Kinematical Hilbert spaces for fermionic and Higgs quantum field theories

The kinematical framework for diffeomorphism invariant theories of connections for compact gauge groups is extended to a diffeomorphism invariant quantum field theory including fermions and Higgs fields.

“We extend the recently developed kinematical framework for diffeomorphism invariant theories of connections for compact gauge groups to the case of a diffeomorphism invariant quantum field theory which includes besides connections also fermions and Higgs fields.”

• AssumptionThe authors state this as an assumption: The four-dimensional manifold is (locally) of the form M = R × Σ, with Σ a smooth 3-manifold admitting smooth Riemannian metrics.

Grassman-valued half-densities enable solving the difficult fermionic adjointness relations.

“so-called Grassman-valued half-densities, which enable us to to solve the difficult fermionic adjointness relations.”

The classical reality conditions can be promoted to quantum adjointness relations, which uniquely fix the gauge and diffeomorphism invariant probability measure underlying the Hilbert space.

“Nevertheless we are able to promote the classical reality conditions to quantum adjointness relations which in turn uniquely fixes the gauge and diffeomorphism invariant probability measure that underlies the Hilbert space.”

The Higgs field framework in the diffeomorphism invariant context is developed by analogy between path holonomy and point holonomy, translating the gauge-field framework to arrive at the Higgs field Hilbert space.

“We develop the framework for the Higgs field, in the diffeomorphism invariant context, by pushing the analogy between a usual path holonomy and a 'point holonomy' constructed from a Higgs Field (see below) and can therefore translate step by step the framework [4, 5, 6, 7, 8] developed for gauge fields to arrive at the Higgs field Hilbert space.”

• cites [4] Ashtekar A, 1992, Class. Quantum Grav. sourceNot checked yet

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• cites [5] Ashtekar A, 1994Not checked yet

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• cites [6] sourceNot checked yet

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• cites [7] sourceNot checked yet

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• cites [8] Marolf D, 1995, Commun. Math. Phys. sourceNot checked yet

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The canonical framework of [3], which applies to any kind of gauge field theory, is extended to fermionic and Higgs field theories (including standard scalar fields not transforming under any gauge group besides the diffeomorphism group).

“In this paper we extend the canonical framework of [3], which applies to any kind of gauge field theory, to the rest of the matter fields, specifically, we are looking at fermionic and Higgs field theories (including standard scalar fields which do not transform under any kind of gauge group besides the diffeomorphism group).”

• cites [3] Ashtekar A, 1995, J. Math. Phys.Not checked yet

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• AssumptionThe authors state this as an assumption: The four-dimensional manifold is (locally) of the form M = R × Σ, with Σ a smooth 3-manifold admitting smooth Riemannian metrics.

The faithful implementation of the ⋆ relations fixes the Hilbert space completely.

“We show that the faithful implementation of the ⋆ relations fixes the Hilbert space completely.”

At finite ε_n, the regulated Fermion measure satisfies the ⋆ relations θ̄̂_n† = θ̂_n for each n.

“It is easy to see that the ⋆ relations ̂ ¯ θ † n = ˆ θ n hold for each n .”

The identity Θ = i/2 ∫Σ d3x [ξ̇ ξ − ξ ξ̇] holds because the time derivative of the half-density factor drops out in the antisymmetrized sum.

“then we have the identity Θ = i 2 ∫ Σ d 3 x [ ξ ˙ ξ -˙ ξξ ] because the time derivative of the halfdensity factor drops out in the antisymmetrized sum.”

The quantities θ_n have a formal limit as ε_n→0 and B_n→v(B_n)=:x, namely θ(x) given by the stated expression, where δ(x,y) is the three-dimensional δ-distribution.

“The quantities θ n have a formal limit as ǫ n → 0 (i.e. the triangulation becomes the continuum) and B n → v ( B n ) =: x : Namely, because χ n ( x ) = √ χ n ( x ) it follows that”

The measure μ_F is gauge invariant because a gauge transformation at x is a unitary transformation on the 2d dimensional vector space with respect to the inner product (3.5).

