
#1
Oct1313, 12:19 PM

P: 270

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 spin1 field, we end up with local gauge symmetry or just gauge symmetry.
My question: can we also start with a pure spin1 Lagrangian (massless or not) and just append an exponential with a complex phase to the spin1 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 spin1 fields have not? Or does it matter whether a field is complex or not? thanks in advance for any anwers! 



#2
Oct1313, 02:20 PM

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PF Gold
P: 2,606

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. 



#3
Oct1413, 02:15 AM

P: 270

Thank you, fzero!!




#4
Oct1413, 05:30 AM

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Thanks
P: 3,846

Do spin1 particles also have phase symmetry? 



#5
Oct1413, 09:52 AM

P: 270

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
Oct1413, 10:17 AM

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P: 11,863





#7
Oct1413, 11:14 AM

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P: 3,846

L = W_{μν} W^{μν} The W bosons form an SU(2) triplet, and the Lagrangian is L = W_{μν} · W^{μν} where the dot product means (W^{1}_{μν})^{2} + (W^{2}_{μν})^{2} + (W^{3}_{μν})^{2} and W^{1}, W^{2}, W^{3} are real. Now we replace W^{1} and W^{2} with complex combinations W^{±} = (W^{1} ∓ i W^{2})/√2. [corrected] The Lagrangian must then be written as L = W_{μν}* · W^{μν} This is invariant under the usual electromagnetic gauge transformation, in which a phase is added to W. 



#8
Oct1413, 12:09 PM

P: 270

Ahh, so the crucial thing is indeed that the field is complex! (scalar, spinor or vector field)



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