On Sun, 23 Nov 2003, Paul wrote:

I could see where one of the string vibrational patterns predicts
the graviton, but I did not see how the theory demonstrates how graviton's
do what they are supposed to do, warp spacetime. Does string theory have an
answer to this particular question, and if so, could someone try to explain
it in a layman's level as Greene does so well in his book for other topics?

Dear Paul, your question is very good and very important: it is obviously
not enough to have the right particle that has the appropriate mass (zero)
and the correct spin (two) - it must also interact in the correct way.
Yes, using the mathematical machinery it is pretty easy to explain that it
is indeed the case of the stringy gravitons.

Unfortunately, the answer (see, for example, chapter 3 of Polchinski's
"String Theory" or chapter 3 of Green+Schwarz+Witten's "Superstring
theory") is much more technical, and this is probably the reason why Brian
Greene has not explained it himself. Second, I may not be as good as Brian
in explaining these concepts so that the readers of sci.physics.relativity
will understand. These two observations might imply that I will be less
successful than Brian. ;-) Let me try anyway.

String theory starts with one-dimensional objects that generate
two-dimensional histories in spacetime (an extra dimension is the time) -
these histories are called the worldsheets (a generalization of the
worldline, which is the path of a point-like particle in spacetime). A
particular history is described by a specific embedding of this worldsheet
into a pre-existing spacetime. This embedding is described by the
functions X_i(sigma,tau) where sigma,tau are two coordinates on the
worldsheet (sigma is space-like and tau is time-like), and X_i - which is
the vector (X_0, X_1 ... X_{d-1}) - tells us where a given point of the
worldsheet was located in spacetime. "d" is the spacetime dimension.

Because the gravitons are a specific vibrational pattern of a *closed*
string - a string with no ends that looks like a wiggly circle - I will
have to explain some details about the closed strings and their
vibrations.

[The open strings are different: their ends are glued to an
object called the D-brane (or Dirichlet brane, because of the Dirichlet
boundary conditions on the string's endpoints) and therefore the open
strings can't be responsible for geometry of spacetime - they are only
responsible for the dynamical properties (like shape or the internal
magnetic fields) of the D-branes onto which they are attached. Gravity
*must* always come from closed strings because it is a consequence of
the geometry of the *whole* spacetime.]

Every string has variables such as X_i(sigma) that describe where the
given point of the string is located in spacetime: the variable "sigma" is
a coordinate along the string, and the index "i" goes between 0 and 9 (for
a 10-dimensional theory such as the superstring theories), and therefore
there are 10 fields defined at each point of the string.

For example, X_3(0.01) is the number that tells us where is the point of
the string sigma=0.01 located in the X_3 direction. For closed strings,
the variable sigma is periodic, and therefore the fields X_i(sigma) must
satisfy the periodicity condition, X_i(sigma+L)=X_i(sigma) where "L" is
the periodicity (the length of the worldsheet spatial coordinate) that
can be taken to be pi or 2.pi by a simple rescaling of sigma.

These fields X_i(sigma) behave like other fields - for example the
electromagnetic fields, although they are defined on a string and not in
spacetime. For example, there are "waves" that can propagate around the
string. Because the string has only two directions (left and right), such
a wave can only move to the left, or to the right. We usually talk about
the "left-moving modes" and the "right-moving modes". These two groups are
more or less decoupled from one another.

Because the dependence of the wave on sigma is something like sin(k.sigma)
where "k" is a wavenumber, and because these waves must be periodic\0
functions of sigma, we see that the wavenumber "k" must be integer. (I
chose the periodicity L of the sigma coordinate to be L=2.pi.) This is
even true for classical waves - "k" is the number of peaks that are
distributed around the string.

Moreover, we want to describe a *quantum* string, and therefore the total
energy carried by any of these waves must be quantized (recall Planck's
E=N.hf): it must essentially be an integer, a generalization of the
"number of photons" in a given state. The waves in quantum theory are made
of quanta (such as photons in the case of electromagnetism).

