Kinematical Hilbert spaces for fermionic and Higgs quantum field theories

In the paper

“The authors of [21] have followed this trend and formulated a kinematical framework for diffeomorphism invariant theories of fermions coupling to real-valued Ashtekar gravity and other gauge fields. That is, they define a natural inner product by means of the Berezin integral, show that a certain class of the operators defined in [15] are well-defined on it and indicate how to define diffeomorphis”

What the sources say

• Partly supported

  Split into 2 atoms — 0 confirmed, 2 partially supported, 0 disputed, 0 misattributed, 0 unverifiable. Expand the claim to see per-atom verdicts.

  cites [21] sourceExperimental — accuracy not yet measured
• Partly supported

  Split into 2 atoms — 0 confirmed, 1 partially supported, 0 disputed, 0 misattributed, 1 unverifiable. Expand the claim to see per-atom verdicts.

  cites [15] Marolf D, 1995, Class. Quantum Grav. sourceExperimental — accuracy not yet measured
• Not checked yetcites [3] Ashtekar A, 1995, J. Math. Phys.

In the paper

“In this paper we define new fermionic canonical variables by casting the fermion field into a half-density . Not only does this trivialize the classical reality conditions on the fermion field, it is also forced on us :”

In the paper

“With the fermionic half-densities on the other hand we can then basically use the natural Berezin integral also employed in [21] to define the inner product. However, now we can rigorously compute adjoints for any, not even gauge-invariant operator constructed from fermion variables and then prove that the inner product is uniquely selected by the reality conditions.”

What the sources say

• Not checked yetcites [21] source

In the paper

“We then have the inclusion Φ F ⊂ H F ⊂ Φ ′ F since || f || 2 2 ≤ || f || 2 1 so that the assertion follows from the Schwarz inequality.”

In the paper

“Thus, we verified that the reality conditions are faithfully implemented.”

In the paper

“Thus || f || = 0 iff f = 0 which would not be the case without the additional 'density' dm/ ( d ¯ θdθ ).”

In the paper

“Notice that the measure µ F is diffeomorphism invariant in the sense that it assigns the same integral to any cylindrical function before and after mapping the points ⃗ v with a diffeomorphism.”

In the paper

“Such diffeomorphisms certainly exist as the reader can convince himself by considering in two dimensions the diffeomorphism ϕ ( x, y ) = ( α 2 x, y ) which leaves, say, the interval [0 , 1] along the y -axis invariant but J ϕ (0 , 0) = | α | .”

In the paper

“Notice that δ x,y is a Kronecker δ not a δ -distribution which is precisely in accordance with the classical symplectic structure { θ ( x ) , θ ( y ) } = -iδ x,y .”

In the paper

“We leave it as anexercise to check that indeed the first condition leads to d = 1, the second to a = 1 , c = 0 and the third to b = 0.”

In the paper

“The formal limit, as ǫ → 0, of this measure is given by”

In the paper

“It turns out that in its manifestly real form (the associated conjugation is just complex conjugation for non-spinorial variables and for spinorial fields it involves a cyclic reversal of order in products) the most convenient description of the constraints is in terms of half-densities ξ := 4 √ det( q ) η . The momentum conjugate to ξ A,µ is then just given by π A,µ = iξ A,µ :”

In the paper

“That is, given an element φ ∈ U and an element f ∈ V we obtain a complex number φ ( f ), moreover, φ ( f + g ) = φ ( f ) + φ ( g ) , φ ( fg ) = φ ( f ) φ ( g ) , φ ( f ⋆ ) = φ ( f ) , φ (1) = 1. This shows that U ⊂ U .”

In the paper

“We see that exp( iφ ( f )) equals the point holonomy exp( i ∑ x ∈ supp ( f ) f ( x )Φ( x )) which therefore comes out of the analysis naturally.”

In the paper

“The trouble with this approach is that multiplication by φ ( x ) but not ˆ p ( x ) are densely defined, not even when smeared with a test function”

In the paper

“They are not orthonormal, but almost : we just need to decompose the fermionic dependence into an orthonormal basis of the Q v .”

