Momentum conservation under a Gauge Parametrization in string theory

This allows us to choose a gauge that conserves momentum on the string, even in the presence of Dirichlet boundary conditions.
  • #1
benbenny
42
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Im reading "A first course in String Theory" by Zwiebach and it says that when applying a gauge parametrization in the form of [tex] n_\mu X^\mu = \lambda \tau[/tex] we can take the vector [tex] n_\mu [/tex] so that for open strings connected to branes (fixed end points), [tex] n^\mu \mathcal{P}^\tau _\mu [/tex] is conserved. But in general momentum is not conserved over the string for dirchlet boundary conditions as I understand, so how does applying the general \tau gauge make it so that it is. How can we chose a gauge that will conserve momentum on the string. The string is still going to be connected to a brane, and without considering the dynamics of the brane I don't see how this can be ensured.

Thanks,

B
 
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  • #2
ranching off from the gauge parametrization, we can look at it from the viewpoint of the equations of motion. By introducing a new vector field n_μ, we can construct a new constraint equation which can be written as n_μX^μ = λτ. This equation enforces that the endpoints of the string must move along trajectories determined by the vector n_μ. Since n_μ is a conserved quantity, this constraint equation implies that momentum is conserved on the string. This is because the constraint equation prevents the string from interacting with external forces or momenta and thus keeps it in equilibrium. In other words, the gauge parametrization provides a way to ensure that the string remains unaffected by external forces or momenta while still remaining connected to the brane.
 

1. What is momentum conservation in string theory?

Momentum conservation in string theory refers to the fundamental principle that the total momentum of a system of strings remains constant over time. This means that the total momentum of all the strings in a given system cannot be created or destroyed, but can only be transferred between strings.

2. What is a Gauge Parametrization in string theory?

A Gauge Parametrization in string theory is a mathematical tool used to describe the dynamics of string interactions. It involves choosing a specific set of coordinates and parameters to represent the physical properties of the strings, and allows for simplification and calculation of string interactions.

3. How does momentum conservation under a Gauge Parametrization work in string theory?

Momentum conservation under a Gauge Parametrization in string theory works by using the chosen set of coordinates and parameters to describe the momenta of the strings in a system. This allows for the conservation of total momentum to be explicitly included in the equations of motion for the system, ensuring that the total momentum remains constant over time.

4. Why is momentum conservation important in string theory?

Momentum conservation is important in string theory because it is a fundamental principle that governs the dynamics of string interactions. It plays a crucial role in understanding and predicting the behavior of strings and their interactions within a system, and is essential for the consistency and validity of the theory.

5. Are there any exceptions to momentum conservation under a Gauge Parametrization in string theory?

While momentum conservation is a fundamental principle in string theory, there are some cases where it may not hold true. For example, in situations involving asymmetrical string interactions or in the presence of certain external forces, momentum may not be conserved. However, these exceptions are typically accounted for and explained within the framework of string theory.

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