# Gauge Choice and Consistency with EOMs

• EuphoGuy
In summary, the conversation discusses the use of the coulomb gauge in gauge transformations and its consistency with equations of motion. It is noted that fixing gauges that are inconsistent with equations of motion may lead to incorrect results. The importance of using a valid gauge condition when working with a gauge dependent Lagrangian is emphasized.
EuphoGuy
Hello, just had a quick silly question about the coulomb gauge here, though I guess it applies for gauge transformations in general. The problem is, I'm concerned about my gauge choice not being consistent with the equations of motion. For example, suppose I'm working with a nonrelativistic theory describing a single complex scalar coupled to a magnetic field. The part referencing A_i is upto constants and signs

$$\mathcal{L} = B_i B_i + A_i ( h^*\partial_i h - \partial_i h^* h ) + A_i A_i h^* h$$

(this isn't gauge invariant because I haven't included the other relevant terms, but these are the only ones that contribute to the EOM)

If I impose the coulomb gauge \partial_i A_i = 0, the gauge fixed lagrangian is

$$\mathcal{L} = - A_i \nabla^2 A_i + A_i h^* \partial_i h + A_i A_i h^* h$$

giving equation of motion

$$\nabla^2 A_i = h^* \partial_i h + 2 A_i h^* h$$

If I took the divergence of this I would get 0 on the left but not on the right. Is this something one has to worry about then? Fixing gauges that are inconsistent with equations of motion? I think I'm rather just making a silly mistake, since given any old A, there's no reason I can't take it to coulomb gauge.

Thanks for the help!

If you work with a gauge dependent Lagrangian, how can you arbitrarily fix a gauge? Either the Lagrangian is valid only in a specific gauge (then you have to impose the gauge condition used to derive the Lagrangian) or the Lagrangian is simply not describing electromagnetic fields correctly and you should not use it.

## 1. What is gauge choice in relation to EOMs?

Gauge choice refers to the freedom to choose a mathematical representation or coordinate system for describing the physical properties of a system. In the context of EOMs, gauge choice can affect the form of the equations and the solutions obtained.

## 2. How does gauge choice impact the consistency of EOMs?

Gauge choice can affect the consistency of EOMs by altering the equations and solutions. In some cases, different gauge choices can lead to inconsistent or contradictory results, while in other cases they may lead to equivalent or even simplified equations.

## 3. What is the role of gauge invariance in EOMs?

Gauge invariance is a property of physical systems that allows for different gauge choices to produce the same physical results. In the context of EOMs, gauge invariance can help ensure the consistency and validity of the equations and solutions.

## 4. Can gauge choice affect the physical interpretation of EOMs?

Yes, gauge choice can affect the physical interpretation of EOMs. The choice of gauge can determine which variables are considered fundamental and which are considered derived, thus influencing the understanding and interpretation of the system's behavior.

## 5. How can one ensure consistency with EOMs when making gauge choices?

To ensure consistency with EOMs when making gauge choices, one must carefully consider the physical properties of the system and the implications of different gauge choices. It is also important to understand the principles of gauge invariance and how they apply to the specific system being studied.

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