Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction
Open accessAbstract
A Planck-scale minimal observable length appears in many approaches to quantum gravity. It is sometimes argued that this minimal length might conflict with Lorentz invariance, because a boosted observer can see the minimal length further Lorentz contracted. We show that this is not the case within loop quantum gravity. In loop quantum gravity the minimal length (more precisely, minimal area) does not appear as a fixed property of geometry, but rather as the minimal (nonzero) eigenvalue of a quantum observable. The boosted observer can see the same observable spectrum, with the same minimal area. What changes continuously in the boost transformation is not the value of the minimal length: it is the probability distribution of seeing one or the other of the discrete eigenvalues of the area. We discuss several difficulties associated with boosts and area measurement in quantum gravity. We compute the transformation of the area operator under a local boost, propose an explicit expression for the generator of local boosts, and give the conditions under which its action is unitary.
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Written from the paper’s full text.
TL;DR
In loop quantum gravity, the Planck-scale minimal area is not a fixed property of geometry but the minimal nonzero eigenvalue of a quantum area observable. The authors argue that a boosted observer sees the same minimal area eigenvalue, while the probability distribution over discrete area eigenvalues changes continuously under the boost.
Key findings
- The minimal nonzero area \(A_0\) observed by one observer remains the same minimal nonzero area for a boosted observer; what changes under the boost is the probability distribution of seeing one or another discrete area eigenvalue.
- The area operators \(A\) and \(A'\) associated with two relatively boosted observers have nonvanishing Poisson brackets, so the corresponding quantum operators cannot commute; therefore if the area is sharp for one observer, it generally cannot be sharp for the other.
- The authors compute the transformation of the area operator under a local boost, propose an explicit expression for the generator of local boosts depending on the Hamiltonian constraint, and state that the transformation is unitary if this generator is Hermitian, in which case the spectrum is preserved.
- They draw an analogy with angular momentum: a rotated observer still sees the same minimal nonzero component \(L_0\), while the distribution of eigenvalues is determined by the rotation group representation theory.
Methods
- The paper studies general relativity coupled to two particles and a two-dimensional surface (“table”), with two observers \(O\) and \(O'\) crossing at a point \(P\); the area is defined as the intersection of the table worldsheet with each observer’s simultaneity surface, assuming a small surface and negligible curvature/acceleration at \(P\).
- Coordinates are gauge-fixed to the matter, and the areas \(A\) and \(A'\) are expressed as functions of the metric and canonical Ashtekar variables; the analysis expands in the boost parameter \(\beta\), computing \(A'\) to first and second order and evaluating the Poisson bracket \(\{A,A'\}\).
- A candidate boost generator is constructed from Lorentz generators and the energy-momentum tensor, with the Minkowski metric replaced by \(g_{\mu\nu}(x)\); its infinitesimal action on \(A\) is checked against the geometric transformation, and unitarity is discussed via Hermiticity of the corresponding quantum operator.
Limitations
- The paper states that the theory as a whole is not Lorentz invariant; Lorentz invariance can only be recovered locally and/or in sufficiently flat regimes, and the viability of expanding around a macroscopically flat quantum state is regarded as an hypothesis because nonperturbative quantum gravity gives less control over the gravitational field state.
- The paper states that the detailed calculation of the boosted simultaneity surface position is not reported, and that the actual definition of the node operator \(g_{1a}x^a_n D_n\) in the quantum theory is left to future investigations.
- Evident from the text: the results rely on perturbative expansion in the boost parameter and on assumptions of small surface size, local flatness, and in part a spatially constant metric simplification, though the authors say the latter is not essential.
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Claims the paper makes
0 of 14 cited sources checked · 0 supported · 0 flagged
- Findingreasoning§ B. Area in general relativity · p. 4
The area A(S) of the 2d surface S given by the intersection of two 3d hypersurfaces Σ and T can be written in terms of the one-forms normal to the two hypersurfaces.
“Here we show that A ( S ) can be written in terms of the one-forms n Σ µ and n T µ normal to the two hypersurfaces.”
- Also stated in: § B. Area in general relativity · p. 5 · plain
“This expression gives us the area directly as a function of surface S , the metric and the normals to the table worldsheet and the observer's simultaneity surface.”
Quote in the paperSubstantial claim - Also stated in: § B. Area in general relativity · p. 5 · plain
- Findingreasoning§ IV. NON COMMUTATIVITY · p. 8
The Poisson brackets between A and A' do not vanish.
“Since the Poisson brackets between A and A ′ do not vanish, the commutator of the corresponding quantum operator cannot vanish either.”
