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Projection evolution of quantum states

A. Góźdź ∗1, M. Góźdź †2, and A. Pȩdrak ‡3

1Institute of Physics, Maria Curie–Skłodowska University inLublin, Poland

2Institute of Computer Science, Maria Curie–SkłodowskaUniversity in Lublin, Poland

3National Centre for Nuclear Research, Warsaw, Poland

Abstract

We discuss the problem of time in quantum mechanics. In thetraditional formulation time enters the model as a parameter, not anobservable, which follows from the famous Pauli theorem. It is nowknown, that Pauli’s assumptions were too strong and that by removingsome of them time can be represented as a quantum observable. Inthis case, instead of the unitary time evolution, other operators whichmap the space of initial states into the space of final states at each stepof the evolution can be used. This allows to treat time as a quantumobservable in a consistent way.

We present the projection evolution model and show how the tra-ditional Schrödinger evolution can be obtained from it. We proposethe form of the time operator which satisfies the energy-time uncer-tainty relation. As an example, we discuss the temporal version of thedouble-slit experiment.

1 IntroductionFor many years time has been treated in physics as a universal parameterwhich allows the observer to divide the reality into past, present, and fu-ture. What is more, time flows always in one direction, called the arrow of∗email: [email protected], ORCID: 0000-0003-4489-5136†email: [email protected], ORCID: 0000-0003-4958-8880‡email: [email protected], ORCID: 0000-0002-8808-3239

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time. This direction implies also the direction of changes that may sponta-neously happen to any physical system, which ultimately leads to the notionof causality. We are used to the fact that past affects future, but futurecannot affect the past, as this will act against the arrow of time.

The development of relativity theory changed this picture in a substantialway. To obtain a consistent model, time had to be treated as a coordinate,forming with the spatial coordinates the spacetime. The metric and othertensors gained their time components which were transforming during thechange of the coordinate system along with the spatial coordinates. Forexample, the four-momentum pµ, µ = 0, 1, 2, 3, takes the form pµ = (p0, ~p),where p0 = E/c, E being the total energy. This feature is absent in thenon-relativistic physics.

One may ask if time behaves in the same way in the macroscopic andmicroscopic scales? We know that both classical and relativistic physicsagree upon the basic properties of time, so if one expects any deviationsfrom the standard picture, one should look at the quantum mechanics.

1.1 The Pauli theorem

In the standard formulation of the quantum theory, any physical quantityis represented by a hermitian operator whose eigenvalues are the possibleoutcomes of its measurement. However, the so-called Pauli theorem [1, 2,3] states, that it is impossible to construct a hermitian time operator. Itfollows that time is not a physical observable but is introduced as a universalnumerical parameter. This approach is inconsistent with what we know fromthe relativity theory, not to mention that it gives very limited means todiscuss quantum events in the time domain.

The Pauli theorem is an observation that it is in general impossible to con-struct a self-adjoint time operator t̂, which would be canonically conjugate toa generic Hamiltonian Ĥ. The justification of this claim is as follows: assumethat we have such a pair of canonically conjugated hermitian operators,

[t̂, Ĥ] = i. (1)

Since t̂† = t̂, one may construct a unitary operator U = exp(−iβt̂), withβ ∈ R being an arbitrary parameter. The commutator [U, Ĥ] can now becomputed using the power expansion of U , yielding [U, Ĥ] = βU . It immedi-ately follows that for a Hamiltonian eigenvector φ, for which Ĥφ = Eφ, wehave

ĤUφ = (E − β)Uφ. (2)Because β is an arbitrary real number, Eq. (2) implies that the spectrumof the Hamiltonian must always be continuous and unbounded from below,

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which is obviously not true. Pauli stated: “We conclude that the introduc-tion of an operator t̂ must fundamentally be abandoned and that the time inquantum mechanics has to be regarded as an ordinary number.” This strongconclusion means that it is impossible to discuss the time structure of eventswithin the quantum theory.

A careful mathematical analysis of this problem was presented by E.A.Galapon in Ref. [4]. By considering the domains of the operators t̂ and Ĥ theauthor showed, that the domain of their commutator does not contain thedomain of Ĥ, contrary to what was silently assumed by Pauli. This impliesthat the β parameter can only have values which appropriately correspondto the eigenvalues of Ĥ and therefore, as it is not arbitrary, the Pauli con-clusion does not hold. It follows that a hermitian time operator canonicallyconjugated to Ĥ can in principle be constructed. What is more, observablesdo not need to be hermitian operators, but are in general represented bypositive operator valued measures (POVM). There are no arguments againstthe construction of a time operator in terms of a POVM – a possibility thatwas not considered by Pauli.

1.2 Theoretical models and experimental work

It has been intensively discussed how to introduce time as an observable inthe theory, as this affects the construction of the arrow of time and clocks (see[5, 6] for recent developments). Related topics include also the problem oftime in entangled systems, the time of decoherence and the role of the energy-time uncertainty relation. The process of quantization can be performed indifferent ways and tested using specially designed experiments, like it hasbeen shown in the example of the time of arrival operator [7, 8]. Sincetime is connected with the energy operator, thermodynamics of quantumprocesses started to be of interest [9]. It has already been shown that dueto the quantum correlations, heat may spontaneously flow from the colderto the hotter subsystem [10], which is not observed in the macroscopic scale.Entanglement and the immediate change of state of both entangled particlesrises also the question how to describe [11] and experimentally investigate[12] this process.

The problem of time appears also in systems performing quantum compu-tation. Most quantum protocols assume that we can neglect the time delaysintroduced by quantum gates and connections in the system, which does nothave to be the case. Another problem arises with the theoretically proposedquantum gates with feedback [13], which are impossible to describe usingstandard tools. Understanding the time structure of quantum operations isalso vital for constructing future quantum neural networks [14].

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Treating time as an observable leads to the problem of time measure-ments [15], also in the context of quantum cosmology [16]. As time becomesa variable, new phenomena start to be possible, like dark matter describedby fields evolving backwards in time [17].

The important role of time in quantum theories is suggested by some ex-periments. In Refs. [18, 19] J.A. Wheeler proposed a Gedankenexperimentbased on the Mach–Zehnder interferometer, consisting of two beamsplittersand two mirrors. A single photon was traveling through the interferometer.During the particle’s flight some changes to the setup were introduced, in-cluding the removal or insertion of the first or the second beamsplitter, evenafter the photon has classically passed that part of the machine. Wheelerargued that the final detection of the photon should be sensitive to thesechanges, mainly due to the spatial width of the photon wave function. Thisidea has been experimentally tested. An analogue of the Mach–Zehnder in-terferometer was used by the group of A. Aspect with the primary intentionto test Bell’s inequalities [20] showing that Wheeler’s predictions were cor-rect. Other groups [21, 22, 23, 24, 25, 26] arrived at similar results. In orderto investigate the problem further, a quantum eraser was used. Its purposewas to remove the information about an additional measurement, which wasdone on the particle during the experiment. It turned out that erasing theinformation recreates the quantum behavior of the system even in the casewhen the eraser worked after the final detection has been performed [27].This setup has been called the delayed choice quantum eraser. In anotherexperiment [28] the interferometer was built between an earth-based stationand a satellite. The photons behaved like particles or waves depending onthe choices made by the investigators on earth. The effect was visible evenwhen the changes introduced to the setup were causally disconnected fromthe particles.

Another experiment was conducted using entangled pairs of photons [29,30]. The pair was created in one laboratory and one of the particles stayedthere, while the other was sent to the second laboratory. The transmissiontook place between two islands, La Palma and Tenerife, with the distancebetween them around 144 km. Even though the particles were causally dis-connected, the changes made in the first laboratory were affecting the secondparticle suggesting, that either we accept a faster-than-light communication,or the notion of the spatial and temporal localization of a quantum objectshould be reformulated.

If time in the quantum regime should be treated as a coordinate, and infact a quantum observable, all physical objects have to have some “width”in the time direction, which is related to the energy-time (more precisely –the temporal component of the four momentum operator versus time) un-

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certainty relation. This means that it should be possible to observe theinterference of quantum objects through their overlap in time. One of thefirst experiments in which such behavior has been observed, was reported inRef. [31]. A single photon was emitted and a spinning chopper in the formof a wheel with slits was placed between the source and the detector. Theenergy spectrum of the detected photons was recorded as a function of thedisc’s rotation frequency. The spectrum clearly showed minima and maximain a way very similar to the interference pattern. The authors were unableto fully explain this behavior, even though they presented a simple analysisbased on the Fourier transform of the energy spectrum. It was pointed outin Refs. [32, 33] that after the Fourier transform of the energy-dependentfunction the authors worked with the time-dependent one and all what theygot could be interpreted as an interference pattern among the temporal partsof the photon wave function. It also means, that time is considered here asa variable canonically conjugated to the energy, and that it cannot be treatedas a parameter. So the most obvious explanation suggests the observationof the interference of the photon with itself in different time instances. Wediscuss this type of interference in Chapter 6.

