- Email: [email protected]
Quantum bubble dynamics in the presence of gravity
Volume 262, number 2,3
PHYSICS LETTERS B
20 June 1991
Quantum bubble dynamics in the presence of gravity Antonio Aurilia
'
Department of Physics, California State Polytechnic University, Pomona, CA 91768, USA
Roberto Balbinot Istituto di Fisica, Universit~di Palermo, Palermo, Italy, and INFN, Sezione di Bologna, 1-40126 Bologna, Italy
and Euro Spallucci 2 Dipartimento di Fisica Teorica, Universitd di Trieste, 1-34014 Trieste, Italy and INFN, Sezione di Trieste, 1-34100 Trieste, Italy Received 7 December 1990
The dynamics of sphericalquantum bubbles in 3 + 1 dimensions is governedby a Klein-Gordon-type equation which simulates the quantum mechanical motion of a relativistic point particle in 1+ 1 dimensions. This dimensional reduction is especiallyclear in the minisuperspace formulation first used in quantum cosmologyand adapted here to quantum bubble dynamics. The payoff of this formulation is the discovery of the gravitational analogue of the Klein effect, namely the crossingof positive and negative energylevelsof the particle spectrum induced by an external gravitational field. This phenomenon givesrise to a finite probability that a vacuum bubble might tunnel from an initial bounded classical trajectory to a final unbounded classical trajectory. This mechanism is believed to be responsiblefor the birth and evolution of a new universe.
1. I n t r o d u c t i o n U n d e r the assumption of spherical symmetry and in the thin wall approximation, the field equations that govern the evolution of a relativistic bubble in a v a c u u m are integrable [ 1 ]. The first integral of motion represents the total mass energy of the bubble, E = c R 3 +a~R 2 X / 1 - ~A_R2 + k z ,
(1.1)
where A C~ ~
-A+ -- 87~2/92GN = g4 7~(E_ - - ~ + ) - - 8 / t 2 p 2 G s , 6GN
a=___l
/~=4rtp, '
R=
--.dR dT
(1.2)
Bitnet address: [email protected]. 2 Bitnet address: [email protected]. 222
In the above expressions p represents the surface tension of the bubble of radius R, GN is Newton's constant of gravitation and z is the proper time, i.e., the time coordinate associated with the intrinsic metric defined on the bubble. Moreover, the interior region of the bubble represents a v a c u u m phase described by the de Sitter geometry with cosmological constant A_ (or v a c u u m energy density E_ ) whereas the exterior region represents a different v a c u u m phase described by the de Sitter-Schwarzschild geometry with cosmological constant A+ (or v a c u u m energy density ~+ ) [ 1 ]. Eq. ( 1.1 ) implicitly defines the classical equation of motion of the bubble radius and in ref. [ 2 ] we have proposed an algorithm to classify the possible trajectories of the bubble in spacetime. In ref. [ 3 ] Berezin, Kozimirov, K u z m i n and Tkachev have suggested that the expression ( 1.1 ) for the mass-energy be inter-
0370-2693/91/$ 03.50 © 1991 - Elsevier SciencePublishers B.V. ( North-Holland )
Volume 262, number 2,3
PHYSICS LETTERSB
preted as the canonical hamiltonian of the bubble at the quantum level. While we have no objection to this proposal for the specific case we have in mind, their suggestion may be questionable, in principle, whenever the spacetime geometry is not asymptotically fiat. We offer an alternative interpretation of eq. (1.1) based, as in classical bubble dynamics [ 1 ], on the analogy of the bubble motion with that of a particle in a potential [ 1,2 ]. The payoff of this new interpretation is the gravitational analogue of the Klein effect familiar from QED; a consequence of the gravitational Klein effect is that there exists a finite probability that a bubble might tunnel through a potential barrier between two disconnected but classically allowed trajectories. Thus the initial classical trajectory may be a bounded one, even a re-collapsing configuration, whereas the final trajectory, after quantum tunneling, may correspond to a classical configuration of arbitrarily large radius. It has been suggested that a mechanism of this type may be responsible for the birth and evolution of a new universe [ 4 ].
