New Equation of Nonrelativistic Physics and Theory of Dark Matter

Having derived the local symmetric and asymmetric equations, their invariance with respect to the Galilean transformations of $\mathcal{G}_{e}$, which involve translations in space and time, rotations and boosts, is now verified.

Let $S$ and $S^{\prime}$ be two intertial frames of reference in Galilean space and time, and let $\mathbf{v}$ = const be the velocity with which $S^{\prime}$ moves with respect to $S$. Then, a boost is given by $\mathbf{x}=\mathbf{x}^{\prime}+\mathbf{v}t$ and $t^{\ \prime}\ =\ t$, which can be written as

and $\nabla=\nabla^{\prime}$ [16]. Applying these transformations to the eigenvalue equations (Eqs. 1 and 2), it is seen that the transformed equations have different forms, which means that they are not Galilean invariant.

It can be shown that all symmetric and asymmetric equations are invariant with respect to the Galilean translations and rotations, which are the subgroups of $\mathcal{G}_{e}$. However, their boost invariance must be investigated independently by using the following transformation law for the wavefunction $\phi(t,\mathbf{x})=\chi(t^{\prime},{\mathbf{x}}^{\prime})\phi^{\prime}(t^{\prime},{\mathbf{x}}^{\prime})$, where $\chi(t^{\prime},{\mathbf{x}}^{\prime})$ is to be determined [15]. The results previously obtained show that none of the symmetric equations given by Eqs. (3) and (4) is Galilean invariant [15].

Galilean invariance of the Schrödinger-like equations given by Eq. (5) requires that the coefficient $C_{s}$ is the same as the coefficient $C^{\prime}_{s}$ in the following transformed Schrödinger-like equations

where $C^{\prime}_{s}=\omega^{\prime}/k^{{\prime}2}$.

The addional requirement for the Galilean invariance is that the wavefunction $\phi(t,\mathbf{x})$ transforms as

where the phase factor $\eta(t^{\prime},{\mathbf{x}^{\prime}})=\left(\mathbf{v}\cdot\mathbf{x}^{\prime}+v^{2}t^{\prime}/2\right)/(2C^{\prime}_{s})$ [16,17]; this standard phase factor for Galilean transformations appears also in the Galilean invariant Newton laws of dynamics [18] from which it can be removed in a self-consistent way [19].

The above two conditions are necessary for all Schrödinger-like equations with arbitrary $C_{s}$ to be Galilean invariant, which means that all extended Galilean observers agree upon the form of the dynamical equation and its wavefunction in their inertial frames of reference.

In previous work [10,15], it was assumed that $C_{s}=C^{\prime}_{s}$ in all inertial frames; a formal proof of this is now presented in Proposition 1.

Proposition 1: The Schrödinger-like equations given by Eq. (5) are Galilean invariant if, and only if, $C_{s}=\omega/k^{2}=C^{\prime}_{s}=\omega^{\prime}/k^{{\prime}2}$ in all inertial frames of reference.

Using the Galilean transformations $t^{\prime}=t$ and $\mathbf{x}^{\prime}=\mathbf{x}-\mathbf{v}t$, and also substituting $C^{\prime}_{s}=C_{s}$, Eq. (12) becomes

which gives

and, taking this result into account, the expression for $\omega^{\prime}$ becomes

Therefore,

and

Since the second term on the LHS of this equation is non-zero, then $C_{s}=\omega/k^{2}$. This conludes the proof.

Corollary 1: The value of $C_{s}$ in the Schrödinger-like equations (Eq. 5) has the same value in all inertial frames of reference.

According to Corollary 1, the constant $C_{s}$ plays the same role in Galilean Relativity as the speed of light in the Special Theory of Relativity; this means that $C_{s}$ remains the same for all Galilean observers.

The main result is that there is an infinite set of Schrödinger-like equations that are Galilean invariant and that they only differ by their different values of $C_{s}$. The significance of this constant in the physics of OM and DM is discussed in Sect. 5.

The new asymmetric equations given by Eq. (6) also form an infinite set because the constant $C_{w}$ may have any real value. Typically, Galilean invariance requires a phase factor in the wavefunction [13,15-17]. However, the results of the following proposition show that no phase factor cannot be defined to make these asymmetric equations Galilean invariant.

Proposition 2: Let $\phi(t,\mathbf{x})$ be the wavefunction of Eq. (8), $\phi^{\prime}({\mathbf{x}^{\prime}},t^{\prime})$ be the transformed wavefunction, and $\eta(t^{\prime},{\mathbf{x}^{\prime}})$ its phase factor given by Eq. (9). Then, no $\eta(t^{\prime},{\mathbf{x}^{\prime}})$ exists to guarantee Galilean invariance of the new asymmetric equations.

Introducing the phase factor $\eta(t^{\prime},{\mathbf{x}^{\prime}})$ in Eq. (9) and some algebric manipulations, Eq. (18) can be written as

which is of the same form as Eq. (18), except the exponential term with the phase factor. The two extra terms remain; thus, there is no $\eta(t^{\prime},{\mathbf{x}^{\prime}})$ that can make Eq. (19) Galilean invariant or of the same form as Eq. (6). This conludes the proof.

