Immirzi parameter in quantum general relativity
Open accessAbstract
Barbero has generalized the Ashtekar canonical transformation to a one-parameter scale transformation $U(\ensuremath{\gamma})$ on the phase space of general relativity. Immirzi has noticed that in loop quantum gravity this transformation alters the spectra of geometrical quantities. We show that $U(\ensuremath{\gamma})$ is a canonical transformation that cannot be implemented unitarily in quantum theory. This implies that there exists a one-parameter quantization ambiguity in quantum gravity, namely, a free parameter that enters the construction of the quantum theory. The purpose of this paper is to elucidate the origin and the role of this free parameter.
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TL;DR
Rovelli and Thiemann argue that loop quantum gravity contains a one-parameter quantization ambiguity, the Immirzi parameter ι, because Barbero’s canonical scaling transformation cannot be implemented unitarily in the quantum theory. This leaves the overall scale of discrete geometrical spectra undetermined unless an additional physical input fixes ι.
Key findings
- The Barbero transformation \(U(\iota): (A,E)\mapsto(A_\iota,E_\iota)\) is a canonical transformation preserving gauge and reality conditions, but it cannot be realized as a unitary transformation between the corresponding quantum theories.
- The ambiguity affects discrete geometrical spectra: quantities homogeneous in \(E\) with dimension \(L^n\) scale as \(\iota^{n/2}\); in particular the area quantum is given as \(A_0 = 8\sqrt{3}\pi \iota l_P^2\).
- The paper rejects several proposed interpretations: ι is not a free constant multiplying the connection in the holonomy, not simply the constant in front of the classical action, and not generally a feature of arbitrary Yang-Mills loop quantization; it arises in general relativity and Maxwell theory.
- Simple models illustrate the point: the harmonic oscillator and particle on a circle show no ι ambiguity, while a constrained \(R^6\) model with holonomy-based quantization does exhibit a one-parameter ambiguity.
Methods
- Theoretical letter analyzing the canonical structure of general relativity in connection variables, using the Barbero/Ashtekar canonical transformation and loop-representation quantization.
- The argument compares quantization based on \((A,E)\) with quantization based on scaled variables \((A_\iota,E_\iota)\), focusing on whether the transformation is unitarily implementable and on the resulting spectra of geometric operators.
- Three illustrative models are discussed: harmonic oscillator, particle on a circle, and a simple \(R^6\) constrained system mimicking the gravity construction.
Limitations
- The paper states that there is no compelling reason for choosing a particular value of ι, and that better knowledge of the theory may indicate such a reason.
- It notes that fixing ι via black-hole entropy has been suggested, but the text does not establish this as a derivation; the relevant numerical formula is not decoded in the provided text.
- Evident from the text: the discussion is a short letter, and the supporting models are analogies rather than full quantum-gravity calculations; no empirical measurement or unique determination of ι is presented.
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Claims the paper makes
0 of 10 cited sources checked · 0 supported · 0 flagged
- Findingnone given§ C. Any Yang-Mills theory has a free ι parameter when quantized in the loop representation · p. 4
In Maxwell theory, the quantity (14) is gauge invariant for any ι.
“In the case of Maxwell, the quantity (14) is gauge invariant for any ι .”
Quote in the paperSubstantial claim - Findingreasoning§ A. Harmonic oscillator: no ι ambiguity · p. 5
In the harmonic oscillator model, quantization based on the scaled variables leaves physical predictions, such as the spectrum of the Hamiltonian, unaltered.
“Therefore, in the quantization based on the scaled variables the wave functions ψ ( q ) have a different interpretation, but physical predictions, such as the spectrum of the Hamiltonian, are not altered.”
Quote in the paperSubstantial claim - Findingnone given§ II. THE IMMIRZI AMBIGUITY · p. 2
Uι preserves the reality conditions.
“iii) Clearly U ι preserves the reality conditions as well.”
Quote in the paperSubstantial claim - Findingreasoning§ II. THE IMMIRZI AMBIGUITY · p. 3
The spectrum of the area in the scaled theory is obtained from the one in the original theory by multiplying it by ι.
