# LOG#116. Basic Neutrinology(I).

This new post ignites a new thread.

Subject: the Science of Neutrinos. Something I usually call Neutrinology.

I am sure you will enjoy it, since I will keep it elementary (even if I discuss some more advanced topics at some moments). Personally, I believe that the neutrinos are the coolest particles in the Standard Model, and their applications in Science (Physics and related areas) or even Technology in the future ( I will share my thoughts on this issue in a forthcoming post) will be even greater than those we have at current time.

Let me begin…

The existence of the phantasmagoric neutrinos ( light, electrically neutral and feebly -very weakly- interacting fermions) was first proposed by W. Pauli in 1930 to save the principle of energy conservation in the theory of nuclear beta decay. The idea was promptly adopted by the physics community but the detection of that particle remained elusive: how could we detect a particle that is electrically neutral and that interact very,very weakly with normal matter? In 1933, E. Fermi takes the neutrino hypothesis, gives the neutrino its name (meaning “little neutron”, since it was realized than neutrinos were not Chadwick’s neutrons) and builds his theory of beta decay and weak interactions. With respect to its mass, Pauli initially expected the mass of the neutrino to be small, but necessarily zero. Pauli believed (originally) that the neutrino should not be much more massive than the electron itself. In 1934, F. Perrin showed that its mass had to be less than that of the electron.

By the other hand, it was firstly proposed to detect neutrinos exploding nuclear bombs! However, it was only in 1956 that C. Cowan and F. Reines (in what today is known as the Reines-Cowan experiment) were able to detect and discover the neutrino (or more precisely, the antineutrino). In 1962, Leon M. Lederman, M. Schwartz, J. Steinberger and Danby et al. showed that more than one type of neutrino species $\nu_e,\nu_\mu$ should exist by first detecting interactions of the muon neutrino. They won the Nobel Prize in 1988.

When we discovered the third lepton, the tau particle (or tauon), in 1975 at the Stanford Linear Accelerator Center, it too was expected to have an associated neutrino particle. The first evidence for this 3rd neutrino “flavor” came from the observation of missing energy and momentum in tau decays. These decays were analogue to the beta decay behaviour leading to the discovery of the neutrino particle.

In 1989, the study of the Z boson lifetime allows us to show with great experimental confidence that only 3 light neutrino species (or flavors) do exist. In 2000, the first detection of tau neutrino ($\nu_\tau$ in addition to $\nu_e,\nu_\mu$) interactions was announced by the DONUT collaboration at Fermilab, making it the latest particle of the Standard Model to have been discovered until the recent Higgs particle discovery (circa 2012, about one year ago).

In 1998, research results at the Super-Kamiokande neutrino detector in Japan (and later, independently, from SNO, Canada) determined for the first time that neutrinos do indeed experiment “neutrino oscillations” (I usually call NOCILLA, or NO for short, this phenomenon), i.e., neutrinos flavor “oscillate” and change their flavor when they travel  “short/long” distances. SNO and Super-Kamiokande tested and confirmed this hypothesis using “solar neutrinos”. this (quantum) phenomenon implies that:

1st. Neutrinos do have a mass. If they were massless, they could not oscillate. Then, the old debate of massless vs. massive neutrinos was finally ended.

2nd. The solar neutrino problem is solved. Some solar neutrinos scape to the detection in Super-Kamiokande and SNO, since they could not detect all the neutrino species. It also solved the old issue of “solar neutrinos”. The flux of (detected) solar neutrinos was lesser than expected (generally speaking by a factor 2). The neutrino oscillation hypothesis solved it since it was imply the fact that some neutrinos have been “transformed” into a type we can not detect.

3rd. New physics does exist. There is new physics at some energy scale beyond the electroweak scale (the electroweak symmetry breaking and typical energy scale is about 100GeV). The SM is not complete. The SM does (indeed) “predict” that the neutrinos are massless. Or, at least, it can be made simpler if you make neutrinos to be massless neutrinos described by Weyl spinors. It shows that, after the discovery of neutrino oscillations, it is not the case. Neutrinos are massive particles. However, they could be Dirac spinors (as all the known spinors in the Standard Model, SM) or they could also be Majorana particles, neutral fermions described by “Majorana” spinors and that makes them to be their own antiparticles! Dirac particles are different to their antiparticles. Majorana particles ARE the same that their own antiparticles.

