## Archive

Archive for the ‘6. Condensed media’ Category

**The Theory of Stochastic Space-Time.**

**1. Gravitation as a Quantum Diffusion****[1]**

*Zahid Zakir* [2]

**Abstract**

The Nelson stochastic mechanics of *inhomogeneous* quantum diffusion in flat spacetime with a tensor of diffusion can be described as a *homogeneous* one in a Riemannian manifold where this tensor of diffusion plays the role of a metric tensor multiplied to the diffusion coefficient. It is shown that such diffusion accelerates both a sample particle and local inertial frames such that their mean accelerations do not depend on their mass. This fact, explaining the principle of equivalence, allows one to represent the curvature and gravitation as consequences of the inhomogeneity of the quantum fluctuations. In this diffusional treatment of gravitation it can be naturally explained the fact that the energy density of the instantaneous Newtonian interaction is negative defined (with respect to a point at spatial infinity).

*PACS*: 04.20.Cv, 03.65.Ta, 05.40.Jc, 04.62.+v

*Key words: stochastic mechanics, tensor of diffusion, quantum fluctuations,** gravitation, curvature*

Vol. **4**, No 1, p. 1 – 5, v1, 19 March 2009

Online: TPAC: 3000-012 v2, 28 September 2012; DOI: 10.9751/TPAC.3000-012

**Download pdf** 259 kb

[2] *Centre for Theoretical Physics and Astrophyics**, Tashkent, Uzbekistan*

zahidzakir@theor-phys.org

**Non-dissipative diffusion in classical and quantum systems**

*Zahid Zakir* [1]

**Abstract**

The theory of non-dissipative diffusion is constructed on an example of diffusion of a light particle in a dilute medium of heavy particles and it is shown that in low dissipative systems there are specific effects of non-dissipativity similar to quantum effects. In a non-dissipativity region mean energy of the light particle is conserved and processes are described by two non-linear diffusion equations with forward and backward time derivatives. Then these two diffusion equations are linearized and give one linear Schrödinger equation for complex amplitudes of probability. As a result, in the non-dissipative classical diffusion should be added probability amplitudes and there holds the superposition principle for these amplitudes. A mean square length of free passage and a mean square momentum define an elementary phase volume and a diffusion constant and they obey the uncertainty relations. It is shown that the formalism of quantum mechanics describes the classical non-dissipative diffusion with a homogeneous diffusion constant and that quantum mechanics is only a particular case when the elementary phase volume of free passage is universal and equal to the Planck constant.

*PACS*: 02.50.Ey, 03.65.Ta , 05.40.Jc,

*Key words: stochastic processes, **quantum mechanics, Brownian motion*

Vol. **3**, No 3, p. 16 – 35, v1, 29 August 2008

Online: TPAC: 2798-011 v2, 28 September 2012; DOI: 10.9751/TPAC.2798-011

**Download pdf** 932 kb

[1] *Centre for Theoretical Physics and Astrophyics**, Tashkent, Uzbekistan*

zahidzakir@theor-phys.org

**Quantization of relativistic fields without zero-point energy **

**2. ****Vector and spinor fields**

*Zahid Zakir* [1]

**Abstract**

The new method of *C-* and *CP-*symmetric quantization of complex fields is applied for spinor, complex vector and electromagnetic fields. It is shown that the constraints imposed by requirements of discrete symmetries on bilinear products of creation-annihilation operators lead to operators of observables in a normal-ordered form without a zero-point energy and a zero-point charge.

