Volume 95B, number 2 PHYSICS LETTERS 22 September 1980

PARTICLE TRANSMUTATIONS IN QUANTUM GRAVITY

Don N. PAGE Department o f Phystcs, The Pennsylvania State University, Umversity Park, PA 16802, USA

Received 9 May 1980

Topologically nontrivial quantum gravitational field configurations may induce transitions between particles and anti- particles. This might explain the change in baryon and other quantum numbers that occur in black-hole formation and evaporation without requiring a loss of information.

Considerations of gravatational collapse have led to the conclusion that new kinds of particle transforma- tions may occur. After Anderson, Stoner, and Chandrasekhar (with greater accuracy) obtained a maximum mass for white dwarf stars [1 -3 ] , Landau showed that nothing in quantum theory would pre- vent a degenerate Fermi gas above the critical mass from collapsing to a point (though he argued quantum mechanics must certainly be violated since heavier stars "do not show any such ridiculous tendencies") [4]. Oppenheimer and Snyder explained how con- tinued gravitational contraction would indeed occur for cold spherical masses above the critical value [5]. Zel'dovich noted that even masses less than the critical value would be unstable to collapse, though with an energy barrier [6]. Wheeler argued that there must be quantum mechanical tunneling through the barrier, so that all matter should manifest a new form of radioac- tivity in which baryon number changes [7]. It was thought that the final state would be a black hole, but Hawking discovered that a hole emits thermal radia- tion and hence presumably evaporates away [8]. Thus gravitation induces transmutations between different species of particles.

However, gravitational collapse followed by black- hole emission has only been calculated in the semiclas- sical approximation. The microscopic details of particle transmutations in quantum gravity have not yet been made clear, though some preliminary estimates of the rates have been given [9,10].

In this paper I will describe how topologically non-

trivial quantum gravitational fields may induce transi- tions between particles and antiparticles. Basically, quantum fluctuations allow gravitational field configu- rations that in certain small regions do not have a well- defined time orientation. A particle traveling through such a region may get its direction of' time reversed, so that an external observer will perceive that it has turned into an antiparticle. The long-range electromag- netic field will prevent individual charged particles from turning into antiparticles, but a neutral particle can turn into an antiparticle or annihilate with another particle by this mechanism (at an extremely slow rate).

Simple examples of topologically nontrivial quan- tum gravitational fields ("gravitational bubbles" in the "spacetime foam" picture [11,12] ) were considered in ref. [10]. We found that they introduced extra sin- gularities in the Feynman propagators which were acausal in the sense that they propagated negative-fre- quency modes forward in time and positive-frequency modes backward in time. We interpreted these singular- ities as giving particle-antiparticle annihilations and creations but no particle-particle or antiparticle-anti- particle scatterings. Then the resulting S-matrix ele- ments violated causality by not having crossing sym- metry [10].

However, I shall now suggest that the "acausal sin- gularities" should not be interpreted as violating cross- ing symmetry but instead as inducing particle-particle or antiparticle-antiparticle annihilations or creations or particle-antiparticle transitions. The idea is that in passing through a gravitational bubble, the direction of

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time is reversed. Hence a particle that passes through the bubble will be traveling backward in time. But a particle traveling backward in time is an antiparticle traveling forward in time [13,14] and will be so seen by an observer that does not pass through the bubble.

Although both the CP 2 and S 2 X S 2 bubbles we considered had acausal singularities [10], the time in- version is most clearly seen for the S 2 × S 2 bubble. In our limiting case the metric was conformal to the flat metric

t ds '2 = dx'U dxu, (1)

with each point x'U identified with an image point

E'u = 2x+U _ x'U, (2)

where x+ u is the fixed point of the identification. For massless particles the asymptotic states are defined on null infinity 9 [15], which in the conformally related metric (1) is the null cone with vertex I at x'U = 0 and the image null cone with vertex '[ at x'U = 2x+ u. The amplitude to propagate from a state on past null infin- ity, 9 - , to a state on future null infinity, 9 +, has a direct piece given by the flat space Feynman propaga- tor with singularities on the null cone of I and an indi- rect piece given by the image propagator with singular- ities on the null cone of"[. The direct piece is the usual causal flat space amplitude, and the indirect piece is the "acausal" amplitude that takes negative frequen- cies from the past to the future.

