Section 1.1: The Klein-Gordon Field

In classical mechanics, a particle is represented as a point with mass m. Let’s denote its position at a given time t as x(t), with potential energy in a region defined as V(x). According to Newton’s second law of motion, we express this relationship mathematically:

Equation 1: Newton’s second law of motion.

The corresponding Lagrangian function L can be expressed as follows:

Equation 2: The Lagrangian of a particle moving in a potential region V(x).

Here, K and V represent the kinetic and potential energies, respectively. We can transition from a point particle at x(t) to a field Φ(x, y, z, t) by substituting x with Φ and t with (x, y, z). The Lagrangian density in this case takes the form:

Equation 3: The Lagrangian density of the real scalar field theory Φ.

It is important to note that Φ is a real scalar field and not a wave function. Equation 3 remains Lorentz invariant, meaning its mathematical structure does not alter for observers in relative inertial frames. The scalar field Φ adheres to the Klein-Gordon equation:

Equation 4: The Klein-Gordon equation for the free field Φ.

Section 1.2: Complex Scalar Fields

To better describe superfluid properties, we need to enhance the complexity of Equation 3. The Lagrangian density for a complex scalar field is given by:

Equation 5: The Lagrangian density for complex scalar field theory Φ.

This formulation encompasses interacting bosons, particularly focusing on the dynamics of slowly moving bosons. Initially, we derive the dynamics of these bosons from the equation of motion in Equation 4 for a free scalar field Φ, where the solutions exhibit time dependence as follows:

Equation 6: Solutions of Eq. 4 as modes.

For our analysis of slowly moving bosons, we express the energy in Equation 6 as follows:

Equation 7: Energy representation in Eq. 6 with m >> ε.

We can separate the mode from Equation 6 into two distinct factors:

Equation 8: Mode separation where φ oscillates slowly compared to the exponential factor.

Using the approximation:

Equation 4 transforms into the Schrödinger equation:

Equation 9: Schrödinger's equation.

By substituting into Equation 5, we obtain the nonrelativistic version of the Lagrangian density. After some straightforward algebra, we arrive at:

Equation 10: Lagrangian density of nonrelativistic bosons with short-range repulsion.

The negative sign in the potential term g²ρ² indicates a repulsive interaction. To mitigate the risk of zero density in Equation 10, a finite density of nonrelativistic bosons is generally introduced into L:

Equation 11: Equation 10 with a chemical potential term.

The potential known as the Mexican hat or champagne-bottle potential is depicted in Fig. 4.

The implications of Equation 11 suggest that φ approaches the expected value of the field φ:

Equation 13: The Mexican hat potential constraining Eq. 12 to be minimal.

To explore spontaneous symmetry breaking, we first express φ in polar coordinates:

Equation 14: The field ϕ in polar coordinates.

We then introduce a minor perturbation to the expected value:

Equation 15: Small perturbation to the expectation value of φ.

Substituting Equation 15 into Equation 11 yields:

Equation 16: Lagrangian density in polar coordinates.

Here, a total divergence term has been omitted. To eliminate h, we integrate the Lagrangian density via a path integral:

Equation 17: Integration of h through a path integral (details omitted).

The resulting Lagrangian density is:

Equation 18: Lagrangian density representing a fluid of bosons with a gapless mode.

The spatial derivative's denominator in Equation 17 can be disregarded. Equation 15 indicates the existence of a fluid of bosons with a gapless mode, and the linear dispersion relation is expressed as follows:

Equation 19: Linear dispersion relation of bosons in a superfluid.

The form of the Lagrangian that emerges post spontaneous symmetry breaking illustrates momentum superflow. The massless bosonic field θ is referred to as a Goldstone boson, a common outcome of spontaneous symmetry breaking as indicated in Equation 15.

Thanks for engaging with this content, and I look forward to your thoughts! For further insights into topics like physics, mathematics, and machine learning, feel free to visit my LinkedIn or personal website at www.marcotavora.me.