Why is spin-1 field described by a vector field?

[ingenue]

It's a famous claim that spin-0, spin-1 and spin-2 fields are described by scalar, vector and second-rank tensor, respectively. My question is: why not other objects? For example, consider spin-1 field, we can use a field that carries two left spinor indexes. From the group-theoretic relation we know it contains a scalar component and a spin-1 component, the same as the vector representation.

This claim is first recognized (to my knowledge) on a classic paper (Fierz 1939 Helv. Phys. Acta, 12, 3), but it's in German and I can't find the English version online. Can anyone help me understand it? Thanks.

 

[humanino]

A vector field carries spin-1. An object with spin-1 is not necessarily a fundamental vector field however. As you said, one can form a scalar and a vector out of two spinors. By the same token, something which has spin-2 could be made out of a pair of vectors.

Given that you seem to realize the above, I am not sure what your question is. If one is interested in fundamental particles (not composite objects) then one has no choice but to use scalar, spinor, vector and symmetric tensor field for spin-0, spin-1/2, spin-1 and spin-2. If one asks the question "what composite objects can make up a spin-1 ?" then there are more possible answers.

 

[ingenue]

Given that you seem to realize the above, I am not sure what your question is. If one is interested in fundamental particles (not composite objects) then one has no choice but to use scalar, spinor, vector and symmetric tensor field for spin-0, spin-1/2, spin-1 and spin-2. If one asks the question "what composite objects can make up a spin-1 ?" then there are more possible answers.

In what sense are scalar, vector and tensor more "fundamental" than spinor? They're both representations of the Lorentz group.

 

[humanino]

In what sense are scalar, vector and tensor more "fundamental" than spinor?

In no sense I can think of and I did not mean to convey such a claim.

 

[kexue]

I remember reading somewhere that you can get a spin-1 particle from an antisymmetric, self-dual tensor field. But such a field theory would not be parity invariant, and so it is unphysical.

So maybe the field you want to construct, is also not physical?

 

[haushofer]

You should read the chapters of Srednicki about the representations of the Lorentz group (chapter 30 and onwards, or something like that). :)

 

[ingenue]

You should read the chapters of Srednicki about the representations of the Lorentz group (chapter 30 and onwards, or something like that). :)

I had read it. It doesn't address my question. For example, the equation (34.19) in the online version of that book shows that a field carries two symmetric spinor indexes is spin-1, so why can't we use this to describe spin-1 fields?

 

[humanino]

I had read it. It doesn't address my question. For example, the equation (34.19) in the online version of that book shows that a field carries two symmetric spinor indexes is spin-1, so why can't we use this to describe spin-1 fields?

You can mathematically do so. The question coming next is then : what does it physically mean ?

One may be tempted to interpret this construction as a photon made-up from a pair of fermions. As you can imagine, this will run into problems.

One may be tempted to reformulate things in terms of twistors. I suspect you would be more interested in this direction.

 

[bobloblaw]

Hey also i think only the field strength (the F tensor) can be described by a symmetric - spinor index - tensor, so you would still have to use a vector to describe the field itself in order to derive the field equations via the Langrangian.