Area, ladder symmetry, degeneracy and fluctuations of a horizon
aa r X i v : . [ h e p - t h ] N ov Area, ladder symmetry, degeneracy and fluctuations of a horizon
Mohammad H. Ansari ∗ University of Waterloo, Waterloo, On, Canada N2L 3G1 andPerimeter Institute, Waterloo, On, Canada N2L 2Y5 (Dated: November 11, 2018)
Abstract
Loop quantum gravity admits a kind of area quantization that is characterized by three quantumnumbers. We show the complete spectrum of area is the union of equidistant subsets and a universalreformulation with fewer parameters is possible. Associated with any area there is also anothernumber that determines its degeneracy. One application is that a quantum horizon manifestsharmonic modes in vacuum fluctuations. It is discussed the physical fluctuations of a space-time horizon should include all the excluded area eigenvalues, where quantum amplification effectoccurs. Due to this effect the uniformity of transition matrix elements between near levels couldbe assumed. Based on these, a modification to the previous method of analyzing the radianceintensities is presented that makes the result one step further precise. A few of harmonic modesappear to be extremely amplified on top of the Hawking’s radiation. They are expected to form afew brightest lines with the wavelength not larger than the black hole size.
PACS numbers: 04.60.Pp, 04.70.Dy ∗ Electronic address: [email protected] ontents I. Introduction II. Area III. Ladder symmetry IV. Degeneracy V. Fluctuations of a horizon References I. INTRODUCTION
So far the main consequence of area quantization in loop quantum gravity has been theremoval of classical gravitational singularities [1] as well as determining the isolated hori-zon entropy [2]. The predicted generic exit of scale factor from an inflation sector into aFriedman universe in a loop quantized minisuperspace is at present in agreement with stan-dard inflation models. This quantum phenomena, which comes from a quantum correctionin the inhomogeneous Hamiltonian constraint, is through elementary area variable whosevalue should be determined by an underlying inhomogeneous state. Area is an elementaryoperator in loop quantum gravity because in the classical limits it is directly related to thedensitized triad as a canonical variable. In this note, we study two previously unknown prop-erties of area quantization that further clarify the understanding of this operator. Firstly,the area eigenvalues possess a symmetry that its spectrum is the union of different evenlyspaced subsets. Secondly, the eigenvalues are substantially degenerate such that in largerarea the degeneracy increases. Due to the presence of a huge class of completely tangentialexcitations on a surface different regions of the surface are distinguishable. These togetherresult in degeneracy increasing in a way that with any eigenvalue a finite exponentiallyproportional to area degeneracy is associated. One application is in area fluctuations of acollapsing star. It is discussed a trustworthy analysis of area fluctuations in a space-timehorizon must include all those excluded quanta from a quantized isolated horizon [2]. Having2ecognized the quantum amplification effect during transitions, the density matrix elementscan be considered uniform in near levels. The black hole undergoes a thermal fluctuationsand harmonic modes resonate. Using these properties, a modification method to the previ-ous analysis of the intensities [4] is introduced that makes the result one step further precise.The major result is that the fluctuations in the dominant configuration with minimal quan-tum of area is mostly amplified by the black hole such that a few sharp and bright linesappear on top of Hawking’s radiation. These modes cannot be seen in the wavelength largerthan the size of black hole. In summary, by the use of a few main assumptions from blackhole studies, loop quantum gravity, a non-perturbative background independent approachto quantum gravity, becomes testable much above the Planck scales if quantum primordialblack holes are ever found.