Another experiment was reported in Ref. [34] where short laser pulsesopened one or two attosecond time periods, during which photoionizationwas possible. A single electron encountered one and two opened slits at thesame time, which was visible in the fringes of the electron’s energy spectrum.The obtained experimental data was qualitatively in agreement with the tem-poral double-slit interpretation. The Authors’ claim about the agreement ofthe data with the numerical solution of the time-dependent Schrödinger equa-tion is unclear. We show in Sec. 4.1 that the solutions of the Schrödingertype equation in the case of time treated as a variable and time considered asa parameter are expected to be numerically similar. However, the physicalinterpretation is very different – only time interpreted as a quantum observ-able allows to explain the temporal interference. The Authors of [34] mentionalso some classical models admitting at the same time that their assumptionsare fulfilled to some extent only. Due to the impossibility of a proper de-scription of the time phenomena in the traditional quantum mechanics, thepaper does not contain, in fact, any theoretical discussion.

It seems to be very difficult to answer the fundamental question: Whatis time? In Ref. [35] the authors propose, that time is a consequence of theentanglement between particles in the universe. If that were true, it would beimpossible to incorporate this notion in any classical theory, as entanglementis a purely quantum effect.

In this paper we present a consistent formulation of the quantum theorywith time being an observable. This allows to interpret the zeroth component

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of the four-momentum as the time translation operator, as well as to definethe time operator. In this approach the evolution of quantum states hasto be reformulated, as time, being a coordinate, cannot act as a universalordering parameter any longer. We show that the traditional Schrödingertime evolution can be obtained as a special case within our model. Theproblem of symmetries and conservation laws during the evolution is alsoshortly discussed. We finish the paper by discussing the temporal double-slitexperiment analogous to the work reported in Ref. [31]. We show that ifa single slit opens during two different time instances at the same spatiallocation, the particle will interfere with itself from a different time instance.

2 Projection evolution of quantum systemsThe probabilistic structure of quantum mechanics allows time to be, undercertain conditions, a quantum observable. First of all it cannot be consid-ered as a parameter which enumerates subsequent events but it has to berepresented by an operator similar to the position operators. It means thatdifferent time characteristics of a given quantum system can be calculated.In general, they are dependent on the state of this system. We start from theassumption that the quantum time, and generally the spacetime, is “created”by changes of the Universe. This requires the modification of some parts ofthe paradigm of science related to the causality and the ordering of quantumevents.

2.1 The changes principle

We start by formulating the fundamental assumption of the projection evo-lution (PEv) principle:

The evolution of a system is a random process caused by spontaneouschanges in the Universe.

We call it the changes principle. It means that we treat the change asthe primary process, which allows to define time (also space and generallyspacetime). This is in contradiction with the usual thinking in which theexistence of time allows the changes to happen. In our approach the changeshappen spontaneously, according to the probability distribution, which isdictated by many factors describing the system and its environment. It doesnot mean that the changes of a quantum state of the subsystems of theUniverse are totally stochastic, without any constraints. They are obviously

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not deterministic, but because of the interactions, symmetries which haveto be conserved, EPR correlations etc., they are related to each other andbound by the rules of their behavior known from our experience.

As a consequence one may expect the existence of a kind of pseudo-causality based on the ordering of the quantum events, which leads to thecausality principle in the case of macroscopic physical systems. In order todescribe this property we introduce a parameter τ which orders quantumevents. This parameter should be common for the whole Universe. It shouldtake values from an ordered set but it does not need to have any metricstructure. The parameter τ is not an additional dimension of our space andit is not a replacement of time. It serves only to enumerate the subsequentsteps of the evolution of the Universe and any of its physical subsystems.The most natural linearly ordered set is any subset of the real numbers.

In what follows we assume that the domain of the evolution parameter τis isomorphic to integers or their subset. In this case we can always use thenotion of “the next step of the evolution,” which may be problematic for thereal numbers.

In the situation of a continuous or dense subset of the real numbers asthe domain for τ , there are some conceptual difficulties which should be, ifneeded, solved in the future.

An additional, very important feature of this approach is that this ideadoes not need the spacetime as the background. Nearly all physical the-ories constructed till now use the spacetime as the primary object, withthe dynamics built on top of it. The projection evolution approach isa background free theory.

2.2 Projection evolution operators

In the standard formulation of quantum physics, there are two kinds of timeevolution: (i) the unitary evolution, which is a deterministic evolution of theactual quantum state, and (ii) the stochastic evolution, which takes placeduring a measurement. The latter process involves the projection of thequantum state onto the measured state and can be described by one of theprojection postulates. There is a common belief that this process can bedescribed by means of the unitary evolution of a larger system. This approachleads, however, to the known quantum measurement problems [36].

The changes principle is incompatible with the unitary evolution, wheretime is considered to be a parameter. The idea of the changes principle sug-gests the opposite scenario – the primary evolution is the stochastic evolutionoffered by a projection postulate. We propose to use the generalized formof the Lüders [37] type of the projection postulate. We show later that the

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Schrödinger type evolution can be obtained as a special case of the projectionevolution.

In the following, we introduce the projection operators which are formallyresponsible for the quantum evolution of a physical object. We expect thatin general these operators will be different for different systems, similarlyto the Hamiltonian, which is a characteristic object for a given quantumsystem. On the other hand, one should in principle be able to construct theprojection evolution operators for the whole Universe which will contain theoperators for any smaller subsystem. It is due to the fact that the proposedformalism does not require any external observer for the evolution.

The projection evolution operator at the evolution step τn, where n ∈ Z, isa family of mappings between the space of quantum states at the evolutionstep τn and the space of quantum states at the evolution step τn+1. Thestate spaces are assumed to be some subspaces of the trace class operators(the space of operators with finite trace, i.e., the set of density operators)defined on a given Hilbert space K. In this case the Hilbert space of a single,nonrelativistic spinless particle is not contained in L2(R3, d3x) but rather inL2(R4, d4x), where the fourth dimension is time, treated here on the samefooting as the positions in the 3D-space.

Let us denote by T +1 (K) the space of positive, trace one operators inK, and by T +(K) the space of finite trace positive operators in K. Here,K = K(τn) denotes the space of states at the evolution step τn.

The projection evolution operators at the evolution step τn are formallydefined as a family of transformations from the quantum state space (densityoperators space) T +1 (K(τn−1)) to the space T +(K(τn)),

F|(τn; ν, ·) : T +1 (K(τn−1))→ T +(K(τn)), (3)

where ν ∈ Qn, with Qn ≡ Qτn being a family of sets of quantum numbersdefining potentially available final states for the evolution from τn−1 to τn.We denote by F|(τn; ν, ρ) the action of the operator F|(τn; ν, ·) on the densityoperator ρ, such that F|(τn; ν, ·)ρ = F|(τn; ν, ρ). The notation F|(τn; ν, ρ) is inmany cases more appropriate because, in general, the evolution operatorsdo not need to be linear, although in the following we assume the operatorsF|(τn; ν, ·) to be linear.

To use the generalized Lüders projection postulate as the principle for theevolution, the operators (3) have to be hermitian, non-negative, and finite:

F|(τ ; ν, ρ)† = F|(τ ; ν, ρ), (4)F|(τ ; ν, ρ) ≥ 0, (5)∑ν∈Qτ

Tr(F|(τ ; ν, ρ))

ν(n−1),1

ν(n−1),3

ν(n+1),2νn,3

ν(n+1),1

ν

νν

ν

ν

n−1ρ(τ

(n−1),2

n,2

n,1

n,4(n+1),3

ρ(τn ρ(τn+1,νn,3 ) ,ν(n+1),1))(n−1),1,ν

Figure 1: The density matrix ρ is randomly chosen at each evolution stepτ from the possible states labeled by Qm = {νm,1, νm,2, . . . }, where m =n− 1, n, n+ 1.

These three conditions allow F| to transform the density operator ρ into an-other density operator, as is shown in Eq. (7) below.

Assume that at the evolution step τn−1 the actual quantum state ofa physical system is given by the density operator ρ(τn−1; νn−1), with νn−1 ∈Qn−1. The changes principle implies that every step of the evolution is simi-lar to the measurement process in the sense that there exists a mechanism inthe Universe, the chooser, which chooses randomly from the set of states de-termined by the projection postulates the next state of the system for τ = τn.With these assumptions, following Ref. [37], we postulate ρ(τn; νn), νn ∈ Qn,in the form

ρ(τn; νn) =F|(τn; νn, ρ(τn−1; νn−1))

Tr (F|(τn; νn, ρ(τn−1; νn−1))). (7)

Because the chooser represents a stochastic process, to fully describe it oneneeds to determine the probability distribution for getting a given state inthe next step of the evolution. An example of an evolution path is presentedin Fig. 1 by the solid line. The dotted lines show other potential paths.

