In this Wikipedia article a relativistic wave equation is derived using the Hamiltonian $$H=\sqrt{\textbf{p}^2 c^2 + m^2 c^4}$$ Substituting this into the Schrödinger equation gives the square root version of the Klein-Gordon equation: $$\left( \sqrt{ (-i \hbar \nabla)^2 c^2 + m^2 c^4 } \right) \psi = i\hbar\frac{\partial}{\partial t} \psi$$ Then the article says:

Another problem, less obvious and more severe, is that it can be shown to be nonlocal and can even violate causality: if the particle is initially localized at a point $r_0$ so that $\psi(r_0 ,t=0)$ is finite and zero elsewhere, then at any later time the equation predicts delocalization $\psi(r,t)\neq 0$ everywhere, even for $r>ct$ which means the particle could arrive at a point before a pulse of light could.

What is this solution explicitly? I have read also this Phys.SE question but there is no clue for my question.


1 Answer 1


Taking from Peskin & Schroeder p.14:
Propagation amplitude

They then calculate it asymptotically, and refer to: Gradshteyn and Ryzhik (1980), #3.914 for an exact solution
Searching that reference, we come across: #3.914, 6: (Available here)
Where $K_2$ is the modified Bessel function.

  • $\begingroup$ that solution is also a solution of the Klein-Gordon equation. So that would imply that the standard Klein-Gordon equation is non-local too. am I right? $\endgroup$
    – Anthonny
    Jul 20, 2015 at 0:18
  • $\begingroup$ "Quantum field theory solves the causality problem in a miraculous way which we will discuss in Section 2.4. We will find that in the multiparticle field theory the propagation of a particle across a spacelike interval is indistinguishable from the propagation of an antiparticle in the opposite direction see (Fig 2.1). When we ask whether an observation made at point x can affect an observation made at point x we will find that the amplitudes for particle and antiparticle propagation exactly cancel--so causality is preserved." - Taken from P&S later on. $\endgroup$
    – Omry
    Jul 20, 2015 at 6:34
  • $\begingroup$ The solution doesn't vanish outside the light cone, and therefore the solution is non-local, but, due to anti-particles, causality is preserved. $\endgroup$
    – Omry
    Jul 20, 2015 at 6:35
  • 1
    $\begingroup$ But this would be a problem for the K-G equation too. $\endgroup$
    – Anthonny
    Jul 20, 2015 at 6:55
  • $\begingroup$ If causality is preserved, it's not really that big of a problem. $\endgroup$
    – Omry
    Jul 20, 2015 at 6:56

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