In the period 2001-2005, neutrino oscillations (NO)/neutrino mixing phenomena(NEMIX) were observed for the first time at a reactor experiment (this type of experiment are usually referred as short baseline experiment in the neutrino community) called KamLAND. They give a good estimate (by the first time) of the difference in the squares of the neutrino masses. In May 2010, it was reported that physicists from CERN and the Italian National Institute for Nuclear Physics, in Gran Sasso National Laboratory, had observed for the first time a transformation between neutrino flavors during an accelerator experiment (also called neutrino beam experiment, a class of neutrino experiment belonging to “long range” or “long” baseline experiments with neutrino particles). It was a new solid evidence that at least one neutrino species or flavor does have mass. In 2012, the Daya Bay Reactor experiment in China, and later RENO in South Korea measured the so called $\theta_{13}$ mixing angle, the last neutrino mixing angle remained to be measured from the neutrino mass matrix. It showed to be larger than expected and it was consistent with earlier, but less significant results by the experiments T2K (another neutrino beam experiment), MINOS (other neutrino beam experiment) and Double Chooz (a reactor neutrino experiment).

With the known value of $\theta_{13}$ there are some probabilities that the $NO\nu A$ experiment at USA can find the neutrino mass hierarchy. In fact, beyond to determine the spinorial character (Dirac or Majorana) of the neutrino particles, and to determine their masses (yeah, we have not been able to “weight” the neutrinos, but we are close to it: they are the only particle in the SM with no “precise” value of mass), the remaining problem with neutrinos is to determine what kind of spectrum they have and to measure the so called CP violating processes. There are generally 3 types of neutrino spectra usually discussed in the literature:

A) Normal Hierarchy (NH): $m_1<. This spectrum follows the same pattern in the observed charged leptons, i.e., $m(e)<. The electron is about 0.511MeV, muon is about 106 MeV and the tau particle is 1777MeV.

B) Inverted Hierarchy (IH): $m_1<. This spectrum follows a pattern similar to the electron shells in atoms. Every “new” shell is closer in energy (“mass”) to the previous “level”.

C) Quasidegenerated (or degenerated) hierarchy/spectrum (QD): $m_1\sim m_2\sim m_3$.

While the above experiments show that neutrinos do have mass, the absolute neutrino mass scale is still not known. There are reasons to believe that its mass scale is in the range of some milielectron-volts (meV) up to the electron-volt scale (eV) if some extra neutrino degree of freedom (sterile neutrinos) do appear. In fact, the Neutrino OScillation EXperiments (NOSEX) are sensitive only to the difference in the square of the neutrino masses. There are some strongest upper limits on the masses of neutrinos that come from Cosmology:

1) The Big Bang model states that there is a fixed ratio between the number of neutrino species and the number of photons in the cosmic microwave background (CMB). If the total energy of all the neutrino species exceeded an upper bound about

$m_\nu\leq 50eV$

per neutrino, then, there would be so much mass in the Universe that it would collapse. It does not (apparently) happen.

2) Cosmological data, such as the cosmic microwave background radiation, the galaxy surveys, or the technique of the Lyman-alpha forest indicate that the sum of the neutrino masses should be less than 0.3 eV (if we don’t include sterile neutrinos, new neutrino species uncharged under the SM gauge group, that could increase that upper bound a little bit).

3) Some early measurements coming from lensing data of a galaxy cluster were analyzed in 2009. They suggest that the neutrino mass upper bound is about 1.5eV. This result is compatible with all the above results.

Today, some measurements in controlled experiments have given us some data about the squared mass differences (from both, solar neutrinos, atmospheric neutrinos produced by cosmic rays and accelerator/reactor experiments):

1) From KamLAND (2005), we get

$\Delta m_{21}^2=0\mbox{.}000079eV^2$

2) From MINOS (2006), we get

$\Delta m_{32}^2=0\mbox{.}0027eV^2$

There are some increasing efforts to directly determine the absolute neutrino mass scale in different laboratory experiments (LEX), mainly:

1) Nuclear beta decay (KATRIN, MARE,…).

2) Neutrinoless double beta decay (e.g., GERDA; CUORE, Cuoricino, NEMO3,…). If the neutrino is a Majorana particle, a new kind of beta decay becomes possible: the double beta decay without neutrinos (i.e., two electrons emitted and no neutrino after this kind of decay).