However, the identification (2) means that the image light cone is inverted. Hence, the direction of time on 9 appears to become inverted when com- pared to a parallel-propagated frame carried from the light cone of I to the light cone of ~. In other words, a particle that passes through the bubble from I to ]~ sees the asymptotically defined direction of time re- versed. Alternatively, an asymptotic observer would see the particle as reyersing its direction of time. Thus the indirect propagation from 9 - to 9 + should be in- terpreted as the amplitude for an incoming particle to turn into an outgoing antiparticle (a particle with the time direction reversed), or for an antiparticle to turn into a particle.

Since the bubble metrics considered here have regu- lar infinity points [10], 9 - and 9 + are identical. This means that the propagation can also be taken from 9 - to 9 - or from 9 + to 9 +. The direct ampli- tude would correspond to particle-antiparticle annihi-

lation or creation. (However, this amplitude is zero for these special bubbles which have self-dual Weyl tensors.) The indirect amplitude corresponds to par- t icle-particle or anti 'particle-antiparticle annihilation or creation. When one integrates over bubble positions, four-momentum conservation requires an annihilation to be accompanied by a creation, but the total number of particles minus antiparticles need not be conserved.

This nonconservation of particle number is reminis- cent of the nonconservation of axial charge in Y a n g - Mills instantons [16,17]. In the present case the global U(1 ) symmetry

~b(X) ~ eia~b(x), ~(x) -~ e - iC~(x ) , (3)

for a constant a gives rise to a Noether current

ju = i('~ Vuc~ _ q~V" ~) , (4)

which is conserved according to the classical equations of motion:

Vuju =0, (5)

where the covariant derivative is taken in the bubble metric. Here there is not an anomaly that arises from zero modes as in the Yang-Mills case [16,17] but rather the nontrivial topology which prevents one from reducing the integral of (5) to a difference between in- coming and outgoing particle numbers. Of course, if the particles are coupled to a long-range gauge field through a local U(1) symmetry, the volume integral of the particle current can be reduced to an asymptot- ic surface integral of the gauge field, which does en- sure charge conservation.

The transmutation amplitudes described here cou- ple only an even number, of particles of each species. Hence they would not lead to the decay of a single baryon into nonbaryonic particles. However, as sug- gested in ref. [9], there could be oscillations between electrically neutral baryon and antibaryon states simi- lar to what the weak interactions induce in the K 0 sys- tem. Using the dimensional dependence of spin-1/2 amplitudes derived in ref. [10], the gravitational'transi- tion amplitude per Planck time between a hydrogen a tom and an antihydrogen atom would be of order (in Planck units with h = c = G = 1, using m~ -1 as the pro- ton size)

(HI O,be C ~e) (t~ 1 C~k 1 ) (~2 C~2)(t~ 3C~ 3 ) I I-l)

m3ot3m 6 ~ 10_193, (6)

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Volume 95B, number 2 PHYSICS LETTERS 22 September 1980

where ~e is the electron field and 41,2,3 the three colored quark fields. If the atom did not decay non- gravitationally, it would have energy eigenstates

IH1)= 2 -1 /2 ( IH)+ i l f i ) ) , [H2)= 2-1/2 (I H) - i I H)), (7)

with an energy difference given by the transition rate (6) or about 10 -165 eV (superfine structure o f hydro- gen). This gives an oscillation period between H and fi of about 10143 yr, which present experiments cannot detect.

