1

The Polyakov action for strings reads

$$ S[X] = -\frac{T}{2} \int d^2\sigma\, \sqrt{h}h^{\alpha\beta} \partial_\alpha X^\mu \partial_\beta X_\mu, $$

from which the Virasoro constraints follow:

$$ T_{\alpha\beta} = -\frac{2}{T} \frac{1}{\sqrt{h}} \frac{\delta S}{\delta h^{\alpha\beta}} = 0. $$

In classic textbooks such as that of Green, Schwarz, and Witten, it is stated that plugging in $T_{\alpha\beta}=0$ to the action transforms the Polyakov action to its alternative, the Nambu-Goto action.

However, according to Dirac's constrained dynamics, it is not automatically justified to put a constraint equation into the action since it may alter the variational condition, potentially leading to nonsensical equations of motion.

To my understanding, such manipulation is allowed only for the first-class constraints, i.e. those with vanishing Poisson bracket with other constraints. In this context, how can one show that using the Virasoro constraints to transform Polyakov action into the Nambu-Goto action?

Qmechanic
  • 220,844
Hyeongmuk LIM
  • 283
  • 1
  • 12

1 Answers1

1

  1. OP is right that premature use of EOMs in an action principle can potentially lead to inconsistencies.

    That said, keep in mind that it is always consistent to integrate out a variable in a path integral. (Of course, that variable will then no longer be an observable in the theory.) In particular, there is a corresponding classical notion of integrating out a variable in an action.

  2. For Dirac's constrained dynamics to be relevant, one should go to the Hamiltonian formulation.

    Concerning OP's last question, one may show that the Hamiltonian formulations of the Nambu-Goto string and the Polyakov string are equivalent, cf. e.g. footnote 1 in my Phys.SE answer here.

  3. And yes, the Virasoro constraints are e.g. obtained as primary constraints from Dirac-Bergmann analysis of the Lagrangian NG string, cf. e.g. my Phys.SE answer here.

Qmechanic
  • 220,844