aa r X i v : . [ phy s i c s . g e n - ph ] O c t New Quantum Structure of the Space-Time
Norma G. SANCHEZ
LERMA CNRS UMR 8112 Observatoire de Paris PSLResearch University, Sorbonne Universit´e UPMC Paris VI,61, Avenue de l’Observatoire, 75014 Paris, France (Dated: October 30, 2019)
Abstract : Starting from quantum theory (instead of general relativity) to ap-proach quantum gravity within a minimal setting allows us here to describe thequantum space-time structure and the quantum light cone. From the classical-quantum duality and quantum harmonic oscillator (
X, P ) variables in global phasespace we promote the space-time coordinates to quantum non-commuting operators.The phase space instanton (
X, P = iT ) describes the hyperbolic quantum space-timestructure and generates the quantum light cone . The classical Minkowski space-timenull generators X = ± T dissapear at the quantum level due to the relevant quantum[ X, T ] conmutator which is always non-zero. A new quantum Planck scale vacuum re-gion emerges. We describe the quantum Rindler and quantum Schwarzshild-Kruskalspace-time structures. The horizons and the r = 0 space-time singularity are quan-tum mechanically erased . The four Kruskal regions merge inside a single quantumPlanck scale ”world”. The quantum space-time structure consists of hyperbolic dis-crete levels of odd numbers ( X − T ) n = (2 n + 1) (in Planck units ), n = 0 , , ... ..( X n , T n ) and the mass levels being p (2 n + 1). A coherent picture emerges: large n levels are semiclassical tending towards a classical continuum space-time. Low n arequantum, the lowest mode ( n = 0) being the Planck scale. Two dual ( ± ) branchesare present in the local variables ( √ n + 1 ± √ n ) reflecting the duality of the largeand small n behaviours and covering the whole mass spectrum: from the largestastrophysical objects in branch (+) to the quantum elementary particles in branch(-) passing by the Planck mass. Black holes belong to both branches (+) and (-)[email protected], https://chalonge-devega.fr/sanchez CONTENTS
I. Introduction and Results 2II. Quantum Space-Time as a Harmonic Oscillator 7III. Quantum Rindler-Minkowski Space-Time 15IV. Quantum Schwarzschild-Kruskal Space-Time 18A. IV. No horizon, no space-time singularity and only one Kruskal world 19V. Mass quantization. The whole mass spectrum 21VI. Conclusions 23References 25
I. INTRODUCTION AND RESULTS
Recently, we extended the known classical-quantum duality to include gravity and thePlanck scale domain ref [1]. This led us to introduce more complete variables O QG fullytaking into account all domains, classical and quantum gravity domains and their dualityproperties, passing by the Planck scale and the elementary particle range.One of the results of such study is the classical-quantum duality of the Schwarzschild-Kruskal space-time.In this paper we go further in exploring the space-time structure with quantum theoryand the Planck scale domain. The classical-quantum duality including gravity and the QGvariables are a key insight in this study. From the usual gravity (G) variables and quantum(Q) variables ( O G , O Q ), we introduced QG variables O QG which in units of the correspondingPlanck scale magnitude o P simply read: O = 12 ( x + 1 x ) , O ≡ O QG o P , x ≡ O G o P = o P O Q (1.1)The QG variables automatically are endowed with the symmetry O (1 /x ) = O ( x ) and satisfy O ( x = 1) = 1 at the Planck scale . (1.2)QG variables are complete or global . Two values x ± of the usual variables O G or O Q arenecessary for each variable QG. The (+) and ( − ) branches precisely correspond to the twodifferent and dual ways of reaching the Planck scale: from the quantum elementary particleside (0 ≤ x ≤
1) and from the classical/semiclassical gravity side (1 ≤ x ≤ ∞ ). There isthus a classical-quantum duality between the two domains. The gravity domain is dual (inthe precise sense of the wave-particle duality) of the quantum elementary particle domainthrough the Planck scale: O G = o P O − Q (1.3)Each of the sides of the duality Eq (1.3) accounts for only one domain: Q or G but not forboth domains together. QG variables account for both of them, they contain the dualityEq.(1.3) and satisfy the QG duality
Eq (1.2).As the wave-particle duality, QG duality is general, it does not relate to the number ofdimensions nor to any other condition.In particular, length and time, basic QG variables (
X, T ) in their respective Planck unitsare: X = 12 ( x + 1 x ) , X ≡ L QG l P (1.4) T = 12 ( t − t ) , T ≡ T QG t P (1.5) l P and t P being the Planck length and time respectively. The usual variables stand here inlowercase letters. QG mass and momemtum variables are similar to ( X, T ): P = 12 ( p − p ) , P ≡ P QG p P (1.6) M = 12 ( x + 1 x ) , M ≡ M QG m P , x ≡ mm P (1.7)These are pure numbers (in Planck units), the space can be parametrized by lengths ormasses. m is the usual mass variable and m P is the Planck mass.The complete manifold of QG variables requires several ”patches” or analytic extensionsto cover the full sets X ≥ X ≤ x ± = X ± √ X − , X ≥ , x ± = X ± √ − X , X ≤ X ≥ X ≤
1) domains being the classical and quantum domains respectivelywith their two ( ± ) branches each, and when x + = x − : X = 1 , x ± = 1, (the Planck scale).The QG variables ( X, T ) satisfy: X ( x ) = X (1 /x ) , X ( − x ) = − X ( x ) , X (1) = 1 (1.9) T ( t ) = − T (1 /t ) , T ( − t ) = − T ( t ) , T (1) = 0 (1.10)QG variables can be also considered in phase-space ( X, P ) with their full global analyticextension as we describe in this paper. Comparison of the QG variables with the completeQ-variables of the harmonic oscillator is enlighting, as we do in section II here.In this paper , by promoting the QG variables (
X, T ) to quantum non-commutativecoordinates, further insight into the quantum space-time structure is obtained and new results do appear.As already mentionned, we take quantum theory as the guide, and start by the ” prototypecase”: the harmonic oscillator.We find the quantum structure of the space-time arising from the relevant non-zero space-time commutator [
