Kinematical Hilbert spaces for fermionic and Higgs quantum field theories

In the paper

“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.”

What the sources say

• Supported

  The citation is attached to the pre-existing framework for gauge fields, not to the citing authors’ proposed point-holonomy construction for Higgs fields. Section 1.2 (p. 6) identifies the holonomy-algebra Hilbert space as L²(∆, dµ), and the Conclusion (p. 39) describes the framework constructed for gauge theories and gravity. (LLM, conf=0.91)

  cites [4] Ashtekar A, 1992, Class. Quantum Grav. source
• Not checked yetcites [5] Ashtekar A, 1994
• Supported

  The citing sentence presents the Higgs-field construction as the citing authors’ own work and cites this paper for the gauge-field framework they adapt. The Abstract (p. 1) describes a background-metric-free geometrical framework for connections suited to diffeomorphism-invariant theories; the Introduction (p. 4) describes holonomies and the associated Hilbert space. (LLM, conf=0.90)

  cites [6] source
• Supported

  The citation identifies a framework developed for gauge fields, which the citing authors say they will adapt to Higgs fields; it does not attribute the Higgs-field construction to this source. The Discussion (p. 33) describes the paper's projective framework for gauge theories, while the Introduction (pp. 3, 6) discusses holonomies, a gauge-theory Hilbert space, and diffeomorphism-invariant measures. (LLM, conf=0.90)

  cites [7] source
• Supported

  The citation is attached to the framework developed for gauge fields, not to the citing authors’ proposed Higgs-field analogy. Section 3 (p. 7) develops the holonomy-based configuration space for gauge connections, and the Introduction (p. 2) discusses its diffeomorphism-invariant measure. The source therefore supports its identification as part of the gauge-field framework being translated. (LLM, conf=0.91)

  cites [8] Marolf D, 1995, Commun. Math. Phys. source

In the paper

“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.”

In the paper

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

In the paper

“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.”

In the paper

“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).”

What the sources say

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

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

“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”

In the paper

“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).”

In the paper

“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.”

In the paper

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

In the paper

“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 ).”

In the paper

“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.”

In the paper

“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.”

In the paper

“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 .”

In the paper

“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 ).”

In the paper

“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.”

In the paper

“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”

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

“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 .”

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

“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 !”

In the paper

“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 .”

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

“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”

In the paper

“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.”

In the paper

“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.”

What the sources say

• Not checked yetcites [6] source

In the paper

“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.”

What the sources say

• Not checked yetcites [6] source

In the paper

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

In the paper

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

In the paper

“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.”

In the paper

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

In the paper

“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.”

In the paper

“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 ).”

In the paper

“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.”

In the paper

“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.”

In the paper

“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 .”

In the paper

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

In the paper

“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.”

In the paper

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

What the sources say

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

In the paper

“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.”

In the paper

“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.”

In the paper

“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”

In the paper

“Then µ U ( F p,c ) = 0 .”

In the paper

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

In the paper

“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.”

In the paper

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

In the paper

“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.”

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

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

In the paper

“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 the paper

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

In the paper

“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.”

In the paper

“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.”

In the paper

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

In the paper

“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.”

What the sources say

• Not checked yetcites Baez and Krasnov
• Not checked yetcites Rovelli and Morales-Tec´ otl

In the paper

“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.”

What the sources say

• Not checked yetcites [1] Glimm J, 1987

In the paper

“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.”

In the paper

“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).”

What the sources say

• Not checked yetcites [1] Glimm J, 1987

In the paper

“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.”

In the paper

“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 ′ .”

What the sources say

• Not checked yetcites [2] Yamasaki Y, 1984

In the paper

“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).”

In the paper

“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.”

What the sources say

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

In the paper

“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.”

What the sources say

• Not checked yetcites [1] Glimm J, 1987

In the paper

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

In the paper

“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].”

What the sources say

• Not checked yetcites [21] source

In the paper

“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.”

What the sources say

• Not checked yetcites [9] Ashtekar A, 1994
• Not checked yetcites [10] source

In the paper

“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.”

In the paper

“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.”

What the sources say

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

In the paper

“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 .”

What the sources say

• Not checked yetcites [2] Yamasaki Y, 1984

In the paper

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

In the paper

“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 .”

What the sources say

• Not checked yetcites [22] source
• Not checked yetcites [16] source
• Not checked yetcites [17] source

In the paper

“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”

In the paper

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

In the paper

“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.”

In the paper

“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”

What the sources say

• 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

“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.”

In the paper

“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.”

In the paper

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

What the sources say

• Not checked yetcites [26] Thiemann T, 1998, Class. Quantum Grav. source

In the paper

“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.”

What the sources say

• Not checked yetcites [2] Yamasaki Y, 1984

In the paper

“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.”

In the paper

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

In the paper

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

What the sources say

• Not checked yetcites [26] Thiemann T, 1998, Class. Quantum Grav. source

In the paper

“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.”