Physicists tried to get rid of ultraviolet divergences in quantum
field theories, in particular in QED, for a long time.
Without much success. The divergences are still there.
I think I know how to eliminate them from the theory.
Wanted to know your opinion.
All this started in the late 1920's when Heisenberg and Pauli
invented QFT. The Hamiltonian H of QED was written and it was
soon discovered that scattering amplitudes are described well in
the 2nd order of the perturbation theory and turn out to be
infinite in the 4th and higher orders.
Then came Tomonaga, Schwinger, and Feynman who said
(that's not what they actually said, but that's a rather
accurate translation of their words to a different language)
that Hamiltonian H is not correct and one must add certain
renormalization corrections R to H in each order of the
perturbation theory. The corrections were chosen so as to
make the calculated mass of the electron equal to the
experimental mass and to fit the lowenergy electronelectron
scattering to experimental data (charge renormalization).
The corrections (or counterterms) R appeared to be infinite,
but the Smatrix calculated with the new Hamiltonian H+R
turned out to be finite and perfectly agreed with experiment.
In calculations of the Smatrix the different infinities
happily cancelled each other.
This is what we have in the renormalized QFT today:
Infinite Hamiltonian and finite and accurate Smatrix.
Theoretically this is a disaster, but from the practical
point of view there is not much trouble, because in
high energy physics we care only about the energies of
bound states (poles of the Smatrix) and scattering crosssections
(squares of the matrix elements of the Smatrix). We will get
in trouble in the future when the resolution of experiments will
allow us to measure the time evolution of interacting particles
during their collisions. The Smatrix is not enough to describe
such time evolution. We need the Hamiltonian, but the Hamiltonian
is bogus.
My idea is that the Hamiltonian with counterterms H+R is
still wrong. We need to find a better Hamiltonian of QED,
let me call it H'. Apparently, this new Hamiltonian must satisfy
at least two conditions:
(1) it must be finite
(2) the Smatrix calculated with H' must be exactly the same as
the accurate Smatrix calculated with H+R.
The condition (2) is satisfied by unitary transforms of H+R. I.e.,
if we choose
H' = U(H+R)U^{1}
where U is a unitary operator satisfying some mild restrictions, we
can guarantee that H' yields exactly the same Smatrix as H+R.
Now, what about the finiteness? It appears that the class of
transformations U preserving the Smatrix is so large that we can
find transformations U which make H' finite.
These operators U are infinite (more
exactly, their phase is infinite), but we couldn't care less, because
U do not have any physical meaning. If all this is done with some
mathematical rigor, we obtain a FINITE Hamiltonian H' describing
interactions of charged particles and yielding a FINITE Smatrix.
Now we can forget about preliminary bad Hamiltonians H and H+R, and
about the infinite transformation operator U. We have a complete
theory with the Hamiltonian H' which allows us to calculate time
evolution, scattering,
bound states, etc. according to usual rules of quantum mechanics
without regularization, renormalization, and other tricks.
The details of the derivation of the true QED Hamiltonian H' are given
in my paper
E. V. Stefanovich, `` Quantum Field Theory without Infinities''
Ann. Phys. (NY) 292 (2001), 139156.
As a sanity check, in the 2nd perturbation order in the v^2/c^2
approximation, I obtained the wellknown Breit Hamiltonian for
the electronelectron interaction. 4th and higher order radiative
corrections can be also readily obtained.
"Eugene" <[Only registered users see links. ]> wrote in message
news:[Only registered users see links. ]...
 Physicists tried to get rid of ultraviolet divergences in quantum
 field theories, in particular in QED, for a long time.
 Without much success. The divergences are still there.
 I think I know how to eliminate them from the theory.
 Wanted to know your opinion.

 All this started in the late 1920's when Heisenberg and Pauli
 invented QFT. The Hamiltonian H of QED was written and it was
 soon discovered that scattering amplitudes are described well in
 the 2nd order of the perturbation theory and turn out to be
 infinite in the 4th and higher orders.

 Then came Tomonaga, Schwinger, and Feynman who said
 (that's not what they actually said, but that's a rather
 accurate translation of their words to a different language)
 that Hamiltonian H is not correct and one must add certain
 renormalization corrections R to H in each order of the
 perturbation theory. The corrections were chosen so as to
 make the calculated mass of the electron equal to the
 experimental mass and to fit the lowenergy electronelectron
 scattering to experimental data (charge renormalization).
 The corrections (or counterterms) R appeared to be infinite,
 but the Smatrix calculated with the new Hamiltonian H+R
 turned out to be finite and perfectly agreed with experiment.
 In calculations of the Smatrix the different infinities
 happily cancelled each other.