X, T ], or non-zero quantum uncertainty ∆ X ∆ T by considering quantum coordinates ( X, T ). All other commutators are zero. The remaining transverse spatialcoordinates X ⊥ have all their commutators zero.The results of this paper are the following: • We find the quantum light cone : It is generated by the quantum Planck hyperbolae X − T = ± [ X, T ] due to the quantum uncertainty [
X, T ] = 1. They replace theclassical light cone generators X = ± T which are quantum mechanically erased . Insidethe Planck hyperbolae there is a enterely new quantum region within the Planck scaleand below which is purely quantum vacuum or zero-point energy. • In higher dimensions, the quantum commuting coordinates (
X, T ) and the transversenon-commuting spatial coordinates X ⊥ j generate the quantum two-sheet hyperboloid X − T + X ⊥ j X j ⊥ = ± ⊥ j = 2 , , ... ( D − D being the total space-timedimensions, D = 4 in particular in the cases considered here. • To quantize Minkowski space-time, we just consider quantum non-commutative coor-dinates (
X, T ) with the usual (non deformed) canonical quantum commutator [
X, T ] =1, (1 is here l P ), and all other commutators zero. In light-cone coordinates U = 1 √ X − T ) , V = 1 √ X + T ) , the quadratic form (symmetric order of operators) s = [ U V + V U ] = X − T =(2 V U + 1) determines the relevant part of the quantum distance. Upon identification T = − iP , the quantum coordinates ( U, V ) for hyperbolic space-time are preciselythe ( a, a + ) operators for euclidean phase space (the phase space instanton ) and as aconsequence V U is the Number operator. The expectation value ( s ) n = (2 n + 1) hasa minimal non zero value: ( s ) n =0 = 1 which is the zero point energy or Planck scalevacuum. Consistently, in quantum space-time:( T − X ) − ≥ timelike ( X − T ) − ≥ spacelike ( T − X ) − , null : the ”quantum light-cone” . This shows that only outside the null hyperbolae, that is outside the Planck scalevacuum region, such notions as distance, and timelike and spacelike signatures, canbe defined, Section III and Figs 3, 4. • Here we quantized the (
X, T ) dimensions which are relevant to the light-cone space-time structure, as this is the case for the Rindler, Schwarzschild - Kruskal and othermanifolds. The remaining spatial transverse dimensions X ⊥ are considered here ascommuting coordinates. For instance, in Minkowski space-time: s = ( X − T + X ⊥ j X j ⊥ ) , ⊥ j = 2 , , ... ( D − . (1.11)[ X ⊥ j , X ] = 0 = [ X ⊥ j , T ] , [ X ⊥ i , X ⊥ j ] = 0 , [ P ⊥ i , P ⊥ j ] = 0 (1.12)for all ⊥ i, j = 1 , ....., ( D − D being the total space-time dimensions.This corresponds to quantize the two-dimensional surface ( X, T ) relevant for the light-cone structure, leaving the transverse spatial dimensions ⊥ essentially unquantized(although they have zero commutators they could fluctuate). This is enough for con-sidering the new features arising in the quantum light cone and in the quantum Rindlerand the quantum Schwarzschild-Kruskal space-time structures, for which as is known,the relevant classical structures are in the ( X, T ) dimensions and not in the transversespatial ⊥ ones. Quantum manifolds where the transverse space X ⊥ coordinates arenon-commuting will be considered elsewhere. • We find the quantum Rindler and the quantum Schwarzschild-Kruskal space-timestructures. At the quantum level, the classical null horizons X = ± T are erased , andthe r = 0 classical singularity dissapears . The space-time structure turns out to be discretized in quantum hyperbolic levels X n − T n = ± (2 n + 1) , n = 0 , , ... . For large n the space-time becomes classical and continuum. Moreover, the classical singular r = 0 hyperbolae are quantum mechanically excluded , they do not belong to any ofthe quantum allowed levels. • We find the mass quantization for all masses. The quantum mass levels are associatedto the quantum space-time structure. The global mass levels are M n = m P √ n + 1for all n = 0 , , , ... . Two dual branches m n ± = m P [ √ n + 1 ± √ n ] do appear forthe usual mass variables, covering the whole mass range : from the Planck mass ( n = 0)till the largest astronomical masses: gravity branch (+), and from zero mass ( n = ∞ )till near the Planck mass: elementary particle branch (-). For large n , masses increase as m P (2 √ n ) in branch (+) while they decrease as m P / (2 √ n ) in branch (-). Forvery large n the spectrum becames continuum. Black holes belong to both branches(+) and (-); quantum strings have similar mass quantization. In the conclusions wecomment on these aspects. • The end of black hole evaporation is not the subject of this paper but our results herehave implications for it. Black hole ends its evaporation in branch (-). We know fromrefs [2],[7],[8] that it decays like a quantum heavy particle in pure (non mixed) states.In its last phase (mass smaller than the Planck mass m P ), the state is not anymorelike a black hole but like a heavy particle . We discuss more on it in the conclusions.This paper is organized as follows: In Section II we describe quantum space-time as aquantum harmonic oscillator and its classical-quantum duality properties. In Section IIIwe describe the quantum Rindler space-time and its structure. Section IV deals with thequantum Schwarzschild-Kruskal space-time and its properties. In Section V we treat thequantized whole mass spectrum. In Section VI we present our remarks and conclusions. II. QUANTUM SPACE-TIME AS A HARMONIC OSCILLATOR
Comparison of the QG variables to the harmonic oscillator variables is enlighting. Letus first consider the complete variables not yet promoted to quantum non-conmuting oper-ators. The oscillator complete variables (
X, P ) containing both the classical and quantumcomponents are: X Q = l l ~ p + ~ l p ) , P Q = ~ l ( l ~ p − ~ l p ) , l = 2 πω being l the length of the oscillator, (also expressed as p ~ m/ω .Or, in dimensionless variables: X = 12 ( p + 1 p ) , P = 12 ( p − p ) , p ≡ l ~ p, X ≡ X Q l , P ≡ l ~ P Q There are two branches p ± for each variable X or P and the two domains X ≥ X ≤ • Classical: X >>