“It is also gauge invariant since a gauge transformation at x is in particular a unitary transformation on the on the 2 d dimensional vector space with respect to the inner product (3.5).”

The cylindrical product measures are consistently defined: m_{v1,...,vn}(f)=m_{v1,...,vm}(f) whenever f depends only on the fermion fields at the first m≤n points.

“These cylindrical product measures are consistently defined, that is, we have m v 1 ,..,v n ( f ) = m v 1 ,..,v m ( f ) whenever f depends only on the fermion fields on the fermion fields at the first m ≤ n points.”

The collection of fermionic vertex functions provides an orthonormal basis for H_F.

“Lemma 3.1 The collection of fermionic vertex functions provides an orthonormal basis for H F .”

The corresponding discrete operators enjoy discrete canonical anti-commutator relations even at finite ε_n because χ_m(x)χ_n(x)=δ_{m,n}χ_n(x).

“enjoy discrete canonical anti-commutator relations (even at finite ǫ n )”

• Also stated in: § 3 Grassman-valued half-densities · p. 11 · plain

  “because χ m ( x ) χ n ( x ) = δ m,n χ n ( x ).”

H_F := L2(S,dμ_F) is the appropriate and unique kinematical Hilbert space on holomorphic Grassman functionals in terms of θ(x) satisfying the appropriate adjointness relations.

“Thus, H F := L 2 ( S , dµ F ) is the appropriate and unique kinematical Hilbert space on holomorphic Grassman functionals (in terms of θ ( x )) satisfying the appropriate adjointness relations.”

For scalar valued fermions θ instead of ξ, the diffeomorphisms just discussed would not alter the orbit of the state and therefore would not contribute to the group average.

“Notice that for scalar valued fermions θ instead of ξ on the other hand the diffeomorphisms just discussed would not alter the orbit of the state and therefore would not contribute to the group average.”

Under a diffeomorphism φ, a cylindrical function with n_v ≤ 2d fermionic half-densities ξ at point v transforms as f_{v1,...,vn} → ∏_{k=1}^n 1/J_φ(v)^{n_v} f_{φ(v1),...,φ(vn)}, where J_φ(v)=|det(∂φ^a/∂x^b)(v)|^{1/2}.

“Due to the half-density weight we get f v 1 ,..,v n → ∏ n k =1 1 J ϕ ( v ) nv f ϕ ( v 1 ) ,..,ϕ ( v n ) where J ϕ ( v ) = | det( ∂ϕ a /∂x b )( v ) | 1 / 2 .”

The measure dμ_F is diffeomorphism invariant.

“By inspection, the measure dµ F is diffeomorphism invariant and obviously one formulates the theory better in terms of the θ ( x ) than in terms of the half-densities ξ ( x ).”

By defining θ̂_i f = θ_i f and θ̄̂_i f = ∂f/∂θ_i, the anti-Poisson algebra is faithfully represented together with the ∗ relations.

“Moreover, by defining ˆ θ i f = θ i f, ̂ ¯ θ i f = ∂f/∂θ i the anti-Poisson algebra is faithfully represented together with the ∗ relations.”

The operators θ̂ f = θ f and θ̄̂ f = df/dθ verify [θ̂, θ̄̂]_+ = 1.

“As usual we now quantize the Grassman algebra by ˆ θf := θf, ̂ ¯ θf := df/dθ which is easily checked to verify [ ˆ θ, ̂ ¯ θ ] + = 1”

The momentum conjugate to ξ is iξ.

“thus displaying as the momentum conjugate to ξ without that A i a gets a correction thus leaving the compactness of the gravitational gauge group intact.”

The anti-Poisson algebra is faithfully represented provided π = iθ̄ is the momentum conjugate to θ, so that π̂ = i d/dθ.

