# LOG#104. Primorial objects.

My post today will be discussing two ideas: the primorial and the paper “The product over all primes is $4\pi^2$ (2003).

The primorial is certain generalization of the factorial, but running on prime numbers. While the factorial is defined as

$n!=n\cdot (n-1)\cdots 3\cdot \cdot 2\cdot 1$

the primorial is defined as follows

$\boxed{p_n\#=\prod_{k=1}^np_k}$

The first primorial numbers for $n=1,2,\ldots$ are

$2,6,30,210,2310,30030,510510,\ldots$

We can also extend the notion of primorial to integer numbers

$\boxed{n\#=\prod_{k=1}^{\pi (n)}p_k}$

where $\pi (n)$ is the prime counting function. The first primorial numbers for integer numbers are

$1,2,6,6,30,30,210,210,210,210,2310,\ldots$

In fact, if you take the limit

$\displaystyle{\lim_{n\rightarrow \infty}(p_n)^{1/p_n}=e}$

since the Chebyshev function provides

$\displaystyle{\lim_{x\rightarrow \infty}\dfrac{x}{\theta (x)}=1}$

By the other hand, the factorial of infinity can be regularized

$\infty ! =1\cdot 2\cdot 3\cdots =\sqrt{2\pi}$

The paper mentioned above provides a set of cool formulae related to infinite products of prime numbers powered to some number! The main result of the paper is that

$\boxed{\prod_p p=4\pi^2}$

If you have an increasing sequence of numbers $0<\lambda_1\leq \lambda_2\leq \ldots$, then we can define the regularized products thanks to the Riemann zeta function (this technique is called zeta function regularization procedure):

$\boxed{\prod_{n=1}^\infty \lambda_n =e^{-\zeta'_\lambda (0)}}$  $\boxed{\zeta_\lambda (s)=\sum_{n=1}^\infty \lambda_n^{-s}}$

and where the $\lambda_n$ is some sequence of positive numbers. The Artin-Hasse exponential can be defined in the following way:

$\boxed{\exp (X)=\prod_{n=1}^\infty (1-X^n)^{-\mu (n)/n}}$

and there $\mu (n)$ is the Möbius function. From this exponential, we can easily get that

$\boxed{e^{p^{-s}}=\prod_{n=1}^\infty (1-p^{-ns})^{-\mu (n)/n}}$

Using the prime zeta function

$\displaystyle{\mathcal{P}(s)= \sum_{p}\dfrac{1}{p^s}}$

we obtain

$\displaystyle{e^{\mathcal{P}(s)}=\prod_p e^{p^{-s}}=\prod_p \prod_{n=1}^\infty (1-p^{-ns})^{-\mu (n)/n}}$

$\displaystyle{e^{\mathcal{P}(s)}=\prod_{n=1}^\infty \prod_p (1-p^{-ns})^{-\mu (n)/n}=\prod_{n=1}^\infty \zeta (ns)^{\mu (n)/n}}$

Therefore

$\boxed{e^{\mathcal{P}(s)}=\prod_{n=1}^\infty \zeta (ns)^{\mu (n)/n}}$

Now, if we remember that

$\displaystyle{\sum_{n=1}^ \infty\dfrac{\mu (n)}{n^{s}}=\dfrac{1}{\zeta (s)}}$

and that

$\displaystyle{\mathcal{P}'(s)=\sum_{n=1}^\infty \dfrac{\mu (n)}{n}\dfrac{n\zeta' (ns)}{\zeta (ns)}=\sum_{n=1}^\infty \mu (n)\dfrac{\zeta' (ns)}{\zeta (ns)}}$

$\displaystyle{\mathcal{P}'(0)=\left(\sum_{n=1}^\infty \mu (n)\right) \dfrac{\zeta' (0)}{\zeta (0)}=\dfrac{\zeta' (0)}{\zeta (0)^2}=-2\log (2\pi)}$

and from this we get

$\displaystyle{\prod_p p=e^{\mathcal{P}'(0)}=(2\pi)^2=4\pi^2}$

Q.E.D.

And similarly, it can be proved the beautiful formula

$\boxed{\displaystyle{\prod_p p^s=(2\pi)^{2s}}}$

Moreover, using the Riemann zeta function

$\displaystyle{\zeta (s)=\prod_p (1-p^{-s})^{-1}=\dfrac{\displaystyle{\prod_p p^s}}{\displaystyle{\prod_p (p^s-1)}}}$

we also have

$\displaystyle{\boxed{\prod_p(p^s-1)=\dfrac{(2\pi)^{2s}}{\zeta (s)}}}$

In particular, e.g., we get

$\displaystyle{\prod_p (p-1)=0}$

and

$\displaystyle{\prod_p (p^2-1)=48\pi^2}$

The final part of the paper is a proof using a “more convergent” series of the previous “prime products”/products of prime numbers. It uses the series

$\displaystyle{e^{\mathcal{P}(s,t)}=\prod_{n=1}^\infty \zeta (ns)^{\mu (n)/n^t}}$

and it converges $\forall Re (s)>1, Re (t)>1$. But then, the series

$\displaystyle{\dfrac{\partial \mathcal{P}}{\partial s}(s,t)=\sum_{n=1}^\infty \dfrac{\mu (n)}{n^{t-1}}\dfrac{\zeta' (ns)}{\zeta (ns)}}$

converges $\forall Re (t)>1, Re (s)> 0$. Then, if $t\in \mathbb{C}$, and $Re (t)>2$, with $s\longrightarrow 0$ we have a limit

$\displaystyle{L=\lim_{Re s>0}\lim _{s\rightarrow 0}\dfrac{\partial \mathcal{P}}{\partial s}(s,t)=\left(\sum_{n=1}^\infty \dfrac{\mu (n)}{n^{t-1}}\right)\dfrac{\zeta' (0)}{\zeta (0)}}$

and thus

$L=\dfrac{\zeta ' (0)}{\zeta (t-1) \zeta (0)}$

Therefore, the meromorphic extension of this formula to the whole complex plane provides that

$\displaystyle{\lim_{t\rightarrow 1} L=\dfrac{\zeta '(0)}{\zeta (0)^2}}$

as we wanted to prove (Q.E.D.).

Let the prime numbers and the primorial be with you!

# LOG#103. Numbers: the list.

Hello, eager earthlings! Today, my list is about fascinating number types. I love them all so much…

1) Natural numbers.

2) Integer numbers.

3) Fractional numbers.

4) Irrational numbers.

5) Real numbers.

6) Complex numbers.

7) Quaternions.

8) Octonions/Octaves/Cayley numbers. (John C. Baez’s favourite numbers!)

9) Sedenions.

10) Hypernumbers (Cayley-Dickson algebras).

11) Grassmann numbers/Grassmannian numbers/anticommuting c-numbers/supernumbers.

12) Clifford numbers.

15) Ternary numbers and n-ary numbers.

16) q-numbers, or xy-qxy, and their q-deformed generalizations.

17) Tropical numbers.

18) Polygonal numbers.

19) Modular numbers.

20) Surreal numbers.

21) Transfinite numbers.

Do you know a cool type of number I should add to “my list”? Let me know…

May the numbers be with you!!!!!

# LOG#102. Superstuff: the list.

Hello, Earth planet! Hello, earthlings!

I want to share with you my list of favourite “superstuff”…Enjoy it:

1) Supersymmetry.

2) Superspace.

3) Supermatrix.

4) Superdeterminant.

5) Supergravity.

6) Superstrings.

7) Super p-branes. $p=-1,0,1,2,\ldots$. Question: What about $p=-1,-2,\ldots$ or “fractional branes”?

8) Supertwistors.

9) Supermatrices.

10) Superstatistics.

11) Supertime.

12) Superheterodyne. (lol, I know, I know)

13) Superextendon.

14) Superconnection.

15) Superconductivity and superconductors.

16) Superfluids and superfluidity.

17) Superinsulators.

18) Superalloy.

19) Supermassive black hole.

20) Supersymmetric object (=algebra, particle, gauge theory, quantum mechanics, field theory, …)

21) Superconformal symmetry/group/algebra.