“But then equations (3) and (10) show that the spectrum of the area in the scaled theory is obtained from the one in the original theory by multiplying it by ι .”
Quote in the paperSubstantial claim - Findingreasoning§ B. Particle on a circle: no ι ambiguity · p. 5
In the particle on a circle model, no free parameter ι appears.
“In any case, no free parameter ι appears.”
Quote in the paperNot a substantial claim
- Findingreasoning§ C. A simple model with the free ι parameter · p. 5
In the simple model with phase space R^6, there is a one-parameter quantization ambiguity.
“Thus, we have a one parameter quantization ambiguity in the theory.”
Quote in the paperSubstantial claim - Findingreasoning§ C. Any Yang-Mills theory has a free ι parameter when quantized in the loop representation · p. 4
In general relativity, where the Hamiltonian gauge group is SU(2), the electric field E has the correct index structure for a connection Γ to be defined via (5).
“In general relativity, where the hamiltonian gauge group is SU (2), the 'electric field' E a i has the correct index structure for a connection Γ i a to be defined via (5).”
Quote in the paperSubstantial claim - Findingreasoning§ V. CONCLUSIONS · p. 6
These elliptic operators have non-vanishing scaling dimension with respect to the affine scaling of the connection.
“But these elliptic operators have non-vanishing scaling dimension with respect to the affine scaling of the connection.”
Quote in the paperSubstantial claim - Findingnone given§ B. ι is the constant that multiplies the classical action · p. 3
The quantity ̃h_e defined in (14) is gauge covariant under the gauge transformation generated by a scaled Gauss constraint.
“The quantity ˜ h e defined in (14) is gauge covariant under the gauge transformation generated by a scaled Gauss constraint”
Quote in the paperNot a substantial claim - Findingnone given§ II. THE IMMIRZI AMBIGUITY · p. 2
The expectation that U(ι) will be implemented unitarily in the quantum theory is false.
“However, this expectation turns out to be false.”
Quote in the paperSubstantial claim - Interpretationreasoning§ C. Any Yang-Mills theory has a free ι parameter when quantized in the loop representation · p. 4
There is no direct analog of this construction, as far as we can see, for an arbitrary Yang-Mills theory.
“There is no direct analog of this construction, as far as we can see, for an arbitrary Yang Mills theory.”
Quote in the paperSubstantial claim - Premisecitation§ The Immirzi parameter in quantum general relativity · p. 1
Barbero generalized the Ashtekar canonical transformation to a one-parameter scale transformation U(ι) on the phase space of general relativity.
“Barbero has generalized the Ashtekar canonical transformation to a one-parameter scale transformation U ( ι ) on the phase space of general relativity.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [4] source
Not checked in the free run.
- Interpretationcitation§ II. THE IMMIRZI AMBIGUITY · p. 3
The constant in the Bekenstein-Hawking entropy derivation is affected by a rescaling of ι.
“This constant is affected by a rescaling of ι , and therefore one might choose a value for ι yielding the correct value const = 1 4 ¯ hG .”
Quote in the paperSubstantial claim - Interpretationreasoning§ V. CONCLUSIONS · p. 6
A free parameter ι appears in the loop quantization of general relativity, where it affects the scale of the discreteness of space.
“A free parameter ι appears in the loop quantization of general relativity, where it affects the scale of the discreteness of space, and in the loop quantization of Maxwell theory, where it has been suggested that it might be related to charge quantization [8].”
Quote in the paperSubstantial claim - Interpretationreasoning§ V. CONCLUSIONS · p. 6
The resulting quantization ambiguity affects the operators with a discrete spectrum that scale with ι.
“The resulting quantization ambiguity, which we have denoted here as Immirzi ambiguity, affects the operators with a discrete spectrum that scale with ι .”
Quote in the paperSubstantial claim - Interpretationcitation§ I. INTRODUCTION · p. 1
The existence of this quantization ambiguity is due to the peculiar kind of representation on which nonperturbative quantum gravity is based.
