Re: GR and quantum mechanics

[Note: Please don't top-post. It interrupts the flow of the thread.]

On 2007-06-22, iuval <clejan@xxxxxxxxxxxxxx> wrote:
Igor Khavkine wrote:

The substitution of a string in place of a particle is motivated by the
removal of divergences present in theories of interacting relativistic
particles (a sector of interacting relativistic field theory). If you
are so inclined, you can think of the introduction of strings as an
alternative regulator (in the sense I described above). This motivation
is described in most (technical as well as popular) introductions to
string theory. To see how strings can be a substitute to field theory,
you should understand how field theory is equivalent to a theory with an
arbitrary number of particles (replace particles by strings and follow
analogous steps).

I used to understand (from a standard proof) how Fock space was
equivalent to the Hilbert space on which field operators operate, but
I don't recall it anymore, except that I have an alternative
understanding using the Schrodinger picture, in which one starts with
functionals of fields and ends up with some functionals being well
localized in spacetime-these are the particles. For free fields the
particles can also be eigenstates of energy and momentum (but then not
localized in spacetime-this is a case where the Schrodinger equation
is analytically soluble). But this understanding so far does not help
me with strings-should I think of functionals which are well localized
on strings as strings? Are particles forbidden in string theory?

The proof of this equivalence (which, for lack of a better name, I like
to call the fundamental theorem of second quantization) is not very
difficult. However, it does not lead to the picture you described, where
localized field configurations describe particles. In fact, localization
in field theory is a subtle topic and does not come to bear on the issue
at hand.

Here's the idea of the proof. Suppose you have a bunch of bosonic
particles. Then their Hilbert space of states is the space of
square-integrable wave functions of as many position variables, which
are symmetric under permutation of their arguments. For fermions, these
wave functions would be antisymmetric instead. If the number of
particles is not fixed, then the total space of states is the direct sum
of these Hilbert spaces for each fixed number of particles (including
zero particles). This total space is the so-called Fock space.

Let H_1 be the single particle subspace. It is special in the sense that
a wave function psi(x) from H_1 defines two operators on the Fock space,
that change the particle number. The creation operator psi+ adds a
particle:

(psi+ phi)(x,y,...,z) = [symmetrized over x,y,...,z] phi(x,y,...)psi(z)

The annihilation operator psi- destroys a particle:

(psi- phi)(x,y,...) = int dz phi(x,y,...,z)psi*(z),

where psi*(z) is the complex conjugate of psi(z). Also, psi- maps the
vector with no particles to 0. These two operators happen to satisfy the
canonical commutation relation [psi-,psi+] = 1 (maybe up to
normalization), which looks suspiciously like the commutation relation
of the aptly named creation-annihilation operators of a quantum harmonic
oscillator.

In fact, if you pick a basis psi_k for the single particle subspace H_1,
then, using these operators, you can express the Fock space as (roughly
speaking) a tensor product of individual simple harmonic oscillator
Hilbert spaces, one for each psi_k. But that is precisely what the
quantum state space of a quantized linear classical field theory looks
like. When quantizing a field theory, we want to find a set of modes
that decouple its Hamiltonian into a sum of independent harmonic
oscillators. The field modes are then more or less identified with a
basis psi_k for the single particle subspace, completing the
equivalence.

For fermions, one would have to replace symmetrization with
antisymmetrization, commutators with anticommutators, and introduce
Grassmann valued fields at the classical level.

Getting closer to the topic of string theory. Suppose that the particles
you are considering have discrete internal degrees of freedom. One
example is a particle with spin (a finite number of independent internal
states). Anothe is a hydrogen atom (an infinite number of independent
internal states, one for each excited stationary bound state). The
single particle space H_1 now has a basis psi_ki, where the index k runs
over a basis of square integrable wave functions, while the index i runs
over a basis of the internal state space. Running through the above
argument again, we find that the corresponding field theory has a field
with multiple component, one for each i. For example, the Fock space of
a particle with spin also corresponds to the quantization of a spinor
field theory. But, as evidenced by the example of the hydrogen atom, the
number of field components need not be finite.

If we take a string and separate its degrees of freedom into its center
of mass coordinate and into internal states corresponding to its
vibration modes, from the above point of view, a string is not much
different from a hydrogen atom. We can construct a Fock space describing
an arbitrary number of bosonic strings, which will correspond to some
field theory with an infinite number of field components. This field
theory is called "string field theory". So, when string theorists talk
about a whole mess of strings moving around and doing something, they
refer to the string Fock space of to string field theory.

When you think of the quantization of a single string, you shoul be
thinking of the single particle Hilbert space H_1, as described above
(actually, single *string* Hilbert space). This corresponds to string
quantization on a world *** that looks like a cylinder for closed
strings, or a ribbon for open strings. The discussion so far has
neglected interactions. Once interactions are included, it makes sense
to consider transition amplitudes between states with different numbers
of strings. In my understanding, these transition amplitudes, in
perturbation theory, are computed by quantizing the string on world
sheets of different topologies. Maybe someone can correct me if I'm off
the mark here.

There are analogous calculations regular particles, which can be done
from first principles or, in principle, obtained from the small diameter
limit of corresponding string calculations.

After all this discussion, a brief answer to your question is that you
can think of strings simply as particles with a very large number of
discrete internal degrees of freedom.

Hope this helps.

Igor

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