Self-Adjoint Time Operator of a Quantum Field

We study the properties of a quantum field with time as a dynamical variable. Temporal vibrations are introduced to restore the symmetry between time and space in a matter field. The system with vibrations of matter in time and space obeys the Klein-Gordon equation and Schrodinger equation. The energy observed is quantized under the constraint that a particle's mass is on shell. This real scalar field has the same properties of a zero-spin bosonic field. Furthermore, the internal time of this system can be represented by a self-adjoint operator without contradicting the Pauli's theorem. Neutrino can be an interesting candidate for investigating the effects of these temporal and spatial vibrations because of its extremely light weight.


Introduction
The asymmetric formulation between time and space in quantum theory has inspired the quest for a time operator. In its formulation, time is postulated as a parameter. There is nothing dynamical about time in quantum theory. On the other hand, spacetime is dynamical and weaved as unity in general relativity. There is no globally de¯ned time in the theory. Therefore, the treatment of time is rather di®erent in quantum theory and general relativity. This con°ict has created constellation of problems when we try to reconcile the two fundamental theories from a single framework. 1,2 The reason why time is not treated as an operator can be traced back to Pauli. 3 Based on his reasonings, widely known as Pauli's theorem, a time operator t and a Hamiltonian operator H should satisfy a commutation relation, ½H; t ¼ Ài. Since the Hamiltonian of a system is typically bounded from below or discrete, time cannot be treated as a self-adjoint operator. On the other hand, time seems to play a more dynamical role in some quantum systems, e.g. tunneling time, 4,5 decay of an unstable particle, 6 dwell time of a particle, 7,8 and others. [9][10][11][12][13][14][15][16] To avoid contradiction with Pauli's theorem, a time operator in most of the studies are formulated by invoking the positive operator valued measures (POVM). [17][18][19][20][21] Apart from these e®orts, many propositions have also been made intending to resolve the dynamical nature of time in quantum theory. For example, Lee advocates that time can be considered as a fundamentally discrete dynamical variable in many classical and quantum models. 22,23 Greenberger infers that a few paradoxical behaviors in classical and quantum mechanics can be cured if mass is an operator and not as a simple parameter. 24 A selfadjoint time operator in the Dirac¯eld that satis¯es a commutation relation with the Hamiltonian analogous to the one between position and momentum has also been introduced by Bauer. 25 In this paper, we show that the properties of a bosonic¯eld can be reconciled from a¯eld with vibrations of matter in time and space. The hypothetical temporal vibrations [26][27][28] are introduced to restore the symmetry between time and space in a matter¯eld. The real scalar¯eld describing this system with temporal and spatial vibrations obeys the Klein-Gordon equation and Schr€ odinger equation. Its energy must be quantized under the constraint that a particle's mass is on shell. Additionally, the internal time of this system can be represented by a self-adjoint operator. The spectrum of this operator spans the entire real line without contradicting Pauli's theorem. The concepts developed are relativistically palatable.