Similarly, if nuclear matter were stable under non- gravitational interactions, the gravitational transmuta- tions of three quarks would give baryon-baryon anni- hilation cross sections of order (in Planck units)

o ~ m 8 ~ 10 -160 ~ 10 -94 cm 2. (8)

3 (e.g., deu- For nuclear matter densities of order n ~ m . terons or heavier nuclei), this gives an annihilation rate per nucleon of order

7 "-1 ~ noc ~ m 11 ~ 10 -220 ~ (10169 yr) -1 . (9) 7r

This rate is much lower than that estimated in refs. [9] and [10] because here only transmutations be- tween particles and antiparticles of the same species occur, so three quark transmutations are needed. This two-baryon annihilation rate is even lower than the hydrogen oscillation frequency because the large num- ber of possible final states do not interfere construc- tively.

The process of particle transmutations described here has the advantage of being described by an S ma- trix (obtained in principle by a path integral over all gravitational and matter field configurations of the appropriate boundary conditions). It does not require the loss of information or unpredictability that Hawking has argued accompanies gravitational collapse [18]. Zel'dovich has questioned the necessity of this indeterminacy [19], and indeed it would be inconsis- tent with CPT invarlance [20,21 ].

It would be o f interest to know whether there are other processes which would greatly dominate the ones considered here. For example, are there gravita-

tional field configurations that can give direct transmu- tations between two particles o f different species, as would be needed to give single baryon decay? Can all processes be described by an S matrix, or are there effects that violate CPT invariance and/or quantum cosmic censorship [20,21] ? Clearly we are only begin- ning to get a hint o f what can happen in quantum grav- ity.

Discussions with G.N. Fleming, E. Kazes, Y. Nambu, M.J. Perry and R.M. Wald have been helpful. This material is based upon work supported in part by the National Science Foundation under Grant No. PHY- 7918430.

Re fe rences

[1] W. Anderson, Z. Phys. 56 (1929) 851. [2] E.C. Stoner, Phil. Mag. 9 (7th Series) (1930) 944. [3] S. Chandrasekhar, Astrophys. J. 74 (1931) 81. [4] L. Landau, Phys. Z. Sowjetunion 1 (1932) 285. [5] J.R. Oppenheimer and H. Snyder, Phys. Rev. 56 (1939)

455. [6] Ya.B. Zel'dovieh, Zh. Eksp. Teor. Fiz. 42 (1962) 641

[Sov. Phys. JETP 15 (1962) 446]. , [7] B.K. Harrison, K.S. Thorne, M. Wakano and

J.A. Wheeler, Gravitation theory and gravitational collapse (Univ. of Chicago Press, 1965).

[8] S.W. Hawking, Nature 248 (1974) 30; Commun. Math. Phys. 43 (1975) 199.

[9] Ya.B. Zel'dovich, Phys. Lett. 59A (1976) 254; Zh. Eksp. Teor. Fiz. 72 (1977) 18 [Soy. Phys. JETP 45 (1977) 9].

[10] S.W. Hawking,D.N. Page and C.N. Pope, Phys. Lett. 86B (1979) 175; Nucl. Phys. B, to be published.

[11] J.A. Wheeler, in: Relativity, groups and topology, eds. C. DeWitt and B. DeWitt (Gordon and Breach, New York, 1964).

[12] S.W. Hawking, Nucl. Phys. B144 (1978) 349. [13] E C.C. Stueckelberg, Helv. Phys. Acta 15 (1942) 23. [14] R.P. Feynman, Phys. Rev. 74 (1948) 939. [15] R. Penrose, Proc. Roy. Soe. London A284 (1965) 159. [16] G. 't Hooft, Phys. Rev. Lett. 37 (1976) 8. [17] R. Jackiw and C. Rebbi, Phys. Rev. Lett. 37 (1976) 172. [18] S.W. Hawking, Phys. Rev. D14 (1976) 2460. [19] Ya.B. Zel'dovich, Usp. Fiz. Nauk 123 (1977) 487 [Soy.

Phys. Usp. 20 (1977) 945]. [20] D.N. Page, Phys. Rev. Lett. 44 (1980) 301. [21] R.M. Wald, Phys. Rev. D21 (1980) 2742.

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