle =\langle\Psi\vert J_+^+J_+\vert\Psi\rangle=\parallel J_+\vert\Psi\rangle\parallel\geq 0

\langle\Psi\vert J_+J_-\vert\Psi\rangle =\langle\Psi\vert J_-^+J_-\vert\Psi\rangle =\parallel J_-\vert\Psi\rangle\parallel\geq 0

The general spectrum of the operators J^2, J_z can be calculated in a completely general way. We have to search for general eigenvalues

J^2\vert\lambda,\mu\rangle=\lambda\vert\lambda,\mu\rangle

J_z\vert\lambda,\mu\rangle=\mu\vert\lambda,\mu\rangle

The general procedure is carried out in several well-defined steps:

1st. Taking into account the positivity of the above operators J^2,J_i^2,J_+J_-,J_-J_+, it means that there is some interesting options

A) J^2 is definite positive, i.e., \lambda \geq 0. Then, we can write for all practical purposes

\lambda=j(j+1)\hbar^2 with j\geq 0

Specifically, we define how the operators J^2  and J_z act onto the states, labeled by two parameters j,m and \vert j,m\rangle in the following way

J^2\vert j,m\rangle =j(j+1)\hbar^2\vert j,m\rangle

J_z\vert j,m\rangle =m\hbar \vert j,m\rangle

B) J_+,J_-,J_+J_- are positive, and we also have

J_-J_+\vert j,m\rangle =\left(J^2-J_z(J_z+I)\right)\vert j,m\rangle =(j-m)(j+m+1)\hbar^2\vert j,m\rangle

J_+,J_-\vert j,m\rangle =\left(J^2-J_z(J_z-I)\right)\vert j,m\rangle =(j+m)(j-m+1)\hbar^2\vert j,m\rangle

That means that the following quantities are positive

(j-m)(j+m+1)\geq 0 \leftrightarrow \begin{cases}j\geq m;\;\; j\geq -m-1\\ j\leq m;\;\; j\leq -m-1\end{cases}

(j+m)(j-m+1)\geq 0 \leftrightarrow \begin{cases}j\geq -m;\;\; j\geq m-1\\ j\leq -m;\;\; j\leq m-1\end{cases}

Therefore, we have deduced that

(1) \boxed{-j\leq m\leq j \leftrightarrow \vert m\vert \leq j} \forall j,m

(2) \boxed{m=\pm j\leftrightarrow \parallel J_\pm \vert j,m\rangle \parallel^2=0}

2nd. We realize that

(1) J_+\vert j,m\rangle is an eigenstate of J^2 and eigenvalue j(j+1). Check:

J^2\left(J_+\vert j,m\rangle \right)=J_+\left(J^2\vert j,m\rangle\right)=j(j+1)\hbar^2\left(J_+\vert j,m\rangle\right)

(2) J_+\vert j,m\rangle is an eigentstate of J_z and eigenvalue (m+1). Check (using \left[J_z,J_+\right]=J_+:

J_z\left(J_+\vert j,m\rangle \right)=J_+(J_z+I)\vert j,m\rangle =(m+1)\hbar \left(J_+\vert j,m\rangle\right)

(3) J_-\vert j,m\rangle is an eigenstate of J^2 with eigenvalue j(j+1). Check:

J^2\left(J_-\vert j,m\rangle \right)=J_-\left(J^2\vert j,m\rangle\right)=j(j+1)\hbar^2\left(J_-\vert j,m\rangle\right)

(4) J_-\vert j,m\rangle is an eigenvector of J_z and (m-1) is its eigenvalue. Check:

J_z\left(J_-\vert j,m\rangle \right)=J_-(J_z-I)\vert j,m\rangle =(m-1)\hbar \left(J_-\vert j,m\rangle\right)

Therefore, we have deduced the following conditions:

1) if m\neq j, equivalently if m\neq -j, then the eigenstates J_+\vert j,m\rangle, equivalently J_-\vert j,m\rangle, are the eigenstates of J^2,J_z. The same situation happens if we have vectors J_+^p\vert j,m\rangle or J_-^q\vert j,m\rangle for any p,q (positive integer numbers). Thus, the sucessive action of any of these two operators increases (decreases) the eigenvalue m in one unit.

2) If m=j or respectively if m\neq -j, the vectors J_+\vert j,m\rangle, respectively J_-\vert j,m\rangle are null vectors:

\exists p\in \mathbb{Z}/\left\{J_+^p\vert j,m\rangle\neq 0,J_+^{p+1}\vert j,m\rangle=0\right\}, m+p=j.

\exists q\in \mathbb{Z}/\left\{J_-^q\vert j,m\rangle\neq 0,J_-^{q+1}\vert j,m\rangle=0\right\}, m-q=-j.

If we begin by certain number m, we can build a series of eigenstates/eigenvectors and their respective eigenvalues

m-1,m-2,\ldots,m-q=-j

m+1,m+2,\ldots,m+q=j

So, then

m+p= j

m-q=-j

2m=q-p

2j=p+q

And thus, since p,q\in\mathbb{Z}, then j=k/2,k\in \mathbb{Z}. The number j can be integer or half-integer. The eigenvalues m have the same character but they can be only separated by one unit.

In summary:

(1) The only possible eigenvalues for J^2 are j(j+1) with j integer or half-integer.

(2) The only possible eigenvalues for J_z are integer numbers or half-integer numbers, i.e.,

\boxed{m=0,\pm \dfrac{1}{2},\pm 1,\pm\dfrac{3}{2},\pm 2,\ldots,\pm \infty}

(3) If \vert j,m\rangle is an eigenvector for J^2 and J_z, then

J^2\vert j,m\rangle=j(j+1)\hbar^2\vert j,m\rangle j=0,1,2,\ldots,

J_z\vert j,m\rangle=m\hbar\vert j,m\rangle -j\leq m\leq j

We have seen that, given an state \vert j,m\rangle, we can build a “complete set of eigenvectors” by sucessive application of ladder operators J_\pm! That is why ladder operators are so useful:

J_+\vert j,m\rangle, J_+^2\vert j,m\rangle, \ldots, J_-\vert j,m\rangle, J_-^2\vert j,m\rangle,\ldots

This list is a set of (2j+1) eigenvectors, all of them with the same quantum number j and different m. The relative phase of J^p_\pm\vert j,m\rangle is not determined. Writing

J_\pm\vert j,m\rangle =c_m\vert j,m+1\rangle

from the previous calculations we easily get that

\parallel J_+\vert j,m\rangle\parallel^2=(j-m)(j+m+1) \hbar^2\langle j,m\vert j,m\rangle

\vert c_m\vert^2=(j-m)(j+m+1)\hbar^2=j(j+1)\hbar^2-m(m+1)\hbar^2

\parallel J_-\vert j,m\rangle \parallel^2=(j+m)(j-m+1)\hbar^2\langle j,m\vert j,m\rangle

\vert c_m\vert^2=(j+m)(j-m+1)\hbar^2=j(j+1)\hbar^2-m(m-1)\hbar^2

The modulus of c_m is determined but its phase IS not. Remember that a complex phase is arbitrary and we can choose it arbitrarily. The usual convention is to define c_m real and positive, so

J_+\vert j,m\rangle =\hbar \sqrt{j(j+1)-m(m+1)}\vert j,m+1\rangle

J_-\vert j,m\rangle =\hbar \sqrt{j(j+1)-m(m-1)}\vert j,m-1\rangle

Invariant subspaces of angular momentum

If we addopt a concrete convention, the complete set of proper states/eigentates is:

B=\left\{ \vert j,-j\rangle ,\vert j,-j+1\rangle,\ldots,\vert j,0\rangle,\ldots,\vert j,j-1\rangle,\vert j,j\rangle\right\}

This set of eigenstates of angular momentum will be denoted by E(j), the proper invariant subspace of angular momentum operators J^2,J_z, with corresponding eigenvalues j(j+1).

The previously studied (above) features tell  us that this invariant subspace E(j) is:

a) Invariant with respect to the application of J^2,J_z, the operators J_x,J_y, and every function of them.

b) E(j) is an irreducible subspace in the sense we have studied in this thread: it has no invariant subspace itself!

