The Nambu-Goto Action - Questions & Answers

[wam_mi]

Hi there,

Recently, I read that the Nambu-Goto action of a free relativistic string is motivated from the study of the relativistic point particle moving sweeping out a world-line parameterised by the proper time. May I ask

(i) Why do we parameterise the world-line of the particle by the proper time? Is it because we want to ensure that the relativistic point particle action remains Lorentz invariant? How important is that?

(ii) A free relativisitc string sweeps out a world-sheet that is parameterised by one time-like and one spatial parameter. Then we can write the action in flat space. But what happens when we have to work in a general curved space-time? How are we sure we are allowed to work with the flat metric, and all the results later (e.g. light-cone gauge quantisation, Virasoro algebra, etc) in flat space-time agrees with that of the general metric?

In other words, how can we assume that flat metric is equivalent to general metric? Has this got something to do with conformal field theory?

Thanks!

 

[curious_mind]

I think, I can answer the first one.

(i) I think parameterizing the world-line is the only way we can explore the dynamics of the point particle. Parameterizing is done through time in point particle case.
Similarly, parameterizing the worldsheet is the way to analyze the dynamics of the string.

Is it so ?

 

[Cepatiti]

(i) Not sure I got the question correctly, but I would say this: proper time corresponds to the length of the worldline, hence it is a very natural parameter to use to parametrize the worldline. As for the Lorentz invariance, observations indicate that Lorentz symmetry is preserved in nature, that's why you want to have Lorentz-invariant theories in physics. Of course, there are also some exotic theories that include Lorentz violation, and there are also on-going searches to look for possible Lorentz violation.

(ii) You can easily generalize the Nambu-Goto action to curved spacetime like so:

$$S[ X] = -T \int d^2\sigma\sqrt{-\text{det}(g_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu)}\,,$$

where $g_{\mu\nu}$ is the metric of the target space.

 

[haushofer]

Hi there,

Recently, I read that the Nambu-Goto action of a free relativistic string is motivated from the study of the relativistic point particle moving sweeping out a world-line parameterised by the proper time. May I ask

(i) Why do we parameterise the world-line of the particle by the proper time? Is it because we want to ensure that the relativistic point particle action remains Lorentz invariant? How important is that?

(ii) A free relativisitc string sweeps out a world-sheet that is parameterised by one time-like and one spatial parameter. Then we can write the action in flat space. But what happens when we have to work in a general curved space-time? How are we sure we are allowed to work with the flat metric, and all the results later (e.g. light-cone gauge quantisation, Virasoro algebra, etc) in flat space-time agrees with that of the general metric?

In other words, how can we assume that flat metric is equivalent to general metric? Has this got something to do with conformal field theory?

Thanks!

(i) We don't need to. The action is manifestly reparametrization invariant. Or, if you want to impress: it's invariant under wordline diffeomorphisms. Choosing the proper time for tau is just one choice.

(ii) Indeed, that's where the magic of conformal invariance kicks in. The worldsheet theory is a 2-dim. conformal field theory. And with the combination of conformal invariance in two dimensions, there is something special going on: the operator-state correspondence. See e.g.

https://physics.stackexchange.com/questions/88773/operator-state-correspondence-in-qft

What this means is the following. In the spectrum you find massless spin-2 particles, which remind you of gravitons. If you want to describe scattering, you can "insert" the corresponding states on the worldsheet via certain operators. The operator-state correspondence translates this into in- and out-states of the scattering process. If you now want to describe a curved background, you can start from flat spacetime and insert as many graviton states on the worldsheet as you like. You get, so to speak, a "coherent state of gravitons", just as a laser is a coherent state of photons. And the result is the same as if you replace the flat metric by a curved metric (which is a solution of the graviton field equations, which on the worldsheet means that the beta-function of the graviton states should disappear to keep conformal invariance after quantizing.)

So simply replacing a flat spacetime by a curved one seems a bit weird and "not in line with the stringy philosophy", but this naive guess turns out to be justifiable by the operator-state correspondence. I believe Green, Schwartz and Witten have a discussion on this, although not with the specific CFT jargon, as the book dates from the 80's.

Hope this helps :)

-edit Wow, only now I see this is a kick of a topic of 12 years old...

 

[haushofer]

(ii) You can easily generalize the Nambu-Goto action to curved spacetime like so:

$$S[ X] = -T \int d^2\sigma\sqrt{-\text{det}(g_{\mu\nu}(X)\partial_aX^\mu\partial_bX^\nu)}\,,$$

where $g_{\mu\nu}$ is the metric of the target space.

Well, yes, but it is highly non-trivial that this works for reasons I described earlier. If you would couple a classical string to a curved background, where the metric is independent from the string, this is the way to go. But in string theory the gravitational interaction arises from the string vibrations itself!

 

[Cepatiti]

But in string theory the gravitational interaction arises from the string vibrations itself!

I know. But I think the original author's concerns were specifically about the Nambu-Goto action, which he seems to believe restricted to flat Minkowski space. For quantization, you would normally use the Polyakov action anyway.
If the question was however about how curved spacetime arises in string theory, then he can find a detailed answer here:
Is string theory formulated in flat or curved spacetime