Current of Complex scalar field

In summary, the conversation discusses deriving the current for a Complex Scalar Field and the use of Noether's Theorem to find the expression for the current. The Lagrangian for the field is shown to be invariant under a particular transformation, and the infinitesimal transformation for the field is derived. The application of Noether's Theorem results in an expression for the current, but it is noted that there is no group parameter in the final equation. It is clarified that this is expected, as the current is defined as the expression proportional to the infinitesimal group parameter.
  • #1
PhyAmateur
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I was trying to derive current for Complex Scalar Field and I ran into the following:So we know that the Lagrangian is:

$$L = (\partial_\mu \phi)(\partial^\mu \phi^*) - m^2 \phi^* \phi$$
The Lagrangian is invariant under the transformation:
$$\phi \rightarrow e^{-i\Lambda} \phi $$ and $$\phi^* \rightarrow e^{i\Lambda} \phi^* $$

Infinitesimal Transformation:
$$\delta \phi = -i\Lambda \phi$$ and $$\delta \phi^* = i\Lambda \phi^*$$

So, applying Noether's Theorem and using the Lagrangian above,

I get

$$J^{\mu} = \frac{\partial L}{\partial (\partial_\mu \phi} (-i\Lambda \phi) + \frac{\partial L}{\partial (\partial_\mu \phi^*} (i\Lambda \phi^*) =$$
$$\partial ^\mu \phi^* (-i\Lambda \phi) + \partial _\mu \phi (i\Lambda \phi) = $$
$$ i(\Lambda \phi^* \partial _\mu \phi - \Lambda \phi \partial ^\mu \phi^*)$$

but as I googled it there is no $$\Lambda$$ in the final equation of the current. What did I do wrong?
 
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  • #2
You did nothing wrong. The current is simply defined as the expression proportional to the infinitesimal group parameter.
 
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  • #3
Ah phew! Thanks!
 

1. What is a complex scalar field?

A complex scalar field is a mathematical concept used in theoretical physics to describe a physical system that has a value at every point in space and time. It is a field that has both magnitude and direction, and its values are represented by complex numbers.

2. What is the difference between a scalar field and a vector field?

A scalar field has a single value at each point in space, while a vector field has both magnitude and direction at each point. In other words, a scalar field is a scalar quantity, while a vector field is a vector quantity.

3. How is complex scalar field used in physics?

Complex scalar fields are used in various areas of physics, such as in quantum field theory and particle physics, to describe the behavior and interactions of particles at a subatomic level. They are also used in cosmology to describe the evolution of the universe.

4. What is the mathematical equation for a complex scalar field?

The mathematical equation for a complex scalar field is represented by a complex-valued function, typically denoted as Φ(x,t), where x represents the position in space and t represents time. It is a solution to the Klein-Gordon equation, which describes the evolution of a scalar field over time.

5. How does the current of a complex scalar field relate to the electromagnetic force?

The current of a complex scalar field is a quantity that describes the flow of energy and momentum of the field. In some theories, this current is related to the electromagnetic force, as the electromagnetic field can be described as the interaction between the current of a charged scalar field and the electromagnetic potential.

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