PlanetPhysics/Metric Superfields

This is a topic entry on metric superfields in quantum [[../AntiCommutationRelations/|supergravity]] and the mathematical cncepts related to [[../ECartan/|spinor]] and [[../HamiltonianAlgebroid3/|tensor fields]].

[[../IsomorphicObjectsUnderAnIsomorphism/|\Section]] {Metric superfields: spinor and tensor fields}

Because in supergravity both spinor and tensor fields are being considered, [[../GravitationalField/|The Gravitational Fields]] are represented in terms of [[../HamiltonianAlgebroid3/|tetrads]], $e_{\mu }^a(x),$ rather than in terms of the general relativistic [[../MetricTensor/|metric]] $g_{\mu \nu }(x)$. The connections between these two distinct [[../CategoricalGroupRepresentation/|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 [[../CosmologicalConstant/|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 [[../AntiCommutationRelations/|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 [[../AntiCommutationRelations/|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 [[../AlbertEinstein/|Einstein]] field equations for a homogeneous, non-zero vacuum [[../CosmologicalConstant2/|energy]] density ${\rho }_V$ that correspond to either a de Sitter space for ${\rho }_V>0$, or an [[../HamiltonianAlgebroid3/|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.

Note The presentation above follows the exposition by S. Weinberg in his book on "Quantum Field Theory" (2000), vol. 3, Cambridge University Press (UK), in terms of both [[../PreciseIdea/|concepts]] and mathematical notations.