Vibrations of Matter in Time and Space
In classical mechanics, matter can have vibration in the spatial directions but not in the temporal direction. If nature has a preference for symmetry, 29 it is not implausible that matter can also have vibration in time. To begin, let us consider a plane wave that has vibrations of matter in time only, as observed in an inertial reference frame O 0 , i.e.
of matter, T 0 is à proper time amplitude', and ! 0 is an intrinsic angular frequency of matter which we will later identify it as the frequency for mass-energy conjectured by de Broglie. 30 The meanings of t 0 f and T 0 will be explained further below. Note that matter in this plane wave has no vibration in the spatial directions, i.e.
where x 0 f is the coordinate of matter displaced from the equilibrium coordinate x 0 due to the vibrations in the wave. In this plane wave with vibrations of matter in proper time, the coordinates x 0 f and x 0 are the same.
`External time' t 0 is measured by clocks that are not coupled to the system under investigation. [31][32][33][34] These clocks are located far away at spatial in¯nity such that the e®ects from our system are negligible. We will use the external time t 0 as a reference to measure the vibrations of time in the system. It is an independent variable in the equations of motion and a parameter used as postulated in quantum theory. There is nothing dynamical about this external time.
Time t 0 f is the`internal time' of matter. It is a function of the external time t 0 and a dynamical variable for the system. Analogous to the amplitude of a classical oscillating system with vibrations in the spatial directions, we will de¯ne the proper time displacement amplitude T 0 as the maximum di®erence between the time t 0 f of matter inside the wave and the external time t 0 . Subsequently, if matter inside the wave carries an internal clock, an inertial observer outside will see the matter's clock measuring a time t 0 f , which is di®erent from the external time t 0 . Consequently, time measured by the matter's internal clock is running at a varying rate relative to the inertial observer's clock, @t 0 which has an average value of 1 over time. Matter in this plane wave will appear to travel along a near time-like geodesic if the magnitude of the vibration is relatively small. On the other hand, the time displacement relative to the external time is, This wave has the semblance of a classical oscillating system except the vibration of matter is in time and not in space. The nature of this internal time and the oscillating time displacement will be elaborated further in Sec. 3. The background coordinates ðt 0 ; x 0 Þ of inertial frame O 0 can be Lorentz transformed to the background coordinates ðt; xÞ for the°at spacetime in another frame of reference O, i.e.
We have assumed O 0 is traveling with velocity v relative to O. Similarly, the displaced coordinates of matter ðt 0 f ; x 0 f Þ can be Lorentz transformed to the displaced coordinates of matter ðt f ; x f Þ as observed in frame O, i.e.
Substitute Eqs. (1), (3), (6) and (7) into Eqs. (9) and (10), the displaced coordinates of matter ðt f ; x f Þ are where t ¼ ÀiTe iðkÁxÀ!tÞ ; ð13Þ x ¼ ÀiXe iðkÁxÀ!tÞ ; ð14Þ Amplitude X is the maximum displacement of matter from its equilibrium coordinate x, and amplitude T is its maximum displacement from the external time t. The proper time displacement T 0 can be seen as a Lorentz transformation of a 4-displacement vector: ðT 0 ; 0; 0; 0Þ ! ðT ; XÞ. Apart from the vibrations in time, matter in the plane wave also has vibrations in space as observed in frame O. At a particular instant, matter is displaced from its equilibrium coordinate x to another coordinate x f as shown in Eq. (12). This spatial vibration of matter is the same as de¯ned in a classical system.
We can write t and x in terms of a plane wave for describing the temporal and spatial vibrations of matter, i.e.
where t ¼ @ 0 ; ð17Þ In the rest of this paper, we will consider T , X and T 0 as complex amplitudes. The plane wave and its complex conjugate Ã satisfy the wave equations: The corresponding Lagrangian density for the equations of motion is and the Hamiltonian density is, where K is a constant of the system to be determined. Therefore, the plane wave satis¯es an equation of motion similar to the Klein-Gordon equation. However, we shall bear in mind that, so far, there is nothing that requires the energy in this wave to be quantized.