 This is what we have in the renormalized QFT today:
 Infinite Hamiltonian and finite and accurate Smatrix.
 Theoretically this is a disaster, but from the practical
 point of view there is not much trouble, because in
 high energy physics we care only about the energies of
 bound states (poles of the Smatrix) and scattering crosssections
 (squares of the matrix elements of the Smatrix). We will get
 in trouble in the future when the resolution of experiments will
 allow us to measure the time evolution of interacting particles
 during their collisions. The Smatrix is not enough to describe
 such time evolution. We need the Hamiltonian, but the Hamiltonian
 is bogus.

 My idea is that the Hamiltonian with counterterms H+R is
 still wrong. We need to find a better Hamiltonian of QED,
 let me call it H'. Apparently, this new Hamiltonian must satisfy
 at least two conditions:

 (1) it must be finite
 (2) the Smatrix calculated with H' must be exactly the same as
 the accurate Smatrix calculated with H+R.

 The condition (2) is satisfied by unitary transforms of H+R. I.e.,
 if we choose

 H' = U(H+R)U^{1}

 where U is a unitary operator satisfying some mild restrictions, we
 can guarantee that H' yields exactly the same Smatrix as H+R.
 Now, what about the finiteness? It appears that the class of
 transformations U preserving the Smatrix is so large that we can
 find transformations U which make H' finite.
 These operators U are infinite (more
 exactly, their phase is infinite), but we couldn't care less, because
 U do not have any physical meaning. If all this is done with some
 mathematical rigor, we obtain a FINITE Hamiltonian H' describing
 interactions of charged particles and yielding a FINITE Smatrix.
 Now we can forget about preliminary bad Hamiltonians H and H+R, and
 about the infinite transformation operator U. We have a complete
 theory with the Hamiltonian H' which allows us to calculate time
 evolution, scattering,
 bound states, etc. according to usual rules of quantum mechanics
 without regularization, renormalization, and other tricks.

 The details of the derivation of the true QED Hamiltonian H' are given
 in my paper

