Quantum Mechanics and Strings: A Puzzling Relationship

[Daveman20]

So fundamental point-like particles are expressed by the vibration of a string... yet a string is two dimensional? or even more? How can quantum mechanics not apply to strings?? I don't understand the size of strings vs the size of the particles they represent? Someone please explain!

 

[haushofer]

Quantum mechanics DOES apply to strings! In "ordinary"QM particles are pointlike, so they are 0-dimensional (you don't need any coordinate to express your position on something pointlike, right?). String theory says that particles are excitations of 1-dimensional objects called strings. Quantizing this theory gives a spectrum of particles which can be generated as excitations of such a string.

However, the string lengt is a very small number, way smaller than we can measure nowadays, so it appears to us that particles are really 0-dimensional. But string theorists know better :P

The reason why we use strings, and not higher dimensional objects to represent particles, has some technical reasons, but is not as arbitrary as may look at first sight.

 

[Daveman20]

Quantum mechanics DOES apply to strings! In "ordinary"QM particles are pointlike, so they are 0-dimensional (you don't need any coordinate to express your position on something pointlike, right?). String theory says that particles are excitations of 1-dimensional objects called strings. Quantizing this theory gives a spectrum of particles which can be generated as excitations of such a string.

However, the string lengt is a very small number, way smaller than we can measure nowadays, so it appears to us that particles are really 0-dimensional. But string theorists know better :P

The reason why we use strings, and not higher dimensional objects to represent particles, has some technical reasons, but is not as arbitrary as may look at first sight.

so yes it is true that particles are not 0 dimensional since they are manifestations of a string which has length in 1 dimension, even though they can be thought of as 0 dimensional because of their mathematical representation? I feel like I am close..

 

[Daveman20]

and if QM applies to strings then how come string theory can marry qm with GR?

I was listening to the elegant universe and it was explained like

point-like particles under QM < plank length where strings lay < and GR where mass behaves classicly

like strings were in this Goldilocks zone

 

[Kevin_Axion]

I believe it has to do with Renormalization, so in current Quantum Field Theory you have a hypothetical spin-2 particle the graviton, say you have a particle of mass and it emits a graviton to interact with another particle with mass. Gravity interacts with the Stress-Energy Tensor I believe or mass and energy via E=mc². Since gravitons have energy you essentially have an infinite amount of self-interactions and the entire situation becomes convoluted with virtual gravitons. Superstring Theory suggests a Supersymmetric 1-dimensional string which is spin-2 - the hypothetical graviton, essentially the self-interactions become more comprehensible and Renormalization is capable. Since Strings are on the order of Planck Length they must obey Quantum Mechanics and since it successfully provides a Quantum Mechanical representation of gravity General Relativity and Quantum Mechanics are fused ingeniously.

 

[Daveman20]

I believe it has to do with Renormalization, so in current Quantum Field Theory you have a hypothetical spin-2 particle the graviton, say you have a particle of mass and it emits a graviton to interact with another particle with mass. Gravity interacts with the Stress-Energy Tensor I believe or mass and energy via E=mc². Since gravitons have energy you essentially have an infinite amount of self-interactions and the entire situation becomes convoluted with virtual gravitons. Superstring Theory suggests a Supersymmetric 1-dimensional string which is spin-2 - the hypothetical graviton, essentially the self-interactions become more comprehensible and Renormalization is capable. Since Strings are on the order of Planck Length they must obey Quantum Mechanics and since it successfully provides a Quantum Mechanical representation of gravity General Relativity and Quantum Mechanics are fused ingeniously.

This was also helpful. So point-like particles were only thought of as zero dimensional because of the math explaining them, mixed with their ambiguous properties below plank level? (uncertainty), but we also see that strings can operate at around this level being under QM, clear up some of the infinities between gravity and would be one dimensional. So can I argue that point-like particles under string theory would actually be 1 dimensional?

 

[Kevin_Axion]

Not necessarily, I'm not sure what you mean by below the Planck scale. The Planck scale is an energy scale approximately 1.22 x 10²⁸ eV derived by fundamental properties in Physics: that is, this is the energy scale in which gravity becomes strong and plays a significant role Quantum Mechanically. Consequently our other Quantum Field Theories: The Standard Model deform and don't make any relevant sense - and gravity becomes non-renormalizable at these great energy scales. Therefore since our theories don't work at those energy scales we must use Superstring Theory/M-Theory to allow General Relativity to be described Quantum Mechanically by the renormalization of the graviton. Yes, all particles are one dimensional "strings" of energy that vibrate at certain frequencies dependent on the geometry of the compactified dimensions. These vibrations generate particles similar to the graviton that allow renormalization of gravity.

 

[Demystifier]

So can I argue that point-like particles under string theory would actually be 1 dimensional?

Yes.

 

[haushofer]

and if QM applies to strings then how come string theory can marry qm with GR?

That has to do indeed with renormalization. The intuitive picture which is often given is the following: if you consider a Feynmandiagram of 2 interacting particles, the interaction vertices are points. These points are mathematically problematic, because they give infinities in your expressions in the integrals. But if you look at the stringy equivalent of this interaction, there is no such point; locally the theory looks like a free theory without any interaction. The point has been "smeared out", and this smearing out has mathematically the result that the problems of these infinitites become much more tractable and controllable.

But the important fact of string theory is that you don't put in gravity by hand! The only thing you do basically is to write down the theory which contains closed strings. After some calculations you'll encounter a massless spin-2 particle in your excitations of these closed strings.

Interacting massless spin-2 particles need a symmetry called "diffemorphism invariance" to make the theory well-defined, which is the symmetry of general relativity. Also, the worldsheet has a symmetry called "conformal invariance". Demanding that this symmetry holds quantum mechanically gives a constraint, which in spacetime gives exactly the equations of motion of general relativity! (Plus higher order corrections, ofcourse).

That's one of the beautiful and remarkable things of string theory, imo. It's the reason that I believe Witten once said that "string theory predicts gravity".

 

[haushofer]

so yes it is true that particles are not 0 dimensional since they are manifestations of a string which has length in 1 dimension, even though they can be thought of as 0 dimensional because of their mathematical representation? I feel like I am close..

Yes, more or less. At "normal length scales" the stringy character isn't apparent. But at very small length scales (which is the same as very high energy scales; think about it! What do you need to probe small length scales with for instance particle accelerators? High energies!) the stringy character becomes important.

That's why the standard model is so succesful: at "low energy scales" it suffices and we can ignore the stringy effects. In much the same way that we can ignore Einstein if we deal with low energies and low velocities: in that case Newton suffices.

That's a very important philosophy in physics: theories are valid in a certain range of energy, or "effective theories". I get the feeling that more and more people are regarding string theory also in this way, eventhough once the claim was that it should be "The Theory of Everything". But that's something I haven't thought about yet ;)

 

[Daveman20]

Yes, more or less. At "normal length scales" the stringy character isn't apparent. But at very small length scales (which is the same as very high energy scales; think about it! What do you need to probe small length scales with for instance particle accelerators? High energies!) the stringy character becomes important.

That's a very important philosophy in physics: theories are valid in a certain range of energy, or "effective theories".

This along with your other posts is very helpful. Thanks to you and everyone who responded