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

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Discussion Overview

The discussion revolves around the characterization of spin-1 fields and the justification for describing them using vector fields. Participants explore the implications of using different mathematical representations, such as scalar, vector, spinor, and tensor fields, in the context of fundamental particles and composite objects.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants assert that spin-1 fields are conventionally described by vector fields, while questioning the necessity of this choice and exploring alternatives such as using two left spinor indices.
  • Others argue that while composite objects can be formed from spinors, fundamental particles must adhere to the established representations of scalar, spinor, vector, and symmetric tensor fields for different spins.
  • A participant mentions the possibility of obtaining a spin-1 particle from an antisymmetric, self-dual tensor field, but notes that such a field theory would lack parity invariance, rendering it unphysical.
  • There is a discussion about the fundamental nature of scalar, vector, and tensor fields compared to spinors, with some participants questioning any claims of superiority among these representations.
  • One participant highlights that while it is mathematically feasible to describe spin-1 fields using symmetric spinor index tensors, the physical interpretation of such constructions may lead to complications, such as the problematic interpretation of a photon as a pair of fermions.
  • Another participant emphasizes that the field strength tensor (F tensor) can be described by a symmetric spinor index tensor, yet a vector is still necessary to describe the field itself for deriving field equations via the Lagrangian.

Areas of Agreement / Disagreement

Participants express differing views on the appropriateness of using vector fields to describe spin-1 fields, with no consensus reached on the necessity or implications of alternative representations.

Contextual Notes

Some discussions hinge on the definitions and interpretations of various mathematical constructs, and the implications of using composite versus fundamental particles remain unresolved.

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

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