- Also stated in: § VI. DISCUSSION AND CONCLUSIONS · p. 11 · boosted
“We have shown that these two observables have nonvanishing Poisson brackets, which implies that the corresponding quantum operators cannot commute.”
Quote in the paperSubstantial claim - Also stated in: § VI. DISCUSSION AND CONCLUSIONS · p. 11 · boosted
- Findingreasoning§ A. The basic idea · p. 1
The area operators A and A' associated to measurements by relatively boosted observers do not commute.
“Now, our main point is the technical observation that A and A ′ do not commute:”
- Also stated in: § A. The basic idea · p. 2 · boosted
“In this paper, we prove equation (1).”
Quote in the paperSubstantial claim- Rests on an assumption: A field operator does not commute with itself at different times.
- Also stated in: § A. The basic idea · p. 2 · boosted
- Findingreasoning§ IV. NON COMMUTATIVITY · p. 7
Even if the dynamics of the gravitational field in these coordinates is assumed unaffected by matter, the right-hand side of the Poisson bracket {A,A'} does not vanish.
“However, one can easily see that even if we assume that in these coordinates the dynamics of the gravitational field is unaffected by the matter, the r.h.s. does not vanish.”
Quote in the paperSubstantial claim - Findingnone given§ B. Area in general relativity · p. 5
The normal one-form of the table worldsheet T does not depend on the metric.
“It does not depend on the metric.”
Quote in the paperNot a substantial claim
- Findingown data§ A. The velocity of the surface · p. 9
The explicit computation of this paper shows that A0 does not always appear as the minimal area of a material object in its own rest frame.
“Similarly, we might imagine that A 0 always appears as the minimal area of a material object in its own rest frame. The explicit computation of this paper shows that this is not the case.”
Quote in the paperSubstantial claim - Findingreasoning§ V. BOOSTS GENERATORS · p. 10
The infinitesimal transformation of A generated by the proposed generator (63) is precisely the second term in the r.h.s. of Equation (50), the infinitesimal transformation of A previously worked out geometrically.
“But this is precisely the second term in the r.h.s of Equation (50), which is the infinitesimal transformation of A we had previously worked out geometrically. This result supports the hypothesis that (63) is the correct generator of the local Lorentz boost.”
Quote in the paperSubstantial claim - Premisecitation§ A. The basic idea · p. 2
The area of a surface defined by coordinate values is not gauge invariant.
“the area of a surface defined by coordinate values is not gauge invariant”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [8] source
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- Interpretationreasoning§ A. Coordinate choice · p. 6
The particle X and the table T are moving with respect to each other even if their coordinate positions have been fixed.
“In other words, the particle X and the table T are moving with respect to each other even is their coordinate positions have been fixed.”
Quote in the paperSubstantial claim - Premisecitation§ Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction · p. 1
In many approaches to quantum gravity, the Planck length LP should play the role of a minimal observable length.
“A Planck-scale minimal observable length appears in many approaches to quantum gravity.”
- Also stated in: § I. INTRODUCTION · p. 1 · hedged
“A large number of convincing semiclassical considerations indicate that in a quantum theory of gravity the Planck length L P should play the role of minimal observable length [1].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [1] source
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- Also stated in: § I. INTRODUCTION · p. 1 · hedged
- Interpretationreasoning§ A. The basic idea · p. 2
A and A' depend on the gravitational field on two distinct 2d surfaces in spacetime.
“This is because A and A ′ depend on the gravitational field on two distinct 2d surfaces in spacetime (see Figure 1)”
Quote in the paperSubstantial claim - Premisecitation§ A. The basic idea · p. 1
In loop quantum gravity, space geometry has no fixed structural property at all.
“Space geometry, indeed, has no fixed structural property at all in this approach.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [3] source
Not checked in the free run.
- Premisecitation§ Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction · p. 1
In loop quantum gravity, the minimal length appears as the minimal nonzero area A0 of a surface, not as a fixed property of geometry.
“In loop quantum gravity the minimal length (more precisely, minimal area) does not appear as a fixed property of geometry, but rather as the minimal (nonzero) eigenvalue of a quantum observable.”
- Also stated in: § I. INTRODUCTION · p. 1 · hedged
“Within loop quantum gravity, a minimal length appears characteristically in the form of a minimal (nonzero) value A 0 of the area of a surface [6,7].”
- Also stated in: § A. The basic idea · p. 1 · plain
“The key to understand how this may happen is the fact that in loop quantum gravity a minimal length does not appear as a fixed structural property of space geometry.”
