Special Relativity | Lecture 4

The speaker begins by transitioning from particle physics to classical field theory, emphasizing the study of fields rather than particles. Fields are observable quantities that vary with space and time, such as the electric field. The lecture introduces the concept of space-time with one time dimension and multiple space dimensions, which is foundational to field theory. Scalar and vector fields are discussed, with examples including temperature and wind velocity, to illustrate the difference between these types of fields.

The discussion shifts to the motion of particles in nonrelativistic physics, using the principle of least action. The action is defined as an integral of the Lagrangian, which is the difference between kinetic and potential energy. The Euler-Lagrange equations are introduced to describe the motion of particles, which are derived from the principle of least action. The speaker also connects the principle of least action to finding the trajectory of least action between two points in space-time.

The lecturer generalizes the concept of field theory to include space-time dimensions beyond a single time dimension. By considering fields as functions of time and space, the speaker explores the idea of fields in various space-time configurations, from zero-dimensional space to higher dimensions. The potential energy is also discussed in the context of fields, highlighting that it can depend on the field's value, even in a space-time with no spatial dimensions.

The action principle is extended to field theory, where the action is considered as a sum over space-time cells. The field's history is represented as a surface in space-time, and the action principle is used to find the field configuration that minimizes the action. The speaker emphasizes that the action principle is a powerful tool that encodes numerous laws of physics and is applicable to field theory in the same way it is to particle mechanics.

The lecture delves into the specifics of the Lagrangian in field theory, which is an integral over space and time. The Lagrangian density is introduced as a function of the field and its derivatives with respect to time and space. The speaker outlines the process of constructing a Lagrangian for field theory by considering all possible scalars that can be made from the field and its derivatives, ensuring that the Lagrangian is a scalar and invariant under Lorentz transformations.

The speaker derives the Euler-Lagrange equations for field theory, which are analogous to those in particle mechanics but include a sum over terms for each direction of space-time. The equations are presented in a form that accounts for both time and space derivatives of the field. The lecturer also touches on the implications of including a potential energy term in the Lagrangian, leading to a wave-like motion described by the equations of motion.

The importance of Lorentz invariance in field theory is emphasized, with the speaker explaining how to construct invariant quantities and the transformation properties of fields. Scalar fields and four-vectors are discussed, along with their transformation under Lorentz transformations. The lecture also covers how to form scalars from four-vectors and the concept of differentiating a scalar to obtain a four-vector.

The speaker outlines the rules for constructing a Lagrangian that is invariant under Lorentz transformations. It is stated that the Lagrangian must be a scalar, and possible scalars that can be included are functions of the field and its derivatives. The lecture also explores the possibility of including higher powers of derivatives and the implications of such inclusions for the complexity of the Lagrangian.

The lecture concludes with a discussion on how a particle might interact with a scalar field. A Lagrangian for a particle moving in a pre-established field is proposed, and it is shown that the particle's motion can be affected by the field. The concept of the Higgs field giving mass to particles is touched upon, and the lecturer suggests that the Higgs mechanism is closely connected to the idea of a scalar field affecting particle mass.

The speaker works through the equations of motion for a particle interacting with a scalar field, using the Euler-Lagrange equations. The process is complex, and the resulting differential equation is not simplified. However, the lecturer suggests that if the field tends to a stable non-zero value, the particle's motion can be described as if it had an effective mass that includes the contribution from the field. The Higgs field is mentioned as a real-world example of such a mechanism.