Fermions in quantum gravity

The complete classification of solutions of the diffeomorphism constraint is given by a generalization of the Knot Classes of pure gravity, with new classes including graphs with an arbitrary number of intersections and open ends.

“The complete classification of the solutions is given by a generalization of the Knot Classes of the pure gravity case - the new classes include graphs with an arbitrary number of intersections and open ends.”

Fermions can be incorporated in the loop representation simply by including open curves into Loop space.

“Not surprisingly, fermions can be incorporated in the loop representation simply by including open curves into 'Loop space', as expected from lattice Yang-Mills theory.”

• Also stated in: § 1.2 Introducing fermions in the loop representation · p. 3 · plain

  “The loop representation of QGD turns out to be a very natural extension of the pure gravity case, obtained by including open curves into Loop Space.”

The same simple action of the hamiltonian constraint, extended to open loops, codes the kinetic fermion energy and the fermions-gravity interaction.

“The surprising result here is that the very same action, extended to open loops, codes the kinetic fermion energy and the fermions-gravity interaction.”

• cites [18] [18] H. Morales-Tecotl. Ph.D. thesis 1993. Unpublished. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The diffeomorphism constraint can be fully solved as in pure gravity.

“The first can be fully solved as in pure gravity.”

• Also stated in: § 1.2 Introducing fermions in the loop representation · p. 3 · plain

  “It is not difficult to solve the diffeomorphism constraint on the resulting state space.”

The hamiltonian constraint operator essentially shifts loops along their tangent.

“In particular, the hamiltonian constraint admits the same simple geometrical interpretation as its pure gravity counterpart: it is the operator that shifts curves along themselves ('shift operator').”

• Also stated in: § 1.2 Introducing fermions in the loop representation · p. 3 · plain

  “The hamilonian constraint operator essentially 'shifts' loops along their tangent.”

The Hamiltonian operator that evolves the system in clock time is finite.

“We show that this Hamiltonian is finite, diffeomorphism invariant, and has a simple geometrical action confined to the intersections and the end points of the 'loops'.”

• Also stated in: § 1.3 Introducing a clock · p. 4 · plain

  “However, now the resulting operator is fully diffeomorphism invariant , and fi nite .”

The X and Y variables are SU(2)-invariant.

“They are clearly SU (2) -invariant.”

The quantum theory of fermions+gravity evolving in clock time is given by the combinatorial and geometrical action of this Hamiltonian on a set of graphs with a finite number of end points.

“The quantum theory of fermions+gravity evolving in the clock time is finally given by the combinatorial and geometrical action of this Hamiltonian on a set of graphs with a finite number of end points.”

• Also stated in: § 1.4 QGD · p. 4 · plain

  “The resulting QGD is then given by a set of quantum states represented by graphs with a finite number of intersections and open ends, and by an Hamiltonian operator that acts in a simple geometrical and combinatorial fashion on these graphs.”

The Hamiltonian operator has a simple geometrical action confined to the intersections and the end points of the loops.

“We show that this Hamiltonian is finite, diffeomorphism invariant, and has a simple geometrical action confined to the intersections and the end points of the 'loops'.”

The loop representation of QGD can be heuristically obtained from a naive Schrödinger-like representation by means of a (ill defined) Loop Transform.

“In analogy with the pure gravity case, we show that the resulting representation can also be heuristically obtained from a naive Shr¨ odinger-like representation by means of a (ill defined) Loop Transform.”

The first δ^3 function in the last equation forces the loop α to have an end-point α_e in x.

“The first δ 3 function in the last equation forces the loop α to have an end-point α e in ⃗ x .”

The integration over the small ball around this point, and the second δ^3 function, pick up a second point α(s)=x+y in α, close to α_e.

“The integration over the small ball (size δ ) around this point, and the second δ 3 function, pick up a second point α ( s ) = ⃗ x + ⃗ y in α , close to α e .”

The quantity (63) provides indeed a regularized form of the fermion hamiltonian constraint.

“We have shown that the quantity (63) provides indeed a regularized form of the fermion hamiltonian constraint.”

The general solution of the diffeomorphism constraint is given by loop functionals constant along the orbits of the action of the Diff group on loop space, namely by generalized knot classes K(α).

“The general solution of the diffeomorphism constraint is the given by the loop functionals constant along the orbits of the action of the Diff group on the loop space, namely, they are given by”

• Also stated in: § 3.1 Diffeomorphisms and diff-invariant states · p. 15 · plain

  “where K ( α ) is a generalized knot class, that is, an equivalent class under diffeomorphisms of sets of graphs formed by open and closed lines.”

