Global symmetry of an N-component Klein-Gordon theory?

In summary, the Lagrangian is invariant under any SO(2N) rotation, but some people say that the symmetry is actually SU(N). There is no formal elaboration for this claim.
  • #1
weejee
199
0
The Lagrangian is given by,
[tex] \sum_{a=1}^N \left[(\partial^{\mu}\phi_{a}^{\ast})(\partial_{\mu}\phi_{a})-m^{2}\phi_{a}^{\ast}\phi_{a}\right][/tex].

Is the symmetry SO(2N), SU(N) or U(N)?

It seemed quite obvious to me and some of my friends that such theory has an SO(2N) symmetry. If we view these N copies of complex K-G fields as 2N copies of real K-G fields, the Lagrangian is invariant under any rotation in the 2N dimensional space. It also seems that there should be N(N-1)/2, which is the number of generators in the SO(2N) group, conserved currents for this theory.

However, I have faced some objections to my claim. What they say is that the actual symmetry is SU(N). The reasoning for this claim was that the real and imaginary parts of these complex K-G fields cannot be considered independent, since they are related by causality. If we allowed an arbitrary SO(2N) rotation, particles and antiparticles would mix each other and the causality would be violated.

What makes me uncomfortable about this statement is that, first, I haven't been able to see any convincing formal development, rather than some hand waving arguments, for it, second, it would mean that one-component complex K-G theory doesn't have a U(1) symmetry.

If somebody said that the symmetry is U(N) due to causality, I would be less unhappy since we can save the 1-component K-G theory from not having even a U(1) symmetry.

I would be very grateful if any of you could clarify this issue, and if this causality argument is right, show me some formal elaboration to it. (or let me know where I can find it)
 
Physics news on Phys.org
  • #2
We don't you just try to implement the action of these symmetries and see if the Lagrangian is unchanged? Consider also the infinitesimal action of such symmetry transformations.
 
  • #3
ansgar said:
We don't you just try to implement the action of these symmetries and see if the Lagrangian is unchanged? Consider also the infinitesimal action of such symmetry transformations.

It is clearly unchanged. The problem is that some say that we should also consider "causality", rather than just the invariance of the Lagrangian, which I haven't been able to understand.
 
  • #4
weejee said:
It is clearly unchanged. The problem is that some say that we should also consider "causality", rather than just the invariance of the Lagrangian, which I haven't been able to understand.

why should we? I have never read that / been taught that - and I have studied 8 or 9 QFT textbooks. As you know, causality is due to existence of antiparticles, see QFT book by peskin chapter 2.
 
  • #5
weejee said:
IThe problem is that some say that we should also consider "causality", rather than just the invariance of the Lagrangian, which I haven't been able to understand.
It's always hard to understand incorrect arguments! :)
 

Related to Global symmetry of an N-component Klein-Gordon theory?

1. What is the concept of global symmetry in a Klein-Gordon theory?

The concept of global symmetry in a Klein-Gordon theory refers to the invariance of the theory under a certain transformation of its fields. This means that the equations of motion and the physical predictions of the theory remain unchanged when the fields are transformed in a particular way.

2. How is global symmetry related to the conservation laws in a Klein-Gordon theory?

Global symmetries in a Klein-Gordon theory are closely linked to conservation laws. For example, a global symmetry under translations in space leads to the conservation of momentum, while a global symmetry under translations in time leads to the conservation of energy.

3. Can global symmetry be broken in a Klein-Gordon theory?

Yes, global symmetry can be broken in a Klein-Gordon theory. This means that the fields do not remain invariant under the symmetry transformation and the equations of motion and physical predictions of the theory are no longer unchanged.

4. What is the significance of global symmetry in particle physics?

Global symmetry plays a crucial role in particle physics as it helps to classify and understand the properties of particles. For example, the conservation of electric charge is a result of a global symmetry in the theory of electromagnetism.

5. How does the number of components in a Klein-Gordon theory affect its global symmetry?

The number of components in a Klein-Gordon theory determines the type of global symmetry that the theory possesses. For an N-component theory, there can be up to (N-1) independent global symmetries. For example, a single-component Klein-Gordon theory has only one possible global symmetry, while a three-component theory can have up to two independent global symmetries.

Similar threads

Replies
24
Views
2K
  • Quantum Physics
Replies
4
Views
1K
Replies
3
Views
3K
  • Quantum Physics
Replies
9
Views
1K
Replies
41
Views
4K
  • Quantum Physics
Replies
2
Views
1K
  • Quantum Physics
Replies
2
Views
2K
Replies
4
Views
2K
  • Quantum Physics
Replies
2
Views
2K
Replies
27
Views
9K
Back
Top