Quantum spin dynamics (QSD): V. Quantum gravity as the natural regulator of the Hamiltonian constraint of matter quantum field theories

May combine two claims.

The electric part of the Yang-Mills Hamiltonian becomes a sum of Laplace-Beltrami operators on G and is therefore sensitive to the colour of the state.

“the electric part of the Yang-Mills Hamiltonian becomes a sum of Laplace-Beltrami operators on G and therefore is sensitive to the colour of the state”

• AssumptionThe authors state this as an assumption: The gauge group G is taken to be semi-simple up to U(1) factors, and Lie algebra indices are raised and lowered with the Cartan-Killing metric.

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

The action of the Higgs field Hamiltonian is analogous to that of the gauge field.

“The action of that Hamiltonian is analogous to that of the gauge field.”

• AssumptionThe authors state this as an assumption: The Higgs field is taken to transform in the adjoint representation of G.

May combine two claims.

When the triangulation is refined indefinitely, the continuum limit exists and yields well-defined operators on the diffeomorphism-invariant Hilbert space.

“When refining the triangulation ad infinitum we find that the continuum limit exists and yields well defined operators on this diffeomorphism-invariant Hilbert space.”

May combine two claims.

This Hamiltonian has no singularities and is completely finite without renormalization.

“This Hamiltonian, surprisingly, does not suffer from any singularities, it is completely finite without renormalization.”

May combine two claims.

The continuum limit of the triangulation regularization for matter Hamiltonians exists, and the action of the continuum operator can be displayed precisely.

“We show that it exists and are able to display the action of the continuum operator precisely.”

May combine two claims.

Because the total measure is a probability measure, gauge-invariant functions remain integrable and the gauge group volume equals unity.

“It turns out that, because our total measure is a probability measure, gauge-invariant functions will still be integrable with respect to it, in other words, 'the gauge group volume' equals unity in our case!”

• AssumptionThe authors state this as an assumption: The gauge group G is taken to be semi-simple up to U(1) factors, and Lie algebra indices are raised and lowered with the Cartan-Killing metric.

May combine two claims.

The fermionic Hamiltonian operator removes fermionic excitations from open string endpoints and creates new excitations on the open string.

“The fermionic Hamiltonian operator removes fermionic excitations from open string endpoints and creates new excitations on the open string.”

May combine two claims.

The matter field theory Hamiltonian constraint is ultraviolet and infrared divergent without gravity and needs renormalization, but is convergent with gravity without renormalization.

“the matter field theory Hamiltonian constraint is ultraviolet and infrared (i) divergent without gravity and needs to be renormalized but are (ii) convergent with gravity without that renormalization is necessary.”

May combine two claims.

No singularities are encountered at any stage, and the final regularized matter operators do not require renormalization.

“At no stage do we encounter any singularities, these final operators do not require any renormalization.”

May combine two claims.

The volume operator has a local action and vanishes everywhere except at points where a string ends or starts.

“This volume operator turns out to have a quite local action, it vanishes everywhere except at points where the string ends or starts.”

May combine two claims.

For a fixed graph, the graph Hilbert space is infinite-dimensional and is the completion of spin-colour-network functions, while each colouring subspace is finite-dimensional with dimension equal to the number of independent gauge-invariant projectors compatible with that colouring.

“The Hilbert space H γ is infinite dimensional and is the completion of the space of functions built from spin-colour-network states on γ while H γ, λ is a finite dimensional vector space with dimension equal to the number of linearly independent projectors on gauge-invariant functions compatible with the colouring λ of γ .”

Classified as probably not a substantial claim.

Every cylindrical function is averaged term-by-term after it is decomposed into spin-colour-network states.

“Furthermore, we have [f ] := I c I [T I ], that is, every f ∈ gets averaged term-wise after decomposing it into spin-colour-network states”

Classified as probably not a substantial claim.

The Einstein contribution to the Hamiltonian constraint of the combined Dirac and Einstein action is H_G as given in equation (A.11).

“Then the Einstein contribution to the Hamiltonian constraint of S Dirac + S Einstein is given by H G := -H E + 2 2κ tr [K a , K b ] E a , E b √ det(q) (A”

Probably not a substantial claim: may not pass S3 (load-bearing).