II. AREA
In this note we choose to define a surface by a coordinate condition. The quantization of a3-manifold is obtained by quantizing the holonomy configuration space on embedded graphsin a spatial manifold. The sub-graphs whose nodes lie on a surface are basis for defining thequantum state of the surface. Densitized conjugate momenta possess full information of thesurface metric and consequently the surface area, [3].Consider a vertex lying on a surface with total upper side spin j u , bottom side spin j d ,and completely tangential edges of total spin j t on the surface. The quantum of area in thisstate depends on the upper and lower spins as well as the total tangent vector induced bythese two on the surface, j u + d , a = a o p f ( j u ) + 2 f ( j d ) − f ( j u + d ) , (1)where f ( x ) = x ( x + 1), a o := 4 πγℓ P , γ the Barbero-Immirzi parameter, and j u + d ∈ [ | j u − j d | , . . . , j u + j d − , j u + j d ]. Note that the completely tangential edges do not contribute inthe area.Consider a closed underlying surface dividing the 3-manifold into two completely disjointsectors and not bounded by a boundary. A few additional vertices are needed in order toclose this quantum state. This introduces two additional constraints on the states, namely: P α j ( α ) u ∈ Z + and P α j ( α ) d ∈ Z + , where α labels all the residing vertices on the surface, [3].3 II. LADDER SYMMETRY
In SO(3) group representation spins are integers. Therefore in (1) the right side canbe written as a positive integer number: m . = f ( j u ) + f ( j d ) − f ( j u + d ) = ( a/a o ) . Thisnumber due to the following proof is in fact any natural number. Suppose j u ≥ j d andthe difference of them is a positive integer n = j u − j d . Restricting to the subset M ∗ of j u + d = j u + j d , it is easy to verify the generator of this subset is n ( n + 1) / j d . The firstterm, a triangular number, is a positive integer. The second term is independent of n andin principle takes any positive integer value. Therefore the set of all M ∗ corresponding tothe states with j d = { , , , , · · · } is equivalent to natural numbers; N ≡ { M ∗ } / R , where / R stands for the modulation of repetition (or in a simple word different copies of onenumber are identified). Since m in general is a positive integer, any other subset fits into N .Consequently, an irreducible reformulation of area when all copies of numbers are identifiedis possible by one quantum number, a = a o √ n , where n ∈ N .The spectrum of area modulo repetitions in SU(2) group representations is impossibleto reformulate by one parameter; however, it is possible by two in the following form: a = a o √ ζ n , for any discriminant of positive definite form ζ and any positive integer n , [6].A universal reformulation is thus possible if one rewrites the SO(3) irreducible reformula-tion as a reducible one by two parameters. In the followings it is shown that any integer c canbe represented uniquely by c = ζ n where ζ is a square-free number and n ∈ N . A positiveinteger that has no perfect square divisors except 1 is called square-free (or quadratfrei) num-ber. In other words it is a number whose prime decomposition contains no repeated factors;for instance 15 is square-free but 18 is not. Now consider an integer c containing s differentprime factors p , p , · · · , p s each repeated n , n , · · · , n s times, respectively; c = Q si =1 ( p i ) n i .The exponents n i are all positive integers and are either even or odd numbers. Considerthe case that the exponents are all odd numbers, n i := 2 m i + 1. Therefore c can be writtenin the form of ( Q sk =1 p k ) . ( Q sl =1 ( p l ) m l ) which shows the integer c is a multiplication of asquare-free part and a square part. This could be redone for any integer number and theresult is the same decomposition. Since the prime factorization of every number is unique, sodoes its decomposition into square and square-free numbers. Therefore, in SO(3) group thecomplete set of quantum area { m } , which fits into natural numbers, is the multiplication ofa square-free and a square number. In other words, the quantum of area can be reformulated4nto a = a o √ ζ n . This makes the universal reformulation of area as a function of ζ and n , a n ( ζ ) = a χ p ζ n (2)for ∀ n ∈ N , where in SO(3) group ζ is any square-free number { , , , , · · · } and χ := √ ζ is the ‘discriminant of any positive definite form’ { , , , , , · · · } and χ := 1 /
2. The parameters χ is ‘the group characteristic parameter.’ Fixing ζ a generation ofevenly spaced numbers is picked out, thus the parameter ζ is the ‘generational number.’ Forthe purpose of making the rest of this note easier to read let us rename the first generationalnumber whose gap between levels is minimal by ζ min and the minimal area a min .Note that the term √ ζ is an irrational number in both groups and in any generation itis unique. Therefore the sum or difference of any two quanta a n ( ζ ) and a n ( ζ ) for ζ = ζ is unique and belongs to none of generations. IV. DEGENERACY