1; Transition: X ≃ p + = p − = 1; Quantum: X ≤ p = exp p ∗ : X = cosh p ∗ , P = sinh p ∗ , X − P = 1 . The value l = 1, ie ~ = mω , (quantum action and classical momentum equal) is here theanalogous of the Planck scale for QG, ie the transition from the classical ( mω >> ~ ) regimeto the quantum ( mω << ~ ) regime. The hyperbolae X − P = 1, or fully dimensional X Q /l − ( l P Q ) / ~ = 1 , are the transition ”boundaries” between the classical or semiclassicaland the quantum regions in the complete analytic extension of the ( X, P ) manifold. This isa hyperbolic phase space structure. Fig. 1 displays the four regions: • Right and left exterior regions to the hyperbola X − P = ± | X | ≥ P and | X | ≤ | P | are classical: X >> mω >> ~ • The hyperbolae X − P = ± l ≃ mω ≃ ~ . Theyseparate the classical from the semiclassical and quantum regions. • ”Future” and ”past” interior regions P >
P <
X << mω << ~ (cid:1)(cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:4)(cid:5)(cid:3)(cid:5)(cid:2)(cid:1)(cid:2) (cid:1)(cid:3) (cid:1)(cid:4) (cid:5)(cid:6) (cid:5)(cid:4) (cid:5)(cid:3) (cid:5)(cid:2) (cid:7) (cid:8) (cid:9)(cid:10)(cid:11)(cid:12)(cid:12)(cid:13)(cid:9)(cid:11)(cid:10)(cid:14)(cid:15)(cid:16)(cid:11)(cid:13)(cid:17)(cid:9)(cid:10)(cid:11)(cid:12)(cid:12)(cid:13)(cid:9)(cid:11)(cid:10)(cid:14)(cid:15)(cid:16)(cid:11)(cid:13)(cid:17) (cid:18) (cid:5) (cid:13) (cid:5) (cid:19)(cid:18) (cid:5) (cid:13)(cid:13) (cid:5) (cid:19)(cid:18) (cid:5) (cid:13)(cid:13)(cid:13) (cid:5) (cid:19) (cid:18) (cid:5) (cid:13)(cid:20) (cid:5) (cid:19) (cid:7) (cid:4) (cid:5)(cid:1)(cid:5)(cid:8) (cid:4) (cid:5)(cid:21)(cid:5)(cid:22)(cid:23)(cid:7) (cid:4) (cid:5)(cid:1)(cid:5)(cid:8) (cid:4) (cid:5)(cid:21)(cid:5)(cid:1)(cid:23) (cid:8) (cid:4) (cid:5) (cid:1) (cid:5) (cid:7) (cid:4) (cid:5) (cid:21) (cid:5) (cid:22) (cid:23) (cid:8) (cid:4) (cid:5) (cid:1) (cid:5) (cid:7) (cid:4) (cid:5) (cid:21) (cid:5) (cid:1) (cid:23) (cid:24)(cid:24) (cid:22) FIG. 1. The complete analytic extension of the (
X, P ) quantum harmonic oscillator variables andits classical and quantum domains:
Hyperbolic phase space . The ( a, a + ) operators are like light-cone coordinates. The instanton P → iP , is the usual (elliptic) phase space with (dimensionless)hamiltonian ( X + P ) = 2 H , (in units of the typical oscillator length). Extension of P to be purely imaginary: P → iP , p ∗ → ip ∗ , (ie instanton ) goes from thehyperbolic to the elliptic phase space structure with the Hamiltonian H = ( X + P ) /
2, orin the dimensionfull variables: H Q = ω ~ X Q l + l P Q ~ ) , H ≡ H Q ω ~ By promoting (
X, P ) to be quantum operators, in terms of the ( a, a + ) representationyields: X = 1 √ a + + a ) , P = i √ a + − a ) , [ a, a + ] = 1 , (2.1)2 H = ( X + P ) = ( aa + + a + a ) = 2 ( a + a + 12 ) , ( X − P ) = ( a + a +2 ) (2.2)[2 H, P ] = iX, [2 H, X ] = − iP, [ X, P ] = i, with the quantum levels ǫ n = ( n + 12 ) , n = 0 , , ... (2.3)These are the dimensionless levels, (otherwise they are multiplied by ω ~ ).The ( a, a + ) operators are the light-cone type quantum coordinates of the phase space( X, P ): a = 1 √ X + iP ) , a + = 1 √ X − iP ) (2.4)The temporal variable T in the space-time configuration ( X, T ) is like the (imaginary)momentum in phase space (
X, P ): The identification P = iT in Eqs (2.1)-(2.3) yields: X = 1 √ a + + a ) , T = 1 √ a + − a ) , [ a, a + ] = 1 , (2.5)2 H = ( X − T ) = 2 ( a + a + 12 ) , ( X + T ) = ( a + a +2 ) , (2.6)[2 H, T ] = X, [2 H, X ] = T, [ X, T ] = 1 , (2.7) a + a = N being the number operator.Regions I, II, III, IV, corresponding to the exterior and interior regions to the hyperbolae X ≥ ( T ± X ≤ ( T ±
1) respectively, are covered by patches similar to the (space-like)Eqs.(2.5)-(2.7). X and T are interchanged in the time-like regions, similar to the globalhyperbolic structure Fig 1.Given the quantum hyperbolic space-time structure above described , we can think thenthe quantum space-time coordinates ( X, T ) as quantum harmonic oscillator coordinates(
X, T = iP ), including quantum space-time fluctuations with length and mass in the Planckscale domain and quantized levels, as described by Eqs (2.5)-(2.7):0 ≤ l ≤ l P , ǫ n = ( n + 12 ) , n = 1 , , ..., ω = 2 π/l Expectation values of Eqs (2.6) yield( X − T ) n = 2 ( n + 12 ) (2.8)The quantum algebra Eqs (2.5)-(2.7) describe the basic quantum space-time structure.0 • When [
X, T ] = 0, they yield the characteristic lines and light cones generators X = ± T of the classical space-time structure and its causal domains, (Fig.2). • At the quantum level, the corresponding characteristic lines and light cone generatorsEqs (2.6)-(2.8) are bent by the relevant [
X, T ] commutator, they do not cross at X = ± T = 0 but are separated by the quantum hyperbolic region ǫ due to the zero pointenergy (or quantum space-time width) ǫ = (1 / X, T ]:( X − T ) = ± [ X, T ] = ± , ǫ , ( n = 0) : the quantum light cone (2.9)[ X, T ] = 0 : X = ± T : the classical light cone . • The hyperbolae Eq.(2.9) are the quantum light cone . They quantum generalize theclassical light cone X = ± T generators when [ X, T ] = 0. The classical generators arethe asymptotes for T → ±∞ . Quantum mechanically, X is always different from ± T since [ X, T ] is always different from zero. Figs 2-3 illustrate these properties: The wellknown classical (non quantum) light cone generators and the new quantum light cone (quantum Planck hyperbolae) due to the 2 ǫ zero-point energy. • Quantum fluctuations and the quantum generated thickness make the space-timestructure spread , and its signature or causal structure is quantum mechanically modi-fied, entangled, or erased in the quantum Planck scale region.The quantization condition Eq.(2.8) yields in this context the quantum levels of the space-time. The space-time hyperbolic structure is discretized in odd number levels, Fig 4. It yieldsfor the global coordinates: X n = p (2 n + 1) for all n = 0, 1, 2, ... X n n>> = √ n + 12 √ n + O (1 /n / ) , large n (2.10) X n = 1 + n + O ( n ) , low n (2.11)In terms of the local coordinates x Eq.(1.4), it translates into the quantization: x n ± = [ X n ± p X n − √ n + 1 ± √ n ] (2.12)The condition X n ≥ n ≥
0: The n = 0 value corresponds to the Planckscale ( X = 1) : x = x − = 1 , n = 0 : Planck scale (2.13)1 (cid:1)(cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:4)(cid:5)(cid:3)(cid:5)(cid:2) (cid:1)(cid:2) (cid:1)(cid:3) (cid:1)(cid:4) (cid:5)(cid:6) (cid:5)(cid:4) (cid:5)(cid:3) (cid:5)(cid:2) (cid:7) (cid:8) (cid:8) (cid:5) (cid:9) (cid:5) (cid:10) (cid:7) (cid:8) (cid:5) (cid:9) (cid:5) (cid:1) (cid:7) (cid:6) FIG. 2.
