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*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*| 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__: Like
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].

@ __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].

main page – abbreviations – journals – comments – other
sites – acknowledgements

send feedback and suggestions to bombelli at olemiss.edu – modified 4
may
2014