“which shows that the anti Poisson algebra is faithfully represented provided that π = i ¯ θ is the momentum conjugate to θ so that ˆ π = id/dθ .”

H := L2(S,dm) is the unique Hilbert space selected by the ⋆ relations.

“Thus H := L 2 ( S , dm ) is the unique Hilbert space selected by the ⋆ relations.”

The reality condition does not lead to the previously displayed inconsistencies because √det(q) does not show up in (3.3).

“which does not lead to the inconsistencies displayed above because the √ det( q ) does not show up in (3.3).”

Using the anti-bracket relation {ξ(x), ξ̄(y)}_+ = -iδ(x,y), one obtains {θ_m, θ̄_n} = -iδ_{m,n}.

“A similar calculation using the anti-bracket relation { ξ ( x ) , ¯ ξ ( y ) } + = -iδ ( x, y ) reveals that { θ m , ¯ θ n } = -iδ m,n .”

The transformation law under diffeomorphisms is compatible with the fact that the θ_i(x) are scalars rather than half-densities.

“This transformation law under diffeomorphisms is compatible with the fact that the θ i ( x ) are scalars rather than half-desities.”

dm is a positive measure.

“Then the integral of positive functions is positive and therefore dm is a positive measure.”

Because the δ distribution is a density of weight one, the square root and the half-density ξ make the quantities θ(x) transform as scalars.

“Notice that the δ distribution is a density of weight one so that due to the square root and the half-density ξ the quantities θ ( x ) transform as scalars !”

Given the representation on holomorphic functions of the anti-Poisson bracket algebra defined by θ̂f=θf and π̂f=i df/dθ, the measure dm is the unique probability measure selected by asking θ̂†=θ̄̂.

“We wish to show now that, given the representation on holomorphic functions of the antiPoisson bracket algebra defined by ˆ θf = θf, ˆ πf = id f /dθ , the measure dm is the unique probability measure selected by asking that ˆ θ † = ̂ ¯ θ .”

Upon dropping a total time derivative, Θ can be written as Θ = ∫Σ d3x [iξ] ξ̇.

“The last equation, upon dropping a total time derivative, can also be written as Θ = ∫ Σ d 3 x [ iξ ] ˙ ξ”

By construction the vertex functions provide a dense set of vectors in H.

“By construction the vertex functions provide a dense set of vectors in H .”

In the field theoretic case, the operators ξ̂_{Aμ}(x) f := ξ_{Aμ}(x) f and π̂_{Aμ}(x) f := iħ δf/δξ_{Aμ}(x) satisfy [ξ̂_{Aμ}(x), π̂_{Bν}(y)]_+ = iħ δ_{AB}δ_{μν}δ(x,y).

“It is easy to see that this satisfies the required anti-commutator relation [ ˆ ξ Aµ ( x ) , ˆ π Bν ( y )] + = i ¯ hδ AB δ µν δ ( x, y ) by using that the functional derivative is also Grassman-valued.”

The spectrum U is in one-to-one correspondence with the set Fun(Σ,G) of G-valued functions on Σ, the correspondence being given by the fundamental representation of G.

“Theorem 4.1 (Characterization of the spectrum) U is in one to one correspondence with the set Fun (Σ , G ) of G -valued functions on Σ , the correspondence being given by”

The Gel'fand topology can be characterized by the fact that the Gel'fand transforms of the vertex functions are continuous.

“Another result that we can copy is the following : the Gel'fand topology can be characterized by the fact that the (Gel'fand transforms of the) vertex functions are continuous.”

The adjointness relation on U_hat is identically satisfied.

“Moreover, by results following from [6] one can show that ˆ p I ( B ) is essentially self-adjoint and that the adjointness relation on ˆ U v,µ,ν is identically satisfied.”

• cites [6] sourceNot checked yet

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p_I(B) is essentially self-adjoint.

