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

\(\square\)*φ* – *V'*(*φ*) = 0 ,

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' *λφ*^{4}-theory with a negative quartic
coupling constant as a valid candidate for an asymptotically free theory of
strong interactions; With positive *λ*,
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.

* __Chameleon field__: A field whose mass depends on the local matter density.

@ __i φ^3 theory__:
Bender et al PRL(04)ht [acceptable
quantum field theory].

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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 [*λφ*^{4}]; in Franklin 10 [IIb].

@ __In a curved background__: Bernardini & Bertolami AP(13)-a1212 [cosmological background, Hamiltonian]; > s.a. critical phenomena; Geometrization.

@ __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].

> __Online resources__: see Wikipedia page.

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