Second route to a relativistic quantum theory

[Orbb]

Hey everyone,

from Srednicki's QFT textbook I've read that there has been work on the following attempt to obtain a relativistic quantum theory: instead of treating space as a label like time, promote time to an operator, such that you get 4-position and -momentum operators. Now I know that there are issues to address, the potential unboundedness of the Hamilton operator and such, but still I would be intrigued to take a look at these attempts. I did a search, but couldn't find anything. So does anybody now of some resources to point me to?

If you're interested, said passage can be found on p.10 of Srednicki's book, which can be found via google book search.

Looking forward to your responses!

 

[Fredrik]

It's not hard to include a link.

I don't have any references for you. I just want to say that these attempts seem misguided to me. I like the approach taken in some books about the more mathematical aspects of quantum theory, where "observables" are defined operationally, as equivalence classes of measuring devices, and we take one of the axioms of QM to be that the observables are represented mathematically by bounded self-adjoint operators. A "measuring device" is supposed to be a physical system that interacts with the system on which we want to perform a measurement, and the "measurement" is such an interaction. (It's an interaction that entangles the eigenstates of the operator representing the observable with macroscopically distinguishable states of the measuring device). My problem with the time-operator approach is that clocks are fundamentally different from measuring devices of the type I just described. The difference is that they don't interact with the system! How can clocks be considered an observable on the system when they don't even interact with it?

(Yes, we can of course set if up so that the clock is started and stopped when the system participates in two specific interactions, but doesn't this just define two measuring devices, each designed to detect a specific interaction, and each one represented by a self-adjoint operator?)

Now Demystifier will disagree with me.

 

[meopemuk]

My problem with the time-operator approach is that clocks are fundamentally different from measuring devices of the type I just described. The difference is that they don't interact with the system! How can clocks be considered an observable on the system when they don't even interact with it?

Fredrik,

well said. I agree with you completely.

Eugene.

 

[genneth]

My problem with the time-operator approach is that clocks are fundamentally different from measuring devices of the type I just described. The difference is that they don't interact with the system! How can clocks be considered an observable on the system when they don't even interact with it?

(Yes, we can of course set if up so that the clock is started and stopped when the system participates in two specific interactions, but doesn't this just define two measuring devices, each designed to detect a specific interaction, and each one represented by a self-adjoint operator?)

If one was interested in a quantum gravity theory, then clocks do interact... But the fact that they don't in normal QFT should not be surprising. After all, the usual formalism treating time and space as commuting numbers can be seen as equivalent to having all other operators commuting with the proposed time and space operators. It is then far easier to treat them as labels then to keep talking about them as operators.

 

[Dickfore]

I had the idea to promote time to an observable before I started reading Srednicki's text. I was also puzzled by his note and could not find any references.

I was thinking about the physical implications behind this and decided that clocks have to be 'quantized' somehow, but I could not define an operational definition of a clock.

This is the idea I had: Using field operators in QFT seems like the Eulerian description of the motion in continuum mechanics. The approach we should follow in the alternative formulation would be analogous to the Lagrangian description. Of course, since the concept of a trajectory has no meaning in QM, we must use some probabilistic description, but this is currently beyond me.

 

[Demystifier]

from Srednicki's QFT textbook I've read that there has been work on the following attempt to obtain a relativistic quantum theory: instead of treating space as a label like time, promote time to an operator, such that you get 4-position and -momentum operators. Now I know that there are issues to address, the potential unboundedness of the Hamilton operator and such, but still I would be intrigued to take a look at these attempts. I did a search, but couldn't find anything. So does anybody now of some resources to point me to?

Here is an example of such an approach:
http://xxx.lanl.gov/abs/0811.1905 [Int. J. Quantum Inf. 7 (2009) 595]

 

[Demystifier]

Now Demystifier will disagree with me.

Am I so predictable?

 

[my_wan]

My problem with the time-operator approach is that clocks are fundamentally different from measuring devices of the type I just described. The difference is that they don't interact with the system! How can clocks be considered an observable on the system when they don't even interact with it?

Fredrik,

well said. I agree with you completely.

Eugene.

A clock may not interact with the system, but in some sense the clock is defined by the system. The clock is merely a separate proxy for the internal clock states of the system, on the assumption that the relative event rates ratios are generally consistent.

 

[LukeD]

A clock itself is a system, so isn't a reading of a clock still a reading of some observable? It is just a system that we understand very well that moves very regularly.

 

[Geigerclick]

Am I so predictable?

You're a Bohmian, so you have a certain trajectory.

Kidding aside, thanks for the excellent link.

 

[A. Neumaier]

Hey everyone,

from Srednicki's QFT textbook I've read that there has been work on the following attempt to obtain a relativistic quantum theory: instead of treating space as a label like time, promote time to an operator, such that you get 4-position and -momentum operators. Now I know that there are issues to address, the potential unboundedness of the Hamilton operator and such, but still I would be intrigued to take a look at these attempts. I did a search, but couldn't find anything. So does anybody now of some resources to point me to?

He probably meant work such as

M. C. Land, L. P. Horwitz
Off-Shell Quantum Electrodynamics
http://arxiv.org/abs/hep-th/9601021

(and much other work by Horwitz). It looks mathematically elegant, but incapable of reproducing the standard results of QED.