Proper Time Oscillator
Let us consider a plane wave from Eq. (16) that has vibrations of matter in time only Substitute 0 into Eq. (24), the Hamiltonian density is This result is similar to the Hamiltonian density of a harmonic oscillating system in classical mechanics, except that the vibrations are in time. Analogous to its classical counterpart, we make an ansatz, for a system that can have multiple number of point particles with mass m in a cube with volume V . Periodic boundary conditions are imposed on the box walls. From Eqs. (26) and (27), the energy inside volume V is, The vibration in proper time is an intrinsic property of matter. Energy E shall, therefore, correspond to certain energy intrinsic to matter. Since the vibration in proper time does not involve any force¯elds, E is not energy resulting from charges. In fact, the only energy present in this system is the matter with mass m. Here, we will consider E as the internal mass-energy resulting from the proper time vibration of matter. The internal mass-energy of matter must be on shell. For a system with only one particle, Eq. (28) becomes or from Eq. (19), This implies only an oscillator with proper time amplitude can be observed. a Additionally, the energy E from Eq. (29) is the internal massenergy of a point mass that is at rest. A particle observed in the plane wave 0 has oscillation in time but with no motion in the spatial directions.
Based on Eqs. (1) and (31), the internal time t 0 f of the particle observed is as follows: The internal time t 0 f is a function of the external time t 0 . It is an intrinsic dynamical property of matter with nothing to do with the relative velocity of the particle nor gravitational e®ects. The varying internal time rate shall have e®ects on the intrinsic properties of matter, e.g. decay rate of an unstable particle. In addition to the classical concepts of mass, 35,36 we suggest here a possibility that a point mass is a temporal oscillator in time with an angular frequency of ! 0 . In the rest of this paper, we will consider the angular frequency ! 0 as the de Broglie's frequency for the massenergy of a particle.
We have assumed matter is not traveling along a true time-like geodesic but with vibration over time. From Eq. (32), the internal time of the particle's clock passes at the rate of @ 0 t 0 f ¼ 1 À cosð! 0 t 0 Þ with respect to the external time. It has an average value of 1 and bounded between 0 and 2. Subsequently, the internal time of the oscillator moves only forward. It cannot go backward to the past. On the other hand, the particle will appear to travel along a time-like geodesic if the observer's clock is not sensitive enough to detect the high frequency of the oscillation. The accuracy of the measuring clock shall be restricted by the energy-time uncertainty relation. 24, 37 From Eq. (32), the time displaced from the external time t 0 of the proper time oscillator is The rate of this oscillating time displacement relative to the external time is The external time t 0 is the`equilibrium position' of this oscillating system. When the internal time is displaced from its`equilibrium position', the system tries to return to its equilibrium. After the internal time reaches its equilibrium, the nonzero oscillating time displacement rate causes the internal time to pass over the equilibrium. This system is analogous to a classical simple harmonic oscillator except the oscillation is in time.
As shown in Eq. (29), the oscillation of matter in time can give rise to the massenergy E of a particle which can also be written in terms of Á t 0 and @ðÁ t 0 Þ=@t 0 , i.e. a A ring symbol on the top denotes the quantity is a property of the observed particle.
The internal mass-energy E is the summation of two parts. The¯rst part is the energy arising from the oscillating time displacement, Á t 0 . The second part is the energy resulting from the oscillating time displacement rate, @ðÁ t 0 Þ=@t 0 . They are analogous to the`potential' and`kinetic' energy components of a classical harmonic oscillator. The total energy of this oscillator is conserved over time. There is no energy°owing in and out.
Next, let us consider a plane wave 0 with amplitude The Hamiltonian density of this plane wave from Eq. (24) is H 0 ¼ m=V . If we probe only a part of the system with volume V 1 (< V ), the energy observable is It is only a fraction of a particle's mass-energy. However, if a particle is point-like and its mass-energy is on shell, how can we explain the presence of a fraction of a particle in volume V 1 ? In a quantum wave, a probability density can be de¯ned for the observation of a particle at a particular location. To explain the presence of a fraction of the particle's mass-energy, we can treat the plane wave 0 as a probabilistic wave. In other words, there is only a probability of observing a particle in the probed volume V 1 . Based on the Hamiltonian density H 0 , we can de¯ne a probability density for the plane wave 0 , i.e.
The probability for observing a particle in volume V 1 is p 1 ¼ V 1 =V . After making many measurements with similar experimental setup, the average mass-energy ob-

Moving Particle with Vibration in Time
In the normalized plane wave 0 from Eq. (36), one particle with rest mass m ¼ ! 0 can be observed when we probe the system as a whole. As discussed, the particle observed has oscillation in proper time with an amplitude jT 0 j ¼ 1=! 0 . From Eq. (16), the Lorentz transformation of the normalized plane wave 0 to another frame O is Frame O 0 is assumed to be traveling at a velocity v ¼ k=! relative to frame O. The particle observed in frame O shall be traveling with an average velocity v, but with oscillations in time and space after the Lorentz transformation of the proper time oscillation; its energy is E ¼ !. However, when we examine the Hamiltonian density of the normalized plane wave obtained from Eq. (24) it is equivalent to one particle with energy ! in a volume Since the system we are studying has a volume V and not V 0 , it is necessary to include the normalization factor with the plane waves when they are used in the superposition for a more general application.
In the rest of this paper, it is more convenient to adopt another plane wave for our analysis, i.e.
Substitute into Eq. (24), the Hamiltonian density is Based on Eqs. (11), (12), (17) and (18), the temporal and spatial vibrations in plane wave aret x ¼ Àr ¼ ÀiXe iðkÁxÀ!tÞ ; ð45Þ For a normalized plane wave with T 0 ¼ 1=! 0 , the Hamiltonian density from Eq. (41) isH ¼ !=V . This is equivalent to one particle with energy ! in a volume V . Since the system that we are investigating has a constant volume V , we shall utilize when the superposition principle is applied. Assuming a particle in the plane wave is¯rst observed at origin of the x coordinates at t ¼ 0. After Lorentz transforming the proper time oscillator observed in plane wave 0 and including the normalization factor with the amplitudes of oscillation, the internal time of the particle following the path x ¼ vt as observed in is The internal time rate relative to the external time is Apart from the oscillation in time, the particle observed also has oscillation in the spatial directions. Its trajectory is, The observed velocity with oscillation is, We shall note that as jvj ! 1, the magnitude of the amplitudes approach in¯nity, i.e. jT j ! 1 and jX j ! 1. On the other hand, ! p is the angular frequency of a moving particle. It is not the angular frequency ! of the plane wave. As jvj ! 1, the angular frequency ! p slows down and approaches zero, ! p ! 0. Subsequently, a particle traveling at a higher speed will have a lower frequency and larger amplitudes of oscillation.
As we have discussed in the previous section, the plane wave 0 with proper time vibrations shall be treated as a probabilistic wave. Similarly, the same concept shall be applied to plane wave . The Hamiltonian density from Eq. (41) can be written in terms of , i.e.H Since the energy of an observed particle in plane wave is E ¼ !, we can de¯ne a probability density of observing a particle as, There is only a probability of observing a particle with energy ! at a particular location in plane wave .