Neutrinos have a unique place among all the SM elementary particles. Their role in the cosmic evolution and the fundamental asymmetries in the SM (like CP violating reactions, or the C, T, and P single violations) make them the most fascinating and interesting particle that we know today (well, maybe, today, the Higgs particle is also as mysterious as the neutrino itself). We believe that neutrinos play an important role in Beyond Standard Model (BSM) Physics. Specially, I would like to highlight two aspects:

1) Baryogenesis from leptogenesis. Neutrinos can allow us to understand how could the Universe end in such an state that it contains (essentially) baryons and no antibaryons (i.e., the apparent matter-antimatter asymmetry of the Universe can be “explained”, with some unsolved problems we have not completely understood, if massive neutrinos are present).

2) Asymmetric mass generation mechanisms or the seesaw. Neutrinos allow us to build an asymmetric mass mechanism known as “seesaw” that makes “some neutrino species/states” very light and other states become “superheavy”. This mechanism is unique and, from some  non-subjective viewpoint, “simple”.

After nearly a century, the question of the neutrino mass and its origin is still an open question and a hot topic in high energy physics, particle physics, astrophysics, cosmology and theoretical physics in general.

If we want to understand the fermion masses, a detailed determination of the neutrino mass is necessary. The question why the neutrino masses are much smaller than their charged partners could be important! The little hierarchy problem is the problem of why the neutrino mass scale is smaller than the other fermionic masses and the electroweak scale. Moreover, neutrinos are a powerful probe of new physics at scales larger than the electroweak scale. Why? It is simple. (Massive) Neutrinos only interact under weak interactions and gravity! At least from the SM perspective, neutrinos are uncharged under electromagnetism or the color group, so they can only interact via intermediate weak bosons AND gravity (via the undiscovered gravitons!).

If neutrino are massive particles, as they show to be with the neutrino oscillation phenomena, the superposition postulates of quantum theory state that neutrinos, particles with identical quantum numbers, could oscillate in flavor space since they are electrically neutral particles. If the absolute difference of masses among them is small, then these oscillations or neutrino (flavor) mixing could have important phenomenological consequences in Astrophysics or Cosmology. Furthermore, neutrinos are basic ingredients of these two fields (Astrophysics and Cosmology). There may be a hot dark matter component (HDM) in the Universe: simulations of structure formation fit the observations only when some significant quantity of HDM is included. If so, neutrinos would be there, at least by weight, and they would be one of the most important ingredients in the composition of the Universe.

Regardless the issue of mass and neutrino oscillations/mixing, astrophysical interests in the neutrino interactions and their properties arise from the fact that it is produced in high temperature/high density environment, such as collapsing stars and/or supernovae or related physical processes. Neutrino physics dominates the physics of those astrophysical objects. Indeed, the neutrino interactions with matter is so weak, that it passes generally unnoticed and travels freely through any ordinary matter existing in the Universe. Thus, neutrinos can travel millions of light years before they interact (in general) with some piece of matter! Neutrinos are a very efficient carrier of energy drain from optically thick objects and they can serve as very good probes for studying the interior of such objects. Neutrino astronomy is just being born in recent years. IceCube and future neutrino “telescopes” will be able to see the Universe in a range of wavelengths and frequencies we have not ever seen till now. Electromagnetic radiation becomes “opaque” at some very high energies that neutrinos are likely been able to explore! Isn’t it wonderful? Neutrinos are high energy “telescopes”!

By the other hand, the solar neutrino flux is, together with heliosysmology and the field of geoneutrinos (neutrinos coming from the inner shells of Earth), some of the known probes of solar core and the Earth core. A similar statement applies to objects like type-II supernovae. Indeed, the most interesting questions around supernovae and the explosion dynamics itself with the shock revival (and the synthesis of the heaviest elements by the so-called r-processes) could be positively affected by changes in the observed neutrino fluxes (via some processes called resonant conversion, and active-sterile conversions).

Finally, ultra high energy neutrinos are likely to be useful probes of diverse distant astrophysical objects. Active Galactic Nuclei (AGN) should be copious emitters of neutrinos, providing detectable point sources and and observable “diffuse” background which is larger in fact that the atmospheric neutrino background in the very high energy range. Relic cosmic neutrinos, their thermal background, known as the cosmic neutrino background, and their detection about 1.9K are one of the most important lacking missing pieces in the Standard Cosmological Model (LCDM).

