A Century of Quantum Theory: Time for a Change in Thinking

Abstract

The aim of science is the explanation of complicated systems by reducing it to simple subsystems. According to a millennia-old imagination this will be attained by dividing matter into smaller and smaller pieces of it. The popular superstition that smallness implies simplicity seems to be ineradicable. However, since the beginning of quantum theory it would be possible to realize that the circumstances in nature are exactly the other way round. The idea “smaller becomes simpler” is useful only down to the atoms of chemistry. Planck’s formula shows that smaller extensions are related to larger energies. That more and more energy should result in simpler and simpler structures, this does not only sound absurd, it is absurd. A reduction to really simple structures leads one to smallest energies and, thus, to utmost extended quantum systems. The simplest quantum structure, referred to as quantum bit, has a two-dimensional state space, and it establishes a cosmological structure. Taking many of such quantum bits allows also for the construction of localized particles. The non-localized fraction of quantum bits can appear as “dark matter”.

Appendix

The construction of quantum particles in Minkowski-space from quantum bits (qubits) requires a distinction between qubits and anti-qubits (termed urs and anti-urs by Weizsäcker).

The operator creating a qubit is denoted by $a_{s}^{\dag }$, the operator destructing one by a _s . To construct particles with mass one has to rely on the Para-Bose commutation rules¹⁶:

$$ \begin{aligned} \frac{1}{2}\left[ {\left\{ {a_{r} ,a_{s}^{\dag } } \right\},a_{t} } \right] & = - \delta_{st} a_{r} \\ \left[ {\left\{ {a_{r} ,a_{s} } \right\},a_{t} } \right] & = \left[ {\left\{ {a_{r}^{\dag } ,a_{s}^{\dag } } \right\},a_{t}^{\dag } } \right] = 0 \\ \end{aligned} $$

Here the subscripts r, s, t refer to the numbers 1, …, 4 and label qubits and anti-qubits.

Acting on the Protyposis vacuum |Ω> we find:

$$ a_{s} a_{r}^{\dag } \left| \varOmega \right\rangle = \delta_{rs} p\left| \varOmega \right\rangle $$

p is the Para–Bose order, p = 1 corresponding to Bose statistics.

The vacuum in Minkowski-space, referred to as Lorentz vacuum |0>, is the state of the one-dimensional representation of the Poincaré-group for mass, energy, and spin equal to zero.¹⁷ This vacuum is characterized by the statement that there is no particle at any of the infinitely many points of the Minkowski-space. This corresponds to an infinite amount of information.

Accordingly, the Lorentz vacuum can be represented as an infinite sum of operator products acting on the vacuum of qubits, |Ω>.

$$ \left| 0 \right\rangle = \sum\limits_{{n_{1} = 0}}^{\infty } {\sum\limits_{{n_{2} = 0}}^{\infty } {\frac{{\left( { - 1} \right)^{{n_{1} + n_{2} }} }}{{n_{1} ! \cdot n_{2} !}}} } \left( {\frac{{a_{1}^{\dag } a_{3}^{\dag } + a_{3}^{\dag } a_{1}^{\dag } }}{2}} \right)\left( {\frac{{a_{2}^{\dag } a_{4}^{\dag } + a_{4}^{\dag } a_{2}^{\dag } }}{2}} \right)\left| \varOmega \right\rangle $$

With regard to the Lorentz-vacuum, the destruction of a qubit effects the creation of the corresponding anti-qubit.

$$ a_{1} \left| 0 \right\rangle = - a_{3}^{\dag } \left| 0 \right\rangle \quad a_{2} \left| 0 \right\rangle = - a_{4}^{\dag } \left| 0 \right\rangle \quad a_{3} \left| 0 \right\rangle = - a_{1}^{\dag } \left| 0 \right\rangle \quad a_{4} \left| 0 \right\rangle = - a_{2}^{\dag } \left| 0 \right\rangle $$

For a computer-based automated formula processing it was useful to introduce the following notations¹⁸:

$$ \begin{aligned} a_{r}^{\dag } & \Rightarrow e\left[ r \right] \, \left( {{\text{Erzeuger}},{\text{creator}}} \right) \\ a_{r} & \Rightarrow v\left[ r \right] \, \left( {{\text{Vernichter}},{\text{destructor}}} \right) \\ \left\{ {a_{r}^{\dag } ,a_{s}^{\dag } } \right\} & \Rightarrow 2f\left[ {r,s} \right] \\ \left\{ {a_{r}^{\dag } ,a_{s} } \right\} & \Rightarrow 2d\left[ {r,s} \right] \\ \left\{ {a_{r} ,a_{s} } \right\} & \Rightarrow 2w\left[ {r,s} \right] \\ \left| 0 \right\rangle & \Rightarrow lvac \\ \end{aligned} $$