Corollary 2: The results of Proposition 2 can be generalized to $\phi(t,\mathbf{x})\ =\ \phi^{\prime}({\mathbf{x}^{\prime}},t^{\prime})\ \chi(t^{\prime},{\mathbf{x}^{\prime}})$, where $\chi$ is any differentiable function.

Since Galilean invariance cannot be guaranteed by introducing a phase factor, a different method is now presented in the following proposition.

Proposition 3: The new asymmetric equation given by Eq. (6) becomes Galilean invariant if, and only if, $\phi({\mathbf{x}},t)=\phi({\mathbf{r}})$ and $\phi^{\prime}({\mathbf{x}^{\prime}},t^{\prime})=\phi^{\prime}({\mathbf{r}^{\prime}})$, where ${\mathbf{r}}=\mathbf{x}+\mathbf{v}t/2$ and ${\mathbf{r}^{\prime}}={\mathbf{x}^{\prime}}+{\mathbf{v}}t^{\prime}/2$.

It must be also noted that after $\phi({\mathbf{x}},t)=\phi({\mathbf{r}})$ is substituted into Eq. (6), its form remains the same as that of Eq. (19); this concludes the proof.

The results of Proposition 3 can be used to derive the following equation for $\phi({\mathbf{x}},t)=\phi({\mathbf{r}})$

or after the integration

where ${\mathbf{r}}=\mathbf{x}+\mathbf{v}t/2$, $C_{w,v}=4C_{w}/v^{2}$ and $C_{0}$ is an integration constant.

Similarly, the corresponding equation for $\phi({\mathbf{x}^{\prime}},t^{\prime})=\phi^{\prime}({\mathbf{r}^{\prime}})$ becomes

or after the integration

where ${\mathbf{r}^{\prime}}={\mathbf{x}^{\prime}}+{\mathbf{v}}t^{\prime}/2$, $C^{\prime}_{w,v}=4C^{\prime}_{w}/v^{2}$ and $C^{\prime}_{0}$ is an integration constant.

Equations (21), (22), (23) and (24) are of the same form if, and only if, $C_{w,v}=C^{\prime}_{w,v}$ and $C_{o}=C^{\prime}_{0}$ have the same values for all extended Galilean observers.

Finding the solutions to Eqs. (21) and (23) is straigthforward; they can be written as

and

where $C_{1}$, $C_{2}$, $C^{\prime}_{1}$ and $C^{\prime}_{2}$ are integration constants. The solutions to Eqs. (22) and (24) are of the same form as those given by Eqs. (25) and (26) if, and only if, $C_{2}=iC_{0}/C_{w,v}$ and $C^{\prime}_{2}=iC^{\prime}_{0}/C^{\prime}_{w,v}$.

For these solutions to be of the same form, it is required that $C_{1}=C^{\prime}_{1}$, $C_{2}=C^{\prime}_{2}$ and $C_{w,v}=C^{\prime}_{w,v}$. The following proposition presents the necessary conditions to guarantee Galilean invariance of Eqs. (21) and (23).

Proposition 4: The conditions for the Galilean invariance $C_{w}=\omega^{2}/k^{2}=C^{\prime}_{w}=\omega^{\prime 2}/k^{{\prime}2}$, $C_{1}=C^{\prime}_{1}$ and $C_{2}=C^{\prime}_{2}$ are satisfied if, and only if

where

and $x=|{\mathbf{x}}|$.

Proof: According to Proposition 3, there is no phase factor; thus, $\phi({\mathbf{k}}\cdot{\mathbf{r}})=\phi^{\prime}({\mathbf{k}^{\prime}}\cdot{\mathbf{r}^{\prime}})$, and Eqs. (25) and (26) give

Taking $C_{1}=C^{\prime}_{1}$, $C_{2}=C^{\prime}_{2}$ and $C_{w}=C^{\prime}_{w}$, Eq. (29) simplifies to

Using the Galilean transformations $\mathbf{x}^{\prime}=\mathbf{x}-\mathbf{v}t$ and $t^{\ \prime}\ =\ t$, Eq. (30) becomes

or

This concludes the proof.

Corollary 3: Since ${\mathbf{v}}=$ const, the condition $C_{w,v}=C^{\prime}_{w,v}$ reduces to $C_{w}=C^{\prime}_{w}$, or more specifically to

The obtained results demonstrate that the new asymmetric equations are Galilean invariant if, and only if, the argument of the wavefunction is $(\mathbf{x}+\mathbf{v}t/2)$, and the wavevector and the transformed wavevector are related to each other by Eq. (33). The method to make equations Galilean invariant presented in Propositions 4 and 5 can also be applied to other second-order equations.

The main results of this paper are the infinite sets of Galilean invariant Schrödinger-like and new asymmetric equations given by Eqs. (5) and (6), respectively, which are local and their Lagrangians are known.

The coefficients $C_{s}$ and $C_{w}$ in the Schrödinger-like and new asymmetric equations, respectively, are called here the Galilean constants as they are the same for all extended Galilean observers; in other words, they resemble the constant speed of light in Special Theory of Relativity but their physical meaning is different, as shown next.

