Factorial Unitary Representations of the Translational Group, Invariant Pure States and the Super Symmetric Graviton

Consider then the automorphic representation g→αg of the translation subgroup T. If we make the minimal assumption that this representation is weakly measurable; i.e. the mapping ( ) , g g A x y α → : is Haar-measurable for all relevant values of A, x and y; then the argument [1] shows that the mapping g→αg is norm continuous. Since the translational group is both abelian and connected, it follows that each αg is an inner automorphism of A and the corresponding unitary Wg has a spectrum contained in the positive half plane. In fact we have; { } ( ) 1 2 2 1 ( ) ;Re where 4 || || 2 g g g g W z z i σ β β α ⊂ ≥ = − −


Introduction and Context
Our focus is on the local fibre algebra A(x) which we assume to be isomorphic to a von Neumann algebra with trivial centre acting on a separable Hilbert space. We also consider the subgroup T of the Poincare group consisting of translations of space-time as a gauge group of automorphisms of A(x). These generate the local diffeomorphisms of General Relativity.
Consider then the automorphic representation g→α g of the translation subgroup T. If we make the minimal assumption that this representation is weakly measurable; i.e. the mapping ( ) , g g A x y α → : is Haar-measurable for all relevant values of A, x and y; then the argument [1] shows that the mapping g→α g is norm continuous. Since the translational group is both abelian and connected, it follows that each α g is an inner automorphism of A and the corresponding unitary W g has a spectrum contained in the positive half plane. In fact we have; Moreover if S denotes the von Neumann subalgebra generated by W g ; g∈T the set of unitary {W g ; g∈T} is actually a commuting set within A(x). Thus S is a commutative subalgebra and contains the identity I of A(x), since if id is the group identity then I=Wid. A(x) is a factor; S thus contains the centre Z of A(x). Such commutative quantum operator algebra is equivalent to the set of continuous functions on a compact space and this equivalence arises through the Gelfand transform; The carrier space Φ S of S is the set of all continuous complex valued homomorphisms on S. and is thus topologically a Stonean space, as is Φ Z and the restriction map π:Φ S →Φ Z is a continuous surjection. We then have, as we will prove, a lifting. : with ( ) | ( ) for For each g∈T we can now define the Gelfand transform of a unitary U g acting on the carrier space Φ S of S as follows;ˆˆˆˆ( = then by the extended form of the Stone-Weierstrass theorem [2] the centre Z corresponds to those elements of Φ S constant on each equivalence class, Applying this to the Gelfand transform ˆ( ) V corresponds to an element V g of the centre. Since A(x) is a factor, this means that with v(g) a complex number defining a coboundary, and that for each g∈T; U g implements α g . In addition, the mapping g→U g is a group homomorphism for if we set; Hence R g,h is a unitary in the centre. In fact R g,h =λ(g,h)I where λ(g,h) is a 2-cocycle.
The fact that the lifting f:Φz→Φs has the property π f(ρ)=f(ρ)| z = for ρ∈Φ z means that , But then we have that; ˆ( ( )) 1 g U f ρ = ∀g∈G, thus , ( ) 1 g h R ρ = ∀ρ∈Φ z ; the 2-cocycle is trivial. Therefore R g,h =I ∀g,h∈G and g→U g is a group representation by unitaries in the fiber algebra, which turns out to be norm continuous, as previously shown.
In summary, the lifting follows from the Axiom of Choice applied to the strange topological properties of Stonean spaces. We can set it in the context of lifting from a totally disconnected, compact Hausdorff base space S into a containing fibre bundle K having a Stonean topology. and short exact sequences in our previous work [1][2][3][4]. To prepare the ground, we need the following seemingly little known result.
Define character acting on a locally compact abelian group G to be a continuous group homomorphism from G to the unit circle. If H is a closed subgroup of G and ρ is a character of H, then ρ extends to a character on G.
Since this useful result plays an important part in our argument, we provide a simple proof as follows, based on Mackey's general approach.
Let Ĝ denote the group of characters on G under point wise multiplication. Pontryagin [5] shows that we can impose a locally compact topology on Ĝ relative to which it is a locally compact abelian group. Define;( Then the mapping ω is a continuous isomorphism between G and Ĝ .