The imaginary unit i = √(−1) will denoted by I, and the normal multiplication by *. Herewith the generators of the Poincaré-group can be written as:

Translations:

$$ \begin{aligned} {\text{P}}1 & = \left( { - {\text{w}}\left[ {2,3} \right] - {\text{f}}\left[ {3,2} \right] - {\text{w}}\left[ {1,4} \right] - {\text{f}}\left[ {4,1} \right] - {\text{d}}\left[ {1,2} \right] - {\text{d}}\left[ {2,1} \right] - {\text{d}}\left[ {4,3} \right] - {\text{d}}\left[ {3,4} \right]} \right)/2 \\ {\text{P}}2 & = {\text{I}}*\left( { - {\text{w}}\left[ {2,3} \right] + {\text{f}}\left[ {3,2} \right] + {\text{w}}\left[ {1,4} \right] - {\text{f}}\left[ {4,1} \right] - {\text{d}}\left[ {1,2} \right] + {\text{d}}\left[ {2,1} \right] - {\text{d}}\left[ {4,3} \right] + {\text{d}}\left[ {3,4} \right]} \right)/2 \\ {\text{P}}3 & = \left( { - {\text{w}}\left[ {1,3} \right] - {\text{f}}\left[ {3,1} \right] + {\text{w}}\left[ {2,4} \right] + {\text{f}}\left[ {4,2} \right] - {\text{d}}\left[ {1,1} \right] + {\text{d}}\left[ {2,2} \right] - {\text{d}}\left[ {3,3} \right] + {\text{d}}\left[ {4,4} \right]} \right)/2 \\ {\text{P}}0 & = \left( { - {\text{w}}\left[ {1,3} \right] - {\text{f}}\left[ {3,1} \right] - {\text{w}}\left[ {2,4} \right] - {\text{f}}\left[ {4,2} \right] - {\text{d}}\left[ {1,1} \right] - {\text{d}}\left[ {2,2} \right] - {\text{d}}\left[ {3,3} \right] - {\text{d}}\left[ {4,4} \right]} \right)/2 \\ \end{aligned} $$

Boosts:

$$ \begin{aligned} {\text{M}}10 & = {\text{I}}*\left( {{\text{w}}\left[ {1,4} \right] - {\text{f}}\left[ {4,1} \right] + {\text{w}}\left[ {2,3} \right] - {\text{f}}\left[ {3,2} \right]} \right)/2 \\ {\text{M}}20 & = \left( {{\text{w}}\left[ {1,4} \right] + {\text{f}}\left[ {4,1} \right] - {\text{w}}\left[ {2,3} \right] - {\text{f}}\left[ {3,2} \right]} \right)/2 \\ {\text{M}}30 & = {\text{I}}*\left( {{\text{w}}\left[ {1,3} \right] - {\text{f}}\left[ {3,1} \right] - {\text{w}}\left[ {2,4} \right] + {\text{f}}\left[ {4,2} \right]} \right)/2 \\ \end{aligned} $$

Rotations:

$$ \begin{aligned} {\text{M}}32 & = \left( {{\text{d}}\left[ {2,1} \right] + {\text{d}}\left[ {1,2} \right] - {\text{d}}\left[ {3,4} \right] - {\text{d}}\left[ {4,3} \right]} \right)/2 \\ {\text{M}}21 & = \left( {{\text{d}}\left[ {1,1} \right] - {\text{d}}\left[ {2,2} \right] - {\text{d}}\left[ {3,3} \right] + {\text{d}}\left[ {4,4} \right]} \right)/2 \\ {\text{M}}31 & = {\text{I}}*\left( {{\text{d}}\left[ {2,1} \right] - {\text{d}}\left[ {1,2} \right] - {\text{d}}\left[ {3,4} \right] + {\text{d}}\left[ {4,3} \right]} \right)/2 \\ \end{aligned} $$

By ** the non-commutative multiplication is denoted. The non-commutative product of n times f[r, s], i.e. ${\text{f}}\left[ {{\text{r}},{\text{s}}} \right]**{\text{f}}\left[ {{\text{r}},{\text{s}}} \right]**\ldots**{\text{f}}\left[ {{\text{r}},{\text{s}}} \right]$ is abbreviated by f[r, s, n].