The so-called matrix elements for angular momentum in these invariant subspaces can be obtained using the ladder opertors. We have

(1) \langle j,m\vert J^2\vert j',m'\rangle = j(j+1)\hbar^2 \delta_{jj'}\delta_{mm'}

(2) \langle j,m \vert J_z\vert j',m'\rangle =m\hbar \delta_{jj'}\delta_{mm'}

(3) \langle j,m\vert J_+\vert j',m'\rangle =\hbar \sqrt{j(j+1)-m'(m'+1)}\delta_{jj'}\delta_{m,m'+1}

(4) \langle j,m\vert J_-\vert j',m'\rangle =\hbar \sqrt{j(j+1)-m'(m'-1)}\delta_{jj'}\delta_{m,m'-1}

Example(I): Spin 0. (Scalar bosons)

If E(0)=\mbox{Span}\left\{ \vert 0\rangle\right\}

This case is trivial. There are no matrices for angular momentum. Well, there are…But they are 1\times 1 and they are all equal to cero. We have

J^2\vert 0\rangle =0\hbar^2=0(0+1)\hbar^2\cdot 1=0

J_x=J_y=J_z=J_+=J_-=0

Example(II): Spin 1/2. (Spinor fields)

Now, we have E(1/2)=\mbox{Span}\left\{\vert 1/2,-1/2\rangle,\vert 1/2,1/2\rangle\right\}

The angular momentum operators are given by multiples of the so-called Pauli matrices. In fact,

J^2=\dfrac{3}{4}\hbar^2\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}=\dfrac{3\hbar^2}{4}I=\dfrac{3\hbar^2}{4}\sigma_0

J_x=\dfrac{\hbar}{2}\begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix}=\dfrac{\hbar}{2}\sigma_x

J_y=\dfrac{\hbar}{2}\begin{pmatrix} 0 & -i\\ i & 0\end{pmatrix}=\dfrac{\hbar}{2}\sigma_y

J_z=\dfrac{\hbar}{2}\begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}=\dfrac{\hbar}{2}\sigma_z

and then J_k=\dfrac{\hbar}{2}S_k=\dfrac{\hbar}{2}\sigma_k and J^2=\dfrac{3}{4}\hbar^2I=\dfrac{3}{4}\hbar^2 \sigma_0.

The Pauli matrices have some beautiful properties, like

i) \sigma_x^2=\sigma_y^2=\sigma_z^2=1 The eigenvalues of these matrices are \pm 1.

ii) \sigma_x\sigma_y=i\sigma_x, \sigma_y\sigma_z=i\sigma_x, \sigma_z\sigma_x=i\sigma_y. This property is related to the fact that the Pauli matrices anticommute.

iii) \sigma_j\sigma_k=i\varepsilon_{jkl}\sigma_l+\delta_{jk}I

iv) With the “unit” vector \vec{n}=\left(\sin\theta\cos\psi,\sin\theta\sin\psi,\cos\theta\right), we get

\vec{n}\cdot \vec{S}=\begin{pmatrix} \cos\theta & e^{-i\psi}\sin\theta\\ e^{i\psi}\sin\theta & -\cos\theta\end{pmatrix}

This matrix has only two eigenvalues \pm 1 for every value of the parameters \theta,\psi. In fact the matrix \sigma_z+i\sigma_x has only an eigenvalue equal to zero, twice, and its eigenvector is:

e_1=\dfrac{1}{\sqrt{2}}\begin{pmatrix} -i\\ 1\end{pmatrix}

And \sigma_z-i\sigma_x has only an eigenvalue equal to zero twice and eigenvector

e_2=\dfrac{1}{\sqrt{2}}\begin{pmatrix} i\\ 1\end{pmatrix}

Example(III): Spin 1. (Bosonic vector fields)

In this case, we get E(1)=\mbox{Span}\left\{\vert 1,-1\rangle,\vert 1,0\rangle,\vert 1,1\rangle\right\}

The restriction to this subspace of the angular momentum operator gives us the following matrices:

J^2=2\hbar^2\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\end{pmatrix}=2\hbar^2I_{3\times 3}

J_x=\dfrac{\hbar}{\sqrt{2}}\begin{pmatrix} 0 & 1 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\end{pmatrix}

J_y=\dfrac{\hbar}{\sqrt{2}}\begin{pmatrix} 0 & -i & 0\\ i & 0 & -i\\ 0 & i & 0\end{pmatrix}

J_z=\hbar\begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0\\ 0 & 0 & -1\end{pmatrix}

and where

A) J^2_x,J^2_y,J^2_z are commutative matrices.

B) J_x^2+J_y^2+J_z^2=J^2=2\hbar^2I=1(1+1)\hbar^2

C) J_x^2+J_y^2 is a diagonarl matrix.

D) J_3+iJ_1=\hbar\begin{pmatrix} 1 & i/\sqrt{2} & 0\\ i/\sqrt{2} & 0 & i/\sqrt{2}\\ 0 & i/\sqrt{2} & -1\end{pmatrix} is a nilpotent matrix since (J_3+iJ_1)^2=0_{3\times 3} with 3 equal null eigenvalues and one single eigenvector

e_1=\dfrac{1}{2}\begin{pmatrix}-1\\ -i/\sqrt{2}\\ 1\end{pmatrix}

Example(IV): Spin 3/2. (Vector spinor fields)

In this case, we have E(3/2)=\mbox{Span}\left\{\vert 3/2,-3/2\rangle,\vert 3/2,-1/2\rangle,\vert 3/2,1/2\rangle,\vert 3/2,3/2\rangle\right\}

The spin-3/2 matrices can be obtained easily too. They are

J_x=\dfrac{\hbar}{2}\begin{pmatrix}0 & \sqrt{3} & 0 & 0\\ \sqrt{3} & 0 & 2 & 0\\ 0 & 2 & 0 & \sqrt{3}\\ 0 & 0 & \sqrt{3} & 0\end{pmatrix}

J_y=\hbar\begin{pmatrix}0 & -i\sqrt{3} & 0 & 0\\ i\sqrt{3} & 0 & -2i & 0\\ 0 & 2i & 0 & -i\sqrt{3}\\ 0 & 0 & i\sqrt{3} & 0\end{pmatrix}

J_z=\hbar\begin{pmatrix}3/2 & 0 & 0 & 0\\ 0 & 1/2 & 0 & 0\\ 0 & 0 & -1/2 & 0\\ 0 & 0 & 0 & -3/2\end{pmatrix}

J^2=J_x^2+J_y^2+J_z^2=\dfrac{15}{4}\hbar^2I_{4\times 4}=\dfrac{3(3+2)\hbar^2}{4}\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix}

The matrix

Z=J_z+iJ_x=\hbar\begin{pmatrix}3/2 & i\sqrt{3}/2 & 0 & 0\\ i\sqrt{3}/2 & 1/2 & i & 0\\ 0 & i & -1/2 & i\sqrt{3}/2\\ 0 & 0 & i\sqrt{3}/2 & -3/2\end{pmatrix}

is nonnormal since \left[Z,Z^+\right]\neq 0 and it is nilpotent in the sense that Z^4=(J_z+iJ_x)^4=0_{4\times 4} and its eigenvalues is zero four times. The only eigenvector is the vector

e_1=\dfrac{1}{\sqrt{8}}\begin{pmatrix}i\\ -\sqrt{3}\\ -i\sqrt{3}\\ 1\end{pmatrix}

This vector is “interesting” in the sense that it is “entangled” and it can not be rewritten as a tensor product of two \mathbb{C}^2. There is nice measure of entanglement, called tangle, that it shows to be nonzero for this state.

Example(V): Spin 2. (Bosonic tensor field with two indices)

In this case, the invariant subspace is formed by the vectors E(2)=\mbox{Span}\left\{\vert 2,-2\rangle,\vert 2,-1\rangle, \vert 2,0\rangle,\vert 2,1\rangle,\vert 2,2\rangle\right\}

For the spin-2 particle, the spin matrices are given by the following 5\times 5 matrices

J_x=\hbar\begin{pmatrix}0 & 1 & 0 & 0 & 0\\ 1 & 0 & \sqrt{6}/2 & 0 & 0\\ 0 & \sqrt{6}/2 & 0 & \sqrt{6}/2 & 0\\ 0 & 0 & \sqrt{6}/2 & 0 & 1\\ 0 & 0 & 0 & 1 & 0\end{pmatrix}

J_y=\hbar\begin{pmatrix}0 & -i & 0 & 0 & 0\\ i & 0 & -i\sqrt{6}/2 & 0 & 0\\ 0 & i\sqrt{6}/2 & 0 & -i\sqrt{6}/2 & 0\\ 0 & 0 & i\sqrt{6}/2 & 0 & -i\\ 0 & 0 & 0 & i & 0\end{pmatrix}

J_z=\hbar\begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & -2\end{pmatrix}

J^2=J_x^2+J_y^2+J_z^2=6\hbar^2I_{5\times 5}=6\hbar^2\begin{pmatrix}1 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1\end{pmatrix}

Moreover, the following matrix

Z=J_z+iJ_x=\hbar\begin{pmatrix}2 & i & 0 & 0 & 0\\ i & 1 & i\sqrt{6}{2} & 0 & 0\\ 0 & i\sqrt{6}/2 & 0 & i\sqrt{6}/2 & 0\\ 0 & 0 & i\sqrt{6}/2 & -1 & i\\ 0 & 0 & 0 & i & -2\end{pmatrix}

is nonnormal and nilpotent with Z^5=(J_z+iJ_x)^5=0_{5\times 5}. Moreover, it has 5 null eigenvalues and a single eigenvector

e_1=\begin{pmatrix}1\\ 2i\\ -\sqrt{6}\\ -2i\\ 1\end{pmatrix}

We see that the spin matrices in 3D satisfy for general s:

i) J_x^2+J_y^2+J_z^2=s(s+1)I_{2s+1} \forall s.

ii) The ladder operators for spin s have the following matrix representation:

J_+=\begin{pmatrix} 0 & \sqrt{2s} & 0 & 0 & \ldots & 0\\ 0 & 0 & \sqrt{2(2s-1)} & 0 & \ldots & 0\\ 0 & 0 & 0 & \sqrt{3(2s-2)} & \ldots & 0\\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots\\ 0 & 0 & 0 & 0 & \ldots & \sqrt{2s}\\ 0 & 0 & 0 & 0 & \ldots & 0\end{pmatrix}

Moreover, J_-=J_+^+ in the matrix sense and the above matrix could even be extended to the case of  a non-bounded spin particle. In that case the above matrix would become an infinite matrix! In the same way, for spin s, we also get that Z=J_z+iJ_1 would be (2s+1)-nilpotent and it would own only a single eigenvector with Z having (2s+1) null eigenvalues. The single eigenvector can be calculated quickly.