“The existence of this quantization ambiguity is due to the peculiar kind of representation on which nonperturbative quantum gravity is based [5,6].”
Quote in the paperSubstantial claim0 of 2 sources checked - Interpretationreasoning§ II. THE IMMIRZI AMBIGUITY · p. 3
Since the spectrum of the area is discrete, it cannot be invariant under a scaling by an arbitrary real parameter.
“Since the spectrum of the area is discrete, it cannot be invariant under a scaling by an arbitrary real parameter.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. ι is a free constant that multiplies the connection in the definition of the holonomy · p. 3
It is not true that in defining the holonomy one is free to add a dimensionless parameter ι in the exponent while keeping everything else fixed in the theory.
“It has been repeatedly suggested that in defining the holonomy one if free to add a dimensionless parameter ι in the exponent of (12) - keeping any other thing fixed in the theory. However, this is not true.”
Quote in the paperSubstantial claim - Premisecitation§ The Immirzi parameter in quantum general relativity · p. 1
Immirzi noticed that in loop quantum gravity the transformation U(ι) alters the spectra of geometrical quantities.
“Immirzi has noticed that in loop quantum gravity this transformation alters the spectra of geometrical quantities.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [3] source
Not checked in the free run.
- Premisecitation§ I. INTRODUCTION · p. 1
This representation is characterized by the fact that the holonomy is a well-defined operator in the quantum theory.
“This representation is characterized by the fact that the holonomy is a well-defined operator in the quantum theory.”
Quote in the paperSubstantial claim0 of 2 sources checked - Interpretationreasoning§ C. Any Yang-Mills theory has a free ι parameter when quantized in the loop representation · p. 4
In U(1) Maxwell theory, the free parameter appears because a scaled abelian connection is still a connection.
“In the ( U (1)) Maxwell theory, the free parameter appears because a scaled abelian connection is still a connection.”
Quote in the paperSubstantial claim - Interpretationreasoning§ D. The algebra has inequivalent representations · p. 4
The Immirzi parameter does not distinguish inequivalent representations of the same algebra.
“It has been suggested that the Immirzi parameter distinguishes inequivalent representations of the same algebra. However, we do not think that this is the case.”
Quote in the paperSubstantial claim - Interpretationreasoning§ C. Any Yang-Mills theory has a free ι parameter when quantized in the loop representation · p. 4
In SU(2) general relativity, the free parameter appears because a second connection for defining an affine transformation is determined by the triad.
“In the ( SU (2)) general relativity it appears because a second connection for defining an affine transformation is determined by the triad.”
Quote in the paperSubstantial claim - Interpretationreasoning§ V. CONCLUSIONS · p. 6
Therefore, the ι quantization ambiguity shows up in the spectrum of the elliptic geometric operators.
“Therefore, the ι quantization ambiguity shows up in the spectrum of the elliptic geometric operators.”
Quote in the paperSubstantial claim - Premisecitation§ I. INTRODUCTION · p. 1
Physical and mathematical arguments indicate that this representation is relevant at the diffeomorphism-invariant and background-independent level.
“But physical and mathematical arguments indicate that this representation is relevant at the diffeomorphism-invariant and background-independent level [6,7].”
Quote in the paperSubstantial claim0 of 2 sources checked - Interpretationreasoning§ II. THE IMMIRZI AMBIGUITY · p. 3
Other geometrical quantities having discrete spectrum, such as volume and length, have different spectra in the scaled theory as well.
“The other geometrical quantities having discrete spectrum, such as volume and length, have different spectra in the scaled theory as well.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ B. ι is the constant that multiplies the classical action · p. 4
The ι ambiguity is not a consequence of the freedom in choosing the constant in front of the action of general relativity.
“Again, this interpretation is wrong.”
Quote in the paperSubstantial claim - Interpretationreasoning§ C. Any Yang-Mills theory has a free ι parameter when quantized in the loop representation · p. 4
We have a free parameter in the loop representations of electromagnetism and general relativity only.
“Thus, we have a free parameter in the loop representations of electromagnetism and general relativity only.”