22) Supergroups.

23) Supermanifolds.

24) Super linear algebra.

25) Superluminal.

27) Supercomputer.

28) Supercapacitor.

29) Supernovae.

30) Superoxide.

31) Superorganism.

32) Superposition principle.

33) Superpower.

34) Superpotential.

35) Superpolynomial.

36) Supergraph.

37) Superreal number.

38) Superresolution.

39) Superring.

40) Superset.

41) SuperEarth.

42) Superstrong force.

43) Superuser.

44) Superunification.

45) Supervisor.

46) Supervectors, supervector spaces.

47) Supervoid.

48) Superworld, superwarp, superforce, supermathematics.

49) Superzoom.

50) (Just for fun) Superman, supergeek, supergeeknerd, superfriends, superstars, superhero, superstudent (übermensch, übergeek, übergeeknerd, überfriends, überstars, überhero, überstudent).

Would you add something else to this list? Let me know…

May the Superforce be with you!

# LOG#101. Hyperstuff: the list.

Hello, world!

This short post is a list with my favourite hyperstuff related to Physmatics. It also begins a new subcategory of (generally short) posts that I have called “Lists” where I will be writing enumerative lists about some “stuff” I love…

1) Hyperpolygons ( a.k.a. polytopes).

2) Hypercubes.

3) Hyperplanes.

4) Hyperspheres.

5) Hyperlogarithms.

6) Hyperdeterminants.

7) Hypermatrices.

8) Hypergraphs.

9) Hypernumbers.

10) Hypersymmetry ( ternary and generalized n-ary supersymmetries, Clifford algebras, etc).

11) Hyperspace.

12) Hypertime.

13) Hypernuclei.

14) Hyperons.

15) Hyperdrive.

16) Hyperhydrogen atoms.

17) Hypercharge.

18) Hyperquarks.

19) Hypercolor.

20) Hyperelliptic functions and integrals.

Do you know some interesting hyperstuff that you would add to this list?

See you in another blog post!

# LOG#100. Crystalline relativity.

CENTENARY BLOG POST! And dedicatories…

My blog is 100 posts “old”. I decided that I wanted a special topic and subject for it, so I have thinking during several days if I should talk about Physmatics, tropical mathematics or polylogarithms, but these topics deserve longer entries, or a full thread to discuss them with details I consider very important, so finally I changed my original mind and I took a different path.

This blog entry is dedicated specially to my friends out there. They are everywhere in the world. And specially to Carlos Castro, M. Pavsic (inventors of C-space, M-space relativity in Clifford spaces and the brane M-space approach to relativity with Clifford Algebras, respectively), my dear friend S.Lukic (now working hard in biomathematics and mathematical approaches to genetics), A. Zinos (a promising Sci-Fi writer), J. Naranja (my best friend, photographer and eclectic man) and all my (reduced) Spanish friends (yes, you know who are you, aren’t you?). I dedicated this special blog entry to my family (even if they don’t know what I am doing with this stuff, likely they have no idea at all…) and those special people who keep me up and make me feel alive (from time to time, when they write me, in russian worlds), even when the thunder sounds and the storm arises and I switch off from almost all the real world. And finally, it is also dedicated to all my unbiased followers around the world… Wherever you are… It is also dedicated to all of you…

Well, firstly I should eat a virtual take, don’t you think so?

1. Serendipitous thoughts about my 100th blog post

Here, in my 100th post, I am going to write about some old fashioned idea/s, likely “crackpot” to some current standards, but it also shares interesting ideas with Sci-Fi and real scientific topics like the recently introduced “time crystals” by Wilczek. The topic today is: a forgotten (likely wrong) multitemporal theory of relativity!

Why did I choose such a crazy topic? Firstly, it is an uncommon topic. Multitemporal theories or theories with extra time-like dimensions are generally given up or neglected by the physics community. The reasons seem to be broad: causality issues (closed time-like curves “are bad”), the loss of experimental evidence (time seems to be 1D, doesn’t it?), vacuum instabilities induced/triggered by QM with extra time-like dimensions and many others (some of them based on phislophical prejudices, I would say). From the pure mathematical viewpoint, extra time-like dimensions are posible and we can handle them in a similar way to space-like dimensions, but some differences arise. Let me remark that there is a complete branch of mathematics (sometimes called semi-riemannian geometry) that faces with spaces with multiple temporal dimensions (spaces with more than one temporal coordinate, generally more than minus, or plus-dependind on your sign convention).

The second reason is that I am very interested in any theory beyond the Standard Model, and particularly, any extension of Special Relativity that has been invented and in any extension that could be built from first principles. Extended theories of relativity beyond Special Relativiy do exist. The first theory Beyond Standard Special Relativity, to my knowledge, was metarelativity, namely: extended special relativity allowing “tachyons”. It was pioneered by Recami, Sudarshan, Pavsic and some other people, to quote only some of the people I have in mind right now. Perhaps, the next (known) trial was Snyder Non-Commutative spacetime. It extends relativity beyond the realm of commutative spacetime coordinates. After these “common” extended relativities, we also have (today): deformed special relativities like Doubly or Triply Special Relativities and q-deformed versions like kappa-Minkovski spacetime and some other models like the de Sitter (dS) relativity. These theories are “non mainstream” today,  but they certainly have some followers (I am one of them) and there are clever people involved in their development. Let me note that Special Relativity seems to hold yet in any High Energy experiment, so extended relativities have to explain the data in such a way that their deformation parameters should approach the Minkonvskian geometry in certain limits. Even the Kaluza-Klein approach to extra spacelike dimensions is “a deformed” version of Special Relativity somehow!

Some more modern versions of extended relativities are the theory of relativity in Clifford spaces ( pioneered by Carlos Castro Perelman and Matej Pavsic, and some other relatively unknown researchers), a theory based in relativity in (generalized) phase spaces with a (generalized) Finsler geometry or the very special relativity.  In fact, Finsler geometries triggered another extension of special relativity long ago. People call this extension VERY SPECIAL relativity (or Born reciprocal relativity in phase space, a Finsler spacetime), and other people anisotropic spacetime relativity (specially some people from Russia and Eastern Europe). Perhaps, there are some subtle details, but they share similar principles and I consider very special relativity and finslerian relativity as “equivalent” models (some precision should be done here from a mathematical perspective, but I prefer an intuitive approach in this post). Remember: all these extensions are out there, irrespectively you believe in them or not, such theories do exist. A different issue IS if Nature obeys them or not closer or not, they can be built and either you neglect them due to some conservative tastes you have (Occam’s razor: you keep Minkovskian/General Relativity since they can fit every observation at a minimum ” theoretical cost”) or you find some experimental fact that can falsify them (note that they can fix their deformation parameters in order you avoid the experimental bounds we have at current time).

My third reason to consider this weird and zenzizenzizenzic post is to be an open mind. String theory or loop quantum gravity have not been “proved” right in the experiments. However, they are great mathematical and theoretical frameworks. Nobody denies that, not me at least. But no new evidences from the alledged predictions of string theory/Loop Quantum Gravity have been confirmed so far. Therefore, we should consider new ideas or reconsider old fashioned ideas in order to be unbiased. Feynman used to say that the most dangerous thing in physics was that everyone were working on the same ideas/theories. Of course, we can coincide in some general ideas or principles, but theory and experiment advances are both necessary. With only one theory or idea in the city, everything is boring. Again, the ultimate theory, if it exists, could be a boring theory, something like SM plus gravity (asymptotically safe) until and even beyond the Planck scale, but some people think otherwise. There are many “dark” and unglued pieces yet in Physmatics…

The final reason I will provide you is that…I like strange ideas! Does it convert me in a crackpot? I wish you think otherwise! I wouldn’t be who I am if I enjoyed dogmatic ideas only. I use to distinguish crackpottery from “non-standard” models, so maybe, a more precise definition or rule should be provided to know what is the difference between them (crackpottery and non-stardardness) but I believe that it is quite “frame dependent” at the end. So…Let me begin now with a historial overview!