The classical light cone. x n ± = 1 ± √ n + n + O ( n ) , low n (2.14) x n + = 2 √ n − √ n + O (1 /n / ) , x n − = 12 √ n + O (1 /n / ) , large n (2.15)Similar analysis holds for T n and the inverse local coordinates t n ± : t n ± = [ T n ± p T n + 1 ] = √ √ n + 1 ± p (2 n + 1) + 1 / X n and T n are exchanged, thus covering the global quantum hyper-bolic structure, as shown in Fig.4.A coherent picture emerges: • The large modes n correspond to the semiclassical or classical states tending towardsthe classical continum space-time in the very large n limit • The low n are quantum, with the lowest mode corresponding to the Planck scale X = 1, x = x = 1.2 (cid:1)(cid:2)(cid:1)(cid:3)(cid:1)(cid:4)(cid:5)(cid:6)(cid:5)(cid:4)(cid:5)(cid:3)(cid:5)(cid:2)(cid:1)(cid:2) (cid:1)(cid:3) (cid:1)(cid:4) (cid:5)(cid:6) (cid:5)(cid:4) (cid:5)(cid:3) (cid:5)(cid:2) (cid:7) (cid:8) (cid:9)(cid:10)(cid:11)(cid:12)(cid:7)(cid:10)(cid:13)(cid:5)(cid:14)(cid:11)(cid:15)(cid:15)(cid:10)(cid:13)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:5)(cid:21)(cid:20)(cid:22)(cid:23)(cid:24)(cid:5)(cid:25)(cid:23)(cid:18)(cid:19)(cid:26)(cid:27)(cid:5)(cid:4)(cid:28) (cid:6) (cid:29)(cid:30)(cid:31)(cid:7) (cid:4) (cid:5)(cid:1)(cid:5)(cid:8) (cid:4) (cid:5)(cid:29)(cid:5) (cid:30)(cid:7) (cid:4) (cid:5)(cid:1)(cid:5)(cid:8) (cid:4) (cid:5)(cid:29)(cid:5)(cid:1)(cid:30) (cid:8) (cid:4) (cid:5) (cid:1) (cid:5) (cid:7) (cid:4) (cid:5) (cid:29) (cid:5) (cid:30) (cid:8) (cid:4) (cid:5) (cid:1) (cid:5) (cid:7) (cid:4) (cid:5) (cid:29) (cid:5) (cid:1) (cid:30) (cid:1)(cid:30) (cid:30) (cid:30)(cid:1)(cid:30) FIG. 3.
The quantum light cone (in units of the Planck length). It is generated by the quantum hyperbolae T − X = ± [ X, T ] = ±
1. For comparison, the classical limit: light conegenerators X = ± T , is shown in Fig. 2. A new quantum region does appear inside the four Planckscale hyperbolae: The Planck scale vacuum due to the zero-point energy 2 ǫ = 1. The four causalregions dissapear inside this Planck scale region. The classical conical vertex X = ± T = 0 spreaded,smeared or erased at the quantum level. This is due to the non-zero quantum commutator [ X, T ]or ∆ X ∆ T uncertainty Eqs (2.7). • The two x n ± values indicate the two different and dual ways of reaching the Planckscale: from the classical/semiclassical side x n + >>
1: the (+) branch, and from thequantum 0 ≤ x n ≤ − ) branch. The large and low n behaviours preciselyaccount for these two dual classical-quantum domains.3 (cid:1) (cid:2)(cid:1)(cid:3)(cid:4)(cid:5)(cid:4)(cid:3)(cid:4)(cid:2) (cid:1)(cid:2) (cid:1)(cid:3) (cid:4)(cid:5) (cid:4)(cid:3) (cid:4)(cid:2) (cid:6) (cid:7) (cid:8) (cid:7) (cid:6) (cid:7)(cid:3) (cid:1)(cid:8) (cid:7)(cid:3) (cid:4)(cid:9)(cid:4)(cid:3)(cid:10) (cid:7) (cid:4)(cid:9)(cid:4)(cid:3)(cid:7)(cid:11)(cid:12) (cid:8) (cid:7)(cid:3) (cid:1)(cid:6) (cid:7)(cid:3) (cid:4)(cid:9)(cid:3)(cid:10) (cid:7) (cid:9)(cid:3)(cid:7)(cid:11)(cid:12)(cid:6) (cid:5)(cid:3) (cid:1)(cid:8) (cid:5)(cid:3) (cid:4)(cid:9)(cid:4)(cid:11)(cid:12)(cid:6) (cid:5)(cid:3) (cid:1)(cid:8) (cid:5)(cid:3) (cid:4)(cid:9)(cid:4)(cid:1)(cid:12) (cid:8) (cid:5)(cid:3) (cid:1) (cid:6) (cid:5)(cid:3) (cid:4) (cid:9) (cid:4) (cid:11) (cid:12) (cid:8) (cid:5)(cid:3) (cid:1) (cid:6) (cid:5)(cid:3) (cid:4) (cid:9) (cid:4) (cid:1) (cid:12) (cid:7) (cid:4) (cid:9) (cid:4) (cid:5) (cid:7) (cid:4) (cid:9) (cid:4) (cid:12) (cid:7) (cid:4) (cid:9) (cid:4) (cid:3) (cid:7) (cid:4) (cid:9) (cid:4) (cid:13) (cid:7) (cid:4) (cid:9) (cid:4) (cid:2) (cid:7) (cid:4) (cid:9) (cid:4) (cid:5) (cid:7) (cid:4) (cid:9) (cid:4) (cid:12) (cid:7) (cid:4) (cid:9) (cid:4) (cid:3) (cid:7) (cid:4) (cid:9) (cid:4) (cid:13) (cid:7) (cid:4) (cid:9) (cid:4) (cid:2) (cid:1)(cid:12) (cid:11)(cid:12)(cid:11)(cid:12)(cid:1)(cid:12) FIG. 4.
