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

Eugene Stefanovich wrote:
> Igor Khavkine wrote:
> > Eugene Stefanovich wrote:

> >>You wrote
> >>
> >>|x> = phi*(x)|0>
> >>
> >>This formula implies that phi*(x) is a creation operator
> >>for the electron with definite position x. This is not true.
> >>The Dirac field phi*(x) cannot be interpreted in this way.
> >
> > It is indeed interpreted this way. I've already explained this in the
> > past. But you don't have to believe me in particular. These people say
> > the same thing as well:
> >
> > Dirac, _Principles of QM_, Sections 59-65
> >
> > Peskin & Schroeder, _Intro to QFT_, Section 2.3,
> > especially around Equation 2.41
> >
> > Mahan, _Many-particle Physics_, Chapter 1
> >
> > Abrikosov et al., _QFT Methods in Statistical Physics_, Chapter 2,
> > Section 6
> >
> > There are many others, these are the ones I could check off hand.
>
> My statement was about relativistic electron-positron quantum field
> psi(x).

While my statement was about the field operators in any field theory.

> Mahan's and Abrikosov's books are about non-relativistic QFT in
> condensed matter physics, so they are not relevant for this discussion.
> I couldn't find Peskin & Schroeder in my library.

QFT has a range of application much wider than just QED. And in all
instances, its application is the same. This includes relativistic,
non-relativistic and condensed matter applications.

> There is no discussion of electron-positron field in sections 59-65
> of the Dirac's books. [...]

That is not why I pointed to those sections. They contain the essense
of second quantization, the translation between wave functions and
field operators. The field operators serve precisely the role of
creation/annihilation operators.

> Besides simply the number of components, there is another reason
> why psi(x,t) cannot be interpreted as an operator analog of
> the position-space wave function.

>>From what you've said, I can only conclude that your definition of wave
function is not the same as other people's. So if you want to
contradict what I wrote at the top of this post, you'd have to specify
what you are actually contradicting.

> As you know, in QM wave functions
> are defined on the sets of eigenvalues of mutually commuting operators.
> Boost transformations of the Dirac's field mix x and t arguments
>
> psi(x,t) --> L psi(x', t') (1)
>
> where L is a 4x4 matrix, and x', t' are related to x,t by Lorentz
> formulas. In QM this implies
> existence of the operator of time T (whose eigenvalue is t) that
> commutes with the operator of position X, but does not commute with
> the generator of boost.
>
> In QM, there is no operator of time. t is just a numerical parameter,
> and correct boost transformation of the position-space wavefunction
> must be
>
> psi(x,t) --> L' psi(x'', t) (2)
>
> Of course, L' and x'' in (2) are different from L and x' in (1).

Of course they are not the same. L and L' differ by a judicious
application of time translation, which is accomplished by the
Hamiltonian. There is no contradiction here.

Igor

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