#### Physics of an Empty Box

Imagine we took a box, removed all matter from it, cooled it to the lowest temperature possible to remove all heat and sealed it perfectly. It appears that we imagined a boring empty box with absolutely nothing in it, but this is not true. Quantum mechanics predicts that in the known universe there is no such thing as an absolutely empty space with zero energy. Inside our box, virtual particle pairs are created for a very short time and disappear when the two collide. Virtual particles arise because the zero-point energy of vacuum, which is the lowest possible energy of empty space, is not zero. To understand why, we need to familiarise ourselves with the Heisenberg Uncertainty Principle and the quantum theory of fields.

#### Heisenberg’s Uncertainty and Quantum Fields** **

The Heisenberg Uncertainty Principle states that the more accurately one measures the position of a particle, the less accurately we can measure its velocity. The same relationship exists for energy and time. The minimum uncertainty, constrained by a special quantity called the Planck constant, is a manifestation of the probabilistic nature of quantum mechanics which comes from the wave-like behaviour of quantum fields. It does not relate the actual ability of the instruments used to measure the quantities in question.

Let us construct a model of a quantum field in our minds. Take an infinite grid and in the middle of each square attach a spring oriented vertically. At the top of each spring attach a ball, which models particles associated with the quantum field. The balls can oscillate up and down, but they cannot interact with each other. This is the simplest quantum field one can make. To make things a bit more interesting, connect all springs to each other with rubber bands. Now the oscillation of one ball can cause the others to oscillate. According to quantum mechanics and the Heisenberg Uncertainty Principle, these springs can never stop oscillating, i.e. minimum energy is not zero, and the amplitude of their oscillation can only take certain values. The size of the spring is also constrained by what is known as the Planck length which in turn constrains the maximum frequency of oscillation called the Planck frequency.

Given all of this, the calculation of the theoretical value of the energy density of vacuum is enormous. The experimental value of the energy density of vacuum is found to be 113 orders of magnitude less of what the theory predicts. That is, the value predicted by the theory is the experimental value multiplied by 10 to the power of 113 (1 followed by 113 zeros). This discrepancy, known as the vacuum catastrophe, is one of the most intriguing unsolved problems in physics. This problem will be discussed more later in the section on the cosmological constant and dark energy.

#### Casimir Effect

The effect of the quantum vacuum fluctuations, which are the appearing and disappearing virtual particles, can be measured in a laboratory. The experiment requires two uncharged metal plates parallel to each other, separated by a tiny distance in a vacuum. In such a setup there should be no force between the plates and they should remain perfectly still. However, in 1948 Hendrik Casimir predicted that quantum fluctuations should cause a net attractive (or in some cases repulsive) force between the plates. This is called the Casimir effect and can be explained using our spring model of quantum fields.

Imagine the balls bouncing between the two plates, as if the two plates were playing tennis with them. By decreasing the distance between the plates, we restrict the possible wavelengths of the oscillations. The balls inside the gap between the plates cannot oscillate with a wavelength that is higher or equal to the size of the gap. Outside the plates, there are no such restrictions. The virtual particles collide with the walls of the plates inside and the outside the gap, but those on the outside exert a higher pressure. This causes the two plates to be pushed together, i.e. there is a net attractive force between the plates. This net force increases as the distance between the plates decreases.

The first attempt to verify Casimir’s prediction was performed in 1958 in Eindhoven in the Netherlands by M. Sparnaay, but the results came with large experimental errors. In 2001, a group at the University of Padua in Italy successfully measured the Casimir effect using microresonators with 15% precision, which gives an unambiguous confirmation of the existence of quantum fluctuations.

#### Cosmological Constant and Dark Energy

One of the most important discoveries in astrophysics was that the expansion of the universe is accelerating. To explain this, Einstein’s cosmological constant was put back into the field equations of the general theory of relativity. The effect of the cosmological constant is to counteract the attractive force of gravity. Physically, it is the value of energy density of vacuum. This mysterious energy was called dark energy and it is estimated to make up 70% of the universe.

The vacuum zero-point energy has some very strange properties. Firstly, the density of vacuum zero-point energy is constant, which means that as the universe expands, more energy is simply created out of nothing. Secondly, the quantum vacuum has negative pressure. According to Einstein, energy and mass are equivalent as sources of gravity, hence an energy density in space will exert a gravitational force. Since the zero-point energy density is not zero, the negative pressure of the vacuum implies that it will exert a repulsive gravitational force. These properties make the vacuum zero-point energy identical to dark energy. However, there is a large discrepancy in their energy densities.

Astrophysicists can estimate the value of the dark energy density from the acceleration of the expansion of the universe. However, as mentioned before, the dark energy density value from experiment is very small compared to the theoretically predicted value of the vacuum zero-point energy density. One proposed solution to this problem is that above a certain frequency, the quantum fluctuations, i.e. virtual particles, are not gravitationally active and hence do not contribute to the acceleration of the expansion of the universe. There should soon be a possibility for an experimental verification of this hypothesis. If it is true, the mystery of the origin of the dark energy would be solved.

#### Discovery in Trinity College Dublin

Earlier this year a group of Trinity scientists in the School of Physics, J.M.D. Coey, K. Ackland and M. Venkatesan, and the School of Mathematics, S. Sen, made a groundbreaking discovery which appeared on the front cover of the prestigious scientific journal Nature Physics. The group studied the magnetism of very tiny particles of cerium oxide (CeO2) with lanthanum (La) impurities. The physicists found that the La impurities in the CeO2 nanoparticles were responsible for the magnetic moment and the magnetisation was temperature-independent. The nanoparticles were subsequently diluted with non-magnetic powders. After the dilution, the magnetisation of the sample decreased dramatically. It is proposed that the magnetism is associated with mesoscale clumps (of approximately 0.1 micron size) of CeO2 nanoparticles.

Upon dilution of CeO2 with non-magnetic powders, the ‘magnetic’ clumps are broken up and the magnetism is dramatically reduced. Furthermore, in their paper, the presence of magnetism in the nanoparticles is proposed to be due to orbital electrons associated with these mesoscale clumps interacting with the zero-point quantum fluctuations of the vacuum. The model accounts for the magnetisation remaining constant for changing temperatures and also explains why it decreases as the nanoparticles are diluted with a powder. This discovery opens experimental possibilities in probing this new observable consequence of the quantum vacuum fluctuations.