PlanetPhysics/Superfields Superspace

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Superspace, superfields, supergravity and Lie superalgebras

In general, a superfield --or quantized gravity [[../CosmologicalConstant2/|field]]- has a highly [[../HamiltonianAlgebroid3/|reducible representation]] of the supersymmetry algebra, and the problem of specifying a supergravity theory can be 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 current theories of [[../LQG2/|quantum gravity]]. Graded `[[../BilinearMap/|Lie' algebras]] (or Lie superalgebras) represent the [[../QuantumSpinNetworkFunctor2/|quantum operator]] supersymmetries by defining these simultaneously for both fermion ([[../QuarkAntiquarkPair/|spin]] $1/2$) and boson (integer or 0 spin [[../Particle/|particles]]).

The quantized physical space with supersymmetric properties is then called a `superspace' , (another name for `quantized space with supersymmetry' ) in Quantum Gravity. The following subsection defines these physical [[../PreciseIdea/|concepts]] in precise mathematical terms.

Mathematical definitions and propagation equations for superfields in superspace: Graded Lie algebras

Supergravity, in essence, is an extended supersymmetric theory of both matter and gravitation (viz . Weinberg, 1995 ). A first approach to supersymmetry relied on a curved `superspace' (Wess and Bagger,1983 ) and is analogous to [[../HamiltonianAlgebroid3/|supersymmetric gauge theories]] (see, for example, [[../IsomorphicObjectsUnderAnIsomorphism/|sections]] 27.1 to 27.3 of Weinberg, 1995). Unfortunately, a complete non--linear supergravity theory might be forbiddingly complicated and furthermore, the constraints that need be made on the [[../BoseEinsteinStatistics/|graviton]] superfield appear somewhat subjective, (according to Weinberg, 1995). In a different approach to supergravity, one considers the physical components of the gravitational superfield which can be then identified based on `flat-space' superfield methods (Chs. 26 and 27 of Weinberg, 1995). By implementing the {\em gravitational weak-field approximation} one obtains several of the most important consequences of supergravity theory, including [[../CosmologicalConstant/|masses]] for the hypothetical `gravitino' and `gaugino particles' whose existence might be expected from supergravity theories. Furthermore, by adding on the higher order terms in the gravitational constant to the supersymmetric transformation, the general coordinate transformations form a closed algebra and the [[../Lagrangian/|Lagrangian]] that describes the interactions of the physical fields is then invariant under such transformations.The first [[../MoyalDeformation/|quantization]] of such a flat-space superfield would obviously involve its `[[../CohomologicalProperties/|deformation',]] and as a result its corresponding supersymmetry algebra becomes non--commutative .

Metric superfield

Because in supergravity both [[../ECartan/|spinor]] and [[../HamiltonianAlgebroid3/|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 [[../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 graviton has a fermionic superpartner, the hypothetical `gravitino' , with helicities $\pm$ 3/2. Such a self-charge-conjugate massless 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 ${\mathrm{\Phi }}_{\mu \nu }(x)<math>andthegravitonfield$\psi _\mu(x)</math> are then incorporated into a term $H_{\mu }(x,\theta )$ defined as the [[../HamiltonianAlgebroid3/|metric superfield]]. The relationships between ${\mathrm{\Phi }}_{{\mu }_{\nu }}(x)<math>and$\psi _\mu(x)</math>, on the one hand, and the components of the metric superfield $H_{\mu }(x,\theta )$, 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]] $T^{\mu \nu }$ represented as a linear combination of components of a real [[../Vectors/|vector]] superfield ${\mathrm{\Theta }}^{\mu }$. 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 [[../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.

