Time Crystals from Minimum Time Uncertainty

Motivated by the Generalized Uncertainty Principle, covariance, and a minimum measurable time, we propose a deformation of the Heisenberg algebra and show that this leads to corrections to all quantum mechanical systems. We also demonstrate that such a deformation implies a discrete spectrum for time. In other words, time behaves like a crystal. As an application of our formalism, we analyze the effect of such a deformation on the rate of spontaneous emission in a hydrogen atom.


Introduction
The Heisenberg uncertainty principle predicts that the position of a particle can in principle be measured as accurately as one wants, if its momentum is allowed to remain completely uncertain.However most approaches to quantum gravity predict the existence of a minimum measurable length scale, usually the Planck length.There are also strong indications from black hole physics and other sources for the existence of a minimum measurable length [1,2,3].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.Similarly, string theory also predicts a minimum length, as strings are the smallest probes [4,5,6,7,8].Also in loop quantum gravity there exists a minimum measurable length scale, which turns the big bang into a big bounce [9].
The existence of a minimum measurable length scale in turn requires the modification of the Heisenberg uncertainty principle into a Generalized Uncertainty Principle (GUP) [4,5,6,7], and a corresponding deformation of the Heisenberg algebra to include momentum dependent terms, and a modified coordinate representation of the momentum operators [8,10,11,12,13,14,15].It may be noted that a different kind of deformation of the Heisenberg algebra occurs due to Doubly Special Relativity (DSR) theories, which postulate the existence of a universal energy scale (the Planck scale) [16,17,18].These are also related to the idea of discrete spacetime [19], spontaneous symmetry breaking of Lorentz invariance in string field theory [20], spacetime foam models [21], spinnetwork in loop quantum gravity [22], non-commutative geometry [23,24,25], ghost condensation in perturbative quantum gravity [26], and Horava-Lifshitz gravity [27].It may be noted that DSR has been generalized to curved spacetime and the resultant theory is called gravity's rainbow [28,29,30,31,32,33].It is interesting to note that the deformation from DSR and the deformation from GUP can be combined into a single consistent deformation of the Heisenberg algebra [34].
A number of interesting quantum systems have been studied using this deformed algebra, such as the transition rate of ultra cold neutrons in gravitational field [35], the Lamb shift and Landau levels [36].There has been another interesting result derived from this deformed algebra, which shows that space needs to be a discrete lattice, and only multiples of a fundamental length scale (normally taken as the Planck length) can be measured [37].Note that minimum length does not automatically imply discrete lengths, or vice-versa.Motivated by this result, in this paper we analyze the deformation of the algebra and the subsequent Schrödinger equation consistent with the existence of a minimum time, and demonstrate that it leads to a discretization of time as well.It may be noted that discretization of time had also been predicted from a deformed version of the Wheeler-DeWitt equation [38].The discretization of time, and the related breakdown of time reparametrization invariance of a system resembles a crystal lattice in time.Time crystals have been studied recently using a very different physical motivation, e.g.analyzing superconducting rings, and the spontaneous breakdown of time-translation symmetry in classical and quantum systems [39,40,41,42,43].

Minimum Time
We start with the modified Heisenberg algebra, the modified expression of the momentum operator in position space, and the GUP consistent with all theoretical models (correct to O(α 2 ).We use c = 1 units) where α = α 0 ℓ P l /h, and ℓ P l ≈ 10 −35 m is the Planck length.It may be noted that it has been suggested that the parameter α 0 could be situated at an intermediate scale between the electroweak scale and the Planck scale, and this could have measurable consequences in the near future [36].Also the apparent non-local nature of operators in Eq.( 2) above poses no problem in one-dimension (space or time).In more than one dimension, the issue was tackled by using the Dirac equation [34].It is also possible to deal with these non-local derivatives, in more than one dimensions, using the theory of harmonic extension of functions [44,45].This modified Heisenberg algebra is consistent with the following Next, from the principle of covariance, we propose the following spacetime commutators We will only use the temporal part of this algebra, which can be expressed as where f (H) is a suitable function of the Hamiltonian of the system.To make this part of the algebra well defined, we consider time as a quantum mechanical observable rather than as a parameter in the theory.It is possible to define time as an observable with reference to the evolution of some non-stationary quantity, if events are characterized by of a specific values of this quantity [46,47,48,49,50].Thus, the temporal part of Eq.( 5) yields the modified Schrödinger equation As can be seen from the above, this deformation of quantum Hamiltonian will produce corrections to all quantum mechanical systems.The temporal part also implies the following time-energy uncertainty