The quantum space-time and its hyperbolic structure . It turns out to be discretized in quantum hyperbolic levels of odd numbers (in units of the Planck length): X n − T n = ± (2 n + 1) (space-like regions), [ T n − X n = ± (2 n + 1) in the timelike regions], n = 0 , , , ... , n = 0being the Planck scale (zero point quantum energy). The n = 0 quantum hyperbolae generate the quantum light cone , Fig. 3. Low n levels are quantum and bent, large n are classical, less benttending asymptotically to a classical continum space-time. For comparison, the classical space timeis shown in Fig.2 We see that in order to gain physical insight in the quantum Minkowski space-timestructure, we can just consider quantum non-commutative coordinates (
X, T ) with usualquantum commutator [
X, T ] = 1, (1 is here l P ), and all other commutators zero. In light-4cone coordinates U = 1 √ X − T ) , V = 1 √ X + T ) , the quadratic form (symmetric order of operators) s = [ U V + V U ] = X − T = (2 V U + 1) , (2.17)determines the relevant component of the quantum distance. This corresponds exactly to theanalytic continuation of the euclidean operator 2 H = ( aa + + a + a ). The quantum coordinates( U, V ) for hyperbolic space-time are the hyperbolic ( T = iP ) operators ( a, a + ) of euclideanphase space and V U ≡ N is the Number operator. The expectation value ( s ) n = (2 n + 1)has as minimal value: ( s ) / n =0 = ±
1. Consistently, in quantum space-time we have:( T − X ) − ≥ timelike ( X − T ) − ≥ spacelike ( T − X ) − ( ±
1) = 0 , null , (the quantum light-cone) . This is so because only outside the null hyperbolae, ie outside the Planck vaccum regionsuch notions as distance, and timelike and spacelike signatures can have a meaning, Figs 1,2. Here we quantized the (
X, T ) dimensions which are relevant to the light-cone space-time structure. The remaining spatial transverse dimensions X ⊥ are considered here ascommuting coordinates, ie having all their commutators zero. For instance, in quantumMinkowski space-time: s = ( X − T + X ⊥ j X j ⊥ ) , ⊥ j = 2 , ... ( D −
2) (2.18)[ X ⊥ j , X ] = 0 = [ X ⊥ j , T ] , [ X ⊥ i , X ⊥ j ] = 0 , [ P ⊥ i , P ⊥ j ] = 0 (2.19)for all ⊥ i, j = 1 , ....., ( D − D being the total space-time dimensions. In particular D = 4in the cases considered here.This corresponds to quantize the two-dimensional surface ( X, T ) relevant for the light-cone structure, leaving the transverse spatial dimensions ⊥ with zero commutators. Thisis enough for considering the new structure arising in the quantum light cone and in thequantum Rindler and quantum Schwarzschild-Kruskal space-times, for which as it is known,5the relevant dimensions for the space-time structure are ( X, T ), (and x ∗ , t ∗ ) and not thetransverse spatial ⊥ dimensions.This is like one harmonic oscillator in the light cone surface ( X, T ), and no oscillator inthe transverse spatial dimensions ⊥ . (Although the X ⊥ j variables have zero commutators,they could fluctuate).Here we focus on the space-time quantum structure arising from the relevant non-zeroconmutator [ X, T ] and the quantum light cone . Thus, to follow on the same line of argument,we will consider below the quantum Rindler and the quantum Schwarzschild-Kruskal space-time structures. Other quantum manifolds where the transverse space X ⊥ coordinates arealso non-commuting can be considered. III. QUANTUM RINDLER-MINKOWSKI SPACE-TIME
The above quantum description is still more illustrative by considering the transforma-tion: X = exp ( κx ∗ ) cos( κp ∗ ) , P = exp ( κx ∗ ) sin( κp ∗ ) (3.1)which is the Rindler phase space representation ( x ∗ , p ∗ ) of the complete Minkowski phasespace ( X, P ). The parameter κ is the dimensionless (in Planck units) acceleration. (Herewe can express κ = l P /l = l P ω ). For classical, ie. non-quantum coordinates ( X, P ) we have:( X + P ) = exp (2 κx ∗ ) = 2 H, ( X − P ) = exp (2 κx ∗ ) cos(2 κp ∗ ) (3.2)We promote now ( X, P ) to be quantum non-commuting operators, as well as ( x ∗ , p ∗ ). Weget: ( X + P ) = exp (2 κx ∗ ) cos( κ [ x ∗ , p ∗ ]) (3.3)( X − P ) = exp (2 κx ∗ ) cos(2 κp ∗ ) (3.4)[ X, P ] = exp (2 κx ∗ ) sin( κ [ x ∗ , p ∗ ]) , (3.5)where we used the usual exponential operator product:exp( A ) exp( B ) = exp( B ) exp( A ) exp([ A, B ]).Eqs (3.3)-(3.5) describe the quantum Rindler phase space structure. The quantumRindler space-time follows upon the identification P = iT, p ∗ = it ∗ : X = exp ( κx ∗ ) cosh( κt ∗ ) , T = exp ( κx ∗ ) sinh( κt ∗ )6( X − T ) = exp (2 κx ∗ ) cosh( κ [ x ∗ , t ∗ ]) (3.6)( X + T ) = exp (2 κx ∗ ) cosh(2 κt ∗ ) (3.7)[ X, T ] = exp (2 κx ∗ ) sinh( κ [ x ∗ , t ∗ ]) (3.8) • We see the new terms appearing due to the quantum conmutators [
X, T ] and [ x ∗ , t ∗ ].At the classical level: [ X, T ] = 0, [ x ∗ , t ∗ ] = 0 and the known classical Rindler-Minkowski equations are recovered. • ( X, T ) and ( x ∗ , t ∗ ) are quantum coordinates and Eqs (3.6)-(3.8) reveal the quantumstructure of the Rindler-Minkowski space-time, their classical, semiclassical and quan-tum regions and the classical-quantum duality between them. Eqs (3.6) and (3.8)yield: ( X − T ) = ± p exp (4 κx ∗ ) + [ X, T ] (3.9) • We see the role played by the quantum non-zero commutators. Also, if the commuta-tors would not be c-numbers, the r.h.s. of Eqs (3.6)-(3.8) would be just the first termsof the exponential operator expansions, but this does not affect the general conclu-sions here. From Eqs (3.6)-(3.8), expectations values and quantum dispersions can beobtained. • Eq (3.9) quantum generalize the classical space-time Rindler ”trajectories”:( X − T ) classical = exp (2 κx ∗ ) , [ X, T ] = 0 classically (3.10)The quantum analogue of the trajectories ( x ∗ = constant) are bendt by the non-zero commutator (quantum uncertainty or quantum width) as well as the generatingRindler’s light-cone. The classical Rindler’s horizons ( x ∗ = −∞ ) X = ± T are quan-tum mechanically erased , replaced by( X − T ) = ± [ X, T ] = ± quantum Planck scale hyperbolae, (3.11)which are the quantum ”light cone”. At the quantum level, the classical null generators X = ± T spread and disappear near and inside the quantum Planck scale vacuum regionEqs (2.9), Fig (3)7 • The quantum algebra Eqs (3.6)-(3.8) and the quantum dispersions and fluctuations im-ply that the four space-time regions (classically I, II, III, IV), are spreaded or ”fuzzy” ,entangled or erased at the quantum level, near and inside the Planck domain delimi-tated by the four Planck scale hyperbolae Eq (3.11), Figs 3 and 4. • Fig 4 shows the quantum discrete levels of Minkowski-Rindler space-time and all theprevious discussion applies here X n − T n = ± (2 n + 1) , n = 0 , , , ... (3.12)”Exterior” Rindler regions to the Planck scale hyperbolaes ( X − T ) n =0 = ± n = 0 and the low n tothe large ones, which became more classical and less bendt, in agreement with theclassical-quantum duality of space-time structure. • The interior region to the n = 0 levels is the full quantum Planck scale domain.The ”future” and ”past” regions are composed by levels from the very quantum (thePlanck n = 0 hyperbolae and low n ), to the semiclassical and classical (large n ) levels( X n , T n ). • The Rindler levels ( x ∗ n ± , t ∗ n ± ) follow from Eqs (2.13)-(2.17) for ( x n ± , t n ± ): x n ± = exp ( κx ∗ n ± ) = [ X n ± p X n − √ n + 1 ± √ n ] (3.13) t n ± = exp ( κt ∗ n ± ) = [ T n ± p T n + 1 ] = [ √ n + 1 ± p (2 n + 1) + 1 / • Due to the quantum space-time width, quantum light-cone or quantum dispersion andfluctuations, and the quantum Planck scale nature of the interior region, the differencebetween the four causal regions I, II, III, IV is quantum mechanically erased in thePlanck scale region. The classical copies or halves (I, II) and (III, IV) became oneonly quantum world . • This provides further support to the antipodal identification of the two space-timecopies which are classically or semiclassically the space and time reflections of eachother and which are classical-quantum duals of each other, and therefore supports theantipodally symmetric quantum theory, refs [3], [4], [5], [6]. The classical/semiclassicalantipodal space-time symmetry and the CPT symmetry belong to the general QGclassical-quantum duality symmetry ref [1].8
IV. QUANTUM SCHWARZSCHILD-KRUSKAL SPACE-TIME
Let us now go beyond the classical Schwarzschild-Kruskal space-time structure and extendto it the findings of the sections II, III above.We have seen in ref [1] that in the complete analytic extension or global structure of theKruskal space-time underlies a classical-quantum duality structure: The external or visibleregion and its mirror copy are the classical or semiclassical gravitational domains while theinternal region is fully quantum gravitational -Planck scale- domain. A duality symmetrybetween the two external regions, and between the internal and external parts shows up asa classical - quantum duality . External and internal regions meaning now with respect tothe hyperbolae X − T = ± quantum Schwarzshild-Kruskal description of space-time, weproceed as with the quantum Minkowski-Rindler space-time variables in previous section.The phase space and space-time coordinate transformations are the same in both Rindlerand Schwarzschild cases. The classical Kruskal phase space coordinates (
X, P ) in terms ofthe Schwarzschild phase-space representation ( x ∗ , p ∗ ) are given by X = exp ( κx ∗ ) cos( κp ∗ ) , P = exp ( κx ∗ ) sin( κp ∗ ) (4.1)( X + P ) = exp (2 κx ∗ ) = 2 H, ( X − P ) = exp (2 κx ∗ ) cos(2 κp ∗ ) (4.2)with the Schwarzschild star coordinate x ∗ :exp( κx ∗ ) = √ κr − κr ) , κr > κ the dimensionless (in Planck units) gravity acceleration or surface gravity. Anotherpatch similar to Eqs (4.1)-(4.3) but with X and P exchanged and x ∗ defined by exp( κx ∗ ) = √ − κr exp( κr ), holds for 2 κr < X, P ) to be quantum coordinates, ie non-commuting operators, and sim-ilarly for ( x ∗ , p ∗ ), yields Eqs (3.3)-(3.5). They provide in this case the quantum Kruskal’sphase space coordinates ( X, P ) in terms of the quantum Schwarzschild coordinates ( x ∗ , p ∗ )with x ∗ given by Eq. (4.3). The corresponding quantum Kruskal’s space-time follow uponthe identification: P = iT, p ∗ = it ∗ . In terms of Schwarzschild’s space-time coordinates9( x ∗ , t ∗ ) it yields: X = exp ( κx ∗ ) cosh( κt ∗ ) , T = exp ( κx ∗ ) sinh( κt ∗ ) (4.4)( X − T ) = exp (2 κx ∗ ) cosh( κ [ x ∗ , t ∗ ]) (4.5)( X + T ) = exp (2 κx ∗ ) cosh(2 κt ∗ ) (4.6)[ X, T ] = exp (2 κx ∗ ) sinh( κ [ x ∗ , t ∗ ]) (4.7)We see the new terms appearing due to the quantum conmutators. At the classical level:[ X, T ] = 0 , [ x ∗ , t ∗ ] = 0 (classically)and the known classical Schwarzschild-Kruskal equations are recovered.Eqs (4.5)-(4.7) describe the quantum Schwarzschild-Kruskal space-time structure and itsproperties, we analyze them below. Upon the identification P = iT , the quantum Kruskalligth-cone variables U = 1 √ X − T ) , V = 1 √ X + T ) (4.8)in hyperbolic space are the ( a, a + ) operators Eqs (2.4). The quadratic form (symmetricorder of operators):2 H = U V + V U = X − T = (2 U V + 1) , U V = N ≡ number operator , yields the quantum hyperbolic structure and the discrete hyperbolic space-time levels: X n − T n = (2 n + 1) and T n − X n = (2 n + 1) , ( n = 0 , , ... ) (4.9)The amplitudes ( X n , T n ) are √ n + 1 and follow the same Eqs (2.10)-(2.12) and Fig 4. Wedescribe the quantum structure below. A. IV. No horizon, no space-time singularity and only one Kruskal world
