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Tagged with constrained-dynamicsstring-theory
19 questions
1vote
1answer
91views
Question about Poisson brackets and classical Virasoro generators in bosonic string
I am reading "String Theory and M theory" by Becker, Becker & Schwartz. I am confused about the following. They state that: "Classically the vanishing of the energy–momentum tensor ...
- 2,664
3votes
2answers
203views
Periodicity of the dual scalar in $T$-duality
This question concerns the dual scalar (Lagrange multiplier) field $\lambda$, which appears in discussions of $T$-duality (e.g. in section 7.5.1 of David Tong's notes where it is called $\tilde\phi$). ...
- 469
1vote
1answer
99views
Are the Virasoro constraints generated from the Polyakov action first-class constraints in Dirac's sense?
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: $$ T_{\...
3votes
1answer
102views
Functional derivative for boson CFT on torus [closed]
Let us consider a bosonic CFT on torus: \begin{eqnarray} S=\int dzd\bar{z}\frac{1}{2}\partial X\bar{\partial} X.\tag{2.1.10} \end{eqnarray} From Page 35 of Polchinski Vol. 1, I do the same function ...
- 823
3votes
0answers
237views
Light-cone quantization of open string as derived in Polchinski
Polchinski uses the following gauge conditions, but I don't follow this procedure of gauge fixing and quantization: \begin{align} X^+ = \tau, \tag{1.3.8a} \\ \partial_\sigma \gamma_{\sigma \sigma} = 0,...
1vote
1answer
328views
Understanding the equations of motion for the Polyakov action in string theory
I want to make a small numerical simulation of how strings in theory move under their equations of motion but I'm getting stuck at implementing the constraints. The Polyakov action reads $$S=-\frac 1{...
5votes
0answers
140views
Has the conjecture of Guillemin-Sternberg been proven for relevant physics cases?
From a working physicist's perspective, the conjecture of Guillemin-Sternberg (and its generalisations) seems to state in a highly technical manner that quantization commutes with gauge-fixing. In ...
5votes
1answer
348views
How to derive the Hamilton-Jacobi equation for the area of a minimal surface on a Riemannian manifold?
The action for a string in this background $$G_{IJ}\tag{1}$$ can be written as the Nambu-Goto action $$S_{NG}=\int d\sigma^1d\sigma^2\sqrt{g}\quad\quad\Rightarrow\quad\mathcal{L}=\sqrt{g}\tag{2}$$ ...
2votes
1answer
462views
Stress-Energy Tensor and Conformal Invariance in String Theory
Since the Euler-Lagrange Equations corresponding to the Polyakov Action implies no dependance on the auxillary metric we arrive at the constraint $T_{ab}=0$. We then change to lightcone coordinates $++...
3votes
2answers
679views
Polyakov Lagrangian and Lagrange multipliers
I'm reading Polchinski's Introduction to String Theory (volume I) and something got me quite puzzled in the beginning (At the top of page 19 to be precise). This part is about the open string and the ...
- 289
3votes
1answer
452views
Membrane Theory
I'm not a physicist or educated mathematician, so please excuse me if my question is scientifically rudimentary. It concerns Membrane Theory. If all open strings are attached to the surface of the D ...
- 141
2votes
1answer
484views
Canonical commutation relations in Light-cone gauge
It seems that when trying to identify the physical degrees of freedom for the string some authors$^1$ use: $$ q^-=\frac{1}{\ell}\int_0^{\ell} X^-(\tau,\sigma)d\sigma$$ Then, the commutation relation ...
- 1,734
6votes
1answer
2kviews
Reduction of Nambu-Goto action to true degrees of freedom
I) First consider the point particle $$S=m\int\sqrt{-\dot{X}^2}d\tau.$$ If you choose the static gauge $$\tau=X^0$$ and replace it in the action you get $$=m\int\sqrt{1-\dot{X}^j\dot{X}^j}d\tau.$$ ...
- 1,734
7votes
3answers
811views
How to find the rank of the matrix $\frac{\partial ^2\mathcal{L}}{\partial \dot{X^\mu} \partial \dot{X^\nu} }$ for the Nambu-Goto string Lagrangian?
In this case $$\mathcal{L}~=~-T\sqrt{-\dot{X^2}X'^2+(\dot{X}\cdot X')^2}.$$ I was reading some books and papers about the constraints in the Nambu-Goto action, and all of them say something like the ...
- 1,734
1vote
1answer
758views
Calculus of variations and string theory
In Polchinski's String theory book, Vol 1., in chapter 1, p. 18, he is deriving the Lagrangian in the light cone gauge (that's not necessary to know in order to answer this question), and he gets $$L~=...
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