The String Coffee Table

Unlooping the LQG string

Posted by Robert H.

Yesterday, we had Thomas Thiemann here at DAMTP for a seminar on his quantization of the bosonic string. Although I did not expect too much, having him explain his paper for nearly two hours on the blackboard (and a real coffee table discussion afterwards) I think I understand now what is going on and why he doesn’t get a central charge.

As a first step let me repeat what he is doing stripping off some of the not so essential parts of his story and using symbols that are closer to the ones more familiar for us. Curiously, in the end, you can get along without mentioning the Pohlmeyer charges at all, so this is not where the disagreement with the usual story is rooted.

For simplicity, let us consider a one-dimensional target space with coordinate $X$. Then, the good objects are the currents $j=\partial X$ and the current in the other chiral half. This is what Thomas calls $Y_\pm$, where $\pm$ indicates left- or right-movers. But as always, it is sufficient to restrict attention to a single chiral half.

The next step is to pick some complete set of functions $(f_n)$ on the circle such that any function on the circle such that any function can be expressed as linear combination of $f_n$’s. Now we can define $j_n = \int f_n j(x) dx$. The usual choice is to take the indices $n$ to be integers and and $f_n=e^{i n x}$. With this choice what I called $j_n$ are actually the $a_n$’s.

Thomas makes a different choice but that shouldn’t affect the physics. He takes the lables to be finite unions of intervals on the circle and the $f_n$ to be characteristic functions of the intervals (that is the functions that are 1 on the interval and 0 outside the interval).

Next we have to work out how the diffeomorphisms of the circle act on the $j_n$’s. We are used to ask this question on the infinitesimal level and use the generators $L_n= -z^{n+1} \partial$ that have the usual simple action on $e^{inx}$ that is expressed in

Thomas prefers to work with finite diffeomorphisms (rather than infinitesimal ones) but again, this is not essential. Again the action is sufficiently simple, the diffeomorphism just moves the intervals around pointwise.

So far everything was classical. Now we promote the $j_n$ to operators. We can be careful and prefer to quantize $e^{i j_n}$ rather than $j_n$ directly because those will be bounded operators. This is similar to quantum mechanics where careful people use Weyl operators $U=e^{i x}$ and $V=e^{i p}$ with commutation relations $U V=e^{i\hbar}V U$ as those are unitary operators while $p$ and $x$ are unbounded. This gives us the “kinematical algebra”.

The next step is the crucial one: One has to chose a Hilbert space for these operators to act on. The standard choice is to take the highest weight representation or Fock space by defining the Hilbert space to be generated by the $a_n$’s with negative $n$ from the vacuum that is annihilated by the $a_n$ with positive $n$.

This step is where Thomas’ construction is different from the usual one. He takes a different Hilbert space. He defines it using the GNS construction (a well established procedure in mathematical physics). This works by selecting a “state” $\omega$ on the algebra (as a reminder, a state is a positive function that assigns to each operator a number, that should be thought of the expectation value of that operator in that state) and defines that as the vacuum on which the Hilbert space is build by acting with all the operators.

In fact, our Fock space can be constructed in that way as well. Just define $\omega(a_n)=p\delta_{n,0}$, just the vacuum expectation value of $a_n$. If you want Weyl operators you have to work a bit more (using the CBH formula) but it can be done. Now, it is essential that the state is invariant under diffeomorphism (or the Virasoro algebra) to obtain a nice, well defined theory. And this is where the central charge comes in. Our Fock space is not invariant, it transforms with a phase given by the exponential of $i$ times the central charge. This is the analogue of the calculation Urs did for us in the old thread. You have to be in the critical dimension and include the reparametrization ghosts to make the Fock vacuum invariant.

But Thomas picks a different vacuum. Remember, his $j_n$ were indexed by intervals and he defines $\omega(e^{i j_n})$ to be 1 if $n$ as an interval is empty and 0 otherwise. This is obviously invariant under diffeomorphisms that move the intervals around. But it is legitimate and you obtain a different Hilbert space (that is is not separable because the orthonormal system is labelled by intervals rather than integers but this is only a consequence but not essential).

