Towards a Non-Perturbative Quantum Gravity

Defining εAB by ( ) 1 0 1 1 0 AB e æ ö ÷ ç ÷ =ç ÷ ç ÷ çè ø we have ( ) 1 0 1 1 0 AB e æ ö÷ ç ÷ =ç ÷ ç ÷ çè ø The matrices εAB and ε AB are tensor objects which can be used to raise and lower indices in the usual vector and tensor calculus, within which left-handed ‘chiral’ Weyl 2-spinors are covariant vectors, and right-handed chiral Weyl 2-spinors are contravariant vectors. Denoting the dual space as F*. The following calculation illustrates this ‘spinor calculus’ for ψ ∈ F* and χ ∈ F;


Spinors and Spin Networks
A left-handed Weyl 2-spinor is an element of a 2-dimensional vector space F with a basis of Clifford variables denoted ψ A (A=1,2).
In this representation, we can show directly that 2 2 0 1 2 3 2 2 0 0 where I 2 × 2 is the 2 × 2 identify matrix 1 0 0 1 This Dirac spin operator representation acts on a 4-dimensional vector space. Each element of this vector space suitably normalized, is a quantum ket state |ψ >; a pure state. Defining g γ 5 to be the composite matrix operator iγ 0 γ 1 γ 2 γ 3 , we can write this state |ψ > in the form Thus, the four vector ψL has only two non-zero elements (the first two) and is of the form similarly, four vector ψR has only two non-zero elements (the last two) and is defined as Assuming planar isotopy, it is possible to associate various locally deformable lines in the plane with Weyl 2-spinor calculations, giving rise to topological structures called spin networks. As a simple example the tensors  Figure 1; 3   1 000  011  110  101  2  1  0  00  11  1 10  01  2  1  0  00  11  1 10  01  2  1  0  1  00  11  0  1 1 10  01  4  1  000  011  100  111  010  001  110  101  4   1 00  0  1  01  1  0  1 This can then be interpreted in terms of the original vector 0 | y > as; Comparing the two expressions implies that; These are all real unitary matrices thus their inverse is simply the transpose matrix in each case; 1 0 0 1 1 0 0 1 00 ; 01 ; 10 ; 11 0 1 1 0 0 Thus, we can transport a Quantum State along such a computational path.
Note that U(00)=σ 0 U(01)=σ 1 U(10)=σ 3 iU(11)=σ 2 where {σ j ; j=0,1,2,3} are the Pauli spin matrices. We identify this transport from node to node of the computational spin network with the action of a sequence of discrete translations in space-time from one node to the next neighbouring node. A discrete path in space-time can be then be considered as a series of applications of the translation subgroup of the Poincare group.

Discrete Paths in Space-Time
We start by considering classical phase space. Given a dynamical system, entropy is defined through considering the phase space of the system. The emergent behaviour of this classical system gives rise to regions of phase space, each corresponding to similar macro-level behaviour. The entropy of such a coarse-grained region is a measure of all the different micro configurations constituting that region. A system starting in a low entropy state will tend to wander into larger coarsegrained volumes; hence thermodynamic entropy tends to increase over time if the system is isolated, giving rise to the second law of thermodynamics. The structure of classical phase space is such that each set of initial conditions (x μ , p μ ) m m generates a unique solution S(x μ , p μ ) m m. For a Hamiltonian system it is possible to reformulate classical mechanics as a symplectic vector space of solutions, or an equivalent set of initial conditions of location and momentum, equipped with a bilinear form Ω which ultimately derives from Hamilton's equations of motion; With ξ≡(x,p) a dimensional vector we have This is of the form of a symplectic vector space where V is a real finite vector space with dual V*. The skew-symmetric rank 2 tensor Ω then takes the general form;  . In our case V is the configuration space, V* the (dual) momentum space and * V V Å the phase space, a roduct vector bundle over V with fibre V*. By choosing values such as (1,0,0,0,0,0) we can pull out elements; Their product is then a closed loop which corresponds to a scalar λ times the identity matrix with λ=1 in this case. The advantage of this approach is that spinor calculations become a sequence of potentially simpler topological transformations with connections to knot theory.

