Time Evolution in Quantum Mechanics with a Minimal Time Scale

Abstract

The existence of a minimum measurable length scale was suggested by various theories of quantum gravity, string theory and black hole physics. Motivated by this, we examine a quantum theory exhibiting a minimum measurable time scale. We use the Page–Wootters formalism to describe time evolution of a quantum system with the modified commutation relations between the time and frequency operator. Such modification leads to a minimal uncertainty in the measurement of time. This causes breaking of the time-translation symmetry and results in a modified version of the Schrödinger equation. A minimal time scale also allows us to introduce a discrete Schrödinger equation describing time evolution on a lattice. We show that both descriptions of time evolution are equivalent. We demonstrate the developed theory on a couple simple quantum systems.

1. Introduction

Various theories of quantum gravity (such as string theory) predict the existence of a minimum measurable length scale, usually on the order of the Planck length [1,2,3,4]. Black hole physics also suggests the existence of a fundamental limit with which we can measure distances [5,6,7]. This is because the energy needed to probe spacetime below the Planck length scale exceeds the energy needed to produce a black hole in that region of spacetime. On the other hand, Doubly Special Relativity theories predict maximum observable momenta [8,9]. All of these theories give rise to the modified commutation relations between position coordinates and momenta, which in turn give rise to the Generalized Uncertainty Principle.

Motivated by these results, we examine a quantum theory exhibiting a minimum measurable time scale. Our starting point is the Page–Wootters formalism of time evolution in quantum mechanics. It is based on the idea that time evolution arises as a result of correlations between the “clock” and the rest of the system. Next, we introduce the minimal time scale into the theory by modifying the commutation relations between the time operator ${\displaystyle \widehat{T}}$ and the operator conjugate to it (the frequency operator ${\displaystyle \widehat{\mathsf{\Omega }}}$). Such a modification causes the time observable to not be represented by a self-adjoint operator but only by a symmetric operator. This is expected as it is no longer possible to consider states of the system at a particular instance of time. As a result we use states of the “clock” system maximally localized around instances of time to construct continuous time representation of the system and we arrive at a modified Schrödinger equation describing time evolution of the system. A minimal time scale also allows us to introduce a discrete time representation of the system and a corresponding discrete Schrödinger equation describing time evolution on a lattice. Interestingly, the received discrete and continuous time representations are equivalent with each other and describe the same time evolution of the system.

According to the received Schrödinger Equations (56) and (57), time evolution is governed by an effective Hamiltonian different than the Hamiltonian of the system. The reason for this is that in the case without a minimal time scale, the isolated system possesses time-translation symmetry. The Hamiltonian is the generator of this symmetry. However, when there exists a minimum measurable time scale, the time-translation symmetry breaks down and the Hamiltonian no longer generates translations in time.

Other approaches to time evolution with a minimal time scale in quantum mechanics can be found in the literature. Worth noting is the paper [10], where the authors deform the Heisenberg algebra of spacetime variables receiving a modified Schrödinger equation

$$i\hslash {\displaystyle \frac{\partial }{\partial t}}|\psi \rangle +{\hslash }^2\alpha {\displaystyle \frac{{\partial }^2}{\partial t^2}}|\psi \rangle =\widehat{H}|\psi \rangle ,$$

(1)

where ${\displaystyle \alpha }$ is the deformation parameter. Interesting results were also received in [11], where a deformation of the Wheeler–DeWitt equation led to a discretization of time. The paper [12] by P. Caldirola should also be mentioned, where the author postulates the existence of a universal interval ${\displaystyle {\tau }_0}$ of proper time. The existence of such a universal interval causes the reaction of a particle to the applied external force to not be continuous. As a result, positions of the particle along its world line are discretized.

Breaking of time-translation symmetry is a basic requirement for the creation of time crystals [13,14,15]. Therefore, minimal time scales can lead to a behavior of the system undergoing time evolution similar to that of a time crystal [10,11].

This paper is structured in the following way. In Section 2, we review the Page–Wootters formalism on which the developed theory will be based. In Section 3, we introduce a minimal time scale into the theory by modifying the commutation relations between the time and frequency operators. Section 4 contains the analysis of the time evolution of a couple simple quantum systems when a minimal time scale is present. The systems of one, two and three spins-${\displaystyle 1/2}$ in an external magnetic field are investigated, as well as a free particle and a harmonic oscillator. The conclusions and a discussion of the received results is given in Section 5.

2. Page–Wootters Formalism

In a standard description of quantum mechanics, time is treated as a background structure with respect to which quantization is performed. In the Schrödinger equation, time is treated as a classical parameter. Physically, it represents the time shown by a “classical” clock in the laboratory. Moreover, in relativistic mechanics, standard quantization techniques are imposing the canonical commutation relations on constant-time hypersurfaces. Furthermore, the Wheeler–DeWitt equation [16] says

$$\widehat{H}|\psi \rangle =0,$$

(2)

where ${\displaystyle \widehat{H}}$ is the Hamiltonian constraint in quantized general relativity and ${\displaystyle |\psi \rangle }$ stands for the wave function of the universe. The universe as a whole is static and does not evolve.

Therefore, it is important to give time a fully quantum description so that time becomes a quantum degree of freedom. In the following section, we will review the Page–Wootters formalism [17,18,19,20], which accomplishes this and which will be particularly suited for the considerations presented in this paper. Other proposals of quantum descriptions of time can be found in the literature [21,22,23,24,25,26,27].

In the Page–Wootters formalism, one assigns to the time degree of freedom a Hilbert space ${\displaystyle {\mathcal{H}}_T}$. This Hilbert space describes a “clock”, which will measure the flow of time. The system undergoing the time evolution will be described by the Hilbert space ${\displaystyle {\mathcal{H}}_S}$. The joint Hilbert space of the “clock” and the system is

$$\mathcal{H}={\mathcal{H}}_T\otimes {\mathcal{H}}_S.$$

(3)

We will assume that ${\displaystyle {\mathcal{H}}_T}$ is isomorphic to the Hilbert space of a particle on a line, i.e., the space ${\displaystyle L^2(\mathbb{R},\mathrm{d}x)}$ of complex-valued functions defined on ${\displaystyle \mathbb{R}}$ and square integrable with respect to the Lebesgue measure. It is worth noting that other choices are also possible [28,29]. On ${\displaystyle {\mathcal{H}}_T}$, we introduce the time operator ${\displaystyle \widehat{T}}$ corresponding to the measurements of time and the frequency operator ${\displaystyle \widehat{\mathsf{\Omega }}}$ conjugate to ${\displaystyle \widehat{T}}$:

$$[\widehat{T},\widehat{\mathsf{\Omega }}]=i\widehat{1}.$$

(4)

When ${\displaystyle {\mathcal{H}}_T\cong L^2(\mathbb{R},\mathrm{d}x)}$, then the operators ${\displaystyle \widehat{T}}$ and ${\displaystyle \widehat{\mathsf{\Omega }}}$ will be represented as the operators of multiplication and differentiation:

$$\widehat{T}\psi \left(t\right)=t\psi \left(t\right),\widehat{\mathsf{\Omega }}\psi \left(t\right)=-i{\partial }_t\psi \left(t\right).$$

(5)

On ${\displaystyle \mathcal{H}}$, we introduce the constraint operator of the model

$$\widehat{J}=\hslash \widehat{\mathsf{\Omega }}\otimes {\widehat{1}}_S+{\widehat{1}}_T\otimes {\widehat{H}}_S,$$

(6)

with ${\displaystyle {\widehat{H}}_S}$ the system Hamiltonian. Such constraint describes the simplest case of the “clock” non-interacting with the system and where the system Hamiltonian is time independent. It is, however, possible to consider more general cases. Since ${\displaystyle \widehat{\mathsf{\Omega }}}$ and ${\displaystyle {\widehat{H}}_S}$ are self-adjoint operators, the constraint operator ${\displaystyle \widehat{J}}$ will be also self-adjoint. If we treat ${\displaystyle \widehat{J}}$ as a total Hamiltonian, then in accordance with the Wheeler–DeWitt Equation (2) we define physical states of the model as vectors ${\displaystyle |\mathsf{\Psi }\rangle \rangle }$ satisfying

$$\widehat{J}|\mathsf{\Psi }\rangle \rangle =0.$$

(7)

We will use the double-ket notation to denote states ${\displaystyle |\mathsf{\Psi }\rangle \rangle }$ from the total Hilbert space ${\displaystyle \mathcal{H}}$. Formula (7) defines physical states as (generalized) eigenvectors of the operator ${\displaystyle \widehat{J}}$ associated with the null eigenvalue. The physical states ${\displaystyle |\mathsf{\Psi }\rangle \rangle }$ provide a complete description of the temporal evolution of the system ${\displaystyle S}$ by representing it in terms of correlations (entanglement) between the latter and the “clock” system.

The conventional state ${\displaystyle {|\psi \left(t\right)\rangle }_S}$ of the system ${\displaystyle S}$ at time ${\displaystyle t}$ can be obtained via projection of ${\displaystyle |\mathsf{\Psi }\rangle \rangle }$ with a generalized eigenstate ${\displaystyle {|t\rangle }_T}$ of the time operator ${\displaystyle \widehat{T}}$:

$${|\psi \left(t\right)\rangle }_S={}_T\langle t|\mathsf{\Psi }\rangle \rangle _{,}$$

(8)

where ${\displaystyle {|t\rangle }_T}$ is the generalized eigenvector of the operator ${\displaystyle \widehat{T}}$ associated with the eigenvalue ${\displaystyle t}$:

$$\widehat{T}{|t\rangle }_T=t{|t\rangle }_T.$$

(9)

By writing (7) in the time representation in ${\displaystyle {\mathcal{H}}_T}$ and with the help of (5):

$${}_T\langle t|\widehat{J}|\mathsf{\Psi }\rangle \rangle =0\iff i\hslash {\displaystyle \frac{\partial }{\partial t}}{|\psi \left(t\right)\rangle }_S={\widehat{H}}_S{|\psi \left(t\right)\rangle }_S$$

(10)

we verify that the vectors ${\displaystyle {|\psi \left(t\right)\rangle }_S}$ obey the Schrödinger equation.

By projecting ${\displaystyle |\mathsf{\Psi }\rangle \rangle }$ on the generalized eigenstates ${\displaystyle {|\omega \rangle }_T}$ of ${\displaystyle \widehat{\mathsf{\Omega }}}$:

$${|\psi \left(\omega \right)\rangle }_S={}_T\langle \omega |\mathsf{\Psi }\rangle \rangle _{,}$$

(11)

where ${\displaystyle {|\omega \rangle }_T}$ is the generalized eigenvector of the operator ${\displaystyle \widehat{\mathsf{\Omega }}}$ associated with the eigenvalue ${\displaystyle \omega }$:

$$\widehat{\mathsf{\Omega }}{|\omega \rangle }_T=\omega {|\omega \rangle }_T,$$

(12)

we obtain the eigenvector equation of the operator ${\displaystyle {\widehat{H}}_S}$:

$${}_T\langle \omega |\widehat{J}|\mathsf{\Psi }\rangle \rangle =0\iff {\widehat{H}}_S{|\psi \left(\omega \right)\rangle }_S=-\hslash \omega {|\psi \left(\omega \right)\rangle }_S.$$

(13)

Specifically, for ${\displaystyle \omega }$ such that ${\displaystyle -\hslash \omega }$ equals an element of the spectrum of ${\displaystyle {\widehat{H}}_S}$, then ${\displaystyle {|\psi \left(\omega \right)\rangle }_S}$ is an eigenvector of ${\displaystyle {\widehat{H}}_S}$ at that eigenvalue; otherwise, ${\displaystyle {|\psi \left(\omega \right)\rangle }_S=0}$.