The action of H[N] on closed loop states is fully equivalent to the action of the pure gravity hamiltonian constraint.

“It follows immediately that the action of ̂ H [ ∼ N ] on the closed loop states is fully equivalent to the action of the pure gravity hamiltonian constraint.”

X is invariant under inversion of the open loop, X[α^-1]=X[α].

“X is invariant under inversion of the open loop, X [ α -1 ] = X [ α ].”

• AssumptionThe authors state this as an assumption: The fermion fields ψ_A(x) anticommute (are Grassmann valued).

A straightforward expansion in τ and δ shows that equation (57) is satisfied, namely that the quantity defined represents a genuine regularization of the Hamiltonian.

“A straigthforward expansion in τ and δ shows that equation (57) is satisfied, namely that the quantity that we have defined represents a genuine regularization of the Hamiltonian.”

By applying the same geometrical operator on open loop states, we have the action of the Weyl-Einstein hamiltonian constraint.

“By applying this same geometrical operator on open loop states, we have the action of the Weyl-Einstein hamiltonian constraint.”

The commutator algebra of the generators of the natural action of the diffeomorphism group on loop space, among themselves and with all other operators, reproduces the corresponding classical Poisson algebra.

“The commutator algebra of these generators among themselves and with all the other operators in the theory, then, reproduces the corresponding classical Poisson algebra.”

• cites Isham [34] [34] C. J. Isham, in Relativity, Groups and Topology II , ed B.S. DeWitt and R. Stora (Amsterdam, Elsevier). -Not checked yet

  Not checked in the free run.

The different strategies for defining the quantum diffeomorphism constraint yield the same quantum diffeomorphism constraint.

“As in pure gravity, it is not difficult to show that these different strategies yeld the same quantum diffeomorphism constraint, and that this is can be expressed as follows.”

By computing the action of S[N], as defined in equation (50), on an open loop state, we get precisely the right hand side of equation (48).

“Indeed, by computing the action of S [ ∼ N ], as defined in equation (50) on an open loop state, we get precisely the right hand side of equation (48).”

• AssumptionThe authors state this as an assumption: In the calculation of the action of the Hamiltonian on an open loop state, α is a single open loop.

In the lattice case, the transform defines a unitary transformation to a new basis in the Hilbert space of the theory.

“In the lattice case it is possible to show that the transform defines a unitary transformation to a new basis in the Hilbert space of the theory.”

The Poisson algebra of the X,Y,T,Ta variables closes.

“Finally, the most important property of the X,Y,T,T a variables is that their Poisson algebra closes.”

The fermionic loop variables X and Y satisfy the retracing and spinor identities that follow from their being defined in terms of parallel propagators of an SU(2) connection.

“As their closed loops counterparts, the fermionic loop variables X and Y satisfies the retracing and spinor identities that follows from their being defined in terms of parallel propagators of an SU (2) connection.”

The Hamiltonian operator that evolves the system in clock time is diffeomorphism invariant.

“We show that this Hamiltonian is finite, diffeomorphism invariant, and has a simple geometrical action confined to the intersections and the end points of the 'loops'.”

• Also stated in: § 1.3 Introducing a clock · p. 4 · plain

  “However, now the resulting operator is fully diffeomorphism invariant , and fi nite .”

The set of operators X, Y, T, Ta provides a representation of the classical algebra (27), (28) and the classical T-algebra.

“The result is that the set of operators ˆ X, ˆ Y , ˆ T, ˆ T a provides a representation of the classical algebra (27), (28) and the classical T -algebra.”

Higher order loop variables have a quantum algebra that reduces to the corresponding classical Poisson algebra in the limit in which the Planck constant goes to zero.

“As their pure gravity counterparts, these have a quantum algebra that reduces to the corresponding classical Poisson algebra in the limit in which the Planck constant goes to zero.”

If α_i=β_i=γ_i, then X[α]X[β]X[γ]=0.

“Thus, for instance, if α i = β i = γ i , then X [ α ] X [ β ] X [ γ ] = 0.”

• AssumptionThe authors state this as an assumption: The fermion fields ψ_A(x) anticommute (are Grassmann valued).

If α has no self-intersections, and up to a divergent factor k, the Shift operator becomes simply the expression given in the paper.

“If α has no self-intersections, and up to a divergent factor k , the Shift operator becomes simply”

The operator F_e acts on end-points by displacing them and possibly changing associated rootings at intersections, but never creates or destroys end-points.

“The operator ˆ F e defined above acts on end-points by displacing them, and possibly by changing the associated rootings at intersections, but never creates or destroys end-points.”

