Spontaneous collapse: a solution to the measurement problem and a source of the decay in mesonic systems

Transcription

1 Spontaneous collapse: a solution to the measurement problem and a source of the decay in mesonic systems Kyrylo Simonov, and Beatrix C. iesmayr, Faculty of Physics, University of Vienna, Boltzmanngasse 5, 9 Vienna, Austria and the CS model Continuous Spontaneous ocalization 5, 6 in its mass-proportional version 7. Both models introduce the spontaneous collapse by a modification of the standard Schrödinger equation by adding specific nonlinear and stochastic terms and introducing two new taking collapse models seriously natural constants: a collapse rate and a coherence length of the localization. Flavor physics is a rich field within physics with many unique features, and new facilities in the near future will tackle very precisely this regime of energy. This contribution investigates how these two collapse models change the flavor changing dynamics of neutral mesons. In particular, we find that the specific link between the position space and the flavor space provides constraints onto the collapse models in general. Moreover, we find also that the noise field underlying any collapse models can be a source of the two different decay constants of mesons. In detail, the non-linear interaction of the quantum system with the noise field implies the dependence on the absolute masses of the lifetime eigenstates, the diagonal states of the amiltonian, such that an effectively non-unitary evolution occurs. The mathematical background are the correlation functions of the stochastic processes in the expansion. The physical background is that a time symmetric approach between the two dual spaces bra/ket is not necessarily required. The QMUP model has been investigated for the spontaneous radiation emission from a non-relativistic free charged particle 8, 9 and put to an intensive experimental test by X-rays. For the mass-proportional CS model experiments with optomechanical cavities have been proposed to detect possible changes in the spectrum of light which drives a mechanical oscillator 3, 4. In another approach a possible increase of equilibrium temperature of a mechanical oscillator proarxiv:66.68v quant-ph 8 Oct 6 Dynamical reduction models propose a solution to the measurement problem in quantum mechanics: the collapse of the wave function becomes a physical process. We compute the predictions to decaying and flavor oscillating neutral mesons for the two most promising collapse models, the QMUP Quantum Mechanics with Universal Position ocalization model and the mass-proportional CS Continuous Spontaneous ocalization model. Our results are showing i a strong sensitivity to the very assumptions of the noise field underlying those two collapse models and ii under particular assumptions the CS case allows even to recover the decay dynamics. This in turn allows to predict the effective collapse rates solely based on the measured values for the oscillation mass differences and the measured values of the decay constants. The four types of neutral mesons Kmeson, D-meson, B d -meson, B s-meson lead surprisingly to ranges comparable to those put forward by Adler 7 and Ghirardi-Rimini-Weber 986. Our results show that these systems at high energies are very sensitive to possible modifications of the standard quantum theory making them a very powerful laboratory to rule out certain collapse scenarios and studying the detailed physical processes solving the measurement problem. PACS numbers: w, 3.65.Tu, 5.4.-a I. INTRODUCTION Quantum mechanics has proven to be an exceedingly successful theory. It covers a plethora of physical phenomena at different energy scales and, up to date, no experiments are in contradiction. owever, quantum theory is very counter-intuitive and meets conceptual problems. Considering quantum mechanics as a fundamental theory also superpositions of macroscopic objects, such as cats, should exist which are obviously not observed in our daily world. In the Copenhagen interpretation during a measurement process a breaking of the superposition is mathematically postulated, but no detailed physical process has been assigned to it. Moreover, separation into macroscopic system measurement apparatus and microscopic system quantum system is utilized but lacks a clear definition. Ruling out unobserved macroscopic superpositions is the heart of the so-called measurement problem or macro-objectification