Scalar Field Theories |

**In General**
> s.a. Kemmer Equation; laplacian operator.

* __Examples__: Dilatons in string
theory; Nambu-Goldstone bosons; Higgs fields; Supersymmetric partners of spin-1/2
particles; Scalar component of gravity; Cosmologically motivated fields such as
quintessence (> s.a. cosmological
models).

* __Field equations__: They are
often taken to satisfy the Klein-Gordon equation, but a more general field
equation with self-interaction is

and they can be described by the Kemmer equation.

* __Scalar components of gravity__:
Scalar fields may couple to gravity in such a way that they give rise to an effective
metric that depends on both the true spacetime metric and on the scalar field and its
derivatives; Such fields can be classified as conformal and disformal, where the
disformal ones introduce gradient couplings between scalar fields and the energy
momentum tensor of other matter fields.

**Types of Scalar Fields** > s.a. parametrized theories.

* __Massless__: In quantum gravity
they cannot exist as elementary particles, because they would acquire a mass
from interactions with topological fluctuations.

* __1+1 dimensions__: There can be
no massless scalar particle even without quantum gravity, because there
could be arbitrarily long wavelength fluctuations, with an arbitrarily small
energy cost – the energy cost, for fixed amplitude, would decrease as the
size of the region increases; This does not happen in higher space dimensions
because the volume integral grows faster with distance; What can exist in 1+1
dimensions is a theory of the gradient, *B*:= ∇*φ*.

@ __Types__: Anco & Wald PRD(89) [Lie algebra-valued];
Unruh & Weiss PRD(89) [massless].

@ __Massive__: Helfer JMP(93) [and null infinity];
Garavaglia ht/01-conf [Green function].

@ __Massless__: Frasca MPLA(09) [mapping to Yang-Mills theory].

@ __Twisted fields__: Isham PRS(78);
Banach & Dowker JPA(79),
JPA(79).

**Specific Theories** > s.a. black holes;
Boson Stars; klein-gordon fields.

* __ λφ^4 theory__:
In 1970 Kurt Symanzik proposed a 'precarious' \(\lambda\phi^4\) theory with a
negative quartic coupling constant as a valid candidate for an asymptotically free
theory of strong interactions; With positive \(\lambda\), the potential is always
positive, therefore it gives rise only to repulsive forces, and we can consider
the theory not to have any bound states; The quantum theory is trivial.

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**References** > s.a. green functions;
thermodynamic systems [thermodynamic quantities and speed of sound].

@ __Overview__: Brans gq/97-fs [in gravity];
Kleinert & Schulte-Frohlinde 01 [\(\lambda\phi^4\)];
in Franklin 10 [IIb].

@ __In a curved background__: Bernardini & Bertolami AP(13)-a1212 [cosmological background, Hamiltonian];
Wohlfarth PRD(18)-a1804 [tangent bundle formalism, and canonical quantization];
> s.a. critical phenomena;
Geometrization; wave phenomena [tails].

@ __Coupled to general relativity__: Christodoulou CMP(86),
CMP(86),
CMP(87) [dynamics];
Faraoni gq/98-proc [value of coupling];
Ayón-Beato et al PRD(05)ht [non-linear fields that do not curve spacetime];
Esposito et al IJGMP(11)-a1009 [complex fields, and cosmology];
Vernov PoS-a1201 [non-local scalar field];
Brax et al JCAP(12)-a1206 [scalar components of gravity, experimental aspects];
Sotiriou ln(14)-a1404.

@ __On generalized spacetime__: Kosiński et al PRD(00) [*κ*-deformed Minkowski];
Schunck & Wainwright JMP(05) [supersphere];
Girelli & Livine AIP(09)-a0910-proc [with coset momentum space, and non-commutativity];
> s.a. fractals in physics; klein-gordon
fields; non-commutative field theory.

@ __Related topics__: Derrick JMP(64),
Adib ht/02 [no stable, \(t\)-independent solutions];
Gudder JMP(94) [non-standard];
Frommert IJTP(97)gq/96 [and relativistic particles];
Faraoni PRD(12)-a1201 [scalar fields and effective perfect fluids];
Ibort et al PLA(12)-a1202 [tomographic description];
Balondo & Govaerts a1906 [global symmetries].

> __Online resources__:
see Wikipedia page.

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

send feedback and suggestions to bombelli at olemiss.edu – modified 18 jun 2019