universes (multiverse) with random laws of physics and set of arbitrary physical constants, but the bubble with our Universe contained laws and constants which ...
“The Ground-State Wave Function of The Universe Closed In A Bubble” Kamil Walczak Department of Chemistry and Physical Sciences, Pace University, 1 Pace Plaza, New York, NY 10038 INTRODUCTION
Decomposition of Basic Equation
The world around us turned out to be rational (logical) and as far as we know the laws of physics are universally applicable, while fundamental physical constants have the same values everywhere at any moment in time. Taking into account a simple formulation of quantum cosmology, we examine unique properties and visualize the ground-state wave function of the Universe closed in a bubble. Our analysis is based on the fact that all matter and energy as well as space-time burst into existence in the event called Big Bang 3.8 billion years ago due to quantum fluctuations (zero-point energy). Such intrinsic energy of vacuum could spontaneously create many universes (multiverse) with random laws of physics and set of arbitrary physical constants, but the bubble with our Universe contained laws and constants which led to the inflation process.
The polar form of the distance-dependent component:
Quantum Cosmology: Formulation The rate at which the wave function of the Universe is changing in time is determined by Hamiltonian operator of its total energy:
(, t ) i ˆ ˆ H Gravity H Matter (, t ) t
i min (, t ) () exp E U t
E
52 7.3 10 J 2TH
min ˆ ˆ HGravity H Matter () E U () 0
Hamiltonian describing gravitational energy may be postulated as:
ˆ H Gravity
1 2M P
2 2 i M P c A LP 2
After mathematical manipulations, we obtain the expression:
1 S() M c P 2 2M P 2
2
A LP
2
2 2 R ( ) R ( ) 2 2M P
i S() S() R () ˆ R () 2 H Matter R () 0 2 2M P 2
The Hamilton-Jacobi equation (term without Planck’s constant):
S() 2 2 A M P c LP
2
2
The diameter of the bubble (max distance in the Universe):
LP t ( t ) exp 1 e 1 TP
Including two different signs in the Hamilton-Jacobi equation:
S MP LP TP e 1
M P 1 2 LP S S0 TP 2 e 1
From the third equation for R, we obtain the following linear relation:
R ( ) 0 2 2
R () const ()
R () R 0
Assuming R is time-dependent, the second equation takes the form:
R ( , t ) i ˆ H R (, t ) Matter R (, t ) i t 2TP
The Ground-State Wave Function
The second equation (with terms proportional to Planck’s constant):
i S ( ) S ( ) ˆ R () H 2 Matter R () 2 2M P 2
R ( ) 0 2 2M P 2
Limiting ourselves only to the ground state (minimum energy in HUP), the spatial component satisfies Wheeler-DeWitt equation:
2 2 2 M c2 i P ˆ A H R ( ) exp S ( ) 0 Matter 2 L 2 M 2 P P
The third equation (proportional to square of Planck’s constant):
The ground-state energy of the Universe and the Hubble lifetime (TH=1/H) should minimize Heisenberg Uncertainty Principle (HUP): min U
Upon substitutions, the Wheeler-DeWitt equation takes the form:
2
The matter included inside the bubble of our Universe is nearly compensated by gravity generated by this matter (its total energy is very close to zero, being limited only by quantum fluctuations). The ground-state wave function of the Universe may be written as a product of a spatial component and exponential phase factor:
E U TH 2
i () R () exp S()
Physical Theories: Cubical Structure
2
Bohr’s Correspondence Principle: in certain limits, the more general physical theory is reducible to its simplified version and must reproduce its flagship results.
Characteristic Parameters From fundamental physical constants and estimated Hubble parameter, we can construct a set of universal parameters: 1/ 2
G LP 3 c
cTP 1.6 10 1/ 2
Substituting for S and minimum energy, the universal wave function:
( / L P ) / LP t / TH exp i (, t ) R (, t ) sin 2 (e 1) 2 2
G TP 5 c
35
m
LP 44 5.4 10 s c
(Planck length)
(Planck time)
1/ 2
c MP G
8
2.2 10 kg
1 a (t) 17 TH 4.55 10 s H( t ) a ( t )
(Planck mass)
(Hubble time)
There are two equivalent definitions of momentum, namely:
S d p M P p dt
CLOSING REMARKS
Hence the general solution of the Hamilton-Jacobi equation (“plus)”:
d cA dt TP
t ( t ) ALP B exp TP
Constants A and B may be evaluated by imposing initial conditions:
(t 0) 0
0 ALP B
A 1 /(e 1)
(t TP ) LP
LP ALP Be
B LP /(e 1)
The wave function of the Universe is normalizable only within a finite region (bubble) and may not be spread out to infinity in order to represent physically acceptable solution. Time evolution of the universal wave function, i.e. dynamics of the Universe, is either generated by Hamiltonian of Matter and/or initial superposition of two or more quantum states. From physical point of view, it is meaningless to discuss the edge of the Universe, the center of the Universe, and what lies beyond the observable part of the Universe.