Example(VI): Rotations and spinors.

We are going to pay attention to the case of spin 1/2 and work out its relation with ordinary rotations and the concept of spinors.

Given the above rotation matrices for spin 1/2 in terms of Pauli matrices, we can use the following matrix property: if M is a matrix that satisfies A^2=I, then we can write that

e^{iAt}=\cos t I+i\sin t A

Then, we write

e^{i\sigma_x t}=\cos t I+i\sin t\sigma_x=\begin{pmatrix} \cos t & i\sin t\\ i\sin t & \cos t\end{pmatrix}

e^{i\sigma_y t}=\cos t I+i\sin t\sigma_y=\begin{pmatrix} \cos t & \sin t\\ -\sin t & \cos t\end{pmatrix}

e^{i\sigma_z t}=\cos t I+i\sin t\sigma_z=\begin{pmatrix} \cos t+i\sin t & 0\\ 0 & \cos t-i\sin t\end{pmatrix}=\begin{pmatrix}e^{it} & 0 \\ 0 & e^{-it}\end{pmatrix}

From these equations and definitions, we can get the rotations around the 3 coordinate planes (it corresponds to the so-called Cayley-Hamilton parametrization).

a) Around the plance (XY), with the Z axis as the rotatin axis, we have

R_z(\theta)=\exp\left(-i\theta \dfrac{J_z}{\hbar}\right)=\exp\left(-i\dfrac{\theta\sigma_z}{2}\right)=\begin{pmatrix}e^{-i\frac{\theta}{2}} & 0\\ 0 & e^{-i\frac{\theta}{2}}\end{pmatrix}

b) Two sucessive rotations yield

R(\theta,\phi)=\exp\left(-i\dfrac{\phi\sigma_z}{2}\right)\exp\left(-i\dfrac{\theta\sigma_y}{2}\right)=\begin{pmatrix}e^{-i\frac{\phi}{2}}\cos\frac{\theta}{2} & e^{i\frac{\phi}{2}}\sin\frac{\theta}{2}\\e^{-i\frac{\phi}{2}}\sin\frac{\theta}{2} & e^{i\frac{\phi}{2}}\cos\frac{\theta}{2}\end{pmatrix}

Remark: R_z(2\pi)=-I!!!!!!!

Remark(II):   R(\phi=0,\theta)=\begin{pmatrix}\cos\frac{\theta}{2} & -\sin\frac{\theta}{2}\\ \sin\frac{\theta}{2} & \cos\frac{\theta}{2}\end{pmatrix}

This matrix has a strange 4\pi periodicity! That is, rotations with angle \beta=2\pi don’t recover the identity but minus the identity matrix!

Imagine a system or particle with spin 1/2, such that the wavefunction is \Psi:

\Psi=\begin{pmatrix}\Psi_1\\ \Psi_2\end{pmatrix}

If we apply a 2\pi rotation to this object, something that we call “spinor”, we naively would expect that the system would be invariant but instead of it, we have

R(2\pi)\Psi=-\Psi

The norm or length is conserved, though, since

\vert \Psi\vert^2=\vert\Psi_1\vert^2+\vert\Psi_2\vert^2

These objects (spinors) own this feature as distinctive character. And it can be generalized to any value of j. In particular:

A) If j is an integer number, then R(2\pi)=I. This is the case of “bosons”/”force carriers”.

B) If j is half-integer, then R(2\pi)=-I!!!!!!!. This is the case of “fermions”/”matter fields”.

Rotation matrices and the subspaces E(j).

We learned that angular momentum operators J are the infinitesimal generators of “generalized” rotations (including those associated to the “internal spin variables”). A theorem, due to Euler, says that every rotation matrix can be written as a function of three angles. However, in Quantum Mechanics, we can choose an alternative representation given by:

U(\alpha,\beta,\gamma)=\exp\left(-\alpha\dfrac{iJ_x}{\hbar}\right)\exp\left(-\beta\dfrac{iJ_y}{\hbar}\right)\exp\left(-\gamma\dfrac{iJ_z}{\hbar}\right)

Given a representation of J in the subspace E(j), we obtain matrices U(\alpha,\beta,\gamma) as we have seen above, and these matrices have the same dimension that those of the irreducible representation in the subspace E(j). There is a general procedure and parametrization of these rotation matrices for any value of j. Using a basis of eigenvectors in E(j):

\boxed{\langle j',m'\vert U\vert j,m\rangle =D^{(j)}_{m'm}\delta_{jj'}}

and where we have defined the so-called Wigner coefficients

D^{(j)}_{m'm}(\alpha,\beta,\gamma)=\langle j'm'\vert e^{-\alpha\frac{iJ_z}{\hbar}}e^{-\beta\frac{iJ_y}{\hbar}}e^{-\gamma\frac{iJ_x}{\hbar}}\vert jm\rangle\equiv e^{-i(\alpha m'+\beta m)}d^{(j)}_{m'm}

The reduced matrix only depends on one single angle (it was firstly calculated by Wigner in some specific cases):

\boxed{d^{(j)}_{m' m}(\beta)=\langle j'm'\vert \exp\left(-\beta \dfrac{i}{\hbar}J_y\right)\vert jm\rangle}

Generally, we will find the rotation matrices when we “act” with some rotation operator onto the eigenstates of angular momentum, mathematically speaking:

\boxed{\displaystyle{U(\alpha,\beta,\gamma)\vert j,m\rangle=\sum_{j',m'}\vert j',m'\rangle \langle j',m'\vert U\vert j,m\rangle=\sum_{m'}D^{(j)}_{m'm}\vert j,m'\rangle}}

See you in my final blog post about  basic group theory!


LOG#050. Why riemannium?

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 s=\sigma+i\tau defined by the next equation:

\boxed{\displaystyle{\zeta (s)=\sum_{n=1}^{\infty}n^{-s}=\sum_{n=1}^{\infty}\dfrac{1}{n^s}=\prod_{p=2}^{\infty}\dfrac{1}{1-p^{-s}}=\prod_{p,\; prime}\dfrac{1}{1-p^{-s}}}}

or

\boxed{\displaystyle{\zeta (s)=\dfrac{1}{1-2^{-s}}\dfrac{1}{1-3^{-s}}\ldots\dfrac{1}{1-137^{-s}}\ldots}}

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. \zeta (1)=\infty 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:

\boxed{\pi^{-\frac{s}{2}}\Gamma \left(\dfrac{s}{2}\right)\zeta (s)=\pi^{-\frac{(1-s)}{2}}\Gamma \left(\dfrac{1-s}{2}\right)\zeta (1-s)}

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

\boxed{\xi (s)=\xi (1-s)}

where we have defined

\xi (s)=\pi^{-\frac{s}{2}}\Gamma \left(\dfrac{s}{2}\right)\zeta (s)

Riemann zeta values are an example of beautiful Mathematics. From \displaystyle{\zeta (s)=\sum_{n=1}^{\infty}n^{-s}}, then we have:

1) \zeta (0)=1+1+\ldots=-\dfrac{1}{2}.

2) \zeta (1)=1+\dfrac{1}{2}+\dfrac{1}{3}+\ldots =\infty. The harmonic series is divergent.

3) \zeta (2)=1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+\ldots =\dfrac{\pi^2}{6}\approx 1.645. The famous Euler result.

4) \zeta (3)=1+\dfrac{1}{2^3}+\dfrac{1}{3^3}+\ldots \approx 1.202. And odd zeta value called Apery’s constant that we do not know yet how to express in terms of irrational numbers.

5) \zeta (4)=\dfrac{\pi^4}{90}\approx 1.0823.

6) \zeta (-2n)=-\dfrac{\pi^{-n}}{2\Gamma (-n+1)}=0,\;\;\forall n=1,2,\ldots ,\infty. Trivial zeroes of zeta.

7) \zeta (2n)=\dfrac{(-1)^{n+1}(2\pi)^{2n}B_{2n}}{2(2n)!}\;\;\forall n=1,2,\ldots ,\infty, where B_{2n} are the Bernoulli numbers. The first 13 Bernoulli numbers are:

B_0=1, B_1=-\dfrac{1}{2}, B_2=\dfrac{1}{6}, B_3=0, B_4=-\dfrac{1}{30}, B_5=0, B_6=\dfrac{1}{42}

B_7=0, B_8=-\dfrac{1}{30}, B_9=0, B_{10}=\dfrac{5}{66}, B_{11}=0, B_{12}=-\dfrac{691}{2730}, B_{13}=0

8) We note that B_{2n+1}=0,\;\; \forall n\geq 1.

9) \zeta (-2n+1)=-\dfrac{B_{2n}}{2n}, \;\; \forall n=1,2,\ldots ,\infty.

