# Do spin-1 particles also have phase symmetry?

• Lapidus
In summary: The W bosons form an SU(2) triplet, and the Lagrangian isL = Wμν · Wμνwhere the dot product means(W1μν)2 + (W2μν)2 + (W3μν)2and W1, W2, W3 are real. Now we replace W1 and W2 with complex combinations W± = (W1 ∓ i W2)/√2. [corrected] The Lagrangian must then beL = Wμν + iWμν+iWμν2

#### Lapidus

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!

Lapidus said:
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.

1 person
Thank you, fzero!

fzero said:
So it is crucial that gauge fields are real.
The W± boson is an example of a gauge field that is complex.

Bill_K said:
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?

Lapidus said:
[...] 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.

Lapidus said:
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.

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

## 1. What is phase symmetry in spin-1 particles?

Phase symmetry in spin-1 particles refers to the property where the wave function of the particle remains unchanged when multiplied by a complex phase factor. This means that the overall behavior and properties of the particle do not change when the phase of the wave function is altered.

## 2. How does phase symmetry affect spin-1 particle interactions?

Phase symmetry plays a crucial role in spin-1 particle interactions as it allows for the conservation of angular momentum and other fundamental symmetries. Without phase symmetry, the behavior of spin-1 particles would be drastically altered and many physical laws and principles would not hold.

## 3. Is phase symmetry unique to spin-1 particles?

No, phase symmetry is not unique to spin-1 particles. It is a fundamental concept in quantum mechanics and applies to all types of particles, including spin-1/2 particles and even composite particles made up of multiple sub-particles.

## 4. Can phase symmetry be broken or violated in spin-1 particles?

While phase symmetry is a fundamental property of spin-1 particles, it can be broken or violated under certain conditions. This can happen in high energy interactions or in the presence of external fields, but the overall symmetry is still preserved in most cases.

## 5. How is phase symmetry related to other symmetries in physics?

Phase symmetry is closely related to other fundamental symmetries in physics, such as translation symmetry and time symmetry. These symmetries work together to form the foundation of many physical laws and principles, and any violation of phase symmetry can have significant implications for our understanding of the universe.