The state |2,2,0;1_0^2> contains two fermions.

“There are two fermions in this quantum state.”

In the δ→0 limit with finite τ, if the end-point is free, the action of the end-point operator adds a small straight line of length τ=εa to the end point of the loop in the direction of the loop tangent.

“It is easy to see that if the end- point is free, the action of the operator is simply to add a small straight line of length τ = ǫa to the end point of the loop, in the direction of the loop tangent (goign out from the intersection).”

λ is a free dimensionless constant that emerges from the regularization procedure.

“λ is a free dimensionless constant that emerges from the regularization procedure.”

There exist solutions and spacetime regions where the inversion T(x,t) to t(x,T) can be made.

“On the other side, there are solutions and spacetime reions where we can make the inversion.”

If αe is not a free end-point, the loop α∗∗γτ αe, ⃗le belongs to a different knot class than α.

“If, on the other way, α e is not a free end-point, then the loop α ∗∗ γ τ α e , ⃗ l e does belong to a different knot class than α .”

• AssumptionThe authors state this as an assumption: Without loss of generality, α is a straight line at the intersection.

The number of end-points N is a conserved quantum number in the theory.

“Since [ ˆ N, ˆ H ] = 0, the number of end-points N is a conserved quantum number in the theory.”

H is a finite operator defined on knot states.

“ˆ H is a finite operator defined on knot states.”

M_i annihilates the state |2,2,0;1_0^2> because there are no intersections or kinks in that state.

“We have that ˆ M i | 2 , 2 , 0; 1 0 2 〉 = 0 because there are no intersections of kinks in | 2 , 2 , 0; 1 0 2 〉 .”

For a diffeomorphism invariant state, if αe is a free end-point, the addition of a small line at the end of a loop does not change the diffeomorphism equivalence class of the loop for small enough τ.

“because for small enough τ the added loop will not interject any other loop, and the addition of a small line at the end of a loop does not change the diffeomorphism equivalence class of the loop.”

The matrix elements of operators M_i and F_e can be directly computed between any two given knot states.

“The matrix elements of the operators ˆ M i ([5]) and ˆ F e can be directly computed between any two given knot states.”

• cites [5] [5] Rovelli C and Smolin L 1993 'The physical hamiltonian in non- perturbative quantum gravity ' Pittburgh University Preprint (1993) gr-qg ?. - sourceNot checked yet

  Not checked in the free run.

The Hamiltonian defined in the paper conserves the number of particles.

“Since the operator ˆ M i too, conserves the number of end-points, it follows that the hamiltonian that we have defined conserves the number of particles.”

The resulting action of F e is well-defined on the diffeomorphism invariant states.

“It is clear that the resulting action of ̂ F e is well-defined on the diffeomorphism invariant states.”

δ must go to zero faster than τ.

“This choice is not completely arbitrary: δ must go to zero faster that τ”

The limit of C(L,τ,δ) can be chosen consistently with all the demands by choosing L(ε), τ(ε), δ(ε) as given in the paper.

“We now claim that this limit can indeed be chosen consistently with all the demands, as follows”

The first significant term is of order zero in δ and first order in τ.

“we see that the first significative term, which is the one that we are considering, is of order zero in δ and of first order in τ .”

The prefactor is finite in the limit.

“Thus, the prefactor is finite in the limit.”

There exists a region Γ in phase space and a corresponding spacetime region S such that for any initial condition in Γ, dot T(x) is positive in S.

“More precisely, we can determine a region Γ in phase space, and a corresponding spacetime region S , such that for any initial condition in Γ, ˙ T ( ⃗ x ) is positive in S .”

Quantum states of QGD admit the same kind of topological description as the states of pure Quantum GR, contrary to the fear that this aspect of the Loop Representation could be lost in the presence of matter couplings.

“Thus, quantum states of QGD admit the same kind of topological description as the states of pure Quantum GR, contrary to the fear that this aspect of the Loop Representation could be lost in presence of matter couplings.”

In pure gravity the action of the hamiltonian constraint on Loop Space can be expressed as the action of the Shift operator (50), which simply shifts non-intersecting loops along their own tangent.

“In pure gravity the action of the hamiltonian constraint on Loop Space can be expressed as the action of the Shift operator (50), which simply shifts non-intersecting loops along their own tangent.”

The action of the Shift operator shifts the loop ahead along its tangent.

“The action of the Shift operator has thus a very simple geometrical interpretation: it shifts the loop ahead along its tangent.”

In pure gravity the Hamiltonian constraint can be simply expressed as the Shift operator.