For instance, \zeta (-1)=-\dfrac{1}{12}=1+2+3+\ldots, \zeta (-3)=\dfrac{1}{120}, and \zeta (-5)=-\dfrac{1}{252}. Indeed, \zeta (-1) arises in string theory trying to renormalize the vacuum energy of an infinite number of harmonic oscillators. The result in the bosonic string is \dfrac{2}{2-D}. 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

\sum \vert n\vert^3=-\dfrac{1}{60}

10) \zeta (\infty)=1. The Riemann zeta value at the infinity is equal to the unit.

11) The derivative of the zeta function is \displaystyle{\zeta '(s)=-\sum_{n=1}^{\infty}\dfrac{\log n}{n^s}}. Particularly important of this derivative are:

\displaystyle{\zeta '(0)=-\sum_{n=1}^\infty \log n=-\log \prod_{n=1}^\infty n=\zeta (0)\log (2\pi)=-\dfrac{1}{2}\log (2\pi)=-\log \sqrt{2\pi}=\log \dfrac{1}{\sqrt{2\pi}}}

or \zeta '(0)=\log \sqrt{\dfrac{1}{2\pi}}

This allow us to define the factorial of the infinity as

\displaystyle{\infty !=\prod_{n=1}^{\infty}n=1\cdot 2\cdots \infty=e^{-\zeta '(0)}=\sqrt{2\pi}}

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

\det _\zeta (A)=a_1\cdot a_2\cdots=\exp \left(-\zeta_A '(0)\right), with \displaystyle{\zeta _A (s)=\sum_{n=1}^\infty \dfrac{1}{a_n^s}}

12) \zeta (1+\varepsilon )=\dfrac{1}{\varepsilon}+\gamma_E +\mathcal{O} (\varepsilon ). This is a result used by theoretical physicists in dimensional renormalization/regularization. \gamma_E\approx 0.577 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:

\boxed{\displaystyle{\eta (s)=\sum_{n=1}^\infty \dfrac{(-1)^{n+1}}{n^s}=\left(1-2^{1-s}\right)\zeta (s)}}

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

\eta (0)=-\zeta (0)=\dfrac{1}{2} \eta (1)=\log 2 \eta (2)=\dfrac{1}{2}\zeta (2)=\dfrac{\pi^2}{12}

\eta (3)=\dfrac{3}{4}\zeta (3)\approx \dfrac{3}{4}(1.202) \eta (4)=\dfrac{7}{8}\zeta (4)=\dfrac{7}{8}\left(\dfrac{\pi^4}{90}\right)

Remark(I): \zeta(2) is important in the physics realm, since the spectrum of the hydrogen atom has the following aspect

E(n)=-\dfrac{K}{n^2}

and the Balmer formula is, as every physicist knows

\Delta E(n,m)=K\left(\dfrac{1}{n^2}-\dfrac{1}{m^2}\right)

Remark (II): The fact that \zeta (2) 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 \zeta (2) is finite.

Remark(III): What about an “atom”/system with spectrum E(n)=\kappa n^{-s}? If s=2, we do know that is the case of the Kepler problem. Moreover, it is easy to observe that s=-1 corresponds to tha harmonic oscillator, i.e., E(n)=\hbar \omega n. We also know that s=-2 is the infinite potential well. So the question is, what about a n^{-3} spectrum and so on?

In summary, does the following spectrum

\boxed{E=\mathbb{K}\dfrac{1}{n^{s}}}

with energy separation/splitting

\boxed{\Delta E(n,m;s)=\mathbb{K}\left(\dfrac{1}{n^{s}}-\dfrac{1}{m^{s}}\right)}

exist in Nature for some physical system beyond the infinite potential well, the harmonic oscillator or the hydrogen atom, where s=-2, s=-1 and s=2 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 s=-2n,\;\forall n=1,2,\ldots,\infty have real part equal to 1/2. In other words, the equation or feynmanity has only the next solutions:

\boxed{\mbox{RH:}\;\;\zeta (s)=0\leftrightarrow \begin{cases} s_n=-2n,\;\forall n=1,\ldots,\infty\;\;\mbox{Trivial zeroes}\\ s_n=\dfrac{1}{2}\pm i\lambda_n, \;\;\forall n=1,\ldots,\infty \;\;\mbox{Non-trivial zeroes}\end{cases}}

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

\boxed{\mbox{PRH:}\;\;\zeta (s)=0\leftrightarrow \begin{cases} s_n=-2n,\;\forall n=1,\ldots,\infty\;\;\mbox{Trivial zeroes}\\ s_n=\dfrac{1\pm i\overline{\lambda}_n}{2}, \;\;\forall n=1,\ldots,\infty \;\;\mbox{Non-trivial zeroes}\end{cases}}

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  \forall x\geq 2 integer numbers, and non-trivial Riemann zeroes in the complex (upper) half-plane with \tau>0:

\boxed{\displaystyle{\pi (x)+\sum_{n=2}^\infty \dfrac{\pi \left( x^{1/n}\right)}{n}=\text{Li} (x)-\sum_{\lambda =\sigma+i\tau }\left(\text{Li}(x^\lambda)+\text{Li}\left( x^{1-\lambda}\right)\right)+\int_x^\infty\dfrac{du}{u(u^2-1)\ln u}-\ln 2}}

and where \pi (x) is the celebrated Gauss prime number counting function, i.e., \pi (x) 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 \zeta (s), the Euler product on one side and the Hadamard product on the other side.

The function \text{Li} (x), sometimes written as \text{li} (x), is the logarithmic integral

\displaystyle{\text{Li} (x) =\text{li} (x)= \int_2^x\dfrac{du}{\ln x}}

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

\pi (x)=\pi_0 (x)+\pi_1 (x)=\pi_{\mbox{smooth}}+\pi_{\mbox{osc-chaotic}}

where

\displaystyle{\pi_0 (x)=\sum_{n=1}^\infty\dfrac{\mu (n)}{n}\left[\mbox{Li}(x^{1/n})-\sum_{k=1}^\infty\mbox{Li}(x^{-2k/n})\right]}

and

\displaystyle{\pi_1 (x)=-2\mbox{Re}\sum_{n=1}^\infty\dfrac{\mu (n)}{n}\sum_{\alpha=1}^\infty\mbox{Li}(x^{(\sigma_\alpha+i\tau_\alpha)/n})}

For large values of x, we have the asymptotics

\pi_0 (x)\approx \mbox{Li} (x)

and

\displaystyle{\pi_1 (x)\approx -\dfrac{2}{\ln x}\sum_{\alpha=1}^\infty\dfrac{x^{\sigma_\alpha}}{\sigma_\alpha^2+\tau_\alpha^2}\left(\sigma_\alpha\cos (\tau_\alpha \ln x)+\tau_\alpha \sin (\tau_\alpha \ln x)\right)}

Remark: Please, don’t confuse the logarithmic integral with the polylogarithm function \text{Li}_x (s).

Gauss also conjectured that

\pi (x)\sim \text{Li} (x)

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 T of the non-trivial zeros

\dfrac{1}{2}+iT

of the Riemann zeta function corresponded to eigenvalues of an unbounded and unknown self adjoint operator \hat{T}. 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:

R_2(x)=1-\left(\dfrac{\sin \pi x}{\pi x}\right)^2

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 H is actually some kind of quantization of the classical Hamiltonian XP where P is the canonical momentum associated with the position operator X. If that Berry-Keating conjecture is true. The simplest Hermitian operator corresponding to XP is

H = \dfrac1{2} (xp+px) = - i \left( x \dfrac{\mathrm{d}}{\mathrm{d} x} + \dfrac{1}{2} \right)

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 f(nx)=f(x) for integer n may help to get the correct asymptotic results valid for big n. That it, in the large n we should obtain

s_n=\dfrac{1}{2} + i \dfrac{ 2\pi n}{\log n}

 

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 H, 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 \infty-dimensional regularized determinant is defined through the Riemann zeta function as follows:

\displaystyle{\det _\zeta \mathcal{P}=e^{-\zeta_{\mathcal{P}}^{'}(0)}}

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 \hbar =1):

f(p)_{MB}=\dfrac{1}{(2\pi)^3}e^{\frac{\mu -E}{k_BT}}

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

\displaystyle{n=\dfrac{1}{(2\pi)^3}\int d^3p e^{\frac{\mu -E}{k_BT}}}

\displaystyle{\varepsilon =\dfrac{1}{(2\pi)^3}\int d^3p Ee^{\frac{\mu -E}{k_BT}}}

\displaystyle{\varepsilon =\dfrac{1}{(2\pi)^3}\int d^3p \dfrac{1}{3}\dfrac{\vert\mathbf{p}\vert^2}{E}e^{\frac{\mu -E}{k_BT}}}

We can introduce the dimensionless variables $late z=\dfrac{mc^2}{k_BT}$, \tau =\dfrac{E}{k_BT}=\dfrac{\sqrt{p^2+m^2c^4}}{k_BT}. In this way,

\vert p\vert=\dfrac{k_BT}{c}\sqrt{\tau^2-z^2}

c^2\vert\mathbf{p}\vert d\vert \mathbf{p}\vert=k_B^2T^2\tau d\tau

c^3\vert\mathbf{p}\vert^2d\vert\mathbf{p}\vert=k_B^3T^3\tau\sqrt{\tau^2-z^2}d\tau

