6. Quantum Tunneling and Vacuum Decay

Let us consider, for example, the classical motion of a shell in the potential of Fig.2.

  

Figure: Graphical representation, in a generic case, of the effective potential described by equation (48). Typically, there are three distinct trajectories. Two of them are classically allowed: (A) bounded, and (C) unbounded. Trajectory (B) is classically forbidden and corresponds to quantum mechanical tunneling through the potential barrier. The exponential of the action evaluated along B gives the WKB approximation to the tunneling amplitude from A to C.

The classically allowed trajectories for the shell are: A (bounded), i.e., the shell starts at R=0, expands reaching a maximum radius at R₁ and then re-collapses to R=0, or C (unbounded), i.e., the shell starts with unbounded radius, contracts reaching a minimum radius at R₂ and then re-expands to infinity. The quantum mechanical amplitude for the shell to tunnel from one classically allowed trajectory (A) to the other (C), is proportional to the exponential of the integral of the Euclidean momentum P_E calculated along the Euclidean trajectory B which interpolates between the two classical ones, namely

$$P \sim e^{- \frac{B}{\hbar}} \quad ,$$ (66)

where

$$B = 2 \int _{R_1} ^{R_2} \left\vert P_E (R) \right\vert d R \quad .$$ (67)

The Euclidean momentum P_E is given by eq.(52). Along the Euclidean trajectory, using the equation of motion H=0, that equation can be rewritten as

$$P_E(R) = - \frac{R}{G_N} \arccos \left( \frac{A_{in} + A_{o... ... \sigma _{in} \sigma _{out} \sqrt{A_{in} A_{out}} } \right)$$ (68)

with

$$- \pi < \arccos < 0 \quad .$$ (69)

Inserting P_E(R) in eq.(67), integrating by parts, taking into account the vanishing of P_E at the turning points R₁ and R₂, one finally obtains

 

$$\frac{1}{G_N} \int dR R^2 \left [ \frac{ 4 \kappa ^2 R^2 A_{in} A... ...A_{in} A_{out} - \left( A_{in} + A_{out} - \kappa ^2 R^2 \right) ^2 } } \right.$$ B =

$$\qquad - \left. \frac{ A^2_{in} A'_{out} + A'_{in} A^2_{out} - \k... ..._{out} - \left( A_{in} + A_{out} - \kappa ^2 R^2 \right) ^2 } } \right] \quad .$$ (70)

where the symbol `` ' '' means differentiation with respect to R. In order to check the consistency of our formulation, in the remainder of this section we shall revisit the problem of vacuum decay and the influence of gravity on it. This subject was discussed in a seminal paper by Coleman and De Luccia [16], and also by Parke [17], using simple semi-classical and geometrical arguments. Our intent, then, is to show how their results follow from the dynamical framework discussed so far.

The system under consideration consists of two de Sitter domains joined along the spherical shell $\Sigma$, which is characterized by a constant surface tension $\rho$. In other words, we assign $\rho = p$ as equation of state; then, equation (17) implies $\rho = const.$, which is assumed to be positive. Therefore, the form of the metric in the interior and exterior regions is

$$ds_1^2 = - \left( 1 - \frac{\Lambda _{in}}{3} r^2 \right) dT^... ...ac{\Lambda _{in}}{3} r^2 \right) ^{-1} dr^2 + r^2 d \Omega ^2$$ (71)

and

$$ds_2^2 = - \left( 1 - \frac{\Lambda _{out}}{3} r^2 \right) dt... ...a _{out}}{3} r^2 \right) ^{-1} dr^2 + r^2 d \Omega ^2 \quad .$$ (72)

The Hamiltonian of the shell is obtained from eq.(37)

$$H = - \frac{R}{G_N} \left[ \sigma _{in} \sqrt{1 - \frac{\L... ... _{out}}{3} R^2 + \dot{R}^2 } - \kappa R \right] \quad .$$ (73)

In order to simplify the notation, let us introduce two parameters $\alpha$ and $\gamma$ such that

  

$$\kappa ^2 \alpha \frac{4}{3} \Lambda _{in}$$ = (74)

$$\kappa ^2 \gamma \frac{\Lambda _{out} - \Lambda _{in}}{3} + \kappa ^2$$ = (75)

and rescale the shell radius and proper time by a factor $\kappa$

$$\xi \kappa R$$ = (76)

$$\bar{\tau} \kappa \tau \quad .$$ = (77)

Then the coefficients of the metric become

$$1 - \frac{\alpha}{4} \xi ^2$$ A_in = (78)

$$1 - \left( \gamma - 1 + \frac{\alpha}{4}\right) \xi ^2$$ A _out = (79)

and the equation of motion, after dividing by R, can be rewritten as

$$\sigma _{in} \sqrt{1 - \frac{\alpha}{4} \xi ^2 + \left( {d\x... ... + \left( {d\xi\over d\bar\tau} \right) ^2 } = \xi \quad .$$ (80)