As an example the state of a massive spinless boson at rest is given. Here the Para-Bose-order has to be greater than one: p[0] > 1. The momentum at rest is P0 = m, P1 = P2 = P3 = 0 and the spin is 0.

$$ \begin{aligned} & \sum\limits_{{{\text{p}}\left[ 3 \right] = 0}}^{\infty } {\sum\limits_{{{\text{p}}\left[ 2 \right] = 0}}^{\infty } {\sum\limits_{{{\text{p}}\left[ 1 \right] = 0}}^{\infty } {\frac{{\left( { - 1} \right)^{{\left( {{\text{p}}[1] + {\text{p}}[2] + {\text{p}}[3]} \right)}} \left( {\text{m}} \right)^{{\left( { 2 {\text{p}}[1] + {\text{p}}[2] + {\text{p}}[3]} \right)}} \left( {{\text{p}}[1] + {\text{p}}[2] + {\text{p}}[3] + {\text{p}}[0] - 2} \right)!}}{{\left( { 2 {\text{p}}[1] + {\text{p}}[2] + {\text{p}}[3] + {\text{p}}[0] - 1} \right)!\left( {{\text{p}}[3] - 1 + {\text{p}}[0] - 1 + {\text{p}}[1]} \right)!\left( {{\text{p}}[2] - 1 + {\text{p}}[0] - 1 + {\text{p}}[1]} \right)!{\text{p}}[1]!{\text{p}}[2]!{\text{p}}[3]!}}} } } \\ & \quad {\text{f}}[4,2,{\text{p}}[3]]**{\text{f}}[4,1,{\text{p}}[1]]**{\text{f}}[3,2,{\text{p}}[1]]**{\text{f}}[3,1,{\text{p}}[2]]**{\text{lvac}} \\ \end{aligned} $$

As another and even more complicated example we present the state of rest (momentum = 0) of a fermion with mass m and spin 1/2 in z-direction. Of course, the Para-Bose order is assumed to be p[0] > 1.

Then the Fermion-state appears as a Fock-State over the Lorentz vacuum, in concrete terms as a two infinite triple-sums of operator products¹⁹:

$$ \begin{aligned} & \sum\limits_{{{\text{p}}\left[ 3 \right] = 0}}^{\infty } {\sum\limits_{{{\text{p}}\left[ 2 \right] = 0}}^{\infty } {\sum\limits_{{{\text{p}}\left[ 1 \right] = 0}}^{\infty } {\frac{{\left( { - 1} \right)^{{\left( {{\text{p}}[1] + {\text{p}}[2] + {\text{p}}[3]} \right)}} \left( {\text{m}} \right)^{{\left( { 2 {\text{p}}[1] + {\text{p}}[2] + {\text{p}}[3]} \right)}} }}{{{\text{p}}[1]!{\text{p}}[2]!{\text{p}}[3]!\left( { 2 {\text{p}}[1] + {\text{p}}[2] + {\text{p}}[3] + {\text{p}}[0]} \right)!}}} } } \\ & \quad *\frac{{\left( {{\text{p}}[1] + {\text{p}}[2] + {\text{p}}[3] + {\text{p}}[0] - 1} \right)!}}{{\left( {{\text{p}}[1] + {\text{p}}[3] + {\text{p}}[0] - 2} \right)\,!\left( {{\text{p}}[1] + {\text{p}}[2] + {\text{p}}[0] - 1} \right)!}} \\ & \quad *{\text{e}}[1]**{\text{f}}[4,2,{\text{p}}[3]]****{\text{f}}[4,1,{\text{p}}[1]]**{\text{f}}[3,2,{\text{p}}[1]]**{\text{f}}[3,1,{\text{p}}[2]]**{\text{lvac}} \\ & \quad + \sum\limits_{{{\text{p}}\left[ 3 \right] = 0}}^{\infty } {\sum\limits_{{{\text{p}}\left[ 2 \right] = 0}}^{\infty } {\sum\limits_{{{\text{p}}\left[ 1 \right] = 0}}^{\infty } {\frac{{\left( { - 1} \right)^{{\left( {{\text{p}}[1] + {\text{p}}[2] + {\text{p}}[3]} \right)}} \left( {\text{m}} \right)^{{\left( { 1 {\text{ + 2p}}[1] + {\text{p}}[2] + {\text{p}}[3]} \right)}} }}{{{\text{p}}[1]!{\text{p}}[2]!{\text{p}}[3]!\left( { 2 {\text{p}}[1] + {\text{p}}[2] + {\text{p}}[3] + {\text{p}}[0] + 1} \right)!}}} } } \\ & \quad *\frac{{\left( {{\text{p}}[1] + {\text{p}}[2] + {\text{p}}[3] + {\text{p}}[0] - 1} \right)!}}{{\left( {{\text{p}}[1] + {\text{p}}[3] + {\text{p}}[0] - 1} \right)\,!\left( {{\text{p}}[1] + {\text{p}}[2] + {\text{p}}[0] - 1} \right)!}} \\ & \quad *{\text{e}}[2]**{\text{f}}[4,2,{\text{p}}[3]]****{\text{f}}[4,1,{\text{p}}[1] + 1]**{\text{f}}[3,2,{\text{p}}[1]]**{\text{f}}[3,1,{\text{p}}[2]]**{\text{lvac}} \\ \end{aligned} $$