With these definitions, the particle density becomes

\displaystyle{n=\dfrac{4\pi k_B^3T^3}{(2\pi)^3}e^{\frac{\mu}{k_BT}}\int_z^\infty d\tau (\tau^2-z^2)^{1/2}\tau e^{-\tau}}

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

K_n (z)=\dfrac{2^nn!}{(2n)!z^n}\int_z^\infty d\tau (\tau^2-z^2)^{n-1/2}e^{-\tau} or equivalently

K_n (z)=\dfrac{2^{n-1}(n-1)!}{(2n-2)!z^n}\int_z^\infty d\tau (\tau^2-z^2)^{n-3/2}\tau e^{-\tau}

K_{n+1} (z)=\dfrac{2nK_n (z)}{z}+K_{n-1} (z)

\displaystyle{K_2 (x)=\dfrac{1}{z^2}\int_z^\infty (\tau^2-z^2)^{1/2}\tau e^{-\tau}d\tau}

And thus, we have the next results (setting c=1 for simplicity):

\mbox{Particle number density}\equiv n=\dfrac{N}{V}=\dfrac{k_B^3T^3}{2\pi^2}z^2K_2 (z)=\dfrac{k_B^3T^3}{2\pi^2}\left(\dfrac{m}{k_BT}\right)^2K_2\left(\dfrac{m}{k_BT}\right)e^{\frac{\mu}{k_BT}}

\mbox{Energy density}\equiv\varepsilon=\dfrac{k_B^4T^4}{2\pi^2}\left[ 3\left(\dfrac{m}{k_BT}\right)^2K_2\left(\dfrac{m}{k_BT}\right)+\left(\dfrac{m}{k_BT}\right)^3K_1\left(\dfrac{m}{k_BT}\right)\right]e^{\frac{\mu}{k_BT}}

\mbox{Pressure}\equiv P=\dfrac{k_B^4T^4}{2\pi^2}\left(\dfrac{m}{k_BT}\right)^2K_2\left(\dfrac{m}{k_BT}\right)e^{\frac{\mu}{k_BT}}

Even entropy density is easiy to compute:

\mbox{Entropy density}\equiv s=\dfrac{m^3}{2\pi^2}e^{\frac{\mu}{k_BT}}\left[ K_1\left(\dfrac{m}{k_BT}\right)+\dfrac{4k_BT-\mu}{m}K_2\left(\dfrac{m}{k_BT}\right)\right]

These results can be simplified in some limit cases. For instance, in the massless limit z=m/k_BT\rightarrow 0. Moreover, we also know that \displaystyle{\lim_{z\rightarrow 0}z^nK_n (z)=2^{n-1}(n-1)!}. In such a case, we obtain:

n\approx \dfrac{k_B^3T^3}{\pi^2}e^{\frac{\mu}{k_BT}}

\varepsilon \approx \dfrac{3k_B^4T^4}{\pi^2}e^{\frac{\mu}{k_BT}}

P\approx \dfrac{k_B^4T^4}{\pi^2}e^{\frac{\mu}{k_BT}}

We note that \varepsilon=3P in this massless limit.

Remark (I): In the massless limit, and whenever there is no degeneracy, \varepsilon =3P holds.

Remark (II): If there is a quantum degeneracy in the energy levels, i.e., if g\neq 1, we must include an extra factor of g_j=2j+1 for massive particles of spin j. For massless photons with helicity, there is a g=2 degeneracy.

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

\mbox{Pressure}\equiv P=\dfrac{s}{d}\dfrac{U}{V}=\dfrac{s}{d}\varepsilon

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

p^1 (5)=7 because 5=4+1=3+2=3+1+1=2+2+1=2+1+1+1=1+1+1+1+1

p^2 (5)=2 because 5=2^2+1^2=1^2+1^2+1^2+1^2+1^2

If E_n=n^s with n\geq 1 and s>0, then x=e^{-\beta} and the partition function is

\displaystyle{Z=\prod_{k}\left( 1+e^{\frac{\mu-E}{k_BT}}\right)}

We will see later that \displaystyle{\sum_{N=0}^\infty x^N=\begin{cases}1+x, FD \\ \dfrac{1}{1-x}, BE\end{cases}}

with \mu =0 is nothing but the generatin function of the partitions p^s (n)

\displaystyle{Z(x=e^{-\beta})=\prod_{n=1}^\infty \dfrac{1}{1-x^{n^s}}=\sum_{n=1}^\infty p^s (n) x^n\approx \int_1^\infty dn p^s (n) e^{-\beta n}}

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

p(n) \approx \dfrac{1}{4\sqrt{3}N}e^{\pi\sqrt{2N/3}}

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

\displaystyle{\int_0^\infty dx \dfrac{x^{n-1}}{e^x\mp 1}=\begin{cases}\Gamma (n) \zeta (n), \;\; BE\\ \Gamma (n)\eta (n)=\Gamma (n) (1-2^{1-n})\zeta (n),\;\; FD\end{cases}}

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

\displaystyle{f(p)=\dfrac{1}{(2\pi)^3}\sum_{n=1}^{\infty}(\mp)^{n+1}e^{-n\frac{(E-\mu)}{k_BT}}}

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

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

\displaystyle{n=\dfrac{1}{(2\pi)^3}\int d^3p \left(e^{\frac{\mu-E}{k_BT}}-1\right)^{-1}=\dfrac{1}{(2\pi)^3}\int d^3p \sum_{n=1}^{\infty}(+1)^{n+1}e^{\frac{n\mu-nE}{k_BT}}}

Introducing a scaled temperature T'=T/n, we get

\displaystyle{n=\sum_{n=1}^{\infty}\left[\dfrac{1}{(2\pi)^3}\int d^3p e^{\frac{n\mu-nE}{k_BT'}}\right]=\sum_{n=1}^{\infty}\dfrac{k_B^3T^3}{2\pi^2}\dfrac{1}{n^3}\left(\dfrac{nm}{k_BT}\right)^2K_2\left(\dfrac{nm}{k_BT}\right)e^{\frac{n\mu}{k_BT}}}

\displaystyle{\varepsilon=\sum_{n=1}^{\infty}\dfrac{k_B^4T^4}{n^4(2\pi^2)}\left[3\left(\dfrac{nm}{k_BT}\right)^2K_2\left(\dfrac{nm}{k_BT}\right)+\left(\dfrac{nm}{k_BT}\right)^3K_1\left(\dfrac{nm}{k_BT}\right)\right]e^{\frac{n\mu}{k_BT}}}

\displaystyle{P=\sum_{n=1}^{\infty}\dfrac{k_B^4T^4}{n^4(2\pi^2)}\left(\dfrac{nm}{k_BT}\right)^2K_2\left(\dfrac{nm}{k_BT}\right)e^{\frac{n\mu}{k_BT}}}

Again, we can study a particularly simple case: the massless limit m\rightarrow 0 with \mu\rightarrow 0. In this case, we get:

\displaystyle{n=\dfrac{k_B^3T^3}{\pi^2}\sum_{n=1}^\infty \dfrac{1}{n^3}=\dfrac{k_B^3T^3}{\pi^2}\zeta (3)\approx 1.202\dfrac{k_B^3T^3}{\pi^2}}

\displaystyle{\varepsilon=\sum_{n=1}^\infty\dfrac{3(k_BT)^4}{\pi^2}\dfrac{1}{n^4}=\dfrac{3(k_BT)^4\zeta (4)}{\pi^2}=\dfrac{\pi^2}{30}(k_BT)^4}

\displaystyle{P=\sum_{n=1}^\infty\dfrac{(k_BT)^4}{\pi^2}\dfrac{1}{n^4}=\dfrac{(k_BT)^4\zeta (4)}{\pi^2}=\dfrac{\pi^2(k_BT)^4}{90}}

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

\displaystyle{n=\sum_{n=1}^\infty (-1)^{n+1}\dfrac{(k_BT)^3}{2\pi^2n^3}\left(\dfrac{nm}{k_BT}\right)^2K_2\left(\dfrac{nm}{k_BT}\right)e^{\frac{\mu n}{k_BT}}}

\displaystyle{\varepsilon= \sum_{n=1}^\infty (-1)^{n+1}\dfrac{(k_BT)^4}{2\pi^2}\left[3\left(\dfrac{nm}{k_BT}\right)^2K_2\left(\dfrac{nm}{k_BT}\right)+\left(\dfrac{nm}{k_BT}\right)^3K_1\left(\dfrac{nm}{k_BT}\right)\right]e^{\frac{\mu n}{k_BT}}}

\displaystyle{P=\sum_{n=1}^\infty (-1)^{n+1}\dfrac{(k_BT)^4}{2\pi^2}\dfrac{1}{n^4}\left(\dfrac{nm}{k_BT}\right)^2K_2\left(\dfrac{nm}{k_BT}\right)e^{\frac{n\mu}{k_BT}}}

and again the massless limit m=0 and \mu\rightarrow 0 provide

\displaystyle{n\approx \dfrac{(k_BT)^3}{\pi^2}\sum_{n=1}^\infty (-1)^{n+1}\dfrac{1}{n^3}=\dfrac{(k_BT)^3}{\pi^2}\eta (3)=\dfrac{(k_BT)^3}{\pi^2}\left(\dfrac{3}{4}\right)\zeta (3)}