After squaring eq.(80) we arrive at

$$\left( {d\xi\over d\bar\tau} \right) ^2 + V \left( \xi \right) = 0 \quad ,$$ (81)

where

$$V \left( \xi \right) = 1 - \left( \frac{\xi}{\xi _N} \right) ^2$$ (82)

and

$$\xi _N = \frac{2}{\displaystyle \sqrt{\alpha + \gamma ^2}} \quad .$$ (83)

Equation(81) is integrated with the initial condition $\xi ( \bar{\tau} = 0 ) = \xi _N$, so that

$$\xi = \xi _N \cosh \left( \frac{\bar{\tau}}{\xi _N} \right) \quad .$$ (84)

The potential $V(\xi )$ is represented in fig.3.

  

Figure: Graphical representation of the effective potential $V(\xi )$ for the situation in which the ``in'' and ``out'' geometries are of de Sitter type. $\xi _N$ will be referred to as the ``nucleation radius''. The diagram displays only the unbounded trajectory, since the bounded one is degenerate ( $R \equiv 0$). Note that only for $\sigma _{in} = \sigma _{out}$ this is a classical solution of eq.(48).

The classical motion corresponds to a shell (separating the two de Sitter spaces) which starts at $\bar{\tau} = -\infty$ with an infinite radius, contracts to a minimum radius $\xi _N$ at $\bar{\tau} = 0$, and then expands again to infinity. From the equation of motion, taking into account eqs.(45-46), we determine the sign of the $\beta$'s as follows

  

$$\sigma _{in} {\rm Sgn} \left( \frac{\Lambda _{out} - \Lambda _{in}}{3} + \kappa ^2 \right)$$ = (85)

$$\sigma _{out} {\rm Sgn} \left( \frac{\Lambda _{out} - \Lambda _{in}}{3} - \kappa ^2 \right)\quad .$$ = (86)

For the purpose of illustrating our method, consider the case $\Lambda _{out} > \Lambda _{in}$, thus fixing $\sigma _{in} = +1$. Then, from eq.(86) and (75), we have

$$\sigma _{out} = {\rm Sgn} \left( \gamma -2 \right) \quad .$$ (87)

Thus, we have to consider two possibilities :

1.

$1 < \gamma < 2$, i.e. $( \sigma _{in}, \sigma _{out} ) = (+1, -1)$

2.

$\gamma > 2$, i.e. $( \sigma _{in}, \sigma _{out} ) = (+1, +1)$

which correspond to the spacetime conformal diagrams shown in fig.4 and fig.5, respectively.

  

Figure: Penrose diagrams corresponding to the ``in'' (a), and the ``out'' (b) domains in the case $1 < \gamma < 2$. The light regions correspond to the physical domains. The side of the diagram in which the trajectories are drawn is determined by the sign of $\sigma _{in}$ and $\sigma _{out}$. Their sign is also related to the direction of the normal to the shell orbit.

  

Figure: Penrose diagrams corresponding to the ``in'' (a), and the ``out'' (b) domains in the case $\gamma > 2$. Comparing with Fig.(4), note that the change of sign of $\sigma _{out}$ is reflected by the fact that the shell trajectory is now drawn in the opposite half of the conformal diagram.

The physical spacetime is obtained by  ``gluing"  along $\Sigma$ the un-dashed regions of each figure.

Let us consider first case # 2, $\gamma > 2$. From the Hamiltonian of eq.(73), we see that the configuration R=0 (i.e., no shell, the whole spacetime is de Sitter ``out'') satisfies the constraint H=0. On the other hand, our previous analysis of the classical motion revealed that classically allowed configurations for the shell exist only for shell radii $R \geq R_N \equiv \xi _N / \kappa$, where R_N represents the turning point.

Let us recall that false vacuum decay is a quantum process which proceeds via the nucleation of true vacuum bubbles of a given non-vanishing radius which then expand filling the surrounding false vacuum region. In our approach, this process (the formation of one bubble) is described by the quantum mechanical tunneling through the barrier represented in fig.3, from the R=0 shell configuration to one for which R=R_N. In terms of conformal diagrams, the process is visualized in fig.6.

  

Figure: Penrose diagram corresponding to tunneling along the classically forbidden trajectory of Fig.(3), in the case $\gamma > 2$. Tunneling occurs at $\bar{\tau} = 0$ and nucleates a true vacuum bubble in the false vacuum background.