Quote in the paperSubstantial claim0 of 2 sources checked- cites [6] source
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- cites [7]
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- Also stated in: § I. INTRODUCTION · p. 1 · hedged
- Interpretationcitation§ I. INTRODUCTION · p. 1
The apparent conflict between Lorentz contraction and Planck scale discreteness is resolved in loop quantum gravity.
“Here, we show how the apparent conflict between Lorentz contraction and Planck scale discreteness is resolved in loop quantum gravity [3]”
- Also stated in: § Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction · p. 1 · plain
“We show that this is not the case within loop quantum gravity.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [3] source
Not checked in the free run.
- Also stated in: § Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction · p. 1 · plain
- Interpretationreasoning§ A. Coordinate choice · p. 5
The physical degrees of freedom of the matter (the two particles and the table) are not killed or frozen by the gauge choice.
“However, this does not mean that the physical degrees of freedom of the matter (the two particles and the table) are being killed or frozen.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. The basic idea · p. 1
A measurement of a length is a measurement in the quantum mechanical sense.
“A measurement of a length is therefore a measurement in the quantum mechanical sense.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ I. INTRODUCTION · p. 1
Planck scale discreteness is compatible with a certain implementation of local Lorentz invariance.
“We show that Planck scale discreteness is compatible with a certain implementation of local Lorentz invariance”
Quote in the paperSubstantial claim - Premisecitation§ A. The basic idea · p. 2
The theory is invariant under diffeomorphisms.
“The theory is invariant under diffeomorphisms”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [8] source
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- Interpretationreasoning§ A. Coordinate choice · p. 6
In this gauge, certain components of the gravitational field that are gauge dependent in pure general relativity become gauge invariant physical quantities.
“As a consequence, certain components of the gravitational field that are gauge dependent quantities in pure general relativity, become gauge invariant physical quantities, precisely as the r.h.s. of (19).”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. Coordinate choice · p. 6
The two descriptions-mass probes do not move and the gravitational field varies, or mass probes oscillate in space-are equivalent descriptions of the same physics.
“These are two equivalent descriptions of the same physics.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ A. The basic idea · p. 2
For observer O', the surface does not have a sharp area.
“That is to say, the theory predicts that, for him, the surface does not have a sharp area.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. The basic idea · p. 1
In the theory there are two distinct operators A and A', associated to the two area measurements by relatively boosted observers.
“Correspondingly, in the theory there are two distinct operators A and A ′ , associated to these two measurements.”
Quote in the paperSubstantial claim - Interpretationreasoning§ IV. NON COMMUTATIVITY · p. 8
The commutator of the quantum operators corresponding to A and A' cannot vanish.
“Since the Poisson brackets between A and A ′ do not vanish, the commutator of the corresponding quantum operator cannot vanish either. Otherwise in the ¯ h → 0 limit the commutator could not reproduce the classical Poisson brackets.”
Quote in the paperSubstantial claim - Interpretationnone given§ A. The basic idea · p. 2
If observer O measures the area and obtains the minimal value A0, the state of the gravitational field will be projected on an eigenstate of A.
“If the observer O measures the area and obtains the minimal value A 0 , the state of the gravitational field will be projected on an eigenstate of A .”
Quote in the paperNot a substantial claim - Interpretationreasoning§ A. The basic idea · p. 1
Observable properties of geometry, such as length or area, are observable properties of a quantum physical system.
“Therefore the observable properties of the geometry, such as, in particular, a length, or an area, are observable properties of a quantum physical system.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. The system · p. 4
A and A' are fully gauge invariant observables in this dynamical system.
“They are fully gauge invariant observables in this dynamical system.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. The basic idea · p. 2
If the experiment is repeated several times, observer O' will observe a probability distribution of area values.
“If the experiment is repeated several times, O ′ will observe a probability distribution of area values.”
Quote in the paperSubstantial claim- Rests on an assumption: Generically, quantum theory does not predict an observable value; it predicts a probability distribution of possible values.
- Premisecitation§ A. The basic idea · p. 1
In loop quantum gravity, the geometry of space comes from a quantum field, the quantum gravitational field.
“The geometry of space comes from a quantum field, the quantum gravitational field.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [3] source
Not checked in the free run.
- Interpretationreasoning§ A. The basic idea · p. 1
Given a surface moving in spacetime, the two measurements of its area performed by two observers O and O' boosted with respect to each other are two distinct quantum measurements.
