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String Quantization

Now, I will show my idea how string quantization is possible, which I mentioned in the previous post (in Japanese).

What will be argued is strings can simply be quantized in any spatial dimensions by complying with the constraints and doing nothing else such as gauge fixing. This quantization in any dimensions (D≠26) has been desired for long time.

Let us consider an open string for simplicity. The string may be described by a variable "X^μ(σ)", where σ is the coordinate on the string. Let σ to range from 0 to π. From now on, please regard, for example, i in Ai to be a subscript, but superscript will be denoted by A^i. The vibration modes will be decomposed into Fourier components.
  X^μ(σ) = Σ_n cos nσ Xn^μ ,
likewise, the momenta may be decomposed as
  Π^μ(σ) = Σ_n cos nσ Πn^μ.
Applying Dirac's canonical formalism for the constrained system to Nambu-Goto action, we get constraints
  X'^μΠ^μ = 0,
  Π^μΠ^μ + X'^μX'^μ= 0
for any σ. The Hamiltonian is H=0 here.

By substituting the expansions, we get, as the first Fourier mode,
  X1・Π0 + ... = 0 .
This equation can be used to determine X1^0. And
  Π1・Π0 + 0×X'・X' + ... = 0
can be used to determine Π1^0. The second Fourier modes of the constraints are
  2 X2・Π0 + X1・Π1 + ... = 0 ,
  Π1・Π0 + 0×X'・X' + ... = 0 .
These conditions may be used to determine X2^0 and Π2^0. So, in general, the original constraints determine all the time components of the string variables. String theory suffered from existence negative norm states. Here the problem is solved.

Therefore, the quantum state space may be constructed only with the space variables Xn^i and Πn^i (i=1, ... , D-1; n=1,...). That is, the wavefunction Ψ is a function of X^1, ... , X^{D-1} but not of "X^0". Then δ/δX(σ) = Π(σ) since [Π^i, X^j] = -δ^{ij}.

Now the vacuum may be defined as α_n^i|0〉= 0 (n=1, ...), and other states may be produced by multiplying α_n^{i†}'s onto the vacuum. It was possible that the condition like (¥ref{2}) will lead to infinite regress but, in this state space, regress will not occur for any finite-particle states (or any finite-energy states).

For the zeroth mode, we have
  Π0・Π0 + Σ_n n αn†・αn = 0 .
This fixes the masses for the excitation modes since Π0^μ is total momentum of a string. Only in this zeroth mode, time component of X^μ is not determined.

In this way, quantization is made possible with the elimination of the negative norm modes.

Further, the interaction vertices may be constructed by writing the wavefunctions before and after a split and taking inner product between them.

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Update (June 7, 2010): The paper based on this idea was posted as
arxiv:1005.2726, Quantization of Nambu-Goto Action in Four Dimensions.

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