{ } ( )
Let ; ( ) 1 and the character group of the quotient group then for if : If we move this up a level and now define S as the character group of Finally we can conclude that, up to isomorphism, If Ω is the quotient mapping: And ξ→ρ is the require extension to the character group G of the character group of the subgroup H. By exploiting Mackey Theory directly, we now aim to prove the following key result by an alternative method to that used in our previous paper [1] in order to help stimulate a broader interest in such group theoretic approaches, which extend the work of Wigner.
Define α:g→α g to be a representation of the translational subgroup T of the Poincare group as a gauge group of automorphisms of the fibre algebra A(x). Provided this group representation α is weakly measurable, then it turns out that it is also norm, weakly and strongly continuous, and is implemented by a norm, weakly and strongly continuous unitary representation by unitaries in the fibre algebra.
We showed in our previous paper [1] that, with these assumptions, the mapping α:g→α is norm continuous and is thus implemented by a set g→W g of commuting unitaries in the fibre algebra.
it follows that the norm continuous mapping α:g→α g is strongly continuous. Our foundational work [6] then shows that we may choose the unitaries in a way that the mapping g→W g is a borel mapping from T to the unitary group of A(x) with the weak operator topology.

Properties of the Projection onto the Base Space B of a Stonean Fibre Bundle K
We prove first the subsidiary fact that the projection π is an open mapping if and only if for each non-trivial open subset E of K, π(E) is not a nowhere dense subset of the base space B.
One way is trivial for if π is an open mapping then π(E) is a nontrivial open set so cannot be nowhere dense. Conversely, the topology of the fibre bundle K is compact and totally disconnected, with a basis of 'clopen' sets (i.e. sets which are both closed and open). Thus every open set is a union of such clopen sets, and it suffices to show that if V is clopen in K, then π(V) is open in B. Since V is clopen, it is a closed and thus compact subset of K, and π is continuous, thus π (V) is compact. If we define Y=π(V)\intπ(V); this a closed set with empty interior thus Y is a nowhere dense set, and is the image of an open set; Y=π(V\π −1 (intπ(V)), using the fact that the projection mapping π is continuous and surjective. It follows that Y is void and π(V)=int π(V).

Lifting from the Base Space B
Consider now the set Φ of all compact subsets S of K such that π (S) = B. Then Φ is a non-void, partially ordered by set inclusion, and every decreasing chain has a lower bound. It follows from Zorn's Lemma that Φ has a smallest element K(0). We show that; | is an open mapping This will prove the result for the lifting we seek is then , the Axiom of Choice selecting out K(0) as a unique minimal section through the fibre bundle K. Consider then V to be a non-trivial open set in K(0). By definition, This is a contradiction due to minimality of the set K(0). Thus π(V) cannot be a nowhere dense set. From our earlier discussion, this is enough to show that is an open mapping Since K(0) is a Hausdorff topological space with a basis of cl open sets, there is a cl open subset V of K(0) containing x 1 but not x 2 . Then; is also clopen and ( ) ( ) is a non-trivial open set, and W and V are disjoint, with This again contradicts the minimality of K(0). Thus

Mackey Theory and a Group Theoretic Proof
For a direct group theoretic proof, avoiding the intricacies of Stonean topology, we have devised the following argument which builds on the discussion of Mackey Theory applied to group extensions Let N=Ker()={g∈T;α g =the identity}. Replacing T by T/N we may assume that; We note that the unitary group of the centre of A(x) is isomorphic to the unit circle O in the complex plane. It is also easy to see that Γ={λWg;λ∈O, and gT} is an abelian subgroup of the unitary group of A(x).
Define the mapping γ(λ)=λI which maps the unit circle O into the subgroup Γ, and the mapping η(λW g )=g which maps the group Γ to the translation group T. Note that the mapping η is well defined here due to (1) above. Then the sequence; We can identify the group Γ with the extension Tη′T where; η′(λ,g)=η(λW g )=g via the mapping; (λ,g)→λW g .
Let the mapping J denote the identification O × T↔ Γ↔ Oη′T then J is a borel mapping. Thus we see again that the mapping W:g→W g is a borel measurable mapping from the translation subgroup T into the group Γ.
The group T is a separable locally compact abelian group and is a standard borel space. The unitary group of A(x) is also a standard borel space, thus W −1 : W g → g is a borel mapping and η′ is a borel system of factors for 'Tη′T. There is thus, as discussed [1,6], a locally compact topology on the group Γ relative to which it is a separable, locally compact abelian group. Γ also contains the unit circle O as a closed subgroup.
With this Mackey topology on Γ, we can now deduce that the identity map e iθ →e iθ : O→O is a character on O and thus lifts to a character σ:Γ→ O.
For each unitary W g ; g∈T define U g =σ(W g ) * W g . Then; Ug implements the automorphism τg and, since σ | o is the identity Since U g ∈Γ∀g, by the definition of Γwe can apply the group homomorphism σ to both sides of this relationship. Noting again that σ | o , is the identity this implies that all such 2-cocycles are trivial, and the mapping g→Ug is a unitary group representation.
From the proof above we can identify the group Γ with the borel system of factors oη′T, the mapping W:g→W g is a borel measurable mapping from the translation subgroup T into the group Γ. Hence the mapping g→〈x, W g x is a measurable mapping for any x in the Hilbert space H. In addition, the characterσ:Γ→ O is a continuous mapping, thus σW:g→(W g ) is borel. Combining these together it follows that the mapping ( ) * , , is borel measurable and is in fact norm continuous due as before to the spectral properties of the {W g ; g∈G} which imply that || Pure

