Justification of Action Integral

[Mike2]

It would seem that the Action integral for the world sheets of string theory are presently justified as a higher dimensional version. But I wonder if this formulation would be a natural description if we were first given the geometry of a world-sheet.

Suppose we understood the necessity of a world-sheet geometry. What would be the most natural mathematical description of what is happening with a world-sheet? We have a sheet that is growing and expanding in time. It adds more surface area with time. This might suggest that we integrate this area from one string to the next. And String theory poses that the Action Integral is protortional to the surface area of the string.

But if the motion of the world-sheet is not arbitrary, then it seems there must be a function at every point on the world-sheet which forms the basis for why the string is there and where it will go in the future. This function would give a wieght to every differential area, or at least a weight for every differential length on a string. This weighting function would determine how the string would oscillate and what direction it will have a tendency to go. Is there any functions along the string or world-sheet that serves this purpose? Does the tension serve this function? Or maybe there is a potential at each point along a string or world-sheet.

I have a justification for a type of world-sheet geometry, a growing surface in space-time as nothing more than a growing "events" in sample space. See here at: http://www.sirus.com/users/mjake/StringTh.html#consider
But I am not sure of the mathematics to descript it? Is this too easy and of course it is the Action integral that describes the situation? Or is it not quite as simple as that? It would be so nice to develop some sort of Action integral for this and then apply symmetries to come up with the same sort of physical laws from Euler-Lagrange equations. But I don't want to simply jump to that conclusion.

Your help would be greatly appreciated. Thanks.

 

[selfAdjoint]

But Mike, the whole point of forming an action integral is to constrain the motion. Alembert's principle requires that the variation of the action vanish; that it be stationary. Euler and Lagrange figured out how to use this to form equations of motion.

So in the case of the world sheet only one "path of sheets" would be physically obtained between any two states.

 

[Mike2]

Originally posted by selfAdjoint
But Mike, the whole point of forming an action integral is to constrain the motion. Alembert's principle requires that the variation of the action vanish; that it be stationary. Euler and Lagrange figured out how to use this to form equations of motion.

So in the case of the world sheet only one "path of sheets" would be physically obtained between any two states.

My post did not take. So I am trying again.

I'm not quite sure what you are addressing, Dick T. (right?) Correct me if I am wrong, but the vanishing of the variation of the Action integral and the symmetries that lead to the Euler-Lagrange equations of motion does not determine where the initial and final string states are. The initial and final position of the string seem to be imposed as limits on the integral in an a priori fashion and then the symmetries and vanishing variation are employed after that.

Is this the same as using the Euler-lagrange equations to determine the final position and velocity of the string given the initial string state?

It seems as though more information is needed other than just the area between initial and final positions of the string. The action integral for strings is a double integral over some function we call the Lagrangian. The Lagrangian is not just one so that the integral is just the area. But the Lagrangian is some scalar function over the surface of the world-sheet. This function gives some weight to area elements on the surface.

The Lagrangian must satisfy the Euler-Lagrange equations. But there may be many functions that satisfy this condition. I wonder if there is any process where we find the final answer as the linear combination of all possible Lagrangians that satisfy the Euler-Lagrange condition. Is this the Feynman path formulation?


I always appreciate your comment, Dick. Thanks.

 

[Mike2]

In other words, I have a function on a surface which grows with time. Is that enough in itself to justify the integration of this function on the surface? Or perhaps I should be looking for a differential equation.

 

[Mike2]

Originally posted by Mike2
In other words, I have a function on a surface which grows with time. Is that enough in itself to justify the integration of this function on the surface? Or perhaps I should be looking for a differential equation.

Let's see. Is geometry a global characteristic best described by some process of integrating over the entire surface? If we know there is a function on the surface but we don't know what it is, then does that require the need of some sort of functional equation?

I was thinking that perhaps I should instead be searching for some sort of differential equation, with some function on the surface serving as boundary conditions. But now I'm thinking that the diff eq I'm looking for are just the equations of motion derived from the invariance of the Action Integral. Does this necessitate such an integration as a first step? Or could there be other diff eqs that could be derive independently. Are diff eqs of the same order on the same geometry of the same space necessarily unique?


I would appreciate your comments. I feel I'm close. Imagine that, Lagrangian mechanics from only logical principles?

 

[selfAdjoint]

Well what will your answers be? If you want numbers, diff eqs will be what you use, but if you want (wave) functions, you have to use functional analysis. Mostly functional analysis gets answers by integrals. Of course you have to integrate diff eqs too, but you usually don't show the sea horses. Superstring books are full of seahorses, and of tricks for turning one integral into another.

 

[Mike2]

Originally posted by selfAdjoint
Superstring books are full of seahorses, and of tricks for turning one integral into another.

By seahorses, I assume you mean integrals.

It seems you get both when dealing with an action integral. You have the functional integral that gives the Euler-Lagrange diff eq. So is the Euler-Lagrange equation just the diff eq version of the functional integral stuff? And does that cover all of my concerns of what approach I use to describe the geometric situation? Thanks for your help.

 

[Mike2]

Also, is the Action Integral over a surface, which is a double integral, equal to one action integral inside another? Is it an action integral over a curve in space inside the actions integral through time?

Part of my concerns about justifying the math to describe the geometry of a world-sheet is whether I should first describe what is happening on the string at one instant of time first and then continue to describe this behavior as time advances. Or should I just describe the world-sheet as a whole. At one instant of time there is a string through space with a function evaluated at each point along the length of the string. This string sweeps out a larger surface as time passes. If I don't know the path it takes or the function on each point of the surface, then it would seem the best I can do is to describe the situation with a surface integral of a scalar function of some unspecified path. This is pretty much what the Action integral is. Then I would naturally apply requirements of invariance with coordinate changes to come up with a Euler-Lagrange vector. Would I then have a reason to assume a vanishing variation to come up with Euler-Lagrange equations (set to zero) and also Noether's theorem of conserved values? Is this last step assuming conserved quantities to begin with? Or is a vanishing variation in and of itself a type of symmetry or invariance that one would naturally expect to employ due to some obvious intrinsic property? I wonder.

 

[Mike2]

Originally posted by Mike2
It would seem that the Action integral for the world sheets of string theory are presently justified as a higher dimensional version. But I wonder if this formulation would be a natural description if we were first given the geometry of a world-sheet.

Does the Action integral describe something real, or is it only a tool of convenience? If it does describe something real, then what is being described, and why is the action integral the best description of it? Are you telling me no one has ever asked this question even in a classical context?

Thanks.

 

[Mike2]

Perhaps I ask too many questions in one post, so I'll just ask one here:

Is the Action Integral over a surface, which is a double integral, equal to one action integral inside another? Is it an action integral over a curve in space only inside the actions integral through time only? Or, in other words, for classical string theory would the string at one particular instant of time be the minimum length across the surface of a world sheet and then also the world sheet a minimum surface area from string to string?

I could use some insights from those more experienced with these matters, thanks.