The unification of fundamental forces in a single theory or unified field theory was Einstein’s biggest dream. After the discovery that there was a pseudoeuclidean 4D geometry and a hidden symmetry in the Maxwell’s equations, Einstein’s quest was to express gravity in way that were consistent with the Minkovskian geometry in certain limit. Maxwell’s equations in 4D can be written as follows in tensor form:

$\partial^\mu F_{\mu\nu}=\mbox{Div} F_{\mu\nu}=J_\nu$

and

$\mbox{Rot}F_{\mu\nu}=\dfrac{1}{2}\epsilon_{\mu\nu\sigma\tau}\partial^\nu F^{\sigma\tau}=0$

where $J_\nu=(-c\rho,\vec{j})$ is the electromagnetic four-current. The symmetry group of these classical electromagnetic equations is the Poincare group, or to be more precise, the conformal group since we are neglecting the quantum corrections that break down that classical symmetre. I have not talked about the conformal group in my group theory thread but nobody is perfect! Eintein’s field equations for gravity are the following equations (they are “common knowledge” in general relativity courses):

$G_{\mu\nu}=\kappa T_{\mu\nu}$

The invariance group of (classical or standard) general relativity is something called the diffeomorphism group (due to general covariance). The diffeomorphism group invariace tells us that every (inertial or not) frame is a valid reference frame to every physical laws. Gravity can be “locally given away” if you use a “free fall” reference frame. The fact that you can “locally” forget about gravity is the content of the Einstein’s equivalence principle. I will discuss more the different classes of existing equivalence principles in a forthcoming thread of General Relativity, but this issue is not important today.

What else? Well, 4D theories seem not to be good enough to explain everything! Einstein’s himself devoted the last years of his life to find the unified theory of electromagnetism and gravity, ignoring the nuclear (quantum) interactions. It was his most famous failure beyond his struggles against the probabilistic interpretation of the  “new” Quantum Mechanics. Eintein’s unification dreams was tried by many others: Weyl, Kaluza, Klein, Eddington, Dirac himself, Heisenberg,…Remember that Faraday himself tried to find out a relation between gravity and electromagnetism! And those dreams continue alive today! In fact, quantum field theory “unifies” electromagnetism and weak nuclear forces with the electroweak theory inside the Standard Model. It is believed  that a Grand Unified Theory(GUT) should unify the electroweak force and the strong (nuclear) interaction at certain energy scale $E_X$. X is called the GUT scale, and it is generally believed that it arises at about $latez 10^{15}$ GeV. Unification with gravity is thought to be “relevant” at Planck scale $E_P$, or about $10^{19}$ GeV. Therefore, we can observe that there are two main “approaches” to the complete unification of the known “fundamental interactions”:

1st. The Particle Physics path. It began with the unification of electricity and magnetism. Then we discovered the nuclear interactions. Electromagnetism and weak interactions were unified in the 70s of the past 20th century. Then, it was conjectured that GUT unification would happen at high energy with Quantum Chromodynamics (the gauge theory of strong nuclear forces), and finally, the unification with gravity at Planck energy. Diagramatically speaking:

$\mbox{EM}\longrightarrow \mbox{Nuclear Forces}\longrightarrow \mbox{EW theory}+\mbox{QCD}\longrightarrow \mbox{EW+QCD}+\mbox{Gravity}$

2nd. The Faraday-Einstein unification path. It begins with the unification of gravity and electromagnetism first! Today, it can be said that the entropic gravity/force approach by Verlinde is a revival of this second path. It is also the classical road followed by Kaluza-Klein theories: gauge fields are higher dimensional components of a “big metric tensor” which becomes “quantized” somehow. Diagramatically:

$\mbox{EM}\longrightarrow \mbox{Gravity}\longrightarrow \mbox{EM theory}+\mbox{Gravity}\longrightarrow \mbox{EM+Gravity}+\mbox{nuclear forces}$

An interesting question is if these two paths are related and how we bring out together the best ideas of both of them. From a purely historical reason, the first path has been favoured and it has succeeded somehow. The classical “second” path is believed to be “wrong” since it neglects Quantum Mechanics and generally it finds issues to explain what Quantum Field Theories do explain. Is it a proof? Of course, it is NOT, but Physics and Physmatics have experimental foundations we can not avoid. It is not only a question of “pure thought” to invent a “good theory”. You have to test it. It has to explain everything you do know until now. That is how the Occam’s razor works in Science. You have experiments to do and observations to explain…You can not come with a new theory if it is in contradiction with well-tested theories. The new theory has to include the previous theories in some limit. Otherwise, you have a completely nonsense theory.

The second path to unification has lots of “hidden” stories and “strange theories”. Einstein’s works about teleparallelism and non-symmetrical metric tensor theories were induced by this road to unification. Has someone else followed this path?

Answer to the last question: Yes! I am going to explain you the generally unknown theory of projective relativity! It was originally created by the italian physicist Fantappie, and it was studied and extended to multiple time-like dimensions via a bulgarian guy called Kalitzin and an italian physicist known as G. Arcidiacono. Perhaps it shares some points with the current five-dimensional theory advocated by P.Wesson, but it is a completely different (parallel likely) history.

Fantappie (1901-1956) built a “projective” version of special relativity the he called “final relativity”. Today, it is known as de Sitter-relativity or de Sitter projective relativity, and according to Levy-Leblond, is one of the maximal deformations of kinematical groups available in classical physics! In fact, we can “see” the Fantappie’s final (projective) relativity as an anticipation of the cosmological constant as a physical reality. The cosmological constant IS a physical parameter in final relativity associated to the radius of the Universe. If you take this statement as “true”, you are driven to think that the cosmological constant is out there as a true “thing”. Giving up the mismatch between our current QFT calculations of vacuum energy, de Sitter relativity/final projective relativity does imply the existence of the cosmological constant! Of course, you should explain why our QFT are wrong in the way they are…But that is a different story. At current time, WMAP/Planck have proved that Dark Energy, a.k.a. the cosmological constant, is real. So, we should rethink about the way in which it enters in physics. Should we include a new symmetry in QFT (de Sitter symmetry) in order to solve the cosmological constant problem? It is a challenge! Usually, QFT are formulated in Minkovski space. But QFT calculations in Minkovski spacetime give no explanation of its cosmological value. Maybe, we should formulate QFT taking into accont the cosmological constant value. As far as I know, QFT defined on de Sitter spaces are much less developed that anti de Sitter spaces, since these ones are popular because of the adS/CFT correspondence. There are some interestings works about QFT in dS spaces in the arxiv. There are issues, though, e.g., the vacuum definition and QFT calculations in dS background are “harder” than the equivalent Minkovskian counterparts! But I believe it is a path to be explored further!

Fantappie had also a hierarchical “vision” on higher dimensional spaces. He defined “hyperspherical” universes $S_n$ contained in rotational groups $R_{n+1}$ with $(n+1)$ euclidean dimensions and $n(n+1)/2$ group parameters. He conjectured that the hierarchy of hyperspherical universes $S_3, S_4, \ldots, S_n$ provided a generalization of Maxwell equations, and with the known connection between $S_n$ and $R_{n+1}$, Fantappie tried the construction of a unified theory with extra dimensions (a cosmological theory, indeed), with the aid of his projective relativity principle. He claimed to be able to generalize Einstein’s gravitational field equations to electromagnetism, following then the second path to unification that I explained above. I don’t know why Fantappie final projective relativity (or de Sitter relativity) is not more known. I am not expert in the History of Physics, but some people and ideas remain buried or get new names (de Sitter relativity is “equivalent” to final relativity) without an apparent reason at first sight. Was Fantappie a crackpot? Something tells me that Fantappie was a weird italian scientist like Majorana but he was not so brilliant. After all, Fermi, Pauli and other conteporary physicists don’t quote his works.