Using D_a E^a_i = 0, the Gauss constraint can be written as equation (A.8).

“Using that D a E a i = 0 we can immediately write G j = 1 κ D a E a j + i det(q)[ψ † τ j ψ + η † τ j η]. (A.8)”

May combine two claims.

The electric piece of the Yang-Mills Hamiltonian and the term bilinear in Higgs momenta are essentially self-adjoint and positive semi-definite on the diffeomorphism-invariant Hilbert space.

“(iii) The electric piece of the Yang-Mills Hamiltonian and the term bi-linear in the Higgs momenta are indeed essentially self-adjoint and positive semi-definite operators on the diffeomorphism-invariant Hilbert space.”

May combine two claims.

The Dirac algebra is not faithfully represented, but the algebra is non-anomalous in the sense that a consistent quantum theory is obtained.

“(i) As we will see in the next section, the Dirac algebra is indeed not faithfully represented (one could, however, use the arguments of [27] to improve this), but still the algebra is non-anomalous in the sense that we obtain a consistent quantum theory”

• cites [27] sourceNot checked yet

  Not checked in the free run.

May combine two claims.

Trivial solutions include distributions defined on graphs whose vertices are all co-planar, because the volume operator annihilates such states so they cannot lie in the image of the Hamiltonian constraint; their fermion and Higgs dependence is arbitrary.

“Trivial solutions are, for instance, distributions which are defined by graphs all of whose vertices are co-planar, just because the volume operator annihilates such states so that they cannot be in the image of the Hamiltonian constraint.The dependence on fermion and Higgs fields of these solutions is completely arbitrary.”

May combine two claims.

The matrix element of the regulated matter Hamiltonian between diffeomorphism-averaged states is independent of δ and constant, so the limit is trivial and equals the regulated matrix element for any positive δ.

“since [T I ] is diffeomorphism invariant, this number, as explained in (4.2), does not depend on δ > 0 (the size or shape of the loop attached) and so is a constant.The limit is therefore trivial and so [T I ], Ĥ m [T J ] Diff = [T I ] Ĥ m δ T J (4.4) for any δ > 0.”

May combine two claims.

The Hilbert space splits into an uncountable direct sum over graphs and, for each graph, a countable direct sum over compatible colorings.

“H = ⊕ γ H γ , H γ = ⊕ λ H γ, λ(4.1)”

• Also stated in: § 4.1. Cylindrical consistency · plain

  “where the first direct sum is an uncountable one while the second is a countable one, running over the possible colourings of the graph with the various compatible irreducible representations”

Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

Equation (3.28) defines a densely defined operator.

“Again, despite its complicated appearance, equation (3.28) defines a densely defined operator.”

May combine two claims.

No compact algorithm for describing S in the matter-coupled case is available, so the paper restricts to trivial solutions, which is enough to show the theory is not empty.

“We see that we know the space of solutions once we know S. A compact algorithm to describe S as for the source-free case is not available at the moment for the matter coupled case so we restrict ourselves to displaying trivial solutions which is enough to show that the theory is not empty.”

May combine two claims.

The two functions appearing in the commutator are related by an analyticity-preserving diffeomorphism, which completes the proof of anomaly-freeness.

“It is then easy to see, by the same argument as in [12], that the two functions Ĥγ (v),v Ĥγ,v f, Ĥγ,v Ĥγ(v ),v f are related by an analyticity preserving diffeomorphism.This furnishes the proof of anomaly-freeness.”

• cites [12] Thiemann T Anomaly-free formulation of non-perturbativeNot checked yet

  Not checked in the free run.

May combine two claims.

The electric piece of the Yang-Mills Hamiltonian is densely defined, positive definite and essentially self-adjoint even on the auxiliary Hilbert space; a similar result holds for the Higgs momentum bilinear.

“Let us then conclude this subsection by showing that the electric piece of the Yang-Mills Hamiltonian is a densely defined, positive definite essentially self-adjoint operator even on the auxiliary Hilbert space H.A similar argument applies to the p I p I part of the Higgs Hamiltonian.”

May combine two claims.

For every Hamiltonian constraint matching (3.30), the construction yields an operator on H that is densely defined, diffeomorphism covariant, and free of anomalies.

