Einbein as Lagrange Multiplier: How Does it Work?

In summary, the conversation discusses the concept of a photon following a curve and the associated action and Hamiltonian equations. The mass-shell equation is derived and it is explained how the einbein field acts as a Lagrange multiplier. The conversation also touches on the application of this concept to massless particles and its relevance in string theory.
  • #1
ergospherical
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Let ##g_{\mu \nu}(x)## be a time-independent metric. A photon following a curve ##\Gamma## has action\begin{align*}
I[x,e]= \dfrac{1}{2} \int_{\Gamma} e^{-1}(\lambda) g_{\mu \nu}(x)\dot{x}^{\mu} \dot{x}^{\nu} d\lambda
\end{align*}with ##e(\lambda)## an independent function of ##\lambda## (an einbein). The canonical momentum is ##\partial L / \partial \dot{x}^{\mu} = e^{-1}(\lambda) \dot{x}_{\mu}## which yields a conserved energy ##-E \equiv -e^{-1}(\lambda) \dot{t}## since the Lagrangian does not depend on time. The Hamiltonian is\begin{align*}
H = \dfrac{1}{2} e^{-1}(\lambda) g_{\mu \nu}(x) \dot{x}^{\mu} \dot{x}^{\nu} = \dfrac{1}{2}e(\lambda)\left( - E^2 + \mathbf{p}^2 \right)
\end{align*}From this follows the mass-shell equation ##E^2 = \mathbf{p}^2##. Why is this? The einbein field ##\dfrac{1}{2}e(\lambda)## is supposedly acting as a Lagrange multiplier but my variational calculus is rusty. (The next step will be to Legendre transform the cyclic variables only to determine the Routhian).
 
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  • #2
Actually I think it makes sense, that ##-E^2 + \mathbf{p}^2 \equiv F = 0## is just a standard constraint between the coordinates and the canonical momentum when re-written.
 
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  • #3
You can make ##e(\lambda)## a Lagrange multiplier. Then you have in addition the variation of the action wrt. ##e##, and this gives the desired constraint ##g_{\mu \nu} \dot{x}^{\mu} \dot{x}^{\nu}## for the motion of a massless particle.
 
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  • #4
Why is this? It's per construction, so you can write down an action which also applies to massless particles and is easy to quantize. In string theory I'd call this trick "switching from Nambu-Goto to Polyakov" :P
 
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Related to Einbein as Lagrange Multiplier: How Does it Work?

1. What is Einbein as Lagrange Multiplier?

Einbein as Lagrange Multiplier is a mathematical technique used to optimize a function subject to one or more constraints. It is named after Joseph-Louis Lagrange, who first introduced the concept in the 18th century.

2. How does Einbein as Lagrange Multiplier work?

Einbein as Lagrange Multiplier works by introducing a new variable, called the Lagrange multiplier, into the function being optimized. This variable is then used to incorporate the constraints into the optimization process, resulting in a set of equations that can be solved to find the optimal solution.

3. What are the advantages of using Einbein as Lagrange Multiplier?

Einbein as Lagrange Multiplier allows for the optimization of functions subject to constraints, which is a common problem in many scientific and engineering fields. It also provides a systematic and efficient way to incorporate constraints into the optimization process.

4. What are some applications of Einbein as Lagrange Multiplier?

Einbein as Lagrange Multiplier has a wide range of applications in various fields, such as economics, physics, and engineering. It can be used to optimize production processes, minimize energy consumption, and find the most efficient solutions to complex problems.

5. Are there any limitations to using Einbein as Lagrange Multiplier?

While Einbein as Lagrange Multiplier is a powerful tool for optimization, it does have some limitations. It may not always provide the global optimal solution, and it may be computationally expensive for complex problems. Additionally, it may not be applicable to all types of constraints.

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