From projective relativity to multitemporal relativity

What about “projective relativity”? It is based on projective geometry. And today we do know that projective geometry is related and used in Quantum Mechanics! In fact, if we take the $r=R\longrightarrow \infty$ limit of “projective” geometry, we end with “classical geometry”, and then $S_n$ becomes $E_n$, the euclidean spaces, when the projective radius tends to “infinity”. Curiously, this idea of projective geometry and projective relativity remained hidden during several decades after Fantappie’s death (it seems so). Only G. Arcidiacono and N. Kalitzin from a completely different multitemporal approach worked in such “absolutely crazy” idea. My next exposition is a personal revision of the Arcidiacono-Kalitzin multitemporal projective relativity. Suppose you are given, beyond the 3 standard spatial dimensions $(n-3)$ new parameters. They are ALL time-like, i.e., you have a $(n-3)$ time vector

$\vec{t}=\left( t_1,t_2,\ldots,t_{n-3}\right)$

We have $(n-3)$ timelike coordinates and $(n-3)$ “proper times” $\tau_s$, with $s=1,2,\ldots,n-3$. Therefore, we will also have $(n-3)$ different notions or “directions” of “velocity” that we can choose mutually orthogonal and normalized. Multitemporal (projective) relativity arise in this $n$ dimensional setting. Moreover, we can introduce $(n-3)$ “different” ( a priori) universal constants/speeds of light $c_s$ and a projective radius of the Universe, R. Kalitzin himself worked with complex temporal dimensions and even he took the limit of $\infty$ temporal dimensions, but we will not follow this path here for simplicity. Furthermore, Kalitzin gave no physical interpretation of those extra timelike dimensions/paramenters/numbers. By the other hand, G. Arcidiacono suggested the following “extension” of Galilean transformations:

$\displaystyle{\overline{X}=f(X)=\sum_{n=0}^\infty \dfrac{X^{(n)}(0)t^n}{n!}}$

$\overline{X}=X(0)+X'(0)t+X''(0)\dfrac{t^2}{2}+\ldots=X(0)+V^{(1)}t+V^{(2)}t^2/2+\ldots$

$\overline{X}=x+Vt+At^2/2+\ldots$

These transformations are nonlinear, but they can be linearized in a standard way. Introduce $(n-3)$ normalized “times” in such a way:

$t_1=t, t_2=t^2/2,\ldots, t_s=t^{s}/s!$

Remark: To be dimensionally correct, one should introduce here some kind of “elementary unit of time” to match the different powers of time.

Remark(II): Arcidiacono claimed that with 2 temporal dimensions $(t,t')$, and $n=5$, one gets “conformal relativity” and 3 universal constants $(R,c,c')$. In 1946, Corben introduced gravity in such a way he related the two speeds of light (and the temporal dimensions) so you get $R=c^2/c'$ when you consider gravity. Corben speculated that $R=c^2/c'$ could be related to the Planck’s legth $L_p$. Corben’s article is titled A classical theory of electromagnetism and gravity, Phys. Rev. 69, 225 (1946).

Arcidiacono’s interpretation of Fantappie’s hyperspherical universes is as follows: the Fantappie’s hyperspheres represent spherical surfaces in n dimensions, and these surfaces are embedded in certain euclidean space with (n+1) dimensions. Thus, we can introduce (n+1) parameters or coordinates

$(\xi_1,\xi_2,\ldots,\xi_n,\xi_0)$

and the hypersphere

$\xi_0^2+\xi_1^2+\ldots+\xi_n^2=r^2=R^2$

Define transformations

$\xi'_A=\alpha_{AB}\xi_B$ with $A,B=0,1,2,\ldots,n$

where $\alpha_{AB}$ are orthogonal $(n+1)\times (n+1)$ matrices with $\det \alpha_{AB}=+1$ for proper rotations. Then, $R_{n+1}\supset R_n$ and rotations in the $(\xi_A,\xi_B)$ plane are determined by $n(n+1)/2$ rotation angles. Moreover, you can introduce (n+1) projective coordinates $(\overline{x}_0,\overline{x}_1,\ldots,\overline{x}_n)$ such as the euclidean coordinates $(x_1,x_2,\ldots,x_n)$ are related with projective coordinates in the following way

$\boxed{x_i=\dfrac{r\overline{x}_i}{\overline{x}_0}}\;\; \forall i=1,2,\ldots,n$

Projective coordinates are generally visualized with the aid of the Beltrami-Reimann sphere, sometimes referred as Bloch or Poincarè sphere in Optics. The Riemann sphere is used in complex analysis. For instance:

This sphere is also used in Quantum Mechanics! In fact, projective geometry is the natural geometry for states in Quantum Physics. It is also useful in the Majorana representation of spin, also called star representation of spin by some authors, and riemann spheres are also the fundamental complex projective objects in Penrose’s twistor theory! To remark these statements, let me use some nice slides I found here http://users.ox.ac.uk/~tweb/00006/

Note: I am not going to explain twistor theory or Clifford algebra today, but I urge you to read the 2 wonderful books by Penrose on Spinors and Spacetime, or, in the case you are mathematically traumatized, you can always read his popular books or his legacy for everyone: The Road to Reality.

Projective coordinates are “normalized” in the sense

$\overline{x}_0^2+\ldots+\overline{x}_n^2=r^2$ or $\overline{x}_A\overline{x}_A=r^2$ $\forall A=0,1,\ldots,n$

It suggests us to introduce a pythagorean (“euclidean-like” ) projective “metric”:

$ds^2=d\overline{x}_Ad\overline{x}_A$

It is sometimes called the Beltrami metric. You can rewrite this metric in the following equivalent notation

$A^4ds^2=A^2(dx^idx^i)-(\alpha_i dx^i)^2$

with

$A^2=1+\alpha_s\alpha_s$ and $\alpha_s=x_s/r$

Some algebraic manipulations provide the fundamental tensor of projective relativity:

$\boxed{A^4 g_{ik}=A^2\delta_{ik}-\dfrac{x_ix_k}{r^2}}$

Here

$\vert g_{ik}\vert =g=A^{-2(n+1)}$ so

$\boxed{g^{ik}=(g_{ik})^{-1}=A^2\left( \delta_{ik}+\dfrac{x_ix_k}{r^2}\right)}$

The D’Alembertian operator is defined to be in this projective space

$\boxed{\square^2 \varphi =\dfrac{1}{\sqrt{g}}\partial_i\left(\sqrt{g}g^{ik}\partial_k \varphi\right)=0}$

Using projective “natural” coordinates with $r=1$ to be simpler in our analysis, we get

$A^{n+1}\partial_i\left[A^{-n-1}A^2(\delta_{ik}+x_ix_k)\partial_k\varphi\right]=0$

or

$A^{n+1}\left[(\partial_iA^{1-n})\partial_i\varphi+(A^{1-n})\partial_i(x_ix_k\partial_k\varphi)\right]=0$

But we know that

$\partial_iA^{1- n}=(1-n)A^{-1-n}x_i$

$\partial_ix_i=n$

$x_sx_s=A^2-1$

And then, if $r\neq 1$, we have the projective D’Alembertian operator

$\boxed{r^2\square^2=A^2\left(r^2\partial_i\partial_i\varphi +x_ix_k\partial_i\partial_k\varphi+2x_k\partial_k\varphi\right)=0}$

Here, $R_{n+1}$ is the tangent space (a projective space) with $\overline{x}'_A=\alpha_{AB}\overline{x}_B$, and where $A,B=0,1,\ldots,n$. We can return to “normal” unprojective relativistic framework choosing

$x'_i=\dfrac{\alpha_{ik}x_k+\alpha_{i0}}{\alpha_{00}+\left(\dfrac{\alpha_{i0}x_i}{r}\right)}$

with $x_i=0$ and $A=1$, and $\overline{x}_A=(r,0,\ldots,0)$. That is, in summary, we have that in projective relativity, using a proper relativistic reference frame, the position vector has NULL components excepting the 0th component $x_0=r=R$! And so, $\overline{x}_A=(r,0,\ldots,0)$ is a “special” reference frame in projective relativity. This phenomenon does not happen in euclidean or pseudoeuclidean relativity, but there is a “similar” phenomenon in group theory when you reduce the de Sitter group to the Poincaré group using a tool named “Inönü-Wigner” group contraction. I will not discuss this topic here!