T-invariant Quantum States
We continue to assume as context that space-time is noncommutative at some energy level such as the Planck regime, with algebraic structure at each event point x of space-time, forming the fibre algebra A(x). This structure then corresponds to the single fibre of a principal fibre bundle. A gauge group of automorphisms corresponding to the translation subgroup T of the Poincare group acts on each fibre algebra locally, while a section through this bundle is then a quantum field of the form {A(x); x∈M} with M the underlying space-time manifold. In addition, we assume a local algebra O(D) corresponding to the algebra of sections of such a principal fibre bundle with base space a finite and bounded subset of space-time, D. The algebraic operations of addition and multiplication are assumed defined fibrewise for this algebra of sections.
We define a separating family of T-invariant quantum states to be a finite or countable subset S of the state space such that the positive kernel of S is zero; i.e. given an observable A in the positive subset of the fibre algebra A(x).
For a general observable A we assume ( ) 0 This reflects the classical case; if a tensor or the difference between two tensors of the same covariant and contravariant class is equal to zero for all local inertial reference frames, then it is zero for all curvilinear reference frames, by the covariance assumptions of General Relativity.
Since the Hilbert space F(x) on which the fibre algebra A(x) acts is separable, by definition, there is a countable dense subset x(n). Defining Since f is separating.
Applying the automorphism If f is a separating T-invariant quantum state, let π be the GNS representation then clearly π is an algebraic isomorphism since the kernel of f is {0}. Let {U g ; g∈G} be the Segal unitaries associated with f; then the mapping g→U g is a unitary representation implementing α:g→α g . If the mapping α:g→ g is weakly measurable in the GNS representation then by our previous results [1] it is norm continuous and we can further assume [1,3,6] that there is a unitary mapping g→U g implementing α:g→α g which is norm continuous with U g ∈π(A(x))∀g∈T.
In previous work [4] we characterised these invariant states in terms of their ergodic action focusing on density matrix states and their von Neumann entropy. From inspection, we can generalize many of these previous results to any (not necessarily normal) quantum state: see final section of the paper. We have, from that analysis; that the automorphic representation α:g→α g of T acts ergodically if and only if is trivial, containing only the projections 0 and I and thus consisting of the set of complex multiples of I.
Since J 2 =1, it follows that the representation α:g→α g of T acts ergodically if and only if is also trivial. But for this case we have also shown that and thus if E is a projection in π(A(x))′ then clearly The GNS representation is thus irreducible in the sense of Murrayvon Neumann, corresponding to f being a pure quantum state.

