Do spin-1 particles also have phase symmetry?

  1. 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!
  2. jcsd
  3. fzero

    fzero 3,120
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    Gold Member

    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.
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  4. Thank you, fzero!!
  5. Bill_K

    Bill_K 4,157
    Science Advisor

    The W± boson is an example of a gauge field that is complex.
  6. 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?
  7. dextercioby

    dextercioby 12,328
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    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.
  8. Bill_K

    Bill_K 4,157
    Science Advisor

    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: Oct 14, 2013
    1 person likes this.
  9. Ahh, so the crucial thing is indeed that the field is complex! (scalar, spinor or vector field)
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