The spacetime [[../TopologicalOrder2/|symmetry groups]], or extended symmetry [[../GroupoidHomomorphism2/|groupoids]], as the case may be-- are different from the `classical' Poincar\'e symmetry group of translations and Lorentz transformations. Such spacetime symmetry groups, in the simplest case, are therefore the Failed to parse (unknown function "\rO"): {\displaystyle \rO(4,1)} [[../TrivialGroupoid/|group]] for the de Sitter space and the Failed to parse (unknown function "\rO"): {\displaystyle \rO(3,2)} group for the anti--de Sitter space . A detailed calculation indicates that the transition from ordinary flat space to a bubble of anti-de Sitter space is not favored energetically and, therefore, the ordinary (de Sitter) flat space is stable (viz. Coleman and De Luccia, 1980), even though quantum fluctuations might occur to an anti--de Sitter bubble within the limits permitted by the [[../QuantumParticle/|Heisenberg uncertainty principle]].

Supersymmetry algebras and Lie, graded superalgebras

It is well known that continuous symmetry transformations can be represented in terms of a [[../TopologicalOrder2/|Lie Algebra]] of linearly independent symmetry [[../Generator/|generators]] $t_j$ that satisfy the commutation [[../Bijective/|relations]]:

$[t_j,t_k]=\iota {\mathrm{\Sigma }}_l{C}_{jk}{t}_l,$

Supersymmetry is similarly expressed in terms of the symmetry generators $t_j$ of a graded (`Lie') algebra which is in fact defined as a superalgebra ) by satisfying relations of the general form:

$t_j{t}_k-(-1{)}^{{\eta }_j{\eta }_k}{t}_k{t}_j=\iota {\mathrm{\Sigma }}_l{C}_{jk}^l{t}_l.$

The generators for which ${\eta }_j=1$ are fermionic whereas those for which ${\eta }_j=0$ are bosonic. The coefficients $C_{jk}^l$ are structure constants satisfying the following conditions:

$C_{jk}^l=-(-1{)}^{{\eta }_j{\eta }_k}{C}_{jk}^l.$

If the generators ${}_j$ are quantum Hermitian [[../QuantumOperatorAlgebra4/|operators]], then the structure constants satisfy the reality conditions $C_{jk}^{*}=-C_{jk}$~. Clearly, such a graded [[../TrivialGroupoid/|algebraic structure]] is a [[../NewtonianMechanics/|superalgebra]] and not a proper Lie algebra; thus graded Lie algebras are often called `Lie superalgebras' .

The standard computational approach in [[../FTNIR/|QM]] utilizes the S-matrix approach, and therefore, one needs to consider the general, graded `Lie algebra' of supersymmetry generators that [[../Commutator/|commute]] with the S-matrix. If one denotes the fermionic generators by $Q$, then $U^{-1}(\mathrm{\Lambda })QU(\mathrm{\Lambda })$ will also be of the same type when $U(\mathrm{\Lambda })$ is the quantum operator corresponding to arbitrary, homogeneous Lorentz transformations ${\mathrm{\Lambda }}^{{\mu }_{\nu }}$~. Such a group of generators provide therefore a representation of the homogeneous [[../CosmologicalConstant/|Lorentz group of transformations]] $𝕃$~. The [[../PureState/|irreducible representation]] of the homogeneous Lorentz group of transformations provides therefore a [[../TrivialGroupoid/|classification]] of such individual generators.

Graded `Lie Algebras'/Superalgebras.

A set of [[../QuantumOperatorAlgebra5/|quantum operators]] $Q_{jk}^{AB}$ form an $𝐀,𝐁<math>representationofthegroup$\mathbf L</math> defined above which satisfy the commutation relations:

$[𝐀,Q_{jk}^{AB}]=-[{{\mathrm{\Sigma }}_j}^{\prime }{J}_{j{j}^{\prime }}^A,Q_{j^{\prime }k}^{AB}],$

and

$[𝐁,Q_{jk}^{AB}]=-[{\mathrm{\Sigma }}_{j^{\prime }}{J}_{k{k}^{\prime }}^A,Q_{j{k}^{\prime }}^{AB}],$

with the generators $𝐀$ and $𝐁$ defined by $𝐀\equiv (1/2)(𝐉\pm i𝐊)$ and $𝐁\equiv (1/2)(𝐉-i𝐊)$, with $𝐉$ and $𝐊$ being the Hermitian generators of rotations and `boosts', respectively.

