# Propagation speeds in literature

1. Sep 17, 2007

### jostpuur

In the beginning of their book, Peskin & Shroeder say that replacing non-relativistic energy $p^2/(2m)$ with the relativistic one $\sqrt{p^2 c^2 + (mc^2)^2}$ does not remove infinite propagation speeds given by the propagator

$$\int\frac{d^3p}{(2\pi\hbar)^3} e^{-i(E_p t - p\cdot(x-y))/\hbar}$$

Does this claim also appear in earlier literature of quantum theory, or is it a new one?

2. Sep 17, 2007

### olgranpappy

I don't think they use the term "propagation speeds"... do they?

Their claims are not new ones. And fear not, causality is preserved in the relativistic theory.

3. Sep 17, 2007

### jostpuur

Not really. They use for example a sentence: "This expression is nonzero for all x and t, indicating that a particle can propagate between any two points in an arbitrarily short time." I'm not aware if "infinite propagation speed" could be understood differently.

This propagator with the relativistic $E_p$ does not allow propagations outside the light cone, and I'm concerned about this belief that it would.

4. Sep 17, 2007

### meopemuk

There are lots of papers about "superluminal spreading" of wavepackets in relativistic quantum mechanics and in QFT. Below is just a sample.

S. N. M. Ruijsenaars, "On Newton-Wigner localization and superluminal propagation speeds", Ann. Phys. 137 (1981), 33

G. C. Hegerfeldt, "Instantaneous spreading and Einstein causality in quantum theory",
Ann. Phys. (Leipzig), 7 (1998), 716; http://www.arxiv.org/abs/quant-ph/9809030

F. Strocchi, "Relativistic quantum mechanics and field theory", Found. Phys. 34 (2004), 501; http://www.arxiv.org/abs/hep-th/0401143

Th. W. Ruijgrok, "On localisation in relativistic quantum mechanics", in Lecture Notes in Physics, Theoretical Physics. Fin de Si\'e cle, vol. 539, edited by A. Borowiec, W. Cegla,
B. Jancewicz, and W. Karwowski (Springer, Berlin, 2000)

Eugene.

5. Sep 17, 2007

### jostpuur

What an interesting paper! He talks about the same square root operator that we did in our debate some time ago. But I must disagree already with his proposition 2.1, though.

6. Sep 18, 2007

### meopemuk

Why do you disagree with proposition 2.1? It basically says that a wave function localized at t=0 spreads out superluminally at later times. The same statement was made by Hegerfeldt, Peskin & Schroeder, and many others. Looks like a proven fact to me.

Eugene.

7. Sep 18, 2007

### Hans de Vries

The analytical Green's function of the Klein Gordon propagator has no propagation
outside the light cone.

$$\Theta(t) \left(\ \frac{1}{2\pi}\delta(s^2)\ + \frac{m}{4\pi s} \Theta(s^2)\ \mbox{\huge J}_1(ms)\ \right), \qquad \mbox{with:}\ \ \ s^2=t^2-x^2$$

Where Theta is the Heaviside step function and J1 is the Bessel J function of the
first order. The Theta at the left selects the forward propagating half while the
other cuts off any propagation outside the light cone.

If you read on a bit in Peskin and Schroeder then you see they later claim that there
is no causal propagation outside the lightcone. Although the argument they use is not
that popular.

This subject has been discussed extensively here, see for instance my posts here:

https://www.physicsforums.com/showpost.php?p=1278078&postcount=6

The latter has some more back ground information.

The propagator (Green's function) for the Klein Gordon equation in any d-dimensional
space can be derived as:

$$\mbox{\Huge G}_d^{KG}{(t,r)}\ =\ \frac{1}{2\pi^a}\ \frac{\partial^a }{\partial (s^2)^a} \left\{\ \Theta(s^2) J_o(ms)\ \right\}$$

Where:

$$a=(d-1)/2, \qquad s^2=t^2-r^2$$.

There is no propagation outside the light cone at any dimension.

Regards, Hans

Last edited: Sep 19, 2007