problem,. Dynamical reduction models, so-called collapse models, introduce an ontologically objective mechanism of the wavefunction collapse. Since collapse models provide definite predictions for the regime between microscopic and macroscopic they are experimentally testable. In 985 the first dynamical reduction model, the GRW model, proposed by Ghirardi, Rimini and Weber 3, appeared on the market. Currently, two sophisticated versions are intensively investigated both from the fundamental point of view as well as from the experimental point of view. These are the QMUP model Quantum Mechanics with Universal Position ocalization 4 Electronic address: Kyrylo.Simonov@univie.ac.at Electronic address: Beatrix.iesmayr@univie.ac.at

2 duced by the spontaneous collapse was revealed 5. For neutral mesons K-, B-, D-meson and neutrinos up to first order in time the effect of the mass-proportional CS model was derived and compared to decoherence models 6, 7 by comparing to experimental data 8 3. Recently, upper bounds on collapse models have been derived for cold-atom experiments 4 and the authors of Ref. 5 have shown that reduction models can lead to nontrivial contribution to an effective cosmological constant. The paper is organized as follows. We start by an introduction into collapse models and the flavor phenomenology and set the stage in Section II. Next we show how for both collapse models the transition probabilities are computed up to second order in time in Section III. These computations are lengthy and involved, therefore, the details are given in Appendix A for the QMUP model and in Appendix B for the CS model, respectively. The correlation functions and their dependence on the physics of the noise field are derived in Appendix C. In Section IV we present the results, the probabilities for the lifetime states and the flavor oscillating probabilities. These are the quantities that are well investigated experimentally. We analyze then different possibilities, one allowing us an independent prediction of the effective collapse rate for the different types of neutral mesons. We proceed by giving a physical meaning to the dependence on the correlation functions of the Wiener process and finalize by developing a decoherence model that leads to the same probabilities as the CS model, however, relies on strictly different physics. ast but not least we provide a summary and an outlook in Section V. II. COAPSE DYNAMICS AND NEUTRA MESON PENOMENOOGY In the QMUP and CS models the collapse is a continuous process which can be described by the following non-linear stochastic modification of a Schrödinger equation for a given amiltonian Ĥ 6 d φ t iĥ dt + λ λ N Âi Âi t dw i,t i N Âi Âi t dt φ t, i with and Âi t : φ t Âi φ t being the standard quantum mechanical expectation value. ere Âi are a set of N self-adjoint commuting operators introducing the collapse in a certain basis choice position basis in most cases, W i,t present a set of N independent standard Wiener processes, one for each collapse operator Âi, and λ quantifies the strength of the collapse processes. The main difference of the two collapse models under investigation is the choice of the localization operators  i. A very useful mathematical property of equation is that its physical predictions concerning the outcomes of measurements are, in terms of statistic expectations or probabilities, invariant under a phase change in the noise 7, 8. In particular, choosing the phase to be i one can rewrite the above equation by d φ t iĥdt + i λ N  i dw i,t λ i N i  i dt φ t. Within collapse models statistics of outcomes of any experiment have to be expressed as averages E φ t Ô φ t TrÔ E φ t φ t TrÔ ˆ ρt, where Ô is a self-adjoint operator. The white noise dw represents the change in time t of the Wiener process W t with the definition W t. The term white uncolored refers to independent and identically distributed growths of dw, with a zero expectation value and a standard deviation proportional to dt. Since the temporal derivative of a Wiener process does not exist only in the sense of distributions the integration of the differential equation depends on the choice of sampling point in the interval t, t + dt. The Itô formalism chooses t left-hand endpoint of each time subinterval, whereas the Stratonovich formalism chooses t + dt/. The advantage of the Stratonovich formalism is that the differential and integration procedures are those familiar from ordinary