Quote in the paperSubstantial claim - Interpretationreasoning§ I. INTRODUCTION · p. 1
There is a one-parameter family of inequivalent quantum theories, which are all, up to additional physical inputs, physically viable.
“This fact gives rise to a one-parameter family of inequivalent quantum theories, which are all, up to additional physical inputs, physically viable.”
Quote in the paperSubstantial claim - Interpretationreasoning§ II. THE IMMIRZI AMBIGUITY · p. 2
Uι leaves the kinematical structure of general relativity invariant.
“Thus U ι leaves the kinematical structure of general relativity invariant.”
Quote in the paperSubstantial claim - Interpretationreasoning§ II. THE IMMIRZI AMBIGUITY · p. 3
Since ι enters linearly in E, which has dimension L^2, the discrete spectrum of a geometrical quantity homogeneous in E with dimension L^n scales as ι^{n/2}.
“Since ι enters linearly in E , which has dimension L 2 , the discrete spectrum of a geometrical quantity homogeneous in E with dimension L n scales as ι n/ 2 .”
Quote in the paperSubstantial claim - Interpretationreasoning§ I. INTRODUCTION · p. 1
The Immirzi parameter appears in the general covariant context.
“Thus, the Immirzi PACS: 04.60.Ds parameter appears in the general covariant context.”
Quote in the paperSubstantial claim - Interpretationreasoning§ II. THE IMMIRZI AMBIGUITY · p. 3
The two quantizations yield unitarily inequivalent theories.
“Therefore the two quantizations yield unitarily inequivalent theories.”
Quote in the paperSubstantial claim - Interpretationreasoning§ The Immirzi parameter in quantum general relativity · p. 1
There exists a one-parameter quantization ambiguity in quantum gravity, namely a free parameter that enters the construction of the quantum theory.
“This implies that there exists a one-parameter quantization ambiguity in quantum gravity, namely a free parameter that enters the construction of the quantum theory.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. Harmonic oscillator: no ι ambiguity · p. 5
In general, a one-parameter scaling of the quantization variables that does not alter the observable algebra will not alter the theory.
“This example shows that in general a one-parameter scaling of the quantization variables that does not alter the observable algebra will not alter the theory.”
Quote in the paperSubstantial claim - Interpretationreasoning§ C. Any Yang-Mills theory has a free ι parameter when quantized in the loop representation · p. 4
In order to be able to write such an affine transformation, we need to have a second, independent connection in the formalism.
“In order to be able to write such an affine transformation, we need to have a second, independent connection in the formalism.”
Quote in the paperSubstantial claim - Premisecitation§ II. THE IMMIRZI AMBIGUITY · p. 2
The case ι = √-1 corresponds to Ashtekar's canonical transformation.
“The case ι = √ -1 corresponds to Ashtekar's canonical transformation [12].”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [12] source
Not checked in the free run.
- Interpretationreasoning§ B. ι is the constant that multiplies the classical action · p. 4
The ratio of a given eigenvalue of the area to the Planck length is not determined by the quantization procedure.
“The point of the Immirzi ambiguity is that the ratio of, say, a given eigenvalue of the area to l P is not determined by the quantization procedure.”
Quote in the paperSubstantial claim - Interpretationreasoning§ I. INTRODUCTION · p. 1
There is a symmetry in the classical theory which is realized as a canonical transformation but cannot be realized as a unitary transformation in the quantum theory.
“Equivalently, there is a symmetry in the classical theory which is realized as a canonical transformation but cannot be realized as a unitary transformation in the quantum theory.”
Quote in the paperSubstantial claim - Interpretationreasoning§ C. Any Yang-Mills theory has a free ι parameter when quantized in the loop representation · p. 4
There is no obvious generalization of this effect to other gauge theories.
“There is no obvious generalization of this effect to other gauge theories.”
Quote in the paperSubstantial claim - Interpretationreasoning§ B. ι is the constant that multiplies the classical action · p. 4
The freedom in the choice of the Immirzi parameter in the quantum theory consists in the fact that the overall scale of the spectra is not determined by low energy physics.