“Given a surface moving in spacetime, the two measurements of its area performed by two observers O and O ′ boosted with respect to each other are two distinct quantum measurements.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. Coordinate choice · p. 6
The relative velocity of the table and particle is given by the time derivative of the metric field.
“The relative velocity of table and particle is given by the time derivative of the metric field.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ A. The basic idea · p. 2
Under certain reasonable assumptions, the area under a Lorentz boost in loop quantum gravity behaves precisely as the L_z component of angular momentum, as illustrated by the analogy with angular momentum.
“In this paper, we analyze all these problems with care, and we show that in spite of all these complications, and under certain reasonable assumptions, what happens to the area under a Lorentz boost in loop quantum gravity is indeed precisely what is described above and is illustrated by the analogy with the angular momentum.”
Quote in the paperSubstantial claim- Rests on an assumption: The theory as a whole is not Lorentz invariant, and a form of Lorentz invariance can only be recovered locally and/or in certain sufficiently flat regimes.
- Rests on an assumption: The area A is a far more complicated function of the basic variables of the theory than L_z.
- Rests on an assumption: Lorentz transformations do not happen at fixed time, so the generators of local Lorentz transformations must know about the dynamics of the theory.
- The authors state this as an assumption: Other physical objects exist in the universe and can be used to perform measurements.
- The authors state this as an assumption: The freedom of choosing the boundary between the observed quantum system and the classical apparatus allows us to treat other physical objects as part of the measuring apparatus.
- The authors state this as an assumption: The dynamics of the system is governed by the Einstein equations and the dynamical equations of the table and the particles.
- The authors state this as an assumption: The matter energy-momentum tensor is negligible in the Einstein equations.
- The authors state this as an assumption: In the quantum theory, the expansion in small boost parameter and small surface remains valid in regimes where the expectation value of the macroscopic curvature is small.
- The authors state this as an assumption: The curvature at and around P and the acceleration of the particles at P are negligible at the scale of the surface.
- Interpretationreasoning§ A. The basic idea · p. 2
A generic eigenstate of A is not an eigenstate of A'.
“It follows that a generic eigenstate of A is not an eigenstate of A ′ .”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. Coordinate choice · p. 6
In this gauge, the areas A and A' are gauge invariant observables expressed solely in terms of g.
“In particular, in this gauge the areas A and A ′ , which are gauge invariant observables, are expressed solely in terms of g , but they still remain, of course, gauge invariant.”
- Also stated in: § A. Coordinate choice · p. 6 · plain
“Similarly, in the coordinates we have chosen the area is given by a function of g alone, and is gauge invariant because it is invariant under coordinate transformations that preserve the coordinate choice made.”
Quote in the paperSubstantial claim - Also stated in: § A. Coordinate choice · p. 6 · plain
- Interpretationreasoning§ A. The basic idea · p. 2
If observer O' then measures the area, he will find the state in a superposition of eigenstates of A'.
“If then the observer O ′ measures the area, he will therefore find the state in a superposition of eigenstates of A ′ .”
Quote in the paperSubstantial claim - Interpretationnone given§ A. The basic idea · p. 2
That eigenstate of A is not an eigenstate of A'.
“This, in turn, is not going to be an eigenstate of A ′ .”
Quote in the paperNot a substantial claim - Interpretationreasoning§ A. Coordinate choice · p. 6
The physical motion of the particles and table is described by components of the metric tensor in this gauge.
“The example illustrate how the physical motion of the particles and table is described by components of the metric tensor in this gauge.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ VI. DISCUSSION AND CONCLUSIONS · p. 11
If the value of the area is sharp for one observer, it cannot be sharp, in general, for the second observer.
“Therefore if the value of the area is sharp for one observer, it cannot be sharp, in general, for the second observer.”
Quote in the paperSubstantial claim - Interpretationreasoning§ VI. DISCUSSION AND CONCLUSIONS · p. 11
The generator transforms the gravitational field that determines the area of the table on the simultaneity surface of the first observer into the field that determines the area of the table on the simultaneity surface of the boosted observer.
“Therefore it transforms the gravitational field that determines the area of the table on the simultaneity surface of the first observer into the field that determines the area of the table on the simultaneity surface of the boosted observer.”
Quote in the paperSubstantial claim - Premisereasoning§ V. BOOSTS GENERATORS · p. 9
The quantities M_ij defined in (54) are generators of spatial rotations.
“and it is easy to verify that these are indeed generators of spatial rotations.”
Quote in the paperNot a substantial claim - Interpretationreasoning§ VI. DISCUSSION AND CONCLUSIONS · p. 11
The minimal area measured by one observer cannot be just Lorentz contracted for the boosted observer.