In general, the probability distribution for the chooser is given by thequantum mechanical transition probability from the previous to the nextstate. This probability for pure quantum states is determined by the appro-priate probability amplitudes in the form of scalar products. The transitionprobability among mixed states, in general, remains an open problem.

We denote the transition probability (or the transition probability den-sity) from the state labelled by the set of quantum numbers νn−1 at τn−1 tothe state labelled by the set of quantum numbers νn at τn for a given evolu-tion process by pev (νn−1 → νn). The arguments of pev indicate the initialand the final state of the transition.

The most important realization of the evolution operators F|(τn; νn, ρ) can

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be constructed from the density matrix ρ and some operators E| in the fol-lowing form: for every νn ∈ Qn we have

F|(τn; νn, ρ) =∑α

E|(τn; νn, α) ρ E|(τn; νn, α)†, (8)

where the summation over α is dependent on the quantum numbers νn. Itis easy to check that the conditions (4) and (5) are automatically fulfilled,namely:

F|(τn; νn, ρ)† =∑α

E|(τn; νn, α) ρ E|(τn; νn, α)† = F|(τn; νn, ρ) (9)

and, since ρ ≥ 0, we have for all φ ∈ K

〈φ|∑α

E|(τn; νn, α) ρ E|(τn; νn, α)†|φ〉

=∑α

(〈φ|E|(τn; νn, α)) ρ (E|(τn; νn, α)†|φ〉) ≥ 0. (10)

Using Eq. (8) and the fact that trace is cyclic, the left hand side of thecondition (6) takes the form∑

νn∈Qn

∑α

Tr(E|(τn; νn, α) ρ E|(τn; νn, α)†)

=∑νn∈Qn

Tr

(∑α

E|(τn; νn, α)†E|(τn; νn, α) ρ

). (11)

It follows that the relation (6) is fulfilled if the transformation∑α

E|(τn; νn, α)† E|(τn; νn, α)

does not lead outside the space T +, i.e.,∑νn∈Qn

∑α

E|(τn; νn, α)† E|(τn; νn, α) : T +1 (K(τn−1))→ T +(K(τn)). (12)

Typical and useful examples of the E| operators are connected with theunitary evolution and the orthogonal resolution of unity. In the first case theoperator is

E|(τn; νn0, α0) = U(τn), (13)

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where νn0 and α0 are some fixed values of νn and α, and U(τn) is a unitaryoperator. In this case, following Eq. (8), the next step of the evolution ischosen uniquely with the probability equal to 1, as

ρ(τn; νn) = U(τn) ρ(τn−1; νn−1) U(τn)†. (14)

One needs to note that the unitary operator (13) is not parametrized bytime but by the evolution parameter τ , even though, in general, it is timedependent.

In the case of the orthogonal resolution of unity with respect to the quan-tum numbers νn the following conditions hold (we have fixed for simplicitythe α parameter and omitted it in the notation, but the more general casecan be written similarly):

E|(τn; νn)† = E|(τn; νn),E|(τn; νn)E|(τn; ν ′n) = δνnν′nE|(τn; νn),∑νn∈Qn

E|(τn; νn) = I, (15)

where I denotes the unit operator. Different alternatives of choices of quan-tum states are described by different sets of quantum numbers νn.

The probability distribution of choosing the next state of the evolutiongenerated by (15) is now given by the known quantum mechanical formula:

pev (νn−1 → νn) = Tr(E|(τn; νn) ρ(τn−1, νn−1) E|(τn; νn)†

). (16)

The above discussed examples, even though generic for many quantummechanical systems, are only special cases of the more general evolutionoperators.

3 The time operatorIn order to introduce the time operator, let us consider, without loss ofgenerality, a single particle described by the state space K ⊂ L2(R4, d4x) ofpure states, i.e., vectors, instead of density operators. The scalar product inthe state space K is given by

〈Φ2|Φ1〉 =∫R4d4xΦ2(x)

∗Φ1(x), (17)

where x denotes the spacetime coordinates x = (x0, ~x). As usually, x0 isthe time coordinate and ~x = (x1, x2, x3) are spatial coordinates. The scalar

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product (17) has the following probabilistic interpretation: the spacetime re-alization Ψ(x) of any pure state |Ψ〉 ∈ K represents the probability amplitudeof finding the particle in the spacetime point x, i.e., 〈Ψ(x)|Ψ(x)〉 = |Ψ(x)|2is the probability density of finding this particle at x.

The PEv approach leads to the breaking of the classically understoodcausality. The functions Ψ(x) ∈ L2(R4, d4x) in their general form connectalso events with space-like intervals (x0)2 − ~x2 < 0. Obviously, this can beeasily removed by assuming that K consists of functions with time-like andzero-like support only, which means that outside the set (x0)2 − ~x2 ≥ 0 thefunctions Ψ(x) are zero. Some experimental works [38] suggest, however,that it is a natural phenomenon that the classical causality is broken in thequantum world. To be more general, we allow for states which break theclassical causality to some acceptable extend. Within the PEv approachthe quantum causality is realized by keeping the correct sequence of thesubsequent steps of the evolution, ordered by the parameter τ .

The PEv formalism allows to construct the time operator, as the Paulitheorem is not applicable in this approach. Let us remind, that all thespacetime components of the position operator x̂ = (x̂0, x̂1, x̂2, x̂3) are consid-ered quantum observables. Using the spacetime representation of the statesΨ(x) = Ψ(x0, x1, x2, x3) ∈ K, the position operators are usually defined asthe multiplication operators:

x̂µΨ(x) = xµΨ(x). (18)

Because they commute, they have a common spectral decomposition,

x̂µ =

∫R4d4x xµMX(x), (19)

where MX(x) = |x〉〈x| stands for the orthogonal spectral measure, formingthe resolution of unity, of the four-vector position operator x̂. In other words,the kets |x〉 are understood as generalized eigenvectors of the four-vectorposition operator x̂. The operator MX(x) = |x〉〈x| itself can be interpretedas the generalized projection operator (in this case it is an operator valueddistribution) which projects onto the state representing the single spacetimepoint x.

In the non-relativistic case the notion of simultaneity is independent ofany observer and one can construct a spectral measure MT (x0), which forany fixed time t = x0 projects onto the space of simultaneous events:

MT (x0) =

∫R3d3xMX(x). (20)

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This allows to built the time operator (a preliminary attempt can be foundin Ref. [39]) in the form

t̂ ≡ x̂0 =∫Rdx0 x0MT (x

0), (21)

which, in fact, is the multiplication operator as mentioned above. Using (20)and (21) we get

t̂Ψ(x) ≡ 〈x|t̂|Ψ〉 =∫R4d4x′t′〈x|x′〉〈x′|Ψ〉 = x0Ψ(x), (22)

where the normalization of the position states |x〉 is given by 〈x|x′〉 = δ4(x−x′).

In the relativistic case the situation is a bit more complicated because thesimultaneity relation is observer dependent. One needs to notice, however,that the time operator is the zeroth component of the four-vector spacetimeposition operator (19), which is a covariant quantity with respect to thePoincaré group,

t̂ =

∫Rdx0 x0

{∫R3d3xMX(x)

}. (23)

This time operator is well determined for every observer but it cannot beconsidered a standalone observable. It has always to be treated as a part ofthe four-vector position operator x̂.

As a by-product one can construct the spectral measure which can beused as a measure of causality of a given state |Ψ〉 at the time x0,

M(C)T (x

0) =

{∫C(x0)

d3xMX(x)

}, (24)

where C(x0) = {~x : (x0)− ~x2 ≥ 0}. The expectation value of this operator,

ProbC [Ψ] = 〈Ψ|M (C)T (x0)|Ψ〉, (25)

gives the probability that the particle described by the state |Ψ〉 is in thelight cone, both in the past and in the future directions, with the vertex atx0.

An important operator related to the time operator is the temporal com-ponent p̂0 of the four-momentum operator p̂ = (p̂0, p̂1, p̂2, p̂3). In the space-time representation, the operator, which is canonically conjugate to the po-sition operator x̂, is the generator of the translations in the spacetime,

p̂µ = i∂

∂xµ. (26)

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The temporal component of the momentum operator measures, similarly tothe spatial components, the value of the product “inertia”×“speed” for theparticle moving along the time direction. At the same time it also allowsto determine the arrow of time: one direction corresponds to p0 > 0, theopposite direction to p0 < 0.