*PACS: 03.70. +* *k, 11.30.Er*

*Key words: quantization, charge conjugation, parity*

Vol. **3**, No 1, p. 1 – 8, v1, 4 March 2008

Online: TPAC: 2620-009 v2, 28 September 2012; DOI: 10.9751/TPAC.2620-009

**Download pdf** 468 kb

[1] *Centre for Theoretical Physics and Astrophyics**, Tashkent, Uzbekistan*

zahidzakir@theor-phys.org

**Quantization of relativistic fields without zero-point energy **

**1. A complex scalar field**

*Zahid Zakir* [1]

**Abstract**

The relativistic fields having opposite sign on frequency modes usually are represented as complex fields only with positive-frequency modes, but with global U(1) symmetry and a charge conjugation (C-symmetry). However, at conventional introduction of operators of antiparticles C-symmetry becomes broken and then a zero-point charge and a vacuum zero-point energy arises. The new method of quantization of relativistic fields without C-symmetry violation is developed. On an example of a complex scalar field it is shown that C-symmetry requirements impose such constraints on bilinear products of ladder operators which lead to new operator identities. They then allow to express the observables through charge-conjugate creation-annihilation operators at once in a normal-ordered form without the zero-point energy and the zero-point charge. It is shown that a self-interaction of a complex scalar field does not change the vacuum energy. As the invariant propagators of particles and antiparticles can be chosen the retarded and advanced propagators accordingly.

*PACS: 03.70.+k; 11.10.-z; 11.30.Er*

*Key words: vacuum energy, vacuum fluctuations, time reversal, charge conjugation*

Vol. **2**, No 3, p. 22 – 31, v1, 2 December 2007

Online: TPAC: 2527-008 v2, 28 September 2012; DOI: 10.9751/TPAC.2527-008

[1] *Centre for Theoretical Physics and Astrophyics**, Tashkent, Uzbekistan*

zahidzakir@theor-phys.org

**Quantization of complex harmonic oscillator**** and the zero-point energy vanishing effect**

*Zahid Zakir* [1]

**Abstract**

The system of two harmonic oscillators with different sign frequencies is presented as a positive-frequency oscillator with a complex generalized coordinate where there are a global U(1) symmetry and a charge conjugation symmetry (C-symmetry). It is shown that two pairs of ladder operators, appearing at the frequency decomposition of canonical variables, are not mutually charge-conjugate and their standard interpretation as operators of charge-conjugate quanta breaks the C-symmetry. Bilinear operator identities for the ladder operators are found allowed the theory to obey the C-symmetry restrictions and to express observables through the mutually charge-conjugate operators. These identities hold for C-symmetric interactions also. It is shown that C-symmetry eliminates a ground state zero-point charge and a zero-point energy. The uncertainty relations are generalized for non-hermitian canonical variables and it is shown that the charge-conjugation symmetric ground states do not quantized. It is shown that the C-symmetric interactions do not contribute to the ground state energy in all orders of perturbation theory.

*PACS: 03.65.Ge, 11.30.Er, 1130.Ly, 11.90. +* *t*

*Key words: Hamiltonian dynamics,* *discrete symmetries, quantization, oscillator, uncertainty relations*

Vol. **2**, No 2, p. 9 – 21, v1, 30 November 2007

Online: TPAC: 2525-007 v2, 18 September 2012; DOI: 10.9751/TPAC.2525-007

[1] *Centre for Theoretical Physics and Astrophyics**, Tashkent, Uzbekistan*

zahidzakir@theor-phys.org

**Quantum field theory and experiments about vanishing of the zero-point vacuum energy**

*Zahid Zakir* [1]

**Abstract**

At quantization of relativistic fields the Hamiltonians symmetrized under the complex conjugate field operators lead to a zero-point vacuum energy. The standard recipe, introducing the operators of antiparticles, breaks the charge conjugation (*C*) symmetry. For the Hamiltonians unsymmetrical under the field operators at exact *C*-symmetry requirement the vacuum does not contain a zero-point energy and zero-point charge. The observable effects are described or as fluctuations of the physical vacuum with loop diagrams (Lamb shift), or as fluctuations of fields of real sources (Casimir effect). Both kind of vacuum fluctuations are created by the interaction Hamiltonians and they have no any relation to the zero-point fluctuations of (external) vacuum fields creating the zero-point vacuum energy in the free Hamiltonians. Therefore, in fact the known experiments exclude the existence of the zero-point fluctuations of the pure vacuum fields and the zero-point vacuum energy and for the relativistic fields allow one to use only Hamiltonians unsymmetrized under the complex conjugate field operators with unbroken *C*-symmetry.