It should be clear enough now that from $(x_1,\ldots,x_n)$, via $\overline{x}_i=x_i/A$ and $\overline{x}_0=r/A$, in the limit of infinite radius $R\longrightarrow \infty$, it reduces to the cartesian euclidean spaces $E_3,E_4,\ldots,E_n$. Nicola Kalitzin (1918-1970), to my knowledge, was one of the few (crackpot?) physicists that have studied multitemporal theories during the 20th century. He argued/claimed that the world is truly higher-dimensional, but ALL the extra dimensions are TIME-like! It is quite a claim, specially from a phenomenological aside! As far as I know he wrote a book/thesis, see here http://www.getcited.org/pub/101913498 but I have not been able to read a copy. I learned about his works thanks to some papers in the arxiv and a bulgarian guy (Z.Andonov) who writes about him in his blog e.g. here http://www.space.bas.bg/SENS2008/6-A.pdf

Arcidiacono has a nice review of Kalitzin multitemporal relativity (in the case of finite $n$ temporal dimensions), but I will modify it a litte bit to addapt the introduction to modern times. I define the Kalitzin metric as the following semiriemannian metric

$\boxed{\displaystyle{ds^2_{KAL}=dx_1^2+dx_2^2+dx_3^2-c_1^2dt_1^2-c_2^2dt_2^2-\ldots -c_{n-3}^2dt_{n-3}^2=\sum_{i=1}^3dx_i^2-\sum_{j=1}^{n-3}c_j^2dt_j^2}}$

Remark (I): It is evident that the above metric reduce to the classical euclidean metric or the Minkovski spacetime metric in the limites where we write $c_j=0$ and $c_1=c, c_{j+1}=0\forall j=1,2,\ldots,n-3$. There is ANOTHER way to recover these limits, but it involves some trickery I am not going to discuss it today. After all, new mathematics requires a suitable presentation! And for all practical purposes, the previous reduction makes the job (at least today).

Remark (II): Just an interesting crazy connection with algebraic “stuff” ( I am sure John C. Baez can enjoy this if he reads it)…

i) If $n-3=0$, then we have $n=3+0$ or 3D “real” (euclidean) space, with 0 temporal dimensions in the metric.

ii) If $n-3=1$, then we have $n=3+1$ or 4D pseudoeuclidean (semiriemannian?) spacetime, or equivalently, the (oldfashioned?) $x_4=ict$ relativity with ONE imaginary time, i.e. with 1 temporal dimension and 1 “imaginary unit” related to time!

iii) If $n-3=2$, then we have $n=3+2=5$ or 5D semiriemannian spacetime, a theory with 2 temporal imaginary dimensions, or 1 complex number (after complexification, we can take one real plus one imaginary unit), maybe related to projective dS/adS relativity in 5D, with $-i_0^2=-1=i_1^2$?

iv) If $n-3=3$, then we have $n=3+3=6$ or 6D semiriemannian spacetime, a theory with 3 temporal dimensions and 3 “imaginary units” related to …Imaginary quaternions $i^2=j^2=k^2=-1$?

v) If $n-3=7$, then we have $n=3+7=10$ or 10D semiriemannian spacetime, a theory with 3 temporal dimensions and 7 “imaginary units” related to …Imaginary octonions $i_1^2=i_2^2=\ldots =i_7^2=-1$?

vi) If $n-3=8$, then we have $n=3+7=11$ or 11D semiriemannian spacetime, a theory with 3 temporal dimensions and 8 “units” related to …Octonions $-i_0^2=i_1^2=i_2^2=\ldots =i_7^2=-1$?

Remark (III): The hidden division algebra connection  with the temporal dimensions of higher dimensional relativities and, in general, multitemporal relativities can be “seen” from the following algebraic facts

$n-3=0\leftrightarrow n=3=3+0\leftrightarrow t\in\mathbb{R}$

$n-3=1\leftrightarrow n=3=3+1\leftrightarrow t\in\mbox{Im}\mathbb{C}$

$n-3=2\leftrightarrow n=5=3+2\leftrightarrow t\in\mathbb{C}$

$n-3=3\leftrightarrow n=6=3+3\leftrightarrow t\in\mbox{Im}\mathbb{H}$

$n-3=4\leftrightarrow n=7=3+4\leftrightarrow t\in\mathbb{H}$

$n-3=7\leftrightarrow n=10=3+7\leftrightarrow t\in \mbox{Im}\mathbb{O}$

$n-3=8\leftrightarrow n=11=3+8\leftrightarrow t\in\mathbb{O}$

Remark (IV): Was the last remark suggestive? I think it is, but the main problem is how do we understand “additional temporal dimensions”? Are they real? Do they exist? Are they a joke as Feynman said when he derived electromagnetism from a non-associative “octonionic-like” multitemporal argument? I know, all this is absolutely crazy!

Remark (V): What about $(n-3)\longrightarrow \infty$ temporal dimensions. In fact, Kalitzin multitemporal relativity and Kalitzin works speculate about having $\infty$ temporal dimensions! I know, it sounds absolutely crazy, it is ridiculous! Specially due to the constants it would seem that there are convergence issues and some other weird stuff, but it can be avoided if you are “clever and sophisticated enough”.

Kalitzin metric introduces $(n-3)$ (a priori) “different” lightspeed species! If you faced problems understanding “light” in 4D minkovskian relativity, how do you feel about $\vec{C}=(c_1,\ldots,c_{n-3})$? Therefore, we can introduce $(n-3)$ proper times ( note that as far as I know at current time, N. Kalitzin introduces only a single proper time; I can not be sure since I have no access to his papers at the moment, but I will in future, I wish!):

$\boxed{-c_s^2d\tau_s^2=dx_1^2+dx_2^2+dx_3^2-c_1^2dt_1^2-\ldots-c_{n-3}^2dt_{n-3}^2}\;\forall s=1,\ldots,n-3$

Therefore, we can define generalized the generalized $\beta_s$ and $\Gamma_s$ parameters, the multitemporal analogues of $\beta$ and $\gamma$ in the following way. Fix some $s$ and $c_s, \tau_s$. Then, we have

$c_s^2d\tau_s^2=-dx_1^2-dx_2^2-dx_3^2+c_1^2dt_1^2+\ldots+c_s^2dt_s^2+\ldots+c_{n-3}^2dt_{n-3}^2$

$c_s^2\dfrac{d\tau_s^2}{dt_s^2}=-\dfrac{(d\vec{x})^2}{(dt_s)^2}+c_1^2\dfrac{dt_1^2}{dt_s^2}+\ldots+c_{n-3}^2\dfrac{dt_{n-3}^2}{dt_s^2}$

$c_s^2\dfrac{d\tau_s^2}{dt_s^2}=-\dfrac{(d\vec{x})^2}{(dt_s)^2}+c_1^2dt_1^2+\ldots+c_s^2+\ldots+c_{n-3}^2dt_{n-3}^2$

$\dfrac{d\tau_s^2}{dt_s^2}=-\dfrac{(d\vec{x})^2}{c_s^2(dt_s)^2}+c_1^2dt_1^2+\ldots+1+\ldots+c_{n-3}^2dt_{n-3}^2$

$\displaystyle{\dfrac{d\tau_s^2}{dt_s^2}=1-\dfrac{(d\vec{x})^2}{c_s^2(dt_s)^2}+\sum_{k\neq s}\dfrac{c_k^2dt_k^2}{c_s^2dt_s^2}}$

Define $B_s= v_{(s)}/c_s$ and $B_s= 1/\Gamma_s$ (be aware with that last notation), where $\Gamma_s, B_s$ are defined via the next equation:

$\boxed{\displaystyle{B_s= \dfrac{1}{\Gamma_s}=\sqrt{1-\beta_s^2+\sum_{k\neq s}\left(\dfrac{c_kdt_k}{c_sd\tau_s}\right)^2}=\sqrt{1-\dfrac{v_{(s)}^2}{c_s^2}+\sum_{k\neq s}\left(\dfrac{c_kdt_k}{c_sd\tau_s}\right)^2}}}$

and where

$\overrightarrow{V}_{(s)}=\vec v_s=\dfrac{d\vec{x}_\alpha}{dt_s}\;\;\forall \alpha=1,2,3$