Wave Function
Based on Eq. (55), we can de¯ne a function in terms of the plane wave in the nonrelativistic limit where and e i is an arbitrary phase factor. From Eqs. (19), (55) and (56), the square modulus of the function , is a probability density; n ¼ a Ã a is the number of particle that can be observed in volume V . As discussed in the previous section, there is only a probability to observe a particle at a location. Function has the basic properties of a wave function in quantum mechanics. Applying the superposition principle, we can write where Boundary condition is imposed such that ðx; tÞ vanishes at the box walls. This wave function ðx; tÞ is a solution of the linear and homogeneous Schr€ odinger equation, i.e. i : ðx; tÞ ¼ Àð2mÞ À1 r 2 ðx; tÞ. The probability amplitude a k of the wave function can be expressed in terms of the proper time vibration amplitude T 0k from Eq. (60). The probability density of observing a particle is the square modulus of ðx; tÞ. As we have illustrated, the properties of the quantum mechanical wave can be reconciled from the system with vibrations of matter in time and space.
In quantum mechanics, it is commonly believed that a matter wave can only have a probabilistic interpretation because of the unobservable phase for the wave function . 37 As we shall note, the introduction of the arbitrary phase factor e i does not change the probability density calculated in Eq. (58). The wave function with the arbitrary phase factor still satis¯es the Schr€ odinger equation. Therefore, the plane wave and the wave function can have an arbitrary phase di®erence but this di®erence will not alter the results obtained in quantum mechanics. As demonstrated in quantum mechanics, the theory developed with wave functions is invariant under global phase transformation but the relative phase factors are physical.
Although the overall phase of the wave function can be unobservable, it serves only as a mathematical tool for describing an underlying wave with vibrations of matter in time and space. Here, we demonstrate a possibility that the matter wave can have a physical interpretation other than the probabilistic one despite the overall phase of the wave function is unobservable.

Bosonic Field
We can obtain a real scalar¯eld by the superposition of the plane waves and their conjugates Ã , i.e.
To adopt the same convention in quantum¯eld theory, we will de¯ne which satis¯es the Klein-Gordon equation. This real scalar¯eld is subject to the boundary condition that '(x) vanishes at the box walls. As discussed in the previous sections, the system with matter vibrating in time and space shall be treated as a probabilistic wave. By allowing matter to vibrate in both the temporal and spatial directions, we can reconcile the properties of a quantum wave in the nonrelativistic limit. The system considered is explained by discrete particles and not a spatially continuous¯eld. As developed in quantum eld theory, the transition of a classical¯eld to a quantum¯eld can be done via canonical quantization. Here, we can also adopt the same concept to obtain a quantum¯eld from the system that has vibrations of mater in time and space. In other words, the¯elds '(x) and (x) are to be promoted to operators. Since the quantization of a real scalar¯eld is a familiar process, we will only outline some of the key results involving the temporal vibrations that are not part of the standard quantum theory.
We can show that '(x) is the same bosonic¯eld in quantum theory after we rewrite Eq. (62) in terms of the creation operator, and the annihilation operator, such that The operators a k , a † k , T 0k and T † 0k shall satisfy the commutation relations, In fact, by expressing the creation and annihilation operators in terms of T 0k and T † 0k , we can rewrite other operators in quantum theory using the temporal and spatial vibrations¯eld. For example, the particle number operator is, The Hamiltonian of the system is, which corresponds to an in¯nite sum of normal mode oscillator excitation. Since the conversion of other operators are straightforward, we will not repeat them in here.
Note that normal ordering shall be taken between T 0k and T † 0k . As we have demonstrated, the formulation of a bosonic¯eld can be expressed in terms of the vibrations of matter in time and space. As a quantum¯eld, the energy in the system considered shall be quantized; the oscillators in proper time are the¯eld quanta. The results obtained here have the familiar properties of a zero-spin bosonic eld except they are obtained from a¯eld that has vibrations of matter in time and space.

