Question

Note: I am not asking about why time is in 2nd derivative: that makes perfect intuitive sense given the relativistic need to treat space and time in equal footing.

We often hear how the Klein-Gordon is the simplest example of equation of motion in (classical/quantum) field theory. So my question is: why is the simplest case of equation of motion in field theory expressed in second derivatives, and not in first derivatives?

My first guess is that it might still be related to Lorentz invariance. Is it so, and why? Do you have physical/intuitive/geometric examples of why a first derivative Klein-Gordon equation does not work?

Answer 1

The Klein-Gordon equation in field theory is expressed in second derivatives because it captures the interplay between space and time, ensuring compatibility with special relativity. First derivative equations do not adequately describe relativistic effects and Lorentz invariance. Waves provide an intuitive example of why a first derivative Klein-Gordon equation is insufficient.

Step-by-step explanation:

In classical field theory, the Klein-Gordon equation is the simplest example of an equation of motion that describes the behavior of a field.

This equation is expressed in second derivatives because it relates the second derivative of the field with respect to both space and time to the field itself.

This second derivative term is necessary to properly describe the dynamics of the field and ensure its compatibility with special relativity.

Expressing the equation in first derivatives alone would not capture the relativistic effects and Lorentz invariance that are important in field theory.

By including both spatial and temporal second derivatives, the equation accounts for the interplay between space and time and allows for consistent descriptions of particle behavior.

An intuitive example of why a first derivative Klein-Gordon equation does not work can be seen in the behavior of waves.

Waves propagate in both space and time, and their dynamics are best described using second derivatives to capture the relationship between oscillations at different points in space and time.