“The freedom in the choice of the Immirzi parameter in the quantum theory consists in the fact that the overall scale of the spectra is not determined by low energy physics.”
Quote in the paperSubstantial claim - Premisecitation§ II. THE IMMIRZI AMBIGUITY · p. 3
It was shown in [15] that one can derive the Bekenstein-Hawking entropy formula Entropy = const × A from loop quantum gravity, with a constant const that is finite but has an incorrect value.
“It was shown in [15] that one can derive the Bekenstein-Hawking entropy formula Entropy = const × A from loop quantum gravity - with a constant const that turns out to be finite, but with an incorrect value.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [15] source
Not checked in the free run.
- Premisenone given§ V. CONCLUSIONS · p. 6
In gravity, the metric information is in the conjugate variable E.
“More precisely, what happens in gravity is that the metric information is in the conjugate variable E .”
Quote in the paperSubstantial claim - Interpretationnone given§ V. CONCLUSIONS · p. 6
The assumption that the elementary physical excitations of the theory are the Wilson loops of Aι turns out to be physically distinct for different values of ι.
“Then, the assumption that the elementary physical excitations of the theory are the Wilson loops of A ι turns out to be physically distinct for different values of ι .”
Quote in the paperSubstantial claim- The authors state this as an assumption: The elementary physical excitations of the theory are the Wilson loops of Aι.
- Interpretationreasoning§ II. THE IMMIRZI AMBIGUITY · p. 3
There is a free parameter ι in the quantization of the theory.
“Thus, there is a free parameter ι in the quantization of the theory.”
Quote in the paperSubstantial claim - Premisereasoning§ A. ι is a free constant that multiplies the connection in the definition of the holonomy · p. 3
The transformation properties of A under an SU(2) gauge transformation are uniquely fixed by the classical action, because from the action one derives the gauge constraint (1).
“The transformation properties of A under an SU (2) gauge transformation are uniquely fixed by the classical action, because from the action one derives the gauge constraint (1).”
Quote in the paperSubstantial claim - Interpretationreasoning§ I. INTRODUCTION · p. 1
The quantum theory is undetermined by one parameter.
“We find that the quantum theory is in fact undetermined by one parameter.”
Quote in the paperSubstantial claim - Interpretationnone given§ I. INTRODUCTION · p. 1
The holonomy algebra on which this approach is based depends on a free parameter.
“This is due to the fact that the holonomy algebra on which this approach is based depends on a free parameter.”
Quote in the paperSubstantial claim - Interpretationreasoning§ II. THE IMMIRZI AMBIGUITY · p. 3
The free parameter ι could be measured, in principle, by measuring the size of the 'quanta' of area.
“It could be measured, in principle, simply by measuring, for instance, the size of the 'quanta' of area.”
Quote in the paperSubstantial claim - Interpretationnone given§ V. CONCLUSIONS · p. 6
There are two length scales in quantum gravity: the Planck constant l_P = √(ħG/c^3) and the quantum of area A0 = 8√3 π ι l_P^2.
“Similarly, there are two length scales in quantum gravity: the Planck constant l P = √ ¯ hG/c 3 and the quantum of area A 0 = 8 √ 3 πιl 2 P .”
Quote in the paperSubstantial claim - Interpretationreasoning§ V. CONCLUSIONS · p. 6
Unless some not yet understood requirement fixes the value of the Immirzi parameter, these two length scales are independent.
“Unless some non yet understood requirement fixes the value of the Immirzi parameter, these two length scales are independent.”
Quote in the paperSubstantial claim
- May combine two claims. 11
- Classified as probably not a substantial claim. 4
- The quote only partly states this claim. 1
May combine two claims.
Findingreasoning§ II. THE IMMIRZI AMBIGUITY · p. 2Aι and Eι transform under gauge transformations as A and E do; in particular, Aι is a connection.
“ii) A ι and E ι transform under gauge transformations as A and E do. In particular, A ι is a connection.”