When the total system is in a state ${\displaystyle |\mathsf{\Psi }\rangle \rangle }$, the probability that the measurement of an observable ${\displaystyle {\widehat{A}}_S}$ (a self-adjoint operator defined on the Hilbert space ${\displaystyle {\mathcal{H}}_S}$) will give a value ${\displaystyle a}$ from the spectrum of ${\displaystyle {\widehat{A}}_S}$ given that the “clock” reads the time ${\displaystyle t}$ is postulated as the conditional probability

$$Prob\left({\displaystyle a}\mathrm{when}{\displaystyle t}\right)={\displaystyle \frac{Prob\left({\displaystyle a}\mathrm{and}{\displaystyle t}\right)}{Prob\left(t\right)}}={\displaystyle \frac{\langle \langle \mathsf{\Psi }|(|t\rangle {\langle t|}_T\otimes |a\rangle {\langle a|}_S)|\mathsf{\Psi }\rangle \rangle }{\langle \langle \mathsf{\Psi }|(|t\rangle {\langle t|}_T\otimes {\widehat{1}}_S)|\mathsf{\Psi }\rangle \rangle }}={\displaystyle \frac{{|}_S{\langle a\left|\psi \right(t)\rangle }_S{|}^2}{{}_S\langle \psi \left(t\right)\left|\psi \right(t)\rangle _S}},$$

(14)

where ${\displaystyle {|a\rangle }_S}$ is an eigenvector of the operator ${\displaystyle {\widehat{A}}_S}$ associated with an eigenvalue ${\displaystyle a}$. We can see that the above postulate reproduces the standard Born rule of computing the probabilities of measurements.

Remark 1. 

The original formulation of the formalism by Page and Wootters was criticized as it seemed that it is not reproducing the correct formulas when calculating the probability of measuring an observable ${\displaystyle {\widehat{B}}_S}$ at time ${\displaystyle t_2}$ when one finds the system at an eigenstate of an observable ${\displaystyle {\widehat{A}}_S}$ at time ${\displaystyle t_1}$. In other words we are not receiving the correct propagators of subsequent measurements of observables ${\displaystyle {\widehat{A}}_S}$ and ${\displaystyle {\widehat{B}}_S}$. For the possible resolution of this problem, see [18,19,20].

Remark 2. 

The Page–Wootters formalism can be extended to incorporate a time-dependent Hamiltonians by replacing the constraint operator (6) with

$$\widehat{J}=\hslash \widehat{\mathsf{\Omega }}\otimes {\widehat{1}}_S+{\widehat{H}}_S\left(\widehat{T}\right),$$

(15)

where ${\displaystyle {\widehat{H}}_S\left(\widehat{T}\right)}$ is an operator on ${\displaystyle \mathcal{H}}$ that explicitly depends on the time operator ${\displaystyle \widehat{T}}$.

Remark 3. 

The generalized eigenvectors of operators appearing in the Page–Wootters formalism are formally defined using rigged Hilbert spaces. A rigged Hilbert space consists of a Hilbert space ${\displaystyle \mathcal{H}}$, together with a dense subspace ${\displaystyle \mathsf{\Phi }}$, such that ${\displaystyle \mathsf{\Phi }}$ is given a topological vector space structure with a finer topology than the one inherited from ${\displaystyle \mathcal{H}}$; i.e., the inclusion map ı${\displaystyle :\mathsf{\Phi }\to \mathcal{H}}$ is continuous. By ${\displaystyle {\mathsf{\Phi }}^{*}}$, we will denote the dual space to ${\displaystyle \mathsf{\Phi }}$, i.e., the space of continuous linear functionals ${\displaystyle \mathsf{\Phi }\to \mathbb{C}}$. Similarly, by ${\displaystyle {\mathcal{H}}^{*}}$ we will denote the dual space to ${\displaystyle \mathcal{H}}$. The adjoint map ı*${\displaystyle :{\mathcal{H}}^{*}\to {\mathsf{\Phi }}^{*}}$ to ı defined by

$$\langle {ı}^{*}\left(\phi \right),\psi \rangle =\langle \phi ,ı\left(\psi \right)\rangle ,\phi \in {\mathcal{H}}^{*},\psi \in \mathsf{\Phi }$$

(16)

is injective by the density of ${\displaystyle \mathsf{\Phi }}$ in ${\displaystyle \mathcal{H}}$. Therefore, ı* provides us with an inclusion of ${\displaystyle {\mathcal{H}}^{*}}$ in ${\displaystyle {\mathsf{\Phi }}^{*}}$. By identifying ${\displaystyle {\mathcal{H}}^{*}}$ with ${\displaystyle \mathcal{H}}$ in accordance to the Riesz representation theorem we obtain the inclusion of ${\displaystyle \mathcal{H}}$ in ${\displaystyle {\mathsf{\Phi }}^{*}}$, i.e.,

$$\mathsf{\Phi }\subset \mathcal{H}\subset {\mathsf{\Phi }}^{*}.$$

(17)

By (16), the duality pairing ${\displaystyle \langle ·,·\rangle }$ between ${\displaystyle \mathsf{\Phi }}$ and ${\displaystyle {\mathsf{\Phi }}^{*}}$ is compatible with the inner product ${\displaystyle (·,·)}$ on ${\displaystyle \mathcal{H}}$:

$$\langle \phi ,\psi \rangle =(\phi ,\psi ),\phi \in \mathcal{H}\subset {\mathsf{\Phi }}^{*},\psi \in \mathsf{\Phi }\subset \mathcal{H}.$$

(18)

The space ${\displaystyle \mathsf{\Phi }}$ plays the role of the space of test functions and ${\displaystyle {\mathsf{\Phi }}^{*}}$ is the space of distributions (generalized vectors).

As an example let ${\displaystyle \mathcal{H}=L^2(\mathbb{R},\mathrm{d}x)}$ and ${\displaystyle \mathsf{\Phi }}$ be the Schwartz space of rapidly decreasing functions. Then, ${\displaystyle {\mathsf{\Phi }}^{*}}$ is the space of tempered distributions. The operators ${\displaystyle \widehat{T}}$ and ${\displaystyle \widehat{\mathsf{\Omega }}}$ given by (5) restricted to ${\displaystyle \mathsf{\Phi }}$ also take values in ${\displaystyle \mathsf{\Phi }}$. They are also continuous with respect to the topology in ${\displaystyle \mathsf{\Phi }}$. Therefore, they can be extended to operators on ${\displaystyle {\mathsf{\Phi }}^{*}}$ and we obtain the following eigenvector equations for these operators

$$\widehat{T}\delta (t-t_0)=t_0\delta (t-t_0),\widehat{\mathsf{\Omega }}{e}^{i\omega t}=\omega e^{i\omega t}.$$

(19)

Both ${\displaystyle \delta (t-t_0)}$ and ${\displaystyle e^{i\omega t}}$ are generalized vectors as they are not elements in ${\displaystyle L^2(\mathbb{R},\mathrm{d}x)}$.

3. Incorporation of a Minimal Time Scale

A minimal time scale can be incorporated into the formalism by modifying the commutation relations for the operators of time and frequency, ${\displaystyle \widehat{T}}$ and ${\displaystyle \widehat{\mathsf{\Omega }}}$:

$$[\widehat{T},\widehat{\mathsf{\Omega }}]=i\left(\widehat{1}+\kappa {\widehat{\mathsf{\Omega }}}^2\right).$$

(20)

Here ${\displaystyle \kappa }$ is a positive constant describing the smallest possible resolution with which we can measure time. For ${\displaystyle \kappa =0}$, we recover the standard commutation relations for ${\displaystyle \widehat{T}}$ and ${\displaystyle \widehat{\mathsf{\Omega }}}$. The commutation relations of the form (20) for the operators of position and momentum were considered in [30] from the point of view of a minimal length scale in quantum mechanics. In the following subsection, we will review some of the results from this paper which will be crucial for our later considerations.

3.1. Minimal Time Scale Uncertainty Relation

The commutation relations (20) imply the following generalized uncertainty principle for the uncertainties ${\displaystyle \mathsf{\Delta }t}$ and ${\displaystyle \mathsf{\Delta }\omega }$ of the measurements of time and frequency:

$$\mathsf{\Delta }t\mathsf{\Delta }\omega \ge {\displaystyle \frac{1}{2}}\left(1+\kappa {\left(\mathsf{\Delta }\omega \right)}^2+\kappa {\langle \widehat{\mathsf{\Omega }}\rangle }^2\right).$$

(21)

From (21), it is not difficult to see that there exists a smallest possible value for the uncertainty ${\displaystyle \mathsf{\Delta }t}$ (see Figure 1):

$$\mathsf{\Delta }{t}_{\min }=\sqrt{\kappa \left(1+\kappa {\langle \widehat{\mathsf{\Omega }}\rangle }^2\right)}.$$

(22)

Every physical state cannot have the uncertainty in time smaller than this value. For states for which the expectation value of the frequency operator ${\displaystyle \widehat{\mathsf{\Omega }}}$ is equal zero, we receive the absolutely smallest uncertainty in time equaling

$$\mathsf{\Delta }{t}_0=\sqrt{\kappa }.$$

(23)

It should be noted that there are states in the Hilbert space ${\displaystyle {\mathcal{H}}_T}$, like, for example, the eigenstates of the time operator ${\displaystyle \widehat{T}}$, for which the uncertainty in time is smaller than ${\displaystyle \mathsf{\Delta }{t}_{\min }}$. Such states are not considered physical.

The modification (20) of the commutation relations for ${\displaystyle \widehat{T}}$ and ${\displaystyle \widehat{\mathsf{\Omega }}}$ is the simplest one leading to a minimal uncertainty in time. Other modifications are possible; however, we will not be dealing with such cases in the current work.

The introduction of a minimal uncertainty in time will cause the time operator ${\displaystyle \widehat{T}}$ to not be essentially self-adjoint but only symmetric. A consequence of this is that eigenvectors of ${\displaystyle \widehat{T}}$ are not physical states. It is no longer possible to consider states of the system ${\displaystyle S}$ at particular instances of time. Therefore, we do not have at a disposal the representation of the Hilbert space ${\displaystyle {\mathcal{H}}_T}$ given by projecting states ${\displaystyle {|\psi \rangle }_T}$ onto eigenvectors ${\displaystyle {|t\rangle }_T}$ of ${\displaystyle \widehat{T}}$. However, the frequency operator ${\displaystyle \widehat{\mathsf{\Omega }}}$ is still self-adjoint, and therefore we can use it to construct a frequency representation of the Hilbert space ${\displaystyle {\mathcal{H}}_T}$. The Hilbert space ${\displaystyle {\mathcal{H}}_T}$ is represented as a space ${\displaystyle L^2(\mathbb{R},\mathrm{d}\mu )}$ of functions defined on ${\displaystyle \mathbb{R}}$ and square integrable with respect to the measure ${\displaystyle \mathrm{d}\mu \left(\omega \right)=\frac{\mathrm{d}\omega }{1+\kappa {\omega }^2}}$. Such functions can be formally constructed by projecting states ${\displaystyle {|\psi \rangle }_T}$ onto eigenvectors ${\displaystyle {|\omega \rangle }_T}$ of ${\displaystyle \widehat{\mathsf{\Omega }}}$:

$$\psi \left(\omega \right)={}_T\langle \omega |\psi \rangle _T.$$

(24)

The scalar product in the Hilbert space ${\displaystyle L^2(\mathbb{R},\mathrm{d}\mu )}$ is given by the formula

$${}_T\langle \phi |\psi \rangle _T={\int }_{-\infty }^{+\infty }{\displaystyle \frac{\mathrm{d}\omega }{1+\kappa {\omega }^2}}\overline{\phi \left(\omega \right)}\psi \left(\omega \right),$$

(25)

and the frequency and time operators ${\displaystyle \widehat{\mathsf{\Omega }}}$, ${\displaystyle \widehat{T}}$ are represented as appropriate multiplication and differentiation operators:

$$\widehat{\mathsf{\Omega }}\psi \left(\omega \right)=\omega \psi \left(\omega \right),\widehat{T}\psi \left(\omega \right)=i(1+\kappa {\omega }^2){\partial }_{\omega }\psi \left(\omega \right).$$

(26)

One can check with a direct calculation that the operators defined by (26) satisfy commutation relations (20).