The probability of this process is given, in the WKB approximation, and apart from an overall factor, by $\exp \left( - B / \hbar \right)$, where B is given by eq.(70). In order to calculate the nucleation coefficient B for the present case, it is useful to define

   

$$- \frac{2}{\alpha + \gamma ^2} \left( \gamma - \gamma ^2 - \frac{\alpha}{2} \right)$$ y₀ ^-1 = (88)

$$\frac{\alpha}{\alpha + \gamma ^2}$$ y₁ ^-1 = (89)

$$\frac{4}{\alpha + \gamma ^2} \left( \gamma - 1 + \frac{\alpha}{4}\right) \quad .$$ y₂ ^-1 = (90)

Note that, since $\gamma > 2$, all three constants are positive. The nucleation coefficient B then becomes

$$B = \frac{\xi _N^2}{2 G_N} \frac{y_1 y_2}{y_0} \int _0 ^1... ...t( y_1 - y \right) ^{-1} \left( y_2 - y \right) ^{-1} \quad .$$ (91)

Performing the integration in the complex plane, as described in Appendix C, we obtain

 

$$\frac{\pi \xi _N^2}{2 G_N} \frac{y_1 y_2}{y_0} \left\{ \left( \fr... ...{\frac{1}{2}} \left( y_1 - y_0 \right) \left( y_1 - y_2 \right) ^{-1} + \right.$$ B =

$$\qquad \qquad \qquad \quad + \left. \left( \frac{y_2}{y_2 - 1} \r... ...rac{1}{2}} \left( y_2 - y_0 \right) \left( y_2 - y_1 \right) ^{-1} - 1 \right\}$$ (92)

$$\frac{\pi \xi _N^3}{2 G_N} \frac{y_1 y_2}{y_0} \left[ - \frac{ 2 ... ...qrt{\alpha + \gamma ^2} } {2 \gamma ^2 - 2 \gamma + \alpha} - 1 \right] \quad .$$ = (93)

Then, with some lengthy algebra, which we confine to Appendix D, one can show that the nucleation radius and the nucleation coefficient B coincide exactly with the expression obtained by Parke [17].

A simpler case worth considering here, is the limiting value of B, given by eq.(92), as $\Lambda _{in} \rightarrow 0$. In this case

$$B \left( \Lambda _{in} = 0 \right ) = \frac{\pi \xi _N^3}{2... ..._2}{y_2 - 1} \right) ^{\frac{1}{2}} \right) \right] \quad ,$$ (94)

which, after unfolding the expression of the constants, takes the form

$$B \left( \Lambda _{in} = 0 \right ) = \frac{\pi ^2 \rho}{2} R_N^2 R_0 \quad ,$$ (95)

where R_N represents the nucleation radius of the bubble,

$$R_N \left( \Lambda _{in} = 0 \right ) = \frac{R_0}{1 + \displaystyle \frac{R_0^2 \Lambda}{12}} \quad .$$ (96)

Equations(95-96) reproduce exactly the results obtained by Coleman and de Luccia[16].

As a final remark, and in preparation of the next case, note that in the previous discussion the initial configuration R=0 consists of the classical spacetime obtained by letting $R(\tau) \rightarrow 0$ in fig.5.b, with $\sigma _{out} = +1$ during the whole process. Therefore, in that limit the shaded region of fig.5.b disappears, leaving as initial configuration for the tunneling process the whole de Sitter spacetime.

Consider now case # 1, i.e., $\gamma < 2$. In the same limit as before, but keeping now $\sigma _{out} = - 1$, we see from fig.4.b that all the un-dashed region (which represents the physical region) disappears. Therefore, in this case we have a peculiar situation in which the initial configuration corresponds to no spacetime at all, so that the tunneling process describes now something rather different from vacuum decay, which we interpret as the creation from nothing of a closed universe formed by two de Sitter cups.

  

Figure: Penrose diagram corresponding to tunneling along the classically forbidden trajectory of Fig.(3), in the case $1 < \gamma < 2$. In this case the configuration $R \equiv 0$ does not represent a classical solution, and one cannot speak of ``bubble nucleation''. One possible quantum mechanical interpretation of this process is simply as the birth ``from nothing'' of a universe composed of two de Sitter domains.

It seems conceivable that this kind of process is characteristic of the foam structure of spacetime at small scales of length at which black holes, wormholes and inflationary domains are continuously created out of the vacuum. The expression for the nucleation coefficient in this case ( $\gamma < 2$), is

$$B = \frac{\pi \xi _N^3}{2 G_N} \frac{y_1 y_2}{y_0} \left[ ... ...pha} \left( 2 \gamma - 2 + \alpha \right)- 1 \right] \quad .$$ (97)

For $\Lambda _{in} = 0$, the above expression reduces to

$$B \left( \Lambda _{in} = 0 \right ) = \frac{\pi \xi _N^3}{2 G_N} {\gamma\over 4}(\gamma + 1)\quad .$$ (98)

By considering a further limit, $\Lambda _{in} = \Lambda _{out} = 0$, corresponding to the creation of a Minkowski pair, we obtain

$$B _{M\,M} = \frac{\pi \xi _N ^3}{2} \left ( \frac{1}{2}- \fr... ...= \frac{\pi}{2} R_0^2 = \frac{1}{8 \pi \rho ^2 G_N^3} \quad ,$$ (99)

in exact agreement with the bounce calculation of ref.[18].