2. Bubble interpretation versus particle interpretation In order to place the particle interpretation of eq. ( 1.1 ) into perspective, and partly to motivate it, it helps to recall first the bubble interpretation of eq. (1.1). The square root term, or surface term, in eq. ( 1.1 ) represents the kinetic term of the bubble obtained by matching the interior metric with the intrinsic metric on the surface of the bubble. The volume term in eq. ( 1.1 ), at least in our theory [ 1 ], originates from two possible sources: (i) it may arise, even in fiat spacetime, from the gauge invariant (minimal) coupling g SINT = ~.. f d3t X~"PA~p
(2.1)
20 June 1991
piing strength in the interaction term (2.1), then one finds A_ =4rtGNg 2 , ~_ = ½g2 ,
(2.2)
so that in the absence of gravity, when G N u 0 but g ~ 0 , one has A _ = 0 but ~ _ ¢ 0 . This limiting case was investigated in ref. [ 5 ], hereafter referred to as I. Presently, even though our formulation applies to the general case, we are primarily interested in the limit g ~ 0 , GN#0, A+ =e+ =0, in which case the volume term in eq. ( 1.1 ) is purely gravitational and our specific objective is to explore the consequences of that term, at least in the semiclassical approximation. Against this background, we wish to show that in both limiting cases considered above, the volume term in eq. ( 1.1 ) acts as a potential barrier for the motion of a relativistic point particle in one spatial dimension. To see this, define R2 (~=_l-~-~c,
( 3 ~ 1/2 Rc=-\~-_j .
(2.3)
Then we can use the matching relation between the interior metric and the intrinsic metric [1 ] to express the mass-energy equation in terms of the interior coordinate time t rather than the proper time z:
E = c R 3 +t~ffR 2
0 x/O_~)-l(dR/dt)2 "
(2.4)
The purpose of switching time is three-fold: (i) First, the expression (2.4) can be interpreted as the hamiltonian derived from the following lagrangian:
L(R,dR/dT) = -cR 3 _ a/~R2 [ ~ _ ~-1 (dR/dt)2] 1/2
(2.5)
Indeed, if one defines the canonical momentum
OL jffR2( d R / dt )O -1 PR- O(dR/dt)=ex/O_(~_l(dR/dt) 2 ,
(2.6)
,rg
between the 3-vector X Izvp tangent to the bubble world-history ~ and a third-rank antisymmetric tensor potential A,,p(X); or (ii) it may arise from the gravitational coupling through the energy-momentum tensor of the bubble. In particular, we recall that i f g represents the cou-
then one verifies that E = P g ( d R / d t ) - L . (ii) Second, in terms of PR we write
H=-E=ex/f~2P 2 +¢(/7R2) 2 +cR 3 ,
(2.7)
which, for an isolated bubble, i.e., when GN = 0, g = 0, or c = 0, ~ = 1 and a = 1, reduces to the free hamiltonian proposed by Collins and Tucker [ 6] in an at223
Volume 262, number 2,3
PHYSICSLETTERSB
tempt to generalize the properties of dual strings to objects of higher dimensionality. To our knowledge, none of the existing formulations of bubble dynamics in the presence of gravity addresses the question of reproducing the results of the quantum theory of membranes in flat spacetime. (iii) Third, the lagrangian (2.5) can be written in a ( 1 + 1 )-dimensional superspace (minisuperspace) [7] with metric GA~-R 4 diag (0, - 0 - l ) and local coordinates yA_ (t, R). Indeed, if s stands for the proper time along the world line in minisuperspace, then the lagrangian (2.5) takes the form
dyA drs~ 1/2 d r S A s L= - ~Otr GAS ds ds ] ------~- '
(2.8)
w h e r e a s - (cR 3, 0). Thus, our lagrangian, and therefore the hamiltonian (2.7), simulates the motion of a point particle in a ( 1 + 1 )-dimensional superspace under the influence of the external potential As. At this point one can derive the Klein-Gordon condition, either from eq. (2.8), as we did in I in the absence of gravity, or directly from the expression (2.7) of the hamiltonian. Both procedures are equivalent. Presently, we shall follow the second route. Eq. (2.7) gives
( E__cR 3) 2=02p2R + O(fiR 2) 2
(2.9)
which we interpret as a constraint equation analogous to the constraint equation for a relativistic particle in an external electrostatic field: (E+eAo)2= 17l+ m 2. In our case, the potential cR 3 originates from the gravitational and gauge couplings and the "mass of the particle" is #R 2, i.e., the proper mass of the bubble. At the quantum level, following the analogy with quantum cosmology [ 7 ], the constraint (2.9) becomes the Klein-Gordon equation in the presence of an external field, or, in natural units
(~2 ~d2 ) 5 + ( E _ c R 3) 2_q~(fiR2 ) z/~u(R)(2.10) ~---0. (2.10) As promised this is the general equation that governs the properties of the wave function of a quantum bubble in the minisuperspace approximation.
