Coupling constatn dimensions

In summary, the dimension of a coupling constant can be obtained from the Lagrangian of a theory. In the case of electromagnetism, the coupling constant is g=1/137, while for gravitation it has units of m**-2. It is possible to rescale the coupling constant to be dimensionless. When considering dimensions, it is important to use c = 1 and h = 1, which leads to M = L^(-1). The action must also be dimensionless, and the Lagrangian density must have dimensions L^(-4) = M^4. The dimension of a scalar field is M, which can be determined from its mass term. The dimensionality of the coupling constant is crucial
  • #1
eljose79
1,518
1
givne a theory with a given lagrangian..how do you obtain the dimension of coupling constatn?..in fact how do you know that for electromagnetism is g=1/137 or that for gravitation have units of m**-2?

Can the coupling constant be rescaled to get always a dimensionless one?
 
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  • #2
Using these dimensions, one has to know that this makes sense when we put c = 1 and h = 1. This makes L = T and M L^2 T^(-1) = 1. So M = L^(-1).
Now, the action needs to be dimensionless (same unit as h = 1). The action is the 4-dim integral of the lagrangian density, so L has to have dimensions L^(-4) = M^4. Mass still has, well, M as dimension. A typical mass term for a scalar field is m^2 phi^2, so it follows that a scalar field has to have dimensions M. As such, you can puzzle together the dimensions of different fields from their "free field" terms. The dimensionality of the interaction term coefficients (the coupling constants) is very important because it determines the renormalizability of the theory.

cheers,
Patrick.
 
  • #3


The dimension of a coupling constant can be obtained by analyzing the Lagrangian of a given theory. The Lagrangian is a mathematical expression that describes the dynamics of a physical system. It contains all the information about the interactions between different particles and fields in the theory.

To obtain the dimension of a coupling constant, one can use the principle of dimensional analysis. This principle states that physical quantities must have consistent dimensions on both sides of an equation. Therefore, by looking at the dimensions of the terms in the Lagrangian, one can determine the dimension of the coupling constant.

For example, in electromagnetism, the coupling constant is given by the fine structure constant, α = e2/4πε0ħc, where e is the elementary charge, ε0 is the permittivity of free space, ħ is the reduced Planck's constant, and c is the speed of light. By analyzing the dimensions of each term, we can see that the fine structure constant has a dimension of length squared, which is consistent with the dimension of a coupling constant.

Similarly, for gravitation, the coupling constant is given by the gravitational constant, G = 1/m2, where m is the mass of the particles involved in the interaction. Again, by analyzing the dimensions of each term, we can see that the gravitational constant has a dimension of length squared, which is consistent with the dimension of a coupling constant.

It is important to note that the numerical value of the coupling constant, such as 1/137 for electromagnetism, is a result of experimental observations and cannot be derived from first principles. However, the dimension of the coupling constant can be determined using dimensional analysis.

It is possible to rescale the coupling constant to obtain a dimensionless one. This is often done in theoretical physics to simplify calculations and make comparisons between different theories. For example, in particle physics, the coupling constants are often rescaled by the energy scale of the interaction. This results in a dimensionless quantity known as the coupling strength, which can be used to compare the strength of different interactions. However, the original dimension of the coupling constant is still important in understanding the physical nature of the interaction.
 

1. What is a coupling constant dimension?

A coupling constant dimension is a measurement of the strength of a fundamental force or interaction between two particles in physics. It is a dimensionless quantity that is used to describe the strength of the force, and it is an important concept in quantum field theory.

2. How is the coupling constant dimension related to the strength of a force?

The coupling constant dimension is directly related to the strength of a force. A larger coupling constant dimension indicates a stronger force, while a smaller coupling constant dimension indicates a weaker force. This relationship is described by the mathematical equations of quantum field theory.

3. What are the units of a coupling constant dimension?

Unlike other physical quantities, the coupling constant dimension does not have any units. This is because it is a dimensionless quantity, meaning it does not have any physical dimensions. Instead, it is a pure number that is used to represent the strength of a force.

4. How does the value of a coupling constant dimension affect particle interactions?

The value of a coupling constant dimension plays a crucial role in determining the strength of particle interactions. A higher value indicates a stronger force between particles, leading to more frequent and stronger interactions. On the other hand, a lower value indicates a weaker force and less frequent interactions between particles.

5. What is the significance of the coupling constant dimension in particle physics?

The coupling constant dimension is a fundamental concept in particle physics that helps us understand the strength of interactions between particles. It is used to describe the behavior and properties of particles, and it is an essential component in many mathematical models and theories, such as the Standard Model of particle physics.

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