LOG#094. Group theory(XIV).

Group theory and the issue of mass: Majorana fermions in 2D spacetime

We have studied in the previous posts that a mass term is “forbidden” in the bivector/sixtor approach and the Dirac-like equation due to the gauge invariance. In fact,


as an operator has an “issue of mass” over the pure Dirac equation i\Gamma^\mu\Psi=0 of fermionic fields. This pure Dirac equation provides

\overline{\Gamma_\mu}\Gamma_\nu\partial_\nu\partial_\mu\Psi=\partial_0^2\Psi+\nabla\times (\nabla\times \Psi)=\partial_0^2-\Delta\Psi+\nabla (\nabla\cdot \Psi)=0

Therefore, \Psi satisfies the wave equation


where \square^2=\Delta-\partial_0^2 if there are no charges or currents! If we introduce a mass by hand for the \Psi field, we obtain

(i\Gamma^\mu\partial_\mu-m )\Psi=0

and we observe that it would spoil the relativistic invariance of the field equation!!!!!!! That is a crucial observation!!!!

A more general ansatz involves the (anti)linear operator V:


A plane wave solution is something like an exponential function \sim e^{\pm ipx} and it obeys:

p^2=p_\mu p^\mu=-m^2

If we square the Dirac-like equation in the next way

i\overline{\Psi}^\nu\partial_\nu (i\Gamma^\mu\partial_\mu\Psi)=-\square \Psi=-m^2\Psi=i\overline{\Psi}^\nu\partial_\nu (mV\Psi)

and a bivector transformation

i\overline{\Gamma}^\mu\partial_\mu (V\Psi)-m\Psi=0

V(i\overline{\Gamma}^\mu\partial_\mu (v\Psi))-mV\Psi=0

Vi\overline{\Gamma}^\mu\partial_\mu (V\Psi)=mV\Psi=i\Gamma^\mu \partial_\mu \Psi

from linearity we get





if a=1,2,3. But this is impossible! Why? The Lie structure constants are “stable” (or invariant) under similarity transformations. You can not change the structure constants with similarity transformations.

In fact, if V is an antilinear operator

V=\tilde{V}\kappa=iV where \kappa is a complex conjugation of the multiplication by the imaginary unit. Then, we would obtain




or equivalently


And again, this is impossible since we would obtain then

\det (V\tilde{V}^*)=\det (V)\det (\tilde{V}^*)=\det \tilde{V}\det \tilde{V}^*>0

and this contradicts the fact that \det (-I_3)=-1!!!

Remark: In 2d, with Pauli matrices defined by


and \epsilon\tilde{\sigma}^*\epsilon^{-1}=-\tilde{\sigma}


\epsilon=\begin{pmatrix}0 & 1\\ -1 & 0\end{pmatrix}

and we have



\det (\epsilon)=\det (-I_2) such that the so-called Majorana equation(s) (as a generalization of the Weyl equation(s) and the Lorentz invariance in 4d) provides a 2-component field equation describing a massive particle DOES exist:


In fact, the Majorana fermions can exist only in certain spacetime dimensions beyond the 1+1=2d example above. In 2D (or 2d) spacetime you write


and it is called the Majorana equation. It describes a massive fermion that is its own antiparticle, an option that can not be possible for electrons or quarks (or their heavy “cousins” and their antiparticles). The only Standard Model particle that could be a Majorana particle are the neutrinos. There, in the above equations,

\sigma^\mu=(I_2,\vec{\sigma}) and \eta is a “pure phase” often referred as the “Majorana phase”.

Gauge formalism and mass terms for field equations

Introduce some gauge potential like

A=A^R+iA^I=\begin{pmatrix}A_1^R+iA_1^I\\ A_2^R+iA_2^I\\ A_3^R+iA_3^I\end{pmatrix}

It is related to the massive bivector/sixtor field with the aid of the next equation

\Psi=i\overline{\Gamma}^\nu\partial_\nu A=(i\partial_0+\nabla\times)(A^R+iA^I)=-\dot{A}^O+\nabla\times A^R+i(\dot{A}^R+\nabla\times A^I)

It satisfies a massive wave equation


This would mean that

i\Gamma^\mu\partial_\mu (i\overline{\Gamma}^\nu \partial_\nu A)=(-\partial_0^2-\nabla\times\nabla\times)A=(-\partial_0^2+\Delta-\nabla (\nabla\cdot))A=- m^2A

and then \nabla (\nabla\cdot A)=0. However, it would NOT BE a Lorentz invariant anymore!

Current couplings

From the Ampère’s law

\partial_t E=\nabla\times B-j

and where we have absorbed the multiplicative constant into the definition of the current j, we observe that i\Gamma^\mu\partial_\mu\Psi can NOT be interpreted as the Dirac form of the Maxwell equations since j=(j_1,j_2,j_3) have 3 spatial components of a charge current 4d density J=j^\mu=(j^0,\mathbf{j})=(j^0,j^1,j^2,j^3) so that

\partial_t\Psi=-i\nabla\times \Psi-\mathbf{j} and

\nabla\cdot (\partial_t \Psi)=\nabla\cdot (-i\nabla\times \Psi-\mathbf{j})


\nabla\cdot \dot{\Psi}=-i\nabla\cdot (\nabla\times \Psi)-\mbox{div}\mathbf{j}=\dot{\rho}

If the continuity equation \dot{\rho}+\mbox{div}\mathbf{j}=0 holds. In the absence of magnetic charges, this last equation is equivalent to \mbox{div} (\dot{E})=\dot{\rho} or \nabla\cdot E=\rho.

Remark: Even though the bivector/sixtor field couples to the spatial part of the 4D electromagnetic current, the charge distribution is encoded in the divergence of the field \Psi itself and it is NOT and independent dynamical variable as the current density (in 4D spacetime) is linked to the initial conditions for the charge distribution and it fixes the actual charge density (also known as divergence of \Psi at any time; \Psi is a bispinor/bivector and it is NOT a true spinor/vector).

