Do spin-1 particles also have phase symmetry?

  • Thread starter Lapidus
  • Start date
  • #1
343
11

Main Question or Discussion Point

In almost every QFT or particle textbook we learn that complex scalar fields or spinor fields (or even multiplets of spinor fields) have a phase symmetry (global gauge symmetry.) You can append to these fields an exponential with a complex phase in the Lagrangian and the dynamics remain the same. If we make the phase depend on spacetime and introduce a massless spin-1 field, we end up with local gauge symmetry or just gauge symmetry.

My question: can we also start with a pure spin-1 Lagrangian (massless or not) and just append an exponential with a complex phase to the spin-1 field? Since the dynamics are described by the square of the field tensor, I don't see how this could work. But what are the deeper reasons that complex scalar fields and spinors have phase symmetries and spin-1 fields have not? Or does it matter whether a field is complex or not?

thanks in advance for any anwers!
 

Answers and Replies

  • #2
fzero
Science Advisor
Homework Helper
Gold Member
3,119
289
In almost every QFT or particle textbook we learn that complex scalar fields or spinor fields (or even multiplets of spinor fields) have a phase symmetry (global gauge symmetry.) You can append to these fields an exponential with a complex phase in the Lagrangian and the dynamics remain the same. If we make the phase depend on spacetime and introduce a massless spin-1 field, we end up with local gauge symmetry or just gauge symmetry.

My question: can we also start with a pure spin-1 Lagrangian (massless or not) and just append an exponential with a complex phase to the spin-1 field? Since the dynamics are described by the square of the field tensor, I don't see how this could work. But what are the deeper reasons that complex scalar fields and spinors have phase symmetries and spin-1 fields have not? Or does it matter whether a field is complex or not?

thanks in advance for any anwers!
Phase symmetries can only be applied to complex fields. Vector fields, except maybe in some speculative exotic scenario, always satisfy some sort of reality condition, typically a Hermitian condition when viewed as a quantum field operator. For ##U(1)## gauge fields, the Hermitian operator condition is appropriate. For ##SU(N)## gauge fields, the generators of the adjoint representation are explicitly Hermitian matrices, then the entries of these matrices are further Hermitian operators.

So gauge fields cannot have a general phase symmetry. The best we can allow is a ##\mathbb{Z}_2## symmetry, but unless this is the same as the parity symmetry, it would forbid the standard formulation of gauge theory in terms of promoting ##\partial_\mu## to ##\partial_\mu + i A_\mu##. Incidentally, the same breakdown of the formalism would be true if we somehow found a way to allow more general phase symmetries. So it is crucial that gauge fields are real.
 
  • Like
Likes 1 person
  • #3
343
11
Thank you, fzero!!
 
  • #4
Bill_K
Science Advisor
Insights Author
4,155
195
So it is crucial that gauge fields are real.
The W± boson is an example of a gauge field that is complex.
 
  • #5
343
11
The W± boson is an example of a gauge field that is complex.
But if reality conditions do not forbid phase symmetry, what then?

Or is it just due to "the trouble with higher spin particles"? The problem that the spin states degree of freedom does not match the Lorentz indices in the spacetimes tensors and we end up with a redundant description (i.e. gauge invariance).

Maybe someone is knowledgeable enough in differential geometry and principal fiber bundles language could help out. Are not there some conditions that might explain why complex scalars, spinors and multiplets of fields have phase symmetry, but vector fields do not seem to have it?
 
  • #6
dextercioby
Science Advisor
Homework Helper
Insights Author
12,985
540
[...] Are not there some conditions that might explain why complex scalars, spinors and multiplets of fields have phase symmetry, but vector fields do not seem to have it?
Sure there are. The complex scalars, vectors and spinors are involuted Grassmann algebra-valued, where involution is complex conjugation. The requirement is to build real Lagrangians/Hamiltonians wrt involution therefore U(1) phase symmetry follows.
 
  • #7
Bill_K
Science Advisor
Insights Author
4,155
195
My question: can we also start with a pure spin-1 Lagrangian (massless or not) and just append an exponential with a complex phase to the spin-1 field? Since the dynamics are described by the square of the field tensor, I don't see how this could work. But what are the deeper reasons that complex scalar fields and spinors have phase symmetries and spin-1 fields have not? Or does it matter whether a field is complex or not?
For a vector particle the Lagrangian is, as you say, the square of the field tensor,

L = Wμν Wμν

The W bosons form an SU(2) triplet, and the Lagrangian is

L = Wμν · Wμν

where the dot product means

(W1μν)2 + (W2μν)2 + (W3μν)2

and W1, W2, W3 are real. Now we replace W1 and W2 with complex combinations W± = (W1 ∓ i W2)/√2. [corrected] The Lagrangian must then be written as

L = Wμν* · Wμν

This is invariant under the usual electromagnetic gauge transformation, in which a phase is added to W.
 
Last edited:
  • Like
Likes 1 person
  • #8
343
11
Ahh, so the crucial thing is indeed that the field is complex! (scalar, spinor or vector field)
 

Related Threads on Do spin-1 particles also have phase symmetry?

Replies
5
Views
3K
Replies
20
Views
12K
  • Last Post
Replies
1
Views
546
  • Last Post
Replies
2
Views
1K
  • Last Post
Replies
14
Views
3K
Replies
6
Views
1K
  • Last Post
Replies
11
Views
3K
  • Last Post
Replies
10
Views
3K
  • Last Post
Replies
3
Views
2K
Replies
19
Views
4K
Top