Computational Spin Networks
Building on these ideas, we define a computational spin network to be a finite quiver consisting of a directed graph with n nodes, where the nodes represent entangled spin inputs and a directed link between two nodes corresponds to a quantum gate, as we now discuss.
If we represent the basis spin ±½ eigenvectors as the column then the NOT quantum gate which switches |0> to |1> and |1> to |0> corresponds to the unitary matrix It is easy to check that U|0>=|1 and U|1>=|0. All quantum gates correspond in this way to multiplying the input state vector by a unitary matrix.
To ease notational clutter, in all that follows we denote the joint tensor product state of the spins |x> and |y> as |xy>.
Given the entangled state, The Dirac canonical quantisation of elements of phase space such as 1 h¢ is equivalent to the canonical quantisation; Ŵ ® W as a (not necessarily bounded) linear operator, and this form of canonical quantisation extends smoothly to countably infinite phase space [1].
Given the canonical quantisation; Ŵ ® W we can form the Weyl unitaries ê xp W i = W. Then closure of linear combinations of these Unitarians and their adjoints in the normed operator topology is the Weyl algebra.
The extension of Ω to the space of solutions allows us to define an inner product on S as where S(1) * is the complex conjugate solution. It turns out [1] that this defines an inner product on S relative to which a one particle Hilbert space can be defined. For a quantum system of bosonic harmonic oscillators, we can then assemble a symmetric tensor product Fock space in the usual way, using creation and annihilation operators.
An example of the Weyl form in a two dimensional locally flat spacetime is now given for a local algebra O(D), having a representation as observables acting on the Hilbert space L 2 (x,t) with Lebesgue measure. For a small increment of space-time (δx,δt) we consider the Poincare Translation subgroup element T (δx, δt): (x,t) → (x +δx, t +δt) and define; Then U is a local group homomorphism of the translation group T as observables acting on L 2 (x,t). Define also; A similar result applies for V δx , by symmetry.

Now introduce a deformation of the form
The mappings V and Z are unitary representations on L 2 (x,t) and so also is their product: V × W: (δx,δt) → T(δx,δt) → V δx ,Z δx Then we have; Then T(δx, δt) → (Zδx, Vδt)is a local Weyl representation of the CCR on L2(x,t). By the Stone-von Neumann theorem, the resulting C*-algebra and its weak closure as a von Neumann algebra must be unitarily isomorphic to Wald's equivalent 'algebraic approach' to quantum field construction and his Weyl Algebra, since we can assume all relevant Hilbert spaces are separable in application to observed real systems [1].
This example indicates that a continuous local group homomorphism from a neighbourhood of the identity of T to a neighbourhood of the identity of the set of observables in O(D) exists as a Weyl algebra. It can be easily extended to 4 dimensions by replacing x by the 3-vector x=(x 1 , x 2 , x 3 ).
A discrete path in space-time, as we have so far generated it, can be considered as a set of linked causally directed intervals each of fractal dimension 1 in renormalized smooth space-time. More formally, we define the path as a series of n linked increments a(j) with varying direction relative to a local forward light cone, such that the path begins at x(0) and ends at x(1), with T (a(j)): x → x+ a(j) elements of the translation subgroup T. The total path is then generated by the product . This we proved in reference [2]; we also proved that the path QP is uniquely determined, and that there is a projection π from the fibre bundle O(D) mapping the quantum field back to the path CP.

Renormalisation of Discrete Paths in Space-Time
The principle of relativity is captured within the assumptions of the Riemannian geometry of 4-manifolds, where formulae equating a tensor expression to zero remain invariant under local diffeomorphism transformations. It is a natural extension of these ideas to additionally postulate that the scales of measurement inscribed on the clocks or measuring rods used by an observer should also not be absolute. Mathematically this can be captured by the additional requirement that the tensor formulae should be invariant under transformations of scale. From this perspective a relativistic quantum system is a scale free system. If Φ is the function transforming system inputs to system outputs, then for a scale-free system, Φ is invariant under a change of scale.
Under the assumptions of such a scale-invariant relativity, let us consider a discrete closed loop in space-time; corresponding to two discrete non-oriented paths sharing the same end points. Then it turns out [2] that this loop is renormalisable and has a finite limit corresponding to the curve fractal dimension as a curve in macroscopic space-time.

Gravity states, the graviton and supersymmetric graviton
We now investigate in more depth the subgroup T of the Poincare group consisting of translations of macroscopic space-time as a gauge group of automorphisms. We define a representation of T as a group of automorphisms of a 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, rooted at the event point x. Consider then the subgroup T acting on A(x). These actions generate the local diffeomorphisms of General Relativity. If we make the minimal assumption that this representation is weakly measurable; i.e. the mapping: is Haarmeasurable for all relevant values of A, x and y; then the argument [3] shows that the mapping g → α g is norm continuous. Since the translational group is both abelian and connected, it follows [4] 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 sub algebra generated by {W g ; g ∈ T}, then the set of unitaries { W g ; g ∈ T } is a commuting set within A(x). Thus, S is a commutative sub algebra and contains the identity I of A(x), since if id is the group identity then id I=W id .A(x) is a factor, S thus contains the centre Z of A(x). Such a 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 such continuous complex valued homomorphism 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 it follows that V g corresponds to an element V g of the centre. Since A(x) is a factor, this means that V g =n (g)I with v(g) a complex number defining a coboundary, and that for each g ∈ T; Ug implements αg. In addition, the mapping g → Ug is a group homomorphism for if we set; 2-cocycle is trivial. Therefore R g,h =I ∀g,h ∈ G, and g → U g is a group representation by unitaries in the fibre algebra, which turns out to be norm continuous, due to their spectral characteristics [4]. We now show that the lifting follows from the Axiom of Choice, in the guise of the equivalent Zorn's Lemma, 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 B into a containing fibre bundle K having a Stonean topology.

