Gauge fixing and residual symmetries

Your Name]In summary, the question is about the relation between the mode expansion coefficients and the residual reparametrization symmetries in the sigma model action. The mode expansion coefficients must vanish due to the gauge fixing of the world sheet metric, and the algebra relation between these coefficients is a result of the residual reparametrization symmetries satisfying the same relation.
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This question comes from reading Schwarz' string theory book, which is why I put it in this section. But it seems like a general QFT question, so maybe this isn't the right forum for it.

Starting with the sigma model action, reparametrization and Weyl invariance allow us to "gauge fix" the auxilliary world sheet metric [itex]h_{\alpha \beta}[/itex] so that [itex]h_{\alpha \beta}=\eta_{\alpha \beta}[/itex], the 2D minkowski metric. This requires retaining the equation of motion of [itex]h_{\alpha \beta}[/itex] as a constraint, which amounts to requiring the world sheet energy momentum tensor to vanish. If we expand the energy momentum tensor in modes with coefficients [itex]L_m[/itex] (which, classically, are functions of the coefficients of the mode expansion of [itex]X^\mu[/itex]), this requires each [itex]L_m[/itex] to vanish.

Here's my question. In section 2.4, it is said that these mode expansion coefficients satisfy the algebra:

[tex] \{L_m, L_n\} = i(m-n) L_{m+n} [/tex]

where the bracket is the poisson bracket (and translates to the commutator after quantization). It is then said this is a result of the fact that the gauge fixing leaves a residual group of reparametrization symmetries whose lie algebra satisfy the same relations. I'm having a difficult time seeing how these two algebras are related. Can someone help me out?
 
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Thank you for your question. I understand your confusion about the relation between the mode expansion coefficients and the residual reparametrization symmetries. Let me try to explain it in a simple way.

In the sigma model action, the world sheet metric h_{\alpha \beta} is a dynamical field, and its equations of motion determine its values at each point on the world sheet. However, as you pointed out, we can use reparametrization and Weyl invariance to "gauge fix" the metric to be the 2D Minkowski metric, h_{\alpha \beta} = \eta_{\alpha \beta}. This means that we are choosing a particular coordinate system on the world sheet and fixing the scale of the metric.

Now, when we expand the energy momentum tensor in modes, we are essentially expanding it in terms of the Fourier modes of the fields X^\mu on the world sheet. These modes are described by the coefficients L_m, which are functions of the mode expansion coefficients of X^\mu. Since we have fixed the world sheet metric to be the Minkowski metric, the equations of motion for h_{\alpha \beta} become constraints on the coefficients L_m. This is why each L_m must vanish.

As for the algebra relation you mentioned, it is a consequence of the fact that the residual reparametrization symmetries also satisfy the same algebra relation, \{L_m, L_n\} = i(m-n) L_{m+n}. This is because these symmetries are related to the residual gauge transformations that we have after fixing the world sheet metric. So, the algebra relation holds for both the mode expansion coefficients and the residual symmetries.

I hope this helps clarify the relation between the mode expansion coefficients and the residual reparametrization symmetries. Please let me know if you have any further questions. Good luck with your studies!
 
  • #3


Gauge fixing is a technique used in many fields of physics, including string theory, to simplify calculations and extract physical results. In this context, gauge fixing refers to choosing a specific set of coordinates or variables that will be used to describe the system, while still preserving the underlying physics. This is often necessary because in many cases, the equations of motion are not explicitly solvable in their most general form.

In the case of the sigma model action, the gauge fixing of the world sheet metric h_{\alpha \beta} allows us to simplify the equations of motion and obtain a more manageable form for the action. This is possible because the reparametrization and Weyl symmetries of the action allow us to choose a specific form for h_{\alpha \beta} without changing the underlying physics. However, this process also introduces constraints, such as the requirement for the world sheet energy momentum tensor to vanish, which results in the vanishing of the mode expansion coefficients L_m.

Now, the algebra of these mode expansion coefficients is related to the algebra of the residual symmetries that are left after gauge fixing. This can be seen by considering the Poisson bracket of two mode expansion coefficients L_m and L_n, which is given by \{L_m, L_n\} = i(m-n) L_{m+n}. This same relation holds for the Lie algebra of the residual symmetries, which can be derived from the gauge fixing procedure. This means that the two algebras are related and can be seen as different manifestations of the same underlying symmetry.

In summary, gauge fixing and residual symmetries are closely connected in the context of the sigma model action. The gauge fixing procedure allows us to simplify the equations of motion and extract physical results, while the residual symmetries are a consequence of this process and are related to the mode expansion coefficients through their algebraic structure. This is a common theme in many areas of physics, where gauge fixing and residual symmetries play important roles in our understanding of physical systems.
 

1. What is gauge fixing and why is it important in scientific research?

Gauge fixing is a mathematical technique used to eliminate redundant degrees of freedom in a physical system. In physics, it is commonly used in the study of gauge theories, such as electromagnetism and quantum chromodynamics. By fixing the gauge, we can simplify the mathematical description of the system and make it more manageable to study. It is important because it allows us to focus on the essential dynamics of the system and make accurate predictions about its behavior.

2. How do residual symmetries play a role in gauge fixing?

Residual symmetries are the remaining symmetries of a system after gauge fixing has been applied. They are important because they can provide insight into the underlying structure of the system and can help us understand its behavior. In some cases, residual symmetries may lead to new insights and predictions that were not apparent before gauge fixing was applied.

3. What are the different types of gauge fixing methods?

There are several different types of gauge fixing methods, including the Coulomb gauge, the axial gauge, and the Lorentz gauge. Each method has its own advantages and disadvantages, and the choice of which method to use depends on the specific system being studied.

4. Can gauge fixing affect the physical predictions of a system?

Yes, gauge fixing can affect the physical predictions of a system. By eliminating redundant degrees of freedom, the dynamics of the system may change, leading to different predictions. However, if the gauge fixing is done correctly, it should not affect the physical observables of the system, which are quantities that can be measured experimentally.

5. What are the potential challenges or limitations of gauge fixing?

One potential challenge of gauge fixing is that it can be a complex and time-consuming process, especially for systems with a large number of degrees of freedom. Another limitation is that the choice of gauge can be somewhat arbitrary and may affect the results in unforeseen ways. Additionally, some systems may not be able to be fully gauge-fixed, leading to residual symmetries that can complicate the analysis of the system.

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