20 June 1991
3. Klein effect and quantum tunnelling We have checked the validity ofeq. (2.10) in various limiting cases. The first case of interest for our subsequent discussion is that of an isolated bubble [5]: in this case, either all parameters in eq. (1.2) are individually set equal to zero with the exception of the surface tension p, or those parameters are so finely tuned that c=0, in which case the effects due to the potential Au~p neutralize the effects due to gravity. In any case, under the sole influence of the surface tension, a classicalbubble always collapses to the central singularity; at the quantum level the collapse is prevented by the uncertainty principle which forbids the exact localization of an extended object at a single point in space and the energy spectrum of the bubble turns out to be discrete [6,5 ]. This interesting effect and the detailed form of the energy spectrum were discussed in I in terms of the WKB solution of the free bubble equation:
5~--~2+EE-~2R 4 ~ ( R ) = 0 dR
which we now regard as a limiting case of eq. (2.10 ) with c = 0 and 0 = 1. For our present purpose it is enough to recall that the asymptotic behavior of the WKB solution is ~(R)--, exp( - ~/~R 3) and one may expect that the quantum production rate of large bubbles with radius R much larger than the "Bohr radius" ao - ( ~ P) -1/3 is essentially vanishing. This overall picture is changed dramatically whenever the interaction, either the gravitational interaction or the gauge interaction (2.1), is switched on. To understand this, note that in the minisuperspace formulation, eq. (3.1) is interpreted as a onedimensional wave equation for a relativistic particle moving along the semi-axis R > 0. In this particle interpretation the relativistic nature of bubble dynamics is reflected by the appearance of positive and negative energy states, as indicated in fig. 1. Compare this case with the situation in which gravity alone is present: in this limiting case c= - c o - -8~2~2GN, ~ = 1 and one readily verifies that eq. (2.10) can be brought to the form
(~-~21+ ( E - V + ) ( E - V - ) ) ~ , ( R )
224
(3.1)
'
=0,
(3.2)
Volume 262, number 2,3
PHYSICS LETTERS B
3.00
20 June 1991
3.00 E>O
states
1.50
1.50
E
E
Idden
0.00
-1.50
...... "'.
E,o s t . t * s
-3.00 0.00
,.~ 0.00
region
. . 0.20
',,,,,
.
. 0.40
-1.50
0.60
0.80
-3.00 0.00
0.50
P o t e n t i a l vs R Fig. 1. Free, spherical bubble: graphs of the function y = + 4/~x 2, where y = V(R )/p 1/3 and x=p ~/3R. Classically, only positive energy states are physically meaningful; the spectrum of positive energy states is quantized; unlike a classical bubble, a q u a n t u m bubble never collapses to a point singularity.
where V +- - ~ - - 8 7 t 2 p 2 G N R 3 F f f R 2 .
1.00
1.50
2.00
P o t e n t i a l vs R
(3.3)
The effect of the volume term, or gravitational term, can be discussed analytically but it is most simply understood in terms of the graphs in fig. 2 which should be compared with the graph in fig. 1: as R-,0, V + --.fir 2and the behavior of the potential is the same with or without gravity. However, at large distances the negative volume term dominates over the fiR 2 term; the positive energy states are pushed "downward" until they eventually overlap with the negative energy states. This phenomenon can be described as the gravitational version of the Klein effect in QED where an external electric field, if strong enough, can shift the Dirac sea of negative energy states "upward" until they overlap with positive energy states. In that case, two degenerate energy states are separated by a potential barrier of width 2m which is the mass gap between a particle-antiparticle pair; similarly, we note that the difference V +- V-=2fir 2 is twice the proper mass of the bubble and represents the "mass gap" in the particle scenario of our minisuperspace formulation. Alternatively, in the bubble scenario the physical consequence of the "level crossing" induced by the
Fig. 2. Spherical bubble coupled to gravity: graphs of the functions y = -xx3+_2nx :, where y = V(R)/p '/3, x=pl/3R and x = Co/p 4/3. At the classical level, depending on the relative weight of the physical parameters, the coupling to gravity may generate a negative pressure between the exterior and interior vacua forcing the bubble to expand indefinitely. At the q u a n t u m level, because of the gravitational coupling, the distinction between positive and negative energy levels of a free bubble becomes ambiguous: there is a "crossing" or degeneracy between positive and negative energy states which are solutions of the one-dimensional KleinGordon equation coupled to an external gravitational field. This is the gravitational analogue of the Klein effect. A consequence of this effect is that, whenever 0 < E < V~+,x,there is a finite probability that a bubble might tunnel from an initial bounded classical trajectory to a final unbounded classical trajectory.