“Theorem 3.1 (structure theorem).Any Hamiltonian constraint of the form (3.30) can be turned into a densely defined operator on H which is diffeomorphism covariantly defined and anomaly free.”

• AssumptionThe authors state this as an assumption: All covariant derivatives are with respect to A_i^a and A_I^a, act only on configuration scalars Q, and are first order only, so no Christoffel connection is needed.
• AssumptionThe authors state this as an assumption: The matter canonical momenta P are scalar densities of weight one and matter configuration variables are scalars, possibly transforming under SU(2)×G.

Classified as probably not a substantial claim.

The Hamiltonian constraint can be written as H = H_G + T, where T is the correction term in equation (A.12).

“H = H G - 2 2κ D a tr K b , E b E a √ det(q) =: H G + T . (A.12)”

May combine two claims.

The electric part no longer depends on the regulator, so its continuum limit is already taken, and it projects to a strongly diffeomorphism-invariant self-adjoint positive semi-definite operator on the Hilbert space defined by the continuum prescription.

“Notice also that the electric part does not depend on the regulator δ any longer so that the continuum limit is already taken.It therefore projects to a strongly diffeomorphisminvariant self-adjoint and positive semi-definite operator on the Hilbert space defined by (4.3) by general theorems proved in [6].”

• cites [6] Ashtekar A, 1995, J. Math. Phys.Not checked yet

  Not checked in the free run.

May combine two claims.

The Hamiltonian can be written as a sum of operators of the form †Â, so its projections are positive semi-definite and symmetric.

“Putting everything together we find that when setting whence we have displayed Ĥ in the form of a sum of operators of the form  † Â, that is, it has positive semi-definite and symmetric projections.”

May combine two claims.

Equation (3.8) does not depend on δ, so taking a limit is unnecessary.

“Now, since (3.8) actually is independent of δ, no limit needs to be taken.”

May combine two claims.

The full matter Hamiltonian operator as constructed in the paper is not symmetric but is diffeomorphism invariant.

“(ii) The full matter Hamiltonian operator, as it stands in this paper, is not symmetric but at least it is diffeomorphism invariant”

Probably not a substantial claim: may not pass S3 (load-bearing).

After lengthy computation, the Dirac action can be written in terms of Weyl spinors as equation (A.2).

“one sees that the action can be written, after lengthy computations, in terms of Weyl spinors as (using the Weyl representation for the Dirac matrices, for instance, to expand out various terms) S Dirac = 1 2 i R dt d 3 x N det(q) T a -N a N ψ † ∇ + a ψ + η † ∇ - a η -CC +e a i ψ † σ i ∇ + a ψ -η † σ i ∇ - a η -CC (A.2)”

May combine two claims.

Every matter Hamiltonian acts discretely on a cylindrical function and carries an overall factor of the Planck mass over a Planck-length power that offsets the volume operator.

“Notice that all Hamiltonians have the same structure, namely an operator which carries out a discrete operation on a cylindrical function, like adding or subtracting lines, fermions or Higgs fields, multiplied by the Planck mass and divided by an appropriate power of the Planck length which compensates the power of the Planck length coming from the action of the volume operator.”

May combine two claims.

The continuum matter Hamiltonian is not even symmetric; a pure Yang-Mills counterexample demonstrates this.

“We will now demonstrate that (4.4) is not even symmetric.We restrict ourselves to the case that we couple only a pure Yang-Mills field.We just need one counterexample.”

• Also stated in: § 4.2. Diffeomorphism-covariance, continuum limit self-adjointness and positive semi-definiteness · boosted

  “It follows that [T J ] Ĥ B Y M,δ T J = 0 thus contradicting symmetry.”

May combine two claims.

Defining the full operator as the sum of the projected regulated operators makes the cylindrical consistency condition hold by construction, which proves consistency.

“But this is the case by construction if we simply define Ĥ := γ, λ Ĥγ, λ pγ, λ .This suffices to prove consistency.”

Probably not a substantial claim: may not pass S4 (based).

The Higgs kinetic operator (3.23) is densely defined.

“The operator (3.23) is certainly quite complicated but it is densely defined!”

May combine two claims.

