Energy of Scalar Field: Evaluating Rubakov's Expression

In summary, there is a confusion between the Lagrangian and the Lagrangian density in the given expression for energy. The Lagrangian density should be used instead of the Lagrangian in order to fix the unit problem. This may be a result of the terminology used, as many field theory texts refer to the Lagrangian density as simply the "Lagrangian".
  • #1
quantum_smile
21
1

Homework Statement


My question is just about a small mathematical detail, but I'll give some context anyways.
(From Rubakov Sec. 2.2)
An expression for energy is given by
[tex] E= \int{}d^3x\frac{\delta{}L}{\delta{}\dot{\phi}(\vec{x})}\dot{\phi}(\vec{x}) - L,
[/tex]
where L is the Lagrangian,
[tex]
L=\int{}d^3{}x(\frac{1}{2}\dot{\phi}^2-\frac{1}{2}\partial_i\phi\partial_i\phi-\frac{m^2}{2}\phi^2).
[/tex]
To derive the expression for energy, Rubakov says that
[tex]
\frac{\delta{}L}{\delta{}\dot{\phi}(\vec{x})}=\dot{\phi}(\vec{x}).
[/tex]
What I want to know is, simply, how does he get this expression for
[tex]
\frac{\delta{}L}{\delta{}\dot{\phi}(\vec{x})}
[/tex]?

Homework Equations

The Attempt at a Solution


If I evaluate the expression, I just get
[tex]
\delta{}L=\int{}d^3x(\dot{\phi}).
[/tex]

Where'd the integral go in Rubakov's expression?
 
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  • #2
I may be wrong, but I believe what is happening is a confusion between the Lagrangian and the Lagrangian density. Look at the expression for the energy, it has an integral in it, so probably the ##L## which appears in there should actually be the Lagrangian density ##\mathcal{L}## defined by ##L=\int d^3x \mathcal{L}##
 
  • #3
Ah, that would make a lot of sense (and fix the weird unit problem). Maybe there's a tiny typo in the text.
 
  • #4
A lot of field theory texts refer to the Lagrangian density as simply the "Lagrangian", so the language might be confusing. Usually the notation is used so that the Lagrangian density is in a calligraphic font though.
 
  • #5


The integral is still there in Rubakov's expression, but it has been absorbed into the functional derivative \frac{\delta{}L}{\delta{}\dot{\phi}(\vec{x})}. This is a common practice in field theory to simplify expressions and make them more compact. In this case, the integral over space has been taken into account in the functional derivative, which is defined as the derivative of the Lagrangian with respect to the field at a specific point in space. So the expression is still valid, but it has been rewritten in a more compact form.
 

1. What is a scalar field?

A scalar field is a physical quantity that has a single numerical value at each point in space. It is a fundamental concept in physics and is used to describe various phenomena, such as temperature, pressure, and energy.

2. What is the energy of a scalar field?

The energy of a scalar field is the total amount of energy contained within the field. It is a measure of the field's strength and is related to the amplitude of the field at each point in space.

3. Who is Rubakov and what is his expression for evaluating the energy of a scalar field?

Valery Rubakov is a Russian theoretical physicist who has made significant contributions to the study of scalar fields. His expression for evaluating the energy of a scalar field is a mathematical formula that takes into account various parameters such as the field's amplitude, its derivatives, and its spatial dimension.

4. How is Rubakov's expression for evaluating the energy of a scalar field derived?

Rubakov's expression is derived from the Lagrangian density, which is a mathematical function that describes the dynamics of a physical system. By applying the Euler-Lagrange equation, which relates the Lagrangian to the system's equations of motion, Rubakov was able to obtain his expression for the energy of a scalar field.

5. What are some applications of Rubakov's expression for evaluating the energy of a scalar field?

Rubakov's expression has been used in various areas of physics, such as cosmology, particle physics, and condensed matter physics. It has been applied to study the behavior of scalar fields in different physical systems, including the early universe, high-energy particle collisions, and superconductors.

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