Dirac spinors under Lorentz transformations

A Lorentz transformation is a map

X'=\Lambda X

A Dirac spinor field is certain “function” transforming under certain representation of the Lorentz group. In particular,

\Psi'(x')=Q_D(\Lambda)\Psi (x) for every Lorentz transformation belonging to the group SO(1,3)^+. Moreover,

x'_\mu=\Lambda^\mu_{\;\;\; \nu}

and Dirac spinor fields obey the so-called Dirac equation (no interactions are assumed in this point, only “free” fields):


This Dirac equation is Lorentz invariant, and it means that it also hold in the “primed”/transformed coordinates since we have

i\gamma^\mu (\partial_{\mu '}\Psi' (x'))=i\gamma^\mu\partial_{\mu '}(Q_D\Psi (x))=mQ_D\Psi


i\gamma^\mu\Lambda^{\;\;\; \nu}_\mu\partial_\nu Q_D\Psi=mQ_D\Psi

Using that \Lambda^\alpha_{\;\;\; \nu}\Lambda^{\nu}_{\;\;\; \mu}=\delta^\alpha_{\;\;\; \mu}

we get the transformation law

\boxed{Q_D^{-1}\gamma^\alpha Q_D=\Lambda^\alpha_{\;\;\; \nu}\gamma^\nu}

Covariant Dirac equations are invariant under Lorentz transformations IFF the transformation of the spinor components is performed with suitable chosen operators Q_D. In fact,

Q^{-1}\Gamma^\alpha Q=\Lambda^\alpha_{\;\;\; \nu}\Gamma^\nu

Q^T\Gamma^\alpha Q=\Lambda^\alpha_{\;\;\; \nu}\Gamma\nu


DOES NOT hold for \Psi bispinors/bivectors. For bivector fields, you obtain



i\Gamma^\mu_{ab}\partial_{\mu '}\Psi '(x')=-iJ'_a (x')

This last equation implies that

i\Gamma_{ab}^\mu\Lambda_{\mu}^{\;\;\; \nu}\partial_\nu Q_{bc}\Psi_c(x)=-i\Lambda^a_{\;\;\; \nu}j^\nu (x)=-i\Lambda^a_{0}\mbox{div} E(x)-i\Lambda^a_ cJ^c(x)



\mbox{div} E=\delta^\nu_c\delta_\nu\Psi_c since \mbox{div} B=\nabla\cdot B=0 because there are no magnetic monopoles.

If \tilde{\Lambda} is the inverse of the 3d matric \Lambda^a_{\;\;\; b}, then we have


In this case, we obtain that

i (\tilde{\Lambda}^d_a\Gamma^\mu_{ab}\Lambda^\nu_\mu Q_{bc}+\tilde{\Lambda}^d_a\Lambda^a_0\delta^\nu_c)\partial_\nu\Psi_c (x)=-i\tilde{\Lambda}^d_a\Lambda^a_cJ^c=-ij^d


\Gamma^\nu_{dc}=\tilde{\Lambda}^d_a\Gamma^\mu_{ab}\Lambda_{\mu}^\nu Q_{bc}+\tilde{\Lambda}^d_a\Lambda^a_0\delta^\nu_c

That is, for rotations we obtain that

\Lambda^a_{\;\;\; b}=Q_{ab} \tilde{\Lambda}^{-1}=Q^{-1} \Lambda^a_{\;\;\; 0}\;\;\forall a=1,2,3

and so

\boxed{\Gamma^\nu=\Lambda_{\mu}^{\;\;\; \nu}Q^{-1}\Gamma^\mu Q}

This means that for the case of pure rotations both bivector/bispinors and current densities transform as vectors under the group SO(3)!!!!

Conclusions of this blog post:

1st. A mass term analogue to the Marjorana or Dirac spinor equation does NOT arise in 4d electromagnetism due to the interplay of relativistic invariance and gauge symmetries. That is, bivector/bispinor fields in D=4 can NOT contain mass terms for group theoretical reasons: Lorentz invariance plus gauge symmetry.

2nd. The Dirac-like equation i\Gamma^\mu \partial_\mu \Psi=0 can NOT be interpreted as a Dirac equation in D=4 due to relativistic symmetry, but you can write that equation at formal level. However, you must be careful with the meaning of this equation!

3rd. In D=2 and other higher dimensions, Majorana “mass” terms arise and you can write a “Majorana mass” term without spoiling relativistic or gauge symmetries. Majorana fermions are particles that are their own antiparticles! Then, only neutrinos can be Majorana fermions in the SM (charged fermions can not be Majorana particles for technical reasons).

4th. The sixtor/bivector/bispinor formalism with F=E+iB has certain uses. For instance, it is used in the so-called Ungar’s formalism of special relativity, it helps to remember the electromagnetic main invariants and the most simple transformations between electric and magnetic fields, even with the most general non-parallel Lorentz transformation.


LOG#041. Muons and relativity.

QUESTION: Is the time dilation real or is it an artifact of our current theories?

There are solid arguments why time dilation is not an apparent effect but a macroscopic measurable effect. Today, we are going to discuss the “reality” of time dilation with a well known result:

Muon detection experiments!

Muons are enigmatic elementary particles from the second generation of the Standard Model with the following properties:

1st. They are created in upper atmosphere at altitudes of about 9000 m, when cosmic rays hit the Earth and they are a common secondary product in the showers created by those mysterious yet cosmic rays.
2nd. The average life span is 2\times 10^{-6}s\approx 2ms
3rd. Typical speed is 0.998c or very close to the speed of light.
So we would expect that they could only travel at most d=0.998c\times 2 \times 10^{-6}\approx 600m
However, surprisingly at first sight, they can be observed at ground level! SR provides a beautiful explanation of this fact. In the rest frame S of the Earth, the lifespan of a traveling muon experiences time dilation. Let us define

A) t= half-life of muon with respect to Earth.

B) t’=half-life of muon of the moving muon (in his rest frame S’ in motion with respect to Earth).

C) According to SR, the time dilation means that t=\gamma t', since the S’ frame is moving with respect to the ground, so its ticks are “longer” than those on Earth.

A typical dilation factor \gamma for the muon is about 15-100, although the value it is quite variable from the observed muons. For instance, if the muon has v=0.998c then \gamma \approx 15. Thus, in the Earth’s reference frame, a typical muon lives about 2×15=30ms, and it travels respect to Earth a distance

d'=0.998c\times 30ms\approx 9000m.

