II.1.4 Spontaneous Symmetry Breaking

The concept of spontaneous symmetry breaking is extremely important to the understanding of electroweak or any further unifications. The weak and electromagnetic fundamental forces seem very different in the present relatively low temperature universe, but in the early history, when the universe was much hotter so that the equilibrium thermal energy was on the order of , these forces may have appeared to be essentially identical parts of the same unified electroweak force. However, since the exchange particle for the electromagnetic part is the massless photon and the exchange particles for the weak interaction are the massive  and  bosons, the symmetry must have been spontaneously broken when the available energy dropped below about  and the weak and electromagnetic forces became distinct. Therefore, according to this model, at an even higher temperature, there is symmetry or unification with the strong interaction, which is called the Grand Unification, and then at even higher energy, the gravity force may join to show the four fundamental forces to be a single unified force, as demonstrated in Figure II.1.

Figure II.1: Unification of the Four Fundamental Interactions. The periods and numbers given are approximate. The Duality of Time Theory predicts that the unified force is the magnetic force itself, which is a property of the pure complex-time geometry that results from the intrinsic spin of its single points. It is also predicted that the number of forces will double once we transform to three spatial dimensions (and we are now in two, but evolving in time), since the number of forces is directly related to the degrees of freedom according to N_F=2^DSo the unified force of magnetism corresponds to D=0, while our current four forces correspond to D=2+T.

Spontaneous symmetry breaking often occurs when there is a phase transition between a high-temperature symmetric phase and a low temperature one in which the symmetry is spontaneously broken. The simplest example is freezing. For example, a round bowl of water sitting on a table has rotational symmetry, because it looks the same from every direction, but when it freezes, the ice crystals form in specific orientations, thus breaking the previous symmetry. In the snowflake, also, both the hydrogen and oxygen molecules are quite symmetric when they are isolated, and the electric force, which governs the structure atoms, is also acting symmetrically. But when the temperature is lowered and these atoms form water, the symmetry of the individual atoms is broken as they form a molecule with  degrees between the hydrogen-oxygen bonds. And then when they freeze to form a snowflake, they form another type of symmetry. So this breaking of the original symmetry is called spontaneous because it occurs without any external intervention. In condensed matter physics, a lot of phases are examples of spontaneous symmetry breaking, which are characterized by an order parameter that describes how the symmetry is being broken. For example, the orientation of spins is the order parameter in a ferromagnetic material, where the symmetry of electron spins is broken below a certain temperature.

Another familiar example is the ferromagnetic materials, where the underlying laws are invariant under spatial rotations, and the order parameter is the magnetization, which measures the magnetic dipole density. Above the Curie temperature, the order parameter is zero, which is spatially invariant, and there is no symmetry breaking, but below this point, the magnetization acquires a constant non-vanishing value, which normally points in a certain direction. The residual rotational symmetries which leave the orientation of this vector invariant remain unbroken, unlike the other rotations which are thus spontaneously broken.

Spontaneous symmetry breaking is a process by which a physical system in a symmetric state ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. When the system goes to one of those vacuum solutions, the symmetry is broken for perturbations around that vacuum even though the entire Lagrangian retains that symmetry. The symmetry group can be discrete or continuous, but if the system contains only a single spatial dimension, then only discrete symmetries may be broken in a vacuum state of the full quantum theory, although a classical solution may break a continuous symmetry too.

The spontaneous symmetry breaking is completely different process than the explicit symmetry breaking, where an external force is required to start the process, but for spontaneous symmetry breaking process, no external force is required. In explicit symmetry breaking, the probability of a pair of outcomes can be different. By definition, spontaneous symmetry breaking requires the existence of a symmetric probability distribution where any pair of outcomes has the same probability where the underlying laws are invariant under a symmetry transformation so that the system, as a whole, changes under such transformations.

Phases of matter, such as crystals, magnets, and conventional superconductors, as well as simple phase transitions, can be described by spontaneous symmetry breaking. This is usually demonstrated through Goldstone s Mexican-hat potential function, which is a symmetric upward dome with a trough circling the bottom, as in Figure II.2. If a ball is put at the peak of the dome, the system is symmetric with respect to a rotation around the center axis, but the ball may spontaneously break this symmetry by rolling down the dome into the trough which has the lowest energy. Afterward, the ball has come to rest at some fixed point on the perimeter, and each of the dome and the ball are retaining their individual symmetry, although the symmetry of the whole system is broken, or hidden. At high energy levels the ball settles at the top, and the result is symmetric, while at lower energy levels, the ball must rest at some random spot on the bottom, resulting in some spontaneous asymmetric outcome.

For example, in particle physics the bosons, which are the force carrier particles, are normally specified by field equations with gauge symmetry, or invariance, so they predict that certain measurements will be the same at any point in the field. For instance, field equations might predict that the total mass of two quarks is constant, so when we solve the equations to find the mass of each quark we may get two equivalent solutions in which the quarks have different mass, though their total mass remain the same. Therefore, the symmetry of the equations is not reflected by the individual solutions, but it is reflected by the range of solutions. An actual measurement reflects only one solution, representing a breakdown in the symmetry of the underlying theory. That s why it is better to say that the symmetry is hidden, rather than broken.

Figure II.2: Graph of Goldstone's Mexican-hat potential function.

For spontaneous symmetry breaking to occur, there must be a system in which there are several equally likely outcomes. The system as a whole is therefore symmetric with respect to these outcomes. However, if the system is sampled, a specific outcome must occur. Though the system as a whole is symmetric, it is never encountered with this symmetry, but only in one specific asymmetric state. Hence, the symmetry is said to be spontaneously broken in that theory. Nevertheless, the fact that each outcome is equally likely is a reflection of the underlying symmetry, which is thus often dubbed hidden symmetry.

When a theory is symmetric with respect to a symmetry group, but requires that one element of the group be distinct, then spontaneous symmetry breaking has occurred. The theory must not dictate which member is distinct, only that one is. From this point on, the theory can be treated as if this element actually is distinct, with the proviso that any results found in this way must be re-symmetrized, by taking the average of each of the elements of the group being the distinct one.

The crucial concept in physics theories is the order parameter. If there is a background field which acquires an expectation value which is not invariant under the symmetry in question, we say that the system is in the ordered phase, and the symmetry is spontaneously broken. This is because other subsystems interact with the order parameter, which specifies a frame of reference to be measured against. In that case, the vacuum state does not obey the initial symmetry, which would keep it invariant, in the linearly realized Wigner mode in which it would be a singlet, and, instead changes under the hidden symmetry, now implemented in the nonlinear Nambu-Goldstone mode. Normally, in the absence of the Higgs mechanism, massless Goldstone bosons arise.