Let’s consider a scalar field, say temperature of a rod varying with time i.e . (something like the following)

We will take this setup and put it on a really fast train moving at a constant velocity *(also known as performing a ‘Lorentz boost’)*.

Now the temperature of the bar in this new frame of reference is given by where,

Temperature is a scalar fiel*d* and therefore irrespective of which frame of reference you are on, the temperature at each point on the rod will remain the same on both the frames i.e

Therefore we can say that Temperature *(a scalar field)* is Lorentz invariant. Now what other quantities can we make from T that would also be Lorentz invariant ?

Is

Well, let’s give it a try:

——–

Clearly,

But just for fun let’s just square the terms and see if we can churn something out of that:

We immediately notice that:

Therefore in addition to realizing that is Lorentz invariant, we have also found another quantity that is also Lorentz invariant. This quantity is also written as .

## Deeper meaning

We know that nature is relativistic and when we are are cooking up a Lagrangian for a theory, we better make sure that it is Lorentz invariant as well. What the above analysis on scalar fields hints us is that the Lagrangian for such a theory can contain terms like in it as the quantity does not change under a Lorentz transformation.

This discussion finds a deeper ground in Quantum Field Theory. For example if is a scalar field, then a Lorentz invariant Lagrangian could take any of the following possible forms:

All of them keep the action invariant under a Lorentz transformation.