Quantum states of hierarchic systems

The density matrix formalism which is widely used in the theory of measurements, quantum computing, quantum description of chemical and biological systems always imply the averaging over the states of the environment. In practice this is impossible because the environment U\setminusS of the system $S$ is the complement of this system to the whole Universe and contains infinitely many degrees of freedom. A novel method of construction density matrix which implies the averaging only over the direct environment is proposed. The Hilbert space of state vectors for the hierarchic quantum systems is constructed.Comment: LaTeX 11 pages, 1 eps figur

Quantum field theory without divergences

It is shown that loop divergences emerging in the Green functions in quantum field theory originate from correspondence of the Green functions to {\em unmeasurable} (and hence unphysical) quantities. This is because no physical quantity can be measured in a point, but in a region, the size of which is constrained by the resolution of measuring equipment. The incorporation of the resolution into the definition of quantum fields $\phi(x)\to\phi^{(A)}(x)$ and appropriate change of Feynman rules results in finite values of the Green functions. The Euclidean $\phi^4$-field theory is taken as an example.Comment: 6 pages, LaTeX, revtex, 2 eps figure

p-Adic wavelet transform and quantum physics

p-Adic wavelet transform is considered as a possible tool for the description of hierarchic quantum systemsComment: 12 pages, LaTeX, 2 eps Figures, Talk at 1st Int. Conf. on $p$-Adic Mathematical Physics, Moscow, Oct 1-5, 200

Consciousness and quantum mechanics of macroscopic systems

We propose the quantum mechanical description of complex systems should be performed using two types of causality relation: the ordering relation ($x\prec y$) and the subset relation ($A\subseteq B$). The structures with two ordering operations, called the causal sites, have been already proposed in context of quantum gravity (Christensen and Crane, 2005). We suggest they are also common to biological physics and may describe how the brain works. In the spirit of the Penrose ideas we identify the geometry of the space-time with universal field of consciousness. The latter has its evident counterparts in ancient Indian philosophy and provides a framework for unification of physical and mental phenomena.Comment: LaTex, 17 pages, 4 eps figure

Wavelet based regularization for Euclidean field theory and stochastic quantization

It is shown that Euclidean field theory with polynomial interaction, can be regularized using the wavelet representation of the fields. The connections between wavelet based regularization and stochastic quantization are considered with $\phi^3$ field theory taken as an example.Comment: LaTeX, 12 pages; 2 eps figures; To appear in "Progress in Field Theory Research" by Nova Science Publishers, In

Unifying renormalization group and the continuous wavelet transform

It is shown that the renormalization group turns to be a symmetry group in a theory initially formulated in a space of scale-dependent functions, i.e, those depending on both the position $x$ and the resolution $a$. Such theory, earlier described in {\em Phys.Rev.D} 81(2010)125003, 88(2013)025015, is finite by construction. The space of scale-dependent functions $\{ \phi_a(x) \}$ is more relevant to physical reality than the space of square-integrable functions $\mathrm{L}^2(R^d)$, because, due to the Heisenberg uncertainty principle, what is really measured in any experiment is always defined in a region rather than point. The effective action $\Gamma_{(A)}$ of our theory turns to be complementary to the exact renormalization group effective action. The role of the regulator is played by the basic wavelet -- an "aperture function" of a measuring device used to produce the snapshot of a field $\phi$ at the point $x$ with the resolution $a$. The standard RG results for $\phi^4$ model are reproduced.Comment: LaTeX, 5 pages, 1 eps figur

Wavelet regularization of gauge theories

Extending the principle of local gauge invariance $\psi(x)\to \exp\left(\imath \sum_A \omega^A(x)T^A \right) \psi(x), x \in \mathbb{R}^d$, with $T^A$ being the generators of the gauge group $\mathcal{A}$, to the fields $\psi(g)\equiv \langle \chi|\Omega^*(g)|\psi\rangle$, defined on a locally compact Lie group $G$, $g\in G$, where $\Omega(g)$ is suitable square-integrable representation of $G$, it is shown that taking the coordinates ($g$) on the affine group, we get a gauge theory that is finite by construction. The renormalization group in the constructed theory relates to each other the charges measured at different scales. The case of the $\mathcal{A}=SU(N)$ gauge group is considered.Comment: 15 LaTeX pages, 8 figure

ϕ4\phi^4 Field theory on a Lie group

The $\phi^4$ field model is generalized to the case when the field $\phi(x)$ is defined on a Lie group: $S[\phi]=\int_{x\in G} L[\phi(x)] d\mu(x)$, $d\mu(x)$ is the left-invariant measure on a locally compact group $G$. For the particular case of the affine group $G:x'=ax+b,a\in\R_+, x,b \in \R^n$ t he Feynman perturbation expansion for the Green functions is shown to have no ultra-violet divergences for certain choice of $\lambda(a) \sim a^\nu$.Comment: 9 pages, AMS-LaTe

On some algebraic problems arising in quantum mechanical description of biological systems

The biological hierarchy and the differences between living and non-living systems are considered from the standpoint of quantum mechanics. The hierarchical organization of biological systems requires hierarchical organization of quantum states. The construction of the hierarchical space of state vectors is presented. The application of similar structures to quantum information processing is considered.Comment: LaTeX, 8 pages. Talk at the Int. Workshop CAAP-2001, June 28-30, 2001, Dubna, Russi

Wavelet based regularization for Euclidean field theory

It is shown that Euclidean field theory with polynomial interaction, can be regularized using the wavelet representation of the fields. The connections between wavelet based regularization and stochastic quantization are considered.Comment: LaTeX, 7 page