 E. V. Stefanovich, `` Quantum Field Theory without Infinities''
 Ann. Phys. (NY) 292 (2001), 139156.
Is this the same as the one at
[Only registered users see links. ] ?
 As a sanity check, in the 2nd perturbation order in the v^2/c^2
 approximation, I obtained the wellknown Breit Hamiltonian for
 the electronelectron interaction. 4th and higher order radiative
 corrections can be also readily obtained.
Why is a cutoff not a _physical_ requirement? QED is not a theory
unto itself. It's a low energy effective theory of E&M.
[...]
It's only theoretically a disaster if you haven't considered the
validity of the assumptions that go into proving the ``disaster''. The
``no interaction'' theorem's (e.g., haag's theorem) depend upon several
assumptions that aren't neccessarily valid assumptions. One is the
unitarity of the Smatrix. Another is that the hamiltonian can be split
into terms containing free fields and interacting fields. Another
is whether or not the infinities are physically meaningful. In QED,
for example, the divergences cancel order by order and the result
of calculations do not depend upon the cut off procedure, only that
a cut off is imposed. If the cut off has a physical basis, then insisting
the theory not depend upon one would be ignoring the physics.
How do you know that? By your hypothesis, we have not made those
measurements. Don't you think that sort of puts the formalism
ahead of the data?
Where is this _general_ theory whose low energy approximation is QED?
Strings? quantum gravity? I do not consider QED as "effective theory"
of something nonexistent yet. I present you a finite, mathematically
sound theory of interacting particles today. We can now calculate
all physical properties (time evolution, scattering, bound states, etc.)
using normal rules of quantum mechanics (no renormalization tricks)
without ever encountering divergent integrals. I didn't write the
Hamiltonian H' in all perturbation orders (I stopped at the 2nd order in
QED and at the 4th order in a simple model theory),
but there is a proof that such a construction is possible, and there are
rules how to do it.
The disaster is that the Hamiltonian is infinite and we cannot
study time evolution.
I presume there is real timedependent dynamics behing scattering
events. I hope that some day we will be able to measure that
(For example, there is a rapid progress in timedependent dynamics of
wave packets made of Rydberg atomic states. Some day, these wave
packets will include lowenergy states affected by the Lamb shifts.
Existing theory cannot predict their time evolution.)
Unitarity is needed to preserve the good Smatrix which we got from
the usual renormalized QED.
The standard model. The electroweak sector is SU(2)_L x U(1)_Y,
where the SU(2)_L is given by a weak isotriplet of vector currents,
W_1, W_2, W_3, and U(1)_Y is a hypercharge singlet, B. QED is the
U(1)_em symmetry obtained from the generator Q = T_3 + Y/2, which
remains invariant in the electroweak ground state. The photon and
Z are linear combinations of W_3 and B. QCD is an asymptotically
free theory, so when you combine the electroweak and the qcd
lagrangian, you have a field theory which is believed (but not
proven) to be asymptotically free.
The standard model has been around for about 2025 years.
Actually, you haven't presented anything. You've just made some rather
general statements to which I've attempted respond in general terms
to the best I have been able to figure out what you are saying.
I think I missed something, since you appear to be claiming here, that
your model, like qed, is also perturbative. Perturbatively, there is no
problem with qed. I have a reference I was going to give that discusses
this, but I just discovered some and perhaps all of both volumes are
available on line from the institute for advanced study. The lecture
notes from Lecture 3 by david gross, esp. ppg 23,5 give the details:
[Only registered users see links. ]
In short, the ``nonexistence'' you are concerned about becomes an
issue for energies in the regime of 10^250 MeV, which is not an issue
for the universe qed inhabits.
You keep saying ``time evolution'', but I don't see the problem.
What do you expect to measure other than scattering amplitudes?
Are you expecting to observe something bizarre like single particle
states off mass shell (which would seem to be unphysical)?
The observables in the experiment being what?
It almost sounds like you are trying to cheat quantum mechanics.
That doesn't mean it's a good _physical_ assumption. It probably is,
at least for this purpose, but it might not be and insisting that it
must be just to preserve a feature in the formalism, seems to invert
the priority a bit. The only justification for anything is that it is
needed to explain real data.
Electroweak theory has the same divergences as QED. It just adds other
particles (W, neutrinos, etc.) and interactions into the mix. QED is not
a lowenergy approximation of the electroweak theory.
Sorry, I have no position on this one. My humble goal is to correctly
describe
the time evolution of two interacting electrons at energies below, say,
10 MeV.
"Eugene" <[Only registered users see links. ]> wrote in message
news:[Only registered users see links. ]...
Bullcrap. The electoweak theory is the theory that is supposed to operate
at energies above which the symmetry between the weak and EM interactions is
no longer broken and are part of a single theory that is not the same as QED
ie it has SU(2) x U(1) symmetry not the U(1) symmetry of QED. Obviously QED
is a low energy approximation to this theory. Also I never claimed it did
not also require renormalization  what I claimed is that your proposal is
for a problem that no longer exists. Of course that does not prove your
proposal wrong (I think they are a crock of crap but I will leave that for
experts to look into before passing a final opinion). Now if your proposing
a way to resolve the divergences of the electroweak theory without
renormalization that would be another matter  I still think your reasoning
is erroneous  but at least it would be addressing a problem that actually
exists.
GV
In that case since you are imposing a cutoff there is no problems with
divergences.
Yes, QED is a part of electroweak theory. Neither QCD nor electroweak
solved the divergence problem. They have the same (or similar)
divergence problems as QED. Actually, my approach to QFT is rather
general and applicable to QCD and electroweak as well.
I thought, when people say that QED is an "effective field theory"
they presume that there is some "grand theory" in which the
divergence problems are fixed, and which reveals new features
somewhere at the "Planck scale". QCD and electroweak are not these
"grand theories". Though, here, I admit, I am leaving my area
of expertise. Please, correct me if you know better.
It is all written in my paper in Ann. Phys. I don't think it is
practical to write here bunch of formulas. The idea of my approach
is very simple: One just need to make a unitary transformation of
the Hamiltonian and infinities are gone!
What do you mean there is no problem? You still need to perform
renormalization. Renormalization means that you introduce infinite
counterterms in the Hamiltonian. This is the problem.
I expect to measure time dependence of particle observables
(position, momenta, energies, spin projections). Currently,
in scattering experiments these observables change abruptly during
the collision. This is because the collisions are too short for
modern experimental techniques. I believe that with much improved
time resolution we will see what exactly happens in the short time
interval of the collision. Current QFT with its emphasis on the Smatrix
cannot predict this time dynamics. My modified version of QFT can.