The spin network states of a surface under the action of area operator manifest a sub-stantial degeneracy. Consider an N -valent vertex lying on a surface, some of the edges arecontained in the upper side, some in the lower, and some lie completely tangential on thesurface. Given the total spin of upper and lower sectors by j u and j d , respectively, a set ofarea eigenvalues are generated from a minimum where j u + d = j u + j d to a maximum where j u + d = | j u − j d | from eq. (1). Changing j u and j d a different finite subset of area is generatedwhose elements may or may not coincide with the elements of the other subset of area eigen-values. Associated with any area eigenvalue there appears unexpectedly a finite numberof completely different eigenstates. For instance, these states | j u = 1 , j d = 0 , j u + d = 1 i , | j u = 0 , j d = 1 , j u + d = 1 i , and | j u = 1 , j d = 1 , j u + d = 2 i correspond to the area a = √ a o .Counting these states for every eigenvalue a power law correlation with the size of area ap-pears such that a larger area possesses a higher degeneracy. This is studied for both SU(2)and SO(3) gauge groups in [6].On a classical surface there are a finite number of area cells and a set of degeneratequantum states could be associated with it. However, this is essential for a backgroundindependent theory to identify only physical states after reducing the redundant gauge-and diffeomorphism-transformed ones. Gauge invariance by definition is satisfied in spin5etwork state, but diffeomorphism invariance should be checked by its imposing on thestates. Consider a surface containing a large number of the same area cell in differentregions. Each cell is a degenerate eigenvalue of area. However, area operator does not ‘see’the completely tangential edges of these degenerate states. By definitions, the number ofcompletely tangential edges at each vertex could vary from zero to infinity and when thereare many of these excitations at one vertex they accept a huge spectrum of spins. Thesevarious states make the identical cell configuration on different regions distinguishable underthe measurements of other observable operators.Note that the area of higher levels can be decomposed precisely into smaller fractions ofthe same generation (without any approximation). For example, a n = na = ( n − a + a = · · · . As it was explained above, these cells are all completely distinguishable. Therefore thedegeneracy of the area eigenvalue a n becomes Ω n = g n + g n − g + · · · + ( g ) n . Obviously thedominant term in the sum belongs to the configuration with maximum number of the areacell a . Therefore the total degeneracy of a n ( ζ ) for n ≫ n ( ζ ) = g ( ζ ) n . (3)In the classical limits, the dominant configuration of a large surface is the one occupyingthe highest possible level of area from the ‘first’ generation ζ min ; i. e. A ≈ na min . Thisdominant degeneracy is g ( ζ min ) n and a kinematic entropy can be associated with it propor-tional to the area; S = A (ln g ( ζ min ) /a min ). Depending on the type of time evolution of thesurface this entropy may vanish, decrease, increase or remains unchanged in the course oftime. In other words, a classical surface characterized by its area at each time slice possessesa finite entropy-like parameter. Space-time horizons as a class of physical surfaces possessa non-decreasing entropy. In other words their kinematical entropy in the course of time,due to the second black hole thermodynamics law, are physical entropy. We will show inthe next section such a horizon carries an entropy whose nature is the total degeneracy ofvacuum fluctuation modes responsible for the thermal radiation of black hole. However, forthe aim of this note on the study of kinematics of fluctuations we disregard here the issuesof defining the Hamiltonian of a quantum horizon based on spin foam, which is still an openproblem. 6 . FLUCTUATIONS OF A HORIZON Having known a suitable definition for the information flow other than expansions ofgeodesic congruences used in general relativity, one can certainly define a quantum blackhole. However, there are different definitions of quantum horizons with different properties,including causal ones. Event horizon is always a null surface by definition, thus it mustsatisfy one-way information transfer, [7]. However an event horizon is not locally defined atall, not even in time. To define it classically, we need the information of the whole manifold.In canonical quantum gravity, we need a definition by which we can look at a place inspace and say those photons reaching to us from there must come from a spatial slice thatintersects a space-time horizon. Such local definitions are in fact those of apparent, trapping,and dynamical horizons, [2]. On the other hand, the space-time horizons are not necessarilynull. They would be so if we have vacuum and absence of gravitational radiation. Vacuumcan easily be achieved for a