Although eigenvectors of the time operator ${\displaystyle \widehat{T}}$ are not physical states, it is possible to define states which are physical and closely resemble eigenstates of ${\displaystyle \widehat{T}}$. These are states of maximal localization around instances of time. A state ${\displaystyle {|{\phi }_{\tau }^{ML}\rangle }_T}$ of maximal localization around an instance of time ${\displaystyle \tau }$ is a state for which the expectation value of ${\displaystyle \widehat{T}}$ is equal ${\displaystyle \tau }$ and the uncertainty of ${\displaystyle \widehat{T}}$ is equal to the smallest possible value ${\displaystyle \mathsf{\Delta }{t}_0=\sqrt{\kappa }}$:

$${}_T\langle {\phi }_{\tau }^{ML}|\widehat{T}|{\phi }_{\tau }^{ML}\rangle _T=\tau ,{\left(\mathsf{\Delta }t\right)}_{{|{\phi }_{\tau }^{ML}\rangle }_T}=\mathsf{\Delta }{t}_0.$$

(27)

The conditions in (27) uniquely determine the state ${\displaystyle {|{\phi }_{\tau }^{ML}\rangle }_T}$, which in frequency representation is given by the formula

$${\phi }_{\tau }^{ML}\left(\omega \right)=\sqrt{{\displaystyle \frac{2\sqrt{\kappa }}{\pi }}}{(1+\kappa {\omega }^2)}^{-1/2}exp\left(-i\tau {\displaystyle \frac{arctan\left(\sqrt{\kappa }\omega \right)}{\sqrt{\kappa }}}\right).$$

(28)

Using the states of maximal localization, it is possible to define a representation of the Hilbert space ${\displaystyle {\mathcal{H}}_T}$, which we will call a continuous time representation. This representation is given by projecting states ${\displaystyle {|\psi \rangle }_T}$ in ${\displaystyle {\mathcal{H}}_T}$ with states of maximal localization

$$\psi \left(\tau \right)={}_T\langle {\phi }_{\tau }^{ML}|\psi \rangle _T.$$

(29)

The received wave functions ${\displaystyle \psi \left(\tau \right)}$ describe the probability amplitude for the system being maximally localized around the instance of time ${\displaystyle \tau }$. In the limit ${\displaystyle \kappa \to 0}$, the wave function ${\displaystyle \psi \left(\tau \right)={}_T\langle \tau |\psi \rangle _T}$ of the ordinary time representation is recovered. From (29), we receive the transformation of a state’s wave function in the frequency representation into its continuous time representation:

$$\psi \left(\tau \right)=\sqrt{{\displaystyle \frac{2\sqrt{\kappa }}{\pi }}}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\mathrm{d}\omega }{{(1+\kappa {\omega }^2)}^{3/2}}}exp\left(i\tau {\displaystyle \frac{arctan\left(\sqrt{\kappa }\omega \right)}{\sqrt{\kappa }}}\right)\psi \left(\omega \right).$$

(30)

The inverse transformation is given by the formula

$$\psi \left(\omega \right)={\displaystyle \frac{1}{\sqrt{8\pi \sqrt{\kappa }}}}{\int }_{-\infty }^{+\infty }\mathrm{d}\tau {(1+\kappa {\omega }^2)}^{1/2}exp\left(-i\tau {\displaystyle \frac{arctan\left(\sqrt{\kappa }\omega \right)}{\sqrt{\kappa }}}\right)\psi \left(\tau \right).$$

(31)

The operators ${\displaystyle \widehat{\mathsf{\Omega }}}$ and ${\displaystyle \widehat{T}}$ in the continuous time representation take the form

$$\widehat{\mathsf{\Omega }}\psi \left(\tau \right)={\displaystyle \frac{tan(-i\sqrt{\kappa }{\partial }_{\tau })}{\sqrt{\kappa }}}\psi \left(\tau \right),\widehat{T}\psi \left(\tau \right)=\left(\tau +i\kappa {\displaystyle \frac{tan(-i\sqrt{\kappa }{\partial }_{\tau })}{\sqrt{\kappa }}}\right)\psi \left(\tau \right).$$

(32)

3.2. Discrete Time Representation

The family of states ${\displaystyle \left\{{|{\phi }_{\tau }^{ML}\rangle }_T\right\}}$ for ${\displaystyle \tau \in \mathbb{R}}$ forms an overcomplete set of vectors. We can choose from this set a smaller countable set forming a basis in the Hilbert space ${\displaystyle {\mathcal{H}}_T}$. Using this basis, we can construct another representation of ${\displaystyle {\mathcal{H}}_T}$ which we will call a discrete time representation. For ${\displaystyle \lambda \in [0,1)}$, let us consider the following set of vectors:

$$\{{|{\phi }_{2\sqrt{\kappa }(\lambda +n)}^{ML}\rangle }_T\mid n\in \mathbb{Z}\}.$$

(33)

The vectors composing this set are linearly independent and the set itself is complete. Thus, we receive a one-parameter family of bases of the Hilbert space ${\displaystyle {\mathcal{H}}_T}$. These bases can be viewed as lattices with spacing ${\displaystyle 2\sqrt{\kappa }=2\mathsf{\Delta }{t}_0}$ shifted by ${\displaystyle 2\sqrt{\kappa }\lambda }$. As we will see, the resulting discrete time representations will describe time evolution on lattices. The vectors from the set (33) are not orthogonal. The scalar product of two states of maximal localization is equal:

$${}_T\langle {\phi }_{{\tau }^{\prime }}^{ML}|{\phi }_{\tau }^{ML}\rangle _T={\displaystyle \frac{1}{\pi }}{\left({\displaystyle \frac{\tau -{\tau }^{\prime }}{2\sqrt{\kappa }}}-{\left({\displaystyle \frac{\tau -{\tau }^{\prime }}{2\sqrt{\kappa }}}\right)}^3\right)}^{-1}sin\left({\displaystyle \frac{\tau -{\tau }^{\prime }}{2\sqrt{\kappa }}}\pi \right).$$

(34)

From this, we can see that

$${}_T\langle {\phi }_{2\sqrt{\kappa }(\lambda +m)}^{ML}|{\phi }_{2\sqrt{\kappa }(\lambda +n)}^{ML}\rangle _T=\left\{\begin{array}{cc}1\hfill & \mathrm{for}{\displaystyle m=n}\hfill \\ \frac{1}{2}\hfill & \mathrm{for}{\displaystyle m=n\pm 1}\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

(35)

so that the vectors from the set (33) are orthonormal except for neighboring vectors.

Using a basis (33), we can construct a discrete time representation of the Hilbert space ${\displaystyle {\mathcal{H}}_T}$. Instead of expanding a state ${\displaystyle {|\psi \rangle }_T}$ in this basis and taking the sequence of coefficients of this expansion as the representation of the vector ${\displaystyle {|\psi \rangle }_T}$, we will represent the state ${\displaystyle {|\psi \rangle }_T}$ with the following sequence:

$${\psi }_n={}_T\langle {\phi }_{2\sqrt{\kappa }(\lambda +n)}^{ML}|\psi \rangle _T.$$

(36)

From this formula, we receive the transformation of a state’s wave function in the frequency representation into its discrete time representation:

$${\psi }_n=\sqrt{{\displaystyle \frac{2\sqrt{\kappa }}{\pi }}}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\mathrm{d}\omega }{{(1+\kappa {\omega }^2)}^{3/2}}}{e}^{2i(\lambda +n)arctan\left(\sqrt{\kappa }\omega \right)}\psi \left(\omega \right).$$

(37)

To calculate the inverse transform, let us rewrite the above formula by changing the variable under the integral sign:

$${\psi }_n={\displaystyle \frac{1}{2\pi }}{\int }_{-\pi }^{+\pi }\sqrt{{\displaystyle \frac{2\pi }{\sqrt{\kappa }}}}\psi \left({\displaystyle \frac{1}{\sqrt{\kappa }}}tan{\displaystyle \frac{x}{2}}\right){\displaystyle \frac{1}{{(1+{tan}^2\frac{x}{2})}^{1/2}}}{e}^{i\lambda x}{e}^{inx}\mathrm{d}x.$$

(38)

From this, we can see that ${\displaystyle {\psi }_n}$ are Fourier coefficients in the expansion of the function

$$f\left(x\right)=\sqrt{{\displaystyle \frac{2\pi }{\sqrt{\kappa }}}}\psi \left({\displaystyle \frac{1}{\sqrt{\kappa }}}tan{\displaystyle \frac{x}{2}}\right){\displaystyle \frac{1}{{(1+{tan}^2\frac{x}{2})}^{1/2}}}{e}^{i\lambda x}$$

(39)

into the Fourier series:

$$f\left(x\right)=\sum _{n=-\infty }^{\infty }{\psi }_n{e}^{-inx}.$$

(40)

The function ${\displaystyle f}$ is square integrable, which follows from the square integrability of ${\displaystyle \psi }$ with respect to the measure ${\displaystyle \mathrm{d}\mu \left(\omega \right)=\frac{\mathrm{d}\omega }{1+\kappa {\omega }^2}}$. Therefore, by virtue of Carleson’s theorem, the above Fourier series converges for almost all ${\displaystyle x}$, where the infinite sum ${\sum }_{n=-\infty }^{\infty }$ is viewed as the limit ${lim}_{N\to \infty }{\sum }_{n=-N}^N$. We can then change the variable again to receive the following inverse transform of (37):

$$\psi \left(\omega \right)=\sqrt{{\displaystyle \frac{\sqrt{\kappa }}{2\pi }}}\sum _{n=-\infty }^{\infty }{(1+\kappa {\omega }^2)}^{1/2}{e}^{-2i(\lambda +n)arctan\left(\sqrt{\kappa }\omega \right)}{\psi }_n.$$

(41)

The transformation of a state’s wave function in the continuous time representation into its discrete time representation is the following:

$${\psi }_n=\psi \left(2\sqrt{\kappa }(\lambda +n)\right).$$

(42)

To calculate the inverse transform, we combine transformations (30) and (41) receiving

$$\begin{array}{cc}\hfill \psi \left(\tau \right)& ={\displaystyle \frac{\sqrt{\kappa }}{\pi }}{\int }_{-\infty }^{+\infty }\sum _{n=-\infty }^{\infty }{\psi }_n{e}^{i(\tau /\sqrt{\kappa }-2(\lambda +n))arctan\left(\sqrt{\kappa }\omega \right)}{\displaystyle \frac{\mathrm{d}\omega }{1+\kappa {\omega }^2}}\hfill \\ \hfill & ={\displaystyle \frac{1}{\pi }}\sum _{n=-\infty }^{\infty }{\psi }_n{\int }_{-\pi /2}^{+\pi /2}{e}^{i(\tau /\sqrt{\kappa }-2(\lambda +n))x}\mathrm{d}x.\hfill \end{array}$$

(43)

Calculating the integral in the above formula, we receive the inverse transform of (42):

$$\psi \left(\tau \right)=\sum _{n=-\infty }^{\infty }{\psi }_{n}sinc\left({\displaystyle \frac{\tau -2\sqrt{\kappa }(\lambda +n)}{2\sqrt{\kappa }}}\right),$$

(44)

where ${\displaystyle sincx=\frac{sin\left(\pi x\right)}{\pi x}}$. The transformations (42) and (44) might seem surprising because they allow us to reproduce the function ${\displaystyle \psi \left(\tau \right)}$ from the knowledge of its values at a countable set of points. This is, however, nothing strange because from (30) after changing the variable under the integral, we can see that ${\displaystyle \psi \left(\tau \right)}$ is a Fourier transform of a square integrable function with compact support. Thus, by virtue of Paley–Wiener theorem, ${\displaystyle \psi \left(\tau \right)}$ extends to an entire function of exponential type and entire functions are completely determined by their values at a countable set of points.

