Lagrangian density for a complex scalar field (classical)


by Trave11er
Tags: classical, complex, density, field, lagrangian, scalar
Trave11er
Trave11er is offline
#1
Dec7-12, 10:44 AM
P: 70
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.
Phys.Org News Partner Physics news on Phys.org
Information storage for the next generation of plastic computers
Scientists capture ultrafast snapshots of light-driven superconductivity
Progress in the fight against quantum dissipation
K^2
K^2 is offline
#2
Dec7-12, 12:27 PM
Sci Advisor
P: 2,470
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.
dextercioby
dextercioby is offline
#3
Dec7-12, 01:02 PM
Sci Advisor
HW Helper
P: 11,863
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.

K^2
K^2 is offline
#4
Dec7-12, 01:46 PM
Sci Advisor
P: 2,470

Lagrangian density for a complex scalar field (classical)


You are right, it does follow from L = L*. I never really thought of it that way.
Trave11er
Trave11er is offline
#5
Dec7-12, 03:56 PM
P: 70
Thank you for the answers - they are very insightful.


Register to reply

Related Discussions
Complex Scalar Field Advanced Physics Homework 0
What's the classical picture of phi^4 scalar field theory? High Energy, Nuclear, Particle Physics 3
complex scalar field Quantum Physics 2
Lagrangian density of the EM field Classical Physics 16
the scalar field lagrangian Quantum Physics 2