Therefore if we want to describe the specific vibrational state of a
string, we must say how many quanta of X_3 is moving as the wave with
momentum k=7 to the right, and we must determine all these numbers, not
only for X_3, and not only for k=7.

So let us classify these states - the possible vibrations of a closed
string in quantum theory. The minimal possible vibration of a string is
the state |0 - the "ground state". According to quantum mechanics - more
precisely the uncertainty principle - it can't happen that the string does
not vibrate at all. If the position of the string were exactly
well-defined, and the velocity would be exactly zero, we would violate the
Heisenberg's insight that the velocity and the position can't be
determined simultaneously.

Nevertheless, there is a state of a closed string that vibrates as little
as possible (it has the minimal energy) - all the "modes" that are
described as harmonic oscillators choose to sit in the ground state. We
usually denote this state |0. Eventually, every possible vibrational
state of the closed string - that is determined by listing all the numbers
of quanta in each possible state - will give us a different type of
particle. A closed string looks like a particle, but precisely which
particle it is depends on the vibrations that we excited (how many
"photons" there are with different momenta etc.). The total energy of the
string's oscillations determines the mass of the resulting particle. You
find out that the masses of the resulting particles are - in this crudest
approximation - quantized - it is the total number of "quanta" moving
around the string (weighted by the wavenumber "k" of each).

The ground state |0 that contains "no" vibrations behaves as a particle
in spacetime called the *tachyon*. (Now I switched to bosonic string
theory that exists in d=26 spacetime dimensions so that I can avoid some
subtleties involving the fermionic, fields on the worldsheet that are
different from X_i.) It turns out that the total energy carried by the
minimal quantum fluctuations is *negative*, and therefore the mass squared
of the tachyon is negative. (The value is naively proportional to
1+2+3+4+5+.... which is infinite, but can be regulated and the answer is
-1/12; I am not kidding, it is minus one twelfth.) In special relativity
we have the "dispersion relation"

m0^2 c^4 = E^2 - p^2.c^2

for the rest mass m0, the energy E, and the momentum "p". If m0^2 is
negative, we see that p^2.c^2 (the momentum squared) will be bigger than
E^2, and therefore the particle should naively move faster than light.

(This violates the rules of causality in special relativity, and it is one
of the reasons why we don't take the 26-dimensional bosonic string theory
seriously. If we move to 10 dimensions, this tachyon disappears. The
tachyon is more usually described as an instability of spacetime because
we can create as many pairs of tachyons with positive+negative energy as
many we want.)

So the ground state |0 is a tachyon (negative mass-squared), which is a
scalar field (the word "scalar" means that it has no indices, i.e. it has
spin zero). What about the higher excitations? We might try to add *one*
photon with the minimal possible energy. Let's add one quantum of the wave
that makes X_i(sigma) oscillating around the string so that only one
peak appears on the string and this whole wave is moving to the right.

This state is usually written as

alpha^i_{-1} |0

Here "i" is an upper index, not a power, that determines the transverse
direction in which the string oscillates. "-1" is a conventional way to
remember that the wave has +1 peak on the string. ;-)

It turns out that the total momentum of the waves moving around the string
must be zero (because the theory describing the worldsheet is a sort of
two-dimensional gravitational theory, and the total momentum must be zero
because otherwise the net gravitational field has nowhere to go on the
closed, compact string). It is a subtle fact that disallows the previous
state, a state with one "photon". The minimal energy state that comes
*after* the tachyonic state |0 must contain one left-moving and one
right-moving wave with momentum one:

alpha^i_{-1} alpha~^j_{-1} |0 (###)

The tilded and untilded alpha's denote the left-moving and the
right-moving waves (one alpha is one quantum of such a wave),
respectively. Note that we could have used two different indices i,j -
different fields X^i and X^j can oscillate in these two "photons". In
other words, the left-moving and the right-moving excitation can oscillate
in different spacetime directions.

Alpha_{-1} is a kind of creation operator, and a string is a collection of
infinitely many quantum harmonic oscillators - for those who know what a
creation operator is.