In what follows, we will derive the representation of the frequency operator ${\displaystyle \widehat{\mathsf{\Omega }}}$ in the discrete time representation. Let us define a discrete derivative with the formula

$$D_n{\psi }_n={\displaystyle \frac{{\psi }_{n+1}-{\psi }_{n-1}}{4\mathsf{\Delta }{t}_0}}.$$

(45)

Using (37) we calculate

$$\begin{array}{cc}\hfill -i{D}_n{\psi }_n& =-i{\displaystyle \frac{{\psi }_{n+1}-{\psi }_{n-1}}{4\sqrt{\kappa }}}\hfill \\ \hfill & ={\displaystyle \frac{-i}{4\sqrt{\kappa }}}\sqrt{{\displaystyle \frac{2\sqrt{\kappa }}{\pi }}}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\mathrm{d}\omega }{{(1+\kappa {\omega }^2)}^{3/2}}}\left(e^{2i(\lambda +n+1)arctan\left(\sqrt{\kappa }\omega \right)}-e^{2i(\lambda +n-1)arctan\left(\sqrt{\kappa }\omega \right)}\right)\psi \left(\omega \right)\hfill \\ \hfill & ={\displaystyle \frac{-i}{4\sqrt{\kappa }}}\sqrt{{\displaystyle \frac{2\sqrt{\kappa }}{\pi }}}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\mathrm{d}\omega }{{(1+\kappa {\omega }^2)}^{3/2}}}{e}^{2i(\lambda +n)arctan\left(\sqrt{\kappa }\omega \right)}\left(e^{2iarctan\left(\sqrt{\kappa }\omega \right)}-e^{-2iarctan\left(\sqrt{\kappa }\omega \right)}\right)\psi \left(\omega \right)\hfill \\ \hfill & =\sqrt{{\displaystyle \frac{2\sqrt{\kappa }}{\pi }}}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\mathrm{d}\omega }{{(1+\kappa {\omega }^2)}^{3/2}}}{e}^{2i(\lambda +n)arctan\left(\sqrt{\kappa }\omega \right)}{\displaystyle \frac{1}{2\sqrt{\kappa }}}sin\left(2arctan\left(\sqrt{\kappa }\omega \right)\right)\psi \left(\omega \right)\hfill \\ \hfill & =\sqrt{{\displaystyle \frac{2\sqrt{\kappa }}{\pi }}}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\mathrm{d}\omega }{{(1+\kappa {\omega }^2)}^{3/2}}}{e}^{2i(\lambda +n)arctan\left(\sqrt{\kappa }\omega \right)}{\displaystyle \frac{\omega }{1+\kappa {\omega }^2}}\psi \left(\omega \right).\hfill \end{array}$$

(46)

From this we get

$${(-i{D}_n)}^k{\psi }_n=\sqrt{{\displaystyle \frac{2\sqrt{\kappa }}{\pi }}}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\mathrm{d}\omega }{{(1+\kappa {\omega }^2)}^{3/2}}}{e}^{2i(\lambda +n)arctan\left(\sqrt{\kappa }\omega \right)}{\left({\displaystyle \frac{\omega }{1+\kappa {\omega }^2}}\right)}^k\psi \left(\omega \right)$$

(47)

and consequently

$${\displaystyle \frac{1}{\sqrt{\kappa }}}f(-i\sqrt{\kappa }{D}_n){\psi }_n=\sqrt{{\displaystyle \frac{2\sqrt{\kappa }}{\pi }}}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\mathrm{d}\omega }{{(1+\kappa {\omega }^2)}^{3/2}}}{e}^{2i(\lambda +n)arctan\left(\sqrt{\kappa }\omega \right)}\omega \psi \left(\omega \right),$$

(48)

where ${\displaystyle f\left(x\right)=\frac{2x}{1+\sqrt{1-x^2}}}$ is a function with an inverse ${\displaystyle f^{-1}\left(x\right)=\frac{x}{1+x^2}}$ and we used the expansion of this function in a Taylor series

$$f\left(x\right)=x+{\displaystyle \frac{1}{4}}{x}^3+{\displaystyle \frac{1}{8}}{x}^5+{\displaystyle \frac{5}{64}}{x}^7+\cdots .$$

(49)

From (48) and the fact that in the frequency representation the operator ${\displaystyle \widehat{\mathsf{\Omega }}}$ is a multiplication operator (see (26)) we receive the discrete time representation of the frequency operator

$$\widehat{\mathsf{\Omega }}{\psi }_n={\displaystyle \frac{f(-i\sqrt{\kappa }{D}_n)}{\sqrt{\kappa }}}{\psi }_n.$$

(50)

Using this result and Formulas (26) and (37) we can also receive the discrete time representation of the time operator

$$\widehat{T}{\psi }_n=\left(2\sqrt{\kappa }(\lambda +n)+i\kappa {\displaystyle \frac{f(-i\sqrt{\kappa }{D}_n)}{\sqrt{\kappa }}}\right){\psi }_n.$$

(51)

3.3. Time Evolution with a Minimal Time Scale

The introduction of a minimal time scale into the formalism will change the time evolution of the system. Since the time operator ${\displaystyle \widehat{T}}$ is no longer self-adjoint, its eigenvectors are not physical states and it is meaningless to talk about the system at particular instances of time. We can only consider the system localized around instances of time with a finite precision. For this reason, as indicators of time of the evolving system, we will take states of maximal localization around instances of time ${\displaystyle {|{\phi }_{\tau }^{ML}\rangle }_T}$ instead of eigenstates ${\displaystyle {|t\rangle }_T}$ of the time operator ${\displaystyle \widehat{T}}$.

The state ${\displaystyle {|\psi \left(\tau \right)\rangle }_S}$ of the system ${\displaystyle S}$ maximally localized around an instance of time ${\displaystyle \tau }$ can be obtained via projection of the physical state ${\displaystyle |\mathsf{\Psi }\rangle \rangle }$ with the maximal localization state ${\displaystyle {|{\phi }_{\tau }^{ML}\rangle }_T}$:

$${|\psi \left(\tau \right)\rangle }_S={}_T\langle {\phi }_{\tau }^{ML}|\mathsf{\Psi }\rangle \rangle _{.}$$

(52)

To receive a modified Schrödinger equation governing the time evolution of states ${\displaystyle {|\psi \left(\tau \right)\rangle }_S}$, we write the constraint equation ${\displaystyle \widehat{J}|\mathsf{\Psi }\rangle \rangle =0}$ in the continuous or discrete time representation in ${\displaystyle {\mathcal{H}}_T}$. Before doing this, first let us use Theorem A1 from Appendix A with the function ${\displaystyle f\left(x\right)=\frac{1}{\sqrt{\kappa }}arctan\left(\sqrt{\kappa }x\right)}$ to rewrite the constraint equation

$$\left(-\widehat{\mathsf{\Omega }}\otimes {\widehat{1}}_S\right)|\mathsf{\Psi }\rangle \rangle =\left({\widehat{1}}_T\otimes {\widehat{H}}_S/\hslash \right)|\mathsf{\Psi }\rangle \rangle$$

(53)

in a different form

$$\begin{array}{c}{\displaystyle \frac{1}{\sqrt{\kappa }}}arctan\left(-\sqrt{\kappa }\widehat{\mathsf{\Omega }}\otimes {\widehat{1}}_S\right)|\mathsf{\Psi }\rangle \rangle ={\displaystyle \frac{1}{\sqrt{\kappa }}}arctan\left(\sqrt{\kappa }{\widehat{1}}_T\otimes {\widehat{H}}_S/\hslash \right)|\mathsf{\Psi }\rangle \rangle \\ \Updownarrow \\ \left({\displaystyle \frac{1}{\sqrt{\kappa }}}arctan\left(-\sqrt{\kappa }\widehat{\mathsf{\Omega }}\right)\otimes {\widehat{1}}_S\right)|\mathsf{\Psi }\rangle \rangle =\left({\widehat{1}}_T\otimes {\displaystyle \frac{1}{\sqrt{\kappa }}}arctan\left(\sqrt{\kappa }{\widehat{H}}_S/\hslash \right)\right)|\mathsf{\Psi }\rangle \rangle .\end{array}$$

(54)

In the continuous time representation, the constraint equation takes the form

$$\begin{array}{c}{}_T\langle {\phi }_{\tau }^{ML}|\widehat{J}|\mathsf{\Psi }\rangle \rangle =0\\ \Updownarrow \\ {}_T\langle {\phi }_{\tau }^{ML}|\left({\displaystyle \frac{1}{\sqrt{\kappa }}}arctan\left(-\sqrt{\kappa }\widehat{\mathsf{\Omega }}\right)\otimes {\widehat{1}}_S\right)|\mathsf{\Psi }\rangle \rangle ={}_T\langle {\phi }_{\tau }^{ML}|\left({\widehat{1}}_T\otimes {\displaystyle \frac{1}{\sqrt{\kappa }}}arctan\left(\sqrt{\kappa }{\widehat{H}}_S/\hslash \right)\right)|\mathsf{\Psi }\rangle \rangle \end{array}$$

(55)

from which and (32) we receive the modified Schrödinger equation

$$i{\displaystyle \frac{\partial }{\partial \tau }}{|\psi \left(\tau \right)\rangle }_S={\displaystyle \frac{1}{\sqrt{\kappa }}}arctan\left(\sqrt{\kappa }{\widehat{H}}_S/\hslash \right){|\psi \left(\tau \right)\rangle }_S.$$

(56)

Similarly, by writing the constraint equation in the discrete time representation and using (50), we receive the discrete Schrödinger equation

$$i\hslash D_n{|{\psi }_n\rangle }_S={\widehat{H}}_S{\left(\widehat{1}+\kappa {({\widehat{H}}_S/\hslash )}^2\right)}^{-1}{|{\psi }_n\rangle }_S.$$

(57)

The solution of (56) subject to an initial condition ${\displaystyle {|\psi (\tau =0)\rangle }_S={|{\psi }_0\rangle }_S}$ is of the form

$${|\psi \left(\tau \right)\rangle }_S=exp\left(-i\tau {\displaystyle \frac{1}{\sqrt{\kappa }}}arctan(\sqrt{\kappa }{\widehat{H}}_S/\hslash )\right){|{\psi }_0\rangle }_S$$

(58)

and the solution of (57) subject to an initial condition ${\displaystyle {|{\psi }_{n=0}\rangle }_S={|{\psi }_0\rangle }_S}$ is of the form

$${|{\psi }_n\rangle }_S=exp\left(-2inarctan(\sqrt{\kappa }{\widehat{H}}_S/\hslash )\right){|{\psi }_0\rangle }_S.$$

(59)

Equations (56) and (57) are equivalent and describe the same time evolution of the system. Knowing the solution of either of these equations, we can reconstruct the solution of the other equation by applying the transform (42) or (44).