Then

$\boxed{d\tau_s=B_sd\tau_s}$ or $\boxed{dt_s=\Gamma_s d\tau_s}$

Therefore, we can define $(n-3)$ different notions of “proper” velocity:

$\boxed{u_i^{(s)}=V^{(s)}=\dfrac{dx_i}{d\tau_s}=\dfrac{1}{B_s}\dfrac{dx_i}{dt_s}=\Gamma_s\dfrac{dx_i}{dt_s}=\Gamma_s \vec v_s}$

In the reference frame where $x_i=0$ AND/IFF $B_s=1$, then $u_i=0$ for all $i=1,2,3$ BUT there are $(s+3)$ “imaginary” components! That is, we have in that particular frame

$\boxed{u_{s+3}^{s}=ic_s} \;\;\forall s$

and thus

$\boxed{u_k^{(r)}u_k^{(s)}=-c_s^2\delta_{rs}}$

This (very important) last equation is strikingly similar to the relationship of reciprocal vectors in solid state physics but extended to the whole spacetime (in temporal dimensions!)! This is what I call “spacetime crystals” or “crystalline (multitemporal) relativity”. Relativity with extra temporal dimensions allows us to define some kind of “relativity” in which the different proper velocities define some kind of (relativistic) lattice. Wilczek came to the idea of “time crystal” in order to search for “periodicities” in the time dimension. With only one timelike dimension, the possible “lattices” are quite trivial. Perhaps the only solution to avoid that would be consider 1D quasicrystals coming from “projections” from higher dimensional “crystals” (quasicrystals in lower dimensions can be thought as crystals in higher dimensions). However, if we extend the notion of unidimensional time, and we study several time-like dimensions, new possibilities arise to build “time crystals”. Of course, the detection of extra timelike dimensions is an experimental challenge and a theoretical one, but, if we give up or solve the problems associated to multiple temporal dimensions, it becomes clear that the “time crystals” in D>1 are interesting objects in their own! Could elementary particles be “phonons” in a space-time (quasi)crystal? Is crystalline (multitemporal) relativity realized in Nature? Our common experience would suggest to the contrary, but it could be interesting to pursue this research line a little bit! What would it be the experimental consequence of the existence of spacetime crystals/crystalline relativity? If you have followed the previous discussion: spacetime crystals are related to different notions of proper velocity (the analogue of reciprocal vectors in solid state physics) and to the existence of “new” limit velocities or “speeds of light”. We only understand the 5% of the universe, according to WMAP/Planck, so I believe that this idea could be interesting in the near future, but at the moment I can not imagine some kind of experiment to search for these “crystals”. Where are they?

Remark: In Kalitzinian metrics, “hyperphotons” or “photons” are defined in the usual way, i.e., $ds_{KAL}^2=0$, so

$\mbox{Hyperphotons}: ds_{KAL}^2=0\leftrightarrow dx_1^2+dx_2^2+dx_3^2=c_1^2dt_1^2+\ldots+c_{n-3}^2dt_{n-3}^2$

Remark(II): In multitemporal or crystalline relativities, we have to be careful with the notion of “point” at local level, since we have different notions of “velocity” and “proper velocity”. Somehow, in every point, we have a “fuzzy” fluctuation along certain directions of time (of course we can neglect them if we take the limit of zero/infinity lightspeed along some temporal direction/time vectors). Then, past, present and future are “fuzzy” notions in every spacetime when we consider a multitemporal approach! In the theory of relativity in Clifford spaces, something similar happens when you consider every possible “grade” and multivector components for a suitable cliffor/polyvector. The notion of “point” becomes meaningless since you attach to the point new “degrees of freedom”. In fact, relativity in Clifford spaces is “more crystalline” than multitemporal relativity since it includes not only vectors but bivectors, trivectors,… See this paper for a nice review: http://vixra.org/pdf/0908.0084v1.pdf

Remark (III):  Define the “big lightspeeds” in the following way

$\boxed{C_s^2=v_s^2=\dfrac{(dx_i)^2}{(dt_s)^2}}\;\;\forall s=1,2,\ldots,n-3$

or

$\boxed{C_s=v_s=\dfrac{dx_i}{dt_s}}\;\;\forall s=1,2,\ldots,n-3$

Then, we have

$C_s^2=\dfrac{c_1^2dt_1^2}{dt_s^2}+\ldots+\dfrac{c_{n-3}^2dt_{n-3}^2}{dt_s^2}$

$C_s^2=c_s^2\dfrac{c_1^2dt_1^2}{c_s^2dt_s^2}+\ldots+c_s^2+\ldots+c_s^2\dfrac{c_{n-3}^2dt_{n-3}^2}{c_s^2dt_s^2}$

$C_s^2=c_s^2\left(\dfrac{c_1^2dt_1^2}{c_s^2dt_s^2}+\ldots+1+\ldots+\dfrac{c_{n-3}^2dt_{n-3}^2}{c_s^2dt_s^2}\right)$

$\displaystyle{C_s^2=c_s^2\left(1+\sum_{k\neq s}^{n-3}\left(\dfrac{c_kdt_k}{c_sdt_s}\right)^2\right)}$

where we note that

$\boxed{\displaystyle{C_s^2=c_s^2\left(1+\sum_{k\neq s}^{n-3}\left(\dfrac{c_kdt_k}{c_sdt_s}\right)^2\right)}\geq c_s^2}$

or

$\boxed{\displaystyle{C_s=c_s\sqrt{\left(1+\sum_{k\neq s}^{n-3}\left(\dfrac{c_kdt_k}{c_sdt_s}\right)^2\right)}}\geq c_s}$

The bound is saturated whenever we have $c_s\longrightarrow\infty$ or $c_k=0$. Such conditions, or the hypothesis of unidimensional time, leave us with the speed of light barrier, but IT IS NO LONGER A BARRIER IN A MULTITEMPORAL SET-UP!

Remark (I): Just for fun…Sci-Fi writers are wrong when they use the “hyperspace” to skip out the lightspeed barrier. What allows to give up such a barrier is MULTITEMPORAL TIME or the hypertime. Of course, if they mean “hyperspacetime”, it would not be so wrong. It is trivial to observe that if you include extra SPACE-LIKE dimensions, and you keep Lorentz Invariance in higher-dimensions, you can NOT scape from the speed of light limit in a classical “way”. Of course, you could use wormholes, Alcubierre drives or quantum “engines”, but they belong to a different theoretical domain I am not going to explain here. Not now.

Remark (II): If we suppose that every speed of light is constant (homogeneity in extradimensional time) and if we suppose, in addition to it, that they are all equal to the same number, say the known $c$, i.e., if we write

$c_1=c_2=\ldots=c_{n-3}=c$

then we can easily obtain that

$\boxed{C_s=c_s\sqrt{1+(n-4)}=c_s\sqrt{n-3}}$

And then, we have

1) n=3 (0 timelike dimensions) implies that $C_s=c_s=0$

2) n=4 (1 timelike dimension) implies that $C_s=c_s=c$

3) n=5 (2 timelike dimensions) implies that $C_s=\sqrt{2}c_s\approx 1.4c$

3) n=6 (3 timelike dimensions) implies that $C_s=\sqrt{3}c_s\approx 1.7c$

4) n=7 (4 timelike dimensions) implies that $C_s=\sqrt{4}c_s=2c_s$

5) n=8 (5 timelike dimensions) implies that $C_s=\sqrt{5}c_s\approx 2.2c$

6) n=9 (6 timelike dimensions) implies that $C_s=\sqrt{6}c_s\approx 2.4c$

7) n=10 (7 timelike dimensions) implies that $C_s=\sqrt{7}c_s\approx 2.6c$

8) n=11 (8 timelike dimensions) implies that $C_s=\sqrt{8}c_s\approx 2.8c$

9) n=12 (9 timelike dimensions) implies that $C_s=\sqrt{9}c_s=3c$

10) $n=\infty$ ($\infty -3=\infty$  timelike dimensions) implies that $C_s=\infty$, and you can travel to virtually any velocity !!!!!!But of course, it seems this is not real, infinite timelike dimensions sound like a completely crazy stuff!!!!! I should go to the doctor…

Remark(III): The main lesson you should learn from this is that spacelike dimensions can not change the speed of light barrier. By the contrary, the true power of extra timelike dimensions is understood when you realize that “higher dimensional” excitations of “temporal dimensions” provide a way to surpass the speed of light. I have no idea of how to manage this, I am only explaining you what are the consequences of the previous stuff.