Note that the previous state (###) has two indices, much like the metric -
two indices is essentially the reason why the gravitons will have spin
two. We're not there yet. Only the ij-symmetric part is the graviton; the
antisymmetric part is called the Kalb-Ramond B-field (not to be confused
with the ordinary magnetic field), a higher-index generalization of the
electromagnetic 4-potential, under which the 1-dimensional strings are
charged.

A detailed stringy calculation shows that the vibrational state (###) of
the closed string, that contains one minimal left-moving wave and
one-minimal right-moving wave, has mass equal to zero. (In superstring
theory in 10 dimensions, the ground state tachyon |0 disappears but the
graviton survives, but let us continue to talk about bosonic string theory
in 26 dimensions because it is a fine toy-model.)

Having the correct spin and the right mass, (###) is a candidate for a
graviton. But does it really interact as a graviton? We're back to the
original question, but I hope that we are better prepared to understand
the answer.

The answer is yes, and we will see it (at least roughly) using the
so-called state-operator correspondence. It is a special dictionary in the
theory that describes stringy worldsheet. This theory is called conformal
field theory which means that the physical properties only depend on the
angles. We can use different coordinates, much like in general relativity,
but we can also multiply all distances (i.e. the whole metric tenor) at a
given point by a constant, without affecting physics. Only angles on the
worldsheet can be measured - not the proper distances. This independence
on the distances is called the "conformal symmetry", and it is very
important for (perturbative) string theory.

What is the dictionary? Imagine a closed string worldsheet. It essentially
looks like an infinitely long cylinder. (It can also look like pants if
the string splits, but let us first consider non-interacting strings.) If
you cut this infinitely long cylinder somewhere, you get a circle - this
is how the closed string looks at time T. A funny thing is that this
collection of circles - the infinite cylinder that is sliced into equally
thick finite cylinders - has the same angles like the infinite plane that
we draw using *concentric* circles, as long as the radii of these
concentric circles increase exponentially, r=exp(T) where T is the time,
the infinite coordinates along the infinite cylinder.

These concentric circles become infinitely dense near the origin of the
coordinate system, r=0. This is where the points from the infinite time
(T=-infinity) of the infinite cylinder are located. The plane must still
be able to remember which stringy vibration was there at T=-infinity. And
it *does* remember it because we must insert a specific operator at r=0.

Which operator?

It turns out that if the closed string is in the ground state, we just
insert the operator "1" as a multiplicative factor to the expressions for
calculating the physical observables. Multiplying by "1" does not change
anything, and therefore we essentially inserted nothing.

If we want to say that the graviton state (###) was coming from r=0
(T=-infinity), the corresponding operator happens to be

partial_z (X_i) partial_{zBAR} (X_j)

This operator is a bilinear expression made of the worldsheet derivatives
of X_i and X_j. The derivatives are taken with respect to "z" and "zBAR"
which is essentially "sigma+i.tau" and "sigma-i.tau".

One can prove that there is a one-to-one correspondence between the
possible vibrational states of the string, and the possible local
operators. Well, it is not hard. The vibrations alpha^i_{-m} are replaced
by the m-th derivative of X^i with respect to "z", and similarly for
"alpha~" (tilded) and "zBAR". The energies of the vibrational states of
the closed strings are mapped to the so-called "dimensions" of the
operators (the dimension is essentially the power of mass that this
operator has according to the dimensional analysis), and almost every fact
about the states can be translated to a statement about the operators and
vice versa.

Let us now calculate some process, for example the probability that a
comet comes close to the Sun and will leave it in a different direction
PHI. (In classical physics, we don't need the probabilities, but in
quantum physics - and we want to study string theory as a quantum theory -
we do. In a correct quantum theory, the probability distribution in the
classical limit becomes concentrated around the correct classical answer.)