If the gamma factor is bigger, the distance d’ grows and so, we can detect muons on the ground, as we do observe indeed!

Remark:  In the traveling muon’s reference frame, it is at rest and the Earth is rushing up to meet it at 0.998c. The distance between it and the Earth thus is shorter than 9000m by length contraction. With respect to the muon, this distance is therefore 9000m/15 = 600m.

An alternative calculation, with approximate numbers:

Suppose muons decay into other particles with half-life of about 0.000001sec. Cosmic ray muons have speed now about v = 0.99995 c.
Without special relativity, muon would travel

d= 0.99995 \times 300000 km/s\times 0.00000156s=0.47 km only!

Few would reach earth’s surface in that case. It we use special relativity, then plugging the corresponding gamma for v=0.99995c, i.e.,  \gamma =100, then muons’ “tics” run slower and muons live 100 times longer. Then, the traveled distance becomes

d'=100\times 0.9995\times 300000000 m/s\times 0.000001s= 30000m

Conclusion: a lot of muons reach the earth’s surface. And we can detect them! For instance, with the detectors on colliders, the cosmic rays detectors, and some other simpler tools.

LOG#006. Lorentz Transformations(II).

\boxed{ \begin{cases} x'_0=ct'=\gamma (ct - \mathbf{\beta} \cdot \mathbf{r}) \\ \mathbf{r'}=\mathbf{r}+(\gamma -1) \dfrac{(\mathbf{\beta}\cdot \mathbf{r})\mathbf{\beta}}{\beta^2} -\gamma \beta ct \\ \gamma = \dfrac{1}{\sqrt{1-\beta^2}}= \dfrac{1}{\sqrt{1-\beta_x^2-\beta_y^2-\beta_z^2}} \end{cases}}

\boxed{\left( \begin{array}{c} ct' \\ x' \\ y' \\ z' \end{array} \right) = \begin{pmatrix} \gamma & -\gamma \beta_x & -\gamma \beta_y & -\gamma \beta_z \\ -\gamma \beta_x & 1+(\gamma -1)\dfrac{\beta_{x}^{2}}{\beta^2} & (\gamma -1)\dfrac{\beta_x \beta_y}{\beta^2} & (\gamma -1)\dfrac{\beta_x \beta_z}{\beta^2} \\ -\gamma \beta_y & (\gamma -1)\dfrac{\beta_y \beta_x}{\beta^2} & 1+(\gamma -1)\dfrac{\beta_{y}^{2}}{\beta^2} & (\gamma -1)\dfrac{\beta_y \beta_z}{\beta^2} \\ -\gamma \beta_z & (\gamma -1)\dfrac{\beta_z \beta_x}{\beta^2} & (\gamma -1)\dfrac{\beta_z \beta_y}{\beta^2} & 1+(\gamma -1)\dfrac{\beta_{z}^{2}}{\beta^2} \end{pmatrix} \left( \begin{array}{c} ct\\ x\\ y\\ z\end{array}\right)}

These equations define the most general (direct) Lorentz transformations  and we see they are not those in the previous post! I mean, they are not the one with the relative velocity in the direction of one particular axis, as we derived in the previous log. We will derive these equations. How can we derive them?

The most general Lorentz transformation involves the following scenario (a full D=3+1 motion):

1st) The space-time coordinates of an event E are described by one observer (and frame) A at rest at the origin of his own frame S. The observer B is at rest at the origin in a second frame S’. S and S’ have parallel axes.

2nd) The origin of the S and S’ frames coincide at t=t’=0.

3rd) B moves relative to A with a velocity  3d vector (space-like) given by \mathbf{v}=(v_x,v_y,v_z).

4th) The position vector of the event in the S-frame is \mathbf{r}=(x,y,z). It is decomposed into  “horizontal/vertical” or parallel/orthogonal pieces as follows

\mathbf{r}=\mathbf{r}_\parallel + \mathbf{r}_ \perp

The following transformation is suitable for the S’-frame, defining \beta=\mathbf{v}/c=(v_x/c,v_y/c,v_z/c):

ct'=x_0=\gamma(ct-\beta r_\parallel)=\gamma (ct - \mathbf{\beta} \cdot \mathbf{r})

\mathbf{r'}_\parallel = \gamma (\mathbf{r}_\parallel - \mathbf{\beta}ct )


where the dot represents scalar product. Using the elementary knowledge and application of the scalar product with projections of vectors, we calculate the projection of the position vector onto the velocity vector in any frame with the scalar product of the position vector with a normalized velocity vector, \mathbf{\hat{v}}=\mathbf{v}/v . In the S’-frame we will get the projection \mathbf{\hat{v}}\cdot \mathbf{r}. Therefore,

\mathbf{r}_\parallel = (\hat{\mathbf{v}}\cdot \mathbf{r})\hat{\mathbf{v}}

and thus, the component of the position vector with respect to the parallel direction to the velocity will be:

\mathbf{r}_\parallel = (\hat{\mathbf{v}}\cdot \mathbf{r})\hat{\mathbf{v}}= \dfrac{(\mathbf{v}\cdot \mathbf{r})\mathbf{v}}{v^2}


\mathbf{r}_\parallel = \dfrac{(\mathbf{\beta}\cdot \mathbf{r})\mathbf{\beta}}{\beta^2}

Then, since \mathbf{r}_\perp = \mathbf{r}-\mathbf{r}_\parallel, we have

\mathbf{r}_\perp = \mathbf{r}- \dfrac{(\mathbf{\beta}\cdot \mathbf{r})\mathbf{\beta}}{\beta^2}

Finally, we put together the vertical/horizontal (orthogonal/parallel) pieces of the general Lorentz transformations:

\mathbf{r'} =\mathbf{r'}_\parallel + \mathbf{r'}_ \perp = \gamma \left( \dfrac{(\mathbf{\beta}\cdot \mathbf{r})\mathbf{\beta}}{\beta^2} -\beta ct \right)+\mathbf{r}- \dfrac{(\mathbf{\beta}\cdot \mathbf{r})\mathbf{\beta}}{\beta^2}

Then, the general 4D=3d+1 Lorentz transformation (GLT) from S to S’ are defined through the equations:



