Why can't SU(2) be the gauge group of electroweak theory?

In summary, the conversation discusses why SU(2) cannot be the electroweak gauge group and focuses on the computation of the commutator between the generators T+, T-, and Q. It is shown that the resulting current does not match the expected form, thus proving that SU(2) is not the correct gauge group.
  • #1
CaptainKidd
1
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I know that there are many reasons why SU(2) can't be the electroweak gauge group, but I want to have some clarifications about the following one, that disergads neutral currents:
in this case the currents are (considering only the lepton sector of the first generation)
[tex]J^{-}_{\mu}=\bar{\nu}_{e}\gamma_{\mu}(1-\gamma_5)e[/tex]

[tex]J^{+}_{\mu}=(J^{-}_{\mu})^\dagger [/tex]

[tex]J^{em}_{\mu}=-\bar{e}\gamma_{\mu}e[/tex]
From these one may hope to build a SU(2) group using as generators the three charges
[tex]T_{+}(t)=\frac{1}{2}\int d^3 x J^-_0(x)=\frac{1}{2}\int d^3 \nu_e^\dagger(1-\gamma_5)e[/tex]

[tex]T_-(t)=T^\dagger_+(t)[/tex]

[tex]Q(t)=\int d^3 x J^em_0(x)=-\int d^3 e^\dagger e[/tex]
At this point one should show that the generators [tex] T_+, \, T_-, \, Q[/tex] do not form a close algebra by computing the commutator [tex]\[T_+,T_-\][/tex] and showing that it is not equal to [tex]Q[/tex] (and this proves that SU(2) is not the right gauge group).
I just want to know how to compute this commutator explicitly. In Cheng and Li's book it is said that it can be computed using the canonical fermion anticommutation relations but I was not able to do it.
Thanks.
 
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  • #2
Well, without actually doing the calculation, it's pretty obvious that it can't possibly work. For one thing, only the left-handed electron field (and its hermitian conjugate) appears in J+ and J-, but J_em also includes the right-handed electron field; this can't be generated out of nothing from the commutator. Also, the neutrino field does not appear in J_em, yet it appears symmetrically with the electron field in J+ and J-, which means that it must appear in the commutator as well. The current you would get from the commutator would be something like

[tex]\bar{\nu}_{e}\gamma_{\mu}(1-\gamma_5){\nu}_{e} - {\bar e}\gamma_{\mu}(1-\gamma_5)e[/tex]
 
  • #3


There are several reasons why SU(2) cannot be the gauge group of electroweak theory. One of the main reasons is that SU(2) does not include the concept of neutral currents, which are an essential part of the electroweak theory.

In the electroweak theory, there are three fundamental interactions: the electromagnetic interaction, the weak interaction, and the strong interaction. The electromagnetic and weak interactions were initially thought to be separate, but it was later discovered that they are actually two aspects of a single unified force. This unification is described by the electroweak theory, which is based on the gauge group U(1)xSU(2).

The SU(2) part of this gauge group is responsible for the weak interaction, which is responsible for processes such as beta decay and neutrino interactions. However, the electromagnetic interaction also plays a role in these processes, through the concept of neutral currents. Neutral currents are interactions between particles that do not involve the exchange of any charged particles, but rather the exchange of Z bosons. These neutral currents are essential for the consistency of the electroweak theory.

In SU(2), the generators are based on the three charges T_+, T_-, and Q, as mentioned in the content. However, these generators do not take into account the concept of neutral currents. In order to incorporate neutral currents, one would need to add an additional generator, which is not possible in SU(2).

To show this, the commutator \[T_+,T_-\] can be computed using the canonical fermion anticommutation relations, as mentioned in Cheng and Li's book. This can be done by considering the commutation relations between the creation and annihilation operators for the fermions involved in the currents. However, this calculation would show that the commutator is not equal to Q, meaning that the generators do not form a closed algebra. This inconsistency proves that SU(2) is not the correct gauge group for the electroweak theory.

In conclusion, SU(2) cannot be the gauge group of electroweak theory because it does not include the concept of neutral currents, which are essential for the consistency of the theory. In order to incorporate neutral currents, an additional generator is needed, which is not possible in SU(2). The calculation of the commutator between the generators also shows that they do not form a closed algebra, further supporting the
 

1. Why is SU(2) not a suitable gauge group for electroweak theory?

The SU(2) gauge group only describes the weak interactions between particles, but does not account for the electromagnetic interactions. Therefore, it is not comprehensive enough to fully explain the electroweak theory.

2. What is the difference between SU(2) and U(1) gauge groups in electroweak theory?

The SU(2) gauge group is a non-Abelian group, meaning that the order of operations matters, while the U(1) gauge group is an Abelian group, meaning that the order of operations does not matter. This difference in group structure leads to different mathematical descriptions of the electroweak theory.

3. Can the electroweak theory be described by multiple gauge groups?

Yes, the electroweak theory is described by a combination of two gauge groups, SU(2) and U(1). This combination is known as the electroweak gauge group.

4. How does the Higgs mechanism relate to the choice of gauge group in electroweak theory?

The Higgs mechanism is responsible for giving mass to the W and Z bosons, which are carriers of the weak force in electroweak theory. This mechanism requires a gauge group with at least three generators, ruling out SU(2) as the sole gauge group for the electroweak theory.

5. Are there any other reasons why SU(2) is not the gauge group of electroweak theory?

Yes, SU(2) does not allow for the unification of weak and electromagnetic forces at high energies, which is a key feature of the electroweak theory. Additionally, experimental evidence, such as the existence of neutral currents, supports the use of the U(1) gauge group in the electroweak theory.

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