Do you understand why neutrinos are my favorite particles? I will devote this basic thread to them. I will make some advanced topics in the future. I promise.

May the Neutrinos be with you!

# LOG#097. Group theory(XVII).

The case of Poincaré symmetry

There is a important symmetry group in (relativistic, quantum) Physics. This is the Poincaré group! What is the Poincaré group definition? There are some different equivalent definitions:

i) The Poincaré group is the isometry group leaving invariant the Minkovski space-time. It includes Lorentz boosts around the 3 planes (X,T) (Y,T) (Z,T) and the rotations around the 3 planes (X,Y) (Y,Z) and (Z,X), but it also includes traslations along any of the 4 coordinates (X,Y,Z,T). Moreover, the Poincaré group in 4D is a 10 dimensional group. In the case of a ND Poincaré group, it has $N(N-1)/2+N$ parameters/dimensions, i.e., the ND Poincaré group is $N(N+1)/2$ dimensional.

ii) The Poincaré group formed when you add traslations to the full Lorentz group. It is sometimes called the inhomogenous Lorentz group and it can be denoted by ISO(3,1). Generally speaking, we will generally have $ISO(d,1)$, a D-dimensional ($D=d+1$) Poincaré group.

The Poincaré group includes as subgroups, the proper Lorentz transformations such as parity symmetry and some other less common symmtries. Note that the time reversal is NOT a proper Lorentz transformation since the determinant is equal to minus one.

Then, the Poincaré group includes: rotations, traslations in space and time, proper Lorentz transformations (boosts). The combined group of rotations, traslations and proper Lorentz transformations of inertial reference frames (those moving with constant relative velocity) IS the Poincaré group. If you give up the traslations in space and time of this list, you get the (proper) Lorentz group.

The full Poincaré group is a NON-COMPACT Lie group with 10 “dimensions”/parameters in 4D spacetime and $N(N+1)/2$ in the ND case.  Note that the boost parameters are “imaginary angles” so some parameters are complex numbers, though. The traslation subgroup of the Poincaré group is an abelian group forming a normal subgroup of the Poincaré group while the Lorentz grou is only a mere subgroup (it is not a normal subgroup of the Poincaré group). The Poincaré group is said, due to these facts, to be a “semidirect” product of traslations in space and time with the group of Lorentz transformations.

The case of Galilean symmetry

We can go back in time to understand some stuff we have already studied with respect to groups. There is a well known example of group in Classical (non-relativistic) Physics.

The Galilean group is the set or family of non-relativistic continuous space-time (yes, there IS space-time in classical physics!) transformations in 3D with an absolute time. This group has some interesting subgroups: 3D rotations, spatial traslations, temporal traslations and proper Galilean transformations ( transformations leaving invariant inertial frames in 3D space with absolute time). Thereforem the number of parameters of the Galilean group is 3+3+1+3=10 parameters. So the Galileo group is 10 dimensional and every parameter is real (unlike Lorentz transformations where there are 3 imaginary rotation angles).

The general Galilean group can be written as follows:

$G\begin{cases} \mathbf{r}\longrightarrow \mathbf{r}'=R\mathbf{r}+\mathbf{x_0}+\mathbf{V}t\\ t\longrightarrow t'=t+t_0\end{cases}$

Any element of the Galileo group can be written as a family of transformations $G=G(R,\mathbf{x_0},\mathbf{v},t_0)$. The parameters are:

i) $R$, an orthogonal (real) matrix with size $3\times 3$. It satisfies $RR^T=R^TR=I$, a real version of the more general unitary matrix $UU^+=U^+U=I$.

ii) $\mathbf{x_0}$ is a 3 component vector, with real entries. It is a 3D traslation.

iii) $\mathbf{V}$ is a 3 component vector, with real entries. It gives a 3D non-relativistic (or galilean) boost for inertial observers.

iv) $t_0$ is a real constant associated to a traslation in time (temporal traslation).

Therefore, we have 10 continuous parameters in general: 3 angles (rotations) defining the matrix $R$, 3 real numbers (traslations $\mathbf{x_0}$), 3 real numbers (galilean boosts denoted by $\mathbf{V}$) and a real number (traslation in time). You can generalize the Galilean group to ND. You would get  $N(N-1)/2+N+N+1$ parameters, i.e, you would obtain a $N(N+3)/2+1$ dimensional group. Note that the total number of parameters of the Poincaré group and the Galilean group is different in general, the fact that in 3D the dimension of the Galilean group matches the dimension of the 4D Poincaré group is a mere “accident”.