\displaystyle{\varepsilon\approx \dfrac{3(k_BT)^4}{\pi^2}\sum_{n=1}^\infty (-1)^{n+1}\dfrac{1}{n^4}=3(k_BT)^4\eta (4)=3(k_BT)^4\dfrac{7}{8}\zeta (4)=\dfrac{\pi^2(k_BT)^4}{30}\left(\dfrac{7}{8}\right)}

\displaystyle{P\approx \dfrac{(k_BT)^4}{\pi^2}\sum_{n=1}^\infty (-1)^{n+1}\dfrac{1}{n^4}=\left(\dfrac{7}{8}\right)\dfrac{\pi^2(k_BT)^4}{90}}

Remark (I): For photons \gamma with degeneracy g=2 we obtain

n_\gamma =\dfrac{2\zeta (3) (k_BT)^3}{\pi^2}

\varepsilon_\gamma= 3P_\gamma =\dfrac{\pi^2 (k_BT)^4}{15}

s_\gamma =P'(T)=\dfrac{4}{3}\left(\dfrac{\pi^2}{15}\right)(k_BT)^3=\dfrac{2\pi^4}{45\zeta (3)}n

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

1eV=1.602\cdot 10^{-19}J

\hbar=1=6.58\cdot 10^{-22}MeVs=7.64\cdot 10^{-12}Ks

\hbar c=1=0.19733GeV\cdot fm=0.2290 K\cdot cm

1 K=0.1532\cdot 10^{-36}g\cdot c^2

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:

T_\gamma \approx 2.725K

Indeed, it also implies that the relic photon density is about n_\gamma =410\dfrac{1}{cm^3}

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:

T_\nu =\left(\dfrac{4}{11}\right)^{1/3}2.7K or equivalently

T_\nu\approx 1.9K

This temperature implies a relic neutrino density (per species, i.e., with g_\nu=1) about

n_\nu=54\dfrac{1}{cm^3}

The cosmological density entropy due to these particles is

s_0=\dfrac{S_0}{V}=\dfrac{4\pi^2}{45}\left[1+\dfrac{2\cdot 3}{2}\left(\dfrac{7}{8}\right)\left(\dfrac{4}{11}\right)\right]T_{0\gamma}^3=2810\dfrac{1}{cm^3}\left( \dfrac{T_{0\gamma}}{2.7K}\right)^3

and then we get

s_0\approx 7.03n_{0\gamma}

Remark (III): In Cosmology, for fermions in 3d ( note that BE implies \varepsilon=3P, and that we must drop the factors \left( 7/8\right), \left( 3/4\right), \left( 7/6\right) in the next numerical values) we can compute

n=\begin{cases}\left(\dfrac{g}{2}\right)\left(\dfrac{3}{4}\right)\dfrac{2\zeta (3)}{\pi^2}(k_BT)^3\\ \left(\dfrac{g}{2}\right)\left(\dfrac{3}{4}\right)31.700\left(\dfrac{k_BT}{GeV}\right)^3\dfrac{1}{fm^3}\\ \left(\dfrac{g}{2}\right)\left(\dfrac{3}{4}\right)20.288\left(\dfrac{T}{K}\right)^3\dfrac{1}{cm^3}\end{cases}

\varepsilon=3P=\begin{cases}\left(\dfrac{g}{2}\right)\left(\dfrac{7}{8}\right)\left(\dfrac{\pi^2}{15}\right)(k_BT)^4\\ \left(\dfrac{g}{2}\right)\left(\dfrac{7}{8}\right)(85.633)\left(\dfrac{k_BT}{GeV}\right)\dfrac{GeV}{fm^3}\\ \left(\dfrac{g}{2}\right)\left(\dfrac{7}{8}\right)\left(0.841\cdot 10^{-36}\right)\left(\dfrac{T}{K}\right)^4\dfrac{g}{cm^3}\end{cases}

s=\dfrac{S}{V}=\left(\dfrac{g}{2}\right)\left(\dfrac{7}{8}\right)\left(\dfrac{4\pi^2}{45}\right)(k_BT)^3=\dfrac{7}{6}\left[\dfrac{2\pi^4}{45\zeta (3)}\right] n

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

g_g=(\mbox{color})(\mbox{spin})=2^3\cdot 2=8\cdot 2=16

g_q=(p\overline{p})(\mbox{spin})(\mbox{color})(\mbox{flavor})=2\cdot 2\cdot 3\cdot N_f=12N_f

Then, the QG plasma degeneracy factor is

g_{QGP}=g_g+\dfrac{7}{8}g_q=16+\dfrac{7}{8}12N_f=16+\dfrac{21}{2}N_f \leftrightarrow \boxed{g_{QGP}=16+\dfrac{21}{2}N_f}

In general, for charged leptons and nucleons g=2, g=1 for neutrinos (per species, of course), and g=2 for gluons and photons. Remember that massive particles with spin j will have g_j=2j+1.

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

\displaystyle{I(T)=\int_0^\infty f(\nu ,T)d\nu=\dfrac{8\pi h}{c^3}\int_0^\infty \dfrac{\nu^3d\nu}{e^{\frac{h\nu}{k_BT}}-1}=\dfrac{8\pi^5k_B^4T^4}{15c^3h^3}}

Remark (VI): Sometimes the following nomenclature is used

i) Extremely degenerated gas if \mu>>k_BT

ii) Non-degenerated gas if \mu <<-k_BT

iii) Extremely relativistic gas ( or ultra-relativistic gas) if p>> mc

iv) Non-relativistic gas if p<<mc

 

 

9. ζ(s) AND GROUP ENTROPIES

Let us define the following shift operator \hat{T}:

\hat{T}f(x)=f(x+\sigma)

where \sigma\in \mathbb{R}. Moreover, there is certain isomorphism  between the shift operator space and the space of functions through the map \hat{T}\leftrightarrow x^\sigma.

We define the generalized logarithm as the image under the previous map of \hat{T}. That is:

\displaystyle{\mbox{Log}_G(x)\equiv \dfrac{1}{\sigma}\sum_{n=l}^{m}k_n x^{\sigma n}}

where l,m\in \mathbb{Z}, with l<m, m-l=r and x>0. Furthermore, the next contraints are also given for every generalized logarithm:

1st. \displaystyle{\sum_{n=1}^m k_n=0}.

2nd. \displaystyle{\sum_{n=l}^m nk_n=c}, k_m\neq 0, and k_l\neq 0.

3rd. \displaystyle{\sum_{n=l}^m\vert n\vert^l k_n=K_l}, \forall l=2,3,\ldots ,m-l and where K_l \in \mathbb{R}.

With these definitions we also have that

A) \mbox{Log}_G(x)=\ln (x)

B) \mbox{Log}_G(1)=0

Examples of generalized logarithms are:

1) The Tsallis logarithm.

\mbox{Log}_T(x)=\dfrac{x^{1-q}-1}{1-q}

2) The Kaniadakis logarithm.

\mbox{Log}_K(x)=\dfrac{x^\kappa-x^{-\kappa}}{2\kappa}

3) The Abe logarithm.

\mbox{Log}_A(x)=\dfrac{x^{\sigma -1}-x^{\sigma^{-1}-1}}{\sigma-\sigma^{-1}}

4) The biparametric logarithm.

\mbox{Log}_B(x)=\dfrac{x^a-x^b}{a-b}

with a=\sigma-1 and b=\sigma^{-1}-1 in the case of the Abe logarithm.

Group entropies are defined through the use of generalized logarithms. Define some discrete probability distribution \left[ p_i\right]_{i=1,\ldots,W} with normalization \displaystyle{\sum_{i=1}^Wp_i=1}. Therefore, the group entropy is the following functional sum:

\boxed{\displaystyle{S_G=-k_B\sum_{i=1}^{W}p_i \mbox{Log}_G \left(\dfrac{1}{p_i}\right)}}

where we have used the previous definition of generalized logarithm and the Boltzmann’s constant k_B is a real number. It is called group entropy due to the fact that S_G 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 x\leftrightarrow e^t is defined to be e_G^t=\dfrac{e^{(1-q)t}-1}{1-q}, and thus we easily get:

\displaystyle{\dfrac{1}{\Gamma (s)}=\int_0^\infty\dfrac{1}{\dfrac{e^{(1-q)t}-1}{1-q}}t^{s-1}dt=\dfrac{\zeta (s)}{(1-q)^{s-1}}}

\forall s such as Re (s)>1 and q<1.

 

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.

The primon gas IS  a quantum field theory (QFT) of a set of non-interacting particles, called the “primons”. 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 (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, H, having eigenstates \vert p\rangle labelled by the prime numbers “p”.

