Why is the Helicity for a Mass-less particle Lorentz invariant?
Below (under my answer) is the technical and very difficult to understand (for someone who is not well versed in advance discourses of maths and physics) explanation given in some place. I could not resist to think about it and share what I found.
Here is my two cents of simpler words, which I am still wanting to understand more.
First I need to define the jargons or the technicalities.
mass-less; I will take just the photons, the only mass-less particles that we have ever found, others being gluon, found but not definitive and gravitons, theorized to exist, but neither found nor definitive.
Photons; no matter where you measure the mass of photon it will always be zero. Thats because Light is a wave it does not have mass and its energy is numerically equal to its momentum. (if expressed in appropriate units and dimensions.) When we consider light as not just pure classical waves (when we follow a method called “quantization of wave fields”) we still have a wave with its associated particles having a zero mass.
Strictly we did not have zero mass particles in classical mechanics (or “pure wave” mechanics) its a quantum mechanical property. where Quantum Mechanics makes sure a weird concept; a particle with zero mass.
As I have explained elsewhere on my web-site in recent articles; waves give to particle properties and particles give their properties to waves, when we theorize wave-particles. There are no pure waves or particles anymore since ~1925. Its for these reason, now particles such as photons, electrons, protons have wiggled existence, they don’t have a definite size or extent that is, because now they boast of wave’s qualities, given that they hang out as the same.
Result; now that photons are mass-less, their energy, momentum, speed, etc are no more variables, in the sense of arbitrariness. They are constants, taking only a few values, but constant in a given situation. But other particles have these properties; arbitrary. So electrons energy and momentum are not fixed, but arbitrary.
But as long as we are considering only elementary particles (that is, we are in a Quantum Zone) eg, electrons, protons, photons, and not nutmegs, soccer balls and airplanes and satellites there is another quantity that is of important consequence that is constant. Spin; whether a mass-less particle or not, spin has the same magnitude for them. that is spin is same for photon, its always 1. Spin for an electron is always 1/2. Spin for proton is always 1/2. Its for this reason photon is called a Boson**. Any thing with spin, 0, 1, 2, etc will be a Boson. Anything with spin 1/2, 3/2 etc will be called Fermion.
**Note that Boson comes from name SN Bose, who discovered this fact about particles with spin 0, 1, 2, 3 etc would follow certain principle. And Fermi did the other part. But its the principle or properties of spin which is named toward Bose, not the particle itself. Just like gravity is named toward Galileo or Newton, for satellites and planets are not. In Particle Physics a particle is named after a scientist only if the scientist has a direct role in the finding of a particular particle. SO its a bunch of attributes and the particle itself which are discovered by a scientist, and still only a fraction of such scientists are lucky to have a name sake. eg Higgs did a great deal of work along with 3 other particular scientists and the particle Higgs is named after him, and not Bose. Plus if a particle were to be named after a scientist Bose, the particle would be called Bose and not Boson. Here Bose is used like Galilean, Newtonian or Einsteinian. Boson depicts a principle and properties and not a particle. A particle is an embodiment of many different principle and certainly Bose did not buy them all. Plus no bunch of particles which includes both discovered and not discovered, found and hypothesized and even possible particles were ever named after a particular scientists. Although Boson sis said in the context of Bose-Einstein principle, Einstein did not have the clout for an arbitrary number of particles to be NAMED after him. Who was Bose? Plus a particle being named after a scientist means, as I just said above, a particular scientist working for a particular particle, and very rarely lucky enough that that particle is named after him, in a particular way, eg Higgs and not Higgsion. Yukawa and not Yukawaion. The analogy is we have Galielan Gravity (as opposed to Einsteinian Gravity) a particular satellite named as Galileo, but not ALL satellites named as Galileo. but one can say a Galielian satellite in certain circumstance to mean a particular principle is in action, but to say that satellite is named after Galileo is to hinge your wishes on illusion and perhaps a very murky sense of entitlement.
Now back to our main contention of this article.
In the quantum zone, all particles (often called elementary particles) have a fixed spin. But only mass-less particles have a fixed momentum as well. (its magnitude is fixed numerically to its energy and energy is a Lorentz Invariant, but we will define the latter first.) But other properties such as energy, momentum, speed etc of the mass-less are also fixed. But for mass-haves these other properties are arbitrary and not constant. Which is why other particles can be brought to rest but not the photons. This is also the reason why God could not intervene in the workings of the Universe. No mayter how much he wishes he cayn’t bring the photon to rest.
So in a sentence one should not say Rest-Frame of Photons, because they don’t exist.
Whats a Lorentz In-variant. A Lorentz invariant is something which is unchanged under something called a Lorentz Transformation. Something is not changing when something else is made to change. The first something is Lorentz Invariant or simply invariant, the second something is Lorentz Transformation or simply transformation. So an invariant is a quantity that did not change and a transformation is a quantity that was made to change. These are important parameters in physics. Lorentz is the name of a scientist after whom these methods and situations are named. Sometimes its called Lorentzian, but still the facts remain which methods are named after him. In more specific ways, A transformation is when you change the speed of an object from where you are making your observations.