“Moreover, by results following from [6] one can show that ˆ p I ( B ) is essentially self-adjoint and that the adjointness relation on ˆ U v,µ,ν is identically satisfied.”

• cites [6] sourceNot checked yet

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μ_U(U)=0.

“Corollary 4.2 Since U ⊂ F p,C for any p, c we have µ U ( U ) = 0 .”

μ_U implements the ⋆ relations.

“This suffices to show that µ U implements the ⋆ relations.”

K_v is gauge invariant.

“Choosing B = B v small enough a neighbourhood of a point v we find that ˆ K ( B v ) =: ˆ K 2 v = -¯ h 2 X I v X J v δ IJ which is gauge invariant and all the ˆ K v actually commute as they should.”

If one assumes {p^I(B),p^J(B')}=0, then the Jacobi identity is violated.

“Otherwise even classically we get in conflict because if one assumes { p I ( B ) , p J ( B ′ ) } = 0 then the Jacobi 'identity' is violated.”

The operator p_I(x) is not densely defined on the L^2 space.

“The operator ˆ p I ( x ) := -i ¯ hδ/δφ I ( x ) on the other hand, which formally implements the canonical commutation relations [ ˆ φ I ( x ) , ˆ p J ( y )] = i ¯ hδ J I δ ( x, y ), is certainly not densely defined on the L 2 space.”

If one assumes φ(x) can be made well-defined operators, then the diffeomorphism invariant measure is automatically supported on those φ only which are of the form ∑_{z∈Σ} δ(x,z)Φ(z).

“But let us further stick with the assumption that the φ ( x ) can be made well-defined operators. Then the diffeomorphism invariant measure automatically is supported on those φ only which are of the form ∑ z ∈ Σ δ ( x, z )Φ( z ).”

• AssumptionThe authors state this as an assumption: The φ(x) can be made well-defined operators.

Gauge invariant functions are still integrable with respect to the total measure.

“It is easy to see that, because our total measure is a probability measure, gauge invariant functions will be still integrable with respect to it.”

All K_v commute.

“Choosing B = B v small enough a neighbourhood of a point v we find that ˆ K ( B v ) =: ˆ K 2 v = -¯ h 2 X I v X J v δ IJ which is gauge invariant and all the ˆ K v actually commute as they should.”

The operator U_hat_{μν}(v) is densely defined on L^2(U,dμ_U).

“The operator ( ˆ U µν ( v ) f )( φ ) := ( U φ ( v )) µν f ( φ ) is densely defined on L 2 ( U , dµ U ), thus the appropriate adjointness relation to be implemented is [ ˆ U φ ( v ) µν ] † = [( ˆ U φ ( v )) -1 ] νµ .”

The group averaging method always results in elements of Φ'_U.

“We quickly verify that this method always results in elements of Φ ′ U .”

The end result (4.8) is independent of the particular function h_ǫ or family of loops α_{r,s} used.

“The end result (4.8) is independent of the particular function h ǫ or family of loops α r,s of loops used and is therefore satisfactory.”

U is contained in a measurable set of μ_U measure zero.

“while U is topologically dense in U it is contained in a measurable set of µ U measure zero.”

• cites [8] Marolf D, 1995, Commun. Math. Phys. sourceNot checked yet

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∑_{z∈Σ}|f(z)|2 does not diverge only if the test function f is not smooth but vanishes at all but a finite number of points.

“But ∑ z ∈ Σ | f ( z ) | 2 does not diverge only if the test function f is not smooth but vanishes at all but a finite number of points.”

Then φ(f)=0 for almost all tempered distributions φ except those of the form φ(x)=∑_{z∈Σ} δ(x,z)Φ(z), where Φ(z) is a scalar.

“But then φ ( f ) = 0 for almost all tempered distributions φ except for those which are of the form φ ( x ) = ∑ z ∈ Σ δ ( x, z )Φ( z ) where Φ( z ) is a scalar.”