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

H\vert p\rangle = E_p \vert p\rangle with E_p=E_0 \log p

Please, note the natural emergence of a “free” scale of energy E_0. 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 k_p of primons in the single-particle states p:

|N\rangle = |k_2, k_3, k_5, k_7, k_{11}, \ldots, k_{137},\ldots, k_p \ldots\rangle

This corresponds to the factorization of N into primes:

N = 2^{k_2} \cdot 3^{k_3} \cdot 5^{k_5} \cdot 7^{k_7} \cdot 11^{k_{11}} \cdots 137^{k_{137}}\cdots p^{k_p} \cdots

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

\displaystyle{E(N) = \sum_p k_p E_p = E_0 \cdot \sum_p k_p \log p = E_0 \log N}

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

\displaystyle{Z_B(T) \equiv\sum_{N=1}^\infty \exp \left(-\dfrac{E(N)}{k_B T}\right) = \sum_{N=1}^\infty \exp \left(-\dfrac{E_0 \log N}{k_B T}\right) = \sum_{N=1}^\infty \dfrac{1}{N^s} = \zeta (s)}

with s=E_0/k_BT=\beta E_0, and where k_B is the Boltzmann’s constant and T is the absolute temperature. The divergence of the zeta function at the value s=1 (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:

T_H=\dfrac{E_0}{k_B}

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

E=-\dfrac{\partial}{\partial \beta}\ln Z_B=-\dfrac{E_0}{\zeta (\beta E_0)}\dfrac{\partial \zeta (\beta E_0)}{\partial \beta}\approx \dfrac{E_0}{s-1}

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 m_i 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

Z_F(T)=\dfrac{\zeta (s)}{\zeta (2s)}

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 \mu (Julia, B. L., 1994, Physica A 203(3-4), 425), and thus, he replaced the primes p with new primes pe^{-\mu}. 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 -1, it shows that its partition function becomes

\overline{Z}_B=\dfrac{\zeta (2s)}{\zeta (s)}

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 E(N)=2E_0\ln p_N. 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, \rho (E)\sim e^E, 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 \kappa, which defines an imaginary particle, the non-interacting parafermions of order \kappa. This order parameter counts how many parafermions can occupy the same state, i.e. the occupation number of any state falls into the interval \left[0,\kappa-1\right], and therefore \kappa=2 belongs to normal fermions, while \kappa\rightarrow\infty 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):

Z_\kappa=\dfrac{\zeta (s)}{\zeta (\kappa s)}

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

\displaystyle{f\star g=\sum_{d\vert n}f(d)g\left(\dfrac{n}{d}\right)}

where the symbol d\vert n means d is a divisor of n. This operation preserves the multiplicative property of the classically defined partition functions, i.e., Z_{\kappa_1\star \kappa_2}=Z_{\kappa_1}\star Z_{\kappa_2}. 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

\displaystyle{f\odot g=\sum_{d\vert n}f(d)g\left( \dfrac{n}{d}\right) K(n,d)}

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

Z_\infty\star Z_\infty=\zeta ^2(s)\zeta(2s)=\dfrac{\zeta (s)}{\zeta(2s)}\zeta (s)=Z_2Z_\infty=Z_FZ_B

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  \star operation (i.e. free mixing), reappears in supersymmetric quantum field theories as a possible representation of the (-1)^F operator, where F is the fermion number operator!  In this context, the fact that \mu (n)=0 for square-free numbers is the manifestation of the Pauli exclusion principle itself! In any QFT with fermions, (-1)^F is a unitary, hermitian, involutive operator where F is the fermion number operator and is equal to the sum of the lepton number plus the baryon number, i.e., F=B+L, 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 2\pi. This splits the Hilbert space into two superselection sectors. Bosonic operators commute with (-1)^F 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 E(n)=E_0\ln (p_n), where p_n, n=2,3,5,\ldots, 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

E=\ln (n) \;\; mod E_0

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 \delta n_0 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 E_N=\ln (N) does not increase with N more rapidly than n^2, 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”:

N=p_1^{n_1}p_2^{n_2}\cdots p_m^{n_m}

Take the logarithm

\log N=\log \left(p_1^{n_1}p_2^{n_2}\cdots p_m^{n_m}\right)=n_1\log p_1+n_2\log p_2+\ldots+n_m\log p_m

\displaystyle{\log N=\sum_{i=1}^{m}n_i\log p_i}

where p_i 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.,

\varepsilon_i=\log p_i with p_i=(1),2,3,5,7,11,13,\ldots,137,\ldots,\infty

with i=0,1,2,3,4,\ldots,\infty. In order to have a “riemann gas”/riemannium, we impose an spectrum labelled in the following fashion

\varepsilon_s =\log (2s+1) \forall s=0,1,2,3,\ldots,\infty

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

\varepsilon_s=\log (s) \forall s=1,2,3,\ldots,\infty

In addition to this, suppose the next quantum postulates:

1st. Logarithmic potential:

V(x)=V_0\ln\dfrac{\vert x\vert}{L} with positive constants V_0, L>0

From the physical viewpoint, the positive constant V_0 means repulsive interaction (force).

2nd. Bohr-Sommerfeld quantization rule:

a) \displaystyle{I=\dfrac{1}{2\pi}\oint pdx=\hbar \left(s+\dfrac{1}{2}\right)}\; \forall s=0,1,\ldots,\infty

or equivalently we could also get

b) \displaystyle{I=\dfrac{1}{2\pi}\oint pdx=\hbar s}\; \forall s=1,2,\ldots,\infty

3rd. Turning point condition:

x_s=L\exp \left(\dfrac{\varepsilon_s}{V_0}\right)

In the case of 2a) we would deduce that

\displaystyle{\dfrac{\hbar \pi}{2}\left(s+\dfrac{1}{2}\right)=\int_0^{x_s}dx\sqrt{2m\left(\varepsilon_s-V_0\ln \dfrac{x}{L}\right)}}

so

\displaystyle{\dfrac{\hbar \pi}{2}\left(s+\dfrac{1}{2}\right)=\int_0^{x_x}dx\sqrt{-\ln \left(\dfrac{x}{x_s}\right)}=\sqrt{2mV_0}x_s\Gamma \left(\dfrac{3}{2}\right)}

and then

x_s=\sqrt{\dfrac{\pi}{2mV_0}}\hbar \left( s+\dfrac{1}{2}\right)

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

\boxed{\dfrac{\varepsilon_s}{V_0}=\ln (2s+1)+\ln \left(\dfrac{\hbar}{2L}\sqrt{\dfrac{\pi}{2mV_0}}\right)} \forall s=0,1,\ldots,\infty

In the case of 2b) we would obtain

\boxed{\dfrac{\varepsilon_s}{V_0}=\ln (s)+\ln \left(\dfrac{\hbar}{L}\sqrt{\dfrac{\pi}{2mV_0}}\right)} \forall s=1,2,\ldots,\infty

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 pc 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 F needed to counteract the centrifugal force F_c=\dfrac{\kappa c^2}{R}. The centrifugal force is equal to pc/R, i.e., the balancing force F is F=pc/R. Then, assuming the two forces are equal in magnitude, we get

F=F_c=\dfrac{A_1}{R}

where A_1 is some constant, and that equation holds regardless the origin of the interaction. The potentail energy U necessary to confine a constituent particle will be, in that case,

\displaystyle{U=\int \dfrac{A_1}{R}dR=A_1\int \dfrac{1}{R}dR=A_1\ln \dfrac{R}{R_\star}}

with R_\star 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:

m=\dfrac{\hbar}{cR}

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

U=\dfrac{A_2 m}{R}

with A_2 is another constant. In fact, A_1, A_2 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 A_2/A_1 to the constant c^2/G_N, where G_N is the gravitational constant. However, such a constraint is not required from first principles or from a clear physical reason. From the following equations:

m=\dfrac{\hbar}{cR} and U=\dfrac{A_2 m}{R}

we get \boxed{U=\dfrac{A_2 \hbar}{cR^2}}

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

U=\dfrac{A_2 m}{cR^2}\rightarrow U_n=\dfrac{A_2 \hbar}{cR_n^2}=\dfrac{A_2 (n+1)\hbar}{cR_0^2}

\forall n=0,1,2,\ldots,\infty

so R_n^2=\dfrac{R_0^2}{n+1}\leftrightarrow R_n=\dfrac{R_0}{\sqrt{n+1}}

Using the previous integral and this last result, we obtain

\ln \left(\dfrac{R_\star}{R_0}\right)=-(n+1)\dfrac{R_\star^2}{R_0^2}

This is due to the fact that U_n=A_2\dfrac{\hbar}{cR_n^2}=\dfrac{A_2\hbar (n+1)}{cR_0^2} and U=A_2\ln \dfrac{R}{R_\star}

Combining these equations, we deduce the value of R_\star as a function of the parameters A_1,A_2

\boxed{R_\star=\sqrt{\dfrac{A_2\hbar}{A_1 c}}}

The ratio R_\star/R_0 can be calculated from the above equations as well, since

\ln \left(\dfrac{R_\star}{R_0}\right)=-(n+1)\dfrac{R_\star^2}{R_0^2} for the case n=0 implies that

\ln \left(\dfrac{R_\star}{R_0}\right)=-\dfrac{R_\star^2}{R_0^2}, and after exponentiation, it yields

\dfrac{R_\star}{R_0}=e^{-\frac{R_\star^2}{R_0^2}}

Introducing the variable x=\dfrac{R_\star}{R_0} we have to solve the equation x=e^{-x^2}

The solution is \phi=\dfrac{1}{x}=1.53158 from which the relationship between R_\star and R_0 can be easily obtained. Indeed, we can make more deductions from this result. From \ln \phi=1/\phi^2, then

R_n=R_\star e^{(n+1)\ln\phi}

If we take R_\star=\alpha R_0, with R_0=\hbar/mc, then

\alpha=m_0\sqrt{\dfrac{A_2 c}{A_1\hbar}} so

R_n=R_0e^{K\varphi_n} with K=\dfrac{1}{2\pi}\ln \phi and \varphi_n=2\pi (n+1)+\varphi_s \varphi_s=2\pi \left(\dfrac{\ln \alpha}{\ln \phi}\right)

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

m_n=\dfrac{\hbar}{R_nc} and m_0=\dfrac{\hbar}{R_0c}

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

\boxed{m_n=m_0e^{K\varphi_n}}

Remark: The shift K\rightarrow -K implies that the spiral would begin with m_0 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

E_n=\hbar\omega n

E(n),\;\forall n=0,1,2,\ldots,\infty, are the H.O. eigenenergies.

