A puzzle of two scalar dynamics

In summary, a mixing term can cause two scalar fields to become infinite if the mixing term is large. If the mixing term is zero, then the fields remain in their independent potentials. Rotating the fields to a basis where the mixing term is absent allows the fields to remain in their potentials. If the cross-term between the fields is not zero, then the behavior of the system is fully-specified by the behavior of one of the fields.
  • #1
Accidently
37
0
I have a puzzle when I study the hybrid inflation model.

Suppose we have two scalar fields, [itex]\phi_1 and \phi_2[/itex]
first, let's consider the situation where they are in their independent potentials
[itex]V(\phi_i)=m_i^2\phi_i^2, i = 1,2[/itex]
with initial value
[itex]\phi_i^{ini}[/itex]
We can solve the scalar dynamic equations for them. And they are both in harmonic oscillation. This is Okay.

But when a 'mixing term' [itex]\lambda^2 \phi_1\phi_2[/itex] is introduced, [itex]\phi_1[/itex] and [itex]\phi_2[/itex] get infinite values, if \lambda is large. This can be showed numerically. What I thought is the large mixing term would lead to [itex]\phi_1 = \phi_2[/itex]. So why it goes to infinite?

And we can rotate [itex]\phi_1[/itex] and [itex]\phi_2[/itex] to a basis where there is no mixing term. In this basis, we would not get infinite values for [itex]\phi_1[/itex] or [itex]\phi_2[/itex]. So it seems I get a different result working in different basis. What is the problem
 
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  • #2
You are confusing scalar quantities with vector quantities.
 
  • #3
Accidently said:
I have a puzzle when I study the hybrid inflation model.

Suppose we have two scalar fields, [itex]\phi_1 and \phi_2[/itex]
first, let's consider the situation where they are in their independent potentials
[itex]V(\phi_i)=m_i^2\phi_i^2, i = 1,2[/itex]
with initial value
[itex]\phi_i^{ini}[/itex]
We can solve the scalar dynamic equations for them. And they are both in harmonic oscillation. This is Okay.

But when a 'mixing term' [itex]\lambda^2 \phi_1\phi_2[/itex] is introduced, [itex]\phi_1[/itex] and [itex]\phi_2[/itex] get infinite values, if \lambda is large. This can be showed numerically. What I thought is the large mixing term would lead to [itex]\phi_1 = \phi_2[/itex]. So why it goes to infinite?

And we can rotate [itex]\phi_1[/itex] and [itex]\phi_2[/itex] to a basis where there is no mixing term. In this basis, we would not get infinite values for [itex]\phi_1[/itex] or [itex]\phi_2[/itex]. So it seems I get a different result working in different basis. What is the problem
How large are we talking? I don't think you can go above [itex]\lambda^2 = m_1^2 + m_2^2[/itex] and have sensible results.
 
  • #4
Chronos said:
You are confusing scalar quantities with vector quantities.

do you mean scalars can not mix? I thought about that. But my understanding is two fields can mix if they have exactly the same quantum number.
 
  • #5
Chalnoth said:
How large are we talking? I don't think you can go above [itex]\lambda^2 = m_1^2 + m_2^2[/itex] and have sensible results.

The limit sounds reasonable. But why do we have this limit? Unfortunately, I am consider some process which can go beyond this limit (for example, a fast scattering between the two scalars, bringing the two fields to equilibrium.)
 
  • #6
Accidently said:
The limit sounds reasonable. But why do we have this limit? Unfortunately, I am consider some process which can go beyond this limit (for example, a fast scattering between the two scalars, bringing the two fields to equilibrium.)
Well, one way to think about this is that the fundamental particles are different from the particles we observe, and that the fundamental particles are mixed, through virtue of some matrix, into the particles we observe. This mixing matrix gives rise to the cross-term interaction.

If your cross term is zero, then the mixing matrix is diagonal, and the particles we observe are the fundamental particles. If, however, the mixing term is at the limit [itex]\lambda^2 = m_1^2 + m_2^2[/itex], then the mixing matrix is saying that there are is in actuality only one fundamental particle that is mixed into these two, and the behavior of the system is fully-specified by the behavior of one of the particles. If you try to get larger off-diagonal terms, the mixing matrix ceases to make any sort of physical sense.
 

1. What is "A puzzle of two scalar dynamics"?

"A puzzle of two scalar dynamics" is a scientific concept that refers to the study of two scalar quantities that interact with each other in a complex and often unpredictable manner. These scalar quantities can be any physical or mathematical values, such as temperature, pressure, or velocity.

2. What is the significance of studying this puzzle?

Studying the puzzle of two scalar dynamics allows scientists to better understand the behavior and interactions of scalar quantities in various systems. This can lead to important insights and advancements in fields such as physics, engineering, and mathematics.

3. How is this puzzle different from other scientific concepts?

This puzzle is unique in that it specifically focuses on the interactions between two scalar quantities, rather than multiple variables or factors. It also involves complex dynamics and nonlinear relationships, making it a challenging and intriguing area of study.

4. What are some real-world applications of this puzzle?

The puzzle of two scalar dynamics has various real-world applications, such as in weather forecasting, financial markets, and population dynamics. Understanding the interactions between scalar quantities can help predict and manage these systems more accurately.

5. Are there any current research developments related to this puzzle?

Yes, there is ongoing research in this field to further understand and unravel the complexities of two scalar dynamics. Some recent developments include the use of machine learning and computational methods to analyze and model these interactions.

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