Re: String Theory, Gravitons, Spacetime (especially for Lubos Motl)

Absurdly elementary question.
Please remember my knowledge level is infinitesimal.

Lubos Motl <[EMAIL PROTECTED]> writes
>In classical physics, the "state" is determined as the information about
>all positions and velocities of all the particles (or objects) in the
>game. A possible state is "the Sun is at (0,0,0) and its speed is zero,
>and the Earth is (1 AU,0,0) and its speed is (0,30 km/s,0)".

Ok. That's *a* possible state. I presume the 'system' is a (massive) sun
and a stably circularly orbiting planet (of negligible mass) and you
chose to describe it 'totally' with a distance and a speed (albeit that
the two are correlated). So the space of all possible states will be
that of all possible stable orbits. Obviously one could extend this.

>In quantum mechanics, the situation changes because only the probabilities
>of the experiments can be predicted, not the actual outcome.  By the
>"state" in quantum physics we mean the wavefunction - the probability wave
>that determines where the particle(s) probably is (are). For example, when
>the electron orbits the proton in the Hydrogen atom, it can be found in
>different states such as the "1s" ground state (whose binding energy is
>-13.6 eV). The "states" are the boxes that you fill with electrons to
>build more complicated atoms. Each state of the Hydrogen atom corresponds
>to some wavefunction;  the wavefunction tells us what is the probability
>distribution - where the electron is more or less likely to be found -
>including the phase.

Ok, that's pretty well identical to the classical example.
Personally I prefer to see it as a wavelike electron occupying the
orbital with the probability assigning the amount of 'electron-ness' at
any point. But no matter.

>The states describing a single particle are functions "psi" of (x,y,z):
>for each point in space, we have one complex number psi. The probability
>to find the particle near (x,y,z) is proportional to |psi(x,y,z)|^2. These
>functions can be added and multiplied by a number, and therefore they form
>a (linear) vector space called the Hilbert space. 

OK, now my questions start (using your hydrogen atom example). 

The hilbert space is a space of *states*??

Now the space of momentum states is (I believe) discontinuous here, so
not all possible momenta will be included in this hilbert state??

I'm by no means convinced that 'the chance of finding an electron just
*here* is a valid question, since the only way to find out *precisely*
where it is, is to interact the atom with something else. This is a
*new* system. In this new system (that is, the atom and the probe) we
will have a whole bunch of 'effectively uninteractive' states, that is
where the atom and the probe can be considered 'separate';
and a whole bunch of 'interactive' states that can only be described by
some wavefunction that describes (simultaneously) the *coupled*
[atom+probe] system. Within the coupled system will be a bunch of states
where the electron is highly localisable (by the probe). This doesn't
tell us where the electron is in the undisturbed atom, but does tell us
where we will 'find it' in the coupled system.
Is this something like right?

Final question here. In any given hilbert space, all possible modes can
be described, or perhaps put another way, the hilbert space is a space
comprising a particular basis. There may be other bases that will also
fully describe the electron (for example momentum) would these comprise
another equivalent (is the tech word isomorphic?) hilbert space?


>A general sum of several
>vectors (wavefunctions) v1,v2,v3 multiplied by some numbers c1,c2,c3 -
>namely c1.v1 + c2.v2 + c3.v3 - is called the "linear combination" or
>"superposition" of the vectors v1,v2,v3. The Hilbert space has many
>dimensions (usually infinitely many), and each dimension corresponds to
>one state, one possible configuration of the system.

Ok, so immediately, even in this very simple system, we see a dichotomy
between GR and QM? At the simplest there will be some states whose
wavefunction is so distant that it will (should be) further away than
the atom's observable universe. Indeed, one could argue convincingly
that the atom's 'observable universe' only extends from when the atom
was created (say a few microseconds ago). That would imply that the
hilbert space must be finite, and further that it is continually
increasing as time goes on. Mind you this would appear to be rather easy
to fix.

>If you study one string using quantum theory, the Hilbert space contains
>the states that I described. A general vector in this Hilbert space - a
>general state - is a linear combination of some fundamental vectors
>(states) chosen from a set called the "basis". One element of a natural
>basis is the lowest energy ground state of the closed string. 

I have some problem visualising this string. I have been told it's
minutely small. This would imply it is highly energetic. I have a
problem making not-at-all-energetic particles from highly energetic
precursors.

>Another
>element contains 3 excitations moving as waves with 13 peaks to the right,
>and so on. 

Eh? Am I to see the closed (presumably 'circular') string as effectively
being 'plucked'?

>A "state" is a different way for the string to vibrate, and
>according to quantum mechanics, the "way" cannot change continuously
>because the energy carried by various types of vibrations can only jump
>discontinuously

Ok, that's not a problem. One imagines some sort of 'tension' in the
string so it can oscillate radially and transversely. That's a
heckuvalota possible modes....

> - this is why quantum theory is called "quantum".

Indeed.

-- 
Oz
This post is worth absolutely nothing and is probably fallacious.
DEMON address no longer in use.