Since the main differences between the equations in each set are their constant coeffcients, it is suggested that only after specific values of these constants are determined by physical properties of matter, such equations with fixed constants are called the fundamental equations of physics. After introducing this new definition, the fundamental equations of physics for ordinary and dark matter are now identified.

The two infinite sets of Galilean invariant equations for the scalar wavefunction cannot be used to describe classical particles of OM as their motion is governed by Newton’s equations, which are Galilean invariant, if the force in the second law of dynamics is also Galilean invariant [11,13,19]. Previous work (e.g., [20-22]) showed some applications of the Schrödinger equation to classical waves. Similar applications of the Schrödinger-like and new asymmetric equations are also possible but they will not be considered here.

Instead, this study concentrates on the microscopic structure of OM and the quantum description of elementary particles, which requires taking into account the wave-particle duality [17]. In the Galilean Relativity the energy-momentum relationship is given by $E=p^{2}/2m$, where $E$, $p$ and $m$ are the kinetic energy, momentum and mass of the free particles, respectively. Based on this relationship, let $C_{s}=\omega/k^{2}=\alpha_{0}/2m$, with $\alpha_{0}$ being a constant of Nature to be determined from the physical properties of OM.

Applying the de Broglie relationship $\mathbf{p}=\hbar\mathbf{k}$ to $C_{s}$, one obtains

This expression is valid if, and only if, $\alpha_{0}=\hbar$, or $\alpha_{0}$ is the same as the Planck constant. Then, Eq. (7) with $C_{s}=\alpha_{0}/2m=\hbar/2m$ becomes the fundamental Schrödinger equation of Quantum Mechanics [17].

Since there are also infinitely many new asymmetric equations, it remains to be determined whether one of them can also describe the microscopic properties of OM; this problem is discussed in Sect. 5.3, where the equations are applied to DM.

As of today, no electromagnetic radiation of any kind has been detected from DM [4-9]. The NASA Bullet Cluster observations show a different behavior of DM when compared to OM [23]. This may imply that DM does not have the same quantum structure as OM and that the fundamental Schrödinger equation with $C_{s}=\hbar/2m$ is not the fundamental equation for DM; nevertheless, other Schrödinger-like equations with different $C_{s}$ are now explored.

Let $C_{s}=\alpha_{1}/2m$, where $\alpha_{1}$ is a constant of Nature whose value is different than the Planck constant, namely, either $\alpha_{1}<\hbar$ or $\alpha_{1}>\hbar$. If this constant exists for DM, then the resulting Schrödinger-like equation would be the fundamental equation for DM. The main problem with this idea is that the dimension of $\alpha_{1}$ is the same ($J\cdot s$) as the dimension $\alpha_{0}=\hbar$ which, from a physical point of view, is highly unlikely that there are two constants of Nature with identical dimensions but different values. Based on this argument, it is concluded that no other Schrödinger-like equation can become the fundamental equation for DM.

The only other possibility is that such fundamenatal equation for DM is among the set of new asymmetric equations discovered in this paper. The coefficient $C_{w,v}$ is given by

and for $C_{w}$ see Eq. (4).

For a DM non-relativistic elementary particle of mass $m$, the coefficient can be written as

where $\beta_{0}$ is a constant of Nature for DM only. Since $\beta_{0}$ is measured in $J$, which is different than the Planck constant, let $\beta_{0}=\varepsilon_{o}$ be a quanta of energy of DM. Then

where $\varepsilon_{v}=mv^{2}/2$ is the kinetic energy of DM particles confined to the intertial frame moving with the velocity $v=|{\mathbf{v}}|$. The energy $\varepsilon_{v}$ normalizes the quanta of energy of DM, so that the coefficient in the fundamental equation for DM (see Eq. 21) and in its solution (see Eq. 25) is dimensionless.

Based on the above results, it is proposed that the fundamental equation for DM is the new asymmetric equation given by

whose Galilean invariant form is

and its solution is

This shows that the fundamental equations for OM and DM are different and that they depend on different constants of Nature. As a result, the Schrödinger equation describes the quantum structure of OM, whereas the quantum structure of DM is described by the new asymmetric equation (see Eqs. 39 and 40).

In the proposed fundamental equation for DM (see Eq. 6 or Eq. 21), the coefficient $C_{w}$ depends on the new universal constant of Nature $\varepsilon_{o}$, which represents the quanta of energy for DM. The role of $\varepsilon_{o}$ is similar to the Planck constant but its physical meaning is different, namely, while the energy for OM is given by $E=\hbar\omega$, the energy for DM is simply $E_{o}=\varepsilon_{o}$. In other words, the main difference between OM and DM is that quantization of DM can only be done using the quanta $\varepsilon_{o}$; this may explain the lack of observed radiation [4-9] and a significantly different behavior of DM than OM [23].

The obtained results imply that DM may be considered as a collection of DM elementary particles with mass $m$ and that these particles may exchange their energy by emitting or absorbing the quanta of energy $\varepsilon_{o}$. Assuming that the particles interact only gravitationally, the process of quanta exchange may produce a detectable gravitational radiation.