The traditional interpretation of p0 as the energy holds only in the casewhen the equations of motion relate p0 directly to the energy of the system,like in the Schrödinger equation p̂0 = Ĥ, Ĥ being the Hamiltonian. Similarrelation is present in the relativistic Klein-Gordon equation, p20 = m20 +~p2. This type of relations can also be found for other physical systems.In general, one can expect that in the spacetime representation, the equationof motion of a free particle relates its four-position to its four-momenta, withthe possibility that also other degrees of freedom, if present, can be involved.

Both the Schrödinger and the Klein-Gordon equations of motion allow toindirectly measure the temporal component p0 of the four-vector momentumoperator p̂. It is traditionally expected that p0 ≥ 0, even though this fea-ture does not follow from the mathematical structure of the model, as thep̂0 operator has the full spectrum R. The condition p0 ≥ 0 can be imposedeither by assuming that the equation of motion allows for real motion only ifp0 ≥ 0, or that this condition is a more fundamental property of our part ofthe Universe. A simple argument supporting the latter possibility is relatedto the initial state of our Universe. Assuming that the four-momentum is aconserved quantity, the initial chaotic motion of matter should have lead tothe situation in which the matter moved in the p0 > 0 and p0 < 0 directionswith the same probability. The spatial components lead to the expansionof matter in the R3 space, the temporal component of the four-momentum,however, lead to the separation of the Universe into two parts: one of whichis moving in the positive direction of time, while the other in the negativedirection of time. Both subspaces of states are orthogonal and cannot com-municate unless an interaction connecting both time directions occurs. Thisimplies that our part of the Universe corresponds to one of the directions ofthe time flow, say, p0 > 0. It does not mean, obviously, that in our part ofthe Universe we do not have the possibility to create particles with p0 < 0.According to common interpretation, such objects are antiparticles. Thisstrongly simplified picture requires further analysis but can provide a possi-ble explanation of the p0 > 0 phenomenon.

An interesting feature of the pair of the operators x̂ and p̂ is that, sincethey fulfill the canonical commutation relations

[p̂µ, x̂ν ] = iδνµ, (27)

14

they obey the Heisenberg uncertainty principle in the Robertson form [40],

Var(pµ) Var(xν) ≥ 1

4〈i[p̂µ, x̂ν ]〉2 ≥ δνµ, (28)

where Var(A) = 〈Â2〉−〈Â〉2 denotes the probabilistic variance of the observ-able Â for a given state ψ. It is interesting to revisit in the future differentforms of the uncertainty principles for time, temporal component of the linearmomentum, and other observables.

An interesting example is the mass operator. Assume that the massoperator for a free particle is given by

m̂2 = p̂µp̂µ. (29)

Then, the uncertainty relation between the invariant mass and the positionin spacetime is given by

Var(m2) Var(xν) ≥ 〈pν〉2. (30)

The width of such a mass is bounded by the ratio of the expectation valueof 〈pν〉 and the variance Var(xν).

In the case when p0 is related to the energy by means of the equations ofmotion for a given system, one obtains in a natural way the uncertainty rela-tion between the energy and time. For example, in the case of the Schrödingertype of motion, described by the equation of motion p̂0 = Ĥ, the Heisenbergrelation (28) can be rewritten as

Var(Ĥ) Var(x̂0) ≥ 14. (31)

This relation is fulfilled in the space of solutions of the Schrödinger equa-tion. Similar relations between time and energy can always be obtained fromappropriate equations of motion of the system under consideration.

4 Generators of the projection evolutionWithin the traditional approach, the evolution of a quantum state is drivenby a Hamiltonian dependent operator e−iĤt. In the projection evolutionmechanism the changes of the system are spontaneous and independent oftime. We assume that a subset of the evolution operators can be obtainedfrom the appropriate operators Ŵ−, the generators of the projection evolution.

For a given evolution step τ the projection evolution generator Ŵ−(τ) isdefined as a hermitian operator whose spectral decomposition gives the or-thogonal resolution of unity, which represents the evolution operators.

15

This generator can be subject to different constraints coming from physicsof the system under consideration. The idea of generators for creation ofthe projection evolution operators can probably be extended to other thanhermitian kinds of operators. Here, however, we restrict ourselves to thehermitian case only.

Let us consider a free single particle with spin equal to zero and no in-trinsic degrees of freedom. In this case the generator Ŵ− can be dependent onthe spacetime position x̂ and the four-momentum p̂ operators only.

Taking into account the translational symmetry, the dependence of Ŵ− onthe position operators disappears. Imposing the additional requirement ofthe rotational symmetry for this evolution generator results in the construc-tion of the operator Ŵ− as a function of the rotational invariants of the formaµp̂µ, a

µν p̂µp̂ν , . . . , where aµ, aµν , . . . are appropriate tensors with respect tothe SO(3) group. Basing on the experience of classical and quantum physicsone can expect that the expansion up to the second order in momenta shouldbe a good approximation, which leaves us with

Ŵ− C= aµp̂µ + aµν p̂µp̂ν , (32)

where C= means that Ŵ− is equal to the right-hand side of Eq. (32) only if theadditional condition C is fulfilled. This condition depends on the physicalproperties of the studied case. We will use that in Sec. 5 where the symmetriesare discussed.

The additional symmetries expected for a free particle are the space in-version and the time reversal. Assuming that aµ, aµν , . . . are invariant withrespect to both of these symmetries, the linear term in momenta reducesto a0p̂0. The quadratic term splits into two parts a00p̂20 + amnp̂mp̂n, wherem,n = 1, 2, 3. The spatial quadratic term has no preferred direction imply-ing, that it can be written in the form amn = Bδmn, which casts Ŵ− in theform

Ŵ− C= a0p̂0 + a00(p̂0)2 +B(p̂21 + p̂22 + p̂23). (33)To compare Eq. (33) with the standard quantum mechanics, one can rescale itsetting a0 = 1. Then, the first and the third term represent the Schrödingerequation for a free particle with mass m = 1

2B. The second term is pro-

portional to the second time derivative (p0)2 ∼ − ∂2

∂t2and is not a part of

the Schrödinger equation in the standard formulation. It is probably highlysuppressed by the a00 coefficient. By setting this coefficient to zero we canremove this term from the equation, recreating the standard Schrödingerevolution.

Imposing in the next step the Lorentz invariance of Ŵ−, one has to rejectthe first order term completely. Setting aµν = gµν = diag(+1,−1,−1,−1)

16

we are left withŴ−KG

C= p̂µp̂

µ, (34)

which leads to the Klein-Gordon equation p̂µp̂µ = m2 with potentially ad-ditional conditions C. Assuming that C stands for positive mass m > 0and positive temporal component of the momentum operator p0 > 0, thegenerator (34) describes the evolution of a free scalar particle. Changing theset of conditions C, one can generate the evolution of other scalar objects.If aµ, aµν , . . . are some tensor operators, one can reproduce other equationsof motion. For example, in the case of spin-1

2particles, assuming aµ = γµ,

where γµ are Dirac matrices, one gets the Dirac equation

Ŵ−DC= γµp̂µ. (35)

We conclude that the known equations, which describe specific quantumparticles, are some special forms of the evolution operator Ŵ−, which allowsalso to describe much more complicated cases.

4.1 The Schrödinger evolution as a special case of PEv

The generator of the Schrödinger evolution can be written as

Ŵ−S = i∂

∂t− Ĥ = p̂0 − Ĥ. (36)

Let us assume that the Hamiltonian Ĥ is independent of time. The eigen-values and the corresponding orthonormal eigenvectors of Ĥ will be denotedby �n and φnµ(~x), respectively, such that

Ĥφnµ(~x) = �nφnµ(~x). (37)

The action of Ŵ−S on the full wave function results in

Ŵ−S ηk0(x0)φnµ(~x) = w(k0, n) ηk0(x0)φnµ(~x), (38)

where

w(k0, n) = k0 − �n, (39)

ηk0(x0) =

1√2πe−ik0x

0

. (40)

The spectral decomposition of the generator Ŵ−S in the form of a Riemann-Stieltjes integral can be written as

Ŵ−S =∫Rw dE

Ŵ−(w), (41)

17

where dEŴ−(w) projects onto the eigenspace of Ŵ−S belonging to the eigenvalue

w. This subspace is spanned by the generalized eigenfunctions of the form

Φw(x0, ~x) =

1√2π

∑n

∑µ

cnµe−i(�n+w)x0φnµ(~x), (42)

with cnµ being c-number coefficients. The scalar product in the state space isgiven by (17). Note that in the traditional three-dimensional scalar productthe integration over time is absent,

〈Φ2|Φ1〉3 =∫R3d3xΦ2(x

0, ~x)∗Φ1(x0, ~x), (43)

because the state space K3 = L2(R3) does not contain time.Using the scalar product (17) we see that the eigenfunctions (42) are

normalized to the Dirac delta functions,

〈Φw′ |Φw〉 =∫R4dx0 dx1dx2dx3Φw′(x

0, ~x)∗Φw(x0, ~x) = δ(w′ − w). (44)

There are a few methods of obtaining vectors belonging to the state space K.For example, one can consider the extended Schrödinger equation which con-tains the temporal part describing the temporal dependencies of the kineticand potential terms. A possible, but not the most general, such extension isgiven by the generator

Ŵ−GS(τ) = p̂0 − Ĥ(τ) +[

1

2B−1T (τ)p̂

20 + VT (τ, x

0)

], (45)

where, in agreement with the PEv approach, the temporal parts of the kineticand potential terms were added. They represent the kinematics and thepossible localization of a physical object in the time axis direction. Theparameter B−1T (τ) represents a kind of temporal inertia of the physical object.