An Example Application; the Supersymmetric Graviton
The Poincare group is a locally compact Lie group with 10 generators, and the translational group is an abelian subgroup generated by the energy-momentum 4-vector P µ . This has the property that its square P 2 =P µ P µ =E 2 lies in the centre of the Lie algebra. If we consider the energy-momentum vector in normalized units (c=1) then P 2 thus has the form P 2 =m 2 I, where m is the mass-energy of the corresponding particle. In other words, a factorial representation of the Translational group corresponds to a particle with fixed mass m and undetermined spin. We can consider two cases; (a) P 2 =m 2 .I; m 2 ≠0 corresponding to a multiple of particles each of the same positive mass but with different spin values; (b) P 2 =0.I. This factorial representation corresponds to a massless particle such as a photon or a graviton, with a Supersymmetric massless fermion partner.
For either case we need to add an additional element, normally denoted Q α , to the Lie algebra, to represent the spread of spin values. In any representation, these are all linear operators, including the identity operator I, and thus form algebra of such operators. Such a factorial representation corresponding to a set of particles must contain equal numbers of bosons and fermions [8]. With certain assumptions, such a representation where the centre of the algebra is non-trivial can be decomposed into a direct integral of factorial representations, as discussed [6].
We can develop the local, linear representation of these operators, which is a faithful representation of the Superspace Lie Algebra [8] by adding a pair of Grassmann variables to the algebraic formulation. We can then generate a standard Lie algebra while mixing commutators anticommutators. For example, if ξ and ξ have the Grassmann property so that ξξ ξξ = − , then;  . Note that we also have to assume that these Grassmann variables commute with the operators Q. A typical element of the corresponding Lie group G is then of the form; Here, x µ is a point in flat space-time, thus we can think of the Grassmann variables as a vector at the point x µ . With this structure, the Superspace is a vector bundle and the locally flat group multiplication structure is of the following form: . This follows from the Grassmannian properties.
From our previous paper [1] we noted that the assumption of local flatness of the space-time manifold is equivalent to the generators P µ (µ=0,1,2,3) commuting and thus corresponds to a factorial representation of the translational group by commuting unitaries. It follows that this factorial representation is norm continuous and is implemented by a norm continuous unitary representation. We can interpret this group product as shifting the locus of the Grassmann vector in space-time from to x x a i i µ µ µ µ µ ξσ θ θσ ξ + − + together with additive change to the Grassmann vector at this point. If this shift is infinitesimal, then, as in normal Lie group theory, we can consider the tangent plane around the group element ( , , ) G a µ ξ ξ , giving the following local representation of the Lie group generators on the tangent plane to the Riemannian space-time manifold at the event point (a v ): In this form the generators all satisfy the algebraic relationships of the Supersymmetric extension of the local translation Lie algebra. Thus this representation is locally an algebraic isomorphism onto the curved Riemannian manifold ℳ and the piecewise local representations; are the generators of the local relativistic diffeomorphisms around the event points of ℳ. The manifold is assumed smoothly differentiable; we focus on the flat tangent plane and assume Dirac relativistic spinor theory applies. We therefore assume that this operator representation acts on a 4 dimensional Hilbert space H of (spinor) wave functions; and we denote by ψ an element of the Hilbert space.
Within these assumptions we have the Dirac matrix γ 5 =iγ 0 γ 1 γ 2 γ 3 , and we can express this state ψ in the form Exploiting the fact that (γ 0 ) 2 =I 4x4 leads to the equation: The matrices (−γ µT ) also satisfy the Clifford algebra relations and there is a 4x4 non-singular matrix C such that C −1µ C=−γ µT . We can thus define the charge conjugate spinor; In the Weyl representation we take; where σ 2 is the second Pauli matrix With ψ the spinor wave function where the ε matrices are the spinor metric 2 × 2 matrices.

The Mass Zero Case as a Graded Lie Algebra
We start with the properties of a Z 2 -graded Lie algebra L=L 0 ⊕L 1 where L 0 is the Lie algebra spanned by the generators P(µ=0, 1,2,3) andL1 is spanned by the spin or charge generators Q α (α=0,1,2,3).
From the definition of a Z 2 -graded Lie algebra L; P µ ∈L 0 , Q α ∈L 1 with grading 0 and 1.
Since Q α is a non-Hermitian operator (by construction), we can also consider the complex conjugate Dirac 4-spinor Q F β ∈  . To resolve differences in the literature we assume Q Q β β =  .
To constrain the number of options it is convenient at this stage to assume; 2 2 Thus ( ) ( ) since 1.
Hence since Following now the logic [8] in general, we consider from the Z 2 grading, Multiplying from the right by C; We assume the operators Q, Q are Marjorana 4-spinors; then in 2-spinor notation we can simply replace the Dirac γ matrices with their equivalent Pauli matrices yielding the following relationship; Since all of the latter equations are relativistically invariant, we can transform them to the rest frame where P µ =(E,0,0,0)=(m,0,0,0) with the speed of light normalized at c=1.
With these values of P µ in eqn. (2) above we have   The left hand side of this expression is a positive definite quantum operator thus for ψ an element of the Hilbert space; If ψ is the vacuum state then <ψ, P0ψ≥0 is equivalent to Thus 2||Q 1 ψ|| 2 +2||Q 2 ψ|| 2 =0, from which we deduce that Q 1 |ψ>=Q 2 |ψ>=0 and also 1 2

Factorial Representations of the Z 2 graded Lie algebra
The extension of the space L 0 to the space L=L 0 ⊕L 1 maintains P 2 as an element of the centre; With P 2 =m 2 I in this representation, and setting P µ =(m,0,0,0) T in the rest frame, we have the following set of identities from the eqn. From these properties we see that the 2-spinors form at least Clifford algebra in this factorial representation, and we see also that 2 0 A Q = and this is in fact Grassmann algebra.