Theories of Everything, Mapped

Explore the deepest mysteries at the frontier of fundamental physics, and the most promising ideas put forth to solve them. Source: Theories of Everything, Mapped

How to quantize a field?

In an earlier blog post, I shared a link to a mind map of important fields in relativistic field theory.  In this mind map, we explored the classical field theory of scalar fields, complex scalar fields, Weyl spinors, Dirac spinors, Majorana spinors and the Vector field. Each of these fields correspond to a simple representation of the Lorentz group. A classical theory of these fields was the first step in going towards a Lorentz invariant theory of particles.

The next step involves going from classical fields to quantum particles. In order to do this, we need to make the fields operator valued and impose suitable commutation or anti-commutation relations on them. This is an elaborate endeavour in itself and hence I decided to dedicate another mind map to this process.

This mind map is about the Quantization of Fields and I am happy to share the link to it here:

https://coggle.it/diagram/VgFKcfyW5qwCs2Va/26b15bd0a9cc5ca6d4fc327af60f2c24fb3e6c5648a344b1840a47bb145e2a22

You may at any time create your own copy of this mind map and customize it for your study. Feel free to suggest changes as well. If you feel that I have missed an important sub-concept associated to the central concept of quantization, drop me a line and I will try my best to include it.

The bizarre parity transformation (in a mind map)

Parity is the operation of flipping the signs of odd numbers of space like co-ordinates.

The operation of parity is not made up of products of many infinitesimal operations starting from identity. It is a discrete transformation and fundamentally different from rotation.

The transformation of Quantum systems under the operation of parity can be quite bizarre. To add to our woes, it also turns out that under certain specific interactions, nature does not preserve parity of physical systems under time evolution! This is known as P violation or Parity violation.

I have summarized the amazing world of Parity in the context of Quantum systems in a mind map drawn in coggle. Take a look 🙂

https://embed.coggle.it/diagram/55e846396a271eae4c8e2d08/758bfbfd53d56bbed23f67acbde5dea16cab14735a38cb7be7d290c43790e888

What the hell is helicity and how is it different from spin?

Some particles have helicity, some have spin

Photons and Neutrinos have helicity while Electrons and Muons have spin. In general, a massless particle has a helicity while a massive particle has spin. In this post, I want to explore the peculiarities of the massive and the massless case which results in this difference.

Spin is a characteristic quantum numbers

Any free particle has a momentum and a set of other characteristics. An electron, for example, has a momentum, a characteristic mass, a characteristic charge etc. Spin is, likewise, a characteristic of a particle. An Electron always has spin $j = \frac{1}{2}$, for example.

There exists something analogous to spin for Photons (not helicity) which we will explore in this post. There’s no name for this quantum number as it is always 0. However, we will see that just like a spin $\frac{1}{2}$ particle has two states $j_z = + \frac{1}{2}$ and $j_z = - \frac{1}{2}$, similarly, Photons have two states with helicities +1 and -1. Therefore, helicity is analogous to the z component of spin and not spin itself.

Where do characteristic quantum numbers come from?

Any symmetry transformation will leave characteristic quantum numbers invariant. If it did not leave it invariant, then we would be able to change the measured charge of an electron by simply rotating our co-ordinate system, for example, which is ludicrous.

In group theoretic terms, this means that the operator, whose eigenvalue gives us the characteristic Quantum numbers, must commute with all generators of the symmetry group. Such operators are called Casimir operators.

The characteristic Quantum numbers are, therefore, Casimir operators corresponding to symmetries. In fact, both spin and helicity originate from the same Casimir operator. This is a Casimir operator of the Poincare group, which is a well known symmetry of nature,

Casimir operators of the Poincare group

The Poincare group has two Casimir operators: the well known $P^{\mu} P_{\mu}$ and the not so well known $W^{\mu}W_{\mu}$, where $P$ is the four-momentum operator and $W$ is the Pauli-Lubansky four-vector given by the expression:

$W^{\mu} = -\frac{1}{2} \epsilon^{\mu\nu\rho\sigma}J_{\nu\rho}P_{\sigma}$

It turns out that spin is the quantum number corresponding to the Casimir operator $W^{\mu}W_{\mu}$ in the case of massive particles. In the case of massless particles $W^{\mu}W_{\mu}$ is taken to be zero. Let us discuss these two cases separately and see how this comes about.

