A Glimpse into Spacetime Diagrams

To comprehend the arguments presented here, it’s critical to understand the geometric features of spacetime diagrams, also known as Minkowski diagrams. These diagrams serve as visual tools to illustrate the movement of particles throughout spacetime.

To clarify the various types of intervals in a spacetime diagram, it is helpful to consider a simplified two-dimensional representation, where one axis represents time and the other represents space. The path of an observer within this diagram is termed a "world line."

In a Minkowski diagram, the velocity corresponds to the inverse of the slope of a line. Photons travel at light speed, represented by a line at a 45-degree angle (for convenience, we set c=1). An observer moving at a consistent speed between rest and light speed will have a world line with a corresponding slope. Any other types of world lines must have slopes greater than 1, restricting them within a wedge formed by the two 45-degree lines. An observer cannot enter the area outside this wedge, as it would lead to superluminal communication and thus violate causality. This topic will be revisited later.

Antiparticles Explained

Every particle in the universe has a corresponding antiparticle, which possesses the same mass but an opposite charge. The most notable example is the positron, the antiparticle of the negatively charged electron. This definition of antiparticles aligns with the standard approach found in quantum field theory texts, though I will reference the refreshing perspective offered by Lancaster and Blundell.

Feynman and Stueckelberg: Time-Reversed Particles

In the 1940s, Richard Feynman and Ernst Stueckelberg independently proposed viewing states with negative energy as particles moving backward through time. These states are now recognized as antiparticles. For instance, in the equations governing charged particles in electromagnetic fields, changing the sign of time and charge yields the same equation, suggesting a particle traveling backward in time appears as an antiparticle with the opposite charge moving forward.

It is important to mention the CPT theorem, a fundamental symmetry in physics. CPT stands for charge conjugation, parity transformation, and time reversal. The theorem asserts that under these transformations, the laws of physics remain invariant. In our discussion, we can focus on the charge and time aspects.

Quantum Mechanics and Perturbation Theory

Let’s consider a particle initially in state ψᵢ. In quantum mechanics, we describe the transition amplitude and its corresponding probability as follows:

K = int d^3 x , e^{i p cdot x} psi_i

This transition relates to a propagator K, which is characterized by the Green's function that integrates causality through the Heaviside function Θ.

When the system maintains time translational invariance, the propagator relies solely on the time difference. The free Green's function, G₀, can be expressed mathematically, illustrating a scenario where a particle transitions from ψᵢ to itself without interaction.

The interaction of one particle with another is represented by a potential V(x,t). In the presence of this interaction, calculating the transition amplitude becomes complex, leading to the use of perturbation theory for small potentials.

Two Interactions in Quantum Mechanics

Now, suppose a particle engages with a potential V(x,t) at two distinct points. We can summarize the process as follows:

1. The particle in state ψᵢ interacts with V at x₁, transitioning to an intermediate state with energy E(m).
2. The particle then evolves freely from x₁ to x₂ for a duration of (t₂ - t₁).
3. Finally, the second interaction with V at x₂ returns the particle to its initial state.

This results in the transition amplitude, which can be expressed using plane waves as intermediate states.

A Quick Note on Fourier Transforms

The Fourier transform of a function decomposes it into constituent frequencies, yielding a complex representation. The inverse Fourier transform reconstructs the original function from its frequency domain.

The Inevitable Presence of Antiparticles

To demonstrate the unavoidable existence of antiparticles, we start with the premise that all energies are positive. Transforming the momentum integral leads us to define a function F(ω), which is zero for ω < m. According to a theorem, if a function can be expressed solely in terms of positive frequencies, it cannot be nonzero over a finite time interval unless it is identically zero everywhere.

Applying this theorem implies that the momentum integral must include nonzero amplitudes representing particles moving faster than light, which leads to the interpretation of some particles appearing as if they are moving backward in time.

In conclusion, the two conditions of energy positivity and adherence to relativity suggest that particle-antiparticle pairs can be created and annihilated under the right circumstances.

Explore More

For additional insights on physics and other subjects, feel free to visit my personal website at www.marcotavora.me.

The first video titled "The reason for antiparticles - Richard P. Feynman" delves into the fundamental concepts behind antiparticles and their significance in the universe.

The second video, "What are Antiparticles? | Anti-Matter Explained - Ep 62 Clips," provides an accessible overview of antiparticles and their role in modern physics.