Time Crystals
Next we note that making a time interval measurement is like putting a particle in a temporal box.This is because in any experiment we first measure an initial state for a system at a given time and then measure the final state of the same system at a later time.This experiment corresponds to analyzing a particle with boundary conditions similar to the ones used for particle in a temporal box, i.e. ψ(0) = ψ(T ) = 0 (this is also consistent with the results of [40], [41]).The temporal part of the deformed Schrödinger equation to first oder in α is given by and it has the solution Applying the boundary condition ψ(0) = 0 leads to B = −A, and the second boundary condition ψ(T ) = 0 leads to which means that either A = B = 0 or both the real and imaginary parts of the above equation are zero.The real part is The imaginary part is If both are zero, then where n ∈ Z.This means that or expanding in terms of α i.e. we can only measure time in discrete steps.It is interesting to note that this discrete interval is dependent on the energy of the system, i.e. the larger the energy the larger will be this discrete interval of time, but since the energy dependence is to second and higher orders, this does not change the time interval by much, except near Planckian energy scales.Further, as can be seen from Eq.( 16) above, the energy of a system is bounded from above by the Planck energy (≈ 1/α), perfectly consistent with DSR type thories.Finally as expected, a continuous time is recovered in the limit in which α → 0. In short, any physical system with finite energy, can only evolve by taking discrete jumps in time rather than continuously.We can also use the lifetime of particles to set bounds on α, for this modified Schrödinger equation.For example, the tau has a lifetime of [51] (290.3 ± 0.5) × 10 −15 s, and since the minimum time from Eq.( 17) must be less than the uncertainty in measuring the tau's life time, then 2πhα < 0.5 × 10 −15 s This means that α 0 < 1.5 × 10 27 .A much lower bound can be found from the lifetime of the Z boson, which is 10 −25 s, leading to the bound α 0 < 10 17 , which agrees with the electroweak scale.

Conclusions
We have shown here that the existence of a minimum measurable time scale in a quantum theory naturally leads to the discretization of time.This is similar to the existence of a minimum measurable length scale leading to a discretization of space.Thus, a crystal in time gets naturally formed by the existence of a minimum measurable time scale in the universe.Time crystals have been studied recently for systems in which time reparametrization is broken, just as spatial translation is broken in regular crystals.Time crystals have also been studied earlier for analyzing superconducting rings [39,40,41,42,43].It would be interesting to analyze a combination of minimum length and minimum time deformations of quantum mechanics to demonstrate a discretization of space and time in four dimensions.We expect to obtain non-local fractional derivative terms in that case, which may possibly be dealt wit using a theory of harmonic extension of functions [44,45], or via the Dirac equation approach [34].It may be noted that it is conceptually useful to view the minimum measurable time as a component of a minimum Euclidean four volume with complex time, and then analytically continue the results to a Lorentz manifold.However, as we analyzed a system with Galilean symmetry, we did not to go through this procedure.
It is expected that the deformation of the Hamiltonian studied here will affect all physical systems.Thus for example, one can study the decay rates of particle and unstable nuclei using this deformed time evolution, which are expected to change as well.In fact, by fixing the value of this deformation parameter just below the experimentally measured limit, it might be possible to devise tests for detecting such deformation of time evolution of quantum mechanics.The deformed Hamiltonian should affect time dependent perturbation theory as well.For example, the out-of-equilibrium Anderson model has been studied using the time-dependent density functional theory [52].This has important applications for time-dependent processes in an open system where different scattering processes take place.This behavior will get modified due to this deformation of quantum mechanics.Similarly the quantum mechanical systems for which the strict adiabatic approximation fails, but which do not escape too far from the adiabatic limit, can be analyzed using a a time dependent adiabatic deformation of the theory [53].It would be interesting to analyze the effect of having a minimum measurable time for such a time dependent adiabatic deformation of the theory.