“This implies that the minimal area measured by one observer cannot be just Lorentz contracted for the boosted observer. This is our main result.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. The velocity of the surface · p. 8
Since the relative velocity between observer and surface does not commute with the area, it does not make sense to start by assuming that the first observer O is at rest with respect to the surface.
“To be more precise, since the relative velocity between observer and surface does not commute with the area, it does not make sense to start by assuming that the first observer O is at rest with respect to the surface.”
Quote in the paperSubstantial claim - Premisecitation§ V. BOOSTS GENERATORS · p. 10
If the gauge N(x)=1, N^a(x)=A^i_0(x)=0 is fixed, the hamiltonian density (58) coincides with the hamiltonian constraint H(x).
“If we can fix the gauge N ( x ) = 1, N a ( x ) = A i 0 ( x ) = 0, ˙ A = { A, H} , the hamiltonian density (58) coincides with the hamiltonian constraint H ( x ) [12].”
Quote in the paperSubstantial claim0 of 1 sources checked- cites [12] source
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- Premisenone given§ VI. DISCUSSION AND CONCLUSIONS · p. 11
The area of a surface is a quantum observable.
“The area of a surface is a quantum observable.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. The velocity of the surface · p. 9
By measuring the area, an observer destroys information on the velocity of the surface.
“Physically, all this means that by measuring the area, an observer destroys information on the velocity of the surface -as measuring the position of a quantum particle destroys information on its momentum.”
Quote in the paperSubstantial claim - Interpretationreasoning§ VI. DISCUSSION AND CONCLUSIONS · p. 11
Diffeomorphism invariance implies that to move the matter with respect to the gravitational field is equivalent to moving the gravitational field with respect to the matter.
“Of course, diffeomorphism invariance implies that to move the matter with respect to the gravitational field is equivalent to moving the gravitational field with respect to the matter.”
Quote in the paperSubstantial claim - Interpretationreasoning§ VI. DISCUSSION AND CONCLUSIONS · p. 11
Since the coordinates are gauge fixed, the physical states are defined by the vanishing of the constraints smeared by generators of diffeomorphisms that send the matter worldhistories into themselves, not by the vanishing of the full constraints.
“Technically, since we have gauge fixed the coordinates, the physical states are not defined by the vanishing of the full constraints, but only by the vanishing of the constraints smeared by generators of diffeomorphisms that send the matter worldhistories into themselves.”
Quote in the paperSubstantial claim - Premisenone given§ VI. DISCUSSION AND CONCLUSIONS · p. 11
The theory is invariant under diffeomorphisms that act simultaneously on the gravitational field and on the matter.
“The theory is invariant under diffeomorphisms that act simultaneously on the gravitational field and on the matter.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. The velocity of the surface · p. 8
The operator A' is not a simple function of the operator A.
“Therefore the operator A ′ is not a simple function of the operator A .”
Quote in the paperSubstantial claim - Interpretationreasoning§ VI. DISCUSSION AND CONCLUSIONS · p. 11
The generator (64) changes the value of the metric field from the one on the t=0 surface to the one of the surface t=βg1a xa, the Lorentz rotated surface.
“At the light of these considerations, the reason for the explicit form of the generator we have considered (see in particular (64)) is transparent: it changes the value of the metric field from the one on the t = 0 surface to the one of the surface t = βg 1 a x a , namely to the Lorentz rotated surface.”
Quote in the paperSubstantial claim - Interpretationreasoning§ VI. DISCUSSION AND CONCLUSIONS · p. 11
The generator of Lorentz transformations introduced defines a genuine transformation in the physical Hilbert space of the theory, despite being a linear function of the Hamiltonian constraint.
“This is why, in spite of being a linear function of the hamiltonian constraint, the generator of Lorentz transformations that we have introduced defines a genuine transformation in the physical Hilbert of the theory.”
Quote in the paperSubstantial claim - Interpretationreasoning§ A. The velocity of the surface · p. 8
The apparent incompatibility between discreteness and Lorentz contraction is resolved by observing that the measurements of area and velocity of a surface are incompatible.
“Therefore one can also say that the apparent incompatibility between discreteness and Lorentz contraction is resolved by observing that the measurements of area and velocity of a surface are incompatible.”
Quote in the paperSubstantial claim
- May combine two claims. 12
- Classified as probably not a substantial claim. 14
May combine two claims.
Findingreasoning§ A. Coordinate choice · p. 6gμν(x) does not depend on α at t=0, while its time derivative does.
