Can Extra Dimensions in String Theory Be Smaller Than a Planck Length?

[ThomasEdison]

Can an extra dimension of empty space be smaller than a Planck length?


Is that why string theory allows those extra dimensions to be smaller than a Planck length? Because they are empty?


I remember reading posters on these forums say that space itself can expand faster than the speed of light and that this is what happened during the big bang.

So exceptions are made for space itself both in quantum physics and in relativity?

Edited 1 hour later :I always assumed that nothing could be shorter than a Planck length.

 

[jtbell]

I forgot to add

To what?

I don't see any other recent posts by you in this particular forum (Quantum Physics). Are you thinking of a thread in some other forum here?

 

[ThomasEdison]

To what?

I don't see any other recent posts by you in this particular forum (Quantum Physics). Are you thinking of a thread in some other forum here?

Corrected.

 

[haael]

I always assumed that nothing could be shorter than a Planck length.

That is not an universally accepted theory and no observations for today have shown any violation of Lorentz invariance.

If the space really was quantized, then yes, there could not be any dimension smaller than Planck length.

 

[tom.stoer]

Can an extra dimension of empty space be smaller than a Planck length?

Is that why string theory allows those extra dimensions to be smaller than a Planck length? Because they are empty?

In ST there is a kind of duality mapping small to large extra dimensions. That means that small extra dimensions of radius R are equivalent to a different ST with larger extra dimensions of radius L/R². Physically there is again a lower limit.

In ST spacetime is smooth, not discrete or quantized; nevertheless it seems that something like a minimum length may emerge from it. I am not sure whether ST really provides a microscopic picture of spacetime or whether this duality tells us that something is still hidden behind the curtain.

I remember reading posters on these forums say that space itself can expand faster than the speed of light and that this is what happened during the big bang.

After the big bang during the inflation. Inflation is not a proven fact, but nevertheless a widely accepted generic scenario emerging from rather different approaches.

So exceptions are made for space itself both in quantum physics and in relativity?

No exception. SR and GR tell us that locally no signal can move faster than the speed of light. Globally it may look like that objects are moving faster than the speed of light, but this is due to the fact that in an expanding (non-static) spacetime there is no unique definition of velocity of distant objects anymore.

Space itself is not a signal, no energy flow, no information. Therefore "not faster than the speed of light" simply does not apply.

Think about the universe (space) as a balloon (that means we are neglecting one space dimension and we are embedding the universe = the balloon into a three dimensional space; this is mathematically incorrect, but visualizes the general idea). Expansion of spacetime is represented by somebody blowing up the ballon, whereas light is represented by ants crawling on the surface of the balloon. The balloon can blow up much faster than an ant can crawl; this is what happened during inflation.

 

[unusualname]

The Generalised Uncertainty Principle ( GUP ) may predict a smallest length, eg see http://arxiv.org/abs/0906.5396, http://arxiv.org/abs/1005.3368

It violates Lorentz Invariance in SR (smallest length in whose frame?), but can be made consistent in Doubly Special Relativity

 

[my_wan]

The Generalised Uncertainty Principle ( GUP ) may predict a smallest length, eg see http://arxiv.org/abs/0906.5396, http://arxiv.org/abs/1005.3368

It violates Lorentz Invariance in SR (smallest length in whose frame?), but can be made consistent in Doubly Special Relativity

The exact uncertainty principle was derived on the assumption of classical ensembles subjected to random momentum fluctuactions, not unlike brownian motion, from which the Schrödinger equation was derived.
http://arxiv.org/abs/quant-ph/0102069"
J. Phys. A 35 (2002) 3289-3303

Could it be that a relativistic spacetime interval covaries with the effective Planck length for a particular observer. This would put the constancy of Planck constant on the same physical footing as the constancy of the speed of light. If there was a fundamental physically defined length scale, then any local variance would be unmeasurable, like trying to measure a change of local time rate. Globally, this would induce GR like effects. It would also wash out the measurability of local mean variations in the energy density of space, thus avoiding a http://en.wikipedia.org/wiki/Vacuum_catastrophe" .

 

[tom.stoer]

The exact uncertainty principle was derived on the assumption of classical ensembles subjected to random momentum fluctuactions, not unlike brownian motion, from which the Schrödinger equation was derived.

I don't understand. The uncertainty principle is simply a theorem derived from non-commuting operators in a Hilbert space.

The problem in SR and GR could be that we are no longer dealing with a Hilbert with positive definite norm i.e. that we have to take into account p² - m² = 0 which is not an operator equation but a constraint on physical states.
(p² does not commute with x^a, but m² does)

 

[my_wan]

I don't understand. The uncertainty principle is simply a theorem derived from non-commuting operators in a Hilbert space.

Not sure what's missing, but I would refer to the original paper for most issues:
http://arxiv.org/abs/quant-ph/0102069
You can also read Reginatto on it here:
http://www.sbfisica.org.br/bjp/files/v35_476.pdf

You can read something about Heisenberg's thought experiment here:
http://en.wikipedia.org/wiki/Heisenberg's_microscope

This[/PLAIN] thought experiment, which began by describing both electrons and photons as though they were discrete entities with exact positions and momenta that could be known and measured, concluded that when all of the operational definitions pertinent to the experiment were completely drawn out it became clear that one could never expect to determine both an exact position and an exact momentum for any electron.

It wasn't originally derived from non-commuting operators in a Hilbert space, though that can certainly be done. Heisenberg considered it a heuristic statement with a quantitative description. It came from a fundamental limitation on measurement accuracy when a measurement entails probing with an effect that approaches the magnitude of the property being probed. In that sense, it is a classical measurement limitation induced by the physical requirement of interacting with the system being measured in order to measure it.

The principle was expanded beyond just a classical measurement limitation because the evolution of quantum systems required a moment to moment stochastic uncertainty, as well defined by the uncertainty principle, to properly describe the probabilistic evolution of the wavefunction. Virtual particle production being a prime example.

This is where the exact uncertainty relation I referenced comes in. M. Hall and M. Reginatto treated the uncertainty terms in the system evolution as random momentum fluctuations, roughly analogous to Brownian motion.

http://arxiv.org/abs/quant-ph/0102069"][/PLAIN] An exact uncertainty principle, formulated as the assumption that a classical ensemble is subject to random momentum fluctuations of a strength which is determined by and scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrödinger equation. Thus there is an exact formulation of the uncertainty principle which precisely captures the essence of what is "quantum" about quantum mechanics.

The problem in SR and GR could be that we are no longer dealing with a Hilbert with positive definite norm i.e. that we have to take into account p² - m² = 0 which is not an operator equation but a constraint on physical states.
(p² does not commute with x^a, but m² does)

Yes, your thinking here doesn't appear to be too far off from mine. I don't think it is explicitly related positive definite norms. It has been suggested that negative probabilities can be used in a wide variety of classical context:
http://www.dersoft.com/negativeprobabilities.pdf"
However, as a linear projection of properties from a space that does not have a linear mapping onto classical linear space, I think you have a point.