Ответ: Yes, I am here. I do not modify the site because it is good enough. :)
As regards divergence and re(?)normalization. Relativistic canonical quantization is a method of quantization of _free_ fields. It can be developed to quantize formal rows of classical perturbation theory. In this context it can be considered as a method of derivation of (formal) Dyson expansion. This formal expansion should be the same as already known. So, it is unlikely that something will substantially change with divergencies. But somebody should demonstrate all these things plainly.

Ответ: Грубо говоря, я лишь утверждаю, что квантовая теория поля, в той части, которую мы умеем математически ясно формулировать, является надстройкой над классической теорией поля. Но это скорее математический факт, чем физическая концепция. (И даже если этот факт кажется "физически абсурдным", он не перестаёт быть фактом.)

Что касается физики. Среди современных физиков считается хорошим тоном "верить", что будто-бы вся квантовая физика "должна" выводиться из квантовой теории поля. Никакого "спокойствия", однако, по данному поводу не наблюдается, т. к. тому нет убедительных математических доказательств. Это лишь, с позволения сказать, некоторая религиозно-философская доктрина, которой лично я не придерживаюсь. (И по этой причине упомянутый выше факт мне "физически абсурдным" не кажется.)

Ответ: PE:
How would you represent Einstein''s imagination (1905) of the electromagnetic field to consist of
spatially and temporarily granular elements? Doesn''t it correspond to the experimental finding of
obtaining spatially bounded spots within bounded time intervals?

DA:
1. As you can see in [VI] ( http://daarb.narod.ru/qed-eng.html ),
there is a one-to-one correspondence between states of the classical e-m field
and one-photon states of the quantized e-m field.
2. For the classical e-m field we have such an approximation as
geometrical optics.
3. According to 1, a similar approximation works for one-photon states.
4. So, if precision of your measurements is not very high, in some experiments
photons can behave like point particles.
5. But when precision of measurements grows, we find that there is nothing
so simple.
6. It is very typical for relativistic quantum systems that they do not
have such an observable as "coordinate in a fixed moment of time". Even
electrons do not have good coordinates.

PE:
The requirement the point-mechanical Hamiltonian to be gauge-invariant
offers a quantization method.
If you replace the momentum p(t) with p(t)+grad chi(x,t) and the potential V(x,t) with
V(x,t)-dchi(x,t)/dt, the Hamiltonian equations of motion remains unchanged, but the Hamilton function
depends on chi(x,t). But a state function should be free of arbitrariness. This dependence is absent
in the state function <psi|H|psi>/<psi|psi>.

DA:
By the way, in the context of my papers "quantization of Hamiltonian"
means the following:
1. A classical field can be moved in time (active transformation). So,
it has the generator of infinitesimal shifts in time - "Hamiltonian".
2. The classical field can be RCQ-quantized. The quantized field can
also be moved in time. So, it has the generator of infinitesimal
shifts in time - "Hamiltonian".
3. The quantum Hamiltonian can be considered as "quantization" of the
classical one, but there is no direct connection between them.

It is also important that in RCQ for construction of the quantized
e-m field we use the requirement of relativistic invariance. And
this requirement turns out to be so strong, that we even do not
need other invariances for quantization. So, the gauge invariance
is not the central part of RCQ. (Though, some difficult questions
about gauge invariance are automatically solved.)

PE:
I''m still studying scalar fields in 1D and the e-m field. The latter with particles, because
they are necessary for axiomatically deriving the Maxwell-Lorentz equations.

DA:
But how do you consider the particles?
If they are classical, they can not properly interact with the
quantized e-m field...
If they are quantum-mechanical, they are not relativistic...

By the way, if the word "axiomatically" refers to something like
axiomatic QFT, I am quite sceptical about that science. People
who work there pretend to be smarter than God. :-)

Ответ: > Dear Dr. Arbatsky,

> I have started to read your articles about RCQ and am very
> enthusiastic w.r.t. the methodological sides of your way of field
> quantization.

> As pointed out by Schleich (Quantum Optics in Phase Space,
> Ch.10), the normal mode quantization involves a different treatment
> of temporal and spatial components, so that one can hardly speak
> about particles within such a framework.

> Following Einstein (Planck''s Radiation Law and the Specific Heat
> of Solids, 1907), quantization is about selecting the set of quantum
> states out off the set of classical states of a system of particles
> (note that Einstein used Newton''s rather than Laplace''s notion of
> state). Can one say the same about fields?

1. "one can hardly speak about particles within such a framework".
Yes. If you imply that the word "particle" means a point object
travelling in space-time, then quantum field theory (and RCQ)
does not describe such objects.

2. "quantization is about selecting the set of quantum states
out off the set of classical states"
I would say that in our days it is misleading to say so.
Quantized field is a more complicated object than classical field.
So, quantizaton implies complication and enrichment of structure,
not selection of a part.
On the other hand, a quantum field has some properties that
are very close to the properties of the corresponding classical
field. So, in some situations you can even pretend that you
study some specific states of the classical field.

