Lorentz Transformation of f(x,t)+g(x,t) - Does the Order Matter?

[Spinnor]

Suppose I have a pair of solutions of the one dimensional massless complex wave equation in 1+1 dimensional spacetime, solutions with equal and opposite momentum in some rest frame,

Ψ+ = exp[-i(Et+px)]
Ψ- = exp[-i(Et-px)]

Does it matter if

1) I sum Ψ+ and Ψ- and then preform a Lorentz Transformation, or
2) preform the Lorentz Transformation, L.T., on Ψ+ and Ψ- and then sum?

L. T. [f(x,t)+g(x,t)] = [L.T. f(x,t)]+[L.T. g(x,t)] ?

I think the answer is it should not matter the order of operation but I am having trouble simplifying my algebra. Just wondering if there is light at the end of the tunnel.

Thanks for any help!

 

[Ambitwistor]

I can confirm that it does not matter which order you perform the operations in. The Lorentz Transformation is a linear operation, which means that it follows the principle of superposition. This means that the transformation of a sum is equal to the sum of the transformations. In other words, the order in which you perform the operations does not change the final result.

To simplify your algebra, you can use the fact that the Lorentz Transformation of a complex exponential function is also a complex exponential function with the same frequency and wave vector, but with a different phase factor. This means that you can write the transformation of Ψ+ and Ψ- as:

L.T. [Ψ+] = exp[-i(E' t'+p'x')] = exp[-i(Et+px+iθ)] = exp[-i(Et+px)]exp[iθ]
L.T. [Ψ-] = exp[-i(E' t'-p'x')] = exp[-i(Et-px+iθ)] = exp[-i(Et-px)]exp[iθ]

Where E' and p' are the transformed energy and momentum, and θ is the phase factor.

Now, if you sum Ψ+ and Ψ- and then perform the Lorentz Transformation, you get:

L.T. [Ψ+ + Ψ-] = L.T. [exp[-i(Et+px)] + exp[-i(Et-px)]] = exp[-i(Et+px)]exp[iθ] + exp[-i(Et-px)]exp[iθ] = exp[-i(Et+px+iθ)] + exp[-i(Et-px+iθ)] = exp[-i(E' t'+p'x')]

Similarly, if you perform the Lorentz Transformation on Ψ+ and Ψ- individually and then sum them, you get:

L.T. [Ψ+] + L.T. [Ψ-] = exp[-i(Et+px)]exp[iθ] + exp[-i(Et-px)]exp[iθ] = exp[-i(Et+px+iθ)] + exp[-i(Et-px+iθ)] = exp[-i(E' t'+p'x')]

As you can see, the final result is the same in both cases. Therefore, you can perform the operations in any order without changing the