In the case of the two-component Weyl-spinors $Q_{jr}$ the Haag--Lopuszanski--Sohnius (HLS) [[../Formula/|theorem]] applies, and thus the fermions form a supersymmetry algebra defined by the anti-commutation relations:

Failed to parse (syntax error): {\displaystyle ~[Q _{jr}, Q _{ks}^*] &= 2\delta _{rs} \sigma^\mu _{jk} P _\mu ~, \\ [Q _{jr}, Q _{ks}] &= e _{jk} Z _{rs} ~, }

where $P_{\mu }$ is the 4--momentum [[../QuantumSpinNetworkFunctor2/|operator]], $Z_{rs}=-Z_{sr}$ are the bosonic symmetry generators, and ${\sigma }_{\mu }$ and $𝐞$ are the usual $2\times 2$ Pauli matrices. Furthermore, the fermionic generators commute with both energy and [[../Momentum/|momentum]] operators:

$[P_{\mu },Q_{jr}]=[P_{\mu },Q_{jr}^{*}]=0.$

The bosonic symmetry generators $Z_{ks}$ and $Z_{ks}^{*}$ represent the set of central charges of the supersymmetric algebra:

$[Z_{rs},Z_{tn}^{*}]=[Z_{rs}^{*},Q_{jt}]=[Z_{rs}^{*},Q_{jt}^{*}]=[Z_{rs}^{*},Z_{tn}^{*}]=0.$

From another direction, the Poincar\'e symmetry mechanism of special relativity can be extended to new [[../CoIntersections/|algebraic]] systems (Tanas\u a, 2006). In Moultaka et al. (2005) in view of such extensions, consider invariant-free Lagrangians and bosonic multiplets constituting a symmetry that interplays with (Abelian) Failed to parse (unknown function "\U"): {\displaystyle \U(1)} --gauge symmetry that may possibly be described in categorical terms, in particular, within the notion of a cubical site (Grandis and Mauri, 2003).

One needs to introduce next further generalizations of the concepts of Lie algebras and graded Lie algebras to the corresponding Lie [[../Algebroids/|algebroids]] that may also be regarded as C*--convolution representations of quantum gravity groupoids and superfield (or supergravity) supersymmetries. This is therefore a novel approach to the proper representation of the \htmladdnormallink{non-commutative geometry {http://planetphysics.us/encyclopedia/NAQAT2.html} of [[../NonAbelianQuantumAlgebraicTopology3/|quantum spacetimes]]}--that are curved (or `deformed') by the presence of intense gravitational fields--in the framework of \htmladdnormallink{non-Abelian {http://planetphysics.us/encyclopedia/AbelianCategory3.html}, graded [[../LieAlgebroids/|Lie algebroids]]}. Their correspondingly \emph{deformed quantum gravity groupoids} (QGG) should, therefore, adequately represent supersymmetries modified by the presence of such intense gravitational fields on the Planck scale. Quantum fluctuations that give rise to quantum `foams' at the Planck scale may be then represented by quantum [[../TrivialGroupoid/|homomorphisms]] of such QGGs. If the corresponding graded Lie algebroids are also integrable , then one can reasonably expect to recover in the limit of $\hslash \to 0$ the [[../NonabelianAlgebraicTopology3/|Riemannian geometry]] of General Relativity and the globally hyperbolic spacetime of Einstein's classical gravitation theory ([[../SR/|GR]]), as a result of such an integration to the quantum gravity [[../CubicalHigherHomotopyGroupoid/|fundamental groupoid]] (QGFG). The following subsection will define the precise mathematical concepts underlying our novel quantum supergravity and extended supersymmetry notions.