Lifting from the base space B of a Stonean fibre bundle K
We prove first the fact that the projection π: K → B 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 non-trivial open set so cannot be nowhere dense.
Conversely, the Stonean 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 (int π(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).

A unique quantum field isomorphic to the base space
Consider now the set Φ of all compact subsets S of K such that π (S)=B. Then Φ is 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; • π| K(0) is an open mapping; • π| K(0) is bijective This will prove the result for the unique quantum field is then f=(π| K(0) ) -1 , the Axiom of Choice selecting out K(0) as a unique minimal section through the fibre bundle K.

(π| K(0) is an open mapping;
Consider then V to be a non-trivial open set in K(0). By definition, π| K(0) is surjective, thus; . Now K(0)\V is a closed thus compact subset of K, and π| K(0) is continuous, thus π(K(0) \V) is compact. It follows that , a closed, compact set. 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 π| K(0) is an open mapping.
Since K(0) is a Hausdorff topological space with a basis of clopen sets, there is a clopen subset V of K(0) containing x 1 but not x 2 . Then; π(V) is also clopen and This again contradicts the minimality of K(0). Thus π| K(0) is a continuous bijection, and the required lifting is given by f=(π| K(0) ) -1 .

Pure gravity states
We define a separating T-invariant quantum state f to be a such that given an observable A in the fibre algebra A(x), We call such states gravity states, motivated by the classical case. If a tensor or the difference between two tensors of the same covariant and contra variant 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 is separating. Applying the automorphism ( ) Additionally, if f n (A*A) → g(A*A) and f n (A*A)=0 then g(A*A). Thus every element of ∆ is separating.
It follows, by applying the Hajian-Kakutani fixed point theorem [6] to the weak*-compact set Δ that it contains an invariant state. Thus, there is a separating T-invariant quantum state. If f is an extreme point of the separating T-invariant quantum states, let π be the GNS representation then clearly π is an algebraic isomorphism since the kernel of f is {0}. Defining; On the pre-Hilbert space of the GNS representation of f; these extend to the 'Segal unitaries' associated with f [4]. The mapping g → U g is a unitary representation implementing α: g → α g . If the mapping α: g → α g is weakly measurable in the GNS representation then by our previous results [3,4] it is norm continuous and the mapping g → U g implementing α: g → α g is also norm continuous with U g ∈ π(A(x))∀g ∈ T We have, from previous work [6,7] 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.
We also have; with 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 Ug ∈π(A(x))∀g ∈ T and thus if E is The GNS representation is thus irreducible in the sense of Murray-von Neumann, corresponding to f being a pure quantum state.

The supersymmetric massless 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 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 multiplet 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.

N=1 Supersymmetry
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 an 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 [9].
We can develop a locally linear representation of these operators, which is a faithful representation of the Super space Lie Algebra [8] by adding a pair of Grassmann variables to the algebraic formulation. We can then generate a standard Lie algebra while mixing commutates anticommutators. For example, if ξ and x have the Grassmann property so that xx xx = -,then; Note that we also require 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 an event point in locally flat space-time, thus we can think of the Grassmann variables as a vector at the point xm. With this structure, the Super space is a vector bundle and the locally flat group multiplication structure is of the following form; 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, from earlier, we can express this state y in the form; where ψ + =(ψ*) T then taking complex conjugates of both sides of the massless Dirac equation followed by transposition yields the identity: Exploiting the fact that (γ 0 ) 2 =I 4 × 4 leads to the equation: The matrices (-γμT) also satisfy the Clifford algebra relations and there is a 4x4 non-singular matrix C such that C -1 γ μ C=-γ μ T. We can thus define the charge conjugate spinor c T C y y = .
In the Weyl representation we take; With ψ the spinor wave function where the ε matrices are the spinor metric