Massive particles have spin

To evaluate the Casimir operator for massive particles, we go to the rest frame of the particle, where its momentum is $\left( m,0,0,0 \right)$ . In this frame, $W^0 = 0$ and $W^i = mJ^i$ and therefore:

$-W^{\mu}W_{\mu} = m^2 \mathbf{J}^2$

This looks like the familiar total spin angular momentum operator.The eigenvalue of $\mathbf{J}^2$ is $j(j+1)$. Therefore massive particles are labeled by their spin value $j$ (which is the characteristic quantum number). States within each representation are labeled by the $j_z = -j, -j+1, \dots , j$.

Therefore, a massive particle can be labeled by the following quantum numbers $m^2, j, p \textrm{ and } j_z$.

One last comment before we close the section on massive particles: Note that we have derived the operators for a certain choice of momentum. These operators are going to look different for different choices of momentum. The Casimir operator will no longer look like $\mathbf{J^2}$ and $J_z$ is not going to look like $J_z$. This is obvious as $J_z$ doesn’t even commute with momentum in general, it only commutes in this special case with a special choice of the momentum.

The subgroup of the Poincare group which keeps a certain choice of momentum unchanged is known as the ‘little group’. The operators that generate the little group commute with momentum and hence can be used to label particle states. $J_z$ is a generator of the little group which keeps the momentum $\left( m, 0, 0, 0 \right)$ unchanged and hence we could use it to label our states.

Massless particles are…umm…different

The situation is different for massless particles. The momentum of a massless particle in its rest frame can be written as $\left( \omega, 0, 0, \omega \right)$. In this frame, a straightforward computation gives $W^0 = W^3 = \omega J^3$, $W^1 = \omega \left( J^1 - K^2 \right) \equiv \omega A$ and $W^2 = \omega \left( J^2 + K^1 \right) \equiv \omega B$. Therefore:

$-W_{\mu}W^{\mu} = \omega^2 \left[ \left( K^2 - J^1 \right)^2 + \left( K^1 + J^2 \right)^2 \right] = \omega \left[ A^2 + B^2 \right]$

These operators $A$ and $B$, along with $J^3$, close an algebra corresponding to the ‘little group’ in the massless case.

$\left[J^3,A \right] = iB, \; \left[ J^3, B \right] = -iA, \;\left[ A,B \right] = 0$

It turns out that $A$ and $B$ commute. Thus we can have states that are simultaneously eigenvalues of both operators.

$A \left| p; a,b \right> = a \left| p;a,b \right>, \; B \left| p;a,b \right> = b \left | p; a,b\right>$

On these states, we can evaluate the Casimir operator, which gives us the characteristic quantum number:

$-W^{\mu}W_{\mu} = \omega^2 ( a^2 + b^2 )$

But it doesn’t end there. To complicate matters, it turns out that if $\left| p;a,b \right>$ is an eigenstate of $A$ and $B$, so is the state:

$\left| p,a,b,\theta \right> \equiv e^{-i\theta J^3} \left| p; a,b \right>$

On this state the eigenvalue of $A$ is $a(\theta) \equiv \left( a \cos \theta - b \sin \theta \right)$ and the eigenvalue of $B$ is $b(\theta) \equiv\left( a \sin \theta + b \cos \theta \right)$!

This means that, unless $a=b=0$, we find a states corresponding with a continuous internal degree of freedom $\theta$! These quantum numbers don’t seem to exist (find physical applications) in Nature. Thus we are forced to choose $a=b=0$, which makes our Casimir operator zero.

$-W^{\mu} W_{\mu} = 0$

The analog of spin for a massless particle is therefore 0.

Massless particles have helicity

We found that the Casimir operator is zero for all massless particles. We still have to label the states though, like we did in the massive case. In order to do this, we need to find the maximal set of commuting operators. We already have $P^{\mu}P_{\mu}$ and $W^{\mu}W_{\mu}$. We have found three operators $A$, $B$ and $J^3$, which commute with $P^{\mu}$. Of course, the operator $A$ and $B$ cannot be used to label the states as we found that all their eigenvalues are zero. We can, however, try $J^3$, which may give us non zero eigenvalues.

$J^3$ generates rotations in the x-y plane which correspond to an Abelian $SO(2)$ algebra. All irreducible representations of this Abelian group (or any Abelian group, for that matter) are one dimensional. Therefore, there is only one state in the representation.