The eigenfunctions (42) considered within the traditional state space K3are general solutions of the Schrödinger equation, where the eigenvalue wdetermines the zero value of the energy represented by the Hamiltonian Ĥ.It follows from the fact that the eigenequation for Ŵ−S, from Eq. (36), can bewritten in the form

i∂

∂tφw = (Ĥ + w)φw, (46)

which means that the arbitrary eigenvalue w shifts the energy spectrum. Ofcourse, the physics in K3 is independent of the chosen value of w.

We conclude that an important difference between the PEv approach andthe traditional formulation of quantum mechanics lies in the interpretation of

18

the wave functions Ψ(x0, ~x). In the PEv formalism the function |Ψ(x0, ~x)|2,where Ψ ∈ K, represents the joined probability density of finding the particlein the four-dimensional spacetime point (x0, ~x). In the traditional form ofquantum mechanics with time being a parameter, the function |Ψ(x0, ~x)|2,where Ψ ∈ K3, represents the conditional probability density of finding theparticle in the three-dimensional space point ~x, assuming that the particle islocalized at time x0.

4.2 Relativistic equations of motion

To see that the PEv approach allows to describe the relativistic evolutionequations in a more natural way than the (1+3)-formalism, it is sufficientto consider the Klein-Gordon equation of motion for a free scalar particle.The generator of the appropriate evolution is given by (34). The mass op-erator m̂2 = p̂µp̂µ has the following continuous spectrum and generalizedeigenvectors:

p̂µp̂µηk(x) = wηk(x), (47)

where ηk(x) = exp(−ikµxµ)/(4π2), pµ = ~kµ, and w ∈ R. Comparing bothsides of Eq. (47) one gets the relation kµkµ = w. This relation determinesthe subspace which is invariant under the Poincaré group, corresponding tostates with definite w. This subspace consists of vectors of the form

Φw(x) =

∫R4d4k δ4(kµk

µ − w)c(k)ηk(x). (48)

Using the usual conditions that the space of states is restricted to the statesfor which m̂2 > 0 and p̂0 > 0, the eigenvalues w can be interpreted as theinvariant mass squared, m2. In this case one gets the known solution for thestandard scalar particle of non-zero mass,

Φw(x) =

∫R3

d3k

k0c(~k)η(k0,~k)(x), (49)

where k = (k0, ~k) = (√m2 + ~k2, ~k). Note that both vectors (48) and (49)

are normalized Dirac delta type distributions.To get the quantum states belonging to the state space K, one may extend

the traditional Klein-Gordon equation by including an appropriate vectorfield Aµ. Using the minimal coupling scheme one gets

Ŵ− = (p̂µ − Aµ)(p̂µ − Aµ). (50)

This vector field plays a role similar to the temporal part of the potential inthe extended Schrödinger PEv generator (45).

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All other relativistic equations of motion can be reproduced in a similarway. One needs, however, to remember that physical consequences of thePEv approach are tremendous. First of all, time becomes a quantum observ-able and it has to be treated on the same footing as the remaining positioncoordinates. This makes a lot of new physical phenomena possible, whichhave to be analyzed.

5 SymmetriesAs it is well known, different kinds of symmetries play a fundamental role inphysics. They are the most important constraints for structure, interactionsand motions of physical objects.

In the case of the PEv formalism one thinks about two distinct types ofsymmetries:

(A) the symmetries for a fixed step of the evolution, i.e., for a constantevolution parameter τ ;

(B) the symmetries related to the transition of the system from one step ofthe evolution to another, i.e., for the case when the evolution parameterchanges, τn−1 → τn.

The first type of symmetries (A) describes structural, spacetime and in-trinsic properties of a quantum system. An important difference is thatthe time is now the quantum observable. Taking this into account, symme-try analysis seems to be similar to those performed in relativistic quantummechanics. Many results remain valid, but most of them require reinterpre-tation. This problem is still open.

The second type of symmetries (B) is different because the evolution op-erators are involved in the symmetry analysis. The operators F|(τ ; ν, ρ) canhave different structures, they can be unitary operators, projection operatorsand other type of operators which allow to transform quantum states intoother quantum states. This opens many mathematical and physical (inter-pretation) problems. In this paper we only introduce the most fundamentalproperties related to symmetries of the type (B). An extended analysis willbe given in a subsequent paper.

In the following we consider only the evolution operators E|(τ ; ν) whichare either a combination of unitary operators or they form an orthogonalresolution of unity.

The problem is to find those properties of a physical object which remaininvariant at subsequent steps of the evolution. In other words, we are looking

20

for these transformations from one evolution step to another which do notchange this object. One of the most interesting problem for studying suchrelations are the relations between the symmetries and the conservation laws.

We start by writing the definition of transformations of the evolutionoperator F|(τ ; ν, ρ). Because F|(τ ; ν, ρ) transforms the quantum state ρ intoanother quantum state (not normalized), the resulting image of F|(τ ; ν, ρ) hasto be consistent with the transformation of its arguments.

To explain this, let us consider the problem of the rotation of some vectorfunction. Let f : R3 → R3. The values of the rotated function f ′ having therotated argument x′ should be equal to the rotation of the value of the originalfunction having the original argument, f ′(x′) = R̂f(x). In an analogical waythe transformation of F|(τ ; ν, ρ) is defined.

Let F|(τn; ν, ρ) be the evolution operator defined in (3). Let G be a groupwith two realizations, S1(g) : T +1 (K)→ T +1 (K) and S(g) : T +(K)→ T +(K).The transformation of the evolution operator F| is defined as:

F|′(τn; ν, ρ′) = S(g) F|(τn; ν, ρ) S(g−1), (51)

i.e., the resultant evolution operator for the transformed state is equal to theappropriately chosen transformation of the evolution operator for the originalstate. This idea can be expressed in a more convenient form:

F|′(τn; ν, ρ) = S(g) F|(τn; ν, S1(g−1)ρS1(g)) S(g−1). (52)

The group G provides the same physical interpretation for the operators S(g)and S1(g) in the above definition. Definition (51) allows to conserve the prob-ability distributions for unitary equivalent images of quantum mechanics.

Let us assume that the transition probability from the evolution step τn−1to τn is given by

pev (ρ(τn−1; νn−1)→ ρ(τn; νn)) = Tr[F|(τn; νn, ρ(τn−1; νn−1))], (53)

as it is for the projection evolution operators represented by an orthogonalresolution of the unit operator. Here, νn denotes the set of quantum numberstaken from Qn ≡ {νn,1, νn,2, . . . , νn,k, . . . }. Similarly, the transition probabil-ity of the state ρ′(τn−1; νn−1) transformed by the group G and, at the sametime, the transformed evolution operator F|′, is given by

pev (ρ′(τn−1; νn−1)→ ρ′(τn; νn)) = Tr[F|′(τn; νn, ρ′(τn−1; νn−1))]. (54)

It follows from equation (51) that both probabilities are equal:

pev (ρ(τn−1; νn−1)→ ρ(τn; νn)) = pev (ρ′(τn−1; νn−1)→ ρ′(τn; νn)) . (55)

21

The consequence of such symmetry for every step of the evolution is the factthat g does not change the structure of the possible evolution paths, whichmeans that the probability distribution of the potential evolution paths isconserved. In this case, the operations g ∈ G can also be interpreted asoperations which transform among equivalent descriptions of a given model(covariance), or as the transformation between equivalent observers (a kindof “relativity”).

In the following, we consider only the most common case when S(g) =S1(g) and the evolution operators are E|(τn; νn). For shortness we use thenotation S(g) = g. In this special case the evolution operators E|(τn; νn)transform as: [

E|(τn; νn)ρ(τn−1; νn−1)E|(τn; νn)†]′ (56)

= gE|(τn; νn)g−1 ρ(τn−1; νn−1) gE|(τn; νn)†g−1.

