Bosonic String Theory

In General > s.a. string theory (including in non-commutative geometry).
* Nambu-Goto action: The geometrical one, equal to the area of the 2D world-sheet in d-dimensional spacetime,

S_NG[x^a] = −T ∫ |h|^1/2 d²y

(by analogy with the relativistic particle); Here, h_ij = η_ab x^a_,i x^b_,j is the metric induced on the world-sheet by the d-dimensional Minkowski metric η_ab, y ⁱ = (σ, τ) coordinates on the world-sheet, and T the string tension.
* Polyakov action: A gauge-fixed version; The energy functional for a harmonic map x^a: (²M, γ) → (^dM, η),

S_P[x^a] = −(T/2) ∫ γ^ij ∂_i x^a ∂_j x^b η_ab |γ|^1/2 d²y ,

which, on variation of γ, gives that γ is the metric induced on ²M by the embedding in d-dimensional Minkowski; This action actually replaces the area of the surface, which is quartic in ∂x, not quadratic,

S_P[x^a] = ∫ [(∂_i x^a ∂_j x^b η_ab) (∂_k x^c ∂_l x^d η_cd) ε^ik ε ^jl]^1/2 d²y .

* Relationships: These two actions are classically equivalent, although S_P is more convenient for calculations.
* Constraints: If σ is the spacelike parameter along the string world-sheet, x^a':= ∂x^a/∂σ, an overdot denotes ∂x^a/∂τ, and P_a:= δS / δ(∂x^a/∂τ),

C₁(σ):= P_a P ^a + x_a' x^a' = 0 ,   C₂(σ):= P_a x^a' = 0 .

* Hamiltonian: As in general relativity, it is a combination of constraints,

H = N₁(σ) C₁(σ) + N₂(σ) C₂(σ) .

* And physics: Not viable because of tachyons; Used as a simplified model, but does not give the standard model.
@ General references: Beig IJTP(91) [geometrical aspects of classical dynamics]; Kachkachi PLB(00) [Polyakov action]; Larrañaga JoT-gq/03 [intro]; Duff PLB(06)ht [Nambu-Goto action symmetries]; Sathiapalan IJMPA(08)-a0712 [gauge-invariant action]; Tseytlin a0808-ln [intro]; Banerjeee et al a2008 [action].
@ Hamiltonian formulation: Kuchař & Torre JMP(89), in(91) [diffeomorphisms]; Materassi ht/99, Montesinos & Vergara RMF(03)ht/01-in [Polyakov].
@ Loop variables: Sathiapalan IJMPA(00)ht, IJMPA(01)ht/00 [mode interaction], MPLA(02)ht, MPLA(04)ht, MPLA(05)ht/04, MPLA(05), MPLA(06)ht [and covariant].
@ Related topics: Lunev TMP(90); Jassal & Mukherjee IJP-ht/01 [propagator in curved spacetime]; Schreiber JHEP(04)mp [Pohlmeyer invariants].

Quantization > s.a. deformation quantization.
* Canonical quantization: One usually quantizes the Polyakov action using a Fock space representation; For bosonic strings, one finds that it is consistent only in d = 26 spacetime dimensions; However, there is an algebra of invariant charges which cannot be consistently quantized in any Fock space representation (D Bahns), indicating the need for a non-standard representation; Also, one doesn't really know how to restore gauge invariance like in gauge theory.
* Other approaches: In the covariant approach, all physical states have positive norm only if d = 26, and in the light-cone gauge approach, one recovers the lost Lorentz invariance at the end only if d = 26 (C Lovelace).
@ Canonical: Marnelius NPB(83), NPB(83) [Polyakov]; Handrich et al MPLA(02)mp; Bahns JMP(04) [algebra of diffeo-invariant charges]; Moncrief GRG(06) [ADM-type].
@ BRST: Hwang PRD(83); Kato & Ogawa NPB(83); Craps & Skenderis JHEP(05).
@ Covariant: Grassi et al CQG(03)ht-in [intro]; Nikolić EPJC(06)ht/05 [De Donder-Weyl covariant canonical formalism].
@ Methods: Mansfield AP(87) [comparison]; Berkovits ht/02-ln [super-Poincaré covariant]; Meusburger & Rehren CMP(03) [algebraic]; Bahns et al CMP(14)-a1204 [Nambu-Goto string effective theory, quantization in arbitrary dimension of the target space].
@ Non-perturbative: Kiritsis AIP(97)ht; D'Appollonio ht/01 (it); Motl PhD(01)ht; Thiemann CQG(06)ht/04 [lqg quantization]; Helling & Policastro ht/04, ht/06 [Fock vs lqg].

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