The Galilean group is completely determined by its “composition rule” or “multiplication operation”. Suppose that:

$G_3(R_3,\mathbf{z_0},\mathbf{V}_3,t_z)=G_2\cdot G_1$

with

$G_1(R_1,\mathbf{x_0},\mathbf{V}_1,t_x)$

and

$G_2(R_2,\mathbf{y_0},\mathbf{V}_2,t_y)$

Then, $G_3$ gives the composition of two different Galilean transformations $G_1, G_2$ into a new one. The composition rule is provided by the following equations:

$R_3=R_2R_1$

$\mathbf{z_0}=\mathbf{y_0}+R_2\mathbf{x_0}+\mathbf{V}_2 t_x$

$\mathbf{V}_3=\mathbf{V}_2+R_2\mathbf{V}_1$

$t_z=t_x+t_y$

Why is all this important? According to the Wigner theorem, for every continuous space-time transformation $g\in G$ should exist a unitary operator $U(g)$ acting on the space of states and observables.

We have seen that every element in uniparametric groups can be expressed as the exponential of certain hermitian generator. The Galilean group or the Poincaré group depends on 10 parameters (sometimes called the dimension of the group but you should NOT confuse them with the space-time dimension where they are defined). Remarkably, one can see that the Galilean transformations also act on “spacetime” but where the time is “universal” (the same for every inertial observer). Then, we can define

$iK_\alpha=\dfrac{\partial G}{\partial \alpha}\bigg| _{\alpha=0}$

These generators, for every parameter $\alpha$, will be bound to dynamical observables such as: linear momentum, angular momentum, energy and many others. A general group transformation for a 10-parametric (sometimes said 10 dimensional) group can be written as follows:

$\displaystyle{G(\alpha_1,\ldots,\alpha_{10}=\prod_{k=1}^{10}e^{iK_{\alpha_k}\alpha_k}}$

We can apply the Baker-Campbell-Hausdorff (BCH) theorem or simply expand every exponential in order to get

$\displaystyle{G(\alpha_1,\ldots,\alpha_{10})=\prod_{k=1}^{10}e^{iK_{\alpha_k}\alpha_k}=\exp \sum_{k=1}^{10}\omega_k (\alpha_1,\ldots,\alpha_{10})K_{\alpha_k}}$

$\displaystyle{G(\alpha_1,\ldots,\alpha_{10})=I+i\sum_{k=1}^{10}\omega_k(\alpha_1,\ldots,\alpha_{10})K_{\alpha_k}+\ldots}$

The Lie algebra will be given by

$\displaystyle{\left[K_i,K_j\right]=i\sum_{k}c_{ij}^kK_k}$

and where the structure constants will encode the complete group multiplication rules. In the case of the Poincaré group Lie algebra, we can write the commutators as follows:

$\left[X_\mu,X_\nu\right]=\left[P_\mu,P_\nu\right]=0$

$\left[M_{\mu\nu},P_\alpha\right]=\eta_{\mu\alpha}P_\nu-\eta_{\nu\alpha}P_\mu$

$\left[M_{\mu\nu},M_{\alpha\beta}\right]=\eta_{\mu\alpha}M_{\nu\beta}-\eta_{\mu\beta}M_{\nu\alpha}-\eta_{\nu\alpha}M_{\mu\beta}+\eta_{\nu\beta}M_{\mu\alpha}$

Here, we have that:

i) $P$ are the generators of the traslation group in spacetime. Note that as they commute with theirselves, the traslation group is an abelian subgroup of the Lorentz group. The noncommutative geometry (Snyder was a pioneer in that idea) is based on the idea that $P$ and more generally even the coordinates $X$ are promoted to noncommutative operators/variables/numbers, so their own commutator would not vanish like the Poincaré case.

ii) $M$ are the generators of the Lorent group in spacetime.