The resulting expression has factor ordering ambiguity but no factor ordering singularity.

“The resulting expression does suffer from a factor ordering ambiguity but not from a factor ordering singularity.”

May combine two claims.

Because the projections preserve each colouring subspace, the family defines a positive symmetric operator on the full Hilbert space, which has self-adjoint extensions; each projection and the whole operator are essentially self-adjoint, making the extension unique.

“Since these projections map H γ, λ into itself (the dependence on the segments s(e) involved in the definition of Ĥ E Y M drops out because of gauge invariance, see [24]), it follows that the family of projections defines a positive and symmetric operator on H which therefore has self-adjoint extensions (actually, by methods similar as for the volume, area and length operators [21][22][23][24] one”

• cites [21][22][23][24]Not checked yet

  Not checked in the free run.

May combine two claims.

The sum in (3.11) runs only over graph vertices and, for each vertex, tetrahedra based there.

“Combining both observations, we find that we need to sum only over vertices of the graph and for each vertex v over those tetrahedra such that v( ) = v.”

May combine two claims.

The commutator of two Hamiltonian constraints on a cylindrical function reduces to the expression (4.7); terms with equal vertices vanish, and for unequal vertices the local character of the vertex operator makes the relevant operators commute.

“In the last step we used that the terms with v = v trivially vanish while for the terms with v = v the local character of the operator Ĥγ,v , performing changes of γ only in a neighbourhood of v makes it commute with Ĥγ (v),v .”

May combine two claims.

The continuum limit is taken after integration; the limit before integration does not exist in the L2 sense.

“Notice that, just as in the continuum limit of a regulated Euclidean path integral of constructive quantum field theory, we take the limit after integrating.Indeed, the limit before integrating does not exist in the L 2 sense (for the same reason that the generator of the diffeomorphism constraint does not exist [6]).”

• cites [6] Ashtekar A, 1995, J. Math. Phys.Not checked yet

  Not checked in the free run.

May combine two claims.

Up to the K_i^a term, the constraint (A.15) is expressed in real-valued quantities and the canonically conjugate pairs (ξ, iξ) and (ρ, iρ).

“This is the form of the constraint that we have been looking for: up to K i a we have expressed everything in terms of real-valued quantities and the canonically conjugate pairs (ξ, i ξ), (ρ, i ρ).”

May combine two claims.

In the source-free case the correction T is proportional to the Gauss constraint and vanishes on the constraint surface; in the matter-coupled case, using the Gauss constraint, T is given by equation (A.13) with current J^j = ψ†σ^jψ + η†σ^jη.

“Notice that in the source-free case the correction T of H to H G is proportional to a Gauss constraint and therefore would vanish separately on the constraint surface.However, in our case, using the Gauss constraint (A.7) we find that T = -1 2 K a , E a j -E a j D a J j (A.13) where we have defined the current J j = ψ † σ j ψ + η † σ j η.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Equation (A.14) can be rewritten in terms of half-densities ξ and ρ as equation (A.15).

“Notice that we can write (A.14) also in terms of the half-densities ξ = 4 √ det(q)ψ, ρ = 4 √ det(q)η by absorbing the √ det(q) appropriately and using that D a det(q) = 0. We find C = H G + E a j 2 √ det(q) D a (ξ † σ j ξ + ρ † σ j ρ) +i[ξ † σ j D a ξ -ρ † σ j D a ρ -CC] -K j a (ξ † ξ -ρ † ρ) . (A.15)”

Probably not a substantial claim: may not pass S3 (load-bearing).

A real-valued Ashtekar connection can be used at the cost of a fairly complicated Lorentzian Hamiltonian constraint.

“one can also use a real-valued Ashtekar connection A i a = i a + K i a at the price of living with a fairly complicated Lorentzian Hamiltonian constraint.”

• cites [14] Thiemann T, 1998, Class. Quantum Grav. sourceNot checked yet

  Not checked in the free run.

May combine two claims.

The theory considered is a particular phase of the standard model coupled to gravity that is entirely non-perturbatively defined and second quantized.

“In contrast to string theory, however, we are dealing with a particular phase of the standard model coupled to gravity which is entirely non-perturbatively defined and second quantized.”

May combine two claims.