“( g µν ( x ) does not depend on α at t = 0, while its time derivative does.)”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Findingreasoning§ A. Coordinate choice · p. 6The relative velocity of the particle X with respect to the center of the table is α.
“Therefore the relative velocity of the particle X with respect to the center of the table is α .”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim.
Findingnone given§ A. The system · p. 4A and A' are diffeomorphism invariant functions of g, X, X', T.
“The quantities A and A ′ are diffeomorphism invariant functions of g , X , X ′ an T .”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
Interpretationreasoning§ A. Coordinate choice · p. 6In the present context, the dynamics is entirely captured by the value of the gravitational field.
“In the present context, we have chosen to attach coordinates to the matter (particles and table). Therefore the dynamics is entirely captured by the value of the gravitational field.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Interpretationnone given§ A. Coordinate choice · p. 5In the chosen gauge, the only remaining degrees of freedom are the ones in the metric tensor gμν(x).
“With these coordinates, namely in this gauge, the only remaining degrees of freedom are the ones in the metric tensor g µν ( x ).”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ I. INTRODUCTION · p. 1In loop quantum gravity, a boosted observer O' does not observe a Lorentz contracted A0; the minimal nonzero area O' can observe is still A0.
“Here we show that in loop quantum gravity a boosted observer O ′ does not observe a Lorentz contracted A 0 . The minimal (nonzero) area that the boosted observer O ′ can observe is still A 0 .”
- Also stated in: § Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction · p. 1 · plain
“The boosted observer can see the same observable spectrum, with the same minimal area.”
Quote in the paperSubstantial claim- Also stated in: § Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction · p. 1 · plain
Probably not a substantial claim: may not pass S3 (load-bearing).
Interpretationreasoning§ A. The system · p. 4A and A' are physical quantities that are in principle observable using appropriate measuring devices.
“They are physical quantities that are in principle observable, by using appropriate measuring devices (formed by light pulses, detectors, clocks, and else).”
Quote in the paperNot a substantial claimProbably not a substantial claim: may not pass S4 (based).
Interpretationnone given§ A. The system · p. 3The system of the gravitational field, two particles, and a two-dimensional surface provides a minimal setting in which to compare the area observed by two observers boosted with respect to each other.
“These are the dynamical quantities of the system we consider. They provide a minimal setting in which we can compare the area observed by two observers boosted with respect each other.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ A. The basic idea · p. 2The mean value of the area can be Lorentz contracted, while the minimal nonzero value of the area can remain A0.
“The mean value of the area can be Lorentz contracted, while the minimal nonzero value of the area can remain A 0 .”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing), S4 (based).
Premisenone given§ A. The system · p. 4Transversal motion does not give rise to Lorentz contraction.
“(We are not interested in transversal motion because it does not give rise to Lorentz contraction.)”
Quote in the paperNot a substantial claimClassified as probably not a substantial claim.
Premisereasoning§ A. Coordinate choice · p. 5The physical, measurable position of an object relative to a reference object is determined by a combination of the coordinate position of matter and the value of the metric tensor.
“The physical, measurable, position of an object (relative to a reference object) is determined by a combinations of the two.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim. May combine two claims.
Premisenone given§ A. Coordinate choice · p. 5The coordinate position of matter and the value of the metric tensor are both gauge dependent quantities due to diffeomorphism invariance.
“Indeed, it is well known that the coordinate position of matter and the value of the metric tensor are both gauge dependent quantities, due to diffeomorphism invariance.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ A. The velocity of the surface · p. 8In general relativity, A becomes a quantum operator because it is a function of the metric, and the velocity V in (51) depends on the metric as well, so V is also an operator in quantum gravity.
“In general relativity, A becomes a quantum operator because it is a function of the metric, namely a function of the quantum field. But the velocity V that appears in (51) depends on the metric as well. Indeed the V that appears in (51) is not a coordinate velocity, it is a physical velocity, and it depends on g µν as well. Thus V as well is an operator in quantum gravity.”
Quote in the paperSubstantial claimProbably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ V. BOOSTS GENERATORS · p. 9In a Lorentz invariant theory, the quantities Mμν defined by (53) are constant.
“In a Lorentz invariant theory, (53) are constant.”
Quote in the paperNot a substantial claim0 of 1 sources checked- cites [11] source
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Probably not a substantial claim: may not pass S3 (load-bearing).
Premisecitation§ A. Unitarity · p. 10The action of the Hamiltonian constraint smeared with a Lapse N is a sum of terms acting on the nodes of the form given in the cited work.