The above equation shows that the transformation (52) for the operatorE|(τn, νn) is equivalent to the transformation by the group element g,

[E|(τn; νn)]′ = gE|(τn; νn)g−1. (57)

Let us consider the invariance of the transition probabilites when the finalstates are generated by the group G. In this case, the symmetry group G isresponsible for creating the sets of states which, at the end, are chosen bythe evolution in a completely random way. This specific conservation of thetransition probability can be defined as:

pev (ρ(τn−1; νn−1)→ ρ(τn; νn)) = pev (ρ(τn−1; νn−1)→ ρ′(τn; νn)) . (58)

The transition probabilities among the transformed states are given by

pev (ρ(τn−1; νn−1)→ ρ′(τn; νn)) (59)= Tr

[gE|(τn; νn)g−1 ρ(τn−1; νn−1) gE|(τn; νn)†g−1

].

It is easy to see that in two special cases, when

[g,E|(τn, νn)] = 0 or [g, ρ(τn−1, νn−1)] = 0, (60)

this probability is the same as for the original states.The next important problem is the relation between symmetries and con-

servation laws. The generally correct formulation of this problem in theprojection evolution model is at present not known. Intuitively one can saythat we are looking for conditions under which the expectation value of agiven observable A is conserved within the PEv approach:

〈A〉ρ(τ1;ν1) = 〈A〉ρ(τ2;ν2) = · · · = 〈A〉ρ(τn;νn) = . . . (61)

22

The required conditions may involve special relations between the evolutionoperators, density operators and quantum observables.

In the case when the evolution is described by the operators E|(τn; νn), theconservation of the expectation value 〈A〉 has, for a series of n, the form

Tr[A ρ(τn−1; νn−1)] =Tr[AE|(τn; νn) ρ(τn−1; νn−1) E|(τn; νn)†]Tr[E|(τn; νn) ρ(τn−1; νn−1) E|(τn; νn)†]

. (62)

Assume that the operator E|(τn; νn) is unitary. Under this assumption thecondition (62) takes the following form:

Tr [Aρ(τn−1; νn−1)] =Tr[AE|(τn; νn) ρ(τn−1; νn−1) E|(τn; νn)†]Tr[E|(τn; νn) ρ(τn−1; νn−1) E|(τn; νn)†]

= Tr[E|(τn; νn)† A E|(τn; νn) ρ(τn−1; νn−1)]. (63)

In this case the expectation value of the observable A is conserved if theoperator A commutes with the evolution operator, i.e., [A,E|(τn; νn)] = 0.This fact has its counterpart in the standard quantum mechanics – if theHamiltonian commutes with the operator A, the expectation value 〈A〉 isconserved during the unitary evolution generated by this Hamiltonian.

The PEv approach allows for the generalization of the idea of the unitaryevolution. For example, it is possible to consider the case when a few differentunitary evolution channels interfere. In this case, the state for the evolutionstep n is a linear combination of the products of different unitary evolutionsof the previous state,

ρ(τn; νn) =

∑Nm=1 Um(τn)ρ(τn−1; νn−1)Um(τn)

†

Tr[∑N

m=1 Um(τn)ρ(τn−1; νn−1)Um(τn)†], (64)

where U †m = U−1m for m = 1, . . . , N .The second example, which leads to a unitary type of evolution, is when

the evolution generator Ŵ−(τn) evolves unitarily, Ŵ−(τn) = U(τn)Ŵ−0U(τn)†.Both examples open a new problem in the analysis of symmetries in the PEvmodel and they require further investigation.

Another problem is the generalization of the situation when both, theevolution operator and the quantum observable, are invariant under a givensymmetry.

Let G be a symmetry Lie group of the evolution generator Ŵ−(τn), i.e.,ĝŴ−(τn)ĝ† = Ŵ−(τn) for every g ∈ G, where the operators ĝ play the roleof a unitary operator representation of this symmetry group in the statespace K. Let us assume that the eigenvectors of Ŵ−(τn) fulfill the followingequations:

Ŵ−(τn)|κnσnΓna〉 = wΓnκn|κσnΓa〉, (65)

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ĝ|κnσnΓna〉 =∑a′

∆Γna′a(g)|κnσnΓna′〉, (66)

where ∆Γn denotes the irreducible representation of the symmetry groupG labelled by Γn, the quantum number a labels vectors within the givenirreducible rpresentation ∆Γn , every pair of quantum numbers (κn, σn) dis-tinguishes among equivalent irreducible representations for fixed Γn and, inaddition, the quantum numbers σn describe possible, not symmetry related,additional degeneracy of the spectrum of the generator Ŵ−(τn). The vectors|κnσnΓna〉 form an ortonormal basis in the state space K.

The elementary, invariant with respect to the symmetry group G, projec-tors can be written as

P Γnκnσn =∑a

|κnσnΓna〉〈κnσnΓna|. (67)

In this case, the definition of the projection evolution generators implies theappropriate evolution operators in the form

E|(τn; Γnκn) =∑σn

P Γnκnσn . (68)

Using the above conditions, the Casimir operator C of the symmetry groupG, which is an observable invariant with respect to this symmetry group,satisfies

CE|(τn; Γnκn) = cΓnE|(τn; Γnκn), (69)where cΓn are eigenvalues of the Casimir operator obtained from

C|κnσnΓna〉 = cΓn|κnσnΓna〉. (70)

Let us assume that the initial state of the evolution is ρ0. For the first stepof the evolution, after ”drawing lots” by Nature, one gets

Tr[Cρ(τ1; Γ1, κ1] =Tr [CE|(τ1; Γ1κ1)ρ0E|(τ1; Γ1κ1)]Tr [E|(τ1; Γ1κ1)ρ0E|(τ1; Γ1κ1)]

= CΓ1 . (71)

However, after the first step of the evolution, the value of the Casimir oper-ator for all subsequent steps is fixed,

Tr[Cρ(τn; Γn, κn)] =

{CΓ1 , for Γn = Γ10, for Γn 6= Γ1

. (72)

We conclude that if the evolution operators are invariant with respect to thegroup G and fulfill the above conditions, the value of the Casimir operator Cof the group G is conserved during the evolution.

24

This special case has its analogy in the standard quantum mechanics.Let us assume that the Hamiltonian H is invariant with respect to a groupG. The eigenvectors of H belong to the invariant subspaces spanned by thebases of the irreducible representations of the group G. In this case theexpectation value of the Casimir operator is conserved during the unitaryevolution generated by this Hamiltonian.

We have presented a short outline of some open problems related to thesymmetry analysis within the projection evolution approach. PEv opensnew areas for applications of symmetries and group theoretical methods inphysics.

6 Interference in time

6.1 Interference on time slits

The Young’s double slit experiment is very well known in both the classicaland quantum versions. Having two wave functions in the spacetime repre-sentation, Ψ1(x) and Ψ2(x), and two slits in the spacetime, one can expectthat under proper conditions one gets an interference of these states. Letus assume that Ψ1(x) is an amplitude, in the spacetime representation, ofpassing the first slit by a particle and that Ψ2(x) is an amplitude of passingthe second slit by the same particle. The resulting probability of passing anyof the slits is given by

|Ψ1(x) + Ψ2(x)|2 = |Ψ1(x)|2 + |Ψ2(x)|2 + 2Re [Ψ1(x)∗Ψ2(x)] . (73)

Treating all the components of the position operator x̂µ on the same footingone can construct devices which open two slits not in the 3D space but intime. The temporal interference, according to Eq. (73) is expected also inthis case and the experiments mentioned in the Introduction seem to confirmthis idea.

The temporal interference is a natural component of the PEv formalism.In the following we describe a relativistic version of the temporal double slitexperiment.

Assume that a relativistic spinless Klein-Gordon particle is emitted froma source and propagates towards the detector. On its way it meets a wallin which a slit may open for a limited time. Let the slit open two times ina fixed spatial location. When the slit is closed, the path to the detector isblocked. The particle, if not observed, will be, after successfully passing theslit, in a superposition of states corresponding to the two time intervals of

25

the opened slit. As a result, the energy spectrum measured by the detectorwill have the form of an interference pattern, as was reported in Ref. [31].

The emission process does not happen in zero time, so the mass m0 ofthe particle will be distributed around some mean value m̄0,

m0 ∈ ∆m̄0 =[m̄0 −

Γ

2, m̄0 +

Γ

2

], (74)

where Γ/2 < m̄0. It follows from the Klein-Gordon equation kµkµ = m20 thatto fulfill (74) the particle’s four-momentum must belong to the set Bm̄0 forwhich

(m̄0 − Γ2

)2 ≤ k2 ≤ (m̄0 + Γ2 )2. We write the initial state of the particlein the form

|ψ0〉 =∫Bm̄0

d4k a(k)|k〉, (75)

where 〈x|k〉 = exp(−ikµxµ)/(4π2) and a(k) denotes the momentum distribu-tion function. We omit for simplicity the normalization factor and introducethe overall normalization in the final formula (93).