> When the harmonic oscillator is quantized, the energy becomes
> proportional to its frequency. The frequency is an internal
> parameter of the oscillator as it does not depend on external
> influences like the initial position and velocity. For a wave,
> however, the frequency is an external parameter, while the internal
> parameter is the speed of propagation, corresponding to possible
> periodicity of the field variable, u(x,t), in both space and time.
> This suggests that du/dx and du/dt rather than u and du/dt should be
> considered as pairs of independent dynamical variables to enter the
> Lagrange and Hamilton formalism, respectively. What do you think?

1. In RCQ we quantize a field as a whole object, not as separate
plane waves. (Though, it turns out to be convenient to use
Fourier transformation for analysis.)
2. If you want to study some properties of a field by studying
the properties of the so called "field oscillator", then you
can use it as it was introduced in
[IV, http://daarb.narod.ru/qed-eng.html]. But you should
understand that in my papers it is a more abstract thing
than a normal mode of the field in a box. (And this abstract
thing better describes symmetries of the field.)
3. The field oscillator, introduced as "inducing Hamiltonian
system", is not a usual object in our space-time. (The "time"
of that oscillator is dimensionless.)

> On your FAQ page you write "that the method of RCQ allows to look
> completely differently at the questions of gauge invariance and at
> the problem of arbitrariness in the choice of Lagrangian". Is this
> related to the use of the Hamilton formalism? How do you look at the
> relationship between Lagrangean (description of motions, incl. their
> symmetries) and Hamiltonian (description of states, incl. their
> symmetries)? Within nonrelativistic wavemechanics (SchrЎdinger
> theory), motion is realized by transitions between stationary states
> (SchrЎdinger 1926). If this is true for relativistic field theory,
> too, then the symmetries of the Lagrangean and of the Hamiltonian
> should be closely related to each other, shouldn''t they?

You are certainly right. This topic was discussed in [II,III].

Note. When we talk about "Hamiltonian" in relativistic field
theory, it is better to imply, generally, a generator of any
infinitesimal transformation from Poincare group, not only
of shift in time.

In [III] I suggested a general formula for such a
generator, where it is expressed by symplectic structure
on invariant phase space. This symplectic structure, on the
other hand, is defined by variations of action, i. e. by
Lagrangian.

So, it is better to say that symmetries of Lagrangian lead
to existence of some generators of infinitesimal canonical
transformations of invariant phase space.

> The point-mechanical Lagrangean is gauge-invariant (modulo total
> time-derivative), while the Hamiltonian is not. The requirement the
> point-mechanical Hamiltonian to be gauge-invariant offers a
> quantization method. Does this work for the field Hamiltonian, too?

"The requirement the point-mechanical Hamiltonian to be
gauge-invariant offers a quantization method."
What do you mean?

And what do you imply when you write here about "the field
Hamiltonian"? Hamiltonian of e-m field? Or electron-positron
field in external classical e-m field? Or e-m field and
electron-positron field interacting with each other?

In my papers, when I talk about a field, I, first of all,
imply the free e-m field. And study how to quantize IT,
even without particles.

> Relativistic and nonrelativistic point mechanics can be
> distinguished through the (in)dependence of the state change on the
> state, v (v - velocity, Euler''s state variable). Euler has derived
> Newton''s equation of motion from the ansatz dv=F*dt/m (F - external
> force, m - mass), ie, for the case of independence of df of v. In
> contrast, the (special-)relativistic equation of motion can be
> obtained from the ansatz d(f(v)v)=F*dt/m, ie, in case of dependence
> of dv on v (Suisky & Enders 2005). Can one carry over this
> difference to field theory?

The answer is "no", but the question is "incorrect".
There is no such thing as "quantization of point particle"
in quantum field theory. All relativistic quantum systems
are quantizations of classical FIELDS, not of point
particles.

On the other hand, if we consider a relativistic field
and consider some well-localized wave-packets, they can
be sometimes well described as point particles. And
equations of motion turn out to be the same as equations
of motion of the corresponding relativistic particles.

> Thank you very much in advance and best wishes for 2006,

The same to you.

> Peter Enders

> Dr. Peter Enders
> Fischerinsel 2
> D - 10179 Berlin
> Germany

> PS: Dirac''s requirement of novel maths for that is preceded by
> SchrЎdinger''s requirement of novel maths for solving the stationary
> SchrЎdinger equation (Quantization as Eigenvalue Problem, Second
> Communication, 1926) (eigenvalue methods belong to the ''old'' maths
> of classical physics).

> PS2: You may write me in Russian and send me articles in Russian
> (if not available on the web - often I face timeout difficulties
> when downloading the pdf versions, however), though your English is
> excellent.