From Eqs (4.5)-(4.7), expectation values and quantum dispersions can be obtained. Forinstance, the equation for the quantum hyperbolic ”trajectories” is( X − T ) = ± p exp (4 κx ∗ ) + [ X, T ] = ± p (1 − κr ) exp (4 κr ) + [ X, T ] (4.10)0The characteristic lines and what classically were the light-cone generating horizons X = ± T (at 2 κr = 1, or x ∗ = −∞ ) are now: X = ± p T + [ X, T ] at 2 κr = 1: X = ± T , no horizons (4.11)We see that X = ± T at 2 κr = 1 and the null horizons are erased .Similarly, in the interior regions the classical hyperbolae ( T − X ) classical = ± r = 0 , ( x ∗ = 0) are now replacedby : ( T − X ) = ± p X, T ] = ±√ r = 0: ( T − X ) = ± no singularity ( T − X ) classical = ± r = 0 classically (4.12)The classical singularity r = 0 = x ∗ is quantum mechanically smeared or erased which iswhat is expected in a quantum space-time description. • The right and left ”exterior” regions to the quantum Planck hyperbolae( X − T ) n =0 = ± n = 0 (Planck scale), low n (quantum) to the intermediate and large n (classical) behaviours. • Similarly, the future and past regions to the quantum Planck hyperbolae( T − X ) n =0 = ±
1, contain all allowed levels and behaviours. There is not r = 0 = x ∗ singularity boundary in the quantum space-time. • ( T − X ) n =0 = ± X = ± T ) classical at x ∗ = −∞ , κr = 1 in the quantum space-time. • ( T − X ) = ±√ T − X ) classical ( r = 0) = ±
1. Moreover, the quantum hyperbolae ( T − X ) = ±√ outside the allowed quantum hyperbolic levels. They do not correspond to any ofthe allowed quantum levels Eqs (4.10), n = 0 , , , ... and therefore, they are excluded at the quantum level: The singularity is removed out from the quantum space-time.1 • There are no singularity boundaries at ( T − X )(2 κr = 1) = ± T − X ) = ±√ extends without boundarybeyond the Planck hyperbolae ( T − X )( n = 0) = ± all levels: from themore quantum (low n ) levels to the classical (large n ) ones, as shown in Fig.4. • The internal region to the four quantum Planck hyperbolae ( T − X )( n = 0) = ± totally quantum and within the Planck scale: this is the quantum vacuum or ”zeropoint Planck energy” region. This confirms and expands our result in ref [1] about the quantum interior region of the black hole. • The null horizons disappeared at the quantum level. Due to the quantum [
X, T ] com-mutator, quantum (
X, T ) dispersions and fluctuations, the difference between the fourclassical Kruskal regions (I, II, III, IV) dissapears in the Planck scale domain.This provides further support to the antipodal identification of the two Kruskalcopies which are classically and semiclassically the space-time reflection of each other,and which translates into the CPT symmetry and antipodally symmetric states refs[3],[4],[5],[6]. • The levels in terms of the Schwarzschild variables ( x ∗ n ± , t ∗ n ± ) follow from Eqs (3.13),(3.14) for ( x n ± , t n ± ), being x = exp ( κx ∗ ), and t = exp ( κt ∗ ): x n ± = [ p κr n ± − κr n ± ) = [ √ n + 1 ± √ n ] (4.13) t n ± = [ √ n + 1 ± p (2 n + 1) + 1 / , (4.14)which complete all the levels. Their large n and low n behaviours follow Eqs (2.14)-(2.16) and their respective clasical-quantum duality properties. V. MASS QUANTIZATION. THE WHOLE MASS SPECTRUM ( X n , T n ) , ( x n , t n ) are given in Planck (length and time) units. In terms of the mass globalvariables X = M/m P , or the local ones x = m/m P , Eqs (1.4), (1.7), it translates into themass levels: M n = m P p (2 n + 1) , all n = 0 , , , .... (5.1) M n n>> = m P [ √ n + 12 √ n + O (1 /n / ) ] , (5.2)2 m n ± = [ M n ± q M n − m P ] , (5.3)The condition M n ≥ m P simply corresponds to the whole spectrum n ≥ m n ± = m P [ √ n + 1 ± √ n ] (5.4) m = m − = M = m P , n = 0 : Planck mass m P (5.5) m n + = m P [ 2 √ n − √ n + O (1 /n / ) ] , large n = > m + larger than m P (5.6) m n − = m P √ n + O (1 /n / ) , large n = > m − smaller than m P (5.7) • The mass quantization here holds for all masses , not only for black holes. Namely, thequantum mass levels are associated to the quantum space-time structure. Space-timecan be parametrized by masses (”mass coordinates”), just related to length and time,as the QG variables, on the same footing as space and time variables. In Planck units,any of these variables (or another convenient set) can be used. • The two ( ± ) dual mass branches (classical and quantum) Eqs (5.4)-(5.7) correspondto the large and small masses with respect to the Planck mass m P , they cover the whole mass range: From the Planck mass: branch (+), and from zero mass till nearthe Planck mass: branch (-). • As n increases, masses in the branch (+) increase from m P covering all the massspectrum of gravitational objects till the largest masses. Masses are quantized as m P (2 √ n ) as the dominant term, Eq (5.6). For very large n the spectrum becamescontinuum. Macroscopic objects and astronomical masses belong to this branch (grav-itationnal branch). • As n increases, masses in the branch (-) decrease : The branch (-) covers the masses smaller than m P from the zero mass to masses remaining smaller than the Planckmass: large n behaviour of branch (-) Eq.