4. Examples

4.1. Spin-${\displaystyle 1/2}$ in an External Magnetic Field

In this section, we will investigate how the minimal time scale influences the time evolution of a spin-${\displaystyle 1/2}$ in an external constant magnetic field. The Hilbert space of the system is ${\displaystyle {\mathcal{H}}_S={\mathbb{C}}^2}$ and the Hamiltonian of the system is

$${\widehat{H}}_S=-\gamma B_0{\widehat{S}}_z=-\gamma B_0{\displaystyle \frac{\hslash }{2}}{\sigma }_z,$$

(60)

where ${\displaystyle \gamma }$ is the gyromagnetic ratio; ${\displaystyle B_0}$ is the strength of the magnetic field, which we assume is directed along the ${\displaystyle z}$-axis; and ${\displaystyle {\widehat{S}}_z}$ is the ${\displaystyle z}$-component of the spin operator expressed by the Pauli matrix

$${\sigma }_z=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right].$$

(61)

Let us assume that the system is initially in an arbitrary state:

$${|\psi \rangle }_S={|\theta ,\phi \rangle }_S=cos(\theta /2)e^{-i\phi /2}{|\uparrow \rangle }_S+sin(\theta /2)e^{i\phi /2}{|\downarrow \rangle }_S,$$

(62)

where ${\displaystyle \theta ,\phi \in \mathbb{R}}$ and ${\displaystyle {|\uparrow \rangle }_S}$, ${\displaystyle {|\downarrow \rangle }_S}$ are spin-up and spin-down eigenstates of the spin operator ${\displaystyle {\widehat{S}}_z}$. In the case without a minimal time scale (${\displaystyle \kappa =0}$), the state ${\displaystyle {|\psi \rangle }_S}$ will evolve in time according to the formula

$$\begin{array}{cc}\hfill {|\psi \left(t\right)\rangle }_S& =e^{-it{\widehat{H}}_S/\hslash }{|\psi \rangle }_S=e^{-it{\omega }_0{\sigma }_z/2}{|\psi \rangle }_S\hfill \\ \hfill & =cos(\theta /2)e^{-i(\phi +{\omega }_{0}t)/2}{|\uparrow \rangle }_S+sin(\theta /2)e^{i(\phi +{\omega }_{0}t)/2}{|\downarrow \rangle }_S\hfill \\ \hfill & ={|\theta ,\phi +{\omega }_{0}t\rangle }_S,\hfill \end{array}$$

(63)

where ${\displaystyle {\omega }_0=-\gamma B_0}$. The spin is precessing around the ${\displaystyle z}$-axis with the frequency ${\displaystyle {\omega }_0}$. This is the usual Larmour precession.

Let us now consider the case of a non-zero minimal time scale (${\displaystyle \kappa >0}$). The state ${\displaystyle {|\psi \rangle }_S}$ will evolve in time according to the Formula (58)

$${|\psi \left(\tau \right)\rangle }_S=exp\left(-i\tau {\displaystyle \frac{1}{\sqrt{\kappa }}}arctan(\sqrt{\kappa }{\omega }_0{\sigma }_z/2)\right){|\psi \rangle }_S.$$

(64)

Using the fact that ${\displaystyle {\sigma }_z^2=I}$, we can calculate explicitly ${\displaystyle arctan}$ of the matrix ${\displaystyle {\sigma }_z}$ receiving

$$\begin{array}{cc}\hfill {|\psi \left(\tau \right)\rangle }_S& =exp\left(-i\tau {\displaystyle \frac{1}{\sqrt{\kappa }}}arctan(\sqrt{\kappa }{\omega }_0/2){\sigma }_z\right){|\psi \rangle }_S\hfill \\ \hfill & =exp\left(-i\tau {\omega }_{\kappa }{\sigma }_z/2\right){|\psi \rangle }_S\hfill \\ \hfill & =cos(\theta /2)e^{-i(\phi +{\omega }_{\kappa }\tau )/2}{|\uparrow \rangle }_S+sin(\theta /2)e^{i(\phi +{\omega }_{\kappa }\tau )/2}{|\downarrow \rangle }_S\hfill \\ \hfill & ={|\theta ,\phi +{\omega }_{\kappa }\tau \rangle }_S,\hfill \end{array}$$

(65)

where ${\displaystyle {\omega }_{\kappa }=\frac{2}{\sqrt{\kappa }}arctan(\sqrt{\kappa }{\omega }_0/2)}$. The spin is now precessing around the ${\displaystyle z}$-axis with the smaller frequency ${\displaystyle {\omega }_{\kappa }}$. Notice, moreover, that since ${\displaystyle arctan}$ is a bounded function, there is an upper limit for the frequency with which a spin can precess. The frequency of precession is necessarily smaller than ${\displaystyle \pi /\sqrt{\kappa }}$ and this upper limit is independent on the strength of the magnetic field. This phenomenon results from the existence of the fundamental limit for the precision with which we can measure time.

4.2. Free Particle

In this section, we will investigate how the minimal time scale influences the time evolution of a free particle moving on a line. The Hilbert space of the system is ${\displaystyle {\mathcal{H}}_S=L^2(\mathbb{R},\mathrm{d}x)}$ and the Hamiltonian of the system is

$${\widehat{H}}_S={\displaystyle \frac{1}{2m}}{\widehat{p}}^2,$$

(66)

where ${\displaystyle m}$ is the mass of a particle. Let us assume that the system is initially in an arbitrary state written in the position representation as

$$\psi \left(x\right)={\displaystyle \frac{1}{\sqrt{2\pi \hslash }}}{\int }_{-\infty }^{+\infty }f\left(p\right)e^{{\displaystyle \frac{i}{\hslash }}px}\mathrm{d}p,$$

(67)

where ${\displaystyle f}$ is an arbitrary function such that ${\displaystyle \psi }$ will be in the domains of the position and momentum operators ${\displaystyle \widehat{x}}$, ${\displaystyle \widehat{p}}$ and their squares ${\displaystyle {\widehat{x}}^2}$, ${\displaystyle {\widehat{p}}^2}$. It is not difficult to see that the function

$$\psi (\tau ,x)={\displaystyle \frac{1}{\sqrt{2\pi \hslash }}}{\int }_{-\infty }^{+\infty }f\left(p\right)e^{{\displaystyle \frac{i}{\hslash }}(px-E\left(p\right)\tau )}\mathrm{d}p,$$

(68)

where

$$E\left(p\right)={\displaystyle \frac{\hslash }{\sqrt{\kappa }}}arctan\left({\displaystyle \frac{\sqrt{\kappa }{p}^2}{2m\hslash }}\right)$$

(69)

is a solution to the modified Schrödinger Equation (56) satisfying the initial condition ${\displaystyle \psi (\tau =0,x)=\psi \left(x\right)}$.

In accordance with (56), the effective Hamiltonian describing the time evolution of the system is

$${\widehat{H}}_{\mathrm{eff}}={\displaystyle \frac{\hslash }{\sqrt{\kappa }}}arctan\left(\sqrt{\kappa }{\widehat{H}}_S/\hslash \right)={\displaystyle \frac{\hslash }{\sqrt{\kappa }}}arctan\left({\displaystyle \frac{\sqrt{\kappa }{\widehat{p}}^2}{2m\hslash }}\right).$$

(70)

By the first equation of Hamilton’s equations in classical Hamiltonian mechanics

$$\dot{q}={\displaystyle \frac{\partial H}{\partial p}},\dot{p}=-{\displaystyle \frac{\partial H}{\partial q}}$$

(71)

we can define a velocity operator by the formula

$$\widehat{v}={\displaystyle \frac{\widehat{p}}{m}}{\left(\widehat{1}+{\displaystyle \frac{\kappa {\widehat{p}}^4}{4m^2{\hslash }^2}}\right)}^{-1}.$$

(72)

The expectation value of the momentum operator ${\displaystyle \widehat{p}}$ in the state ${\displaystyle \psi (\tau ,x)}$ is equal

$${\langle \widehat{p}\rangle }_{\psi \left(\tau \right)}={\int }_{-\infty }^{+\infty }\overline{\tilde{\psi }(\tau ,p)}p\tilde{\psi }(\tau ,p)\mathrm{d}p={\int }_{-\infty }^{+\infty }p{\left|f\left(p\right)\right|}^2\mathrm{d}p={\langle \widehat{p}\rangle }_{\psi \left(0\right)},$$

(73)

where ${\displaystyle \tilde{\psi }(\tau ,p)=f\left(p\right)e^{-\frac{i}{\hslash }E\left(p\right)\tau }}$ is the Fourier transform of ${\displaystyle \psi (\tau ,x)}$. Notice that the expectation value of the momentum is the same as in the ${\displaystyle \kappa =0}$ case and is independent of time.

The expectation value of the position operator ${\displaystyle \widehat{x}}$ in the state ${\displaystyle \psi (\tau ,x)}$ is equal to

$$\begin{array}{cc}\hfill {\langle \widehat{x}\rangle }_{\psi \left(\tau \right)}& ={\int }_{-\infty }^{+\infty }\overline{\tilde{\psi }(\tau ,p)}i\hslash {\partial }_p\tilde{\psi }(\tau ,p)\mathrm{d}p\hfill \\ \hfill & =i\hslash {\int }_{-\infty }^{+\infty }\overline{f\left(p\right)}{f}^{\prime }\left(p\right)\mathrm{d}p+\tau {\int }_{-\infty }^{+\infty }{\left|f\left(p\right)\right|}^2{E}^{\prime }\left(p\right)\mathrm{d}p\hfill \\ \hfill & ={\langle \widehat{x}\rangle }_{\psi \left(0\right)}+{\displaystyle \frac{\tau }{m}}{\int }_{-\infty }^{+\infty }{\left|f\left(p\right)\right|}^2{\displaystyle \frac{p}{1+\kappa p^4/4m^2{\hslash }^2}}\mathrm{d}p\hfill \\ \hfill & ={\langle \widehat{x}\rangle }_{\psi \left(0\right)}+\tau {\langle \widehat{v}\rangle }_{\psi \left(0\right)},\hfill \end{array}$$

(74)

which shows that indeed ${\displaystyle \widehat{v}}$ can be interpreted as the velocity operator.

The expectation value of the velocity operator ${\displaystyle \widehat{v}}$ in the state ${\displaystyle \psi (\tau ,x)}$ is equal to

$${\langle \widehat{v}\rangle }_{\psi \left(\tau \right)}={\displaystyle \frac{1}{m}}{\int }_{-\infty }^{+\infty }{\left|f\left(p\right)\right|}^2{\displaystyle \frac{p}{1+\kappa p^4/4m^2{\hslash }^2}}\mathrm{d}p$$

(75)

and is independent of time. Notice that the expectation value of the velocity is smaller than in the ${\displaystyle \kappa =0}$ case, where it is equal to the expectation value of the momentum divided by the mass ${\displaystyle m}$. Since the state ${\displaystyle \psi (\tau ,x)}$ is normalized to unity and the function ${\displaystyle \frac{p/m}{1+\kappa p^4/4m^2{\hslash }^2}}$ is bounded with a maximum value equal

$$v_{\max }=\sqrt{{\displaystyle \frac{3\sqrt{3}\hslash }{8m\sqrt{\kappa }}}}$$

(76)

this value will be the upper limit on the speed of propagation of wave packets. Notice that ${\displaystyle v_{\max }}$ does not depend on the initial state ${\displaystyle \psi \left(x\right)}$, so in particular on the expectation value of the momentum, but it depends on the mass ${\displaystyle m}$ of the particle. The heavier the particle, the smaller this speed limit is. The existence of the upper limit with which wave packets can propagate is another consequence of the fundamental limit for the precision with which we can measure time.

As an example, let us calculate ${\displaystyle v_{\max }}$ for the proton assuming that ${\displaystyle \mathsf{\Delta }{t}_0=\sqrt{\kappa }}$ is equal to the Planck’s time ${\displaystyle t_p}$. We obtain the value

$$v_{\max }\approx 8.72·{10}^{17}{\displaystyle \frac{\mathrm{m}}{\mathrm{s}}},$$

(77)

which is much bigger than the speed of light. It should be noted that we were not considering the relativistic mechanics, which is the reason we received a speed limit bigger than the speed of light. However, for an object with high enough mass, ${\displaystyle v_{\max }}$ can be much smaller than ${\displaystyle c}$. For example, when ${\displaystyle m}$ is equal to the Planck’s mass ${\displaystyle m_p}$, then

$$v_{\max }=\sqrt{{\displaystyle \frac{3\sqrt{3}}{8}}}c\approx 0.8c$$

(78)

and when ${\displaystyle m=150\sqrt{3}{m}_p\approx 5.65\mathrm{mg}}$, then ${\displaystyle v_{\max }=0.05c}$.

