Consciousness theory development versus David Bohm quantum

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One of the fundamental presentation of human consciousness is human memory. In the language of D. Bohm interpretation of the quantum mechanics is the ...
Consciousness theory development versus David Bohm quantum mechanics J. Marciak-Kozłowska Institute of Electron Technology Warsaw, Poland M.Kozłowski Warsaw University, Warsaw, Poland

Abstract One of the fundamental presentation of human consciousness is human memory. In the language of D. Bohm interpretation of the quantum mechanics is the pilot wave ( pilot consciousness wave) which followed of “particles” including humans. In this paper we developed the modified Schrodinger equation (MSE) which contained the consciousness term. We solved the MSE and obtained formula for pilot wave of human neuron Key words: Modified Schrodinger equation, pilot wave, Planck mass, neuron mass

1. Quantum systems with memory (consciousness) In this paper we study the D. Bohm quantum equation and its relation to consciousness ( e.g. memory). We show that the D. Bohm equation ( modified Schrodinger equation ) is the result of the memory of quantum system. When M. Planck made the first quantum discovery he noted an interesting fact [1]. The speed of light, Newton’s gravity constant and Planck’s constant clearly reflect fundamental properties of the world. From them it is possible to derive the characteristic mass MP, length LP and time TP with approximate values LP = 10-35 m TP = 10-43 s MP = 10-5 g . Nowadays much of cosmology is concerned with “interface” of gravity and quantum mechanics. After the Alpha moment – the spark in eternity [1] the space and time were created by “Intelligent Design” [2] at t = TP. The enormous efforts of the physicists, mathematicians and philosophers investigate the Alpha moment. Scholars seriously discuss the Alpha moment – by all possible means: theological and physico-mathematical with growing complexity of theories. The most important result of these investigation is the anthropic principle and Intelligent Design theory (ID). . The thermal history of the system (heated gas container, semiconductor or Universe) can be described by the generalized Fourier equation [3]-[5] t

q(t )    K (t  t ' ) T (t ' )dt '.      thermal history

diffusion

(3.1) In Eq. (3.1) q(t) is the density of the energy flux, T is the temperature of the system and K(t – t ') is the thermal memory of the system K (t  t ' ) 

K  (t  t ' )  exp  , τ τ  

where K is constant, and τ denotes the relaxation time. As was shown in [3]-[5]

(3.2)

  Kδ (t  t ' )  K (t  t ' )   K  constant K (t  t ' )   exp    τ τ  

diffusion wave damped wave or hyperbolic diffusion.

The damped wave or hyperbolic diffusion equation can be written as:  2T 1 T DT 2    T. t 2 τ t τ For τ  0 , Eq. (3.3) is the Fourier thermal equation T  DT  2T t

(3.3)

(3.4) and DT is the thermal diffusion coefficient. The systems with very short relaxation time have very short memory. On the other hand for τ   Eq. (3.3) has the form of the thermal wave (undamped) equation, or ballistic thermal equation. In the solid state physics the ballistic phonons or electrons are those for which τ   . The experiments with ballistic phonons or electrons demonstrate the existence of the wave motion on the lattice scale or on the electron gas scale.  2T DT 2   T. t 2 τ

(3.5)

For the systems with very long memory Eq. (3.3) is time symmetric equation with no arrow of time, for the Eq. (3.5) does not change the shape when t  t . In Eq. (3.3) we define: D  υ   T ,  τ 

(3.6)

velocity of thermal wave propagation and λ  υτ,

(3.7) where λ is the mean free path of the heat carriers. With formula (3.6) equation (3.3) can be written as 1  2T 1 T  2   2T . 2 2 υ t τυ t

(3.8)

From the mathematical point of view equation: 1  2T 1 T    2T 2 2 υ t D t

is the hyperbolic partial differential equation (PDE). On the other hand Fourier equation 1 T   2T D t

and Schrödinger equation

(3.9)

i

 2 2   t 2m

(3.10) are the parabolic equations. Formally with substitutions t  it ,   T

(3.11) Fourier equation (3.9) can be written as i

   D 2  t

(3.12) and by comparison with Schrödinger equation one obtains DT  

2 2m

(3.13) and DT 

 . 2m

(3.14) Considering that DT  τυ2 (3.6) we obtain from (3.14) τ

 . 2mυh2

(3.15) Formula (3.15) describes the relaxation time for quantum thermal processes. Starting with Schrödinger equation for particle with mass m in potential V: i

 2 2     V t 2m

(3.16) and performing the substitution (3.11) one obtains T  2 2    T  VT t 2m

(3.17)