The volume is quantized with a discrete spectrum, and the quantum of volume is of order the Planck volume.

“the volume is quantized with a discrete spectrum, the quantum of volume being indeed of order 3 p (see [20,21])!”

• cites [20] Thiemann T Reality conditions inducing transforms for quantum gauge field theoryNot checked yet

  Not checked in the free run.

• cites [21] Thiemann T, 1996, Class. Quantum Grav. sourceNot checked yet

  Not checked in the free run.

May combine two claims.

The gravitational connection is real only when fermions are represented by half-density variables; using scalar variables makes the connection complex.

“As we will see in the appendix, the connection is real only if we use the quantities ξ with density weight 1 2 , if we would use the scalar variables η as in [9] then the gravitational connection becomes by the argument given in [18] A”

• cites [9] sourceNot checked yet

  Not checked in the free run.

• cites [18] F Barbero, 1995, Phys. Rev. sourceNot checked yet

  Not checked in the free run.

May combine two claims.

The fermionic excitations produced by this regularization differ greatly from those in the cited earlier work, because the reality conditions are faithfully implemented.

“Quite surprisingly, these fermionic excitations are very different from those discussed in [8,9], the underlying reason being the faithful implementation of the reality conditions.”

• cites [8] sourceNot checked yet

  Not checked in the free run.

• cites [9] sourceNot checked yet

  Not checked in the free run.

May combine two claims.

Because the volume operator is densely defined on the Hilbert space, the ultraviolet singularity is removed by coupling quantum gravity without renormalization, restoring diffeomorphism covariance.

“Since this volume operator turns out to be densely defined on the Hilbert space the UV singularity 1/ 3 is removed by coupling quantum gravity, without renormalization, thereby reinstalling diffeomorphism covariance.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The canonical action principle for general relativity coupled to Dirac fields in manifestly real form and in terms of the canonical pair A_i^a, E^a_i had not been derived in the literature before this work.

“In this section we wish to derive the canonical action principle for general relativity coupled to Dirac fields in manifestly real form and in terms of the canonical pair A i a = i a +K i a , E a i introduced in section 2. This has not been done so far in the literature because either one was interested in the standard Palatini formulation or in the Ashtekar formulation, the latter”

Classified as probably not a substantial claim.

The 3+1 split Dirac action is to be combined with the 3+1 split Einstein action to obtain the desired form in terms of A_i^a and E^a_i.

“This is the 3 + 1 split Dirac action that we are going to combine with the 3 + 1 split Einstein action to obtain the desired form in terms of A i a , E a i .”

May combine two claims.

Achieving the more ambitious goal that the commutator equal an operator proportional to a diffeomorphism generator is difficult because the diffeomorphism generator does not exist in the chosen representation and because an operator corresponding to q^ab is not obvious to construct.

“The difficulty in achieving this more ambitious goal is twofold as pointed out already in [12]: first, not even the generator of a one-parameter family of diffeomorphisms exists in the representation that we have chosen since the associated representation of the diffeomorphism group does not act strongly continuously on the Hilbert space.Secondly, since {H (M), H (N )} = V a q ab (M ,b N -MN ,b )”

• cites [12] Thiemann T Anomaly-free formulation of non-perturbativeNot checked yet

  Not checked in the free run.

May combine two claims.

The diffeomorphism-averaged spin-colour-network object is a well-defined distribution and the averaged states form an orthonormal basis of the diffeomorphism-invariant Hilbert space.

“As shown in [6,7], the following object makes sense as a distribution on : [T I ] := T ∈{T I } T , where {T I } := { Û(ϕ)T I : ϕ ∈ Diff( )} is the orbit of the vector T I under the diffeomorphism group.Moreover, as one can show, the [T I ] provide an orthonormal basis on the diffeomorphism-invariant Hilbert space”

• cites [6] Ashtekar A, 1995, J. Math. Phys.Not checked yet

  Not checked in the free run.

• cites [7] Thiemann T Kinematical Hilbert spaces for fermionic, Class. Quantum Grav.Not checked yet

  Not checked in the free run.

May combine two claims.

The theorem is inapplicable in one spatial dimension, where all tensors are densities, and in zero spatial dimension, where there is no field theory.