calculus. Therefore, we will stick to this formalism. Then equation becomes a Schrödinger-like equation linear with a random amiltonian i d dt φ t Ĥ N λ  i w i,t φ t : i Ĥ + ˆNt φ t, where w i,t : d dt W i,t. Flavor oscillating systems such as the neutral meson systems are described by two-state phenomenological amiltonians. Mesons are massive systems that decay with two different decay constants, consequently, giving rise to a non-unitary time evolution. To account for the decay one adds a non-ermitian part ˆΓ to the amiltonian. Since it is clear how to add the decay constants to the final formulae, we will later omit them in the computation of the effect of a spontaneous collapse. In Ref. 9 the authors succeeded to show that effect of the non-hermitian part of the amiltonian decay can be understood if the system is considered to be an open quantum system. Then the Schrödinger equation is turned to a Gorini-Kossakowski-indblad-Sudarshan master equation 3, where a indblad operator implies the transition from the surviving part to the decaying part of the system under investigation. Consequently, the decay property can be incorporated via a indblad operator into the quantum system and can be physically understood as an interaction with a virtual environment in quantum field theory it would refer to the QCD

3 3 vacuum. This in turn shows that the total time evolution is a completely positive map. We will give the details in Section IV D. A neutral meson M consists of quark-antiquark pair, and both the particle state M and the antiparticle state M can decay into the same final states. Therefore, neutral mesons have to be considered as a two-state system. The dynamics of a M M oscillating system is covered by an effective Schrödinger equation d dt ψ t i Ĥeff ψ t, 3 ψ t at M + bt M, 4 where the phenomenological effective amiltonian Ĥ eff ˆM + i ˆΓ is non-hermitian, ˆM ˆM is the mass operator which describes the unitary part of the dynamics of a neutral meson, and ˆΓ ˆΓ covers the decay non-unitary part. Diagonalizing the phenomenological amiltonian leads to two different mass eigenstates c Ĥ M i m i + i Γ i M i, 5 These two states M and M have distinct masses and without loss of generality m denotes the lower one... light,... heavy. For all types of neutral mesons the decay rates Γ, Γ are approximately equal, except for K-mesons whose decay rates differ by a huge factor 6. For the sake of simplicity we assume that mass eigenstates are orthogonal, M M, herewith we neglect a small violation of the charge-conjugation parity CP symmetry. The relation between the flavor eigenstates and mass eigenstates is then given by introducing without loss of generality a particular phase convention M M + M, 6a M M M, 6b et us here also remark that temporal part of the evolution of mesons is not normalized in time due to the non- ermitian part of the effective amiltonian, i.e. for clarity we add here the natural constants which we else set to one ψt e i mc t e Γ t ψt dt Γ. Obviously, a normalization of the temporal part by Γ would give a similar expression as the Born rule for the spatial part ψ x d 3 x and allow for a definition of a time operator owever, recently, it was shown that this formal normalization leads to contradiction with experimental data if the small violation of the charge-conjugation parity CP symmetry is taken into account 37. This expresses the strikingly different roles of time and space in the quantum theory and the importance of discrete symmetries. et us now move to the two dynamical reduction models, the QMUP model for one particle N and the mass-proportional CS model, and apply them to meson systems. In the QMUP model the operators Âi introduce the collapse and are chosen to be the position operators ˆq i. In order to describe the collapse dynamics in the case of neutral mesons we extend the collapse operators Âi by a flavor part 7 Â QMUP ˆq m m M M + m m M M, 8a and, consequently, the ˆNt operator of the Schrödinger-like equation 3 becomes ˆN QMUP t λ wt ˆq m m M M + m m M M, 8b where m is a reference mass which is taken usually to be the nucleon mass. We consider wt : dwt dt as a white uncolored noise field, where Wt W t,..., W d t} and the corresponding correlation function is Ewt ws δt s. ere d denotes the dimension, i.e. d,, 3 in general. In the case of the mass-proportional CS model the collapse operator acts in a Fock space, so we replace Âi by a continuous set of operators Âx, one for each point in space, i.e.