The inverse GLT (IGLT) will be:

\boxed{IGLT(S'\rightarrow S) \begin{cases} x_0=ct=\gamma (ct' + \mathbf{\beta} \cdot \mathbf{r}') \\ \mathbf{r}=\mathbf{r}'+(\gamma -1) \dfrac{(\mathbf{\beta}\cdot \mathbf{r}')\mathbf{\beta}}{\beta^2} +\gamma \beta ct' \\ \gamma = \dfrac{1}{\sqrt{1-\beta^2}}= \dfrac{1}{\sqrt{1-\beta_x^2-\beta_y^2-\beta_z^2}} \end{cases}}

Indeed, these transformations allow a trivial generalization to D=d+1, i.e., these transformations are generalized to d-spatial dimensions simply allowing a d-space velocity and beta parameter, while time remains 1d. Indeed, the Lorentz transformations form a group. A group is a mathematical gadget with certain “nice features” that physicists and mathematicians love. You can imagine the Lorentz group in D=d+1 dimensions as a generalization of the rotation group  called Lorentz group. The Lorentz group involves rotations around the spatial axes plus the so-called “boosts”, transformations involving mixing of space and time coordinates. Indeed, the Lorentz transformations involving relative motion along one particular axis IS a (Lorentz) boost! That is, the simplest Lorentz transformations like the one in the previous posts are “boosts”.

With the above transformations, the GLT can be easily written in components:



These transformations can be written in a symbolic way using matrix notation as \mathbb{X}'=\mathbb{L}\mathbb{X} or using tensor calculus:

x^{\mu'}=\Lambda^{\mu'}_{\;\nu} x^\nu

The inverse GLT (IGLT) will be in component way:


and they can be written as \mathbb{X}=\mathbb{L}^{-1}\mathbb{X'}, or using tensor notation

x^\rho=(\Lambda^{-1})^\rho_{\;\mu'} x^{\mu'}

in such a way that

x^{\mu'} = \Lambda^{\mu}_{\; \nu} x^\nu \rightarrow (\Lambda^{-1})^{\rho}_{\; \mu'}x^{\mu'} = (\Lambda^{-1})^{\rho}_{\;\mu'}(\Lambda)^{\mu'}_{\;\nu} x^{\nu} = x^{\rho} = \delta ^{\rho}_{\; \nu}x^\nu

Thus, (\Lambda^{-1})^{\rho}_{\;\mu'}(\Lambda)^{\mu'}_{\;\nu} = \delta ^{\rho}_{\; \nu}

or equivalently \mathbb{L}^{-1}\mathbb{L}=\mathbb{L}\mathbb{L}^{-1}=\mathbb{I}.

\delta ^{\rho}_{\; \nu} is the “unity” tensor, also called Kronecker delta, meaning that its components are 1 if \rho = \nu and 0 otherwise (if \rho \neq \nu). The Kronecker delta is therefore the “unit” tensor with two indexes.

NOTATIONAL CAUTION: Be aware, some books and people use to assume you know when you need the matrix \mathbb{L} or its inverse \mathbb{L}^{-1}. Thus, you will often read and see this

 x^{\mu'}=\Lambda^{\mu'}_{\;\nu} x^\nu \rightarrow x^\nu= \Lambda^\nu_{\;\mu'} x^{\mu'}

where certain abuse of language since it implies that

\Lambda^\nu_{\;\mu'} = (\Lambda^{-1})^{\mu'}_{\;\nu}

and because we have  to be mathematically consistent, the following  relationship is required to hold


or more precisely, taking care with the so-called free indexes

\Lambda^\rho_{\;\mu'} \Lambda^{\mu'}_{\;\sigma}=\delta^\rho_\sigma

as before.

LOG#005. Lorentz transformations(I).

For physicists working with objects approaching the light speed, e.g., handling with electromagnetic waves, the use of special relativity is essential.

The special theory of relativity is based on two single postulates:

1st) Covariance or invariance of all the physical laws (Mechanics, Electromagnetism,…) for all the inertial observers ( i.e. those moving with constant velocity relative to each other).

It means that there is no preferent frame or observer, only “relative” motion is meaningful when speaking of motion with respect to certain observer or frame. Indeed, unfortunately, it generated a misnomer and a misconception in “popular” Physics when talking about relativity (“Everything is relative”). What is relative then? The relative motion between inertial observers and its description using certain “coordinates” or “reference frames”. However, the true “relativity” theory introduces just about the opposite view. Physical laws and principles are “invariant” and “universal” (not relative!).  Einstein himself was aware of this, in spite he contributed to the initial spreading of the name “special relativity”, understood as a more general galilean invariance that contains itself the electromagnetic phenomena derived from Maxwell’s equations.

2nd) The speed of light is independent of the source motion or the observers. Equivalently, the speed of light is constant everywhere in the Universe.

No matter how much you can run, speed of light is universal and invariant. Massive particles can never move at speed of light. Two beams of light approaching to each other does not exceed the speed of light either. Then, the usual rule for the addition of velocities is not exact. Special relativity provides the new rule for adding velocities.

In this post, the first of a whole thread devoted to special relativity, I will review one of the easiest ways to derive the Lorentz transformations. There are many ways to “guess” them, but I think it is very important to keep the mathematics as simple as possible. And here simple means basic (undergraduate) Algebra and some basic Physics concepts from electromagnetism, galilean physics and the use of reference frames. Also, we will limite here to 1D motion in the x-direction.