spin network case, but we cannot prevent the local gravitationaldegrees of freedom to be excited in the neighborhood of a space-time horizon. With thesegravitational radiation across the horizon and with positive energy conditions (or vacuum)the horizon will be space-like rather than null. Moreover, the energy conditions in quantumgravity could not be taken for granted, even for semiclassical states, as long as violationsoccur on small length scales, [12]. Thus, quantum space-time horizons can become even timelike with a two-way information transfer. As a consequence, one cannot restrict thequantum fluctuations of horizon area to the subset that is considered in the trapping-basedtheories of horizon because the basic assumption underneath those theories is that a quantumhorizon is the extention of a classical null boundary of space-time in a quantum theory, [2].Physical fluctuations of space-time horizons, in fact, occurs in a wider spectrum that includesall excluded quanta of area.Note that in the Hawking’s conception of a black hole radiation, those modes createdin vacuum at past null infinity pass through the center of a collapsing star, hover aroundit and come out of it at future infinity. The outgoing quanta get a thermal statistics fromthis incipient (about-to-be-formed) black hole. Quantum fluctuations of the horizon changethis simple picture because the Hawking quanta will not be able to hover at a nearly fixeddistance from the fluctuating horizon. Bekenstein and Mukhanov postulated an equidistantspectrum for the horizon area fluctuations in [9] and showed concentrating of radiance modes7n discrete lines. In loop quantum gravity as a fundamental candidate theory of quantumgravity, quantum of area is different and here its emissive pattern is work out.During the latest stages of gravitational collapse of a neutral non-rotating spherical star,all radiatable multipole perturbations in the gravitational fields are radiated away such thatits classical physics is described only by its horizon area. The energy associated with thisobject depends on the area by the relation A = πG c M . The energy fluctuations of alarge space-time horizon are easy to find δM = γχ M Pl M √ ζδn . Ladder symmetry classifiesthe transitions between levels into: 1) ‘generational transitions’, those with both initial andfinal levels belonging to the same generation, or 2) ‘inter-generational transitions’, withinitial and final levels belonging to two different generations. The generational transitionsproduce ‘harmonic’ frequencies proportional to a fundamental frequency by an integer. Inter-generational transitions produce ‘non-harmonic modes’.In generational transitions, the fundamental frequency is the jump between two consec-utive levels with frequency ̟ ( ζ ) = ( γχ √ ζ ) ω o , where ω o := c GM is the so-called ‘frequencyscale’. For instance, a black hole of mass 10 − M ⊙ has a horizon of area about 10 − m and a temperature about 10 K. The frequency scale is thus of the order of ∼
10 keV.Such a typical hole has a horizon 40 order of magnitude larger than the Planck length area.Therefore from each harmonic mode there are many copies emitted in the different levels;or in other words these modes are amplified. On the other hand, since the difference of twolevels of different generations is a unique number, there exists only one copy from each non-harmonic mode in all possible transitions. This quantum amplification effect makes a blackhole condensate its particles production mostly on harmonic modes. One important con-sequence is the density matrix elements of non-harmonic modes can be regarded negligibleand therefore the generational transitions matrix elements can be assumed to be uniform.In a transition down the level of a generation, there are two weight factors: the transitionand the population weights. Assume a hole of large area A . When the hole jumps f stepsdown the ladder of levels in the generation ζ , it emits a quanta of the frequency f ̟ ( ζ ). Thismuch of radiance energy could also be emitted in the dominant configuration by radiating f a ( ζ ) a min quanta of the fundamental frequency ̟ ( ζ min ). These two transitions, although areof the same radiance frequency, appear with different possibilities. The degeneracy ratioof these two is Ω( f ̟ ( ζ )) / Ω( f a ( ζ ) a min ̟ ( ζ min )) that gives rise to the definition of ‘transitionweight’ θ ( ζ , f ) = g ( ζ ) f g ( ζ min ) − fa ( ζ ) /a min . The second weight is the population one that8omes from a different root. Due to quantum amplification effect, from each harmonicfrequency there produced many copies in different levels on the generation. This weight isin fact the number of possible quanta emitting from different levels with the same frequency.It is easy to verify this number is N ̟ ( ζ ) − f + 1 where N ̟ ( ζ ) is the number of copies from thefundamental frequency, and for near level modes ( f ≪ N ̟ ( ζ ) ) it is Aa ( ζ ) . We absorb constantsin normalization factors and the population weight in near levels becomes ρ ( ζ ) := 1 / √ ζ .Finally notice that within one generation when a space-time hole jumps f steps downthe ladder of levels, the degeneracy decreases by a factor of g ( ζ ) f . Having defined thetransition and the population weights, the conditional probability of ω f ( ζ ) emission afterusing (2) becomes P ( ω f ( ζ ) |