The slit is open at certain spatial location during two time periods. Wedenote the spacetime regions of the opened slit by ∆1 and ∆2. The evolutionoperator for the first evolution step τ1 is parametrized by the quantum num-ber ν1 having two values: (i) ν1 = 1 representing the passing of the particlethrough the slits and (ii) ν1 = 0 for the complementary case in which theparticle did not pass the slits. For the first case the evolution operator takesthe form of a projection of the state onto the region ∆T = ∆1 ∪∆2, and forthe second case the corresponding operator is the projection operator ontothe complementary space:

E|S(τ1; ν1 = 1) =∫

∆T

d4x |x〉〈x|, (76)

E|S(τ1; ν1 = 0) = I− E|S(τ1; ν1 = 1). (77)

The operator E|S(τ1; ν1 = 0) can be decomposed into more detailed evolutionoperators which will describe the particle which did not manage to pass theslit, but in what follows we are interested in the form (76) only. The actionof (76) on |ψ0〉 is given by

E|S(τ1)|ψ0〉 =∫

∆T

d4x |x〉∫Bm̄0

d4k a(k)〈x|k〉. (78)

In the next step the particle propagates freely from the slits to the detec-tor. During this step the Klein-Gordon equation is fulfilled, so the evolution

26

operator projects onto the momentum space Bm̄0 ,

E|F (τ2; ν2 = 1) =∫Bm̄0

d4k′ |k′〉〈k′|. (79)

In what follows we neglect the off-shell case, which is described by the pro-jection operator E|F (τ2; ν2 = 0) = I − E|F (τ2; ν2 = 1), as it should be onlya small correction to the main effect. The unnormalized state of the particleat this step reads

E|F (τ2; ν2 = 1)E|S(τ1; ν1 = 1)|ψ0〉 (80)

=

∫Bm̄0

d4k′ |k′〉∫Bm̄0

d4k a(k)

∫∆T

d4x 〈k′|x〉〈x|k〉.

The detector measures the four-momentum κ of the particle, so the lastevolution operator will be the following set of projections:

E|D(τ3; ν3 = κ) = |κ〉〈κ|, (81)

which results in the final state:

E|D(τ3; ν3 = κ)E|F (τ2; ν2 = 1)E|S(τ1; ν1 = 1)|ψ0〉

= |κ〉∫Bm̄0

d4k′ 〈κ|k′〉∫Bm̄0

d4k a(k)

∫∆T

d4x 〈k′|x〉〈x|k〉

= |κ〉∫Bm̄0

d4k′ δ4(κ− k′)∫Bm̄0

d4k a(k)

∫∆T

d4x 〈κ|x〉〈x|k〉, (82)

where the Dirac delta appears due to the orthogonality of the momenta. Itis worth mentioning that Eq. (82) represents only one, chosen by us, possiblepath of evolution.

To evaluate the expression (82) we notice, that the integration over k′ isequal to zero if κ 6∈ Bm̄0 and is equal to one if κ ∈ Bm̄0 . We account for thisfact introducing the function idBm̄0 (κ) defined as

idBm̄0 (κ) =∫Bm̄0

d4k′ δ4(κ− k′) =

{1 if κ ∈ Bm̄00 if κ 6∈ Bm̄0

. (83)

Let the spacetime coordinates of the opened slit be in the form

∆i =

(ti −

δT2, ti +

δT2

)(84)

×(x1s −

δ12, x1s +

δ12

)×(x2s −

δ22, x2s +

δ22

)×(x3s −

δ32, x3s +

δ32

).

27

Since the scalar product 〈x|k〉 = exp(−ikµxµ)/(4π2), the integration over xin (82) takes the form∫

∆T

d4x 〈κ|x〉〈x|k〉 =(

1√2π

)8 ∫∆T

d4x e−i(kµ−κµ)xµ

=

(1

2π

)4(∫∆1


+

∫∆2


). (85)

The integrals in (85) can be evaluated using (84),∫∆i


= δT δ1δ2δ3e−i(k0−κ0)tie−i(k1−κ1)x

1se−i(k2−κ2)x

2se−i(k3−κ3)x

3s

×j0(k0 − κ0

2δT

)j0

(k1 − κ1

2δ1

)j0

(k2 − κ2

2δ2

)j0

(k3 − κ3

2δ3

), (86)

where j0(z) = sin(z)/z is the spherical Bessel function of the first kind.The probability density Prob(κ) of detecting a particle with the four-

momentum κ is given by the modulus squared of the expression (82). Usingequations (83), (85) and (86) we obtain

Prob(κ) = (δT δ1δ2δ3)2(

1

2π

)8idBm̄0 (κ)

×

∣∣∣∣∣∫Bm̄0

d4k a(k)(e−i(k0−κ0)t1 + ei(k0−κ0)t2

)ei(

~k−~κ)~xs (87)

× j0(k0 − κ0

2δT

)j0

(k1 − κ1

2δ1

)j0

(k2 − κ2

2δ2

)j0

(k3 − κ3

2δ3

)∣∣∣∣2 .The interference term e−i(k0−κ0)t1 + ei(k0−κ0)t2 can be rewritten after the vari-able change,

t1 = Ts −�T2, t2 = Ts +

�T2, (88)

which leads to

Prob(κ) = 4(δT δ1δ2δ3)2(

1

2π

)8idBm̄0 (κ)

×

∣∣∣∣∣∫Bm̄0

d4k a(k)e−i(k0−κ0)Tsei(~k−~κ)~xs cos

(k0 − κ0

2�T

)j0

(k0 − κ0

2δT

)× j0

(k1 − κ1

2δ1

)j0

(k2 − κ2

2δ2

)j0

(k3 − κ3

2δ3

)∣∣∣∣2 . (89)28

We evaluate the exact expression (89) assuming that the spatial momen-tum of the particle is directed along the z axis, ~k = (0, 0,−kz). The profilea(k) takes in this case the form

a(k) = ã(m2)δ(k1)δ(k2)δ(k3 − kz). (90)

By changing the integration variables, the modulus squared in (89) reducesto [

j0

(κ12δ1

)j0

(κ22δ2

)j0

(kz + κ3

2δ3

)]2 ∣∣∣∣∣∫ (m̄0+ Γ2 )2

(m̄0−Γ2 )2d(m2)

ã(m2)

2√m2 + k2z

× e−i√m2+k2zTs cos

(√m2 + k2z − κ0

2�T

)j0

(√m2 + k2z − κ0

2δT

)∣∣∣∣∣2

.(91)

This expression can be simplified further if we assume that the mass spreadis small, i.e., Γ ≈ 0. In this case the integration can be approximated usingthe mean value theorem. As a result we obtain

Prob(κ) ≈ N[j0

(κ12δ1

)j0

(κ22δ2

)j0

(kz + κ3

2δ3

)]2(92)

× idBm̄0 (κ)

[cos

(√m̄20 + k

2z − κ0

2�T

)j0

(√m̄20 + k

2z − κ0

2δT

)]2,

where we have introduced N as the overall normalization factor. In our con-struction this factor does not normalize (93) to 1 because not all particles areassumed to pass the slit, see Eqs. (76) and (77). We may however normalize(93) for all particles measured by the detector, in which case we get∫

R4d4κ Prob(κ) = 1. (93)

6.2 A numerical example

As an example, let us discuss the π+ particle in a setup with no sources of theelectromagnetic interactions. During the numerical calculations the naturalunits c = ~ = 1 will be used.

The particle is produced with the initial three-momentum ~k = (0, 0,−kz).On its way to the detector it has to pass a slit which opens twice in thesame spatial location. The slit has spatial widths δ1 and δ2 in the planeperpendicular to the direction of motion. The width δ3 is less important,because the highest probability for the detector to register the pion is for the

29

incoming momentum κ3 = −kz, for which the term j0((kz + κ3)δ3/2) = 1drops out. We denote the time width of the opened slit by δT , and the timebetween the two openings by �T .