The minimal time scale will also influence the wave packet spreading. Calculating the uncertainty in position for the state (68), we obtain

$$\mathsf{\Delta }{x}_{\psi \left(\tau \right)}=\sqrt{{\left(\mathsf{\Delta }{x}_{\psi \left(0\right)}\right)}^2+\tau \left({\langle \widehat{v}\widehat{x}+\widehat{x}\widehat{v}\rangle }_{\psi \left(0\right)}-2{\langle \widehat{x}\rangle }_{\psi \left(0\right)}{\langle \widehat{v}\rangle }_{\psi \left(0\right)}\right)+{\tau }^2{\left(\mathsf{\Delta }{v}_{\psi \left(0\right)}\right)}^2},$$

(79)

which is a similar looking formula as in the ${\displaystyle \kappa =0}$ case except that the velocity operator ${\displaystyle \widehat{v}}$ is expressed by a different formula. Therefore, the uncertainty in position will increase in time, which is interpreted as the spreading of the wave packet evolving in time. However, ${\displaystyle \mathsf{\Delta }{x}_{\psi \left(\tau \right)}}$ will change in time in a slightly different way than in the ${\displaystyle \kappa =0}$ case. Formula (79) simplifies for states (68) for which the function ${\displaystyle f}$ is real-valued. Then, by integration by parts, it is easy to see that ${\displaystyle {\langle \widehat{x}\rangle }_{\psi \left(0\right)}=0}$ and ${\displaystyle {\langle \widehat{v}\widehat{x}+\widehat{x}\widehat{v}\rangle }_{\psi \left(0\right)}=0}$ from which we get

$$\mathsf{\Delta }{x}_{\psi \left(\tau \right)}=\sqrt{{\left(\mathsf{\Delta }{x}_{\psi \left(0\right)}\right)}^2+{\tau }^2{\left(\mathsf{\Delta }{v}_{\psi \left(0\right)}\right)}^2}.$$

(80)

As an example, we can take the state (68) in the form of a Gaussian wave packet by taking the function ${\displaystyle f}$ in the form of a Gaussian function:

$$f\left(p\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^{1/4}{\left(\mathsf{\Delta }p\right)}^{1/2}}}exp\left(-{\displaystyle \frac{{(p-p_0)}^2}{4{\left(\mathsf{\Delta }p\right)}^2}}\right).$$

(81)

In Figure 2, plots of the probability density of the state (68) in the form of a Gaussian wave packet are presented. From these plots, we can see that the velocity and spreading of a Gaussian wave packet are smaller for the ${\displaystyle \kappa >0}$ case. In Figure 3, a plot of the expectation value of velocity as a function of the expectation value of momentum and a plot of the uncertainty in position as a function of time are given. From these plots, we can see that in the ${\displaystyle \kappa >0}$ case, the expectation value of velocity attains a maximum and approaches ${\displaystyle 0}$ as ${\displaystyle p_0\to \pm \infty }$ and that the spreading of the Gaussian wave packet is slower than in the ${\displaystyle \kappa =0}$ case.

4.3. Harmonic Oscillator

In this section, we will investigate how the minimal time scale influences the time evolution of a harmonic oscillator initially in a coherent state. The Hilbert space of the system is ${\displaystyle {\mathcal{H}}_S=L^2(\mathbb{R},\mathrm{d}x)}$ and the Hamiltonian of the system is

$${\widehat{H}}_S={\displaystyle \frac{1}{2m}}{\widehat{p}}^2+{\displaystyle \frac{1}{2}}m{\omega }^2{\widehat{x}}^2,$$

(82)

where ${\displaystyle m}$ is the mass of a particle and ${\displaystyle \omega }$ is a frequency of oscillations. Let us assume that the system is initially in a coherent state ${\displaystyle {|\alpha \rangle }_S}$. A coherent state ${\displaystyle {|\alpha \rangle }_S}$ is defined to be the eigenstate of the annihilation operator

$$\widehat{a}=\sqrt{{\displaystyle \frac{m\omega }{2\hslash }}}\widehat{x}+{\displaystyle \frac{i}{\sqrt{2m\omega \hslash }}}\widehat{p}$$

(83)

with a corresponding eigenvalue ${\displaystyle \alpha \in \mathbb{C}}$:

$$\widehat{a}{|\alpha \rangle }_S=\alpha {|\alpha \rangle }_S.$$

(84)

The expectation values ${\displaystyle x_0,p_0}$ of the position and momentum operators in the coherent state ${\displaystyle {|\alpha \rangle }_S}$ are equal

$$x_0={}_S\langle \alpha |\widehat{x}|\alpha \rangle _S=\sqrt{{\displaystyle \frac{2\hslash }{m\omega }}}\mathrm{Re}\left(\alpha \right),p_0={}_S\langle \alpha |\widehat{p}|\alpha \rangle _S=\sqrt{2m\omega \hslash }Im\left(\alpha \right).$$

(85)

Therefore, ${\displaystyle \alpha }$ is expressed by the expectation values ${\displaystyle x_0,p_0}$ of position and momentum operators in the following way:

$$\alpha =\sqrt{{\displaystyle \frac{m\omega }{2\hslash }}}{x}_0+{\displaystyle \frac{i}{\sqrt{2m\omega \hslash }}}{p}_0.$$

(86)

In the case without a minimal time scale (${\displaystyle \kappa =0}$), the coherent state ${\displaystyle {|\alpha \rangle }_S}$ will evolve in time according to the formula

$${|\alpha \left(t\right)\rangle }_S=e^{-it{\widehat{H}}_S/\hslash }{|\alpha \rangle }_S.$$

(87)

Expanding ${\displaystyle {|\alpha \rangle }_S}$ in the basis of eigenstates ${\displaystyle {|n\rangle }_S}$ of the Hamiltonian ${\displaystyle {\widehat{H}}_S}$, we obtain

$${|\alpha \left(t\right)\rangle }_S=e^{-it{\widehat{H}}_S/\hslash }\sum _{n=0}^{\infty }{|n\rangle }_{SS}{\langle n|\alpha \rangle }_S=\sum _{n=0}^{\infty }{}_S\langle n|\alpha \rangle _S{e}^{-it{E}_n/\hslash }{|n\rangle }_S.$$

(88)

Using the fact that

$${}_S\langle n|\alpha \rangle _S=e^{-{\displaystyle \frac{1}{2}}{\left|\alpha \right|}^2}{\displaystyle \frac{{\alpha }^n}{\sqrt{n!}}}$$

(89)

and the energy eigenvalues ${\displaystyle E_n=\hslash \omega (n+\frac{1}{2})}$, we can write (88) in the form

$$\begin{array}{cc}\hfill {|\alpha \left(t\right)\rangle }_S& =e^{-{\displaystyle \frac{1}{2}}{\left|\alpha \right|}^2}\sum _{n=0}^{\infty }{\displaystyle \frac{{\alpha }^n}{\sqrt{n!}}}{e}^{-i(n+{\displaystyle \frac{1}{2}})\omega t}{|n\rangle }_S=e^{-i\omega t/2}{e}^{-{\displaystyle \frac{1}{2}}{\left|\alpha \right|}^2}\sum _{n=0}^{\infty }{\displaystyle \frac{{\left(\alpha e^{-i\omega t}\right)}^n}{\sqrt{n!}}}{|n\rangle }_S\hfill \\ \hfill & =e^{-i\omega t/2}{|\alpha e^{-i\omega t}\rangle }_S.\hfill \end{array}$$

(90)

Therefore, time-evolved coherent states are also coherent with a phase shift. The position representation of the coherent state ${\displaystyle {|\alpha \rangle }_S}$ is equal to

$${\psi }_{\alpha }\left(x\right)={}_S\langle x|\alpha \rangle _S={\left({\displaystyle \frac{m\omega }{\pi \hslash }}\right)}^{1/4}{e}^{-{\displaystyle \frac{m\omega }{2\hslash }}{(x-x_0)}^2}{e}^{{\displaystyle \frac{i}{\hslash }}(x-{\displaystyle \frac{1}{2}}{x}_0)p_0}.$$

(91)

From (90) and (91), we obtain

$${\psi }_{\alpha }(t,x)={}_S\langle x\left|\alpha \right(t)\rangle _S={\left({\displaystyle \frac{m\omega }{\pi \hslash }}\right)}^{1/4}{e}^{-{\displaystyle \frac{m\omega }{2\hslash }}{(x-x_0\left(t\right))}^2}{e}^{{\displaystyle \frac{i}{\hslash }}(x-{\displaystyle \frac{1}{2}}{x}_0\left(t\right))p_0\left(t\right)}{e}^{-i\omega t/2},$$

(92)

where ${\displaystyle x_0\left(t\right)}$ and ${\displaystyle p_0\left(t\right)}$ are expectation values of position and momentum operators in the time-evolved coherent state ${\displaystyle {|\alpha \left(t\right)\rangle }_S}$ given by

$$\begin{array}{cc}\hfill x_0\left(t\right)& =\sqrt{{\displaystyle \frac{2\hslash }{m\omega }}}\left|\alpha \right|cos(\omega t-\varphi ),\hfill \\ \hfill p_0\left(t\right)& =-\sqrt{2m\omega \hslash }\left|\alpha \right|sin(\omega t-\varphi ),\hfill \end{array}$$

(93)

where ${\displaystyle \alpha =\left|\alpha \right|e^{i\varphi }}$.

In the case with a minimal time scale (${\displaystyle \kappa >0}$), the coherent state ${\displaystyle {|\alpha \rangle }_S}$ will evolve in time according to the formula

$${|\alpha \left(\tau \right)\rangle }_S=exp\left(-i\tau {\displaystyle \frac{1}{\sqrt{\kappa }}}arctan(\sqrt{\kappa }{\widehat{H}}_S/\hslash )\right){|\alpha \rangle }_S.$$

(94)

Performing similar calculations as before, we obtain

$${|\alpha \left(\tau \right)\rangle }_S=e^{-{\displaystyle \frac{1}{2}}{\left|\alpha \right|}^2}\sum _{n=0}^{\infty }{\displaystyle \frac{{\alpha }^n}{\sqrt{n!}}}{e}^{-i\tau {\displaystyle \frac{1}{\sqrt{\kappa }}}arctan\left(\sqrt{\kappa }\omega (n+{\displaystyle \frac{1}{2}})\right)}{|n\rangle }_S.$$

(95)

From this, we can see that the state ${\displaystyle {|\alpha \left(\tau \right)\rangle }_S}$ is no longer coherent for ${\displaystyle \tau \ne 0}$. In the position representation, the eigenstates ${\displaystyle {|n\rangle }_S}$ of the Hamiltonian take the form

$${\psi }_n\left(x\right)={}_S\langle x|n\rangle _S={\displaystyle \frac{1}{\sqrt{2^{n}n!}}}{\left({\displaystyle \frac{m\omega }{\pi \hslash }}\right)}^{1/4}{e}^{-{\displaystyle \frac{m\omega x^2}{2\hslash }}}{H}_n\left(\sqrt{{\displaystyle \frac{m\omega }{\hslash }}}x\right),$$

(96)

where ${\displaystyle H_n}$ are Hermite polynomials. Hence, the time-evolved coherent state ${\displaystyle {|\alpha \left(\tau \right)\rangle }_S}$ takes the form

$$\begin{array}{cc}\hfill {\psi }_{\alpha }(\tau ,x)& ={}_S\langle x\left|\alpha \right(\tau )\rangle _S\hfill \\ \hfill & ={\left({\displaystyle \frac{m\omega }{\pi \hslash }}\right)}^{1/4}{e}^{-{\displaystyle \frac{m\omega x^2}{2\hslash }}}{e}^{-{\displaystyle \frac{1}{2}}{\left|\alpha \right|}^2}\sum _{n=0}^{\infty }{\displaystyle \frac{1}{n!}}{\left({\displaystyle \frac{\alpha }{\sqrt{2}}}\right)}^n{H}_n\left(\sqrt{{\displaystyle \frac{m\omega }{\hslash }}}x\right)e^{-i\tau {\displaystyle \frac{1}{\sqrt{\kappa }}}arctan\left(\sqrt{\kappa }\omega (n+{\displaystyle \frac{1}{2}})\right)}.\hfill \end{array}$$

(97)

In Figure 4, plots of the probability density of the state (97) for particular values of time ${\displaystyle \tau }$ are presented. These plots illustrate how the coherence of the state is destroyed during time evolution as a result of the existence of the fundamental limit for the precision with which we can measure time.