The functional derivative of quantity (4.6) is well-defined.

“The functional derivative of the quantity (4.6) is well-defined : We get just zero unless there is a solution α r x ,s x ( t x ) = x with ( r x , s x , t x ) ∈ (0 , 1) 2 × S 1 in which case we find”

For any point p and curve c starting at p, μ_U(F_{p,c})=0.

“Then µ U ( F p,c ) = 0 .”

The vertex functions provide an orthonormal basis for L^2(U,dμ_U).

“Corollary 4.1 The vertex functions provide an orthonormal basis for L 2 ( U , dµ U ) .”

• AssumptionRests on an assumption: The gauge group G is compact.

The set of φ's which are continuous at any point and in any direction is of measure zero.

“To see this we show a stronger result : the set of φ 's which are continuous at any point and in any direction is of measure zero.”

The operator p_I(B) is densely defined.

“Let us check that ˆ p I ( B ) is indeed densely defined.”

The spectrum is homeomorphic with the uncountable direct product X=∏_{v∈Σ} X_v of copies of G equipped with the Tychonov topology.

“Now we can see that the spectrum is homeomorphic with the uncountable direct product X := ∏ v ∈ Σ X v of copies of G (that is, X v = G ) equipped with the Tychonov topology.”

The measure μ_U is gauge invariant.

“It is immediate from this definition that the measure µ U is diffeomorphism invariant and gauge invariant.”

The measure μ_U is diffeomorphism invariant.

“It is immediate from this definition that the measure µ U is diffeomorphism invariant and gauge invariant.”

The measure μ is diffeomorphism invariant if and only if χ(φ⋆ f)=χ(f) for any φ.

“Therefore, the measure µ is diffeomorphism invariant if and only if χ ( ϕ ⋆ f ) = χ ( f ) for any ϕ .”

Spin-colour-network states are a basis for the subspace of gauge invariant functions.

“These spin-colour-networks turn out to be a basis for the subspace of gauge invariant functions.”

The measure is an infinite product of Gaussian measures, one for each point in Σ.

“We see that the measure is an infinite product of Gaussian measures, one for each point in Σ.”

The only quantity merely built from the coordinates that transforms like a double density and is diffeomorphism invariant is of the form C(x,y)=∑_{z∈Σ} δ(z,x)δ(z,y), involving a sum rather than an integral over Σ.

“The only quantity merely built from the coordinates, which transfroms like a double density and is diffeomorphism invariant is of the form C ( x, y ) = ∑ z ∈ Σ δ ( z, x ) δ ( z, y ) (notice that this involves a sum rather than an integral over Σ).”

In this case a representation of the diffeomorphism group is given by U(φ) φ(f)=φ([φ^{-1}]^* f).

“In this case we have a representation of the diffeomorphism group given by ˆ U ( ϕ ) φ ( f ) = φ ([ ϕ -1 ] ∗ f ).”

Diffeomorphism invariance imposes C(x,y)=|det(∂φ(x)/∂x) det(∂φ(y)/∂y)| C(φ(x),φ(y)), which is the transformation law of a double density.

“C ( x, y ) = | det( ∂ϕ ( x ) /∂x ) det( ∂ϕ ( y ) /∂y ) | C ( ϕ ( x ) , ϕ ( y )) which is the transformation law of a double density.”

Equation (4.9) is covariant under gauge transformations.

“This is an important consistency check because it implies that X I ( hgh -1 ) = Ad ( h ) IJ X J ( g ) transforms according to adjoint representation as well and so (4.9) is covariant under gauge transformations.”

The extended framework is appropriate for coupling matter to quantum gravity.

“This framework is appropriate for coupling matter to quantum gravity.”

Most of the fermionic part of this work is independent of the recent preprint by Baez and Krasnov and earlier work by Rovelli and Morales-Tecotl because new canonical fermionic variables (Grassman-valued half-densities) are used.