If we study the Galilean group, there are some interesting commutation relationships fo the corresponding generators (rotations and traslations). There are 6 “interesting” operators:

$K_{i}\equiv \overrightarrow{J}$ if $i=1,2,3$

$K_{i}\equiv \overrightarrow{P}$ if $i=,4,5,6$

These equations provide

$\left[P_\alpha,P_\beta\right]=0$

$\left[J_\alpha,J_\beta\right]=i\varepsilon_{\alpha\beta}^\gamma J_\gamma$

$\left[J_\alpha,P_\beta\right]=i\varepsilon_{\alpha\beta}^\gamma P_\gamma$

$\forall\alpha,\beta=1,2,3$

The case of the traslation group

In Quantum Mechanics, traslations are defined in the space of states in the following sense:

$\vert\vec{r}\rangle\longrightarrow\vert\vec{r}'\rangle =\exp\left(-i\vec{x_0}\cdot \vec{p}\right)\vert \vec{r}\rangle=\vert\vec{r}+\vec{x_0}\rangle$

Let us define two linear operators, $R$ and $R'$ associated, respectively, to initial position and shifted position. Then the transformation defining the traslation over the states are defined by:

$R\longrightarrow R'=\exp\left(-i\vec{x_0}\cdot\vec{p}\right)R\exp \left(i\vec{x_0}\cdot \vec{p}\right)$

where

$R_i\vert\vec{r}'\rangle=\vec{r}_i\vert\vec{r}'\rangle$

Furthermore, we also have

$\left[\vec{x_0}\cdot \vec{p},\vec{y_0}\cdot R\right]=-i\vec{x_0}\cdot\vec{y_0}$

$\left[R_\alpha,p_\beta\right]=i\delta_{\alpha\beta}I$

The case of the rotation group

What about the rotation group? We must remember what a rotation means in the space $\mathbb{R}^n$. A rotation is a transformation group

$\displaystyle{X'=RX\longrightarrow \parallel X'\parallel^2=\parallel X\parallel^2 =\sum_{i=i}^n (x'_i)^2=\sum_{i=1}^n x_i^2}$

The matrix associated with this transformation belongs to the orthogonal group with unit determinant, i.e., it is an element of $SO(N)$. In the case of 3D space, it would be $SO(3)$. Moreover, the ND rotation matrix satisfy:

$\displaystyle{I=X^TX=XX^T\leftrightarrow \sum_{i=1}^N R_{ik}R_{ij}=R_{ik}R_{ij}=\delta_{kj}}$

The rotation matrices in 3D depends on 3 angles, and they are generally called the Euler angles in some texts. $R(\theta_1,\theta_2,\theta_3)=R(\theta)$. Therefore, the associated generators are defined by

$iM_j\equiv\dfrac{\partial R}{\partial \theta_j}\bigg|_{\theta_j=0}$

Any other rotation matric can be decomposed into a producto of 3 uniparametric rotations, rotation along certain 2d planes. Therefore,

$R(\theta_1,\theta_2,\theta_3)=R_1(\theta_1)R_2(\theta_2)R_3(\theta_3)$

where the elementary rotations are defined by

Rotation around the YZ plane: $R_1(\theta_1)=\begin{pmatrix} 1 & 0 & 0\\ 0 & \cos\theta_1 & -\sin\theta_1\\ 0 & \sin\theta_1 & \cos\theta_1\end{pmatrix}$

Rotation around the XZ plane: $R_2(\theta_2)=\begin{pmatrix} \cos\theta_2 & 0 & \sin\theta_2\\ 0 & 1 & 0\\ -\sin\theta_2 & 0 & \cos\theta_2\end{pmatrix}$

Rotation around the XY plane: $R_3(\theta_3)=\begin{pmatrix} \cos\theta_3 & -\sin\theta_3 & 0\\ \sin\theta_3 & \cos\theta_3 & 0\\ 0 & 0 & 1\end{pmatrix}$

Using the above matrices, we can find an explicit representation for every group generator (3D rotation):

$M_1=-i\begin{pmatrix}0 & 0 & 0\\ 0 & 0 & 1\\ 0 & -1 & 0\end{pmatrix}$

$M_2=-i\begin{pmatrix}0 & 0 & -1\\ 0 & 0 & 0\\ 1 & 0 & 0\end{pmatrix}$

$M_3=-i\begin{pmatrix}0 & 1 & 0\\ -1 & 0 & 0\\ 0 & 0 & 0\end{pmatrix}$

and we also have

$\left[M_j,M_k\right]=i\varepsilon^{m}_{jk}M_{m}$

where the $\varepsilon^m_{jk}=\varepsilon_{mjk}$ is the completely antisymmetry Levi-Civita symbol/tensor with 3 indices. There is a “for all practical purposes” formula that represents a rotation with respect to some axis in certain direction $\vec{n}$. We can make an infinitesimal rotation with angle $d\theta$, due to the fact that rotation are continuous transformations, it commutes with itself and it is unitary, so that:

