Re: How active is research on other quantum gravity theories than

On Tue, 13 Sep 2005 06:17:21 +0000, Eugene Stefanovich wrote:

> J. Horta wrote:
>> On Sat, 10 Sep 2005 15:10:29 +0000, Eugene Stefanovich wrote:
>>
>>
>>>In my opinion, the fundamental problem facing modern theoretical physics is
>>>the deep contradiction between quantum mechanics and Einsteinian
>>>relativity (both special and general relativity). In Einsteinian relativity
>>>space and time are interchangeable. In quantum mechanics,
>>>position is an observable that depends on the state of the system, and time
>>>is a numerical parameter. A big difference.
>>>
>>>Eugene.
>>
>>
>> Is this strickly true in Quantum Field Theory? As far as I can tell
>> quantum fields are defined as operator valued distributions parameterized
>> by space and time. Space and time appear on an equal footing in QFT and
>> are not operators as say partical position x in (first quantized) QM.
>> I don't see the conflict.
>
> There is a conflict. Just try to follow this logic:
> 1. We are trying to describe observables (like position) in
> multiparticle systems. This is true for both QM (=fixed number
> of pareticles) and QFT (=variable number of particles).
> 2. Then we need to have a position operator R_i for each particle i in
> the system. Then, the state of an n-particle system can be described
> as a position-space wave function, i.e., a complex function
> on the common spectrum of eigenvalues of n commuting operators R_i:
> phi(r_1, r_2, ..., r_n)
> 3. Question: how these position operators are defined in QFT?
>

Thanks for the reply. First I don't see the burning need for R_i operators
in the field theory case. Also, it is less clear that such operators have
an unambiguous definition given particle creation and annihilation. At low
energies where it makes sense to talk about a fixed number of particles
let me write the QM expression on the left and the corresponding QFT one
on the right. The QM 1 particle wave function is

<x|psi> = <0|phi(x)|psi>

where phi(x) is the annihilation operator and |psi> is a single particle
state vector. Wouldn't a location operator for this single particle be
something like

<x|x|psi> = <0|x phi(x)|psi> ?

For two particles we would have

<x_1,x_2|psi_1,psi_2> = <0| x_1 psi(x_1) x_2 psi(x_2) |psi_1,psi_2>

<snip>

Let me respond to the rest when I get longer to think.

Thanks
Paul C.

> Eugene.

.

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