gravitational potential can be understood as follows: ifRmax =2fi/3Co is the value of R for which V ÷ (R) attains its maximum value Vmax + =4p/27Co, 2 then for a given positive energy E < Vma x+ one has a spectrum of quasi-stationary states (classicallybounded trajectories) which can decay through quantum tunneling to states representing classical trajectories of bubbles whose radius can grow without bound. The decay rate is given by the transmission coefficient T=exp(-B) R2
=exp(-2~x/l(E+coR3)2-(fiR2)21),
(3.4)
which we have calculated analytically only in the special case E = 0. In this case, if R, = 0 and RE = fi/Co = 1/ 225
Volume 262, number 2,3
PHYSICS LETTERS B
2npGN are the turning point roots of the equation V ÷ = 0, then we find 1 M6j B = 16np2G------~N--16ztp2.
(3.5)
This result tells us that the production rate of quantum gravitational bubbles is appreciable only when the value of the surface tension is comparable with M ~ . Thus if this effect has any relevance at all, it must be associated with the birth of the universe itself. Finally, we shall briefly mention the last limiting case ofeq. (2.10): in this case GN=0 but g # 0 and the vacuum energy density inside the bubble originates from the minimal coupling (2.1). This is the gauge theory discussed in I with A_ =A+ = E+ = 0 and E_ = ½ g2. The quantum effects of the potential A~vp are essentially polarization effects: Au,,p can create (or destroy) bubbles just as the electromagnetic field can materialize particle-antiparticle pairs under the time constraint of the uncertainty principle; furthermore, switching on the interaction (2.1) has once again the consequence of shifting or "crossing" bubble energy levels [ 5 ] and the analogy with the Klein effect in QED is even more stringent than in the gravitational case since the effective potential in the minisuperspace formulation is attractive rather than repulsive. The slightly more general case in which there is no gravity but both e_ and ~+ are different from zero,
226
20 June 1991
follows from eq. (2.10) with GN = 0, ~ = 1 and c = 4n(~_ - E+ ). The bubble nucleation radius in this case is found to be Rr~ = 3l)/I e_ - E+ I and the rate of bubble production by quantum tunneling is given by T = e x p ( - B ) , B= ½n2pR~ in agreement with the resuits of Coleman and De Luccia [ 8 ], and of Teitelboim [9 ].
References [ 1 ] A. Aurilia, G. Denardo, F. Legovini and E. Spallucci, Phys. Lett. B 147 (1984) 303; Nucl. Phys. B 252 (1984) 523; A. Aurilia, R. Kissack, R. Mann and E. Spallucci, Phys. Rev. D 3 5 (1987) 2961. [2] S. Blau, G. Guendelmann and A. Guth, Phys. Rev. D 35 (1987) 1474; A. Aurilia, M. Palmer and E. Spallucci, Phys. Rev. D 40 (1989) 2511. [3]V.A. Berezin, N.G. Kozimirov, V.A. Kuzmin and I.I. Tkachev, Phys. Lett. B 212 (1988) 415. [ 4 ] A. Vilenkin, Phys. Rev. D 27 ( 1983 ) 2848; E. Farhi and A.H. Guth, Phys. Lett. B 183 (1987) 149; E. Farhi, A.H. Guth and J. Guven, Nucl. Phys. B 339 (1990) 417. [ 5 ] A. Aurilia and E. Spallucci, Phys. Lett. B 251 ( 1991 ) 39. [6] P.A. Collins and R.W. Tucker, Nucl. Phys. B 112 ( 1976 ) 150. [ 7 ] B. DeWitt, Phys. Rev. 160 ( 1977 ) 1113. [ 8 ] S. Coleman and F. De Luccia, Phys. Rev. D 21 (1980) 3305. [9] C. Teitelboim, Phys. Lett. B 167 (1986) 63.
