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];
Wohlfarth a1804 [tangent bundle formalism, and canonical quantization];
> 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__:
s see Wikipedia page.

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

send feedback and suggestions to bombelli at olemiss.edu – modified 22 jun 2018