4.4. System of Two Coupled Spins-${\displaystyle 1/2}$ in an External Magnetic Field

In this section, we will investigate a system composed of two spins-${\displaystyle 1/2}$ interacting with each other and with an external constant magnetic field. The interaction between spins will cause the system initially in a non-entangled state to entangle over time. We will explore how the minimal time scale influences the entanglement.

The Hilbert space of the system is ${\displaystyle {\mathcal{H}}_S={\mathcal{H}}_1\otimes {\mathcal{H}}_2={\mathbb{C}}^2\otimes {\mathbb{C}}^2={\mathbb{C}}^4}$, where ${\displaystyle {\mathcal{H}}_1={\mathbb{C}}^2}$ and ${\displaystyle {\mathcal{H}}_2={\mathbb{C}}^2}$ are Hilbert spaces of individual spins. We will take the following Hamiltonian of the system:

$$\begin{array}{cc}\hfill {\widehat{H}}_S& ={\widehat{H}}_0+{\widehat{H}}_1=-\gamma B_0({\widehat{S}}_z\otimes \widehat{1}+\widehat{1}\otimes {\widehat{S}}_z)+\lambda ({\widehat{S}}_x\otimes {\widehat{S}}_x+{\widehat{S}}_y\otimes {\widehat{S}}_y)\hfill \\ \hfill & ={\displaystyle \frac{\hslash {\omega }_0}{2}}({\sigma }_z\otimes I+I\otimes {\sigma }_z)+{\displaystyle \frac{{\hslash }^2\lambda }{4}}({\sigma }_x\otimes {\sigma }_x+{\sigma }_y\otimes {\sigma }_y),\hfill \end{array}$$

(98)

where ${\displaystyle \gamma }$ is the gyromagnetic ratio; ${\displaystyle B_0}$ is the strength of the magnetic field, which we assume is directed along the ${\displaystyle z}$-axis; ${\displaystyle \lambda }$ is the strength of the interaction between spins; ${\displaystyle {\omega }_0=-\gamma B_0}$ is the frequency of precession of the spins; and ${\displaystyle {\widehat{S}}_x}$, ${\displaystyle {\widehat{S}}_y}$, ${\displaystyle {\widehat{S}}_z}$ are components of the spin operator expressed by the Pauli matrices

$${\sigma }_x=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],{\sigma }_y=\left[\begin{array}{cc}0& -i\\ i& 0\end{array}\right],{\sigma }_z=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right].$$

(99)

The operator ${\displaystyle {\widehat{H}}_0}$ describes non-interacting spins precessing around the ${\displaystyle z}$-axis with the frequency ${\displaystyle {\omega }_0}$. The operator ${\displaystyle {\widehat{H}}_1}$ describes the interaction between spins.

First, let us notice that ${\displaystyle {\widehat{H}}_0{\widehat{H}}_1={\widehat{H}}_1{\widehat{H}}_0=0}$. Because of this, ${\displaystyle {\widehat{H}}_S^n={\widehat{H}}_0^n+{\widehat{H}}_1^n}$ for ${\displaystyle n\in \mathbb{N}}$ and

$${\displaystyle \frac{1}{\sqrt{\kappa }}}arctan(\sqrt{\kappa }{\widehat{H}}_S/\hslash )={\displaystyle \frac{1}{\sqrt{\kappa }}}arctan(\sqrt{\kappa }{\widehat{H}}_0/\hslash )+{\displaystyle \frac{1}{\sqrt{\kappa }}}arctan(\sqrt{\kappa }{\widehat{H}}_1/\hslash ).$$

(100)

Thus

$$\begin{array}{c}exp\left(-i\tau {\displaystyle \frac{1}{\sqrt{\kappa }}}arctan(\sqrt{\kappa }{\widehat{H}}_S/\hslash )\right)=exp\left(-i\tau {\displaystyle \frac{1}{\sqrt{\kappa }}}arctan(\sqrt{\kappa }{\widehat{H}}_1/\hslash )\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times exp\left(-i\tau {\displaystyle \frac{1}{\sqrt{\kappa }}}arctan(\sqrt{\kappa }{\widehat{H}}_0/\hslash )\right).\hfill \end{array}$$

(101)

Next, observe that for ${\displaystyle n\in \mathbb{N}}$

$$\begin{array}{cc}\hfill {\widehat{H}}_0^{2n+1}& ={\displaystyle \frac{1}{2}}{(\hslash {\omega }_0)}^{2n+1}({\sigma }_z\otimes I+I\otimes {\sigma }_z),\hfill \\ \hfill {\widehat{H}}_1^{2n+1}& ={\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{{\hslash }^2\lambda }{2}}\right)}^{2n+1}({\sigma }_x\otimes {\sigma }_x+{\sigma }_y\otimes {\sigma }_y).\hfill \end{array}$$

(102)

From this,

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\sqrt{\kappa }}}arctan(\sqrt{\kappa }{\widehat{H}}_0/\hslash )& ={\displaystyle \frac{1}{2\sqrt{\kappa }}}arctan\left(\sqrt{\kappa }{\omega }_0\right)({\sigma }_z\otimes I+I\otimes {\sigma }_z),\hfill \\ \hfill {\displaystyle \frac{1}{\sqrt{\kappa }}}arctan(\sqrt{\kappa }{\widehat{H}}_1/\hslash )& ={\displaystyle \frac{1}{2\sqrt{\kappa }}}arctan\left({\displaystyle \frac{\hslash \sqrt{\kappa }}{2}}\lambda \right)({\sigma }_x\otimes {\sigma }_x+{\sigma }_y\otimes {\sigma }_y).\hfill \end{array}$$

(103)

Ultimately, the time evolution of the system initially in a non-entangled state ${\displaystyle {|\psi \rangle }_S=|{\psi }_1\rangle \otimes |{\psi }_2\rangle }$ is given by the formula

$${|\psi \left(\tau \right)\rangle }_S=e^{-i\tau \hslash {\lambda }_{\kappa }({\sigma }_x\otimes {\sigma }_x+{\sigma }_y\otimes {\sigma }_y)/4}{e}^{-i\tau {\omega }_{\kappa }({\sigma }_z\otimes I+I\otimes {\sigma }_z)/2}|{\psi }_1\rangle \otimes |{\psi }_2\rangle ,$$

(104)

where ${\displaystyle {\omega }_{\kappa }=\frac{1}{\sqrt{\kappa }}arctan\left(\sqrt{\kappa }{\omega }_0\right)}$ and ${\displaystyle {\lambda }_{\kappa }=\frac{2}{\hslash \sqrt{\kappa }}arctan\left(\frac{\hslash \sqrt{\kappa }}{2}\lambda \right)}$.

Let us assume that the initial states ${\displaystyle |{\psi }_1\rangle }$ and ${\displaystyle |{\psi }_2\rangle }$ are the same and equal

$$|{\psi }_1\rangle =|{\psi }_2\rangle =|\theta ,\phi \rangle =cos(\theta /2)e^{-i\phi /2}|\uparrow \rangle +sin(\theta /2)e^{i\phi /2}|\downarrow \rangle ,$$

(105)

where ${\displaystyle \theta ,\phi \in \mathbb{R}}$ and ${\displaystyle |\uparrow \rangle }$, ${\displaystyle |\downarrow \rangle }$ are spin-up and spin-down eigenstates of the spin operator ${\displaystyle {\widehat{S}}_z}$. The action of the operator ${\displaystyle e^{-i\tau {\omega }_{\kappa }({\sigma }_z\otimes I+I\otimes {\sigma }_z)/2}}$ on ${\displaystyle |{\psi }_1\rangle \otimes |{\psi }_2\rangle }$ will result in each of the states ${\displaystyle |{\psi }_1\rangle }$ and ${\displaystyle |{\psi }_2\rangle }$ evolving in time according to the Formula (65), so only the angle ${\displaystyle \phi }$ will change in both of these states:

$$e^{-i\tau {\omega }_{\kappa }({\sigma }_z\otimes I+I\otimes {\sigma }_z)/2}|{\psi }_1\rangle \otimes |{\psi }_2\rangle =|\theta ,\phi +{\omega }_{\kappa }\tau \rangle \otimes |\theta ,\phi +{\omega }_{\kappa }\tau \rangle .$$

(106)

We can calculate that

$$\begin{array}{c}{e}^{-i\tau \hslash {\lambda }_{\kappa }({\sigma }_x\otimes {\sigma }_x+{\sigma }_y\otimes {\sigma }_y)/4}={\displaystyle \frac{1}{2}}(I\otimes I+{\sigma }_z\otimes {\sigma }_z+cos\left({\displaystyle \frac{\hslash {\lambda }_{\kappa }\tau }{2}}\right)(I\otimes I-{\sigma }_z\otimes {\sigma }_z)\hfill \\ \hfill -isin\left({\displaystyle \frac{\hslash {\lambda }_{\kappa }\tau }{2}}\right)({\sigma }_x\otimes {\sigma }_x+{\sigma }_y\otimes {\sigma }_y)).\end{array}$$

(107)

From this, we obtain the following expression for the time evolution of the state ${\displaystyle {|\psi \rangle }_S=|{\psi }_1\rangle \otimes |{\psi }_2\rangle }$

$$\begin{array}{c}{|\psi \left(\tau \right)\rangle }_S={cos}^2{\displaystyle \frac{\theta }{2}}{e}^{-i(\phi +{\omega }_{\kappa }\tau )}|\uparrow \rangle \otimes |\uparrow \rangle +{sin}^2{\displaystyle \frac{\theta }{2}}{e}^{i(\phi +{\omega }_{\kappa }\tau )}|\downarrow \rangle \otimes |\downarrow \rangle \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{1}{2}}sin\theta e^{-i\hslash {\lambda }_{\kappa }\tau /2}(|\uparrow \rangle \otimes |\downarrow \rangle +|\downarrow \rangle \otimes |\uparrow \rangle ).\hfill \end{array}$$

(108)

A useful measure of the degree of quantum entanglement between two subsystems is the entropy of entanglement ${\displaystyle S}$. It is defined as the Von Neumann entropy of the reduced density matrix for any of the subsystems:

$$S=-k_{B}Tr({\widehat{\rho }}_{1}ln{\widehat{\rho }}_1)=-k_{B}Tr({\widehat{\rho }}_{2}ln{\widehat{\rho }}_2),$$

(109)

where ${\displaystyle k_B}$ is the Boltzmann’s constant, ${\displaystyle {\widehat{\rho }}_1={Tr}_2\left(\widehat{\rho }\right)}$ and ${\displaystyle {\widehat{\rho }}_2={Tr}_1\left(\widehat{\rho }\right)}$ are the reduced density matrices of each subsystem, and ${\displaystyle \widehat{\rho }}$ is the density matrix of the whole system. If the subsystems are not entangled, then ${\displaystyle S=0}$; otherwise, ${\displaystyle S>0}$, and the bigger the entropy ${\displaystyle S}$ is, the more entangled the subsystems are.