Internal Time Operator
After quantization, the¯eld (x) from Eq. (61) can be rewritten in terms of the temporal vibration amplitude operatorsT k and their Hermitian conjugatesT † k , i.e. whereT the semi-bounded Hamiltonian that restricts the spectrum of the temporal vibration operator [using t (x) or (x)] to be bounded. Based on Eq. (11), the internal time in the real scalar¯eld at a particular time t is where t ðt; xÞ is real as shown in Eq. (77). The internal time t f ðt; xÞ is the summation of the temporal vibration t ðt; xÞ and the external time t. Since the temporal vibration t ðt; xÞ is a self-adjoint operator and the external time t is a parameter, the internal time t f ðt; xÞ is also a self-adjoint operator. The symmetry between time and space is restored in the system with both the internal time and position of matter can be treated as self-adjoint operators.

Conclusions and Discussions
Unlike the asymmetrical formulation of time in the classical and quantum theory, time and space have a more symmetrical treatment in the system considered. Matter in this system can vibrate not only in the spatial directions but also in the temporal direction. The symmetry between time and space is restored in the system with both the internal time and position of matter can be treated as self-adjoint operators. Here, we demonstrate a possibility that time can have a more dynamical role in the quantum¯eld. By allowing matter to vibrate in both time and space, we have reconciled the properties of a bosonic¯eld. The reason why the temporal vibration t can be treated as a self-adjoint operator is that we are considering an oscillating system. An oscillator with temporal vibration can have displacement either in the positive or negative direction relative to the external time t. Therefore, its spectrum can span the whole real line. As we have pointed out in the previous section, the internal time t f can also be treated as a selfadjoint operator since it is the summation of a parameter t and a self-adjoint operator t . This internal time operator does not form a conjugate pair with the Hamiltonian of the system. Therefore, the restriction imposed by Pauli's theorem does not apply in this case. The inclusion of time oscillation in a matter¯eld is a new approach that we have adopted in this paper.
The temporal oscillation can theoretically a®ect the rate of change for the intrinsic properties of a particle (e.g. decay rate of an unstable particle). Additionally, a particle has oscillation in the spatial directions as it propagates through space. To examine the magnitude of these oscillations, let us consider a neutrino, which is the lightest known elementary particle. Assuming the mass of a neutrino b is m ¼ 2 eV (! 0 ¼ 3:04 Â 10 15 s À1 and T 0 ¼ 1=! 0 ¼ 3:29 Â 10 À16 s), the amplitudes and b At present, the absolute mass scale of neutrino has not yet been determined. The assumed mass m ¼ 2 eV is the upper limit of the electron-neutrino mass determined by direct measurements as shown in Refs. 38 and 39. frequency of a moving neutrino can be obtained from Eqs. (48), (49) and (52), e.g.
E ¼ 1 Gev ) T ¼ 7:4 Â 10 À12 s; jX j ¼ 0:22 cm; ! p ¼ 6:1 Â 10 6 s À1 : The e®ects of the temporal and spatial oscillations shall be considered when we are taking measurements of a neutrino. As discussed in Sec. 4, a particle traveling at a higher speed will have a lower frequency and larger amplitudes of oscillation. Consequently, it will be easier to detect the hypothetical e®ects of the additional oscillations at a higher speed. For instance, the spatial oscillation of a particle is along the direction of propagation. A neutrino will propagate with oscillating motions along its path. Therefore, two neutrinos with the same initial velocity can reach a target at slightly di®erent times depending on the relative phase of their oscillations. In theory, this deviation can be observable by repeated measurements of the neutrinos' arrival times at a detector. Because of their extremely light weight, neutrinos can be projected to a very high speed which can amplify the oscillations for measurements. Neutrino can be an interesting candidate for investigating the e®ects of these temporal and spatial oscillations.