Let me begin! We assume we have two different observers and frames, denoted by S and S’. The observer in S is at rest while the observer in S’ is moving at speed v with respect to S. Classical Physics laws are invariant under the galilean group of transformations:


We know that Maxwell equations for electromagnetic waves are not invariant under Galileo transformations, so we have to search for some deformation and generalization of the above galilean invariance. This general and “special” transformation will reduce to galilean transformations whenever the velocity is tiny compared with the speed of light (electromagnetic waves). Mathematically speaking, we are searching for transformations:

x'=\gamma (x-vt)


x=\gamma (x'+vt')

for the inverse transformation. There \gamma=\gamma(c,v) is a function of the speed of light (denoted as c, and constant in every frame!) and the relative velocity v of the moving object in S’ with respect to S. The small velocity limit of special relativity to galilean relativity imposes the condition:

\displaystyle{\lim_{v \to 0} \gamma (c,v) =1}

By the other hand, according to special relativity second postulate, light speed is constant in every reference frame. Therefore, the distance a light beam ( or wave packet) travels in every frame is:

x=ct in S, or equivalently x^2=c^2t^2


x'=ct' in S’, or equivalently x'^2=c^2t'^2

Then, the squared spacial  separation between the moving light-like object at S’ with respect to S will be


Squaring the modified galilean transformations, we obtain:

x'^2=\gamma ^2(x-vt)^2 \rightarrow x'^2=\gamma ^2 (x^2+v^2t^2-2xvt) \rightarrow x'^2-\gamma ^2x^2+2\gamma ^2xvt=\gamma ^2 v^2t^2

x^2=\gamma ^2 (x'+vt')^2 \rightarrow x^2-\gamma ^2x'^2-2\gamma ^2x'vt'=\gamma ^2v^2t'^2

The only “weird” term in the above last two equations are the mixed term with “xvt” (or the x’vt’ term). So, we have to make some tricky algebraic thing to change it. Fortunately for us, we do know that x'=\gamma(x-vt), so

x'=\gamma x -\gamma vt \rightarrow \gamma x'=\gamma ^2 x-\gamma ^2 vt \rightarrow \gamma xx'=\gamma ^2 x^2-\gamma ^2 xvt

and thus

2\gamma xx'=2 \gamma ^2 x^2-2\gamma ^2 xvt \rightarrow 2\gamma ^2 xvt =2\gamma ^2x^2-2\gamma xx'

In the same way,  we proceed with the inverse transformations:

x=\gamma x'+\gamma vt' \rightarrow \gamma x=\gamma ^2x'+\gamma ^2vt' \rightarrow \gamma xx'=\gamma ^2x'^2-\gamma ^2x'vt'

and thus

2\gamma xx'=2\gamma^2x'^2+2\gamma^2x'vt' \rightarrow 2\gamma^2x'vt'=2\gamma^2x'^2-2\gamma^2xx'

We got it! We can know susbtitute the mixed x-v-t and x’-v-t’ triple terms in terms of the last expressions. In this way, we get the following equations:

x'^2=\gamma ^2(x-vt)^2 \rightarrow x'^2=\gamma ^2(x^2+v^2t^2-2xvt) \rightarrow x'^2-\gamma ^2x^2+2\gamma ^2x^2-2\gamma ^2xx'=\gamma ^2v^2t^2 \rightarrow x'^2+\gamma ^2x^2-2\gamma ^2xx'=\gamma ^2v^2t^2

x'^2=\gamma ^2(x'+vt')^2 \rightarrow x^2=\gamma ^2(x'^2+v^2t^2+2x'vt') \rightarrow x^2-\gamma ^2x'^2+2\gamma ^2x'^2-2\gamma ^2xx'=\gamma ^2v^2t'^2 \rightarrow x^2+\gamma ^2x'^2-2\gamma ^2xx'=\gamma ^2v^2t'^2

And now, the final stage! We substract the first equation to the second one in the above last equations:

x^2-x'^2+\gamma ^2(x'^2-x^2)=\gamma ^2v^2(t'^2-t^2) \rightarrow (x'^2-x^2)(\gamma ^2-1)= \gamma ^2v^2(t'^2-t^2)

But we know that x^2-x'^2=c^2(t^2-t'^2), and so

(x'^2-x^2)(\gamma ^2-1)= \gamma ^2v^2(t'^2-t^2) \rightarrow c^2(x'^2-x^2)(\gamma ^2-1)= \gamma ^2v^2(x'^2-x^2)


c^2(\gamma ^2-1)= \gamma ^2v^2 \rightarrow -c^2= -\gamma ^2c^2+\gamma ^2v^2 \rightarrow \gamma ^2=\dfrac{c^2}{c^2-v^2}

or, more commonly we write:

\gamma ^2=\dfrac{1}{1-\dfrac{v^2}{c^2}}

and therefore

\gamma =\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}

Moreover, we usually define the beta (or boost) parameter to be

\beta = \dfrac{v}{c}

To obtain the time transformation we only remember that x'=ct' and x=ct for light signals, so then, for time we  obtain:

x'=\gamma (x-vt) \rightarrow t' =x' /c= \gamma (x/c-vt/c)=\gamma ( t- vx/c^2)

Finally, we put everything together to define the Lorentz transformations and their inverse for 1D motion along the x-axis:

x'=\gamma (x-vt)



t'=\gamma \left( t-\dfrac{vx}{c^2}\right)

and for the inverse transformations

x=\gamma (x'+vt)



t=\gamma \left( t'+\dfrac{vx'}{c^2}\right)


If you don’t like those long calculations, there is a trick to simplify the “derivation” above.  The principle of Galilean relativity enlarged for electromagnetic phenomena implies the structure:

x'=\gamma (x-vt) and x=\gamma (x'+vt') for the inverse.

Now, the second postulate of special relativity says that light signals travel in such a way light speed in vacuum is constant, so t=x/c and t'=x'/c. Inserting these times in the last two equations:

x'=\gamma (1-v/c)x and x=\gamma (1+v/c)x'

Multiplying these two equations, we get:

x'x =\gamma ^2(1+v/c)(1-v/c)xx'.

If we consider any event beyond the initial tic-tac, i.e., if we suppose t\neq 0 and t'\neq 0, the product xx' will be different from zero, and we can cancel the factors on both sides to get what we know and expect: \gamma^2(1-v^2/c^2)=1 i.e.


LOG#004. Feynmanity.

The dream of every theoretical physicist, perhaps the most ancient dream of every scientist, is to reduce the Universe ( or the Polyverse if you believe we live in a Polyverse, also called Multiverse by some quantum theorists) to a single set of principles and/or equations. Principles should be intuitive and meaningful, while equations should be as simple as possible but no simpler to describe every possible phenomenon in the Universe/Polyverse.

