Instantons| Instantons |
In General > s.a. gravitational
instantons [including supergravity].
@ References: Jentschura & Zinn-Justin JPA(01)mp [higher-order
corrections];
Paradis et al PRA(05)qp/04 [and
tunneling].
In Gauge Theories > s.a. QCD
phenomenology; quantum particles.
* Idea: Instantons are
(anti)self-dual solutions of the Yang-Mills field equations in Euclideanized
spacetime, satisfying certain boundary conditions that physically guarantee
that they will have a
finite action, and they can formally be extended from R4 to
S4.
* History: Initially
called also pseudo-particles; They have become very useful in the theory of
4-manifolds.
* Boundary conditions:
We impose Aa 
g–1 
a g (pure
gauge) and Fab → 0
at infinity.
* Field equations: The
(anti)self-duality conditions guarantee that the field equations D*F =
0 are satisfied, since in this case they are equivalent to the Bianchi identities DF =
0; Recall also that these equations depend only on the conformal structure
of
the metric; But it can also be shown that (anti)self-dual
solutions are
global
minima
of the action for a given topological sector, characterized by the integer k = –C2(P), P being
the bundle over S4 to which the gauge field
belongs.
* Reason for Euclideanization:
The reason why Euclideanized spacetime is used (besides convenience for calculations)
is that in Lorentzian spacetime the (anti)self-duality
condition (which would read F = 
i
*F, instead of F =
*F) would
mean i 
=
, where is
the Lie algebra of the gauge group, and this requires that the Lie group be
non-compact, contrary
to what is usually assumed.
* Applications: They
are used in calculating tunneling probability amplitudes by the semiclassical
(WKB) formula 
exp(–I/
),
with I the
Euclidean action.
@ In Yang-Mills theory: Schwarz PLB(77),
Jackiw & Rebbi PLB(77)
[SU(2), degrees of freedom]; Ford et al NPB(01)ht/00 [ADHM
construction for SU(2) on torus]; Tyurin RMS(02)
[rev, mathematical]; Bonora et al JPA(03)ht/02 [2D,
stringy]; Forkel ht/04 [closed
FRW models]; Colladay & McDonald
JMP(04)
[and Lorentz violation]; Vandoren & van Nieuwenhuizen a0802-ln
[in super-Yang-Mills theory]; > s.a. self-dual
solutions.
@ In QCD: Shuryak NPB(82)
+ next two; Brown et al PRD(99)hp/98 [and
QCD vacuum]; Schäfer & Shuryak RMP(98);
't Hooft ht/99; > s.a. U(1) problem.
@ Related topics: Garland & Murray CMP(89)
[relation with monopoles]; Demetrian PS(06)
[Coleman-de Luccia, second-order].
References > s.a. 4D manifolds;
Quaternion; riemannian
geometry.
@ Books and reviews: Rajaraman 82; Coleman 85(77); Freed & Uhlenbeck
90; Dorey et al PRP(02)ht [multi-instanton
calculus]; Ritter mp/03;
Jardim m.DG/05-in
[topological aspects].
@ Original papers: Belavin et al PLB(75); Brown et al PRD(77).
@ Mathematical: Atiyah & Ward CMP(77); Atiyah et al PRS(78).
@ Instanton moduli space: Blau et al ht/01 [information
metric and AdS-cft].
@ As a tool in quantum field theory: Mueller-Kirsten et al JHEP(01)
[and WKB].
@ In curved spacetime: Gibbons [Taub-NUT]; Kim & Kim NCB(99)
[in de Sitter]; Etesi & Hausel JGP(01)ht/00 [abelian,
euclidean Schwarzschild].
@ Fuzzy / non-commutative: Balachandran & Vaidya IJMPA(01)ht/99;
Schwarz CMP(01)ht; > s.a. non-commutative
fields, spacetime foam.
@ Related topics: Lo et al PRD(79) [multi-instantons]; Landi & Marmo
PLB(88) [algebraic]; Shuryak NPB(88)
+ next three [ensemble]; Moch et al NPB(97)
[and deep inelastic scattering];
García Pérez et al PLB(00)
[size distribution]; Jardim JGP(04)
[Nahm transform].
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