In string theory, such probabilities can be calculated as the (Feynman's)
path integral - the sum over all histories of the string and all possible
ways how it could have vibrated, joined and get split - and we insert the
"vertex operator" of the initial comet and the initial Sun with the right
momentum (or position) as multiplicative factors to the path integral;
and the same things with the final states whose probability we want to
know. The vertex operator is the local operator that corresponds to the
vibrational state of a string by the state-operator correspondence
described above.

The funny thing is that we saw that the vertex operator of the graviton was

partial_z (X_i) partial_{zBAR} (X_j) exp(i.k.X(z)).

I added exp(i.k.X(z)) which is a way how the state-operator correspondence
remembers the d-momentum "k" of the graviton. This operator looks much
like the two-dimensional worldsheet action that is essential to define the
path integral: it is also bilinear in the derivatives of the fields "X"
with respect to the worldsheet.

In fact, this fact can be made precise: the derivative of the path
integral - mainly the two-dimensional action - with respect to the
*background metric* equals the original path integral (without
any derivatives) *with* the graviton vertex operator inserted.

Because the previous paragraph sounds too technical, let us use a more
friendly language: changing the geometry of the original spacetime - so
that we want the strings to propagate on a slightly curved background - is
fully equivalent to keeping the metric unchanged, and adding an extra
external string in the vibrational state that we call the graviton.

This fact guarantees that the physical influences of this vibrational
state on other strings (and other objects in general) will be
indistinguishable from curving the metric of the spacetime. If a comet
moves in a background that is filled with a "condensate" of strings that
vibrate in the gravitonic vibrational pattern, the comet will do
everything like a comet that moves in a curved space.

Therefore, the stringy vibrations called the gravitons affect the matter
exactly in the same way as a gravitational field - i.e. as a curved
spacetime geometry. It can essentially be reduced to the fact that the
operators associated with the gravitons look like the derivative of the
two-dimensional action with respect to the spacetime metric.

Therefore if we want to describe spacetime physics of string theory using
the effective action, we know that the "number of strings in the graviton
mode" will behave as the field h_{ij} that measures the deviation of the
spacetime metric from the original reference metric that we picked:

gFULL_{ij} = eta_{ij} + h_{ij}

Therefore this action will simply be an expansion of the Einstein-Hilbert
action around a chosen solution eta_{ij}. All the nonlinear terms (recall
the nontrivial definition of the Riemann tensor) are also guaranteed to
come out correctly, and they do. We can also derive that it is only
consistent to define the propagation of strings on a spacetime that
satisfies the correct (Einstein's or generalized) equations.

It really does not matter that we started with eta_{ij} - I was thinking
about the flat spacetime metric eta_{ij}. We could have started with
another metric, too. The physical results would be identical because
changing the initial metric can be exactly mimicked by condensing the
closed strings that vibrate in the gravitational mode on another geometry.

Once we can derive the effective spacetime action and once we realize that
it is just the expansion of the full Einstein-Hilbert action (plus other
terms corresponding to other fields and interactions that string theory
predicts) around a specific classical solution (this solution includes the
background metric), we can guarantee that not only the graviton-moded
strings will affect other objects like regular gravity - which means just
like spacetime curvature - but they will be also influenced by other
matter according to the Einstein's equations (the matter will curve the
geometry properly).

The graviton is a small perturbation of the spacetime geometry - all of
its effects coincide with the changed spacetime geometry. Once we talk
about the gravitons, we *must* adopt this perturbative point of view.
However, an important fact is that all nonlinear terms in the Einstein's
equations etc. are guaranteed to come out correctly. It is possible to
prove that the propagation of strings can only be defined on the
backgrounds that satisfy the right Einstein's equations. (Wrong
backgrounds imply "conformal anomaly", a quantum violation of the
conformal symmetry - the independence of the scaling factor of the
worldsheet coordinate, and this conformal anomaly is proportional to
the difference "LHS - RHS" of the Einstein's equations.)