Quote in the paperSubstantial claimMay combine two claims.
Findingnone given§ II. THE IMMIRZI AMBIGUITY · p. 2Aι and Eι are canonically conjugate; the map Uι: (A,E) ↦ (Aι,Eι) is a canonical transformation.
“i) A ι and E ι are canonically conjugate, namely the map U ι : ( A,E ) ↦→ ( A ι , E ι ) is a canonical transformation.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ V. CONCLUSIONS · p. 6Geometrical quantities which are functions of E turn out to be elliptic operators over the group manifold, hence their discrete spectrum.
“Geometrical quantities which are functions of E turn out to be elliptic operators over the group manifold. Hence their discrete spectrum.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S4 (based).
Premisenone given§ II. THE IMMIRZI AMBIGUITY · p. 2In general, starting a quantization from scaled variables has no effect on the physical predictions of the quantum theory.
“In general, starting a quantization from scaled variables has no effect on the physical predictions of the quantum theory.”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based). The quote only partly states this claim.
Interpretationnone given§ V. CONCLUSIONS · p. 6The dependence of some physical spectra upon ι manifests the physical distinctness of the Wilson-loop excitations for different ι.
“This is manifested in the dependence of some physical spectra upon ι .”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationnone given§ The Immirzi parameter in quantum general relativity · p. 1U(ι) is a canonical transformation that cannot be implemented unitarily in the quantum theory.
“We show that U ( ι ) is a canonical transformation that cannot be implement unitarily in the quantum theory.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ C. A simple model with the free ι parameter · p. 5The ι parameter is intimately linked with the affine structure of the configuration space and with the choice of the holonomies as basic operators in quantum gravity.
“This shows how the ι parameter is intimately linked with the affine structure of the configuration space and with the choice of the holonomies as basic operators in quantum gravity.”
Quote in the paperSubstantial claimMay combine two claims.
Premisenone given§ I. INTRODUCTION · p. 1Conventional perturbative Maxwell and Yang-Mills theories are not defined using this kind of representation and the ι parameter does not appear in that context.
“Conventional perturbative Maxwell and Yang-Mills theories are not defined using this kind of representation and the ι parameter does not appear in that context.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Interpretationreasoning§ B. ι is the constant that multiplies the classical action · p. 3The scaled Gauss constraint can be obtained by scaling the action of the theory.
“This constraint, in turn, can be obtained by scaling the action of the theory.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ C. Any Yang-Mills theory has a free ι parameter when quantized in the loop representation · p. 4The Immirzi parameter is not related to a multiplicative scaling of the connection; rather, it is generated by the affine transformation (6).
“As the previous comments indicate, the Immirzi parameter is not related to a multiplicative scaling of the connection. Rather, it is generated by the affine transformation (6).”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ V. CONCLUSIONS · p. 6The origin of the Immirzi ambiguity is not tied to the infinite dimensionality of the phase space, nor, at least to a first analysis, to the existence of inequivalent representations of the observables algebra.
“Its origin is not tied to the infinite dimensionality of the phase space, nor (at least to a first analysis) to the existence of inequivalent representations of the observables algebra.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ II. THE IMMIRZI AMBIGUITY · p. 3The two theories give different physics; for instance, the predicted spectrum of the area differs.
“The two theories give different physics. For instance, the predicted spectrum of the area differ.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S4 (based).
Interpretationnone given§ III. INCORRECT INTERPRETATIONS · p. 3Some proposed interpretations of the appearance of the free parameter ι in quantum gravity are incorrect.
“Several interpretations of the appearance of the free parameter ι in quantum gravity have been recently proposed. Some of these are incorrect.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ A. ι is a free constant that multiplies the connection in the definition of the holonomy · p. 3If we multiply a connection by a real number, we do not obtain a quantity that transforms as a connection; connections form an affine space, not a linear space.
“In other words, if we multiply a connection by a real number, we do not obtain a quantity that transforms as a connection: connections form an affine space, not a linear space.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ V. CONCLUSIONS · p. 6The indeterminacy is given by a single parameter, and therefore it does not reduce the predictive power of the theory more than λ_qcd in quantum chromodynamics, or the string constant in string theory.
