• Support PF! Buy your school textbooks, materials and every day products Here!

4-derivative kinetic term Lagrangian

  • Thread starter kelly0303
  • Start date
  • #1
246
12

Homework Statement


Show that $$L=\phi\Box^2\phi$$ generates negative energy density.

Homework Equations




The Attempt at a Solution


The energy density is $$E=\frac{\partial L}{\partial \dot{\phi}}\dot{\phi}-L$$ Also the Lagrangian can be rewritten (using divergence theorem) as $$L=-\partial_\mu\phi\partial_\mu(\Box\phi)$$ So I would get $$E=-\partial_0\phi\partial_0(\Box\phi)+\partial_\mu\phi\partial_\mu(\Box\phi)$$ $$E=-\partial_x\phi\partial_x(\Box\phi)-\partial_y\phi\partial_y(\Box\phi)-\partial_z\phi\partial_z(\Box\phi)$$. Why is this negative necessarily? Thank you!
 

Answers and Replies

  • #2
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,690
6,465
Also the Lagrangian can be rewritten (using divergence theorem) as
L=−∂μϕ∂μ(□ϕ)​
No, it cannot. You need to remove the d’Alembertian from this expression.
 
  • #3
246
12
No, it cannot. You need to remove the d’Alembertian from this expression.
I am not sure what you mean. I used $$\int(\phi\Box^2\phi)=\int\partial_\mu(\phi\partial_\mu \Box \phi)-\int(\partial_\mu\phi\partial_\mu \Box \phi)$$ What is wrong? What do you mean by getting rid of the d'Alembertian?
 
  • #4
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,690
6,465
I am not sure what you mean. I used $$\int(\phi\Box^2\phi)=\int\partial_\mu(\phi\partial_\mu \Box \phi)-\int(\partial_\mu\phi\partial_\mu \Box \phi)$$ What is wrong? What do you mean by getting rid of the d'Alembertian?
This is simply not true. What you are looking for is
$$
\phi \Box \phi = \partial_\mu (\phi \partial^\mu \phi) - (\partial_\mu \phi)(\partial^\mu\phi)
$$
 
  • #5
Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Insights Author
Gold Member
16,690
6,465
Ah, sorry. You have ##\Box^2##, not ##\Box##. In that case you cannot use the expression for the energy that you have given since your Lagrangian will contain derivatives of order higher than one.
 

Related Threads on 4-derivative kinetic term Lagrangian

Replies
3
Views
5K
Replies
0
Views
1K
Replies
0
Views
2K
Replies
1
Views
2K
  • Last Post
Replies
0
Views
3K
  • Last Post
Replies
7
Views
405
  • Last Post
Replies
11
Views
2K
Replies
13
Views
2K
  • Last Post
Replies
3
Views
4K
Replies
1
Views
3K
Top