“We recall that the action of the Hamiltonian constraint smeared with a Lapse N is a sum of terms acting on the nodes of the form [15,16]”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [15]
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- cites [16]
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Premisecitation§ A. Unitarity · p. 10A basis of area eigenstates is given by spin network states.
“Recall that a basis of area eigenstates is given by spin network states [13,14].”
Quote in the paperNot a substantial claim0 of 2 sources checked- cites [13]
Not checked in the free run.
- cites [14]
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Probably not a substantial claim: may not pass S4 (based).
Premisenone given§ VI. DISCUSSION AND CONCLUSIONS · p. 11In loop quantum gravity the metric is an operator.
“In loop quantum gravity the metric is an operator.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationreasoning§ VI. DISCUSSION AND CONCLUSIONS · p. 11The Lorentz transformations considered act on the matter at fixed field, or equivalently on the field leaving the matter fixed, and this is why they are not part of the gauge.
“The Lorentz transformations we have considered act on the matter at fixed field, or, equivalently, on the field leaving the matter fixed. This is why they are not part of the gauge.”
Quote in the paperSubstantial claimMay combine two claims.
Premisenone given§ V. BOOSTS GENERATORS · p. 9In gravity, the quantities (53) are formal objects; they are not tensorial, not defined for all values of the fields, not defined on the entire spacelike surface.
“In gravity, we can still write the quantities (53). These are formal objects. They are not tensorial, not defined for all values of the fields, not defined on the entire spacelike surface.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ VI. DISCUSSION AND CONCLUSIONS · p. 11The theory is not invariant under a diffeomorphism that acts on the matter leaving the gravitational field untouched, nor under one that acts on the gravitational field leaving the matter untouched.
“However, it is not invariant under a diffeomorphism that acts on the matter leaving the gravitational field untouched. Nor under a diffeomorphism that acts on the gravitational field leaving the matter untouched.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim. May combine two claims.
Premisenone given§ V. BOOSTS GENERATORS · p. 10The momentum generates spatial translations, and in general relativity spatial translations are generated by the momentum constraint H_a(x).
“The momentum, generates spatial translations. Spatial translations are generated by the momentum constraint H a ( x ) in general relativity.”
Quote in the paperSubstantial claimMay combine two claims.
Premisenone given§ VI. DISCUSSION AND CONCLUSIONS · p. 11At the Planck scale, the area is quantized and there is a finite nonzero minimal value.
“At the Planck scale, this area is quantized and there is a finite nonzero minimal value.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ VI. DISCUSSION AND CONCLUSIONS · p. 11In the presence of matter, or in the gauge-fixed coordinates chosen, an active Lorentz transformation rotating the gravitational field is a change of physical state, whereas it would be a gauge transformation in the absence of matter and in arbitrary coordinates.
“Concretely, we have gauge fixed the coordinate position of the matter, and considered an active Lorentz transformation rotating (in spacetime) the gravitational field. While this would be a gauge transformation in the absence of matter and in arbitrary coordinates, it is, instead, a change of physical state in the presence of matter, or, equivalently, in the gauge fixed coordinates we have chosen.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationreasoning§ A. Unitarity · p. 10Assuming the operator M(λ) is hermitian, the finite transformation it generates is unitary, and the spectrum of areas A and A' is the same.
“This operator is unitary if M ( λ ) is hermitian. This is the condition under which the Lorentz transformation is unitary in the quantum theory. Assuming it is satisfied, the spectrum of the areas A and A ′ is the same.”
Quote in the paperSubstantial claim- The authors state this as an assumption: The quantum operator M(λ) corresponding to the classical observable (61) is well defined in the theory.
- The authors state this as an assumption: The operator M(λ) is hermitian.
The paper relies on this without support.
A field operator does not commute with itself at different times.
“a field operator does not commute with itself at different times.”
The paper relies on this without support.
Generically, quantum theory does not predict an observable value; it predicts a probability distribution of possible values.
“Generically, quantum theory does not predict an observable value: it predicts a probability distributions of possible values.”
The paper relies on this without support.
The theory as a whole is not Lorentz invariant, and a form of Lorentz invariance can only be recovered locally and/or in certain sufficiently flat regimes.
“The theory as a whole is not Lorentz invariant, and a form of Lorentz invariance can only be recovered locally and/or in certain ('sufficiently flat') regimes.”
The paper relies on this without support.
The area A is a far more complicated function of the basic variables of the theory than L_z.
