Lecture 7 | Quantum Entanglements, Part 3 (Stanford)

This paragraph introduces the topic of charged particles moving in electric and magnetic fields. It discusses the impact of relativity on these particles and fields, the Lorentz force law, and the need to reconcile it with the principles of relativity. The paragraph also touches on the concept of force in different contexts, from non-relativistic to relativistic physics, and hints at the differences between the laws of gravitation and electrical forces under the special theory of relativity.

The second paragraph delves into the definitions of momentum and force within the framework of Newtonian and relativistic physics. It explains the time rate of change of momentum equals force, and how this definition extends to special relativity. The paragraph also introduces the Lorentz force law, which incorporates electric and magnetic forces on a charged object, and challenges the audience to reintroduce the speed of light into the equations for a nonrelativistic limit.

This section explores the cross product of vectors and how it can be represented as an anti-symmetric tensor. It discusses the properties of tensors, their transformation under rotations, and the relationship between the cross product of two vectors and the components of an anti-symmetric tensor. The paragraph establishes the groundwork for understanding how the magnetic field can be represented in terms of tensors, which is crucial for generalizing to other dimensions.

The fourth paragraph continues the discussion on the magnetic field, emphasizing its dual representation as both a vector and an anti-symmetric tensor. It corrects a previous statement about the components of the cross product and their correspondence to the tensor components. The paragraph also highlights the unique aspects of three-dimensional space that allow for this dual representation and introduces the concept of the magnetic field's geometry in relation to an anti-symmetric tensor.

This paragraph examines the geometrical interpretation of anti-symmetric tensors, particularly in the context of the magnetic field. It discusses how each component of the tensor is associated with two axes and the implications of describing a vector by the plane it's perpendicular to. The paragraph also touches on the transformation properties of the tensor under rotations and the distinction between describing a vector by its components versus the planes perpendicular to those components.

The fifth paragraph works through the components of the cross product V cross B, focusing on the third component. It demonstrates how to express this component using the identification of the magnetic field as a tensor. The paragraph also discusses the theorem that the force on a charged particle in a magnetic field is given by V cross B and the implications of this for the particle's motion, including the constancy of speed in a static magnetic field.

This section focuses on the differentiation of the square of the velocity with respect to time in a magnetic field. It establishes that the magnitude of the velocity (or speed) of a particle does not change when it is solely under the influence of a static magnetic field. The paragraph uses the Lorentz force law to demonstrate this property and highlights the difference between a time-dependent and time-independent magnetic field's impact on the particle's velocity.

The seventh paragraph discusses how electric and magnetic fields transform under different reference frames moving with relative velocity. It explains that a pure magnetic field in one frame can appear as a combination of electric and magnetic fields in another frame. The paragraph also touches on the concept of four-vectors and the need to work with four-dimensional space-time to accurately describe the transformation properties of these fields.

This paragraph introduces the electromagnetic field tensor, an anti-symmetric tensor that combines the electric and magnetic fields. It discusses the structure of this tensor and how it fills the six components of the electromagnetic field. The paragraph also explores the historical context of the tensor's development and its role in the force experienced by charged particles within the framework of special relativity.

The ninth paragraph defines the four-velocity and four-momentum in the context of special relativity. It explains how the components of these four-vectors are derived from the particle's ordinary velocity and mass. The paragraph also establishes the relationship between the force experienced by a particle and the anti-symmetric tensor, leading to the conclusion that the force must be orthogonal to the four-velocity.

The tenth paragraph works through the derivation of the Lorentz force law using the four-vector notation. It shows how the force experienced by a charged particle is related to the electric and magnetic fields and the particle's velocity. The paragraph also discusses the transformation properties of the electromagnetic field tensor and how these properties ensure that the equations of motion for a charged particle remain consistent across different reference frames.

The final paragraph shifts the focus to wave equations and scalar waves. It introduces the concept of a scalar field and its differentiation to form covariant components of a four-vector. The paragraph also discusses the transformation properties of these derivatives and sets the stage for a future discussion on Maxwell's equations and their Lorentz invariance.

This paragraph discusses the importance of the invariance of wave equations under Lorentz transformations. It presents a simple wave equation and explains that for an equation to be invariant, it must result in a scalar, which remains the same across all reference frames. The paragraph concludes with a note on the upcoming discussion on Maxwell's equations in a covariant form, which will be explicitly Lorentz invariant.