B) Consider the Tsallis partition function

\displaystyle{Z_q (\beta )=\sum_{n=0}^{\infty}e_q^{-\beta E_n}=\sum_{n=0}^{\infty}e_q^{-\beta\hbar\omega n}}

where q>1 and the deformed q-exponential is defined as

e_q^z\equiv \left[1+(q-1)z\right]_+^{\frac{1}{1-q}}

and \left[\alpha\right]=\begin{cases}\alpha, \alpha>0\\ 0,otherwise\end{cases}

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

\ln_q z=\dfrac{z^{1-q}-1}{1-q}

It implies that

\boxed{\displaystyle{Z_q=\sum_{n=0}^{\infty}\dfrac{1}{\left[1+(q-1)\beta\hbar\omega n\right]^{\frac{1}{q-1}}}=\dfrac{1}{\left[(q-1)\beta\hbar \omega\right]^{\frac{1}{q-1}}}\sum_{n=0}^{\infty}\dfrac{1}{\left[\left(\dfrac{1}{(q-1)\beta\hbar\omega}\right)+n\right]^{\frac{1}{q-1}}}}}

Now, defining the Hurwitz zeta function as:

\displaystyle{\zeta (s,Q)=\sum_{n=0}^{\infty}\dfrac{1}{\left(Q+n\right)^{s}}=\dfrac{1}{Q^s}+\sum_{n=1}^{\infty}\dfrac{1}{\left(Q+n\right)^{s}}}

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

\boxed{\displaystyle{Z_q=\dfrac{1}{\left[(q-1)\beta\hbar\omega\right]^{\frac{1}{q-1}}}\zeta \left(\dfrac{1}{q-1},\dfrac{1}{(q-1)\beta\hbar\omega}\right)}}

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 q\rightarrow 1.

In the limit Q\rightarrow 1 we get the Riemann zeta function from the Hurwitz zeta function:

\displaystyle{\zeta (s,1)\equiv \zeta (s)=\sum_{n=1}^{\infty}n^{-s}=\sum_{n=1}^{\infty}\dfrac{1}{n^s}=\prod_{p=2}^{\infty}\dfrac{1}{1-p^{-s}}=\prod_{p}\dfrac{1}{1-p^{-s}}}

or

\displaystyle{\zeta (s)=\dfrac{1}{1-2^{-s}}\dfrac{1}{1-3^{-s}}\ldots\dfrac{1}{1-137^{-s}}\ldots}

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.

\boxed{\displaystyle{S_q=k_B\dfrac{1-\displaystyle{\sum_{i=1}^W p_i^q}}{q-1}=-k_B\sum_{i=1}^Wp_i^q\ln_q p_i=k_B\sum_{i=1}^Wp_i\ln_q \left(\dfrac{1}{p_i}\right)}}

plus the normalization condition \boxed{\displaystyle{\sum_{i=1}^Wp_i=1}}

2nd. Continuous case.

\boxed{\displaystyle{S_q=-k_B\int dX\left[p(X)\right]^q\ln_q p(X)=k_B\int dX p(X)\ln_q\dfrac{1}{p(X)}}}

plus the normalization condition \boxed{\displaystyle{\int dX p(X)=1}}

3rd. Quantum case. Tsallis matrix density.

\boxed{\displaystyle{S_q=-k_BTr\rho^q\ln _q\rho\equiv k_BTr\rho \ln_q\dfrac{1}{\rho}}}

plus the normatlization condition \boxed{Tr\rho=1}

In all the three cases above, we have defined the q-logarithm as \ln_q z\equiv\dfrac{z^{1-q}-1}{q-1}, \ln_1 z\equiv \ln z, and the 3 Tsallis entropies satisfy the non-additive property:

\boxed{\dfrac{S_q(A+B)}{k_B}=\dfrac{S_q (A)}{k_B}+\dfrac{S_q (B)}{k_B}+(1-q)\dfrac{S_q (A)}{k_B}\dfrac{S_q (B)}{k_B}}

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 \vert x+y\vert \leq \vert x\vert +\vert y\vert.

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

\vert x+y\vert _p\leq \mbox{max}\{\vert x\vert_p ,\vert y \vert_p\}

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,

\mathbb{A}=\left( x_\infty, x_2,x_3,x_5,\ldots,x_p,\ldots\right)

where x_\infty is a real number and the x_p are p-adic numbers living in the p-adic field \mathbb{Q}_p. 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 x_p lie in the entire p-adic set \mathbb{Z}_p. The adele ring is therefore a restricted direct (cartesian) product. The idele group is defined as the essentially invertible elements of the adelic ring:

\mathbb{I}=\mathbb{A}^\star =\{ x\in \mathbb{A}, \mbox{where}\;\; x_\infty \in \mathbb{R}^{\star} \;\; \mbox{and} \;\; \vert x_p\vert _p=1,\; \mbox{for all but finitely many primes p.}\}

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, e^X, logarithms \log (x) 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 \mathbb{R} and \mathbb{Q}_p for all primes p. Furthermore, the real and p-adic norms are related to each other by the remarkable adelic product formula/identity:

\displaystyle{\vert x\vert_\infty \prod_p\vert x\vert_p=1}

and where x 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 D=1.5 for the Riemann zeroes and D=1.8 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

Dx=dx_\infty dx_2dx_3\cdots dx_p\cdots and Dx^\star=dx_\infty^\star dx_2^\star dx_3^\star\cdots dx_p^\star\cdots

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:

\displaystyle{\int_{\mathbb{Q}_p}\chi_\infty (ax_\infty^2+bx_\infty)dx_\infty \prod_p \int_{\mathbb{Q}_p}\chi_p (ax_p^2+bx_p)dx_p=1}

where \chi is an additive character from the adeles to complex numbers \mathbb{C} given by the map:

\displaystyle{\chi (x)=\chi_\infty (x_\infty)\prod_p \chi_p (x_p)\rightarrow e^{-2\pi ix_\infty}\prod_p e^{2\pi i\{p\}_p}}

and  \{x_p\}_p is the fractional part of x_p in the ordinary p-adic expression for x. This can be thought of as a strong generalization of the homomorphism \mathbb{Z}/\mathbb{Z}_n\rightarrow e^{2\pi i/n}.Then, the adelic path integral, with input parameters in the adelic ring \mathbb{A}  and generating complex-valued wavefunctions follows up:

\displaystyle{K_{\mathbb{A}} (x'',t'';x',t') =\prod_\alpha \int_{(x' _\alpha ,t' _\alpha)}^{(x'' _\alpha ,t'' _\alpha)}\chi_\alpha \left(-\dfrac{1}{h}\int_{t' _\alpha}^{t''_\alpha}L(\dot{q} _\alpha ,q_\alpha ,t_\alpha )dt_\alpha \right) Dq_\alpha}

The eigenvalue problem over the adelic ring is given by:

U(t) \psi_\alpha (x)=\chi (E_\alpha (t))\psi_\alpha (x)

where U is the time-development operator, \psi_\alpha are adelic eigenfunctions, and E_\alpha is the adelic energy. Here the notation has been simplified by using the subscript \alpha, which stands for all primes including the prime at infinity. One notices the additive character \chi 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

\displaystyle{\vert 0\rangle =\Psi_0 (x)=2^{1/4}e^{-\pi x_\infty^2}\prod_p \Omega (\vert x_p\vert_p)}

where \Omega \left(\vert x_p \vert _p\right)  is 1 if \vert x_p\vert_p 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

\Phi (\alpha)=\sqrt{2}\Gamma \left(\dfrac{\alpha}{2}\right)\pi^{-\alpha/2}\zeta (\alpha)

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

\Phi (\alpha)=\Phi (1-\alpha).

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 amplitudesA(a,b) 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:

A_\infty (a,b)=g_\infty^2 \dfrac{\zeta (1-a)}{\zeta (a)}\dfrac{\zeta (1-b)}{\zeta (b)}\dfrac{\zeta (1-c)}{\zeta (c)}

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

\displaystyle{A_\infty (a,b)\prod_p A_p (a,b)=1}

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:

\boxed{\displaystyle{\sum_{n\in \mathbb{Z}}\vert n\vert^d =2\zeta (-d)}}

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 (-1)^F 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!