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

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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.
 

1. What does it mean to restore S.I. units to a Lagrangian in natural units?

Restoring S.I. units to a Lagrangian that has been expressed in natural units involves reintroducing the fundamental constants such as the speed of light (c), Planck's constant (ħ), and the gravitational constant (G), which are typically set to 1 in natural units. This process is necessary for translating theoretical predictions into experimental measurements that can be compared with real-world data, as experimental results are often measured in S.I. units.

2. Why are natural units used in theoretical physics?

Natural units simplify many of the equations in theoretical physics by setting constants like the speed of light (c) and Planck's constant (ħ) to 1. This not only reduces clutter in the equations but also highlights the fundamental relationships and scales of the processes involved. Using natural units can clarify the mathematical structure of theories and make them easier to manipulate and solve.

3. How do you convert a Lagrangian from natural units back to S.I. units?

To convert a Lagrangian from natural units back to S.I. units, you need to reintroduce the correct dimensions by multiplying terms by appropriate powers of the constants that were set to 1. This often involves dimensional analysis to determine the correct factors of c, ħ, and other relevant constants. The process ensures that each term in the Lagrangian has the correct units of action, typically joule-seconds (J·s) in S.I. units.

4. What are the typical challenges when restoring S.I. units in a Lagrangian?

One of the main challenges is correctly identifying the powers of the constants that were set to 1 to ensure dimensional consistency across the Lagrangian. Errors in this process can lead to incorrect predictions and interpretations. Another challenge is managing the resulting complexity in the equations, as reintroducing units can make them less tractable and more difficult to solve analytically.

5. How does restoring S.I. units affect the interpretation of a Lagrangian?

Restoring S.I. units to a Lagrangian affects its interpretation by making it directly applicable to experimental and real-world scenarios. It allows physicists to calculate observable quantities in units that can be measured experimentally. While natural units provide a cleaner theoretical framework, S.I. units facilitate the practical application and validation of theoretical predictions through experiments.

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