PlanetPhysics/Superfields2

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Metric superfields

In general, superfields are physically understood as quantized gravity [[../CosmologicalConstant/|fields]] that admit a highly reducible representation of a [[../AntiCommutationRelations/|supersymmetry]] algebra. The problem of specifying a [[../AntiCommutationRelations/|supergravity]] theory can be then defined as a search for those [[../CategoricalGroupRepresentation/|representations]] that allow the construction of consistent local actions, perhaps considered as either [[../QuantumGroup4/|quantum group]], or [[../WeakHopfAlgebra/|quantum groupoid]], actions. Extending [[../HilbertBundle/|quantum symmetries]] to include quantized gravity fields--specified as `superfields'-- is called supersymmetry in [[../SpaceTimeQuantizationInQuantumGravityTheories/|quantum gravity theories]]. A first approach to supersymmetry relied on a curved `[[../AntiCommutationRelations/|superspace]]' (Wess and Bagger,1983 ) and is analogous to supersymmetric gauge theories (see, for example, [[../IsomorphicObjectsUnderAnIsomorphism/|sections]] 27.1 to 27.3 of Weinberg, 1995).

Metric superfield

Because in supergravity both [[../ECartan/|spinor]] and tensor fields are being considered, [[../GravitationalField/|The Gravitational Fields]] are represented in terms of tetrads , $e_{\mu }^a(x),$ rather than in terms of [[../AlbertEinstein/|Einstein's]] general relativistic [[../MetricTensor/|metric]] $g_{\mu \nu }(x)$. The connections between these two distinct representations are as follows:

$g_{\mu \nu }(x)={\eta }_{ab}{e}_{\mu }^a(x)e_{\gamma }^b(x),$

with the general coordinates being indexed by $\mu ,\nu ,$ etc., whereas local coordinates that are being defined in a locally inertial coordinate [[../SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence/|system]] are labeled with superscripts a, b, etc.; ${\eta }_{ab}$ is the diagonal [[../Matrix/|matrix]] with elements +1, +1, +1 and -1. The tetrads are invariant to two distinct [[../Bijective/|types]] of symmetry transformations--the local [[../CosmologicalConstant/|Lorentz transformations]]:

$e_{\mu }^a(x)⟼{\mathrm{\Lambda }}_b^a(x)e_{\mu }^b(x),$

(where ${\mathrm{\Lambda }}_b^a$ is an arbitrary real matrix), and the general coordinate transformations:

$x^{\mu }⟼(x^{\prime }{)}^{\mu }(x).$

In a weak gravitational field the tetrad may be represented as:

$e_{\mu }^a(x)={\delta }_{\mu }^a(x)+2\kappa {\mathrm{\Phi }}_{\mu }^a(x),$

where ${\mathrm{\Phi }}_{\mu }^a(x)$ is small compared with ${\delta }_{\mu }^a(x)$ for all $x$ values, and $\kappa =\surd 8\pi G$, where G is Newton's gravitational constant. As it will be discussed next, the supersymmetry algebra (SA) implies that the [[../BoseEinsteinStatistics/|graviton]] has a fermionic superpartner, the hypothetical `gravitino' , with helicities $\pm$ 3/2. Such a self-charge-conjugate massless [[../Particle/|particle]] as the `gravitiono' with helicities $\pm$ 3/2 can only have low-energy interactions if it is represented by a Majorana field ${\psi }_{\mu }(x)$ which is invariant under the gauge transformations:

${\psi }_{\mu }(x)⟼{\psi }_{\mu }(x)+{\delta }_{\mu }\psi (x),$

with $\psi (x)$ being an arbitrary Majorana field as defined by Grisaru and Pendleton (1977). The tetrad field </math>\Phi _{\mu \nu}(x)$andthegravitonfield$\psi _\mu(x)$arethenincorporatedintoaterm<math>H_{\mu }(x,\theta )$ defined as the metric superfield . The relationships between </math>\Phi _{\mu _ \nu}(x)$and$\psi _\mu(x)$,ontheonehand,andthecomponentsofthemetricsuperfield<math>H_{\mu }(x,\theta )$, on the other hand, can be derived from the transformations of the whole metric superfield:

$H_{\mu }(x,\theta )⟼H_{\mu }(x,\theta )+{\mathrm{\Delta }}_{\mu }(x,\theta ),$

by making the simplifying-- and physically realistic-- assumption of a weak gravitational field (further details can be found, for example, in Ch.31 of vol.3. of Weinberg, 1995). The interactions of the entire superfield $H_{\mu }(x)$ with matter would be then described by considering how a weak gravitational field, $h_{{\mu }_{\nu }}$ interacts with an [[../PrincipleOfCorrespondingStates/|energy-momentum tensor]] </math>T^{\mu \nu}$representedasalinearcombinationofcomponentsofareal[[../Vectors/|vector]]superfield<math>{\mathrm{\Theta }}^{\mu }$. Such interaction terms would, therefore, have the form:

$I_{ℳ}=2\kappa \int d{x}^4[H_{\mu }{\mathrm{\Theta }}^{\mu }{]}_D,$

($ℳ$ denotes `matter') integrated over a four-dimensional (Minkowski) [[../SR/|spacetime]] with the metric defined by the superfield $H_{\mu }(x,\theta )$. The term ${\mathrm{\Theta }}^{\mu }$, as defined above, is physically a supercurrent and satisfies the conservation conditions:

${\gamma }^{\mu }𝐃{\mathrm{\Theta }}_{\mu }=𝐃,$

where $𝐃$ is the four-component super-derivative and $X$ denotes a real chiral [[../Vectors/|scalar]] superfield. This leads immediately to the calculation of the interactions of matter with a weak gravitational field as:

$I_{ℳ}=\kappa \int d^{4}x{T}^{\mu \nu }(x)h_{\mu \nu }(x),$

It is interesting to note that the gravitational actions for the superfield that are invariant under the generalized gauge transformations $H_{\mu }⟼H_{\mu }+{\mathrm{\Delta }}_{\mu }$ lead to solutions of the Einstein field equations for a homogeneous, non-zero vacuum [[../CosmologicalConstant/|energy]] density ${\rho }_V$ that correspond to either a de Sitter space for ${\rho }_V>0$, or an anti-de Sitter space for ${\rho }_V<0$. Such spaces can be represented in terms of the hypersurface equation

$x_5^2\pm {\eta }_{\mu ,\nu }{x}^{\mu }{x}^{\nu }=R^2,$

in a quasi-Euclidean five-dimensional space with the metric specified as:

$d{s}^2={\eta }_{\mu ,\nu }{x}^{\mu }{x}^{\nu }\pm d{x}_5^2,$

with '$+$' for de Sitter space and '$-$' for anti-de Sitter space, respectively.