PHYSICS LETTERS B
20 June 1991
Quantum bubble dynamics in the presence of gravity Antonio Aurilia
'
Department of Physics, California State Polytechnic University, Pomona, CA 91768, USA
Roberto Balbinot Istituto di Fisica, Universit~di Palermo, Palermo, Italy, and INFN, Sezione di Bologna, 1-40126 Bologna, Italy
and Euro Spallucci 2 Dipartimento di Fisica Teorica, Universitd di Trieste, 1-34014 Trieste, Italy and INFN, Sezione di Trieste, 1-34100 Trieste, Italy Received 7 December 1990
The dynamics of sphericalquantum bubbles in 3 + 1 dimensions is governedby a Klein-Gordon-type equation which simulates the quantum mechanical motion of a relativistic point particle in 1+ 1 dimensions. This dimensional reduction is especiallyclear in the minisuperspace formulation first used in quantum cosmologyand adapted here to quantum bubble dynamics. The payoff of this formulation is the discovery of the gravitational analogue of the Klein effect, namely the crossingof positive and negative energylevelsof the particle spectrum induced by an external gravitational field. This phenomenon givesrise to a finite probability that a vacuum bubble might tunnel from an initial bounded classical trajectory to a final unbounded classical trajectory. This mechanism is believed to be responsiblefor the birth and evolution of a new universe.
1. I n t r o d u c t i o n U n d e r the assumption of spherical symmetry and in the thin wall approximation, the field equations that govern the evolution of a relativistic bubble in a v a c u u m are integrable [ 1 ]. The first integral of motion represents the total mass energy of the bubble, E = c R 3 +a~R 2 X / 1 - ~A_R2 + k z ,
(1.1)
where A C~ ~
-A+ -- 87~2/92GN = g4 7~(E_ - - ~ + ) - - 8 / t 2 p 2 G s , 6GN
a=___l
/~=4rtp, '
R=
--.dR dT
(1.2)
Bitnet address: [email protected]. 2 Bitnet address: [email protected]. 222
In the above expressions p represents the surface tension of the bubble of radius R, GN is Newton's constant of gravitation and z is the proper time, i.e., the time coordinate associated with the intrinsic metric defined on the bubble. Moreover, the interior region of the bubble represents a v a c u u m phase described by the de Sitter geometry with cosmological constant A_ (or v a c u u m energy density E_ ) whereas the exterior region represents a different v a c u u m phase described by the de Sitter-Schwarzschild geometry with cosmological constant A+ (or v a c u u m energy density ~+ ) [ 1 ]. Eq. ( 1.1 ) implicitly defines the classical equation of motion of the bubble radius and in ref. [ 2 ] we have proposed an algorithm to classify the possible trajectories of the bubble in spacetime. In ref. [ 3 ] Berezin, Kozimirov, K u z m i n and Tkachev have suggested that the expression ( 1.1 ) for the mass-energy be inter-
0370-2693/91/$ 03.50 © 1991 - Elsevier SciencePublishers B.V. ( North-Holland )
Volume 262, number 2,3
PHYSICS LETTERSB
preted as the canonical hamiltonian of the bubble at the quantum level. While we have no objection to this proposal for the specific case we have in mind, their suggestion may be questionable, in principle, whenever the spacetime geometry is not asymptotically fiat. We offer an alternative interpretation of eq. (1.1) based, as in classical bubble dynamics [ 1 ], on the analogy of the bubble motion with that of a particle in a potential [ 1,2 ]. The payoff of this new interpretation is the gravitational analogue of the Klein effect familiar from QED; a consequence of the gravitational Klein effect is that there exists a finite probability that a bubble might tunnel through a potential barrier between two disconnected but classically allowed trajectories. Thus the initial classical trajectory may be a bounded one, even a re-collapsing configuration, whereas the final trajectory, after quantum tunneling, may correspond to a classical configuration of arbitrarily large radius. It has been suggested that a mechanism of this type may be responsible for the birth and evolution of a new universe [ 4 ].
