In this note we'll discuss classical bosonic strings, and briefly introduce the quantization procedure.
This note follows Superstring Theory by Edward Witten, and Introduction to String Theory and M-Theory by Michio Kaku.
Bosonic string actions
Before considering the string action, we first study the action of a relativistic point particle, it shares many properties with the bosonic string action.
Point particle action
The action is given by the length of the world line:
Notice that it has reparameterization invariance:
the action changes as
In infinitesimal form, the reparameterization is
This invariance is gauge invariance, so gauge fixing procedure is required, which is simple in classical case, but quit non-trivial in quantum theory.
Note that the cannonical momentum
leads to a vanishing Hamitonian
which is not only nonsense in classical theory, but also would result in serious problems in quantization. This is a result of gauge invariance, the four components of the momentum are not all physical, there are redundant degrees of freedom.
Dirac resolved this problem by treating as an constraint, and write the Lagrangian as
where is a Lagrangian multiplier. There's no kinetic terms in , so we can solve for gives
substitute back to Lagrangian gives
The equation for reads
solve for and substitute it into above, we recover the original Lagrangian. Note that this new Lagrangian still has reparameterization invariance:
Notice changes covariantly, so can be regarded as some sort of "metric" on the world line, and reparameterization is a change of coordinate, similiar to the diffeomorphism invariance of general relativity. Since has one gauge freedom, we can make a gauge choice by setting , the momentum and equation for becomes
the former is the familiar definition of 4-momentum, and the latter is the normalize condition of 4-velocity.
The Nambu-Goto action
Now consider string case. We use to parameterize the world sheet sweeped by the string, where is the timelike direction, and is the spacelike, ranging from to . Thus the spacetime position of a point on the string can be denoted by .
By anology from point particle, whose action is the length of the world line, the action of a string can be defined by the area of its world sheet. We can write the action, which is proportional to the world sheet area, as
where (we use capital Latin letters to number the world sheet coordinate) is the induced metric on the world sheet
This is the Nambu-Goto action. is a constant which we'll later see is the string tension.
Similiar to particle case, this action has reparameterization invariance
The canonical momentum
also leads to a vanishing Hamitonian
As in the particle case, we introduce by constraints. But now we need two Lagrange multipliers:
Solve the equation for , we get
substitute back to gives
If we write the Lagrangian in the form
then the matrix is
which satisfies . So we can introduce a "metric" on the world sheet by setting
the Nambu-Goto action then becomes
Again, this action also posesses reparametrization invariance
or in infinitesimal form
Since the action has gauge freedom, we can fix the gauge by setting
the action then becomes
Since the coordinate has finite range, the variation of the action would give a surface term
This term automatically vanishes in the closed string case, but for open string, we need to impose either the Neumann boundary condition
or the Dirichlet boundary condition
For now, we'll only consider the Neumann boundary condition.
The equation of motion is then
Introduce a new set of coordinate , so that , the above can be written as
so that the solution is
But in addition to the equation of motion, we still need to impose the equation of :
which is equivalent to the vanishing of the energy-momentum tensor
Using the fact that
the energy-momentum tensor is given by
Its components are
we can express the components of in coordinate as
So the constraint from becomes
It is easy to see that the string action has Poincaré invariance
so there are associated Neother currents, just like in field theory. Using similiar method, we can find the Neother currents associated with translation and rotation are
They can be interpreted as momentum and angular momentum respectively.
We now consider the solution of an open string rotating in the plane:
It is easy to verify this solution satisfies the equation of motion and constraints. The total energy and angular momentum are given by
we found that is proportional to
This relation is known as the Regge trajectory, is the Regge slope.
For closed strings, since is periodic in , we can write the solution as
where constant is given by
and are Fourier coefficients, requiring to be real gives
Solution for open string can be obtained by rewriting the closed string solution for , and setting
so that the Neumann boundary condition is automatically satisfied, and we also get
the open string solution is then
The constraint condition for closed string reads
where we have set . Similiarly for open string
We can now express the constraints in terms of the Fourier components (evaluated at )
for closed string. For open string, since it's not periodic in but in , we have
We begin by imposing the canonical commutation relation
is the canonical momentum. This commutator implies
while all others vanish. The operator thus can be regarded as mode ladder operators. We may denote the vacuum state with momentum by , satisfies
Excited states can be created by action the vacuum state by or .
The Hamitonian is then given by
for closed string, and
for open string.
Constraints and Virasoro algebra
Not all the states created by are physical, they must also satisfy constraints. If we directly regard the constraint as an operator relation, this would be too restricted. So, like the weak Lorentz condition imposed in the quantization of Maxwell field, we impose the constraint by
where and are physical states, these's a similiar constraint about for closed strings. Since , this condition is true if
But since and do not commute, the expression for should be considered carefully. This commuting ambiguity only arises for , so we define (similiarly for ) to be
and write the constraint for as , where is an undetermined constant. Since is proportional to the momentum operator, this constraint gives the mass-shell condition
for closed string, and
for open string.
The commutator between generates an important algebra called the Virasoro algebra.
No commutation ambiguity arise if , so that
this is the Witt algebra. The case should be carried out more carefully. We first introduce a regulator for :
When , the first line of the last step becomes , the divergent term is actually a result of normal ordering:
So we get
The commutator thus can be summarized as
This is the Virasoro algebra.
Consider the first exited open string state where is the polarization vector. Its norm is then given by
which might be negative, since is not positive definite. Negative norm states are called ghost, they're unfavourable in a sensible theory.
But under certain conditions, the constraints would exclude all the ghosts from the physical space. Consider again, mass-shell condition gives , while constraint of gives
This implies and , leaves only possible polarizations. We may let the polarization vectors to be othogonal to the plane so they're all spacelike, giving a positive norm. While the last polarization is in the plane, must not be spacelike otherwise the polarization is timelike and has a negative norm. So for the theory to be ghost-free, must satisfies
Under this condition, the ground state , whose mass-shell condition is , becomes a tachyon state.
We'll prove in later notes that the condition of no-ghost is and or and .