(5.7). The quantum elementary particlemasses belong to this branch (quantum particle branch). • Black hole masses belong to both branches (+) and (-). Branch (+) covers all macro-scopic and astrophysical black holes as well as semiclassical black hole quantization √ n till masses nearby the Planck mass.3 • The microscopic black holes, quantum black holes (with masses near the Planck massand smaller till the zero mass, ie as a consequence of black hole evaporation), belongto the branch (-). The branches (+) and (-) cover all the black hole masses. Theblack hole masses in the process of black hole evaporation go from branches (+) to(-). Black hole ends its evaporation in branch (-) decaying as a pure quantum state. • Black hole evaporation is not the subject of this paper but our results here haveimplications for it. The last stage of black hole evaporation and its quantum decaybelong to the quantum branch (-). Black hole evaporation is thermal (ie a mixed state)in its semiclassical gravity phase (Hawking radiation) and it is non thermal in its lastquantum stage (pure quantum decay) refs [2], [7], [8]. In its last phase (mass smallerthan the Planck mass m P ), the state is not anymore a black hole, but a pure (nonmixed) quantum state, decaying like a quantum heavy particle. VI. CONCLUSIONS • We have investigated here the quantum space-time structure arising from the relevantnon-zero space-time commutator [
X, T ], or non-zero quantum uncertainty ∆ X ∆ T byconsidering quantum coordinates ( X, T ). The remaining transverse spatial coordinates X ⊥ have all their commutators zero. This is enough to capture the essential featuresof the new quantum space-time structure. • We found the quantum light cone : It is generated by the quantum Planck hyperbolae X − T = ± [ X, T ] due to the quantum uncertainty [
X, T ] = 1 They replace theclassical light cone generators X = ± T which are quantum mechanically erased . Insidethe four Planck hyperbolae there is a enterely new quantum region within the Planckscale and below which is a purely quantum vacuum or zero-point Planck energy region • The quantum non-commuting coordinates (
X, T ) and the transverse commuting spa-tial coordinates X ⊥ j generate the quantum two-sheet hyperboloid X − T + X ⊥ j X j ⊥ = ± • We found the quantum Rindler and the quantum Schwarzschild-Kruskal space-timestructures: we considered the relevant quantum non-commutative coordinates and the4quantum hyperbolic ”light cone” hyperbolae. They generalize the classical knownSchwarzschild-Kruskal structures and yield them in the classical case (zero quan-tum commutators). At the quantum level, the classical null horizons X = ± T are erased , and the r = 0 classical singularity dissapears . Interestingly enough, theKruskal space-time structure turns out to be discretized in quantum hyperbolic levels ( X n − T n ) = ± (2 n + 1) , n = 0 , , ... . Moreover, the r = 0 singular -hyperbola isquantum mechanically excluded , it does not belong to any of the quantum allowed levels. • The quantum Schwarzschild-Kruskal space-time extends without boundary and with-out any singularity in quantum discrete allowed levels beyond the quantum Planckhyperbolae X − T = ±
1, from the Planck scale ( n = 0) and the very quantumlevels (low n ) to the quasi-classical and classical levels (intermediate and large n ), andasymptotically tend to a continuum classical space-time for very large n . • The quantum mass levels here hold for all masses. The two ( ± ) dual mass branchescorrespond to the larger and smaller masses with respect to the Planck mass m P respectively, they cover the whole mass range from the Planck mass in branch (+) untillthe largest astronomical masses, and from zero mass in branch (-) in the elementaryparticle domain till near the Planck mass. As n increases, masses in the branch (+) increase (as 2 √ n ). For very large n the spectrum becames continuum. Masses in thebranch (-) decrease in the large n behaviour, precisely as 1 / (2 √ n ), the dual of branch(+). The whole mass levels are provided in Section V above. Black hole masses belongto both branches (+) and (-). • The quantum end of black hole evaporation is not the central issue of this paper, butour results here have consequences for this problem which we will discuss elsewhere:The quantum black hole decays into elementary particle states, that is to say pure(non mixed) quantum states, in discrete levels and other implications ref [9]. • We can similarly think in quantum string coordinates (collection of point oscillators) todescribe the quantum space-time structure, (which is different from strings propagatingon a fixed space-time background). This yields similar results to the results foundhere with a quantum hyperbolic space-time width and hyperbolic structure for the5characteristic lines and light cone generators, or for the space-time horizons ref [9].Moreover, we see that the mass quantization m P √ n we found here, ie Eq (5.1), Eq(5.4), is like the string mass quantization M n = m s √ n , n = 0 , , ... with the Planckmass m P instead of the fundamental string mass m s , ie G/c instead of the stringconstant α ′ . • Here we focused on the space-time quantum structure arising from the relevant non-zero commutator [
X, T ]: the quantum light cone which is relevant for the Minkowski,Rindler and the Schwarzschild-Kruskal quantum space-time structures.Quantizing the higher dimensional transverse dimensions X ⊥ j does not change thebasic new quantum structure here. In another manifolds, there will be specific ( D −
2) spatial transverse contributions. Quantum non-commuting transverse coordinatesimportant for another type of manifolds will be considered elsewhere, ref [9].
ACKNOWLEDGEMENTS
The author thanks G.’t Hooft for interesting and stimulating communications on severaloccasions, M. Ramon Medrano for useful discussions and encouredgement and F.Sevre forhelp with the figures. The author acknowledges the French National Center of ScientificResearch (CNRS) for Emeritus contract. This work was performed in LERMA-CNRS-Observatoire de Paris- PSL Research University-Sorbonne Universit´e Pierre et Marie Curie.
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