Remark (IV): Just for fun (or not). I am a big fan of Asimov’s books. Specially the Foundation series and the Robot stories. When I discovered these facts, long ago, I wondered myself if Isaac Asimov met Kalitzin/Arcidiacono (I think he could not meet Fantappie or Fantappie’s works about projective relativity but I am sure he knew a little bit about hyperspace and hypertime, despite the fact he, as many others at current time, confused the idea of hyperspace and hypertime, but sometimes he seemed to know more than he was explaining. I am not sure. I am not a Sci-fi writer…But I suppose he knew “something”…But not exactly these facts). I think to remember a quote from one of his books in which a character said something like “(…)One of the biggest mistakes of theoretical physicists is to confuse the hyperspace unlimited C with the bounded velocity c in usual relativity(…)”. I think these are not the exact words, but I remember I read something like that in some of his books. I can not remember what and I have no time to search for it right now, so I leave this activity to you…To find out where Asimov wrote something very close to it. Remember my words are not quite exact, I presume…I have not read a “normal” Sci-Fi book since years ago!

Arcidiacono worked out a simple example of multitemporal theory. He formulated the enhacen galilean group in the following way

$x'=x+V_1t+V_2t^2/2+\ldots+V_{n-3}t^{n-3}/(n-3)!$

$t'=t$

with $V_1$ the velocity, $V_2$ the acceleration, $V_3$ the jerk,…$V_{n-3}$ the (n-3)th order velocity. He linearized that nonlinear group using the transformations

$t_s=t^s/s!$ $\forall s=1,2,\ldots,n-3$

and it gives

$x'=x+V_1t_1+V_2t_2+\ldots+V_{n-3}t_{n-3}$

$t'_1=t_1$

$t'_2=t_2$$t'_{n-3}=t_{n-3}$

So we have a group matrix

$G=\begin{pmatrix}1 & V_1 & \cdots & V_{n-3}\\ 0 & 1 & \cdots & 0\\ \cdots & \cdots & \cdots & \cdots\\ \cdots & \cdots & \cdots & 1\end{pmatrix}$

The simplest case is usual galilean relavity

$x'=x+Vt$

$t'=t$

The second simpler example is two time enhaced galilean relativity:

$x'=x+Vt_1+V_2t_2$

$t'_1=t_1$ $t'_2=t'_2$

If we use that $V_1=V$ and $t_s=t^s/s!$, then we have

$\dfrac{dx}{dt_s}=\dfrac{s-1}{t^{s-1}}\dfrac{dx}{dt}$

and then

$V_s=(s-1)t^{1-s}V$

With 2 times, we have $V_2=V/t$, and moreover, the free point particle referred to $t_s$ satisfies (according to Arcidiacono)

$\dfrac{d^2x}{dt_s^2}=0\leftrightarrow \dfrac{d^2x}{dt^2}-\left(\dfrac{s-1}{t}\right)\dfrac{dx}{dt}=0$

Let us work out this case with more details

$X'=X+Vt+At^2/2$

$T=t$

where we have 3 spatial coordinates (x,y,z) and two times (t,t’). Performing the above transformations

$X'=x+Vt+At'$

$T=t$ $T'=t'$

with velocities

$V=\dfrac{dx}{dt}$ and $V'=\dfrac{dx}{dt'}$, and with $V'=V/t$. If $V=At$, then $V'=A$, so a second order velocity becomes the constant acceleration in that frame. Furthermore

$\dfrac{d^2x}{dt'^2}=0$

implies that

$\dfrac{dV}{dt}=\dfrac{V}{t}$ and $x=At^2/2$

That is, invariant mechanics under uniformly accelerated motion with “multiple” velocities is possible! In fact, in this framework, uniformly accelerated motion seems to be “purely inertial”, or equivalently, it seems to be “fully machian”!!!!

If uniformly accelerated gravitational field is applied to the point particle, then, in this framework, it seems to suggest that it “changes” the time scale a quantity

$t'=t^2/2$

and it becomes a uniform motion! If a body moves unofrmorly, changing the scale of time, in multitemporal relativity, ib becomes uniformaly accelerated! I don’t understand this claim well enough, but it seems totally crazy or completely …Suggestive of a purely machian relativity? Wilczek called it “total relativity” long ago…

A conformal relativity with two time dimensions and two time dimensions was also studied by Arcidiacono (quite naively, I believe). He studied also a metric

$ds^2=dx^2+dy^2+dz^2-c^2dt^2-c'^2dt'^2$

with a conformal time

$t'=\dfrac{c^2t^2-x^2}{2c^2}$

Note that $c\longrightarrow \infty$ implies that $t'=t^2/2$. It implies some kind of hyperbolic motion

$V=\dfrac{At}{\sqrt{1+\dfrac{A^2t^2}{c^2}}}$

and

$x=\dfrac{c^2}{A^2}\left[-1+\sqrt{1+\dfrac{A^2t^2}{c^2}}\right]$

Remark: $Ax^2+2c^2x-Ac^2t^2=0\leftrightarrow x=\dfrac{A}{2c^2}\left(c^2t^2-x^2\right)$. Introductin a second time $x=At'$, then $V'=A$, where

$V'=\dfrac{V}{t-\dfrac{Vx}{c^2}}$

and again $V'=A$ produces the “classical relativity”.

Remark(II): Projective special relativity should produce some kind of “projective general relativity” (Arcidiacono claimed). This is quite a statement, since the diffeomorphism group in general relativity contains “general coordinate transformations”. I am not sure what he meant with that. Anyway, a projective version of “general relavity” is provided by twistor theory or similar theories, due to the use of complex projective spaces and generalizations of them. Conformal special relativity should imply some class of conformal general relativity. However, physical laws are not (apparently) invariant under conformal transformations in general. What about de Sitter/anti de Sitter spaces? I have to learn more about that and tell you about it in the future. Classical electromagnetism and even pure Yang-Mills theories at classical level can be made invariant under conformal transformations only with special care. Quantum Mechanics seems  to break that symmetry due to the presence of mass terms that spoil the gauge invariance of the theory, not only the conformal symmetry. Only the Higgs mechanism and “topological” terms allow us to introduce “mass terms” in a gauge invariant way! Any way, remember that Classical Mechanics is based on symplectic geometry, very similar to projective geometry in some circumstances, and Classical Field Theories also contain fiber fundles and some special classes of field theories, like Conformal Field Theories or even String Theory, have some elements of projective geometry in their own manner. Moreover, conformal symmetries are also an alternative approach to new physics. For instante, Georgi created the notion of a “hidden conformal sector” BSM theory, something that he called “unparticles”. People generalized the concept and you can read about “ungravity” as well. Unparticles, ungravity, unforces…Really weird stuff!!! Did you think multiple temporal dimensions were the only uncommon “ugly ducks” in the city? No, they weren’t…Crazy ideas are everywhere in theoretical physics. The real point is to find them applications and/or to find them in real experiments! It happened with this Higgs-like particle about 127GeV/c². And I think Higgs et alii will deserve a Nobel Prize this year due to it.

Remark (III): Final relativity, in the sense of Fantappie’s ideas, has to own a different type of Cosmology… In fact it has. It has a dS relativity Cosmology! The Stantard Cosmological Model fits the vacuum energy (more precisely we “fit” $\Omega_\Lambda$). It is important to understand what $\Lambda$ is. The Standard Cosmological Model does not explain it at all. We should explore the kinematical and cosmological models induced by the de Sitter group, and its associated QFT. However, QFT on dS spaces are not fully developed. So, that is an important research line for the future.

