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 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.

 

[strangerep]

I have been trying to work out the mathematical details of H Snyder's 1947 paper, titled 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?

I attempted an answer to your question over on sci.physics.research. But later, I doubted
whether I had properly understood the real point of your question. Maybe if you
elaborate your question a bit more, better answers might be forthcoming.

 

[arkobose]

Thanks. OK, I shall try to better elaborate my point.

The Lorentz transformation equations are, with proper choice of axes are given as in this page: http://en.wikipedia.org/wiki/Lorent...ormation_for_frames_in_standard_configuration.

The presence of the parameter v for speed is conspicuous in the equations.

My question is, how do I define v in the context of Snyder's paper, to verify that the paramters x, y, z and t are Lorentz invariant?

Hope this helps.

Thanks.

 

[strangerep]

The Lorentz transformation equations are, with proper choice of axes are given as in this page: http://en.wikipedia.org/wiki/Lorent...ormation_for_frames_in_standard_configuration.

The presence of the parameter v for speed is conspicuous in the equations.

That Wiki page doesn't explain clearly that Lorentz transformations are defined by the
property of preserving the spacetime interval. Look a bit further down that page and
you'll see:

$$s^2 = -c^2(\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2$$

A Lorentz transformation preserves $s^2$. Only a subset of these transformations
(the "boosts") involve velocity. Unfortunately, those are the ones that appear first on the
Wiki page.

My question is, how do I define v in the context of Snyder's paper, to verify that the parameters x, y, z and t are Lorentz invariant?

Snyder is trying to define operators on a Hilbert space which could correspond to the
ordinary notion of position -- in some physically sensible way. In particular, he is trying
to find position-time operators, which I'll call X,Y,Z,T (even though Snyder calls them x,y,z,t
which are then too easy to confuse with their eigenvalues).

You don't need to "verify that the parameters x, y, z and t are
Lorentz invariant". I suspect you're mis-reading Snyder's sentence just before his
eqn(2) where he says "To find operators x, y, z and t possessing
Lorentz invariant spectra, we consider [...]". The key word here is "spectra", i.e: the
set of eigenvalues. The set of all the eigenvalues must be closed under the action
of the Lorentz generators on the corresponding operators X,Y,Z,T.

If $L_{\mu\nu}$ are the generators of a Lorentz transformation, Snyder
must show that $[L_{\mu\nu}, S^2] = 0$, where $S^2 := -c^2T^2+X^2+Y^2+Z^2$.
He must also show that

$$[L_{\rho\sigma}, X_\mu] = i(g_{\mu\sigma}X_\rho - g_{\mu\rho}X_\sigma)$$

while also having a similar commutation relation between $L_{\mu\nu}$ and
$P_\mu$ (the 4-momentum translation generator). He also needs a commutation
relation like $[X_\mu, P_\nu] = i g_{\mu\nu} I$ -- to make contact with ordinary QM.
Oh, and he also needs to show that the operators are Hermitian (if they are to
represent observable quantities).

That's enough to show that one has a set of operators that form a plausible quantum
version of the usual Minkowski space. You don't need an explicit representation of
the velocity to achieve this.

BTW, the above is called the "Heisenberg-Poincare" group, and there are far more
modern treatments. Snyder's tedious treatment is based on representation by
differential operators on a DeSitter space. For more modern papers, see for example:

hep-th/0410212 (Chryssomalakos & Okon) and also the Mendes references therein.

If you google for "Heisenberg-Poincare" you'll probably find more stuff. You
could also use Google Scholar to find more modern papers which cite
Synder's paper in their references.

Related work is known by the (dreadfully misleading) names of "doubly-special"
and "triply-special" relativity.

That's the limit of the help I can offer on this subject. If you need more info about
the Lorentz group, such questions should probably be asked over on the relativity
forum, or maybe the quantum physics forum.