How string theory gets around the mass-paradox (Thanks Lubos)

In summary, Lubos explains that the uncertainty in the position and momentum of a string is due to its internal degrees of freedom, similar to ordinary particles. The minimum size of the string is determined by a balance between its kinetic and potential energy, and this minimum size is on the order of the string length, which is around the Planck scale. The momentum uncertainty of the string is not a problem for the mass paradox because the particles are not made up of the string - it is a fundamental object with different vibrational modes.
  • #1
bananan
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Dear Lubos,

Heisenberg's uncertainty princple states that xp>=h bar/2

a string is smaller than the particles they make up, so x is smaller than the particles of the standard model.

The momentum of uncertainty p should be greater than the particles themselves.

How does string theory get around the mass-paradox:

The momentum uncertainty of a string confined in such a small location would be much larger than the particles they supposedly make up.


Lubos kindly replies:

Dear Dan,

the center-of-mass position X0 and the total momentum of a string, P0, behave just like for pointlike particles. They don't commute and follow the uncertainty principle. A certain X0 means uncertain P0 and vice versa, the product being above hbar/2.

Besides the zero modes (center-of-mass degrees of freedom), every string has infinitely many internal degrees of freedom. That's like an atom with many electrons, but you have infinitely many arranged along a string. The relative motion of pieces of string gives energy - expressed as the usual sum of the kinetic and potential contributions. And because they're relativistic strings, energy also means mass via E=mc^2.

The result is that the minimal size of the string - in the lowest-energy or lowest-mass states - is a compromise in which the kinetic terms from the internal degrees of freedom contribute the same as the potential terms, just like for X and P in the harmonic oscillator, attempting to minimize the energy while satisfying the uncertainty relation for the internal X,P degrees of freedom. This minimum occurs if the typical size of the string is comparable to a typical distance scale derivable from the tension of the string - the string length - that is conventionally believed to be close to the Planck scale 10^{-35} meters (a bit longer than that).

The actual numerical coefficient of the string is actually logarithmically divergent but this fact doesn't affect any finite-energy experiments.

Your question has the same answer in string theory just like for ordinary particles because it is the zero modes that matter here. The internal degrees of freedom are only relavant for experimental uncertainty considerations if your probe the internal structure of a string, and indeed, you will always find out that the "radius" of it seems to be of order the string length.

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  • #2
wishes
Lubos


Dear Lubos,

Thank you for your explanation. So, if I understand correctly, the uncertainty in the position and momentum of the string is due to its internal degrees of freedom, just like for ordinary particles. And the minimum size of the string is determined by a balance between its kinetic and potential energy, similar to the harmonic oscillator. And this minimum size is on the order of the string length, which is around the Planck scale.

But how does this address the mass paradox? If the string has a minimum size on the order of the Planck scale, wouldn't its momentum uncertainty be much larger than the particles it is supposed to make up?


Dan

Dear Dan,

Yes, the momentum uncertainty of the string is much larger than the particles it is supposed to make up. But that's not a problem because the particles are not really "made up" of the string in the sense that the string would be a composite object. The string is a fundamental object and the particles are just different vibrational modes of the string. So the momentum uncertainty of the string is not really a problem for the mass paradox.

Best wishes,
Lubos
 

FAQ: How string theory gets around the mass-paradox (Thanks Lubos)

What is the mass-paradox and how does string theory address it?

The mass-paradox refers to the discrepancy between the observed mass of particles and the predicted mass based on the Standard Model of particle physics. String theory proposes that particles are actually tiny strings vibrating at different frequencies, which can account for the difference in mass.

How does string theory explain the existence of multiple dimensions?

String theory suggests that there are more than three spatial dimensions, which can help reconcile the inconsistencies between the theory of general relativity and quantum mechanics. The strings in string theory vibrate in these extra dimensions, giving rise to different types of particles.

Can string theory be tested experimentally?

At this point, string theory is still a theoretical framework and has not been confirmed through experiments. However, some scientists are working on ways to test certain predictions of string theory, such as the existence of extra dimensions, through particle accelerators and other experiments.

How does string theory unify gravity and the other fundamental forces?

In string theory, gravity is not treated as a force like the other three fundamental forces (electromagnetism, strong nuclear force, and weak nuclear force). Instead, it is seen as a manifestation of the geometry of space-time, which is influenced by the vibrations of strings. This unifies gravity with the other forces.

What are some potential implications of string theory?

String theory has the potential to explain various phenomena in the universe that are currently unexplained, such as the origin of the universe, dark matter, and the behavior of black holes. It could also lead to advancements in technology, such as quantum computing and space exploration.

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