What is the most fundamental equation?What is the equation of everything? Does it exist? Indeed, this question was already formulated by Feynman himself  in his wonderful Lectures on Physics! Long ago, Feynman gave us other example of his physical and intuitive mind facing the First Question in Physics (and no, the First Question is NOT “(…)Dr.Who?(…)” despite many Doctors have faced it in different moments of the Human History).

Today, we will travel through this old issue and the modest but yet intriguing and fascinating answer (perhaps too simple and general) that R.P. Feynman found.

Well, how is it?What is the equation of the Universe? Feynman idea is indeed very simple. A nullity condition! I call this action a Feynmann nullity, or feynmanity ( a portmanteau), for brief. The Feynman equation for the Universe is a feynmanity:


Impressed?Indeed, it is very simple. What is the problem then?As Feynman himself said, the problem is really a question of “order” and a “relational” one. A question of what theoretical physicists call “unification”. No matter you can put equations together, when they are related, they “mix” somehow through the suitable mathematical structures.  Gluing “different” pieces and objects is not easy.  I mean, if you pick every equation together and recast them as feynmanities, you will realize that there is no relation a priori between them. However, it can not be so in a truly unified theory. Think about electromagnetism. In 3 dimensions, we have 4 laws written in vectorial form, plus the gauge condition and electric charge conservation through a current. However, in 4D you realize that they are indeed more simple. The 4D viewpoint helps to understand electric and magnetic fields as the the two sides of a same “coin” (the coin is a tensor). And thus, you can see the origin of the electric and magnetic fields through the Faraday-Maxwell tensor F_{\mu \nu }. Therefore, a higher dimensional picture simplifies equations (something that it has been remarked by physicists like Michio Kaku or Edward Witten) and helps you to understand the electric and magnetic field origin from a second rank tensor on equal footing.

You can take every equation describing the Universe set it equal to zero. But of course, it does not explain the origin of the Universe (if any), the quantum gravity (yet to be discovered) or whatever. However, the remarkable fact is that every important equation can be recasted as a Feynmanity! Let me put some simple examples:

Example 1. The Euler equation in Mathematics. The most famous formula in complex analysis is a Feynmanity e^{i\pi}+1=0 or e^{2\pi i}=1+0 if you prefer the constant \tau=2\pi.

Example 2. The Riemann’s hypothesis. The most important unsolved problem in Mathematics(and number theory, Physics?) is the solution to the equation \zeta (s)=0, where \zeta(s) is the celebrated riemann zeta function in complex variable s=\kappa + i \lambda, \kappa, \lambda \in \mathbb{R}. Trivial zeroes are placed in the real axis s=-2n \forall n=1,2,3,...,\infty. Riemann hypothesis is the statement that every non-trivial zero of the Riemann zeta function is placed parallel to the imaginary axis and they have all real part equal to 1/2. That is, Riemann hypothesis says that the feynmanity \zeta(s)=0 has non-trivial solutions iff s=1/2\pm i\lambda _n, \forall n=1,2,3,...,\infty, so that

\displaystyle{\lambda_{1}=14.134725, \lambda_{2}= 21.022040, \lambda_{3}=25.010858, \lambda _{4}=30.424876, \lambda_{5}=32.935062, ...}

I generally prefer to write the Riemann hypothesis in a more symmetrical and “projective” form. Non-trivial zeroes have the form s_n=\dfrac{1\pm i \gamma _n}{2} so that for me, non-trivial true zeroes are derived from projective-like operators \hat{P}_n=\dfrac{1\pm i\hat{\gamma} _n}{2}, \forall n=1,2,3,...,\infty. Thus

\gamma_1 =28.269450, \gamma_2= 42.044080, \gamma_3=50.021216, \gamma _4=60.849752, \gamma_5=65.870124,...

Example 3. Maxwell equations in special relativity. Maxwell equations have been formulated in many different ways along the history of Physics. Here a picture of that. Using tensor calculus, they can be written as 2 equations:

\partial _\mu F^{\mu \nu}-j^\nu=0


\epsilon ^{\sigma \tau \mu \nu} \partial _\tau F_{\mu\nu}=\partial _\tau F_{\mu \nu}+ \partial _\nu F_{\tau \mu}+\partial_\mu F_{\nu \tau}=0

Using differential forms:



d\star F-J=0

Using Clifford algebra (Clifford calculus/geometric algebra, although some people prefer to talk about the “Kähler form” of Maxwell equations) Maxwell equations are a single equation: \nabla F-J=0 where the geometric product is defined as \nabla F=\nabla \cdot F+ \nabla \wedge F.

Indeed, in the Lorentz gauge  \partial_\mu A^\mu=0, the Maxwell equations reduce to the spin one field equations:

\square ^2 A^\nu=0

where we defined

\square ^2=\square \cdot \square = \partial_\mu \partial ^\mu =\dfrac{\partial^2}{\partial x^i \partial x_i}-\dfrac{\partial ^2}{c^2\partial t^2}

Example 4. Yang-Mills equations. The non-abelian generalization of electromagnetism can be also described by 2 feynmanities:

The current equation for YM fields is (D^{\mu}F_{\mu \nu})^a-J_\nu^a=0

The Bianchi identities are (D _\tau F_{\mu \nu})^a+( D _\nu F_{\tau \mu})^a+(D_\mu F_{\nu \tau})^a=0

Example 5. Noether’s theorems for rigid and local symmetries. Emmy Noether proved that when a r-paramatric Lie group leaves the lagrangian quasiinvariant and the action invariant, a global conservation law (or first integral of motion) follows. It can be summarized as:

D_iJ^i=0 for suitable (generally differential) operators D^i,J^i depending on the particular lagrangian (or lagrangian density) and \forall i=1,...,r.

Moreover, she proved another theorem. The second Noether’s theorem applies to infinite-dimensional Lie groups. When the lagrangian is invariant (quasiinvariant is more precise) and the action is invariant under the infinite-dimensional Lie group parametrized by some set of arbitrary (gauge) functions ( gauge transformations), then some identities between the equations of motion follow. They are called Noether identities and take the form:

\dfrac{\delta S}{\delta \phi ^i}N^i_\alpha=0

where the gauge transformations are defined locally as

\delta \phi ^i= N^i_\alpha \epsilon ^\alpha

with N^i_\alpha certain differential operators depending on the fields and their derivatives up to certain order. Noether theorem’s are so general that can be easily generalized for groups more general than those of Lie type. For instance, Noether’s theorem for superymmetric theories (involving lie “supergroups”) and many other more general transformations can be easily built. That is one of the reasons theoretical physicists love Noether’s theorems. They are fully general.