2. Bubble interpretation versus particle interpretation In order to place the particle interpretation of eq. ( 1.1 ) into perspective, and partly to motivate it, it helps to recall first the bubble interpretation of eq. (1.1). The square root term, or surface term, in eq. ( 1.1 ) represents the kinetic term of the bubble obtained by matching the interior metric with the intrinsic metric on the surface of the bubble. The volume term in eq. ( 1.1 ), at least in our theory [ 1 ], originates from two possible sources: (i) it may arise, even in fiat spacetime, from the gauge invariant (minimal) coupling g SINT = ~.. f d3t X~"PA~p
(2.1)
20 June 1991
piing strength in the interaction term (2.1), then one finds A_ =4rtGNg 2 , ~_ = ½g2 ,
(2.2)
so that in the absence of gravity, when G N u 0 but g ~ 0 , one has A _ = 0 but ~ _ ¢ 0 . This limiting case was investigated in ref. [ 5 ], hereafter referred to as I. Presently, even though our formulation applies to the general case, we are primarily interested in the limit g ~ 0 , GN#0, A+ =e+ =0, in which case the volume term in eq. ( 1.1 ) is purely gravitational and our specific objective is to explore the consequences of that term, at least in the semiclassical approximation. Against this background, we wish to show that in both limiting cases considered above, the volume term in eq. ( 1.1 ) acts as a potential barrier for the motion of a relativistic point particle in one spatial dimension. To see this, define R2 (~=_l-~-~c,
( 3 ~ 1/2 Rc=-\~-_j .
(2.3)
Then we can use the matching relation between the interior metric and the intrinsic metric [1 ] to express the mass-energy equation in terms of the interior coordinate time t rather than the proper time z:
E = c R 3 +t~ffR 2
0 x/O_~)-l(dR/dt)2 "
(2.4)
The purpose of switching time is three-fold: (i) First, the expression (2.4) can be interpreted as the hamiltonian derived from the following lagrangian:
L(R,dR/dT) = -cR 3 _ a/~R2 [ ~ _ ~-1 (dR/dt)2] 1/2
(2.5)
Indeed, if one defines the canonical momentum
OL jffR2( d R / dt )O -1 PR- O(dR/dt)=ex/O_(~_l(dR/dt) 2 ,
(2.6)
,rg
between the 3-vector X Izvp tangent to the bubble world-history ~ and a third-rank antisymmetric tensor potential A,,p(X); or (ii) it may arise from the gravitational coupling through the energy-momentum tensor of the bubble. In particular, we recall that i f g represents the cou-
then one verifies that E = P g ( d R / d t ) - L . (ii) Second, in terms of PR we write
H=-E=ex/f~2P 2 +¢(/7R2) 2 +cR 3 ,
(2.7)
which, for an isolated bubble, i.e., when GN = 0, g = 0, or c = 0, ~ = 1 and a = 1, reduces to the free hamiltonian proposed by Collins and Tucker [ 6] in an at223
Volume 262, number 2,3
PHYSICSLETTERSB
tempt to generalize the properties of dual strings to objects of higher dimensionality. To our knowledge, none of the existing formulations of bubble dynamics in the presence of gravity addresses the question of reproducing the results of the quantum theory of membranes in flat spacetime. (iii) Third, the lagrangian (2.5) can be written in a ( 1 + 1 )-dimensional superspace (minisuperspace) [7] with metric GA~-R 4 diag (0, - 0 - l ) and local coordinates yA_ (t, R). Indeed, if s stands for the proper time along the world line in minisuperspace, then the lagrangian (2.5) takes the form
dyA drs~ 1/2 d r S A s L= - ~Otr GAS ds ds ] ------~- '
(2.8)
w h e r e a s - (cR 3, 0). Thus, our lagrangian, and therefore the hamiltonian (2.7), simulates the motion of a point particle in a ( 1 + 1 )-dimensional superspace under the influence of the external potential As. At this point one can derive the Klein-Gordon condition, either from eq. (2.8), as we did in I in the absence of gravity, or directly from the expression (2.7) of the hamiltonian. Both procedures are equivalent. Presently, we shall follow the second route. Eq. (2.7) gives
( E__cR 3) 2=02p2R + O(fiR 2) 2
(2.9)
which we interpret as a constraint equation analogous to the constraint equation for a relativistic particle in an external electrostatic field: (E+eAo)2= 17l+ m 2. In our case, the potential cR 3 originates from the gravitational and gauge couplings and the "mass of the particle" is #R 2, i.e., the proper mass of the bubble. At the quantum level, following the analogy with quantum cosmology [ 7 ], the constraint (2.9) becomes the Klein-Gordon equation in the presence of an external field, or, in natural units
(~2 ~d2 ) 5 + ( E _ c R 3) 2_q~(fiR2 ) z/~u(R)(2.10) ~---0. (2.10) As promised this is the general equation that governs the properties of the wave function of a quantum bubble in the minisuperspace approximation.