So is he allowed to do this? Why doesn’t this happen in ordinary quantum mechanics? Well, it happens as well (this example is also due to Thomas): There you usually take the Hilbert space $L_2(R)$ on which $e^{i a x}$ as a multiplication operator and $e^{i b p}$ by translations by $b$. This is the coordinate representation. There are other choices, for example the momentum representation or the oscillator representation, but they are related by a change of basis (a unitary equivalence in techspeak) so they are not really different.

In fact, there is the Stone-von-Neumann theorem, that tells you that up to unitary equivalence this is your only choice. So you usually don’t think about this. However, there is a technical assumption in this theorem, namely that the representation should be weakly continuous in $a$ and $b$ meaning that any matrix elements of $e^{i a x}$ and $e^{i b p}$ are continuous in $a$ and $b$. If you drop that assumption there is another possibility: As your Hilbert space you take what is generated by all $T_s=e^{i s}$ for real $s$ and $U$ and $V$ act the same way as before but you take a different scalar product: You define $\langle T_s,T_{s'}\rangle = \delta(s-s')$. Again, this Hilbert space is not separable, the ON system is labelled by continuous $s$ rather than by integers, so it cannot be unitary equivalent to the standard Hilbert space and in fact the representation is not continuous: The expectation value (a special matrix element) for the translation by $a$ is $\delta(a)$ which is not continuous in $a$.

Obviously, in this Hilbert space it’s also a bad idea to take the derivative with respect to $a$, so $p$, the infinitesimal operator of translations, is not well defined. So usually this pathological Hilbert space realization is ruled out by the assumption of weak continuity but Thomas has promised to give me a reference to a paper that discusses the relevance of this pathological Hilbert space in some condensed matter system.

Going back to the string case, we can see that Thomas’ vacuum (and thus Hilbert space) is similar in spirit to this pathological Hilbert space in this quantum mechanical example. Again, it is not weakly continuous and the scalar product that is induced by his state has the same delta function characteristic. I understand he agrees that if one requires continuity one would probably be left only with the usual Fock space representation that gives rise to the critical dimension.

He is not saying that the rest of the world is doing something wrong but only that there is another, pathological possibility to quantize the string if one uses weak enough rules for the game called quantization.

There are two straight forward calculations that one might do if interested: The first is to work out what $\omega$ really is in the Fock space state and check that the diffeomorphism group is really only represented projectively (that is with the phase from the central charge) thus establishing that as one would expect the usually construction can also be rewritten in this GNS formalism. This should be an easy calculation that is basically the exponentiated version of Urs’ calculation and I have no doubts that it will work out in the end.

The other calculation is in fact a bit more challenging: The algebra of the $a_n$’s is obviously an infinite copy of Heisenberg algebras (the algebra of $p$’s and $x$’s). One could try to use the pathological quantum mechanical Hilbert space for each oscillator and check what one would get. My bet would be that it yields something closely related to Thomas’ Hilbert space representation.

Finally note that the Pohlmeyer charges didn’t play a role: All really needed was the trivial observation that Thomas’ state was invariant under diffeomorphisms and from that on we could use the $j_n$ fields that by themselves are not diffeomorphism invariant.

Remains one question: Having now established that this construction is not really in conflict with what we usually do in string theory (in fact adding a slight additional technical assumption probably rules out the pathological state and brings us back to our beloved Fock space construction) what does it teach us about loop quantum gravity. There similar tricks are played: From what I know, first the kinematical algebra is constructed (basically the parallel transporters along edges and their momenta, the “electric” fields, please correct me if I’m wrong) and then a diffeomorphism invariant state is constructed on which the Hilbert space is build. This state is the Ashtekar-Lewandowsky measure (or Abhay-Jurek measure for friends) that is also very singular similar to Thomas’ string vacuum. I vaguely remember somebody making the statement that this is the only choice there.

So maybe in the end, there is nevertheless some fruitful interplay between strings and LQG: It seems that in the critical dimension and including ghosts the usually Fock vacuum is in fact another, much better behaved, diffeomorphism invariant state, at least in 1+1 dimensions!