9

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.

Community
  • 1
Anthonny
  • 1,714

1 Answers1

2

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)
Calculation
Where $K_2$ is the modified Bessel function.

Omry
  • 1,315
  • 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? – Anthonny Jul 20 '15 at 00:18
  • "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. – Omry Jul 20 '15 at 06:34
  • The solution doesn't vanish outside the light cone, and therefore the solution is non-local, but, due to anti-particles, causality is preserved. – Omry Jul 20 '15 at 06:35
  • 1
    But this would be a problem for the K-G equation too. – Anthonny Jul 20 '15 at 06:55
  • If causality is preserved, it's not really that big of a problem. – Omry Jul 20 '15 at 06:56
  • But (acording to the book) causality problem is solved using QFT. My quetion is just about the equations. K-G equation (which is second order in time derivatives and space derivatives) and the square root version. Some people call them local and non-local respectively. – Anthonny Jul 20 '15 at 07:02
  • So, K-G equation admits a nonlocal solution too. – Anthonny Jul 20 '15 at 07:04
  • The KG equation itself has this problem, that much is true. However, when using it in QFT, causality is preserved. So while the equation itself has this problem, it is later solved. – Omry Jul 20 '15 at 07:04