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False vacuum

In quantum field theory, a false vacuum is a metastable sector of space which appears to be a perturbative vacuum but is unstable to instanton effects which may tunnel to a lower energy state. This tunneling can be caused by quantum fluctuations or the creation of high energy particles. Simply put, the false vacuum is a local minimum, but not the lowest energy state, even though it may remain stable for some time. This is analogous to metastability for first order phase transitions.

Vacuum metastability event

In their paper, Coleman and de Luccia noted:

The possibility that we are living in a false vacuum has never been a cheering one to contemplate. Vacuum decay is the ultimate ecological catastrophe; in the new vacuum there are new constants of nature; after vacuum decay, not only is life as we know it impossible, so is chemistry as we know it. However, one could always draw stoic comfort from the possibility that perhaps in the course of time the new vacuum would sustain, if not life as we know it, at least some structures capable of knowing joy. This possibility has now been eliminated.

The possibility that we are living in a false vacuum has been considered. If a bubble of lower energy vacuum were nucleated, it would approach at nearly the speed of light and destroy the Earth instantaneously, without any forewarning. Thus, this vacuum metastability event is a theoretical doomsday event. This was used in a science-fiction story in 1998 by Geoffrey A. Landis, and in 2000 by Stephen Baxter.

Particle accelerator

One scenario is that, rather than quantum tunnelling, a particle accelerator, which produces very high energies in a very small area, could create sufficiently high energy density as to penetrate the barrier and stimulate the decay of the false vacuum to the lower energy vacuum. Hut and Rees, however, have determined that because we have observed cosmic ray collisions at much higher energies than those produced in terrestrial particle accelerators, that these experiments will not, at least for the foreseeable future, pose a threat to our vacuum. Particle accelerations have reached energies of only approximately four thousand billion electron volts (4 ×10³ GeV). Cosmic ray collisions have been observed at and beyond energies of 10¹¹ GeV, the so-called Greisen-Zatsepin-Kuzmin limit. John Leslie has argued that if present trends continue, particle accelerators will exceed the energy given off in naturally occurring cosmic ray collisions by the year 2150.

This event would be contingent on our living in a metastable vacuum, an issue which is far from resolved. Worries about the vacuum metastability event are reminiscent of the controversy about turning the Relativistic Heavy Ion Collider on.

Bubble nucleation

In a physical theory in a false vacuum, the system moves to a lower energy state – either the true vacuum, or another, lower energy vacuum – through a process known as bubble nucleation. In this, instanton effects cause a bubble to appear in which fields have their true vacuum values inside. Therefore, the interior of the bubble has a lower energy. The walls of the bubble (or domain walls) have a surface tension, as energy is expended as the fields roll over the potential barrier to the lower energy vacuum. The most likely size of the bubble is determined in the semiclassical approximation to be such that the bubble has zero total change in the energy: the decrease in energy by the true vacuum in the interior is compensated by the tension of the walls.

Expansion of bubble

Any increase in size of the bubble will decrease its potential energy, as the energy of the wall increases as the area of a sphere $4pi r^2$ but the negative contribution of the interior increases more quickly, as the volume of a sphere $4/3pi r^3$ . Therefore, after the bubble is nucleated, it quickly begins expanding at very nearly the speed of light. The excess energy contributes to the very large kinetic energy of the walls. If two bubbles are nucleated and they eventually collide, it is thought that particle production occurs where the walls collide.

The tunneling rate is increased by increasing the energy difference between the two vacua and decreased by increasing the height or width of the barrier.

Gravitational effects

The addition of gravity to the story leads to a considerably richer variety of phenomena. The key insight is that a false vacuum with positive potential energy density is a de Sitter vacuum, in which the potential energy acts as a cosmological constant and the Universe is undergoing the exponential expansion of de Sitter space. This leads to a number of interesting effects, first studied by Coleman and de Luccia:

Space with zero potential energy

Tunnelling from a space with zero potential energy (e.g. Minkowski space) to negative potential energy leads to the following. The walls of the bubble grow at the speed of light, as described above. However, the interior of the bubble rapidly collapses, as anti-de Sitter space and the universe ends (see ultimate fate of the universe and vacuum metastability event, below).

Space with positive potential energy

Tunneling from a space of positive potential energy (de Sitter space) to one of vanishing potential energy (Minkowski space) leads to the following. The volume of the bubble continues to grow at the speed of light. However, since the exterior of the bubble is expanding exponentially while the Minkowski space is not—unlike the non-gravitational case—the whole of space time need never be dominated by the lower energy vacuum. If the tunnelling rate is slow enough, the exponentially expanding space in the false vacuum state can expand sufficiently quickly so that the bubbles of lower-energy space never begin to collide and convert all of space-time to the lower energy state. That is, the tunnelling is competing with rapid expansion, and the exponential expansion can be so rapid that the tunneling effect is overwhelmed.

From positive potential energy to lower, positive potential energy

Tunnelling from positive potential energy to lower, positive potential energy leads to the following. Just as for the above case, the more rapid exponential expansion of the higher energy false vacuum can continue to dominate.

From positive potential energy to negative potential energy

Tunneling from positive potential energy to negative potential energy leads to the following. This effect is highly suppressed: the expansion of the positive energy vacuum dominates the contraction of the negative energy vacuum.

A final kind of tunnelling

A final kind of tunnelling is the Hawking-Moss instanton. This occurs when the size of the Coleman–de Luccia bubble is larger than the size of the universe, in a closed universe, or of the horizon. In this case, the entire universe tunnels from the false vacuum to the true vacuum at once.

Developments of theory

Alan Guth in his original proposal for cosmic inflation proposed that inflation could end through quantum mechanical bubble nucleation of the sort described above. See History of Chaotic inflation theory. It was soon understood that a homogeneous and isotropic universe could not be preserved through the violent tunneling process. This led Andrei Linde and, independently, Andreas Albrecht and Paul Steinhardt to propose "new inflation" or "slow roll inflation" in which no tunnelling occurs, and the inflationary scalar field instead rolls down a gentle slope.

String landscape

A more recent application of these tunnelling phenomena in cosmology and particle physics is the string landscape in which string theory is conjectured to be populated by an exponentially large "discretuum" of false vacua, and the small observed value of the cosmological constant (see dark energy) can be explained by the anthropic principle and quantum mechanical tunnelling to the lowest positive energy vacuum.