Bosonic
String Theory| 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,
SNG[xa]
= –T 
|h|1/2 dy
(by analogy with the relativistic particle); Here, hij = 
ab xa,i xb,j is
the metric induced on the world-sheet by the d-dimensional
Minkowski metric ab,
y i = (
,
)
coordinates on the world-sheet,
and T the string tension.
* Polyakov action: A
gauge-fixed version; The energy functional for a
harmonic map xa: (2M,
) →
(dM,),
SP[xa]
= –(T/2)
ij
i xa j xb ab || d2y ,
which, on variation of ,
gives that 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,
SP[xa] = [(i xa j xb ab)
(k xcl xd cd)
ik jl]1/2 d2y .
* Relationships: These two
actions are classically equivalent, although SP is more convenient
for calculations.
* Constraints: If is
the spacelike parameter along the string world-sheet, xa':=
xa/,
an overdot denotes xa/,
and Pa:=
S/(xa/),
C1():=
Pa P a +
xa' xa' = 0
, C2():=
Pa xa' = 0
.
* Hamiltonian: Like in general relativity, it is a combination of constraints,
H = N1()
C1()
+ N2()
C2()
.
* 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 gq/03 [intro];
Duff PLB(06)ht [Nambu-Goto
action symmetries]; Sathiapalan a0712 [gauge-invariant action].
@ Hamiltonian formulation: Kuchar & 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 ht/01/IJP
[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]; Nikolic 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].
@ Non-perturbative: Kiritsis ht/97-ln;
D'Appollonio ht/01 (it);
Motl ht/01-PhD;
Thiemann CQG(06)ht/04 [lqg
quantization]; Helling & Policastro ht/04,
ht/06 [Fock
vs lqg].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
30 may 2008