$R(d\theta)\vec{r}=\vec{r}+d\theta(\vec{n}\times\vec{r}+\mathcal{O}(d\theta^2)=\vec{r}-id\theta M_\alpha\vec{r}+\mathcal{O}(d\theta^2)$

In the space of physical states, with $\vec{k}=\theta\vec{n}$ some arbitrary vector

$\vec{r}'=R\vec{r}\longrightarrow\vert\vec{r}'\rangle=\vert R\vec{r}\rangle=U(R)\vert\vec{r}\rangle=e^{-i\vec{k}\cdot\vec{J}}\vert\vec{r}\rangle=e^{-i(k_xJ_x+k_yJ_y+k_zJ_z)}\vert \vec{r}\rangle$

Here, the operators $J=(J_x,J_y,J_z)$ are the infinitesimal generators in the space of physical states. The next goal is to relate these generators with position operators $Q$ through commutation rules. Let us begin with

$Q\longrightarrow Q'=e^{-i\vec{k}\cdot{J}}Qe^{i\vec{k}\cdot\vec{J}}$

$Q'\vert\vec{r}'\rangle =\vec{r}\vert\vec{r}'\rangle$

Using this last result, we can calculate for any 2 vectors $\vec{k},\vec{n}$:

$\left[\vec{k}\cdot\vec{J},\vec{n}\cdot\vec{Q}\right]=i(\vec{k}\times\vec{n})\cdot\vec{Q}$

or equivalent, in component form,

$\left[J_j,Q_k\right]=i\varepsilon_{jkm}Q_m$

These commutators complement the above commutation rules, and thus, we have in general

$\left[\vec{k}\cdot\vec{J},\vec{n}\cdot\vec{Q}\right]=i(\vec{k}\times\vec{n})\cdot\vec{Q}$

$\left[J_j,J_k\right]=i\varepsilon_{jkm}J_m$

$\left[J_j,Q_k\right]=i\varepsilon_{jkm}Q_m$

In summary: a triplet of rotation operators generates “a vector” somehow.

The case of spinning particles

In fact, these features provide two different cases in the case of a single particle:

i) Particles with no “internal structure” or “scalars”/spinless particles. A good example could it be the Higgs boson.

ii) Particles with “internal” degrees of freedom/structure/particles with spin.

In the case of a particle without spin in 3D we can define the angular momentum operator as we did in classical physics ($L=r\times p$), in such a way that

$J=Q\times P$

Note that the “cross product” or “vector product” in 3D is generally defined if $C=A\times B$ as

$C=A\times B=\begin{vmatrix}i & j & k\\ A_x & A_y & A_z\\ B_x & B_y & B_z\end{vmatrix}$

or by components, using the maginal word XYZZY, we also have

$C_x=A_yB_z-A_zB_y$

$C_y=A_zB_x-A_xB_z$

$C_z=A_xB_y-A_yB_x$

Remember that the usual “dot” or “scalar” product is $A\cdot B=A_xB_x+A_yB_y+A_zB_z$

Therefore, the above operator $J$ defined in terms of the cross product satisfies the Lie algebra of $SO(3)$.

By the other hand, in the case of a spinning particle/particle with spin/internal structure/degrees of freedom, the internal degrees of freedom must be represented by some other operator, independently from $Q,P$. In particular, it must also commute with both operators. Then, by definition, for a particle with spin, the angular momentum will be a sum with two contributions: one contribution due to the “usual” angular momentum (orbital part) and an additional “internal” contribution (spin part). That is, mathematically speaking, we should have a decomposition

$J=Q\times P+S$

with $\left[Q,S\right]=\left[P,S\right]=0$

If $S$, the spin operator, satisfies the above commutation rules (in fact, the same relations than the usual angular momentum), we must impose

$\left[S_j,S_k\right]=i\varepsilon_{jkm}S_m$

The case of Parity P/Spatial inversions

This special transformation naturally arises in some applications. From the pure geometrical viewpoint, this transformation is very simple:

$\vec{r}'=-\vec{r}$

In coordinates and 3D, the spatial inversion or parity is represented by a simple matrix equals to minus the identity matrix

