Heisenberg Quantization in Physical Space Based on Established Experiments

TL;DR

This paper extends Heisenberg quantization from empty space to physical space, demonstrating noncommutative space, a minimal length scale, and a link between space properties and system motion based on experimental data.

Contribution

It introduces a generalized Heisenberg quantization in physical space, revealing noncommutative geometry and a minimal volume, grounded in established experiments.

Findings

Physical space is noncommutative.

Existence of a non-zero minimal length scale.

Space non-commutativity correlates with system momentum.

Abstract

It is clarified that Heisenberg quantization was proposed in empty space. Based on established experiments, the generalized Heisenberg quantization in physical space is obtained. Physical space quantization includes important new physics: Proving that physical space is noncommutative space; Exploring the existence of a non-zero minimal length scale, which leads to new space structures and the existence of the space minimal finite volume; Finding a new correlativity of the property of space with the motion status of the system: space non-commutativity is determined by the momentum non-commutativity.

Equations

$$[x_i,p_j]=i\hslash {\delta }_{ij},[x_i,x_j]=0,[p_i,p_j]=0,(i,j=1,2),$$

$${\mathbf{B}}^{\mathbf{c}}\sim 10^{-8}\mathbf{Gauss}=10^{-12}\mathbf{T}$$

$$[{\tilde{x}}_i,{\tilde{p}}_j]=i\hslash {\delta }_{ij},[{\tilde{p}}_i,{\tilde{p}}_j]=i{\xi }^2{\eta }_c{ϵ}_{ij},[{\tilde{x}}_i,{\tilde{x}}_j]=i{\xi }^2{\theta }_c{ϵ}_{ij},(i,j=1,2),$$

$${\tilde{p}}_i=\xi [p_i+{\eta }_c{ϵ}_{ij}{x}_j/(2\hslash )],{\tilde{x}}_i=\xi [x_i-{\theta }_c{ϵ}_{ij}{p}_j/(2\hslash )].$$

$$[{\tilde{x}}_i,{\tilde{p}}_j]=i\hslash {\delta }_{ij},[{\tilde{p}}_i,{\tilde{p}}_j]=i{\eta }_c{ϵ}_{ij},[{\tilde{x}}_i,{\tilde{x}}_j]=0,$$

$${\tilde{p}}_i=p_i+{\eta }_c{ϵ}_{ij}{x}_j/(2\hslash ),{\tilde{x}}_i=x_i,(i,j=1,2),$$

$${\eta }_c=\hslash q{B}^c\sim 10^{-65}k{g}^2\cdot m^2\cdot s^{-2},$$

$${\tilde{a}}_i^{†}=\sqrt{\frac{\mu \omega }{2\hslash }}\left({\tilde{x}}_i-\frac{i}{\mu \omega }{\tilde{p}}_i\right),(i=1,2).$$

$$\begin{gathered} [{\tilde{a}}_i,{\tilde{a}}_j^{†}]={\delta }_{ij}+i{\xi }^{-2}\mu \omega {\theta }_c{ϵ}_{ij},[{\tilde{a}}_i,{\tilde{a}}_j]=[{\tilde{a}}_i^{†},{\tilde{a}}_j^{†}]=0, \\ (i,j=1,2), \end{gathered}$$

$${\theta }_c={\eta }_c/(\mu \omega {)}^2.$$

$${\theta }_c\ne 0.$$

$$(\Delta \tilde{p}{)}_{min}=\sqrt{{\eta }_c/2};$$

$$(\Delta \tilde{x}{)}_{min}=\sqrt{{\theta }_c/2}.$$

$$[{\tilde{a}}_1,{\tilde{a}}_2^{†}]=i{\xi }^2\mu \omega {\theta }_c/\hslash .$$

Full text

∗ E-mail address: jzzhang@ecust.edu.cn

Keywords: Physical space quantum mechanics; Minimal length scale;

Correlativity of space property with motion status

Quantum mechanics based on Heisenberg quantization (HQM) is one of the most successful physical theories. It has obtained highly accurate test of all experiments. It is the theoretical basis of modern physics, chemistry, information technology and so on. Heisenberg quantization was established in empty space (pure vacuum). It reads (we consider two dimensional space)

[TABLE]

where it is supposed that both position-position and momentum-momentum are commuting. Eq.(1) is the basis of HQM (Tree equations all together in Eq.(1) is also called Heisenberg-Weyl algebra).

In the real universe, space and the cosmic magnetic field are simultaneously exist in a very large cosmic scale since the early Universe. In the reality, empty space does not exist; Only physical space (space filled by the cosmic magnetic field) exists. The cosmic magnetic field has a role of the intrinsic magnetic field of physical space. The present intergalactic magnetic field $\bf B^{c}$ has an intensity of

[TABLE]

and a dominant scale-length $\sim$ 10 kpc [1]. In the following, we mean that the cosmic magnetic field is this $\bf B^{c}$. Although $\bf B^{c}$ is so weak, its role in quantum mechanics is much more important than our previous understanding. In the traditional way of HQM, $\bf B^{c}$ is treated as perturbation in the Schrödinger equation. However, there is another way: this intrinsic magnetic field $\bf B^{c}$, according to the minimal coupling of the electromagnetic interaction, is included in the definition of mechanical momentum, it follows that important new physics emerge, which are not noticed before.

