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I will ask my question about Lagrange multipliers by using an example in string theory, as this question was inspired by my string theory course, but it applies to every theory with Lagrange multipliers I guess. In string theory, we are taught that one can obtain the Nambu-Goto action from the string sigma-model action by varying the latter with respect to the induced metric $h_{\alpha\beta}$ from $$S_{\sigma}=-\frac{T}{2}\int d^2\sigma\sqrt{-h}h^{\alpha\beta}\partial_{\alpha}X \cdot\partial_{\beta}X$$ (where I use the notation of "String theory and M-theory" written by Becker, Becker and Schwarz), and requiring the variation of the action $S_{\sigma}$ with respect to the induced metric $h^{\alpha\beta}$ be zero. This yields the classical equations of motion and when they satisfied, we obtain the Nambu-Goto action. So far so good...

However, what happens if for some reason the equations of motion are not satisfied? Can something like that take place for whatever reason? For instance one could argue that if we were to promote the components of the induced metric to quantum operators, then the latter components would not have to obey the classical equation of motion. This is an example coming to mind for a circumstance in which the induced metric does not satisfy the equation of motion and, therefore, one can meaningfully demand that the equations of motion be satisfied.

But, the induced metric components are not dynamical. So, there is no way of having a conjugate momentum to promote to quantum operators with. Am I lead to believe that the equations of motion must be satisfied, or am I lead to believe that the equations of motion can not be satisfied for some other reason (and which reason is that)?

I guess the same question applies every time we introduce a Lagrange multiplier into a theory, regardless whether this is string theory or not, but nevertheless...

Any help will be appreciated.

schris38
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  • To take a much simpler example - from discrete mechanics, not field theory, but you can still quantize it - mass is a Lagrange multiplier in the kinetic sector of $L=-\frac12mx(\ddot{x}+\omega^2x)$. It ends up quantized. – J.G. Mar 24 '23 at 17:57
  • What is this link about? I do not understand... Do you mean that one can quantize a Lagrange multiplier by making it obey some commutation relations? And if so, what is the conjugate momentum? One must have a conjugate momentum, right? – schris38 Mar 24 '23 at 18:01
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    Related: https://physics.stackexchange.com/q/17349/2451 – Qmechanic Mar 24 '23 at 18:43
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    This question is not really about Lagrange multipliers, but about gauge constraints I think, see for example https://en.wikipedia.org/wiki/Dirac_bracket. – Non-invertibleSym Mar 25 '23 at 02:22
  • I think there exists a kinetic term regarding the additional metric $h_{\alpha\beta}$, it is just that it is equal to something called the Euler characteristic, which is a number, and therefore we neglect it if I understand correctly... – schris38 Mar 27 '23 at 12:37

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