By the state-operator correspondence, this requirement that the background
satisfies the right equations guarantees that the higher-order
interactions of the gravitons will follow Einstein's prescription
(including all nonlinear terms) as well. (This is true at long distances;
at distances comparable to the typical size of the string, string theory
predicts important corrections that - however - are too small to be
measured directly with the technology of 2003.) Computing the nonlinear,
many-graviton corrections to the anomaly is related to computing the
scattering of many gravitons - something that even classical general
relativity predicts to some extent.

There are many other backgrounds and descriptions of string/M-theory, and
we can show that the canonical Einstein's picture of gravity appears in
all of them. For example, the AdS/CFT correspondence says that a
d-dimensional field theory on the boundary of the anti de Sitter space is
equivalent to the full gravitational (string) theory on a d+1-dimensional
background. The appearance of 5-dimensional gravity (=Einstein's rules of
curved geometry) from a 4-dimensional gauge theory is highly nontrivial
and the explanation is very different from the explanation why a specific
stringy mode interacts as a graviton, but at any rate, it is true that in
any description of string/M-theory we are always guaranteed that gravity,
as described by Einstein, is a part of the game. We can show that all
these nontrivial ways how gravity can appear are related - they are
equivalent in a broader sense.

Note that this massless spin-two particle was one of the reasons why
people hated string theory in 1973 or so. It just was not seen in nuclear
physics. A closer inspection shows that by conformal invariance and the
state-operator correspondence, inserting such a particle (a vibrating
string) is equivalent to curving the spacetime, and therefore the exchange
of these virtual strings will be indistinguishable from the gravitational
force predicted by Einstein.

It is not true anymore that this specific vibration of a string is the
only way how can we obtain Einsteinian gravity in a quantum theory, but it
is still true that all the known different ways to achieve this goal (and
sometimes they are very different) are still connected to one theory, and
we continue to call this theory "string theory" even though it contains
much more stuff beyond strings. It is the theory formerly known as strings
(as Michael Duff liked to say), but in practice, it is the theory that
contains - by definition - all good ideas about
quantum-gravity-and-high-energy-related physics (as Joe Polchinski would
say).

Best wishes
Lubos
__________________________________________________ ____________________________
E-mail: fax: +1-617/496-0110 Web: http://lumo.matfyz.cz/
phone: work: +1-617/496-8199 home: +1-617/868-4487
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Superstring/M-theory is the language in which God wrote the world.

Lubos gave an execellant post but Paul or other people might
understand it better after reading my articles and papers on my
website, such as my paper Beyond the Standard Model.

http://www.geocities.com/jefferywinkler

Jeffery Winkler

Lubos Motl wrote:
It really does not matter that we started with eta_{ij} - I was thinking
about the flat spacetime metric eta_{ij}. We could have started with
another metric, too. The physical results would be identical because
changing the initial metric can be exactly mimicked by condensing the
closed strings that vibrate in the gravitational mode on another geometry.

Interesting post. Now I have not gone in any real detail into string theory
(as an initial attempt I purchased Kaku's book but did not like it much - a
bit too handwavy and disjointed for my liking) but my concern with the above
comment has to do with QFT. AFAIKS string theory still uses the same old
QFT as espoused by Weinberg is his book Quantum Theory of Fields (a book I
have studied in detail, really like, and fully agree with). Under this view
QFT is the way it is because it is the only way of reconciling SR and QM.
This would seem to rule out backgrounds other than flat space time if you
use QFT. Also AFAICS this is not really a problem because spin 2 particles
coupled to mass/energy inevitably leads to curved space-time - only the
interpretation is slightly different - the spin 2 particles affect rulers
and clocks in such a way that they behave as if space time is curved - with
no observational difference. However fundamentally we have space time as
flat. Hence I can not see any advantage in even trying to do what your
suggesting - and considering Weinberg's view - it seems to have fundamental
problems.

Now I know people say why should nature single out a particular space time -
to which I reply why not? - especially since flat space time has such nice
symmetry properties. To me each approach is just as elegant - in one we
have this nice symmetrical simple flat space time in the other approach we
say no space time is better or worse than any other.

Thanks
Bill