When the cosmic magnetic field $\bf B^{c}$ is considered, there are two possibilities for generalizing Heisenberg quantization to physical space:

1. Both commutators of the momentum-momentum and the position-position are non-commuting. In this case, the generalized Heisenberg quantization (GHQ) reads

[TABLE]

where ($\widetilde{\bf p},\widetilde{\bf x}$) are expressed by (${\bf x},{\bf p}$):

[TABLE]

In the above, $\epsilon_{ij}$ is an antisymmetric unit tensor, $\epsilon_{12}=-\epsilon_{21}=1,$ $\epsilon_{11}=\epsilon_{22}=0;$ $\xi=[1+\theta_{c}\eta_{c}/(4\hbar^{2})]^{-1/2}$ is the scaling factor , which is demanded by the consistence of Eq.(3). $\eta_{c}$ and $\theta_{c}$ are, respectively, the momentum-momentum noncommutative parameter and the position-position noncommutative parameter.

In the tilde system ($\widetilde{\bf p},\widetilde{\bf x}$), Heisenberg quantization [the first equation of Eq.(1)] is maintained.

2. Only the commutator of the momentum-momentum is non-commuting, but the commutator of the position-position is commuting. In this case, $\theta_{c}=0$, thus $\xi=1$, $\widetilde{x}_{i}=x_{i}$, then Eq.(3) and Eq.(4) reduce to

[TABLE]

[TABLE]

which also maintains Heisenberg quantization.

Proof of Eq.(4). When $\bf B^{c}$ is considered, the minimal coupling of the electromagnetic interaction demands that the canonical momentum ${\bf p}$ should be replaced by $\widetilde{p}_{i}\sim p_{i}-qA_{i}^{c}(x)$, where $\bf A^{c}$ is the vector potential of $\bf B^{c}$. At the micro-scale, $\bf B^{c}$ can be considered as a constant field. Taking the direction of $\bf B^{c}$ as the $z$ direction, it follows that $A_{i}^{c}=-B^{c}\epsilon_{ij}x_{j}/2.$ Thus, we obtain the expression of $\widetilde{p}_{i}$ by $p_{i}$ and $x_{j}$ in Eq.(4), there the momentum-momentum noncommutative parameter

[TABLE]

where fundamental physical constants are taken from [2].

The Physical principles of determining $\widetilde{x}_{i}$. In the system described by $\widetilde{p}_{i}$, the position $x_{i}$ should be replaced by $\widetilde{x}_{i}$ accordingly. To find its expression, Heisenberg quantization in Eq.(1) and Bose-Einstein (B-E) statistics have to be Maintained in the system $(\widetilde{p}_{i},\widetilde{x}_{i})$.

Conditions of Maintaining B-E statistics. When state vector space of identical bosons is constructed by generalizing one-particle quantum mechanics, for two dimensional harmonic oscillator, the annihilation-creation operators $(\widetilde{a}_{i},\widetilde{a}_{i}^{\dagger})$ are:

[TABLE]

where $\mu$ is mass of the considered particle and $\omega$ is its characteristic frequency [3].

The closed and complete B-E algebra of $\widetilde{a}_{i}$ and $\widetilde{a}_{i}^{\dagger}$ are [4]

[TABLE]

Eq.(4) is a linear transformation between two sets of phase space variables $(\widetilde{p}_{i},\widetilde{x}_{i})$ and $(x_{i},p_{i})$. It is proved that except Eq.(4) any other type of such linear transformations cannot maintain both the physical space Heisenberg-Weyl algebra Eq.(3) and the physical space bosonic algebra Eq.(8) [3].

To maintain B-E statistics the condition is that $\widetilde{a}_{i}^{\dagger}$ and $\widetilde{a}_{j}^{\dagger}$ should be commuting, $[\widetilde{a}_{i}^{\dagger},\widetilde{a}_{j}^{\dagger}]=0.$ This leads to two results:

(i) The consistency expression of $\widetilde{x}_{i}$ by $x_{i}$ and $p_{j}$ is obtained. By such ( $\widetilde{p}_{i},\widetilde{x}_{i}$) in Eq.(4), Heisenberg quantization is maintained in physical space.

