Передайте математикам, что они могут спать спокойно. Ничего интересного в статье товарища Дынина нет. Я в деталях не разобрался (там много жаргона из функционального анализа, с которым я незнаком), но никаких новых физических идей там нет. Первая половина статьи - обзор, потом идет обычное каноническое квантование, но с ошибками (игнорируя связи, т.е. закон Гаусса, и калибровочную инвариантность).
Comrade Leblon admits and shows that he does not understand my paper. He has no idea what is and what is not there. Large part of the paper is exactly about constraints and gauge invariance translated to the initial data of Yang-Mills equations.
I have had rather hot exchanges with Faddeev, Slavnov and other serious physicists during my April visit to Moscow and St. Petersburg (to the both Steklov divisions). But their concerns have been different though also caused by misapprehension. I have placed a streamlined 2nd version of my paper on arXive to address their concerns. (The bracketed numbers below are references from the 2nd version.) Also I have added an appendix about quantum Yang-Mills dynamics and a discussion of asymptotic freedom coming from
The first concern was how I break the conformal invariance of the Yang-Mills action and equations to get the physical mass dimension.
Conformal symmetry can be broken, e. g., by Higgs mechanism or by quantization In particular, in the perturbation quantization this is done via "dimensional transmutation" of the coupling constant [13]. However my proof is both Higgsless and non-perturbative. Thus my answer is that there is no coupling constant in my paper.
I follow I. Segal's quantization of (constrained) Cauchy data for Yang-Mills fields along with Schwinger's postulate to quantize Noether functionals, in particular, the conserved energy-momentum relativistic vector. The latter has the mass physical dimension. (So in some sense I got a "dimensionful costant" which is the conserved energy-mass.)
The second concern was about the quantization per se.
I adopt P. Kree's quantization scheme [21] where quantum operators are continuous linear operators in a 2nd quantized configuration Gelfand triple but not fields of selfadjoint operators in an unknown Hilbert space. However, I differ from Kree in two ways:
1. I use the Bargmann-Segal transform of Hida Gelfand triple of white noise calculus [24].
2. I consider the Noether functionals as the anti-Wick (aka Berezin) symbols of an infinite-dimensional "pseudodiferential operator" in a Bargmann-Fock Gelfand triple over the standard Schwartz triple on $\mathbb{R}^3$. Characteristically, the quantized operators are not selfadjoint in Bargmann Hilbert space (the middle space in the triple): they are even not densely defined there and von Neumann theory of self-adjoint operators is not applicable. Still the operators are symmetric.
Now, whatever your definition of the spectrum of a symmetric operator would be, it should imply the variational mini-max property. So I define the spectrum of operators in Gelfand triples by this property.
The anti-Wick quantization is very compatible with such definition, in contrast with the Wick quantization where a quantized operator with non-negative symbol may be unbounded from below [16].
There is a serious obstacle though. Even for the number operator all positive eigenvalues have infinite multiplicity. To proceed further than the 2nd eigenvalue I use in the mini-max principle the space dimensions relative the abelian von Neumann algebra generated by spectral projectors of the number operator.
The von Neumann spectrum of the Yang-Mills energy-mass operator is infinite and discrete. Finally, the usual mass gap equals to von Neuman mass gap and so the former is positive. QED.
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branchandroot
Off-topic
Date: 2009-04-17 12:52 am (UTC)А то математики вдруг заволновались.
Re: Off-topic
Date: 2009-04-17 01:24 pm (UTC)Re: Off-topic
Date: 2009-04-17 08:30 pm (UTC)Re: Off-topic
Date: 2009-04-18 07:02 am (UTC)Re: Off-topic
Date: 2009-04-18 07:09 am (UTC)Re: Off-topic
Date: 2009-04-18 08:23 am (UTC)Re: Off-topic
Date: 2009-04-18 07:10 am (UTC)(На всякий случай: одним из этих физиков был Фейнман.)
Re: Off-topic
Date: 2009-04-18 08:24 am (UTC)Re: Off-topic
Date: 2009-06-14 03:16 pm (UTC)Comrade Leblon admits and shows that he does not understand my paper. He
has no idea what is and what is not there. Large part of the paper is
exactly about constraints and gauge invariance translated to the initial
data of Yang-Mills equations.
I have had rather hot exchanges with Faddeev, Slavnov and other serious
physicists during my April visit to Moscow and St. Petersburg (to the both
Steklov divisions). But their concerns have been different though also
caused by misapprehension. I have placed a streamlined
2nd version of my paper on arXive to address their concerns. (The
bracketed numbers below are
references from the 2nd version.) Also I have added an appendix about
quantum Yang-Mills dynamics and a discussion of asymptotic freedom coming
from
The first concern was how I break the conformal invariance of the
Yang-Mills action and equations to get the physical mass dimension.
Conformal symmetry can be broken, e. g., by Higgs mechanism or by
quantization In particular, in the perturbation quantization this is
done via "dimensional transmutation" of the
coupling constant [13]. However my proof is both Higgsless and
non-perturbative.
Thus my answer is that there is no coupling constant in my paper.
I follow I. Segal's quantization of (constrained) Cauchy data for
Yang-Mills fields along with Schwinger's postulate to quantize Noether
functionals, in particular, the conserved energy-momentum relativistic
vector. The latter has the mass physical dimension. (So in some sense I
got a "dimensionful costant" which is the conserved energy-mass.)
The second concern was about the quantization per se.
I adopt P. Kree's quantization scheme [21] where quantum operators
are continuous linear operators in a 2nd quantized configuration Gelfand
triple but not fields of selfadjoint operators in an unknown Hilbert
space. However, I differ from Kree in two ways:
1. I use the Bargmann-Segal transform of Hida Gelfand triple of white
noise calculus [24].
2. I consider the Noether functionals as the anti-Wick (aka Berezin)
symbols of an infinite-dimensional "pseudodiferential operator" in a
Bargmann-Fock Gelfand
triple over the standard Schwartz triple on $\mathbb{R}^3$.
Characteristically, the quantized operators are not selfadjoint in
Bargmann Hilbert space (the middle space in the triple): they are even not
densely defined there and von Neumann theory of self-adjoint operators is
not applicable.
Still the operators are symmetric.
Now, whatever your definition of the spectrum of a symmetric operator
would be, it should
imply the variational mini-max property. So I define the spectrum of
operators in Gelfand triples by this property.
The anti-Wick quantization is very compatible with such definition, in
contrast with the Wick quantization where a quantized operator with
non-negative symbol may be unbounded from below [16].
There is a serious obstacle though. Even for the number operator all
positive eigenvalues have infinite multiplicity. To proceed further than
the 2nd eigenvalue I use in the mini-max principle the space dimensions
relative the abelian von Neumann algebra generated by spectral projectors
of the number operator.
The von Neumann spectrum of the Yang-Mills energy-mass operator is
infinite and discrete. Finally, the usual mass gap equals to von Neuman
mass gap and so the former is positive. QED.