An energy is always a Lorentz Invariant, a fact said as “conservation of energy”. When you are measuring the energy of a photon from earth which is lets say moving around sun at 30 kms/ second you will get the same value that if you measure from a satellite in free-fall around sun, at a different speed than earth. (Earth is in free fall around sun at 30 kms/ sec) You can also assign a great value for speed, lets say 300 km/sec or 30000 kms/sec but still the speed of photon will be the same. Also the energy of the photon is same under such alternative conditions. These alternative conditions are called as Transformation. And energy and speed-of-photon turned out to be Lorentz Invariant or invariant under Lorentz transformation. That is they are the same, no matter how you changed the speed of objects from where to measure the values.
Energy is Lorentz Invariant for all the particles we have named above, in Quantum Zone. But speed is only for mass-less particles a Lorentz Invariant Quantity.
Now you might have gotten your clue as to why Helicity would be a Lorentz Invariant for mass-less particles. Its because spin and momentum are both constant for the mass-less particle. No mass is spent, because no mass is there to begin, hence momentum and energy are not traded among each other, or rather traded in an infinitesimal way, which is not affected by how fast you are moving wrt it, because your motion is not changing the mass, but for other quantum-zone particles, energy is traded between mass and momentum. Which is why you have the relation E(squared)=m(squared)+p(squared) in appropriate c-units. When you are measuring this relation from an object which has same velocity as that of the particle whose energy and momentum you are measuring, the p=zero and you have E=m or E=mc(square), the famous Einstein Relation. So not only energy is equivalent to mass but also its equivalent to momentum. Its traded between both, in general. Since for a quantum-zone particle other than mass-less photon, one (eg electron) has a mass which is not zero, depending on where you are measuring its physical quantities, it will have arbitrary velocities and hence an arbitrary trade-off between energy, mass and momentum. That simply means mass also becomes arbitrary even if its fixed when measured when the particle is at rest. Hence the concept of rest-mass vs general mass is sensible only for mass-have particles such as electrons, but for photons its nonsensical, rest-mass and rest-frames are non-existent parameters for photons. But all in all even if electron has arbitrary mass and velocity (when we apply Lorentz transformation to it) its energy turn out to be invariant or unchanged under such transformations. Thats why we say energy is conserved.
Now Lorentz Transformation is basically a change in velocity , hence change in speed and direction. Rather a change in what direction we are measuring the various particles motion and at what rate that position and direction is changing. I just change my angles and I change the rate at which the angle is changing. For the mass-less photon (and for any particle) helicity is H = S.p(cap) where S is spin and p(cap) unit direction of momentum. For the mass-less S is spin is constant and changing direction will change direction of both spin and momentum in the same way. But for mass-haves the spin is constant but the momentum is arbitrary. So depending on speed at which we are measuring the particle its helicity will differ as spin and momentum will be differently changed.
IF there is something thats technically inconsistent you are welcome to point out. This freedom is towards those who really understand the situation, and not towards “I feel like I have my own theory of Physics, and you are wrong no matter what you say”. Sorry, go elsewhere. By the way there is a discount on door locks today. Get one and lock yourself and throw the key and call your ISP to disconnect you from hellow world.
For the ans I copied from Quora, I am not denouncing it, its just too technical and too long. Well thats what I thought, of not denouncing it, rather I changed my mind after reading the first sentence; the first sentence is kind of very murky, so I am denouncing it. How is photon momentum zero again, because mass is zero? Is energy also zero, not?
Answer from Quora from someone who proclaims he is a physicist
It is a consequence of the fact that for mass-less particles of arbitrary spin the momentum is a null vector – since the momentum squared, which is the mass squared, is zero.
The little group for a null vector, the little group being that subgroup of all Lorentz transformations that leaves some given four momentum invariant, in four dimensions, is just E2 – it is isomorphic to the group of translations of the two dimensional Euclidean plane.
It is a smaller and simpler subgroup than is the little group for a massive particle, which is SU(2). It is not hard to imagine why this is the case for massless particles: it’s because all null vectors have at least two non-vanishing components that cannot be rotated or boosted away, while for massive particles, the four momentum is timelike, and all but one component of a timelike vector can always be rotated or boosted away.
So the little group for a massive particle is less constrained than it is for a massless particle.
For tachyonic representations it is a very nice excercise to work out what the little group is.
To construct the unitary irreducible representations of the Lorentz group for a particle with general spin it suffices to construct the representations of the little group – then a general boost generates an irreducible representation of the full Lorentz group.
The representations of E2 are of two types: there are both discrete and continuous spin representations. The discrete spin representations are the normal ones considered as single particle states.
For the discrete spin representations it’s found that general boosts leave the helicity invariant – this is almost immediately clear since the helicity is by definition that component of the spin that lies along the momentum vector of the particle: but for a massless particle the momentum vector is a null vector.
It’s really a consequence of the definition of helicity and the fact that a massless particle moves on the light cone, to put things as simply as is possible.
You can easily verify that it’s the case from the definition of helicity and the form of a general Lorentz transformation.
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