In the paper

“It is easy to check that Definition (4.3) is equivalent to the following cylindrical definition of µ U :”

In the paper

“The isomorphism is in fact an isometry since on the one hand”

• Also stated in: § 4 Point Holonomies · p. 17 · plain

  “Therefore || ˆ f || ≤ || f || .”

In the paper

“The corresponding Gaussian measure is formally given by [ d Φ 2 π ] exp( -1 / 2 ∑ z ∈ Σ Φ( z ) 2 ), a 'discrete white noise measure' which is not even expressible in terms of φ ( x ).”

In the paper

“Therefore the pairing φ ( f ) only has a covariant meaning (observing that Σ is invariant under diffeomorphisms) provided that φ is a scalar density ! This can be achieved by performing the classical canonical transformation ( φ, p ) → ( p, -φ ) and so does not pose any immediate problem.”

In the paper

“which implies that [ X I , X J ] = f IJK ( -X K R + X K L ) / 4 so that the algebra of the X I is not closed while the algebra of the ( -X R + X L ) / 2 is closed.”

In the paper

“In any case we are now in the position to show that the adjointness relations ˆ U v,µν ( φ ) † = ˆ U -1 v,νµ ( φ ) , ( p I ( B )) † = p I ( B ) ∀ v, B hold on H = L 2 ( U , dµ U ).”

In the paper

“The so defined family of measures { µ ⃗ v } is of course consistently defined because the Haar measure is a probability measure.”

In the paper

“That this completely defines Γ U follows from the fact that finite linear combinations of vertex functions are dense in V and thus by the Gel'fand isomorphism dense in C ( U ).”

In the paper

“Since the adjointness relations on an inner product are the only conditions that we can impose in lack of a background structure (in constructive field theory it is an axiom that the inner product be Poincar´ e invariant which usually fixes it uniquely) it is unclear from their framework whether their inner product is the appropriate one.”

In the paper

“This choice is radical because the fields are not smeared with test functions in three space directions. In fact, the phase of the theory that we obtain will be very different from the Fock phase, a phase appropriate for a manifestly diffeomorphism invariant description of the theory !”

In the paper

“The gravitational sector can be described [14] in terms of 'real-valued Ashtekar variables', that is, by a canonical pair ( A i a , E a i /κ ) where a, b, c etc. are tensorial indices for Σ, i, j, k etc. are su (2) indices, A i a is an SU (2) connection and E a i is its conjugate momentum, an ad SU (2) transforming vector density of weight one.”

What the sources say

• Not checked yetcites [14] Marolf D, 1995, Class. Quantum Grav. source

In the paper

“If one does not work with half densities then it turns out that a) The gravitational connection becomes complex valued. Thus, the gravitational gauge group would become become SL (2 , C) which is non-compact and would make the arsenal of techniques developed in [4, 5, 6, 7, 8, 3] inapplicable.”

What the sources say

• Not checked yetcites [4] Ashtekar A, 1992, Class. Quantum Grav. source
• Not checked yetcites [5] Ashtekar A, 1994
• Not checked yetcites [6] source
• Not checked yetcites [7] source
• Not checked yetcites [8] Marolf D, 1995, Commun. Math. Phys. source
• Not checked yetcites [3] Ashtekar A, 1995, J. Math. Phys.

In the paper

“The kinematical framework developed in this paper is applied in [22] which extends the dynamical framework of [16, 17, 18, 19, 20] to the standard model coupled to gravity and other kind of matter.”

What the sources say

• Not checked yetcites [22] source
• Not checked yetcites [16] source
• Not checked yetcites [17] source
• Not checked yetcites [18] Barbero F, 1995, Phys. Rev. source
• Not checked yetcites [19] source
• Not checked yetcites [20] source

In the paper

“The fact that for d = 2 also Yang-Mills theory is represented on this Hilbert space [10] can be traced back to the fact that this theory is almost diffeomorphism invariant, it is invariant under area-preserving diffeomorphisms.”