The density matrix of the system of two spins is equal ${\displaystyle \widehat{\rho }=|\psi \left(\tau \right)\rangle {\langle \psi \left(\tau \right)|}_S}$, where ${\displaystyle {|\psi \left(\tau \right)\rangle }_S}$ is given by (108). By taking the partial trace ${\displaystyle {Tr}_2}$ of ${\displaystyle \widehat{\rho }}$ over the Hilbert space of the second subsystem, we receive the reduced density matrix ${\displaystyle {\widehat{\rho }}_1}$ of the first subsystem. It is given by the equation

$$\begin{array}{c}{\widehat{\rho }}_1={cos}^2{\displaystyle \frac{\theta }{2}}|\uparrow \rangle \langle \uparrow |+{sin}^2{\displaystyle \frac{\theta }{2}}|\downarrow \rangle \langle \downarrow |\hfill \\ \hspace{1em}\hspace{1em}+{\displaystyle \frac{1}{2}}sin\theta e^{-i(\phi +{\omega }_{\kappa }\tau )}\left({cos}^2{\displaystyle \frac{\theta }{2}}{e}^{i\hslash {\lambda }_{\kappa }\tau /2}+{sin}^2{\displaystyle \frac{\theta }{2}}{e}^{-i\hslash {\lambda }_{\kappa }\tau /2}\right)|\uparrow \rangle \langle \downarrow |\\ \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{1}{2}}sin\theta e^{i(\phi +{\omega }_{\kappa }\tau )}\left({cos}^2{\displaystyle \frac{\theta }{2}}{e}^{-i\hslash {\lambda }_{\kappa }\tau /2}+{sin}^2{\displaystyle \frac{\theta }{2}}{e}^{i\hslash {\lambda }_{\kappa }\tau /2}\right)|\downarrow \rangle \langle \uparrow |.\end{array}$$

(110)

The matrix ${\displaystyle {\widehat{\rho }}_1}$ has the following eigenvalues

$$p_1={\displaystyle \frac{1+\sqrt{1-{sin}^4\theta {sin}^2(\hslash {\lambda }_{\kappa }\tau /2)}}{2}},p_2={\displaystyle \frac{1-\sqrt{1-{sin}^4\theta {sin}^2(\hslash {\lambda }_{\kappa }\tau /2)}}{2}}.$$

(111)

Thus, the entropy of entanglement is equal to

$$\begin{array}{cc}\hfill S& =-k_{B}Tr({\widehat{\rho }}_{1}ln{\widehat{\rho }}_1)=-k_B(p_{1}ln{p}_1+p_{2}ln{p}_2)\hfill \\ \hfill & =-k_B(ln{\displaystyle \frac{{sin}^2\theta \left|sin(\hslash {\lambda }_{\kappa }\tau /2)\right|}{4}}\hfill \\ \hfill & +\sqrt{1-{sin}^4\theta {sin}^2(\hslash {\lambda }_{\kappa }\tau /2)}ar\; tanh\sqrt{1-{sin}^4\theta {sin}^2(\hslash {\lambda }_{\kappa }\tau /2)}).\hfill \end{array}$$

(112)

From the above formula, we can see that ${\displaystyle S}$ is a periodic function of ${\displaystyle \tau }$. The entanglement of the spins will periodically increase reaching its maximum, then decrease until the spins become non-entangled. The period of these oscillations is equal to ${\displaystyle \frac{2\pi }{\hslash {\lambda }_{\kappa }}}$. In the case without a minimal time scale (${\displaystyle \kappa =0}$), this period is equal to ${\displaystyle \frac{2\pi }{\hslash \lambda }}$ and is smaller than in the case with a minimal time scale (${\displaystyle \kappa >0}$); see Figure 5. Therefore, the existence of the fundamental limit for the precision with which we can measure time influences the entanglement.

Notice that from (104) we can see that two non-interacting spins will precess around the ${\displaystyle z}$-axis with a smaller frequency than only one spin, as described in Section 4.1. This result is quite peculiar because it means that even though the spins are not interacting, they still influence each other in some way which causes the decrease in the frequency of precession.

4.5. System of Three Spins-${\displaystyle 1/2}$ in an External Magnetic Field

In this section, we will investigate a system composed of three spins-${\displaystyle 1/2}$ interacting with an external constant magnetic field. We will show that the minimal time scale will cause the spins to entangle over time even though the spins are not interacting with each other.

The Hilbert space of the system is ${\displaystyle {\mathcal{H}}_S={\mathcal{H}}_1\otimes {\mathcal{H}}_2\otimes {\mathcal{H}}_3={\mathbb{C}}^2\otimes {\mathbb{C}}^2\otimes {\mathbb{C}}^2={\mathbb{C}}^6}$, where ${\displaystyle {\mathcal{H}}_1={\mathbb{C}}^2}$, ${\displaystyle {\mathcal{H}}_2={\mathbb{C}}^2}$ and ${\displaystyle {\mathcal{H}}_3={\mathbb{C}}^2}$ are Hilbert spaces of individual spins. The Hamiltonian of the system is equal

$$\begin{array}{cc}\hfill {\widehat{H}}_S& =-\gamma B_0({\widehat{S}}_z\otimes \widehat{1}\otimes \widehat{1}+\widehat{1}\otimes {\widehat{S}}_z\otimes \widehat{1}+\widehat{1}\otimes \widehat{1}\otimes {\widehat{S}}_z)\hfill \\ \hfill & ={\displaystyle \frac{\hslash {\omega }_0}{2}}({\sigma }_z\otimes I\otimes I+I\otimes {\sigma }_z\otimes I+I\otimes I\otimes {\sigma }_z),\hfill \end{array}$$

(113)

where ${\displaystyle \gamma }$ is the gyromagnetic ratio; ${\displaystyle B_0}$ is the strength of the magnetic field, which we assume is directed along the ${\displaystyle z}$-axis; ${\displaystyle {\omega }_0=-\gamma B_0}$; and ${\displaystyle {\widehat{S}}_z}$ is the ${\displaystyle z}$-component of the spin operator expressed by the Pauli matrix ${\displaystyle {\sigma }_z}$.

We can calculate that for ${\displaystyle n\in \mathbb{N}}$

$$\begin{array}{c}{\widehat{H}}_S^{2n+1}={\displaystyle \frac{1}{4}}{\left({\displaystyle \frac{\hslash {\omega }_0}{2}}\right)}^{2n+1}((3^{2n+1}+1)({\sigma }_z\otimes I\otimes I+I\otimes {\sigma }_z\otimes I+I\otimes I\otimes {\sigma }_z)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+(3^{2n+1}-3){\sigma }_z\otimes {\sigma }_z\otimes {\sigma }_z)\hfill \end{array}$$

(114)

for which we obtain

$$\begin{array}{cc}\hfill {\widehat{H}}_{\mathrm{eff}}& ={\displaystyle \frac{\hslash }{\sqrt{\kappa }}}arctan(\sqrt{\kappa }{\widehat{H}}_S/\hslash )\hfill \\ \hfill & ={\displaystyle \frac{\hslash {\omega }_{\kappa }}{2}}({\sigma }_z\otimes I\otimes I+I\otimes {\sigma }_z\otimes I+I\otimes I\otimes {\sigma }_z)+\hslash {\lambda }_{\kappa }{\sigma }_z\otimes {\sigma }_z\otimes {\sigma }_z,\hfill \end{array}$$

(115)

where

$$\begin{array}{cc}\hfill {\omega }_{\kappa }& ={\displaystyle \frac{1}{4\sqrt{\kappa }}}\left(arctan\left({\displaystyle \frac{3\sqrt{\kappa }{\omega }_0}{2}}\right)+arctan\left({\displaystyle \frac{\sqrt{\kappa }{\omega }_0}{2}}\right)\right),\hfill \\ \hfill {\lambda }_{\kappa }& ={\displaystyle \frac{1}{4\sqrt{\kappa }}}\left(arctan\left({\displaystyle \frac{3\sqrt{\kappa }{\omega }_0}{2}}\right)-3arctan\left({\displaystyle \frac{\sqrt{\kappa }{\omega }_0}{2}}\right)\right).\hfill \end{array}$$

(116)

We can see that the effective Hamiltonian ${\displaystyle {\widehat{H}}_{\mathrm{eff}}}$ describing the time evolution of the system contains a term ${\displaystyle \hslash {\lambda }_{\kappa }{\sigma }_z\otimes {\sigma }_z\otimes {\sigma }_z}$ which will cause the entanglement of the system over time. Taking the system initially in a state

$${|\psi \rangle }_S=|{\theta }_1,{\phi }_1\rangle \otimes |{\theta }_2,{\phi }_2\rangle \otimes |{\theta }_3,{\phi }_3\rangle$$

(117)

and noticing that

$$e^{-i\tau {\lambda }_{\kappa }{\sigma }_z\otimes {\sigma }_z\otimes {\sigma }_z}=cos\left({\lambda }_{\kappa }\tau \right)I\otimes I\otimes I-isin\left({\lambda }_{\kappa }\tau \right){\sigma }_z\otimes {\sigma }_z\otimes {\sigma }_z$$

(118)

we obtain the following expression for the time evolution of the state ${\displaystyle {|\psi \rangle }_S}$:

$$\begin{array}{c}{|\psi \left(\tau \right)\rangle }_S=cos\left({\lambda }_{\kappa }\tau \right)|{\theta }_1,{\phi }_1+{\omega }_{\kappa }\tau \rangle \otimes |{\theta }_2,{\phi }_2+{\omega }_{\kappa }\tau \rangle \otimes |{\theta }_3,{\phi }_3+{\omega }_{\kappa }\tau \rangle \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-isin\left({\lambda }_{\kappa }\tau \right)|-{\theta }_1,{\phi }_1+{\omega }_{\kappa }\tau \rangle \otimes |-{\theta }_2,{\phi }_2+{\omega }_{\kappa }\tau \rangle \otimes |-{\theta }_3,{\phi }_3+{\omega }_{\kappa }\tau \rangle .\hfill \end{array}$$

(119)

From the above formula, we can see that the existence of the fundamental limit for the precision with which we can measure time will cause the system initially in a non-entangled state to evolve in time into an entangled state.

5. Conclusions and Discussion

We developed a theory of non-relativistic quantum mechanics exhibiting a non-zero minimal uncertainty in time. The theory was based on the Page–Wootters formalism with a modified commutation relations between the time and frequency operators. We received a modified version of the Schrödinger equation in a continuous and discrete time representation. Interestingly, both representations are equivalent, so, in particular, by solving the discrete Schrödinger equation, we receive a time evolution on a lattice of a state vector, which we can then transform to the continuous time representation. It should be noted that even though we have a discrete time evolution, it does not mean that fundamentally time is discrete. It only means that, using a particular representation, we can describe time evolution on a lattice.

The developed formalism was used to describe time evolution of couple simple quantum systems. The received results could be tested experimentally, at least in principle. In particular, we found that in the system of spins-${\displaystyle 1/2}$ in an external magnetic field, the frequency of precession of the spins depends on the number of spins composing the system. Furthermore, the existence of a fundamental limit with which we can measure time causes the entanglement of the system of spins. These results are quite peculiar, especially that the spins do not need to explicitly interact with each other, but still there exists some kind of influence between them. The interpretation of these results can be difficult and may suggest that perhaps the considered modification of commutation relations is not physically correct and some different modification will be more suitable.

It should be noted that we have not explicitly assumed that the parameter ${\displaystyle \kappa }$ describing the minimal time scale is dependent on the system undergoing time evolution. However, in view of the comment from the previous paragraph it might be possible that ${\displaystyle \kappa }$ will depend on the system. Such a possibility will require further investigation.

In this paper, we only considered time-independent Hamiltonians. It is possible to generalize the theory to include time-dependent Hamiltonians by considering a constraint operator ${\displaystyle \widehat{J}}$ as in (15). However, in this case, it will not be possible to use Theorem A1 when deriving time evolution equations because operators ${\displaystyle \hslash \widehat{\mathsf{\Omega }}\otimes {\widehat{1}}_S}$ and ${\displaystyle {\widehat{H}}_S\left(\widehat{T}\right)}$ composing the constraint operator ${\displaystyle \widehat{J}}$ do not commute.

Worth investigating will also be a relativistic quantum mechanics exhibiting a non-zero uncertainty in time and space. In the Page–Wootters formalism, time is treated as a quantum degree of freedom resulting from a correlation between the “clock” and the rest of the system. Interestingly the same approach can be performed to describe space degrees of freedom [31]. Such formalism will have an advantage of treating time and space on an equal footing. Lorentz transformations can be defined as an appropriate unitary transformation. It is then possible to consider different modifications, commonly found in the literature, of the commutation relations between ${\displaystyle \widehat{T}}$ and ${\displaystyle \widehat{\mathsf{\Omega }}}$ operators, as well as position and momentum operators.