Restoring S.I. units to a Lagrangian in natural units

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Precious Adegbite
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Restoring the Lagrangian units
If we have a natural unit Lagrangian, where some fundamental quantities have been excluded to ease calculations...and aim to restore it's S.I units back, do we just have to plug back the fundamental quantities that were initially excluded Into the Lagrangian...or we use some specific scaling factors corresponding to that field that has the same S.I units of Lagrangian density, and scale the unitless Lagrangian with it.
 
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"fundamental quantities" are not excluded in using natural units.
Only quantities relating two different dimensions are excluded.
 

Related to Restoring S.I. units to a Lagrangian in natural units

What are natural units, and why are they used in physics?

Natural units are a system of units where certain fundamental constants are set to 1 to simplify equations. In particle physics, for example, the speed of light (c) and the reduced Planck constant (ℏ) are often set to 1. This simplifies equations and calculations by eliminating these constants, making it easier to focus on the relationships between physical quantities.

How do you restore S.I. units to a Lagrangian expressed in natural units?

To restore S.I. units to a Lagrangian expressed in natural units, you need to reintroduce the fundamental constants that were set to 1. This typically involves reintroducing factors of the speed of light (c), the reduced Planck constant (ℏ), and sometimes other constants like the gravitational constant (G) or the Boltzmann constant (k). The specific factors depend on the dimensions of the quantities involved in the Lagrangian.

What is the significance of the Lagrangian in physics?

The Lagrangian is a function that summarizes the dynamics of a physical system. In classical mechanics, it is defined as the difference between the kinetic and potential energies of the system. In field theory and particle physics, the Lagrangian density is used to describe the dynamics of fields and particles. The equations of motion for the system can be derived from the Lagrangian using the principle of least action.

Can you provide an example of converting a term in the Lagrangian from natural units to S.I. units?

Consider a simple term in the Lagrangian for a free scalar field: \( \mathcal{L} = \frac{1}{2} (\partial^\mu \phi)(\partial_\mu \phi) - \frac{1}{2} m^2 \phi^2 \). In natural units, \( c = 1 \) and \( \hbar = 1 \). To restore S.I. units, reintroduce these constants: \( \mathcal{L} = \frac{1}{2} \hbar c (\partial^\mu \phi)(\partial_\mu \phi) - \frac{1}{2} \frac{m^2 c^2}{\hbar} \phi^2 \). This ensures that the dimensions of each term are consistent with S.I. units.

Why is it important to be able to switch between natural units and S.I. units?

Switching between natural units and S.I. units is important because natural units are convenient for theoretical calculations, while S.I. units are the standard in experimental and practical applications. Being able to convert between the two allows for a better understanding and communication between

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