*PACS: 03.70. + k; 11.10.-z; 11.30. Er*

*Key words*: *vacuum energy, vacuum fluctuations, Lamb shift, Casimir effect, time reversal, charge conjugation, cosmological constant*

Vol. **1**, No 4, p. 61 – 72, v1, 28 November 2006

Online: TPAC: 2158-005 v2, 28 September 2012; DOI: 10.9751/TPAC.2158-005

[1] *Centre for Theoretical Physics and Astrophyics**, Tashkent, Uzbekistan*

zahidzakir@theor-phys.org

*T***– ****и**** ***CP***–****symmetric quantization of fields**** without zero-point energy**

*Zahid Zakir* [1]

**Abstract**

The standard replacement of negatively-frequency operators of relativistic fields to operators of antiparticles leads to incomplete time-reversal in field’s observables. In fact, a correct transition to the operators of antiparticles occurs only at a full* T- *or* CP-*transformation of matrix elements. Then the operators of observables of *T* – or *CP*-symmetric fields with standard Lagrangians appear as normal-ordered and the zero-point energy and the zero-point charge do not arise. The rules* *for the *CP-*symmetric quantization of fields are formulated and also the theorem about vanishing of the zero-point energy is proved. The theorem is in agreement with all known experiments since in fact the contributions of real sources are observed only. The divergent zero-point energy and zero-point charge appear only at using of the Lagrangians symmetrized under the complex-conjugate field operators, while the standard Lagrangians without such symmetrization lead to the required vanishing vacuum energy. At spontaneous symmetry breaking the vacuum zero-point energy and zero-point charge do not appear only if the remained scalar field is complex one and in this case the Standard Model predicts a pair boson-antiboson with hypercharge.

*PACS*: 03.70.+k; 11.10.-z; 11.30.Er

*Key words*: *vacuum energy, vacuum fluctuations, Casimir effect, time reversal, charge conjugation, cosmological constant*

Vol. **1**, No 1 , p. 11 – 27, v1, 22 September 2006

Online: TPAC: 2091-002 v2, 28 September 2012; DOI: 10.9751/TPAC.2091-002

[1] *Centre for Theoretical Physics and Astrophyics**, Tashkent, Uzbekistan*

zahidzakir@theor-phys.org

**Symmetries of harmonic oscillator generating**** the zero-point and negative energies**

*Zahid Zakir* [1]

**Abstract**

The Hamiltonian of harmonic oscillator is symmetric under the replacement of canonically conjugate variables and a canonical transformation to ladder operators maintains this symmetry. The Hamiltonian is expressed through a symmetrized product of the ladder operators and, as a result, at quantization there arise a zero-point energy. Therefore, for quantized fields, canonically conjugate variables of which enter into the Hamiltonian unsymmetrically, the zero-point energy could not arise. A new symmetry of harmonic oscillator is found: a wave equation and its solutions do not vary at a joint changing of signs of frequency, energy and mass of a particle. It is shown that the problem of the negative norm for negative-frequency states appears at taking positive mass at negative energy and, contrary, the problem disappears at taking the same sign of mass and energy as it is required by relativistic kinematics. In the nonrelativistic theory, considered as a limiting case of relativistic theory, the states of a particle with negative frequency, energy and mass are described consistently as evolving only backward in time and representing the states of its antiparticle with positive frequency, energy and mass evolving forward in time. For such charge-conjugation symmetric system of oscillators an extended space of states with generalized operators is constructed.

* PACS: 03.65.Ge, 11.30.Er, 1130.Ly, 11.90.+t*

* Key words: Hamiltonian dynamics,* *discrete symmetries, time reversal,* *quantization*

Vol. **1**, No 1 , p. 1 – 10, v1, 22 September 2006

Online: TPAC: 2091-001 v2, 28 September 2012; DOI: 10.9751/TPAC.2091-001

[1] *Centre for Theoretical Physics and Astrophyics**, Tashkent, Uzbekistan*

zahidzakir@theor-phys.org

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