The mass of π+ ismπ ≈ 139 MeV. Its half-life is tπ = 3.95·107 eV−1, whichimplies the mass spread of the order of Γ ∼ 1/tπ ≈ 2.5 ·10−8 eV. Because mπand Γ differ by sixteen orders of magnitude, the pion is almost exactly on itsmass shell; as a consequence, Bm̄0 is just the Klein-Gordon condition, andthe function idBm̄0 (κ) = 1 becomes trivial. If the initial Klein-Gordon stateof the particle is given by k20 = m2π + k2z , the state seen by the detector willbe κ20 = m2π + κ21 + κ22 + k2z . Taking all this into account, the unnormalizedprobability, as a function of κ1 and κ2, is given by the expression:

Prob(κ1, κ2) ≈ j20(κ1

2δ1

)j20

(κ22δ2

)× cos2

(√m2π − k2z −

√m2π + κ

21 + κ

22 − k2z

2�T

)

×j20

(√m2π − k2z −

√m2π + κ

21 + κ

22 − k2z

2δT

). (94)

The detection probability (94) is plotted on Fig. 2. The slits are 0.01 mmwide in the x and y directions. The time parameter �T takes values from10−7 s to 10−12 s whereas δT is set to δT = �T/3. The momentum kz isa constant number and does not play any significant role. For long openingtimes, the detection is possible for very small perpendicular momenta κ1 andκ2 only. For smaller �T the inner region widens and starting from �T = 10−10 shigher order maxima start to appear. They are clearly visible for �T = 10−10 sand shorter times.

The interference effect present on Fig. 2 comes from two sources – thespatial diffraction on the slits and the temporal interference. On Fig. 3 wehave drawn density plots of the temporal part of Eq. (94). For shorter open-ing times the axes have been rescaled to show, that the maxima appear forhigher values of the momenta. Even though we have not used the Heisenberg-like condition for the energy and time, high momenta, and thus high energy,are needed in the case of short times. It follows from Eq. (94) that the tem-poral part is dependent on κ21 + κ22, therefore circles appear on the plots onFig. 3. The comparison of these diagrams with those on Fig. 2 reveals, thatthe spatial part forces the maxima to appear along the κ1 = 0 and κ2 = 0directions, dominating the temporal effects.

One may make the temporal effect visible by manipulating the spatialwidths of the slit. Looking at Eq. (94) one sees, that for small δ1 and δ2 the

30

-300 -150 0 150 300κ1 [eV]-300

-150 0

150 300

κ2 [eV]

0.2 0.4 0.6 0.8

1

Prob(κ1,κ2), εT=10-7s

-300 -150 0 150 300κ1 [eV]-300

-150 0

150 300

κ2 [eV]

0.2 0.4 0.6 0.8

1


-300 -150 0 150 300κ1 [eV]-300

-150 0

150 300

κ2 [eV]

0.2 0.4 0.6 0.8

1


-300 -150 0 150 300κ1 [eV]-300

-150 0

150 300

κ2 [eV]

0.2 0.4 0.6 0.8

1


-300 -150 0 150 300κ1 [eV]-300

-150 0

150 300

κ2 [eV]

0.2 0.4 0.6 0.8

1


-300 -150 0 150 300κ1 [eV]-300

-150 0

150 300

κ2 [eV]

0.2 0.4 0.6 0.8

1


Figure 2: The detection probability as a function of κ1 and κ2 for differentvalues of the opening times.

31

-300 -200 -100 0 100 200 300

κ1 [eV], εT=10-7s

-300

-200

-100

0

100

200

300

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-300 -200 -100 0 100 200 300

κ1 [eV], εT=10-8s

-300

-200

-100

0

100

200

300

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-300 -200 -100 0 100 200 300

κ1 [eV], εT=10-9s

-300

-200

-100

0

100

200

300

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-600 -400 -200 0 200 400 600

κ1 [eV], εT=10-10s

-600

-400

-200

0

200

400

600

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-3000 -2000 -1000 0 1000 2000 3000

κ1 [eV], εT=10-11s

-3000

-2000

-1000

0

1000

2000

3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-3000 -2000 -1000 0 1000 2000 3000

κ1 [eV], εT=10-12s

-3000

-2000

-1000

0

1000

2000

3000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 3: The temporal part of Prob(κ1, κ2) for different values of the openingtimes �T .

32

-30 -15 0 15 30κ1 [keV]-15

015

30

κ2 [keV]

0.2

0.6

1

Prob(κ1,κ2), d=10-8mm

-30 -15 0 15 30κ1 [keV]-15

015

30

κ2 [keV]

0.2

0.6

1

Prob(κ1,κ2), d=3*10-8mm

-30 -15 0 15 30κ1 [keV]-15

015

30

κ2 [keV]

0.2

0.6

1

Prob(κ1,κ2), d=5*10-8mm

-30 -15 0 15 30κ1 [keV]-15

015

30

κ2 [keV]

0.2

0.6

1


-3.0 -1.5 0 1.5 3.0κ1 [keV]-1.5

01.5

3.0

κ2 [keV]

0.2

0.6

1

Prob(κ1,κ2), d=5*10-7mm

-3.0 -1.5 0 1.5 3.0κ1 [keV]-1.5

01.5

3.0

κ2 [keV]

0.2

0.6

1


-1.0 -0.5 0 0.5 1.0κ1 [keV]-0.5

00.5

1.0

κ2 [keV]

0.2

0.6

1

Prob(κ1,κ2), d=5*10-6mm

-1.0 -0.5 0 0.5 1.0κ1 [keV]-0.5

00.5

1.0

κ2 [keV]

0.2

0.6

1

Prob(κ1,κ2), d=7*10-6mm

-1.0 -0.5 0 0.5 1.0κ1 [keV]-0.5

00.5

1.0

κ2 [keV]

0.2

0.6

1


Figure 4: The detection probability as a function of κ1 and κ2 for differentspatial widths of the slit. Here d = δ1 = δ2. The time between the openingsis set to �T = 10−14 s and the slit is open for δT = �T/3.

33

Bessel functions are close to one and do not suppress the temporal part. Weshow this on Fig. 4 where the temporal parameters are set to �T = 10−14 sand δT = �T/3. The spatial size of the slit is δ1 = δ2 = d. For d = 10−8 mmthe circular maxima are clearly visible. For larger slits, d = 10−7 mm, thischanges into the four-fold shape dictated by the Bessel functions.

7 Concluding remarksThe discussion about the structure and the role of time is as long as thehistory of physics. A collection of papers devoted to different aspects ofthe physical time from the modern perspective can be found, among others,in [33, 41]. In Ref. [33] the paper by P. Busch mentiones three types oftime. The most popular one is the time considered as a parameter whichis measured by an external laboratory clock, uncoupled from the measuredsystem. This time is called the external time. Time can be defined alsothrough the dynamics of the observed quantum systems, in which case wedeal with the dynamical (or intrinsic) time. Lastly, time can be consideredon the same footing as other quantum observables, especially as positionsin space. This is called by P. Busch the observable (or event) time and itrepresents the approach discussed in the present paper.

In the experimental practice the external time is usually used. It is intro-duced by constructing different kinds of clocks uncoupled from the analyzedphysical phenomenon. In this case, the clocks are defined by some processeswhich can be parameterized by the parameter θ (usually the label t is usedinstead of θ). One can expect that for the external clocks, θ is approximatelya monotonic function of the evolution parameter τn, e.g., θ = Tr(t̂ρ(τn; ν))and Tr(t̂ρ(τn+1; ν ′)) > Tr(t̂ρ(τn; ν)). The trace Tr(t̂ρ(τn; ν)) denotes, in anal-ogy to the average position of an object in space, the average time on theclock being in the state ρ(τn; ν) at the step evolution τn. Having one clock,one can treat it as the standard clock. All other clocks can be constructedand synchronized to this standard clock. In this context the external time,though very useful, is a conventional rather than physical entity.

The intrinsic time, or times, to be more precise, is determined by anyarbitrary set of dynamical variables. It is compatible with our “changesprinciple”, i.e., that changes of states or observables are more fundamentalthan the time itself. However, because in our approach the physical timeis a quantum observable, the required characteristic times (intrinsic times)for a given physical process can be directly calculated. In this context, theintrinsic times are not fundamental but derivable temporal observables.

The observable time can, in a natural way, account for many quantum

34

mechanical effects regarded as paradoxes. It is also important that it allowsto calculate temporal characteristics of a quantum system on the same basisas it can be done for other observables. It introduces through the equationsof motion the time-energy uncertainty relation on the same basis as for theposition–momentum observables. The time operator and the correspondingconjugate temporal momentum operator are the very natural complements ofthe covariant relativistic four-position and four-momentum operators. A fewexamples of processes analyzed in terms of the observable time can be foundin [42, 43, 44, 45, 46, 47, 48, 49, 50]. The cited papers show some steps inthe historical development of the PEv idea.

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38

1 Introduction1.1 The Pauli theorem1.2 Theoretical models and experimental work

2 Projection evolution of quantum systems2.1 The changes principle2.2 Projection evolution operators

3 The time operator4 Generators of the projection evolution4.1 The Schrödinger evolution as a special case of PEv4.2 Relativistic equations of motion

5 Symmetries6 Interference in time6.1 Interference on time slits6.2 A numerical example

7 Concluding remarks

projection evolution of quantum states. part i.projection evolution of quantum states. part i. ......

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