4 4 Â CS x ˆN CS t γ m dy gy x m dy wy, t m m ˆψ y ˆψ y + m ˆψ m y ˆψ y, 9a ˆψ y ˆψ y + m m ˆψ y ˆψ y, 9b where ˆψ j y and ˆψ j y are creation and annihilation operators of a particle of type j, in a point y. The smearing function gy x is usually taken to be of a Gaussian type gy x πr C d e y x /r C, where r C is a spatial correlation length and represents one of the two phenomenological constants γ, r c of the mass-proportional CS model. The correlation functions of the mass-proportional CS noise are given by Ewx, twy, s F x yδt s, where F x / 4πr C d e x C. Note that we have substituted the rate λ by γ which has now the units m d /s. A characteristic of the CS model is that all observable results will be proportional to the ratio γ/rc d being a rate or by including all units the strength of the interaction. III. DERIVATION OF TE TRANSITIONS PROBABIITIES Accelerator facilities have intensively studied and will study the transition probabilities from mass eigenstates to mass eigenstates, P Mµ M ν t, and from flavor eigenstates to flavor eigenstates, P M M / M t. For the QMUP model we need to define the initial spatial state. We will assume a wave packet in position picture with a width α in d-dimensional space d,, 3 and with a momentum p i. Further we assume that the final state is a momentum eigenstate most common scenario in typical accelerator facilities. The probabilities of interest are P Mµ M ν α; t p f E M ν, p f M µ t, p i, α, P M M / M α; t p f E M / M, p f M t, p i, α, where E denotes the noise average and µ, ν,. For the QMUP model we start with the -dimensional case and then generalize the results to the d-dimensional case. In the case of the mass-proportional CS model we start directly with the d-dimensional case. To obtain the probabilities of interest we need to compute first the transition amplitudes for all mass eigenstates. For that we move to the interaction picture and treat the noise term ˆNt as a perturbation T µν p f, p i, α; t : M ν, p f M µ t, p i, α 3 e imµt M ν, p f ÛIt M µ, p i, α, where the evolution operator ÛIt is the corresponding one in the interaction picture. The evolution operator is then expanded into a Dyson series up to fourth perturbative order with T T µν p f, p i, α; t e imµt T µν p f, p i, α; t 4 + T µν p f, p i, α; t + T µν p f, p i, α; t + T 3 µν p f, p i, α; t + T 4 µν p f, p i, α; t, µν p f, p i, α; t M ν, p f M µ, p i, α, 5a T n µν p f, p i, α; t i n M ν, p f n j... n dt n 5b ˆNI t j M µ, p i, α for n,, 3, 4, where ˆN I t is the noise term in the interaction picture. Each term in 4 is in detail computed in the appendix A. The crucial derivations of the correlation functions are given in appendix C. For the CS model we follow a similar strategy, however, we can immediately consider the d-dimensional case and the method introduced in Ref. 6, 8. All details are summarized in appendix B. IV. RESUTS We first give the general result and analyze its features. Then we present a decoherence model within standard quantum mechanics resulting in the same probabilities as those of the CS model. A. General results Putting all pieces together we obtain the desired probabilities up to second order in time and collapse parameters

5 5 P QMUP M µ/ M ν/ t δ µν Λ QMUP µ t + 3 ΛQMUP P CS M µ/ M ν/ t δ µν Γ CS µ t + ΓCS µ t + Ot 3 µ t + Ot 3 e Γµt, 6 e Γµt, 7 with Λ QMUP µ αλ m µ m θ, Γ CS µ γ 4πr C m µ d m θ, 8 where θ is the eaviside function at zero. The result of the QMUP model agrees with those in Ref. 38. et us mention here a couple of comments. Firstly, the mass eigenstates do not oscillate as it is the case in the standard approach. Secondly, we find that the effect of the collapse in position space leads to a term that is proportional to the mass squared per unit mass squared. These masses never appear in standard quantum theory. Moreover, it gives an inverted ordering, namely the decay rate that is bigger than the other one is connected to the heavier mass. This in turns means that the eigenstate of the heavier mass decays earlier. We reconsider this point in section IV B. Thirdly, the result of the QMUP model is independent of the number of dimensions d see appendix B. Fourthly, there is an additional factor 3 independent of the number of dimensions d in second order of time. This is a bit surprising. aving a closer look into the computations summarized in appendix A we observe that the factor 3 is a product of choosing Gaussian wave functions and their integration over all final momenta. As a consequence the effect of the collapse on the meson time