20 June 1991
3. Klein effect and quantum tunnelling We have checked the validity ofeq. (2.10) in various limiting cases. The first case of interest for our subsequent discussion is that of an isolated bubble [5]: in this case, either all parameters in eq. (1.2) are individually set equal to zero with the exception of the surface tension p, or those parameters are so finely tuned that c=0, in which case the effects due to the potential Au~p neutralize the effects due to gravity. In any case, under the sole influence of the surface tension, a classicalbubble always collapses to the central singularity; at the quantum level the collapse is prevented by the uncertainty principle which forbids the exact localization of an extended object at a single point in space and the energy spectrum of the bubble turns out to be discrete [6,5 ]. This interesting effect and the detailed form of the energy spectrum were discussed in I in terms of the WKB solution of the free bubble equation:
5~--~2+EE-~2R 4 ~ ( R ) = 0 dR
which we now regard as a limiting case of eq. (2.10 ) with c = 0 and 0 = 1. For our present purpose it is enough to recall that the asymptotic behavior of the WKB solution is ~(R)--, exp( - ~/~R 3) and one may expect that the quantum production rate of large bubbles with radius R much larger than the "Bohr radius" ao - ( ~ P) -1/3 is essentially vanishing. This overall picture is changed dramatically whenever the interaction, either the gravitational interaction or the gauge interaction (2.1), is switched on. To understand this, note that in the minisuperspace formulation, eq. (3.1) is interpreted as a onedimensional wave equation for a relativistic particle moving along the semi-axis R > 0. In this particle interpretation the relativistic nature of bubble dynamics is reflected by the appearance of positive and negative energy states, as indicated in fig. 1. Compare this case with the situation in which gravity alone is present: in this limiting case c= - c o - -8~2~2GN, ~ = 1 and one readily verifies that eq. (2.10) can be brought to the form
(~-~21+ ( E - V + ) ( E - V - ) ) ~ , ( R )
224
(3.1)
'
=0,
(3.2)
Volume 262, number 2,3
PHYSICS LETTERS B
3.00
20 June 1991
3.00 E>O
states
1.50
1.50
E
E
Idden
0.00
-1.50
...... "'.
E,o s t . t * s
-3.00 0.00
,.~ 0.00
region
. . 0.20
',,,,,
.
. 0.40
-1.50
0.60
0.80
-3.00 0.00
0.50
P o t e n t i a l vs R Fig. 1. Free, spherical bubble: graphs of the function y = + 4/~x 2, where y = V(R )/p 1/3 and x=p ~/3R. Classically, only positive energy states are physically meaningful; the spectrum of positive energy states is quantized; unlike a classical bubble, a q u a n t u m bubble never collapses to a point singularity.
where V +- - ~ - - 8 7 t 2 p 2 G N R 3 F f f R 2 .
1.00
1.50
2.00
P o t e n t i a l vs R
(3.3)
The effect of the volume term, or gravitational term, can be discussed analytically but it is most simply understood in terms of the graphs in fig. 2 which should be compared with the graph in fig. 1: as R-,0, V + --.fir 2and the behavior of the potential is the same with or without gravity. However, at large distances the negative volume term dominates over the fiR 2 term; the positive energy states are pushed "downward" until they eventually overlap with the negative energy states. This phenomenon can be described as the gravitational version of the Klein effect in QED where an external electric field, if strong enough, can shift the Dirac sea of negative energy states "upward" until they overlap with positive energy states. In that case, two degenerate energy states are separated by a potential barrier of width 2m which is the mass gap between a particle-antiparticle pair; similarly, we note that the difference V +- V-=2fir 2 is twice the proper mass of the bubble and represents the "mass gap" in the particle scenario of our minisuperspace formulation. Alternatively, in the bubble scenario the physical consequence of the "level crossing" induced by the
Fig. 2. Spherical bubble coupled to gravity: graphs of the functions y = -xx3+_2nx :, where y = V(R)/p '/3, x=pl/3R and x = Co/p 4/3. At the classical level, depending on the relative weight of the physical parameters, the coupling to gravity may generate a negative pressure between the exterior and interior vacua forcing the bubble to expand indefinitely. At the q u a n t u m level, because of the gravitational coupling, the distinction between positive and negative energy levels of a free bubble becomes ambiguous: there is a "crossing" or degeneracy between positive and negative energy states which are solutions of the one-dimensional KleinGordon equation coupled to an external gravitational field. This is the gravitational analogue of the Klein effect. A consequence of this effect is that, whenever 0 < E < V~+,x,there is a finite probability that a bubble might tunnel from an initial bounded classical trajectory to a final unbounded classical trajectory.