Example 6. Euler-Lagrange equations for a variational principle in Dynamics take the form \hat{E}(L)=0, where L is the lagrangian (for a particle or system of particles and \hat{E}(L) is the so-called Euler operator for the considered physical system, i.e., if we have finite degrees of freedom, L is a lagrangian) and a lagrangian “density” in the more general “field” theory framework( where we have infinite degrees of freedom and then L is a lagrangian density \mathcal{L}. Even the classical (and quantum) theory of (super)string theory follows from a lagrangian (or more precisely, a lagrangian density). Classical actions for extended objects do exist, so it does their “lagrangians”. Quantum theory for p-branes p=2,3,... is not yet built but it surely exists, like M-theory, whatever it is.

Example 7.  The variational approach to Dynamics or Physics implies  a minimum ( or more generally a “stationary”) condition for the action. Then the feynmanity for the variational approach to Dynamics is simply \delta S=0. Every known fundamental force can be described through a variational principle.

Example 8. The Schrödinger’s equation in Quantum Mechanics H\Psi-E\Psi=0, for certain hamiltonian operator H. Note that the feynmanity is itself H=0 when we studied special relativity from the hamiltonian formalism. Even more, in Loop Quantum Gravity, one important unsolved problem is the solution to the general hamiltonian constraint for the gauge “Wilson-like” loop variables, \hat{H}=0.

Example 9. The Dirac’s equation (i\gamma ^\mu \partial_\mu - m) \Psi =0 describing free spin 1/2 fields. It can be also easily generalized to interacting fields and even curved space-time backgrounds. Dirac equation admits a natural extension when the spinor is a neutral particle and it is its own antiparticle through the Majorana equation

i\gamma^\mu\partial_\mu \Psi -m\Psi_c=0

Example 10. Klein-Gordon’s equation for spin 0 particles: (\square ^2 +m^2 )\phi=0.

Example 11. Rarita-Schwinger spin 3/2 field equation: \gamma ^{\mu \nu \sigma}\partial_{\nu}\Psi_\sigma+m\gamma^{\mu\nu}\Psi_\nu=0. If m=0 and the general conventions for gamma matrices, it can be also alternatively writen as

\gamma ^\mu (\partial _\mu \Psi_\nu -\partial_\nu\Psi_\mu)=0

Note that antisymmetric gamma matrices verify:

\gamma ^{\mu \nu}\partial_{\mu}\Psi_\nu=0

More generally, every local (and non-local) field theory equation for spin s can be written as a feynmanity or even a theory which contains interacting fields of different spin( s=0,1/2,1,3/2,2,…).  Thus, field equations have a general structure of feynmanity(even with interactions and a potential energy U) and they are given by \Lambda (\Psi)=0, where I don’t write the indices explicitely). I will not discuss here about the quantum and classical consistency of higher spin field theories (like those existing in Vasiliev’s theory) but field equations for arbitrary spin fields can be built!

Example 12. SUSY charges. Supersymmetry charges can be considered as operators that satisfy the condicion \hat{Q}^2=0 and \hat{Q}^{\dagger 2}=0. Note that Grassman numbers, also called grassmanian variables (or anticommuting c-numbers) are “numbers” satisfying \theta ^2=0 and \bar{\theta}^2=0.

The Feynman’s conjecture that everything in a fundamental theory can be recasted as a feynmanity seems very general, perhaps even silly, but  it is quite accurate for the current state of Physics, and in spite of the fact that the list of equations can be seen unordered of unrelated, the simplicity of the general feynmanity (other of the relatively unknown neverending Feynman contributions to Physics)

something =0

is so great that it likely will remain forever in the future of Physics. Mathematics is so elegant and general that the Feynmanity will survive further advances unless  a  Feynman “inequality” (that we could call perhaps, unfeynmanity?) shows to be more important and fundamental than an identity. Of course, there are many important results in Physics, like the uncertainty principle or the second law of thermodynamics that are not feynmanities (since they are inequalities).

Do you know more examples of important feynmanities?

Do you know any other fundamental physical laws or principles that can not be expressed as feynmanities, and then, they are important unfeynmanities?

LOG#002. Information and noise.

We live in the information era. Read more about this age here. Everything in your sorrounding and environtment is bound and related to some kind of “information processing”. Information can also be recorded and transmitted.  Therefore, being rude, information is something which is processed, stored and transmitted. Your computer is now processing information, while you read these words. You also record and save your favourite pages and files in your computer. There are many tools to store digital information: HDs, CDs, DVDs, USBs,…And you can transmit that information to your buddies by e-mail, old fashioned postcards and letters, MSN, phone,…You are even processing information with your brain and senses, whenever you are reading this text. Thus, the information idea is abstract and very general. The following diagram shows you how large and multidisciplinary information theory(IT) is:

I  enjoyed as a teenager that old game in which you are told a message in your ear, and you transmit it to other human, this one to another and so on. Today, you can see it at big scale on Twitter. Hey! The message is generally very different to the original one! This simple example explains the other side of communication or information transmission: “noise”.  Although efficiency is also used. The storage or transmission of information is generally not completely efficient. You can loose information. Roughly speaking, every amount of information has some quantity of noise that depends upon how you transmit the information(you can include a noiseless transmission as a subtype of information process in which,  there is no lost information). Indeed, this is also why we age. Our DNA, which is continuously replicating itself thanks to the metabolism (possible ultimately thanksto the solar light), gets progressively corrupted by free radicals and  different “chemicals” that makes our cellular replication more and more inefficient. Don’t you remember it to something you do know from High-School? Yes! I am thinking about Thermodynamics. Indeed, the reason because Thermodynamics was a main topic during the 19th century till now, is simple: quantity of energy is constant but its quality is not. Then, we must be careful to build machines/engines that be energy-efficient for the available energy sources.