The mass zero case as a graded lie algebra
We start with the properties of a Z 2 -graded Lie algebra 0 1 L L L = Å where L 0 is the Lie algebra spanned by the generators P μ (μ=0,1, 2,3) and L 1 is spanned by the spinor charge generators Q α (α=0,1,2,3).
From the definition of a Z 2 -graded Lie algebra L; the anticommutator. Since Q α is a non-Hermitian operator (by construction), we can also consider the complex conjugate Dirac 4-spinor Q F b Î  . To resolve differences in the literature we assume To constrain the number of options it is convenient at this stage to assume; Thus ( ) ( ) Hence T Q C Q = since C T =-C 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; Similarly, we can show that, for 4-spinors, Hence in 2-spinor notation, Since all 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 normalised at c=1.
With these values of P μ in equation (1) For consistency with the current literature, we assume the constant a=-1.
Thus, we have the quantum operator equality; The left-hand side of this expression is a positive definite quantum operator thus for y an element of the Hilbert space; If ψ is the vacuum state then, <ψ,P 0 ψ>0 is equivalent to

Factorial representations of the 2 Z graded lie algebra
The extension of the space L 0 to the space , In a factorial representation π of the Z 2 graded algebra, it follows that π(P 2 )=m 2 I, fixing the mass m of all particles in this representation. However, the spins of the particles in this representation are not fixed at a common value. In this factorial 2-spinor representation, we have, from earlier, the algebraic identity; 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 equation (1) From these properties we see that the 2-spinors form at least a Clifford algebra in this factorial representation, and we see also that Q A 2 =0 and this is in fact a Grassmann algebra.
The corresponding (pure) vector state is invariant under the translational group since we have 0=|P μ Q A |p,λ>=Q A P μ |p,λ>=Q A P μ | p,λ>=P μ Q A |p,λ We can thus always choose a minimum energy pure vector state ω |p,λ> which is translation invariant and with ω| |p,λ> (Q A )=<p,λ|Q A |p,λ>=0. It is specified by its mass-energy p and its spin value λ. Thus ω |p,λ> is a translation invariant ergodic pure state.
If we now, exploiting local special relativistic covariance, choose an inertial reference frame in which the Wigner little group contains the spin generating 2x2 rotation matrices in the x-y plane, we have; Pμ=(E,0,0,E) therefore; Hence applying this to our translation invariant vacuum state, ω |p,2> we have;

Quantum States Invariant under the Action of Compact Lie Groups
The weak topology σ(A(x) * ,A(x)) can be defined on the predual A(x) * as the coarsest topology for which elements of the predual are continuous [10]. It is defined by a set of semi-norms p=|f | for f a density matrix linear functional which as a set are separating for A(x) * . Making minimal assumptions we let α: g → α g be a weakly measurable representation of the compact Lie group G as automorphisms of A(x). By this we mean that the induced mapping 2 ( ) 1 : : is measurable for Haar measure on G and the σ(A(x) * ,A(x)) topology on A(x) * Since every positive element of A(x) * is a countable sum of vector states this is equivalent to the definition that ( ) : : is measurable for all x in the fibre Hilbert space F(x).
Given that the induced mapping ( ) : : is measurable in the sense now defined above we have previously [11]; This demonstrates the following result, which allows the extension of continuous gauge automorphic representations of compact Lie groups to their cross-product such as the Standard Model gauge group SU(3) × SU(2) × S(1); For the induced representation ( ) : : on the predual of A(x), weak measurability is equivalent to weak continuity.
We have shown, as for local diffeomorphism-invariant quantum states [3,7], that quantum states invariant under the action now of compact Lie groups are common in the sense that the weakly closed convex hull of every normal state contains such a state. We are now dealing with groups such as SU(n) which are both compact and nonabelian thus different techniques are required. To achieve this result, we developed a new idea based on group stabilizer theory which we called Wigner sets [12]. These are complementary to little groups.

Wigner sets and the finite intersection property
Given a density matrix quantum state f, and a weakly measurable representation g → α g of a compact Lie group G as gauge automorphisms of the fibre algebra A(x); define the closed convex hull; ( )

{ }
; g X f co f g G a = Î  with closure in the σ(A(x) * ,A(x))-topology.
Define the group of isometric and σ(A(x) * ,A(x))continuous transformations mapping X(f) → X(f) by Mathematically, we note that since G is compact and * is weakly measurable and thus weakly continuous; this implies that ( ) f G a  is σ(A(x) * ,A(x))-compact. The Krein-Smulian theorem [13], then shows that X(f) is also a σ(A(x) * ,A(x))-compact set. Thus X(f) is a non-void σ(A(x) * ,A(x))-compact convex subset of the locally convex Hausdorff linear topological space of ultraweakly continuous linear functionals acting on the fibre algebra A(x). ; , is a non-contracting (semi)-group of weakly continuous affine maps of X(f) onto itself. We can therefore, apply the Ryll Nardzewski fixed point theorem [14] to establish the existence of an invariant normal state contained in X(f). The physical implications highlight the role of what we have termed Wigner sets.
Given, define the Wigner set of the mapping v(g):X(f) → X(f) as the stabiliser set;