T  2 V   T  T. t 2m 

(3.18) Equation (3.18) is Fourier equation (parabolic PDE) for τ = 0. For τ  0 we obtain τ

 2T T V  2   T  T, 2 t t  2m

(3.19) τ

(3.20) or

 2mυ 2

1  2T 2m T 2Vm   2 T   2T . 2 2 υ t  t 

With the substitution (3.11) equation (3.19) can be written as i

 2 2  2  V      2 . t 2m t

(3.21) The new term, relaxation term 

 2 t 2

(3.22) describes the interaction of the particle with mass m with space-time. The relaxation time τ can be calculated as: 1 ,  1   e1p  ...   Planck (3.23) where, for example τe-p denotes the scattering of the particle m on the electronpositron pair (  e p ~ 10 17 s) and the shortest relaxation time τPlanck is the Planck time ( τ Planck ~ 1043 s). From equation (3.23) we conclude that τ  τ Planck and equation (3.21) can be written as i

 2 2  2  V     τ Planck 2 , t 2m t

(3.24) where 1

τ Planck

1  G  2    5   . 2 c  2M p c 2

(3.25) In formula (3.25) Mp is the mass Planck. Considering Eq. (3.25), Eq. (3.24) can be written as 2 2 2  1 2 2 2 2 i    V     (   2 2 ). t 2m 2M p 2M p c t

(3.26)

The last two terms in Eq. (3.26) can be defined as the Bohmian pilot wave 2  2  2 2   0, 2M p 2M p c 2 t 2

(3.27) i.e. 2 

(3.28)

1  2  0. c 2 t 2

Equations (3.27) and (3,28) constitute D. Bohm type quantum equation, Modified Schrodinger Equation (MSE) It is interesting to observe that pilot wave  does not depend on the mass of the particle. With postulate (3.28) we obtain from equation (3.26)  2 2 2 i     V  2 t 2m 2M p

(3.29) and simultaneously 2  2  2 2   0. 2M p 2M p c 2 t 2

(3.30) In the operator form Eq. (3.21) can be written as pˆ 2 1 Eˆ   Eˆ 2 , 2 2m 2 M p c

(3.31) where Eˆ and pˆ denote the operators for energy and momentum of the particle with mass m. Equation (3.31) is the new dispersion relation for quantum particle with mass m. From Eq. (3.21) one can concludes that Schrödinger quantum mechanics is valid for particles with mass m « MP. But pilot wave exists independent of the mass of the particles. For particles with mass m « MP Eq. (3.29) has the form i

 2 2     V. t 2m

(3.32) In the case when m  M p Eq. (3.29) can be written as i

 2   2   V, t 2M p

(3.33) but considering Eq. (3.30) one obtains   2  2 i   V t 2M p c 2 t 2

(3.34) or  2  2   i  V  0. 2 2 2M p c t t

(3.35) We look for the solution of Eq. (3.35) in the form ( x, t )  eiωt u( x).

(3.36) After substitution formula (3.36) to Eq. (3.35) we obtain

2 ω2  ω  V ( x)  0 2 2M p c

(3.37) with the solution  M pc2  M pc2 1  ω1 

2V M pc2

  M pc2  M pc2 1 

ω2 

2V M pc2



(3.38) for

M pc2 2

 V and

Considering that Mp equals the mass of human neuron ( 10 -5g)  M p c 2  iM p c 2 ω1 

2V 1 M pc2

  M p c 2  iM p c 2

ω2 

2V 1 M pc2



(3.39) for

M pc2 2

 V.

Both formulae (3.38) and (3.39) describe the string oscillation, formula (3.27) damped oscillation and formula (3.28) over damped string oscillation. Considering that Mp equals the mass of human neuron ( 10 -5g) both equation describe the human neuron oscillations –emission of brain waves

References [1] Julian Barbour, The End of Time, Oxford University Press, 2000. [2] Jeffrey F. Addicot, Ohio State Law Journal, 2003. [3] M. Kozlowski, J. Marciak-Kozlowska, The time arrow in a Planck gas, Foundations of Physics Letters, 10, p. 295,(1997). [4] M. Kozlowski, J. Marciak-Kozlowska, The smearing out of the thermal initial conditions created in a Planck Era, Foundation of Physics Letters, 10, p. 599,(1997). [5] M. Kozlowski, J. Marciak-Kozlowska, Klein-Gordon thermal equation for Planck gas, Foundations of Physics Letters, 12, p. 93,(1999).