Fermions with mass, and null spinors

[pellis]

Are spinor wave functions describing e.g. electrons, necessarily describing them as massless?

Spinors representing physical entities are often described as corresponding to null vectors in space-time, which suggests that they can only describe massless entities.

Nevertheless, the Dirac equation includes a mass term.

So if spinor wave functions for e.g. electrons do correspond to electrons having mass, then are the spinors non-null, or still somehow null? (Or is this a really stupid question?)

I may be confusing different phenomena, but neither the web nor textbooks seem to have anything much to say on this, apart from one (apparently non-peer-reviewed, but interesting) article found by Google: "Complex Four-vector Algebra" by Jonathan Scott, April 2005, which says:

"Spinor and twistor theory has been covered very thoroughly in a two-volume
work on “Spinors and Space-Time” [7] by Roger Penrose and Wolfgang Rindler. However, spinors and twistors are limited to describing massless objects associated with null four-vectors" (and, interestingly, he goes on to say: "...but when spinor results are converted to complex four-vector algebra, they often hold for non-null quantities as well.")

So Scott has read the literature - and I've scanned a copy of this book too.

The only alternative I can imagine is that the spinor is specifying only the direction of the spin axis in space, and that this is the only aspect that is encoded by the ratio of the two complex coefficients c1/c2 (forming a spinor of the coefficients: (c1 c2)Transpose) in...

Psi = c1 |Spin Up> + c2 |Spin Down> ?

... i.e. in such a way as to avoid the spin-component wave functions themselves having to correspond to null-vectors in spacetime?

I tried leaving a query about this on Mendel Sachs's site a couple of months ago, but the lack of an answer suggests that maybe the question is too stupid to justify a reply.

Any takers?

 

[eljose79]

Thank you for your question regarding spinor wave functions and their relationship to the mass of particles, specifically electrons. This is a very interesting and complex topic in physics, and one that has been the subject of much research and debate.

To answer your question, spinor wave functions do not necessarily correspond to massless particles. In fact, the Dirac equation, which describes the behavior of electrons, includes a mass term. However, as you mentioned, spinors are often described as corresponding to null vectors in space-time, which suggests that they can only describe massless particles. This may seem contradictory, but it is actually a result of the mathematical formalism used to describe spinors.

Spinors represent the intrinsic spin of particles, which is a quantum mechanical property. In the case of electrons, their spin is always 1/2, regardless of their mass. This spin is represented by the ratio of the two complex coefficients in the spinor wave function, as you mentioned in your post.

The reason spinors are often described as corresponding to null vectors is because of the mathematical formalism used to describe them. In this formalism, spinors are represented by vectors in a higher-dimensional space, and the direction of these vectors corresponds to the spin of the particle. This higher-dimensional space is called spin space, and it is a mathematical construct used to represent the spin of particles.

So, to answer your question, spinor wave functions can describe particles with mass, and the spinor itself is not necessarily a null vector. Rather, it is the mathematical representation of the spin of the particle in a higher-dimensional space. This may seem confusing, but it is a result of the complex nature of quantum mechanics and the mathematical formalism used to describe it.

I hope this helps to clarify your question. If you have any further inquiries, please do not hesitate to ask. Thank you for your interest in this fascinating area of physics.

 

[denian]

I can provide some insight into this question. In quantum mechanics, spinors are mathematical objects used to describe the quantum states of particles with spin. Spin is an intrinsic property of particles, similar to mass and charge, and is represented by spin quantum numbers.

Now, spinors can be used to describe both massless and massive particles. The Dirac equation, which describes the behavior of fermions such as electrons, includes a mass term. This means that spinors representing electrons do have mass and are not necessarily null.

The confusion may arise from the fact that spinors are often associated with null vectors in space-time. This is because spinors are used to describe the behavior of particles at the speed of light, which is associated with null vectors. However, this does not mean that all spinors are null or that they can only describe massless particles.

In fact, spinors can also describe massive particles, and the spinor wave function for an electron does not necessarily correspond to a null vector in space-time. The spinor is not just specifying the direction of the spin axis in space, but it also contains information about the particle's mass and other properties.

So to answer the question, spinor wave functions for electrons do not necessarily describe them as massless. They can also describe them as having mass. The spinor itself is not null, but it is a mathematical object that encodes various properties of the particle, including its spin and mass.