“However, in one spatial dimension all tensors are densities and in zero spatial dimension we do not have a field theory so that the theorem does not apply in these cases anyway.”

May combine two claims.

An operator corresponding to q^ab V_b has been constructed in prior work, and part of it generates diffeomorphisms.

“Indeed, in [26] such an operator is constructed and part of it actually does generate diffeomorphisms!”

• cites [26] sourceNot checked yet

  Not checked in the free run.

May combine two claims.

The δ-independence is due to the diffeomorphism covariance of the graph-dependent triangulation prescription, a result proved for the gravitational part and asserted to carry over to the matter-coupled case.

“The underlying reason for this is the diffeomorphism covariance of the graph-dependent triangulation prescription.This is proved in [12,26] for the gravitational part of the Hamiltonian constraint and the same reasoning applies to the matter coupled case as well and will not be repeated here.”

• cites [12] Thiemann T Anomaly-free formulation of non-perturbativeNot checked yet

  Not checked in the free run.

• cites [26] sourceNot checked yet

  Not checked in the free run.

• AssumptionRests on an assumption: The reasoning that proves δ-independence for the gravitational Hamiltonian constraint also applies to the matter-coupled case.

May combine two claims.

The construction applies in every spatial dimension d at least 2, since the inequality dk > n+m+k−1 admits a k for arbitrary n,m exactly when d exceeds 1.

“The theorem works the same way in any dimension d 2 because the critical condition dk > n + m + k -1 can be satisfied by some k for any value of n, m only for d > 1.”

May combine two claims.

A symmetric Hamiltonian constraint operator risks conflicting with faithful implementation of the Dirac constraint algebra, and a quantum anomaly is expected in that case.

“Now, by arguments explained in [25], a symmetric Hamiltonian constraint operator is in danger of being in conflict with the task of faithfully implementing the Dirac constraint algebra.Most certainly, one expects a quantum anomaly in this case.”

• cites [25] Choquet-Bruhat Y, 1989Not checked yet

  Not checked in the free run.

May combine two claims.

Because V and H_E(1) admit well-defined quantizations, equation (A.15) also admits a well-defined quantization despite its complicated appearance.

“Then it is still true that K j a = -A j a , {V , H E (1)} and since V , H E (1) admit well defined quantizations [12] we conclude that despite its complicated Quantum spin dynamics (QSD): V 1313 appearance (A.15) admits a well defined quantization as well.”

• cites [12] Thiemann T Anomaly-free formulation of non-perturbativeNot checked yet

  Not checked in the free run.

May combine two claims.

If one used ψ and η instead of half-densities, the gravitational connection would acquire a correction proportional to i e_i^a(ψ†ψ + η†η), forcing a complex connection that would be disastrous because the Hilbert space techniques would not be available.

“Note that if we had not worked with half-densities ξ, ρ but with the ψ, η then, while i √ det(q) ψ is the momentum conjugate to ψ, the gravitational connection would get a correction proportional to ie i a [ψ † ψ + η † η].Thus we would have had to admit a complex connection which would be disasterous as the Hilbert space techniques [1][2][3][4][5][6] would not be at our disposal.”

• cites [1][2][3][4][5][6]Not checked yet

  Not checked in the free run.

May combine two claims.

The strategy of working with ψ as advertised in prior work is not suitable for quantizing the Einstein-Dirac theory, so the framework developed in another prior work is forced.

“Therefore the strategy of working with ψ as advertised in [9] is not suitable for quantizing the Einstein-Dirac theory and we are forced to adapt the framework developed in [7].”

• cites [9] sourceNot checked yet

  Not checked in the free run.

• cites [7] Thiemann T Kinematical Hilbert spaces for fermionic, Class. Quantum Grav.Not checked yet

  Not checked in the free run.

May combine two claims.

The classical limit schemes lim_{ε→0} lim_{δ→0} and (lim_{δ→0})|_{ε=δ} are identical and both give back the classical magnetic Hamiltonian constraint.

“This is justified because the classical limit schemes lim →0 lim δ→0 and (lim δ→0 ) | = (δ), (0)=0 are identical, both give back the classical magnetic Hamiltonian constraint.”

• cites [27] sourceNot checked yet

  Not checked in the free run.