Arcidiacono generalizes electromagnetism to multitemporal dimensions (naively he “wrongly” thought he had unified electromagnetism and hydrodynamics) with the followin equations

$\mbox{Rot}H_{AB}=J_{ABC}$

$\mbox{Div}H_{AB}=I_A$

where $A,B=0,1,\ldots, n$. The tensor $H_{AB }$ have $n(n +1)/2$ components. The integrability conditions are

$\mbox{Rot}J_{ABC}=0$

and

$\mbox{Div}I_A=0$

We can build some potentials $U_A$, and $V_{ABC}$, so

$\mbox{Div}U_A=0$

$\mbox{Rot}V_{ABC}=0$

with $H_{AB}=\mbox{Div}V_{ABC}+\mbox{Rot}U_A$

we have

$\square^2V_{ABC}=J_{ABC}$ and $\square^2 U_A=I_A$

A generalized electromagnetic force is introduce

$2f_A=J_{ABC}H_{BC}-2I_BH_{AB}$

If $f_A=\mbox{Div}T_{AB}$, then the energy-momentum tensor will be

$T_{AB}=H_{AS}H_{SB}+\dfrac{1}{4}H_{RS}H_{RS}\delta_{AB}$

For position vectors $\overline{x}_A$, we have $(n-3)$ projectie velocities \$late \overline{u}_A^s, such as

$\overline{u}^s_A=\dfrac{d\overline{x}_A}{d\tau_s}$

$\boxed{\overline{u}_A^{(r)}\overline{u}_A^{(s)}=-c_s^2\delta_{rs}}$

where $\overline{x}_A\overline{x}_A=r^2$ and $\overline{x}_A\overline{u}_A^s=0$. From $H_{AB}$ we get

(1) $c_A$ hydrodynamics vector plus (n-3) magnetic vectors $h_A^s$ such as

$c_A=H_{AB}x_B$

$h_A^s=H_{AB}u^s_B=H_{AB}u^s_B$

and where

$c_Ax_A=0$ and $h_A^su^s_A=0$.

(2) Fluid indices for

$f^s=H_{AB}x_Au_B^s$

$f^{rs}=H_{AB}u_A^ru_B^s$

with

$(n-3)+\begin{pmatrix}n-3\\ 2\end{pmatrix}=\begin{pmatrix}n-2\\ 2\end{pmatrix}=\dfrac{(n-2)(n-3)}{2}$ total components. Note that if you introduce n=4 you get only 1 single independent component.

(3) The dual tensor $\star H_{ABC\ldots D}$ to $H_{AB}$ has (n-1) undices, so we can make

$K_{AB}=\star H_{ABC\ldots D}u_A^1u_B^2\ldots u_C^{n-3}$ and then $K_{AB}u_B^s=0$. The generalized electric field reads

$e=K_{AB}x_B$

so $e_Ax_A=e_Au_A^s=0$

Note that in this last equation, projective relativity means a total equivalence in a transformation changing position and multitemporal velocities, i.e., invariance under $x_A\leftrightarrow u_A^s$ is present in the last equation for electric fields in a multitemporal setting.

1) Multitemporal theories of relativity do exist. In fact, Dirac himself and De Donder studied this types of theories. However, they did not publish too much papers about this crazy subject.

2) Fantappie’s final relativity is an old idea that today can be seen as de Sitter Relativity. The contraction of the de Sitter group provides the Lorentz groupo. Final relativity/de Sitter relativity is based on “projective geometry” somehow.

3) Kalitzin’s and Arcidiacono’s ideas, likely quite naive and likely wrong, does not mean that multitemporal dimensions don’t exist. The only problem is to explain why the world is 3+1 if they exist or, equivalently, just as the space-like dimensions, the perception of multiple temporal dimensions is an experimental issue.

4) The main issues for extra timelike dimensions are: closed time-like curves, causality and vacuum instabilities (“imposible” processes) when Quantum Mechanics is taken into account in multi-time setting.

5) Beyond multi-time theories, there are interesting extensions of special relativity, e.g., C-space relativity.

6) Multiple temporal dimensions make the notion of point and event a little “fuzzy”.

7) Multiple time-like dimensions are what make possible to overpass the invariant speed of light. I am not going to prove it here, in the case of $c_k=c\forall k$ the maximum invariant velocity is equal to $\sqrt{n-3}c$. When the speeds of light are “different” the invariant velocity is a harder formula, but it does exist. From this viewpoint, it is hypertime dimensions and not hyperspace dimensions what make possible the faster than light travel (Giving up CTC, causality issues and vacuum instabilities triggered by quantum theories).

8) Hyperphotons are the equivalent concept of photons in multitemporal relativities and they are not tachyons, but they have a different invariant speed.

9) Philosophers have discussed the role of multitemporal dimensions. For instance, I read about Bennett 3d time, with 3 components he called time, hyparxis and eternity long ago, see here http://en.wikipedia.org/wiki/John_G._Bennett.

10) Isaac Asimov stories, beyond the imagination and intuition Asimov had, match the theory of relavity with extra time-like and space-like dimensions. I don’t know if he met Kalitzin, Dirac or some other physicist working on this field, but it is quite remarkable from the purely layman approach!

11) Theories with extra temporal dimensions have been studied by both mathematicians and physicists. At current time, maybe I can point out that F-theory has two timelike dimensions, Itzhak Bars has papers about two-time physics, semiriemannian (multitemporal) metrics are being studied by the balkan and russian schools and likely many others.

12) The so-called problem of time is even more radical when you deal with multi-time theories because the relation of multitemporal coordinates with the physical time is obscure. We don’t understand time.

13) We can formulate theories in a multi-time setting, but it requires a harder framework than in normal relativity: velocity becomes “a matrix”, there are different notions of accelerations, energy becomes a vector, “mass” is a “tensor”, multi-time electrodynamics becomes more difficult and many other issues arise with a multi-time setting. You have to study: jet theory, Finsler spaces, nonlinear connections, and some more sophisticated machinery in order to understand it.

14) Are multi-time theories important? Maybe…The answer is that we don’t know for sure, despite the fact that they are “controversial” and “problematic”. However, if you think multi-time theories are “dark”, maybe you should thing about that “dark stuff” forming the 95% of the Universe. However, Irina Aref’eva and other authors have studied the physical consequences of multi-time theores. Aref’eva herself, in collaboration with other russian physicists, proved that an additional timelike dimension can solve the cosmological constant problem (giving up any issue that an additional time dimension produces).

15) The idea of “time crystals” is boring in 1d time. It becomes more interesting when you thing about multi-time crystals as some of the ingredients of certain “crystalline relativity”. In fact, a similar idea has been coined by P. Jizba et alii, and it is known as “World Crystal”.

16) Final questions:

i) Can multi-time relativity be used by Nature? The answer can only be answered from an experimental viewpoint!

ii) Do we live in an anisotropic spacetime (quasi)crystal? I have no idea! But particles theirselves could be seen as (quantum) excitations of the spacetime crystal. In fact, I am wondering if the strange spectrum of the Standard Model could be some kind of 3d+1 time quasicrystal. If it is so, it could be that in certain higher dimensions, the spectrum of the SM could be more “simple”. Of course, it is the idea of extra dimensions, but I have not read any paper or article studying the SM particle spectrum from a quasicrystal viewpoint. It could be an interesting project to make some investigations about this idea.

iii) How many lightspeeds are there in the Universe? We can put by hand that every “lightspeed” species is equal to the common speed of light, but is it right? Could exist new lightspeed species out there? Note that if we considered those “higher lightspeeds” very large numbers, they could be unnoticed by us if the “electromagnetism” in the extra temporal dimensions were far different than the known electromagnetism. That is, it could be that $c=c_1< or that some of them were very small constants…In both cases, normal relativity could be some kind of “group” reduction.

iv) Could the time be secretly infinite-dimensional? Experiments show that the only invariant speed is c, but could it be an illusion?

v) Can we avoid the main problems of multi-time theories? I mean causality, Closed Timelike Curves (CTC), and vacuum instabilities as the most important of all of them.

vi) Is the problem of time related to the the multitemporality of the world?

# LOG#099. Group theory(XIX).

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).

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 $J^2=J_x^2+J_y^2+J_z^2$

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

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

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

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

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

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!