$P=\begin{pmatrix} -1 & 0 & 0\\ 0 & -1 & 0\\ 0& 0 & -1\end{pmatrix}$

This operator correspods, according to the theory we have been studying, to some operator P (please, don’t confuse P with momentum) that satisfies

$PqP^{-1}=-q$

$PpP^{-1}=-p$

and where $q, p$ are the usual position and momentum operators. Then, the operator

$L=q\times p$ is invariant by parity/spatial inversion P, and thus, this feature can be extended to any angular momentum operator like spin S or angular momentum J. That is,

$PJP^{-1}=J$ and $PSP^{-1}=S$

The Wigner’s theorem implies that corresponding to the operator P, a discrete transformation, must exist some unitary or antiunitary operator. In fact, it shows that P is indeed unitary

$P\left[Q_i,P_j\right]P^{-1}=\left[Q_i,P_j\right]=P(i\hbar\delta_{ij})P^{-1}$

If P were antiunitary we should get

$P\left[Q_i,P_j\right]P^{-1}=\left[Q_i,P_j\right]=P(i\hbar\delta_{ij})P^{-1}=-i\hbar\delta_{ij}$

Then, the parity operator P is unitary and $P^{-1}=P$. In fact, this can be easily proved from its own definition.

If we apply two succesive parity transformations we leave the state invariant, so $P^2=I$. We say that the parity operator is idempotent.  The check is quite straightforward

$\vert\Psi\rangle\longrightarrow\vert\Psi'\rangle=PP\vert\Psi\rangle\longrightarrow\vert\Psi\rangle=e^{i\omega}\vert\Psi\rangle$

Therefore, from this viewpoint, there are (in general) only 2 different ways to satisfy this as we have $PP=e^{i\omega}I$:

i) $e^{i\omega}=+1$. The phase is equal to $0$ modulus $2\pi$. We have hermitian operators

$P=P^{-1}=P^+$

Then, the effect on wavefunctions is that $\Psi (P^{-1}(\vec{r}))=\Psi (-\vec{r})$. That is the case of usual particles.

ii) The case $e^{i\omega}=-1$. The phase is equal to $\pi$ modulus $2\pi$. This is the case of an important class of particles. In fact, Steven Weinberg has showed that $P^2=(-1)^F$ where F is the fermion number operator in the SM. The fermionic number operator is defined to be the sum $F=L+B$ where L is now the leptonic number and B is the baryonic number. Moreover, for all particles in the Standard Model and since lepton number and baryon number are charges Q of continuous symmetries $e^{iQ}$  it is possible to redefine the parity operator so that $P^2=I$. However, if there exist Majorana neutrinos, which experimentalists today believe is quite possible or at least it is not forbidden by any experiment, their fermion number would be equal to one because they are neutrinos while their baryon and lepton numbers are zero because they are Majorana fermions, and so $(-1)^F$ would not be embedded in a continuous symmetry group. Thus Majorana neutrinos would have parity equal to $\pm i$. Beautiful and odd, isnt’t it? In fact, if some people are both worried or excited about having Majorana neutrinos is also due to the weird properties a Majorana neutrino would have under parity!

The strange case of time reversal T

In Quantum Mechanics, temporal inversions or more generally the time reversal is defined as the operator that inverts the “flow or direction” of time. We have

$T: t\longrightarrow t'=-t$ $\vec{r}'(-t)=\vec{r}(t)$

And it implies that $\vec{p}(-t)=-\vec{p}(t)$. Therefore, the time reversal operator $T$ satisfies

$TQT^{-1}=Q$

$TPT^{-1}=-P$

In summary: T is by definition the “inversion of time” so it also inverts the linear momentum while it leaves invariant the position operator.

Thus, we also have the following transformation of angular momentum under time reversal:

$TJT^{-1}=-J$

$TST^{-2}=-S$

Time reversal can not be a unitary operator, and it shows that the time reversal T is indeed an antiunitary operator. The check is quite easy:

$T\left[Q,P\right]T^{-1}=\left[TQT^{-1},TPT^{-1}\right]=-\left[Q,P\right]=Ti\hbar T^{-1}$

This equation matches the original definiton if and only if (IFF)

$TiT^{-1}=-i \leftrightarrow TT^{-1}=-1$

Time reversal is as consequence of this fact an antiunitary operator.