(ii) The position-position noncommutative parameter $\theta_{c}$ must be a non-zero constant. Using (7) we obtain $[\widetilde{a}_{1}^{\dagger},\widetilde{a}_{2}^{\dagger}]=i\xi\mu\omega[\theta_{c}-\eta_{c}/(\mu\omega)^{2}]/(2\hbar).$ From $[\widetilde{a}_{i}^{\dagger},\widetilde{a}_{j}^{\dagger}]=0$, it follows that [5]

[TABLE]

If $\theta_{c}$ was zero, $[\widetilde{a}_{1}^{\dagger},\widetilde{a}_{2}^{\dagger}]=-i\mu\omega[\eta_{c}/(\mu\omega)^{2}]/(2\hbar)\neq 0$, thus B-E statistics could not be maintained. Therefore, the maintenance of B-E statistics leads to that $\theta_{c}$ being a non-zero constant,

[TABLE]

Although Eq.(10) is obtained in an example, it is enough to ascertain that the existence of the non-zero $\theta_{c}$ is of universal significance.

Eq.(10) clarifies that the case 2) is excluded by maintaining B-E statistics. It concludes that physical space is noncommutative space (the case $[\widetilde{x}_{i},\widetilde{x}_{j}]=0$ cannot exist.)

Eq.(3) is the basis of physical space quantum mechanics (PSQM).

The position-position minimal uncertainty of GHQ. In GHQ of Eq.(3) there are two new minimal uncertainties: The second equation in Eq.(3) indicates that the momentum-momentum minimal uncertainty $(\Delta{\tilde{p}})_{min}\equiv(\Delta{\tilde{p}}_{1})_{min}=(\Delta{\tilde{p}}_{2})_{min}$ of ${\bf{\tilde{p}}}$ reads (Neglect the high order term $\eta_{c}\theta_{c}$, thus $\xi\sim 1$)

[TABLE]

From the third equation in Eq.(3), it follows the position-position minimal uncertainty

$(\Delta{\tilde{x}})_{min}\equiv(\Delta{\tilde{x}}_{1})_{min}=(\Delta{\tilde{x}}_{2})_{min}$ of $\bf{\tilde{x}}$ is

[TABLE]

$(\Delta{\tilde{x}})_{min}$ provides a fundamental minimal length scale $(\Delta{\tilde{x}})_{min}$, which leads to new space structures in noncommutative space.

The existence of the non-zero minimal length scale $(\Delta{\tilde{x}})_{min}$ in Eq.(12) indicates that the procedure of dividing a finite area in the $\tilde{x}_{1}-\tilde{x}_{2}$ plane is not infinitive, it stopes at a minimal area, in which lengths of any two orthogonal dimensions are not less than the minimal length scale. Generalizing to three dimensional system, the procedure of dividing a finite volume of space is not infinitive, it stopes at a minimal volume, in which lengths of any three orthogonal dimensions are not less than the minimal length scale.

Summary and Discussions. New finds in this paper are: (i) Heisenberg quantization Eq.(1) and HQM were proposed in empty space, but empty space does not exist.

(ii) Based on established experiments, GHQ is obtained, and PSQM based on GHQ has not free parameters [6],[?–?].

(iii) By using Eq.(4), all calculations of PSQM are realized by using variables ($x,p$). In the zero order of $\eta_{c}$ and $\theta_{c}$, PSQM recovers all predictions of HQM. First orders of $O(\eta_{c})$ and $O(\theta_{c})$ give perturbation corrections of PSQM to HQM. Furthermore, GHQ and PSQM include important new physics.

(iv) It explores the existence of a non-zero position-position noncommutative parameter $\theta_{c}$ in Eq.(10), and the relevant non-zero minimal length scale in Eq.(12). The existence of such a minimal length scale leads to essentially new space structure.

(v) It clarifies that physical space is noncommutative space. GHQ and PSQM are quantum theories in noncommutative space.

(vi) The correlativity of the property of space with the motion status. In the special theory of relativity, Einstein found that the length scale of space depends on the velocity of the inertial frame in which measurements of the length scale are made. This is the first discovery of the space property related to the motion status. Here, Eq.(9) explores a new correlativity of the property of space with the motion status of the system: space non-commutativity is determined by the momentum non-commutativity.

The physical space algebra Eq.(3) is related to the empty space algebra Eq.(1) by a similarity transformation. However, Eq.(3) and Eq.(1) are un-equivalency [10]. Because Eq.(1) and Eq.(3) are, respectively, the foundations of HQM and PSQM, based on such a un-equivalency, one expects essentially new physics emerged from GHQ and PSQM. So the above new physics can be understood.

The existence of a minimal length scale leads to that the space structure of physical space is essentially different from one of empty space. From the existence of the minimal length scale, it follows the existence of the minimal space volume. It indicates that point particle does not exist, any massive particle, including the electron [11], [?–?], has a minimal volume.

Acknowledgements

This work has been supported by NNSFC (the National Natural Science Foundation of China) for financial support under grant numbers: 10575037, 10074014, 19674014, 19274017, 1880126 and SEDF (the Shanghai Education Development Foundation). The author would like to thank Huashan Hospital very much for the outstanding medical service. The harmonious atmosphere in Ward 27 and Ward 28 of Inpatient lets author possible to clarify physics of the paper and to organize the draft during his stay in Huashan Hospital.