In two-dimensional spacetime, things are very simple. Indeed, many quantum field theories are exactly solvable. I want to exploit this simplicity and better understand the kinematic invariants that appear. My hope is that some of these lessons can be applied in higher-dimensional kinematics.

This program began with an observation in non-relativistic elastic collisions:

• Two-Dimensional Elastic Kinematics

Energy-momentum vectors are very simple in two-dimensional spacetime:

• Two-Dimensional Energy-Momentum

The most interesting thing that I learned was that the inner product of either two slow or two fast vectors is proportional to a hyperbolic cosine, but the inner product of a slow and a fast vector is proportional to a hyperbolic sine. This has some consequences when interpreting 2-Gram invariants. Indeed, you can use a permanent:

• Encounter with Permanents

You can also get results involving speed and rapidity:

• Two-Dimensional Speed and Rapidity

One of the most important problems involves four external states. You can have a mixture of slow, null, and fast momenta:

• Two-Dimensional Four-Point Kinematics (1)
• Two-Dimensional Four-Point Kinematics (2)
• Two-Dimensional Four-Point Kinematics (3)
• Two-Dimensional Four-Point Kinematics (4)
• Two-Dimensional Four-Point Kinematics (5)

You can study the two-body Sudakov decomposition:

• Two-Dimensional Sudakov Decomposition (1)
• Two-Dimensional Sudakov Decomposition (2)

The two-body Sudakov decomposition can be useful when working with 2-to-\(N\) scattering. The conjugate problem, \(N\)-to-2 scattering is also interesting because you can use energy-momentum conservation to solve for the outgoing rapidities in terms of the incoming rapidities:

• Two-Dimensional Three-Point Slow Kinematics
• Two-Dimensional Four-Point Slow Kinematics
• Two-Dimensional Six-Point Slow Kinematics