“Thus in pure gravity the Hamiltonian constraint can be simply expressed as the Shift operator.”

The solutions of the gauge-fixed Einstein-Weyl-scalar-field theory are related to the solutions of the full Einstein-Weyl-scalar-field theory via equation (53).

“It is shown in [5] that the solutions of this theory, A ( ⃗ x, T ) , ψ ( ⃗ x, T ), are related to the solutions of the full Einstein-Weyl-scalar-field theory via equation (53).”

• cites [5] [5] Rovelli C and Smolin L 1993 'The physical hamiltonian in non- perturbative quantum gravity ' Pittburgh University Preprint (1993) gr-qg ?. - sourceNot checked yet

  Not checked in the free run.

If Ψ[A,ψ,T] satisfies the Wheeler-DeWitt constraint, then Ψ[A,ψ](T) satisfies equation (54).

“Moreover, if Ψ[ A,ψ,T ] satsfies the Wheeler-DeWitt constraint, it is shown in [5] that Ψ[ A,ψ ]( T ) satisfies the equation where the operator ̂ H will be defined in a moment.”

• cites [5] [5] Rovelli C and Smolin L 1993 'The physical hamiltonian in non- perturbative quantum gravity ' Pittburgh University Preprint (1993) gr-qg ?. - sourceNot checked yet

  Not checked in the free run.

The constraints form a Poisson algebra.

“This is shown, together with the fact that they form a Poisson algebra, in [Ashtekar et.al. ] by smearing the constraints as”

• cites [Ashtekar et.al. ]Not checked yet

  Not checked in the free run.

Loop states corresponding to smooth non-intersecting loops are in the kernel of the Hamiltonian constraint.

“This simple action is one of the ways in which one can (formally) understand the well known result that loop states corresponding to smooth non intersecting loops are in the kernel of the Hamiltonian constraint.”

The theory retains the clean geometrical features of pure quantum gravity.

“The theory retains the clean geometrical features of the pure quantum gravity.”

By a suitable gauge fixing one can express the dynamics of gravity as intrinsic evolution with respect to the intrinsic time defined by the scalar field.

“It is shown in Ref. [5] that by a suitable gauge fixing one can express the dynamics of gravity as intrinsic evolution with respect to the intrinsic time (or physical time) defined by the scalar field.”

• cites [5] [5] Rovelli C and Smolin L 1993 'The physical hamiltonian in non- perturbative quantum gravity ' Pittburgh University Preprint (1993) gr-qg ?. - sourceNot checked yet

  Not checked in the free run.

The quantum diffeomorphism constraint defined as the generator of the natural action of the diffeomorphism group on loop space represents a consistent quantization of its classical counterpart.

“Since the correct classical limit is the sole requirement we have on the theory, the quantum diffeomorphism constraint defined in this way represents a consistent quantization of its classical counterpart.”

If divergences are disregarded, the action of the Hamiltonian operator on a pure loop state can be written as the recalled expression from ref. [1].

“We now recall from ref.[1] that, if we disregard divergences, the action of the Hamiltonian operator on a pure loop state can be written as”

• cites [1] [1] C. Rovelli and L. Smolin, Phys. Rev. Lett. 61, 1155 (1988); Nucl. Phys. B133 (1990) 80. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The fact that the action of the hamiltonian constraint on the kernel of the Loop Transform can be expressed in terms of an operator acting on the loop variable enables us to interpret this operator as the operator that represents the Hamiltonian constraint in the Loop Representation.

“The fact that the action of the hamiltonian constraint on the kernel of the Loop Transform can be expressed in terms of an operator acting on the loop variable enables us to interpret this operator as the operator that represents the Hamiltonian constraint in the Loop Representation [1].”

• cites [1] [1] C. Rovelli and L. Smolin, Phys. Rev. Lett. 61, 1155 (1988); Nucl. Phys. B133 (1990) 80. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The coupled Einstein-Weyl equations can be coded in the extremely simple Shift operator.

“We consider the fact that the coupled Einstein-Weyl equations can be coded in the extremely simple Shift operator as a striking fact.”

The dynamics is given by the hamiltonian constraint, which includes the fermion kinetic energy and the fermion-gravity interaction.

“The dynamics is given by the hamiltonian constraint, which includes the fermion kinetic energy and the fermion-gravity interaction.”

The classical vector constraint also generates spatial diffeomorphisms in the quantum theory, provided the correct ordering of the vector constraint quantum operator is chosen.

“This is also true in the quantum theory providing the correct ordering of the vector constraint quantum operator is chosen.”

The coupling of a scalar clock field and the replacement of the hamiltonian constraint with an hamiltonian is a way to achieve a diffeomorphism invariant operator.