“Most of the fermionic part of this work is independent of the recent preprint by Baez and Krasnov and earlier work by Rovelli and Morales-Tec´ otl because we use new canonical fermionic variables, so-called Grassman-valued half-densities, which enable us to to solve the difficult fermionic adjointness relations.”

• cites Baez and KrasnovNot checked yet

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• cites Rovelli and Morales-Tec´ otlNot checked yet

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In constructive quantum field theory, the measure on the space of quantum fields is required to be supported on the space of tempered distributions S′ on R^{d+1}.

“This follows from the fact that right from the beginning the measure on the space of quantum fields is required to be supported on the space of tempered distributions S ′ on R d +1 where d is the dimension of a spacelike hypersurface in Minkowski space.”

• cites [1] Glimm J, 1987Not checked yet

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Diffeomorphism invariance forces a non-Fock-like representation in which connection excitations are along open or closed strings and fermion or Higgs excitations are at string endpoints.

“The presence of diffeomorphism invariance forces us to choose a representation which is a rather non-Fock-like one : the elementary excitations of the connection are along open or closed strings while those of the fermions or Higgs fields are at the end points of the string.”

The Osterwalder-Scrader axioms of constructive quantum field theory are only appropriate for kinematically linear theories, where the space of quantum fields is a vector space before imposing quantum dynamics.

“The Osterwalder-Scrader axioms of constructive quantum field theory [1] are first of all only appropriate for kinematically linear theories, that is, the space of quantum fields is a vector space (before imposing the quantum dynamics).”

• cites [1] Glimm J, 1987Not checked yet

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The Hilbert space of this class of theories, i.e. the measure on A, is uniquely determined by promoting the reality structure of the classical phase space into adjointness relations with respect to the inner product.

“The Hilbert space of this class of theories, in other words the measure on A , is then uniquely determined by promoting the reality structure of the classical phase space into adjointness relations with respect to the inner product.”

Bochner-Minlos theory guarantees that under natural assumptions about the characteristic functional of any measure (satisfied by free scalar field theories) the support is contained in S′.

“by the Bochner-Minlos theory [2] which tells us that under natural asuumptions about the characteristic functional of any measure (satisfied by free scalar field theories) the support is guaranteed to be contained in S ′ .”

• cites [2] Yamasaki Y, 1984Not checked yet

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The quantum configuration space of distributional gauge fields A (or A/G after moding by gauge freedom) is naturally identified with the compact Hausdorff space obtained from W by the Gel'fand isomorphism.

“The quantum configuration space of distributional gauge fields A (or A / G after moding by the gauge freedom) is naturally identified with the compact Hausdorff space that one gets from W by the Gel'fand isomorphism (recall that an Abelian C ⋆ algebra is isometric isomorphic with the algebra of continuous functions on a compact Hausdorff space).”

Diffeomorphism invariant theories of connections for a compact gauge group in d+1 dimensions provide an example of departure from S′.

“An already well-established example for such a departure from S ′ is provided by diffeomorphism invariant theories of connections for a compact gauge group in d +1 dimensions.”

• cites [3] Ashtekar A, 1995, J. Math. Phys.Not checked yet

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The discrete time reflection subgroup enables the condition of reflection positivity, by which one constructs a Hamiltonian and a Hilbert space.

“In particular, the discrete time reflection subgroup enables us to state the condition of 'reflection positivity' [1] by means of which one makes contact with physics, that is, one constructs a Hamiltonian and a Hilbert space.”

• cites [1] Glimm J, 1987Not checked yet

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The Gaussian measure underlying the Klein-Gordon field is supported on S′.

“by the fact that the Gausian measure underlying, say, the Klein-Gordon field is actually supported on S ′”

If one does not work with half densities, the faithful implementation of the reality conditions on scalar fermion fields at the quantum level is inconsistent with the implementation of the canonical Anti-Poisson brackets, as observed in [21].

