Help with basic tensor algebra

In summary, the book uses an operator which is not quite what I got. Or is it the same after all? If not, what did I do wrong in the calculation? Or is it perhaps a typo in the book?
  • #1
Spinny
20
0
Hi, I need some help understanding basic tensor algebra, especially differentiation. The subject I'm studying is quantum field theory, so I'll use examples from there.

First let's start with a real scalar field. This has a Lagrangian density given by

[tex]\mathcal{L} = \frac{1}{2}\partial_{\mu}\phi \partial^{\mu}\phi - \frac{1}{2}m^2\phi^2-\frac{\lambda}{4!}\phi^4[/tex]

where [tex]\lambda[/tex] is just a (coupling) constant. We must then have that the Euler-Lagrange equation

[tex]\frac{\partial \mathcal{L}}{\partial \phi}-\partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi)} = 0[/tex]

coincides with the dynamic equation

[tex](\square + m^2)\phi = -\frac{\lambda}{6}\phi^3[/tex]

The first part of the Euler-Lagrange equation is rather easy, differentiate with respect to [tex]\phi[/tex], and this gives

[tex]\frac{\partial \mathcal{L}}{\partial \phi} = -m^2\phi - \frac{\lambda}{3!}\phi^3[/tex]

Then, the second part, which I get to be

[tex]\partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi)} = \frac{1}{2}\partial_{\mu}\partial^{\mu}\phi[/tex]

Combining these gives

[tex]\left( \frac{1}{2}\partial_{\mu}\partial^{\mu}+m^2 \right) \phi = -\frac{\lambda}{6}\phi^3[/tex]

This is an example from the textbook (Elementary Particles and Their Interactions by Quand Ho-Kim and Pham Xuan Yem), although the calculation has not been carried out explicitly. In this book the operator [tex]\square[/tex] is also defined as [tex]\square = \partial^{\mu}\partial_{\mu}[/tex], which is not quite what I got. Or is it the same after all? If not, what did I do wrong in the calculation? Or is it perhaps a typo in the book? It would be nice if someone could enlighten me on this. I have more examples (with a vector field), but I'll post that after I've, hopefully, gotten some respons to this problem.
 
Physics news on Phys.org
  • #2
Saw your similar thread on the Homework: College Level forum. If no one else replies, I will help you either later today or tomorrow. I'm a bit pressed for time right now.

Regards,
George
 
  • #3
You are mistaken:smile:

[tex]\partial_{\mu}\frac{\partial \mathcal{L}}{\partial(\partial_{\mu}\phi)} = \partial_{\mu}\partial^{\mu}\phi[/tex]
 
  • #4
Here are some tips.

1) Make sure that the indices in the symbol with respect to which you differentiate don't match *any* of the indices in the terms that get differentiated. If the indices match, bad things happen - either an idex that was originally free becomes repeated and thus summed over, or you end up with a term with 3 repeated indices, which is meaningless.

2) Using 1) will introduce expressions like

[tex]\frac{\partial a_{\mu}}{\partial a_{\alpha}} = \delta_{\mu}^{\alpha}[/tex],

and like

[tex]\delta_{\mu}^{\alpha} b_{\alpha} = b_{\mu}[/tex].

3) Use the metric to make sure that all indices are in appropriate upstairs or downstairs locations. For example, if you want to differentiate an expression that contains

[tex]\partial^{\mu} \phi[/tex]

with respect to

[tex]\partial_{\alpha} \phi[/tex],

then the substitution

[tex]\partial^{\mu} \phi = g^{\mu \beta} \partial_{\beta} \phi[/tex]

should be made.

Regards,
George
 
Last edited:
  • #5
Hi, thank you so much for your insight Mr. Jones. I can imagine it wasn't very easy to help when I have such general questions, but I must say, your tips were a very good start.

Now I'll need to take a look at the examples and exercises again with this in mind, but have no fear, I'll probably be back with more questions, and hopefull they'll be more specific.

And also thanks to gvk for pointing out that it was indeed I who was mistaking, and not the book. And now that we've located where the problem is... :smile:
 

1. What is tensor algebra?

Tensor algebra is a branch of mathematics that deals with the manipulation and study of mathematical objects called tensors. Tensors are multi-dimensional arrays that can represent physical quantities such as force, velocity, and stress in a coordinate-independent manner.

2. What are the basic operations in tensor algebra?

The basic operations in tensor algebra include addition, multiplication, and contraction. Addition involves adding tensors of the same order, while multiplication involves multiplying tensors of different orders. Contraction is the process of summing over repeated indices in a tensor expression.

3. How is tensor algebra used in science?

Tensor algebra has various applications in physics, engineering, and other sciences. It is used to solve problems involving the manipulation of multi-dimensional quantities, such as stress and strain in materials, fluid flow dynamics, and electromagnetic fields. It is also used in data analysis and machine learning.

4. What are the common notations used in tensor algebra?

The most commonly used notations in tensor algebra are the index notation, Einstein summation convention, and the matrix notation. Index notation involves using indices to represent the components of a tensor, while the Einstein summation convention simplifies tensor expressions by summing over repeated indices. Matrix notation uses matrices to represent tensors and their operations.

5. Is it necessary to have a strong background in mathematics to understand tensor algebra?

While a strong foundation in mathematics is helpful, it is not necessary to understand tensor algebra. A basic understanding of linear algebra and calculus is sufficient to grasp the concepts and applications of tensor algebra. With practice and patience, anyone can learn and apply this mathematical tool to solve problems in science and engineering.

Similar threads

Replies
5
Views
251
  • Advanced Physics Homework Help
Replies
1
Views
204
  • Classical Physics
Replies
4
Views
129
  • Advanced Physics Homework Help
Replies
10
Views
943
  • Differential Geometry
Replies
7
Views
2K
  • Differential Geometry
Replies
4
Views
3K
  • Calculus
Replies
1
Views
903
Replies
5
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
548
  • High Energy, Nuclear, Particle Physics
Replies
1
Views
980
Back
Top