What the sources say

• Not checked yetcites [10] source

In the paper

“Again, all those properties are satisfied by the free scalar field theories and even some interacting scalar field theories such as for instance the P ( φ ) 2 theories.”

In the paper

“In [3] such theories were canonically quantized and the representation of the canonical commutation relations that was chosen is generated by the Abelian C ⋆ algebra W of Wilson loop observables via the GNS construction [4].”

What the sources say

• Not checked yetcites [3] Ashtekar A, 1995, J. Math. Phys.
• Not checked yetcites [4] Ashtekar A, 1992, Class. Quantum Grav. source

In the paper

“Notice that [3] also applies to quantum gravity when formulated as a dynamical theory of connections [13].”

What the sources say

• Not checked yetcites [3] Ashtekar A, 1995, J. Math. Phys.
• Not checked yetcites [13] Marolf D, 1995

In the paper

“We need it in its manifestly real formulation as advocated, for the Lorentzian signature, first in [14] because [3] only applies when the gauge group is compact (in this case SU (2)).”

What the sources say

• Not checked yetcites [14] Marolf D, 1995, Class. Quantum Grav. source
• Not checked yetcites [3] Ashtekar A, 1995, J. Math. Phys.

In the paper

“The framework was incomplete because no inner product for complex Ashtekar gravity was known at that time.”

In the paper

“Secondly, the Euclidean group of spacetime plays an important role in that the measure is required to be invariant under the motions of the Euclidean group and that the time translation subgroup acts ergodically on the measure space.”

In the paper

“the space A is 1) not a vector space, 2) not a subset of of S ′ (R d ) and 3) S ′ (R d ) is not contained in A .”

In the paper

“Moreover, the same techniques can be employed to quantize a) Euclidean 2+1 gravity in the real-valued Ashtekar- (rather than Witten-) formulation [19] giving the expected results and b) the ADM energy [20].”

What the sources say

• Not checked yetcites [19] source
• Not checked yetcites [20] source

In the paper

“in [15] the authors introduce an algebra of observables for complex Ashtekar gravity and the Maxwell field coupled to fermions. The fermions couple by making insertions at the endpoints of Wilson lines. The authors compute the Poisson and commutator algebra of this (overcomplete) set of gauge invariant observables.”

What the sources say

• Not checked yetcites [15] Marolf D, 1995, Class. Quantum Grav. source

In the paper

“However, certain difficulties having to do with the awkward reality structure of the fermion field make their framework incomplete in the sense that most of the interesting operators defined in [15] do not have an adjoint on the Hilbert space defined. In particular, this prevented the authors from proving that their inner product is selected by the classical reality conditions.”

What the sources say

• Not checked yetcites [21] source
• Not checked yetcites [15] Marolf D, 1995, Class. Quantum Grav. source

In the paper

“Notice that there is no minus sign missing in this canonical transformation because we take the fermion fields to be anti-commuting, the action is invariant under this transformation [23, 22].”

What the sources say

• Not checked yetcites [23] Thiemann T, 1998, Class. Quantum Grav. source
• Not checked yetcites [22] source

In the paper

“Namely, assume, for instance, that we have managed to construct an inner product such that ˆ π † = -i √ ̂ det( q ) ˆ ξ and if f = f ( A ) is any non-trivial real-valued function of the gravitational connection A then 0 = 0 † = ([ˆ π, f ( A )]) † = i [ √ ̂ det( q ) , f ( A )]ˆ η = 0 which is a contradiction unless the commutator vanishes which would be unnatural because the classical Poisson”

In the paper

“This is rather important because as we will see later, there is no obvious way to group average the ξ i ( x ) such that the result is a distribution.”

In the paper

“Without loss of generality we can take these as real valued by suitably raising the number of Higgs families.”

In the paper

“In particular, the probability measure µ AL is very natural and gauge and diffeomorphism invariant.”