“As shown in ref.[5], the coupling of a scalar clock field and the replacement of the hamiltonian constraint with an hamiltonian is a way to acheive this result.”

• cites [5] [5] Rovelli C and Smolin L 1993 'The physical hamiltonian in non- perturbative quantum gravity ' Pittburgh University Preprint (1993) gr-qg ?. - sourceNot checked yet

  Not checked in the free run.

The Hamiltonian constraint, being a scalar density, is diffeomorphism covariant, not diffeomorphism invariant.

“the Hamiltonian constraint, being a scalar (density) is diffeomorphism covariant, not diffeomorphism invariant”

The minimally coupled action without quartic terms yields the Einstein-Cartan theory, in which fermions act as a source of (non-propagating) torsion.

“As made clear in [25], the minimally coupled action without quartic terms, yelds the EinsteinCartan theory, in which fermions act as a source of (non-propagating) torsion.”

• cites [25] [25] esposito morales -Not checked yet

  Not checked in the free run.

Equation (54) can be viewed as a genuine Schrödinger equation which evolves in the time T.

“We can view equation (54) as a genuine Schr¨ odinger equation which evolves in the time T .”

This evolution is explicitly generated on the state space by a Hamiltonian operator H.

“This evolution is explicitely generated on the state space by a hamitonian operator ˆ H .”

• cites [5] [5] Rovelli C and Smolin L 1993 'The physical hamiltonian in non- perturbative quantum gravity ' Pittburgh University Preprint (1993) gr-qg ?. - sourceNot checked yet

  Not checked in the free run.

Ashtekar et al. added quartic terms to the minimally coupled fermion-gravity action to get the equations of motion of the Einstein-Weil torsion-free theory.

“Ashtekar et.al. have added a quartic terms to the minimally coupled fermion-gravity action in order to get the equations of motion of the Einstein-Weil torsion-free theory.”

• cites [16] [16] A. Ashtekar, J. D. Romano, and R. S. Tate. New variables for gravity: Inclusion of matter. Phys.Rev.D D40, 2572-2587 (1989). -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

States with eigenvalues of H that are not real positive numbers must be excluded from the state space of QGD as formulated here.

“We must therefore exclude from the state space of QGD, as formulated here, the graphs that are eigenvalues of ˆ H witha an eigenvalue that is not a real positive number.”

The dimensionless nature of C(L,τ,δ) is a necessary condition for having a finite limit.

“This is a necessary condition for having a finite limit.”

Imaginary eigenvalues of H signal that the formalism is exiting its regime of validity, namely the clock regime.

“Thus, imaginary eigenvalues of ˆ H signal that we are exiting the regime of validity of the formalism we have developed here: the object we have chosen as clock is running backward.”

In the Einstein-Weyl theory, the quantity N commutes with all constraints, including the Hamiltonian constraint.

“This can be seen classically from the fact that the quantity <!-- formula-not-decoded --> commutes with all the constraints, including the hamiltonain constraint [30].”

• cites [30] [30] Kim -Not checked yet

  Not checked in the free run.

Since all these definitions correspond to the same classical limit, the choice amounts to a choice of different orderings of the quantum hamiltonian.

“Since all these definitions correspond to the same classical limit, the choice amount to a choice of different orderings of the quantum hamiltonian.”

When the Weyl system is coupled with gravitation, the particle-antiparticle distinction becomes local.

“In flat space one can globally distinguish particles from antiparticles; but when the Weyl system is coupled with gravitation, something curious happens: the particle antiparticle distinction becomes local.”

The matrix elements of H can be interpreted as first order transition amplitudes in a time dependent perturbation expansion in the clock time T.

“We can interpret the matrix elements of ˆ H as first order transitions amplitudes in a time dependent perturbation expansion in the clock time T .”

The action of the Hamiltonian constraint on generalized knot classes can be represented in a fully geometrical way in terms of the action of the operator on single intersections.

“It is clear from the discussion above that the action of the Hamiltonian constraint on the generalized knot classes can be represented in a fully geometrical way in terms of the action of the operator on single intersections.”

The quantum formalism constructed in the paper is meaningful only in the clock regime.

“The quantum formalism that we have constructed is meanigfull in the clock regime.”

The action of the Hamiltonian opens up the intersection between β and the end point of α.

“The action of the Hamiltonian, therefore, has opened up the intersection between β and the end point of α .”

The Weyl field theory describes a particle-antiparticle couple.

“The Weyl field theory describe a particleantiparticle couple (say a neutrino and its anti- neutrino).”