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}
d^{2}*y*

(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}:
(^{2}*M*, *γ*) →
(^{d}*M*, *η*),

*S*_{P}[*x*^{a}]
= −(*T*/2) ∫ *γ*^{ij}
∂_{i} *x*^{a}
∂_{j} *x*^{b}
*η*_{ab} |*γ*|^{1/2}
d^{2}*y* ,

which, on variation of *γ*, gives that *γ* is
the metric induced on ^{2}*M* 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}
d^{2}*y* .

* __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*),

*C*_{1}(*σ*):=
*P*_{a} *P*^{ a} +
*x*_{a}' *x*^{a}'
= 0 , *C*_{2}(*σ*):=
*P*_{a}
*x*^{a}' = 0 .

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

*H* = *N*_{1}(*σ*)
*C*_{1}(*σ*)
+ *N*_{2}(*σ*)
*C*_{2}(*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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