“b) The faithful implementation of the reality conditions on scalar fermion fields at the quantum level is inconsistent with the implementation of the canonical Anti-Poisson brackets as observed in [21].”

• cites [21] sourceNot checked yet

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For the free Maxwell field (scattering processes), this Hilbert space is worthless except in d=2 because it does not coincide with Fock space.

“Of course, from the point of view of, say, the free Maxwell field (scattering processes) this Hilbert space is worthless (except in d = 2 [9, 10]) because it does not coincide with Fock space.”

• cites [9] Ashtekar A, 1994Not checked yet

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• cites [10] sourceNot checked yet

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Taking the non-linear structure of the quantum configuration space seriously yields a fairly non-standard Hilbert space in any spatial dimension d which still represents the canonical commutation relations faithfully.

“Thus, as argued above, taking the non-linear structure of the quantum configuration space seriously we obtain a fairly non-standard Hilbert space in any spatial dimension d which still represents the canonical commutation relations faithfully.”

The result that the inner product is uniquely selected by reality conditions can be established at the non-gauge-invariant level, without dealing with complicated path observables defined in [15,21].

“We do not need to deal with the complicated path observables defined in [15, 21] in order to establish this result because we can prove everything at the non-gauge-invariant level.”

• cites [15] Marolf D, 1995, Class. Quantum Grav. sourceNot checked yet

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• cites [21] sourceNot checked yet

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By the Kolmogorov theorem for the uncountable direct product, the consistent family {m_{v1,...,vn}} has a unique σ-additive extension to the infinite product measurable space S:=×_x S_x.

“Since then we have an uncountable product of probability measures the Kolmogorov theorem for the uncountable direct product [2] tells us that the consistent family { m v 1 ,..,v n } has the unique σ -additive extension (3.15) to the infinite product measurable space S := × x S x .”

• cites [2] Yamasaki Y, 1984Not checked yet

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In an attempt to resolve the problems that occur, one is naturally led to the consideration of point holonomies.

“In an attempt to resolve the problems that occur we will see that we are naturally lead to the consideration of point holonomies.”

For the Einstein-Dirac theory in variables (A^i_a, E^a_i), the symplectic structure for each fermionic degree of freedom is Θ = i/2 ∫ d^3x √det(q)[η ̇η - ̇ηη] with scalar-valued η.

“The second reason becomes evident when recalling that for the Einstein-Dirac theory, formulated in terms of the variables ( A i a , E a i ) [22] which have proved to be successful for quantizing the Wheeler-DeWitt constraint [16, 17], the symplectic structure for each of the fermionic degrees of freedom is given by Θ = i 2 ∫ Σ d 3 x √ det( q )[ η ˙ η -˙ ηη ] with scalar-valued η .”

• cites [22] sourceNot checked yet

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• cites [16] sourceNot checked yet

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• cites [17] sourceNot checked yet

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A kinematical measure of Gaussian type is in conflict with diffeomorphism invariance when one implements the canonical commutation relations and the adjointness relations.

“We will now argue that this sort of a kinematical measure is in conflict with diffeomorphism invariance when one implements the canonical commutation relations and the adjointness relations”

• AssumptionThe authors state this as an assumption: The basic configuration field variables for scalar fields are φ^I(x), and a Gaussian measure is a natural choice for the inner product.

The shift from ξ(x) to θ(x) becomes mandatory in the diffeomorphism invariant context.

“We conclude this section with showing why the shift from ξ ( x ) to θ ( x ) becomes mandatory in the diffeomorphism invariant context :”

It is no lack of generality to restrict to just one helicity because the canonical transformation (i ̄ξ, ξ) → (iεξ, εξ) can always be performed.

“Notice that it is no lack of generality to restrict ourselves to just one helicity as we can always perform the canonical transformation ( i ¯ ξ, ξ ) → ( iǫξ, ǫξ ) where ǫ is the spinor-metric, the totally skew symbol in two dimensions.”