“Finally, the indeterminacy is given by a single parameter, and therefore it does not reduce the predictive power of the theory more than, say, λ qcd in quantum cromodynamics, or the string constant in string theory.”
Quote in the paperSubstantial claim
The authors state this as an assumption.
The quantization is based on the physical assumption that the Wilson loops, or the 'Faraday lines' of the theory, are the physical elementary quantum excitations and thus correspond to finite norm states.
“The quantization is based on the physical assumption that the Wilson loops, or the 'Faraday lines' of the theory are the physical elementary quantum excitations, and thus correspond to finite norm states.”
The authors state this as an assumption.
The holonomy operator is well-defined.
“Equivalently, the holonomy operator is well-defined.”
The authors state this as an assumption.
The elementary physical excitations of the theory are the Wilson loops of Aι.
“Then, the assumption that the elementary physical excitations of the theory are the Wilson loops of A ι turns out to be physically distinct for different values of ι .”
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Ashtekar, A., & Lewandowski, J. (2004) · Classical and Quantum Gravity · doi:10.1088/0264-9381/21/15/r01 · Cited by 1,951 · see citing works
Quantum geometry and black hole entropy
Ashtekar, A., Baez, J., Corichi, A., & Krasnov, K. (1998) · Physical Review Letters · doi:10.1103/physrevlett.80.904 · Cited by 1,105 · see citing works
Quantum geometry of isolated horizons and black hole entropy
Ashtekar, A., Baez, J. C., & Krasnov, K. (2000) · Advances in Theoretical and Mathematical Physics · doi:10.4310/atmp.2000.v4.n1.a1 · Cited by 580 · see citing works
Symmetry reduction for quantized diffeomorphism-invariant theories of connections
Bojowald, M., & Kastrup, H. A. (2000) · Classical and Quantum Gravity · doi:10.1088/0264-9381/17/15/311 · Cited by 168 · see citing works
Alexandrov, S., & Livine, E. R. (2003) · Physical Review D · doi:10.1103/physrevd.67.044009 · Cited by 141 · see citing works
<i>SO</i> (4, <i>C</i> )-covariant Ashtekar-Barbero gravity and the Immirzi parameter
Alexandrov, S. (2000) · Classical and Quantum Gravity · doi:10.1088/0264-9381/17/20/307 · Cited by 112 · see citing works
Quantum spin dynamics: VIII. The master constraint
Thiemann, T. (2006) · Classical and Quantum Gravity · doi:10.1088/0264-9381/23/7/003 · Cited by 99 · see citing works
Self-citationArea spectrum in Lorentz covariant loop gravity
Alexandrov, S., & Vassilevich, D. (2001) · Physical Review D · doi:10.1103/physrevd.64.044023 · Cited by 68 · see citing works
Livine, R. E., & Oriti, D. (2002) · Physical Review D · doi:10.1103/physrevd.65.044025 · Cited by 46 · see citing works
- Authors
- Rovelli, C., & Thiemann, T.
- Affiliations
- Harvard University (US)
- ORCID
- Thiemann, Thomas
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 1998-01-15
- Volume
- 57
- Issue
- 2
- Pages
- 1009-1014
- ISSN
- 0556-2821 (ISSN-L)
- 1089-4918
- OpenAlex
- W2098788693
- Semantic Scholar
- e24bafa326d923d1311fa96d2ab880ff302f339a
- Cited by
- 171
- References
- 5
- Citations by year
- 2026: 4 · 2025: 4 · 2024: 2 · 2023: 2 · 2022: 4 · 2021: 4 · 2020: 2 · 2019: 10 · 2018: 4 · 2017: 6 · 2016: 4 · 2015: 3 · 2014: 4 · 2013: 5 · 2012: 6
- FWCI
- 9.12
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Keywords
- Immirzi parameter, loop quantum gravity, quantum general relativity, canonical transformations, unitarity, quantization ambiguity, scale transformation
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