1) = C − ρ ( ζ ) g ( ζ min ) − f √ ζ/ζ min , where C is the normalizationfactor, [13].One can consider a successive emissions and associates a probability to it as the mul-tiplication of the probability of each emission. The conditional probability of a j dimen-sional sequence of different frequencies becomes Q ji =1 P ( ω f i ( ζ i ) | k emissions out of j to be of the frequency ω f ∗ ( ζ ∗ ) (in no matter whatorder) while the rest of accompanying emissions are of any value except this frequency, is P ( k, ω f ∗ ( ζ ∗ ); { ω f ( ζ ) , · · · }| j ) = ( jk ) [ P ( ω f ∗ ( ζ ∗ ) | k × Q j − ki =1; ζ = ζ ∗ P ( ω f i ( ζ i ) | ω f ∗ ( ζ ∗ ) and therefore the prob-abilities of any accompanying frequency should sum. From the definition of C , it is easyto find out in each sum over accompanying modes instead of P ω = ω ∗ P ( ω f i ( ζ i ) |
1) we can re-place C − P ( ω f ∗ ( ζ ∗ ) |
1) that simplifies the probability to P ( k, ω f ∗ ( ζ ∗ ) | j ) = ( kj ) [ P ( ω f ∗ ( ζ ∗ ) | k × [ C − P ( ω f ∗ ( ζ ∗ ) | j − k .Note that a black hole radiates in a ‘time’ sequential order, [11]. The probabilitiesof zero and one jump (of no matter what frequency) in the time interval ∆ t are P ∆ t (0)and P ∆ t (1), respectively. In the time interval 2∆ t , the probabilities of zero, one, and twojumps are P ∆ t (0) , 2 P ∆ t (0) P ∆ t (1), and 2 P ∆ t (0) P ∆ t (2) + P ∆ t (1) , respectively. By inductionthis is found for higher number of jumps in an interval and for longer time. A generalsolution for the equations the probability of j time-ordered decays in an interval of time∆ t is P ∆ t ( j ) = j ! ( ∆ tτ ) j exp( ∆ tτ ). Multiplying this probability with P ( k, ω f ∗ ( ζ ∗ ) | j ) and thensumming over all sequence dimensions j ≥ k , it is easy to manipulate the total probabilityof k emissions with frequency ω f ∗ ( ζ ∗ ) to be P ∆ t ( k, ω f ∗ ( ζ ∗ )) = k ! ( x ∗ f ) k exp( − x ∗ f ), where x ∗ f = ∆ tCτ ρ ( ζ ∗ ) g ( ζ min ) − f √ ζ/ζ min . This indicates the distribution of the number of quanta emitted9n harmonic modes is Poisson-like.Let us now look at the distribution of the number of quanta emitted from a blackbody radiation. The probability of one emission of frequency ω ∗ is Boltzmann-like; π ω ∗ = B exp( − ~ ω ∗ kT ) where B is normalization factor B = P ω π ω . Successive emissionsoccurs independently and therefore the probability of a j dimensional sequence in which p emissions are of the frequency ω ∗ is (cid:0) jp (cid:1) ( π ω ∗ ) k Q j − pi P ω i = ω ∗ π ω i . The last summation termcan be replaced from the normalization relation by B − π ω ∗ . This makes the probabilityequivalent with the black hole emission probability P ( k, ω f ∗ ( ζ ∗ ) | j ) when g ( ζ min ) − f ∗ √ ζ ∗ /ζ min (i.e. exp( − S )) is replaced with exp( − ~ ω ∗ kT ). The analogy indicates that the hole radiationis characterized by Planck’s black body radiation and the temperature matches the blackhole temperature when the Barbero-immirzi parameter is properly defined for getting theBekenstein-Hawking entropy. In fact the black hole is hot and the thermal character ofthe radiation is entirely due to the degeneracy of the levels, the same degeneracy (3) thatbecomes manifest as black hole entropy.By definition, the intensity of a mode is the total energy emitted in that frequency perunit time and area. The average number of emissive quanta at a typical harmonic frequencyis k = P ∞ k =1 kP ∆ t ( k | ω f ( ζ )). Calculating this summation gives rise to the intensity I ( ω f ( ζ )) I o = f g ( ζ min ) − f √ ζ/ζ min (4)where I o is constant.To estimate the width of lines, we need to compare the average loss of collapsing starmass in late times with a black body. The average of time elapsing between two decays is¯ t = R dt tP t (1) = 2 τ and its uncertainty is (∆ t ) = R dt ( t − ¯ t ) P t (1) = 3 τ . The averagefrequency emitted from a black hole can be shown to be ¯ ω = ω o γχ , [14] Moreover, themean value of the number of jumps in ∆ t is ¯ j = P j jP ∆ t ( j ), which becomes ∆ tτ . As aconsequence, a black hole losses the ratio of mass ∆ ¯ M ∆ t = − ~ ω o γχc τ on average. On the otherhand, the nature of a black hole radiation is the same as a black body where the loss ofmean energy is described by Stephan-Boltzman law, ∆ ¯ M ∆ t = − ~ c πG M . Comparing thesetwo, one finds τ = πγχω o . According to the uncertainty principle ∆ E ∆ t ∼ ~ , the frequencyuncertainty becomes of the order of a thousandth of the frequency scale ω o . This shows thatthe spectrum lines are indeed very narrow and the various black hole lines of one generationare unlikely to overlap. 10 IG. 1: The intensity envelope of some generations.