The eigenvalue of $J^3$ on this one dimensional state is known as helicity. It can be shown (using topological arguments) that the eigenvalue is quantized and can take values $h = 0, \pm \frac{1}{2}, \pm 1, \dots$.

Helicity and parity

We also remember that the choice of the momentum was arbitrary. We could have chosen the three momentum to be in any direction we wanted.  Therefore, the following is a more general and accurate representation of the helicity operator:

$H = \hat{\mathbf{p}} \cdot \mathbf{J}$

From this expression we can immediately see that helicity is a pseudoscalar, i.e, it changes sign under a  parity transformation. Thus, even though particles with $h = 1$ and $h= -1$ are logically two different species of particles under the Poincare algebra, parity exchanges them. If parity is a symmetry of the theory under consideration, then it is more convenient to group together particles with opposite parity and call them ‘left handed’ and ‘right handed’ variants of the same particle.

Parity is a symmetry of both electromagnetism and gravitation. Thus we have two polarization states of the Photon ($h = \pm 1$) and two polarization states of the Graviton ($h = \pm 2$).  On the other hand, Neutrinos, which only interact via the parity violating weak interaction,  are given different names according to their helicity. The name Neutrino is reserved for $h = -\frac{1}{2}$ and the name Antineutrino is reserved for $h = +\frac{1}{2}$.

A mind map of Quantum Fields

What is a mind map?

A mind map is a diagram used for visually organizing connected information. A mind map often revolves around a central concept or idea which is represented at the centre of the diagram. The representation can be in the form of text, images, links equations etc. Associated ideas or concepts branch out from the central concept. The branching can go on indefinitely until all the information associated with the central concept has been represented on the diagram.

Mind maps make complicated ideas look simple by providing a bird’s eye view of the inter-relationship between a central idea and associated ideas. This helps in memorization, seeing the big pictures, and in some cases, connecting the dots.

Studying Quantum Field Theory can be frustrating

I had decided earlier that I would study Physics using mind maps at some point in my life. In particular, I was frustrated at my inability to memorize the concepts of Quantum field theory. QFT is a complex theory with hundreds of associated concepts which find application, not only in the theory of elementary particles, but also in Condensed matter physics and Cosmology.  I also felt the need to eliminate the noise and look at the big picture at times. Therefore I decided to start making mind maps to make sense of things.

Mind mapping Quantum Fields with Coggle

Recently I came across Coggle,  a beautiful web based mind mapper with LaTeX support and I decided to start making mind maps on QFT and Elementary Particle Physics.

I made a mind map on Quantum Fields and I wanted to share it on my blog. Here it is (still a bit incomplete though):

https://embed.coggle.it/diagram/55dc56db241048026b544f39/eba0b19af09909f844d1d3838c74de17bc8b050078d95bf10622c0224d14c8f3

Compton wavelength via dimensional analysis

Dimensional analysis is cool because it gets wonderful results out of literally nothing. In this post, we figure out the Compton wavelength of an electron by simply talking and doing little to zero maths.

The cool way of finding the Compton wavelength of an electron

1. Suppose we want to find the Compton wavelength of the electron.
2. We know that the Compton effect is seen when a photon scatters off an electron. Therefore the physical entities that play a role in Compton scattering are the electron and the photon. An electron has a fixed mass and charge. The photon has a fixed velocity. We argue, therefore, that the Compton wavelength, which is a fixed quantity, must depend on these other fixed quantities and nothing else.
3. We reduce the set of quantities that might make up the Compton wavelength by further noting that the Compton effect occurs because of relativistic energy momentum conservation in an inelastic scattering. Since the photon wavelength is determined purely by energy momentum conservation, the electric charge plays no role : it could have been anything and the results would have been the same. Also, we can forget about $c$ and $\hbar$ because we will be working in natural units, which includes these two quantities by default. This is explained in detail in another blog post.
4. We conclude that the only quantity that might make up the Compton wavelength is the electron mass, which is 0.5 MeV.
5. We notice that $\frac{1}{m_e}$ has the units of length.
6. We conclude that the Compton wavelength must be in the scale of $\frac{1}{m_e} \sim 2 \times 10^{-12} \text{m}$
7. 😎

What is Theoretical Physics?

Theoretical physics is defined as a sequence of courses, all of which discuss the Harmonic Oscillator.