Kinematical Hilbert Spaces for Fermionic and Higgs Quantum Field Theories

Thomas Thiemann

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Kinematical Hilbert Spaces for Fermionic and Higgs Quantum Field Theories

Thomas Thiemann

General Relativity and Quantum Cosmology · Submitted 10 May 1997

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The authors' main claim 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.

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What the abstract itself states, as probabilities: claims first 0.10 · states limitation 0.44 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.86 · mentions code or data availability 0.02.

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

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	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.	contribution 1.00	0.91	0.28	asserted 1.00
s1	This framework is appropriate for coupling matter to quantum gravity.	interpretation 0.88	0.18	0.09	asserted 0.99
s2	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.	result 0.77	0.80	0.20	asserted 1.00
s3	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.	result 0.54	0.93	0.23	asserted 1.00
s4	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.	contribution 0.64	0.85	0.07	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.82 · evidence from proof or derivation 0.99.

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Original abstract

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

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References

Forward citations

Cited by 8 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Coarse-grained models for loop quantum gravity and their renormalization
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The paper constructs coarse-grained Hilbert spaces and effective Hamiltonians for loop quantum gravity and writes down non-perturbative renormalization flow equations for them.
Deformed General Relativity and Effective Actions from Loop Quantum Gravity
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Reconstructing effective actions from deformed constraint algebras shows that loop quantum gravity's inverse-triad corrections reshape the action at second order, and that holonomy corrections turn the high-density re...
Inflation in unimodular loop quantum cosmology
gr-qc 2026-07 conditional novelty 5.0 of 10

In unimodular loop quantum cosmology, an α-attractor potential allows a potential-energy-dominated bounce at the Planck scale that is consistent with CMB observations and may show quantum-gravity imprints.
Non-perturbative, background independent canonical quantum gravity in Fock representations
gr-qc 2026-06 unverdicted novelty 5.0 of 10

Existence of background-independent Fock representations for canonical quantum gravity with matter, producing a separable Hilbert space unlike LQG.
Bell states for fermions in loop quantum gravity
gr-qc 2025-08 unverdicted novelty 5.0 of 10

In loop quantum gravity, a surface-normal fermion spin observable can be used to construct CHSH-type correlations that are violated by fermion states coupled to quantum geometry.

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Consistency of modified gauge transformations in loop quantum gravity forces a sign-flipping factor in the effective metric's time-time component, producing a Euclidean black hole core.
Quantum black holes in Loop Quantum Gravity
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Two inequivalent loop-quantization schemes for spherically symmetric vacuum spacetimes both remove the black hole singularity and produce the same set of Dirac observables.
Introduction to Modern Canonical Quantum General Relativity
gr-qc 2001-10 conditional novelty 2.0 of 10

A structured, self-contained survey of canonical quantum general relativity in connection variables, presenting the classical Ashtekar-Barbero formulation, the distributional-connection Hilbert space, and the status o...

How later papers used it

48 later papers cite it

Evidence: substantial. 41 of the citing papers share no author with it, and 14 of those carry a confident label from Pith's citation classifier.

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5 later papers by other authors are listed below: 1 built on it and 4 used its method or tool. 1 more paper by the same authors is listed: 1 used its method or tool.

Builds on it 1 of 2 papers with this label

This is a natural generalization of the construction of [ref] .

Kinematics of arbitrary spin matter fields in loop quantum gravity (2020) · Loop Quantum Gravity with Matter Fields · 1.00

Uses its method or tool 5 of 16 papers with this label · 1 by the same authors

Following this paper, first exploited that the fields are half densities and introduced new Grassmann-valued variables.

$\mathcal{N}=1$ Supergravity with LQG methods and quantization of the SUSY constraint (2020) · 0.96 · paraphrased: the licence does not allow quoting
Following this paper's strategy for quantising point particles, exponentiated the momentum and defined point holonomies.

On the relation between reduced quantisation and quantum reduction for spherical symmetry in loop quantum gravity (2015) · 0.88 · paraphrased: the licence does not allow quoting
Due to the fact that $𝐏^αβ_ij$ defines a projection operator allows to quantize the fermionic fields by adapting standard tools of LQG as developed in [ref] .

Hamiltonian Theory: generalizations to higher dimensions, supersymmetry and modified gravity (2024) · Quantization of the SUSY constraint · 1.00
As proposed in this paper for real Ashtekar variables, changed the fermion fields' density weight by going over to half-densities to simplify the reality conditions.

Supersymmetric minisuperspace models in self-dual loop quantum cosmology (2020) · 0.86 · paraphrased: the licence does not allow quoting

By the same authors

Quantised the fermionic contributions to the classical constraints using standard methods and using this paper's representation.

New Variables for Classical and Quantum Gravity in all Dimensions IV. Matter Coupling (2011) (same authors) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

35 arXiv papers cite this in 92 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 59 sentences without a confident label

Uses its method or tool 12 papers

After having obtained the fermionic contributions to the classical constraints we quantise them using standard methods [ref] and using the representation [ref] .

New Variables for Classical and Quantum Gravity in all Dimensions IV. Matter Coupling · Introduction · 0.99
This can be done in full analogy to [ref] by a rigging map construction, see also [ref] .

An embedding of loop quantum cosmology in (b, v) variables into a full theory context · Symmetry reduced quantum states · 0.97
Following the construction in [ref] we obtain the combined Hilbert space for the scalar field and the connection simply by the tensor product, $ℋ_T = ℋ_grav ⊗ℋ_N = L_2(𝒜^SO(D+1), dμ_AL^SO(D+1)) ⊗L_2(N, dμ_N)$.

Towards Loop Quantum Supergravity (LQSG) I. Rarita-Schwinger Sector · Kinematical Hilbert Space for $N^I$ · 0.85

Compares with it 1 paper

Notice that $f∈V$ depends on all real Graßmann coordinates, it is not holomorphic as in the case of the Dirac spinor field [ref] .

Towards Loop Quantum Supergravity (LQSG) I. Rarita-Schwinger Sector · Background Independent Hilbert Space Representations for Majorana Fermions · 0.83

Cites it as background 9 papers

The quantisation of fermionic degrees of freedom was described in detail in [ref] , which we assume the reader to be familiar with.

New Variables for Classical and Quantum Gravity in all Dimensions IV. Matter Coupling · Implementation of the Hamiltonian Constraint Operator · 0.83
The standard results of loop quantum gravity (LQG) approach show that it enters the spectra of geometrical operators as a scaling factor thus becoming an observable parameter [ref] .[An attempt to consistently include the minimally coupled fermions in this framework can be found in [ref] .] On the other hand, the approach called covariant loop quantum gravity suggests a quantization with results independent of $β$ and argues that the standard LQG is anomalous [ref] .

Immirzi parameter and fermions with non-minimal coupling · Introduction · 0.91
In the formulation of general relativity in terms of real SU(2) connection, Thiemann discussed loop quantization of standard model fields [ref] .

Matter in Loop Quantum Gravity · Fermions · 0.88

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