gravitational potential can be understood as follows: ifRmax =2fi/3Co is the value of R for which V ÷ (R) attains its maximum value Vmax + =4p/27Co, 2 then for a given positive energy E < Vma x+ one has a spectrum of quasi-stationary states (classicallybounded trajectories) which can decay through quantum tunneling to states representing classical trajectories of bubbles whose radius can grow without bound. The decay rate is given by the transmission coefficient T=exp(-B) R2
=exp(-2~x/l(E+coR3)2-(fiR2)21),
(3.4)
which we have calculated analytically only in the special case E = 0. In this case, if R, = 0 and RE = fi/Co = 1/ 225
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PHYSICS LETTERS B
2npGN are the turning point roots of the equation V ÷ = 0, then we find 1 M6j B = 16np2G------~N--16ztp2.
(3.5)
This result tells us that the production rate of quantum gravitational bubbles is appreciable only when the value of the surface tension is comparable with M ~ . Thus if this effect has any relevance at all, it must be associated with the birth of the universe itself. Finally, we shall briefly mention the last limiting case ofeq. (2.10): in this case GN=0 but g # 0 and the vacuum energy density inside the bubble originates from the minimal coupling (2.1). This is the gauge theory discussed in I with A_ =A+ = E+ = 0 and E_ = ½ g2. The quantum effects of the potential A~vp are essentially polarization effects: Au,,p can create (or destroy) bubbles just as the electromagnetic field can materialize particle-antiparticle pairs under the time constraint of the uncertainty principle; furthermore, switching on the interaction (2.1) has once again the consequence of shifting or "crossing" bubble energy levels [ 5 ] and the analogy with the Klein effect in QED is even more stringent than in the gravitational case since the effective potential in the minisuperspace formulation is attractive rather than repulsive. The slightly more general case in which there is no gravity but both e_ and ~+ are different from zero,
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follows from eq. (2.10) with GN = 0, ~ = 1 and c = 4n(~_ - E+ ). The bubble nucleation radius in this case is found to be Rr~ = 3l)/I e_ - E+ I and the rate of bubble production by quantum tunneling is given by T = e x p ( - B ) , B= ½n2pR~ in agreement with the resuits of Coleman and De Luccia [ 8 ], and of Teitelboim [9 ].
References [ 1 ] A. Aurilia, G. Denardo, F. Legovini and E. Spallucci, Phys. Lett. B 147 (1984) 303; Nucl. Phys. B 252 (1984) 523; A. Aurilia, R. Kissack, R. Mann and E. Spallucci, Phys. Rev. D 3 5 (1987) 2961. [2] S. Blau, G. Guendelmann and A. Guth, Phys. Rev. D 35 (1987) 1474; A. Aurilia, M. Palmer and E. Spallucci, Phys. Rev. D 40 (1989) 2511. [3]V.A. Berezin, N.G. Kozimirov, V.A. Kuzmin and I.I. Tkachev, Phys. Lett. B 212 (1988) 415. [ 4 ] A. Vilenkin, Phys. Rev. D 27 ( 1983 ) 2848; E. Farhi and A.H. Guth, Phys. Lett. B 183 (1987) 149; E. Farhi, A.H. Guth and J. Guven, Nucl. Phys. B 339 (1990) 417. [ 5 ] A. Aurilia and E. Spallucci, Phys. Lett. B 251 ( 1991 ) 39. [6] P.A. Collins and R.W. Tucker, Nucl. Phys. B 112 ( 1976 ) 150. [ 7 ] B. DeWitt, Phys. Rev. 160 ( 1977 ) 1113. [ 8 ] S. Coleman and F. De Luccia, Phys. Rev. D 21 (1980) 3305. [9] C. Teitelboim, Phys. Lett. B 167 (1986) 63.