On Snyder's paper on quantized space-time

arkobose

Hello,
I have been trying to work out the mathematical details of H Snyder's
1947 paper, titled Quantized Space-Time
(http://prola.aps.org/abstract/PR/v71/i1/p38_1),
and I am stuck at something.

When the space-time variables are considered as Hermitian operators,
and we need to verify that they satisfy Lorentz invariance, I believe
we need the quantity speed in the Lorentz transformation equations. My
question is, in the context of Snyder's paper, how do we define speed?

Further, if speed is not required, then how do we prove the Lorentz
invariance of these operators?

Please do guide me on this, if you have an idea of what I am talking
about.

Thank you.

sr

arkobose wrote:

> I have been trying to work out the mathematical details of H Snyder's
> 1947 paper, titled Quantized Space-Time
> (http://prola.aps.org/abstract/PR/v71/i1/p38_1),
> and I am stuck at something.
>
> When the space-time variables are considered as Hermitian operators,
> and we need to verify that they satisfy Lorentz invariance, I believe
> we need the quantity speed in the Lorentz transformation equations. My
> question is, in the context of Snyder's paper, how do we define speed?
>
> Further, if speed is not required, then how do we prove the Lorentz
> invariance of these operators?

Lorentz transformations are defined by their property of preserving
the usual indefinite quadratic form, i.e: eqn(1) in Snyder's paper.

S^2 = c^2 t^2 - x^2 - y^2 - z^2 (1)

In ordinary SR, we find that some of these transformations (the
boosts)
can be expressed in terms of velocity. But the fundamental property
is preservation of the above form (1).

Snyder takes exactly this approach. He considers a larger De Sitter
space with an extra "\eta_4" variable, and looks for transformations
which leave "\eta_4" invariant, as well as preserving his eqn(2)
(i.e., the quadratic form in De Sitter space).

HTH.