The intensity envelope of the first three generations is plotted in Fig. (1), where theenvelope (a), (b) and (c) belongs to the intensity of harmonics in the first, second and thirdgenerations, respectively. It becomes clear that in a generation with the least gap betweenlevels, the strongest harmonic modes are amplified. The brightest lines belong to a few ofthe first harmonics of the generation ζ min . Other than these lines, the intensity of the restof harmonics in other generations are suppressed exponentially. We expect in a low energyspectroscopy a clear observation of only a few narrow and unblended lines highly on top ofother harmonics. Also we expect these brightest lines appear in the wavelength not largerthan the size of black hole M in Planck units.In summary: we showed the quantum of area are substantially degenerate. The completespectrum is possible to reformulate into a universal form with two parameters and moreimportantly it is the union of exactly equidistant subsets. The spectrum of radiation due tothese new properties reveals a clear discretization on a few brightest lines which cannot blendinto one another. The most notable point is that loop quantum gravity as one fundamentaltheory of quantum gravity is substantially testable with an observational justification ifprimordial black holes are ever found.This research was supported by Perimeter Institute for Theoretical Physics. Research atPerimeter Institute is supported by the Government of Canada through Industry Canada11nd by the Province of Ontario through the Ministry of Research and Innovation. [1] M. Bojowald, Phys. Rev. Lett. , 5227 (2001) [arXiv:gr-qc/0102069].[2] A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett. , 904 (1998)[arXiv:gr-qc/9710007].[3] A. Ashtekar and J. Lewandowski, Class. Quant. Grav. , A55 (1997). [arXiv:gr-qc/9602046];S. Frittelli, L. Lehner and C. Rovelli, Class. Quant. Grav. , 2921 (1996)[arXiv:gr-qc/9608043]; C. Rovelli and L. Smolin, Nucl. Phys. B , 593 (1995) [Erratum-ibid.B , 753 (1995)]. [arXiv:gr-qc/9411005].[4] M. H. Ansari, Nucl. Phys. B , 179 (2007) [arXiv:hep-th/0607081].[5] C. Rovelli, Phys. Rev. Lett. , 3288 (1996) [arXiv:gr-qc/9603063].[6] M. H. Ansari, arXiv:gr-qc/0603121.[7] D. Beckman, D. Gottesman, M. A. Nielsen and J. Preskill, Phys. Rev. A , 052309(2001) [arXiv:quant-ph/0102043]; T. Eggeling, D. Schlingemann, and R. F. Werner,[arXiv.org:quant-ph/0104027]; B. Schumacher, M. D. Westmoreland, Quant. Info. Proc. 4,13 (2005) [arXiv.org:quant-ph/0406223].[8] R. D. Sorkin, Ten theses on black hole entropy, Stud.Hist. Philos. Mod. Phys. , 291 (2005)[9] Bekenstein et. al. Phys. Lett. B , 7 (1995) [arXiv:gr-qc/9505012].[10] S. Hawking, Nature , 30 (1974), S. W. Hawking, Commun. Math. Phys. , 199 (1975).[11] U. Gerlach, preceding paper, Phys. Rev. D , 1479 (1976).[12] There are also examples in quantum field theory on a curved background for how energyconditions can be violated locally.[13] From normalization C = P ζ ρ ( ζ ) / | − g ( ζ min ) − √ ζ/ζ min | .[14] By definition ¯ ω = P ζ P f ω f P ∆ t ( { ω f | } ). After using P n nx n = x/ (1 − x ) for x < ζ , the integral gives thesame result when the sum in the definition of C is approximated sum by integral.is approximated sum by integral.