Lagrangian density for a complex scalar field (classical)

In summary, the conversation discusses the separation of a complex scalar field into real and imaginary parts and the resulting Lagrangian density. It is then questioned why the imaginary part is treated equally with the real part in the density summation. This is explained by the superposition principle and U(1) symmetry, which are true in quantum mechanics but not necessarily in general field theory. The latter is required for the Lagrangian to be real under complex conjugation and for the action to contain matched products of the field and its complex conjugate. The conversation concludes with the realization that the U(1) symmetry follows from these requirements and the appreciation for the insightful answers.
  • #1
Trave11er
71
0
Hi.
Let's say we have a complex scalar field [itex]\varphi[/itex] and we separate it into the real and the imaginary parts:
[itex]\varphi[/itex] = ([itex]\varphi1[/itex] + i[itex]\varphi2[/itex])
It's Lagrangian density L is given by:
L = L([itex]\varphi1[/itex]) + L([itex]\varphi1[/itex])
Can you tell the argument behind the idea that in summing the densities of cpts. we treat the imaginary part on equal basis with the real.
 
Physics news on Phys.org
  • #2
Do you mean to say L([itex]\varphi[/itex]) = L([itex]\varphi1[/itex]) + L([itex]\varphi2[/itex])? That's because L([itex]\varphi[/itex]) = L([itex]\varphi1[/itex]) + L([itex]i\varphi2[/itex]) due to superposition principle, and L([itex]i\varphi2[/itex])=L([itex]\varphi2[/itex]) due to U(1) symmetry. Neither are absolutely generally true. Former requires a linear Lagrangian, later requires it to be symmetric under U(1) transformations. Both of these are true in Quantum Mechanics, but not necessarily in general field theory.
 
  • #3
U(1) symmetry follows from the general requirements for a Lagrangian field theory. The action must be real under complex conjugation, hence the lagrangian density must contain matched products of phi and phi star and subsequent spacetime derivatives.
 
  • #4
You are right, it does follow from L = L*. I never really thought of it that way.
 
  • #5
Thank you for the answers - they are very insightful.
 

1. What is a Lagrangian density for a complex scalar field?

A Lagrangian density for a complex scalar field is a mathematical function that describes the dynamics of a complex scalar field in classical physics. It is derived from the Lagrangian formalism, which is a mathematical framework used to describe the behavior of physical systems.

2. How is the Lagrangian density for a complex scalar field different from a Lagrangian for a real scalar field?

The Lagrangian density for a complex scalar field includes both the scalar field and its complex conjugate, while the Lagrangian for a real scalar field only includes the scalar field itself. This allows for more complex and realistic descriptions of physical systems.

3. What are the equations of motion derived from the Lagrangian density for a complex scalar field?

The equations of motion derived from the Lagrangian density for a complex scalar field are the Klein-Gordon equation and its conjugate. These equations describe the time evolution of the complex scalar field and its complex conjugate, respectively.

4. How is the Lagrangian density for a complex scalar field used in quantum field theory?

In quantum field theory, the Lagrangian density for a complex scalar field is used to construct the Lagrangian for the quantum field theory. This allows for the calculation of particle interactions and the prediction of physical observables.

5. What are the physical implications of the Lagrangian density for a complex scalar field?

The Lagrangian density for a complex scalar field has important physical implications, such as describing the behavior of particles in a quantum field theory, predicting particle interactions, and providing a deeper understanding of the underlying dynamics of physical systems.

Similar threads

Replies
3
Views
1K
  • Quantum Physics
Replies
5
Views
886
  • Quantum Physics
Replies
13
Views
697
Replies
1
Views
823
  • High Energy, Nuclear, Particle Physics
Replies
2
Views
464
Replies
7
Views
887
Replies
3
Views
897
Replies
29
Views
1K
  • Special and General Relativity
Replies
2
Views
862
Back
Top