In the paper

“The first reason comes from the fact that in the latter case the momentum conjugate to ξ would be the density π = i √ det( q ) ξ [22]. The density weight itself is not troublesome, what is troublesome is that the density weight comes from the function √ det( q ) which in quantum gravity becomes itself an operator. This immediately leads to inconsistencies as already stated in [21] if one wants to”

What the sources say

• Not checked yetcites [22] source
• Not checked yetcites [21] source

In the paper

“In this section we develop the integration theory for Grassman-valued half-densities since to our knowledge this is a situation usually not considered in the literature.”

In the paper

“Assume now that we would define π := i √ det( q ) η to be the momentum conjugate to η . Upon an integration by parts we see that the symplectic structure becomes up to a total time derivative equal to ∫ Σ d 3 x [ π ˙ η + 1 4 πηE i a ˙ E a i ] where E i a denotes the inverse of E a i . But since E a i is the momentum conjugate to A i a this implies that the real-valued gravitational connection A i”

What the sources say

• Not checked yetcites [4] Ashtekar A, 1992, Class. Quantum Grav. source
• Not checked yetcites [5] Ashtekar A, 1994
• Not checked yetcites [6] source
• Not checked yetcites [7] source
• Not checked yetcites [8] Marolf D, 1995, Commun. Math. Phys. source
• Not checked yetcites [3] Ashtekar A, 1995, J. Math. Phys.

In the paper

“Clearly, the fermion species η transforms like a scalar and according to an irreducible representation of SU (2) × G .”

In the paper

“Moreover, we will show now that if one does not work with half-densities, then the gravitational connection becomes actually complex-valued so that one could not apply the techniques associated with A / G which hold only for compact gauge groups.”

In the paper

“In fact, both problems are related because if one uses ψ as a canonical variable then A is not real-valued as we just proved and then 0 = [ˆ π, f ( A )] † = i [ √ ̂ det( q )ˆ η, f ( A ) † ] is not a contradiction but very messy and it will be hard to write f ( A ) † for general f .”

In the paper

“In [21] the authors partly solve the problem by restricting the adjoint operation to a certain sub-algebra of 'real elements' of the (gauge-invariant) observable algebra. However, as one can show, the action of the Hamiltonian constraint derived in section 3 does not preserve any cyclic subspace generated by that subalgebra.”

What the sources say

• Not checked yetcites [21] source

In the paper

“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.”

In the paper

“Finally, U is densely embedded into U by standard Gel'fand theory.”

In the paper

“The Gel'fand isomorphism V → C ( U ); f → ˆ f ; ˆ f ( φ ) := φ ( f ) shows that V is isomorphic with the set of continuous functions on its spectrum.”

In the paper

“Recall [1] that a measure for a field whose quantum configuration space is a vector space is rigorously defined in terms of its characteristic functional.”

What the sources say

• Not checked yetcites [1] Glimm J, 1987

In the paper

“We see that the assumption that φ was a density has lead to a more natural description in terms of scalar Φ so let us start with scalar φ := Φ right from the beginning and that the Gaussian measure is the discrete white noise measure in terms of φ ( x ).”

In the paper

“This is the precise analogue of the result [8] that A / G is the projective limit (rather than direct product) of spaces ( A / G ) γ , one for each graph γ , which is isomorphic with G n , n the number of edges of γ .”

What the sources say

• Not checked yetcites [8] Marolf D, 1995, Commun. Math. Phys. source

In the paper

“Since the spectrum is a compact Hausdorff space, positive linear functionals Γ on C ( U ) are in one to one correspondence with regular Borel probability measures on U , the correspondence being given by Γ( f ) = ∫ U dµf .”

In the paper

“We conclude that nothing is inconsistent.”

In the paper

“Recall that the Gel'fand spectrum of an (Abelian) C ⋆ algebra is the set of its characters, that is, the set of all ⋆ homomorphisms from V into the complex numbers.”