# Dual of the inertial observer

1. Aug 16, 2009

### rwestafer

If an inertial "observer" or state has mass and no rotation, then a massless state with rotation (i.e. having maybe a generalized rotation such as "spin," e.g. a photon) seems to be dual to that state.

Would this viewpoint then take the photon as a "matching" channel or "process" for communication among inertial observers? This seems something like Lie algebra mapping to operator; like a gauge boson. If not a photon, perhaps a graviton. I am not well accustomed to the intricacies of group theory, but I mention it in an effort to be constructive and prompt discussion.

In communications and transmission line theory, for the optimum exchange of energy the channel must be "matched" to the source and receiver. This is why I stretched a bit to think of the dual of the inertial observer, representing a mapping between two observers.

References which might help clarify the point:
(Link to the Physics Forums definition of "inertial observer")
https://www.physicsforums.com/library.php?do=view_item&itemid=35
(Link to mappings of Lie algebras, Chapter 6)
http://www.physics.drexel.edu/~bob/LieGroups.html
http://en.wikipedia.org/wiki/Gauge_boson

Last edited: Aug 16, 2009
2. Aug 16, 2009

### JesseM

What does "dual intermediate" mean? And what do you mean when you call a massless particle a "representation"? Finally, what do you mean by "suitable"--do you think there are methods that inertial observers could use to communicate that would be "unsuitable" in some sense?

3. Aug 16, 2009

### rwestafer

JesseM,

I updated the post to clarify, and thanks for calling attention to detail.

4. Aug 17, 2009

### JesseM

Still not sure what you mean by "dual", or what you mean when you ask if the photon is the "matching" channel for communication among inertial observers. And you say:
Can you give some link or other reference on what you're talking about here? Does communication and transmission line theory say anything about the type of particle that should be used "for the optimum exchange of energy"?

5. Aug 23, 2009

### rwestafer

Maybe a more precise definition of "inertial observer" would help, but is there one? My assumption was the two characteristics given elsewhere on the forum: (a) the observer's frame does not accelerate or rotate, and (b) the observer has mass.
The term "dual" can be used in the mathematical sense if observer and observed are QM states. Otherwise the term is an abstraction of that concept suggesting the photon as an entity which neatly counters both criteria, as I think any gauge boson should do.
By the term "matching" I'm referring to the coupling of energy between source and receiver, the observed and the observer. Again, to briefly refer to QM measurement, consider the overlap integral (IUPAC) representing the probability of an interaction. If the probability is 1, the states are entirely "matched."

No, I don't have any references for this; it's a personal thought and interpretation.
Let's relate this to common ideas. Yes, communication theory does say something about a quite related point, the optimum channel "for the optimum exchange of energy." The optimum channel will provide maximum capacity (Shannon-Hartley theorem) to "symbols" (think of particles) of the correct type.
$$Capacity=Bandwidth\:\log\left(1 + \frac{Signal}{Noise}\right)$$​
For fixed bandwidth (i.e. relevant energies or frequency range), maximum signal-to-noise ratio (SNR) achieves the greatest capacity. That's the ratio of signal power to noise power. OK, so the desired channel for a source and receiver is one which achieves a large SNR. Now we've defined some things and should work toward the analogy of "particle" and "channel."

Though the source and receiver both appear as particles (matter-like) in this example, we often think of the channel as particle(s) itself. Conveniently, I think it can be shown that a state and its dual can form a particle-like operator (think of a virtual particle) which can mediate two other states as a sort of "channel."