The techniques of [5,6,7,8,3] can be applied to write down a kinematical inner product for the gravitational and Yang-Mills sector that faithfully incorporates all the reality conditions, yielding a Hilbert space L^2(A_SU(2) × A_G, dμ_AL,SU(2) ⊗ dμ_AL,G).

“We can immediately apply the techniques of [5, 6, 7, 8, 3] to write down a kinematical inner product for the gravitational and YangMills sector that faithfully incorporates all the reality conditions. We get a Hilbert space L 2 ( A SU (2) ×A G , dµ AL,SU (2) ⊗ dµ AL,G ) where the index 'AL' stands for AshtekarLewandowski measure and the group index indicates to which gauge group the AshtekarLewand”

• cites [5] Ashtekar A, 1994Not checked yet

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• cites [6] sourceNot checked yet

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• cites [7] sourceNot checked yet

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• cites [8] Marolf D, 1995, Commun. Math. Phys. sourceNot checked yet

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• cites [3] Ashtekar A, 1995, J. Math. Phys.Not checked yet

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• AssumptionThe authors state this as an assumption: The gauge group G is an arbitrary compact gauge group.

The gravitational connection of the Einstein-Dirac theory is real valued (an su(2) rather than sl(2,C) connection) only if one uses half-densities.

“Namely, as we will show in the next section, the gravitational connection of the Einstein-Dirac theory is real valued (that is, it is an su (2) rather than sl (2 , C) connection) only if one uses half-densities.”

The vector fields X^I_R and X^I_L on G are respectively right and left invariant.

“The vector fields X I R and X I L on G are respectively right and left invariant in the sense X R ( gh ) = X R ( g ) , X L ( hg ) = X L ( g ) for each h ∈ G as one can easily check.”

The anomaly that [p^I(B),p^J(B)] is not expressible in terms of a p^I(B') is due to not properly smearing the φ's.

“The reason for this 'anomaly' is due to the fact that we did not properly smear the φ 's as explained in [26] :”

• cites [26] Thiemann T, 1998, Class. Quantum Grav. sourceNot checked yet

  Not checked in the free run.

The cylindrical definition of μ_U is rigorous because a measure which on cylindrical subspaces is a product of probability measures corresponds uniquely to a σ-additive measure on the uncountable product measurable space.

“Notice that this definition is rigorous : As is shown in [2], a measure which on cylindrical subspaces is a product of probability measures corresponds uniquely to a σ -additive measure on the uncountable product measurable space.”

• cites [2] Yamasaki Y, 1984Not checked yet

  Not checked in the free run.

The label f(x) is a function of the coordinates whose only meaningful transformation law under diffeomorphisms is that it is a scalar, and therefore f is a scalar.

“The point is now that the label f ( x ) is a function of the coordinates x a whose only meaningful transformation law under diffeomorphisms amounts to saying that they are scalars and therefore f is a scalar.”

The interpretation of U(v) as a point holonomy is correct.

“which is exactly the result (4.9), thus reassuring that our interpretation of U ( v ) as a point holonomy is correct.”

The correspondence {.,.}→1/(iħ)[.,.] holds everywhere only if p's and φ's are smeared in three directions.

“we can expect the correspondence { ., . } → 1 i ¯ h [ ., . ] to hold everywhere only if p 's and φ 's are smeared in three directions.”

• cites [26] Thiemann T, 1998, Class. Quantum Grav. sourceNot checked yet

  Not checked in the free run.

The Gaussian measure defined above, which comes from the assumption that the φ are well-defined operators, leads to rather difficult mathematical problems given the adjointness relations.

“Thus, we conclude that the Gaussian measure defined above and which comes from the assumption that the φ are well-defined operators leads to rather difficult mathematical problems given our adjointness relations.”

• AssumptionThe authors state this as an assumption: The φ(x) can be made well-defined operators.