“The area A is a far more complicated function of the basic variables of the theory than L z .”
The paper relies on this without support.
Lorentz transformations do not happen at fixed time, so the generators of local Lorentz transformations must know about the dynamics of the theory.
“Lorentz transformations, unlike rotational symmetry, do not happen at fixed time. Therefore the generators of the (local) Lorentz transformations have to know about the dynamics of the theory, which is highly nontrivial in quantum gravity.”
The authors state this as an assumption.
Other physical objects exist in the universe and can be used to perform measurements.
“Besides these dynamical quantities, we assume that all sort of other physical objects exist in the universe. These can be used to perform measurements (for instance, light pulses traveling along geodesics, apparatus that detect the arrival of these light pulses, clocks that measure proper time along world lines, recording devices and so on).”
The authors state this as an assumption.
The freedom of choosing the boundary between the observed quantum system and the classical apparatus allows us to treat other physical objects as part of the measuring apparatus.
“To be precise, we assume that the well known freedom of choosing the boundary between the observed quantum system and the classical apparatus -emphasized by Von Neumann- allows us to do so in this context.”
The authors state this as an assumption.
The dynamics of the system is governed by the Einstein equations and the dynamical equations of the table and the particles.
“We assume the dynamics of this system to be governed by the the Einstein equations and the dynamical equations of the table and the particles.”
The authors state this as an assumption.
The matter energy-momentum tensor is negligible in the Einstein equations.
“For simplicity, we assume that the matter energy-momentum tensor is negligible in the Einstein equations, but this is not essential in what follows.”
The authors state this as an assumption.
In the quantum theory, the expansion in small boost parameter and small surface remains valid in regimes where the expectation value of the macroscopic curvature is small.
“We then assume that in the quantum theory the expansion remains valid in the regimes where the expectation value of the macroscopic curvature is small.”
The authors state this as an assumption.
The curvature at and around P and the acceleration of the particles at P are negligible at the scale of the surface.
“Second, (in the classical analysis) we assume that the curvature at and around P and the acceleration of the particles at P are negligible at the scale of the surface. That is, we take the surface small enough that we can expand around P and keep the lowest terms only.”
The authors state this as an assumption.
The relative three-velocity of the two observers is tangent to the table.
“For simplicity, we also assume that the relative three-velocity of the two observers is tangent to the table.”
The authors state this as an assumption.
The metric is spatially constant at t=0.
“Let us now make the additional assumption that the metric is spatially constant at t = 0.”
The authors state this as an assumption.
The quantum operator M(λ) corresponding to the classical observable (61) is well defined in the theory.
“We assume that this operator is well defined in the theory.”
The authors state this as an assumption.
The operator M(λ) is hermitian.
“This operator is unitary if M ( λ ) is hermitian. This is the condition under which the Lorentz transformation is unitary in the quantum theory. Assuming it is satisfied, the spectrum of the areas A and A ′ is the same.”
The paper relies on this without support.
Under a Lorentz transformation, the minimal area does not change, so two observers boosted with respect to each other see the same spectrum.
“Under a Lorentz transformation, we expect this minimal value not to change. That is, we expect that two observers boosted with respect to each other, see the same spectrum.”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
A discrete geometry is compatible with Lorentz invariance.
“on the one hand, discreteness of geometry can coexist with Lorentz invariance [331]”
From The Spin-Foam Approach to Quantum Gravity (2013) · cites this paper as [331]
The citation lists haven’t been fetched yet
References 18 · Cited by 136
Research works printed but not linked: 1
- C. Rovelli, 1995, Nucl. Phys., B456, 734
- Authors
- Rovelli, C., & Speziale, S.
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 2003-03-27
- Volume
- 67
- Issue
- 6
- Pages
- 064019
- ISSN
- 0556-2821 (ISSN-L)
- 1089-4918
- OpenAlex
- W2016972835
- Semantic Scholar
- c3ebd4ed5ac833f55d5c51fcf62ff9746b8f7f7f
- Cited by
- 136
- References
- 18
- Citations by year
- 2026: 2 · 2025: 4 · 2024: 6 · 2023: 3 · 2022: 2 · 2021: 6 · 2020: 5 · 2019: 1 · 2018: 6 · 2017: 6 · 2016: 6 · 2015: 2 · 2014: 4 · 2013: 14 · 2012: 9
- FWCI
- 6.39
- 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)
- HAL (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
- Planck scale discreteness, minimal length, Lorentz invariance, length contraction, loop quantum gravity, minimal area, area operator, unitarity, quantum gravity
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