Before going into further details, you are likely wondering about what information is! It is a set of symbols, signs or objects with some well defined order. That is what information is. For instance, the word ORDER is giving you  information. A random permutation of those letters, like ORRDE or OERRD is generally meaningless. I said information was “something” but I didn’t go any further! Well, here is where Mathematics and Physics appear. Don’t run far away!  The beauty of Physics and Maths, or as I like to call them, Physmatics, is that concepts, intuitions and definitions, rigorously made, are well enough to satisfy your general requirements. Something IS a general object, or a set of objects with certain order. It can be certain DNA sequence coding how to produce certain substance (e.g.: a protein) our body needs. It can a simple or complex message hidden in a highly advanced cryptographic code. It is whatever you are recording on your DVD ( a new OS, a movie, your favourite music,…) or any other storage device. It can also be what your brain is learning how to do. That is  “something”, or really whatever. You can say it is something obscure and weird definition. Really it is! It can also be what electromagnetic waves transmit. Is it magic? Maybe! It has always seems magic to me how you can browse the internet thanks to your Wi-Fi network! Of course, it is not magic. It is Science. Digital or analogic information can be seen as large ordered strings of  1’s and 0’s, making “bits” of information. We will not discuss about bits in this log. Future logs will…

Now, we have to introduce the concepts through some general ideas we have mention and we know from High-School. Firstly, Thermodynamics. As everybody knows, and you have experiences about it, energy can not completely turned into useful “work”. There is a quality in energy. Heat is the most degradated form of energy. When you turn on your car and you burn fuel, you know that some of the energy is transformed into mechanical energy and a lof of energy is dissipated into heat to the atmosphere. I will not talk about the details about the different cycles engines can realize, but you can learn more about them in the references below. Simbollically, we can state that

\begin{pmatrix} AVAILABLE \\ENERGY\end{pmatrix}=\begin{pmatrix}TOTAL \;\;ENERGY \\SUPPLIED\end{pmatrix} - \begin{pmatrix}UNAVAILABLE \\ENERGY\end{pmatrix}

The great thing is that an analogue relation in information theory  does exist! The relation is:

\boxed{\mbox{INFORMATION} = \mbox{SIGNAL} - \mbox{NOISE}}

Therefore, there is some subtle analogy and likely some deeper idea with all this stuff. How do physicists play to this game? It is easy. They invent a “thermodynamic potential”! A thermodynamic potential is a gadget (mathematically a function) that relates a set of different thermodynamic variables. For all practical purposes, we will focus here with the so-called Gibbs “free-energy”. It allows to measure how useful a “chemical reaction” or “process” is. Moreover, it also gives a criterion of spontaneity for processes with constant pressure and temperature. But it is not important for the present discussion. Let’s define Gibbs free energy G as follows:

G= H - TS

where H is called enthalpy, T is the temperature and S is the entropy. You can identify these terms with the previous concepts. Can you see the similarity with the written letters in terms of energy and communication concepts? Information is something like “free energy” (do you like freedom?Sure! You will love free energy!). Thus, noise is related to entropy and temperature, to randomness, i.e., something that does not store “useful information”.

Internet is also a source of information and noise. There are lots of good readings but there are also spam. Spam is not really useful for you, isn’t it? Recalling our thermodynamic analogy, since the first law of thermodynamics says that the “quantity of energy” is constant and the second law says something like “the quality of energy, in general, decreases“, we have to be aware of information/energy processing. You find that there are signals and noise out there. This is also important, for instance, in High Energy Physics or particle Physics. You have to distinguish in a collision process what events are a “signal” from a generally big “background”.

We will learn more about information(or entropy) and noise in my next log entries. Hopefully, my blog and microblog will become signals and not noise in the whole web.

Where could you get more information? 😀 You have some good ideas and suggestions in the following references:

1) I found many years ago the analogy between Thermodynamics-Information in this cool book (easy to read for even for non-experts)

Applied Chaos Theory: A paradigm for complexity. Ali Bulent Cambel (Author)Publisher: Academic Press; 1st edition (November 19, 1992)

Unfortunately, in those times, as an undergraduate student, my teachers were not very interested in this subject. What a pity!

2) There are some good books on Thermodynamics, I love (and fortunately own) these jewels: 

Concepts in Thermal Physics, by Stephen Blundell, OUP. 2009.

A really self-contained book on Thermodynamics, Statistical Physics and topics not included in standard books. I really like it very much. It includes some issues related to the global warming and interesting Mathematics. I enjoy how it introduces polylogarithms in order to handle closed formulae for the Quantum Statistics.

Thermodynamcis and Statistical Mechanics. (Dover Books on Physics & Chemistry). Peter T. Landsberg

A really old-fashioned and weird book. But it has some insights to make you think about the foundations of Thermodynamics.

Thermodynamcis, Dover Pub. Enrico Fermi

This really tiny book is delicious. I learned a lot of fun stuff from it. Basic, concise and completely original, as Fermi himself. Are you afraid of him? Me too! E. Fermi was a really exceptional physicist and lecturer. Don’t loose the opportunity to read his lectures on Thermodynamcis.

Mere Thermodynamics. Don S. Lemons. Johns Hopkins University Press.

Other  great little book if you really need a crash course on Thermodynamics.

Introduction to Modern Statistical Physics: A Set of Lectures. Zaitsev, R.O. URSS publishings.

I have read and learned some extra stuff from URSS ed. books like this one. Russian books on Science are generally great and uncommon. And I enjoy some very great poorly known books written by generally unknow russian scientists. Of course, you have ever known about Landau and Lipshitz books but there are many other russian authors who deserve your attention.

3) Information Theory books. Classical information theory books for your curious minds are 

An Introduction to Information Theory: Symbols, Signals and Noise. Dover Pub. 2nd Revised ed. 1980.   John. R. Pierce.

A really nice and basic book about classical Information Theory.

An introduction to Information Theory. Dover Books on Mathematics. F.M.Reza. Basic book for beginners.

The Mathematical Theory of Communication. Claude E. Shannon and W.Weaver.Univ. of Illinois Press.

A classical book by one of the fathers of information and communication theory.

Mathematical Foundations of Information Theory. Dover Books on Mathematics. A.Y.Khinchin.

A “must read” if you are interested in the mathematical foundations of IT.