evolution cannot be assumed to be an exponential effect in general. In strong contrast to the CS model, where we can expect that the dynamics of a mass eigenstate propagating in free space is exponential P CS M µ/ M ν/ t δ µν e ΓCS µ +Γ µt. 9 ast but not least the choice of θ, coming from the correlation functions of the Wiener processes leads to positive θ, }, zero θ or negative θ, } values of ΓCS. Before we proceed in discussing which value θ should be taken, let us present the results for the transition of the flavor eigenstates flavor oscillation P QMUP M M / M 4 P CS M M / M 4 i, e Γit Λ QMUP i t + 3 ΛQMUP i t + Ot 3 ± cos mt e Γ +Γ t + 3 i, αλ m m e Γit Γ CS i ± cos mt e Γ +Γ t + γ 4πr C d αλ m m θ + m m m θ θ + m m m θ t + Ot 3 m t + ΓCS i t + Ot 3 m γ 4πr C d m m t } θ + m m m θ t } θ + m m m θ t + Ot 3,, where m m m. Again, for the CS model we assume that the higher orders in time lead to an exponential behavior P CS M M / M t } e Γ +Γ CS t + e Γ +Γ CS t ± cos mt e Γ +Γ t e ΓCS +ΓCS t e γ m 4πr C d m t. 4

6 6 This we can rewrite in the following form P CS M M / M t e Γ CS +Γ t + e Γ +Γ CS t ± 4 cosh Γ +Γ CS cos mt e γ Γ +Γ CS t m 4πr C d m } t.3 This is an interesting result since it disentangles two effects of the collapse model. A damping of the interference term proportional to the mass difference squared m is independent of the choice of the eaviside function θ and additional energy terms Γ CS i proportional to the absolute masses depend on the eaviside function. These additional energy terms play the same role as the decay constants added by hands in standard quantum theory. In the next step we investigate whether the collapse dynamics leading to the above result can explain the full dynamics of the neutral meson systems without defining decay constants by hands due to Wigner-Weisskopf approximation. B. Impact and observability of the CS model prediction At accelerator facilities the following asymmetry term At is experimentally well investigated At P M CS M t P CS M M t PM CS M t + P CS M M t cos mt e γ cosh Γ +Γ CS Γ +Γ CS t 4 m 4πr C d m t. From that we observe that the damping term proportional to γ m is in principle measurable. The 4πr C d m standard proposed value for the mass-proportional CS γ model is λ CS : 4πr C d 8± s Adler 39 or 6 s GRW 3. ere the coherence length is assumed to be of the order 5 cm and d 3 and from that the collapse strength γ can be deduced. For more details on the allowed parameter space for r C and γ consider, e.g., Ref. 4. et us also note that the best experimental upper bound is currently obtained by X- rays being five orders away from the proposed value of Adler, i.e. s. Plugging in these two values Adler/GRW and the measured mass differences we find damping rates of the order 38 s / 46 s for K-mesons, 3 s / 38 s for B d -mesons, 3 s / 38 s for B s mesons and 34 s / 4 s for D mesons see also Ref. 7. The choice of the reference mass m being either the neutron mass or the rest mass of the respective neutral meson does not affect the values considerably. This is not directly observable since it corresponds to a lifetime much greater than the decay rates of the respective neutral meson. Consequently, the effect of the spontaneous collapse on the interference can be safely neglected. The idea behind the choice of m, being generally a free parameter of the CS model, is that for ordinary matter the mass ratio corresponds to an average number of constituents of the composite object 4; the bigger the object, the stronger the effect of spontaneous localization. The choice in the meson system stems from our assumption that if collapse models are relevant in Nature then they have to hold for all physical systems. For the meson system this mass ratio mµ m decreases if m is of the order of a nucleon or the rest mass of the mesons system, i.e. has the opposite behavior. Thus it may seem more reasonable to have for particles lighter than those that make up the ordinary matter the inverted ratio. If we do so then the damping factor of the interference term becomes λ CS m m, which is only computable if we m m know the absolute masses. The second modification due to the mass-proportional CS model compared to the standard approach is for the decay rates, i.e. Γ µ + Γ CS µ. ere Γ µ are the standard decay rates introduced to the system by the Wigner- Weisskopf approximation. The collapse contribution is connected to the absolute mass playing no role in the standard approach and the value of the eaviside function at zero, i.e. Γ CS µ λ CS m µ θ or in m the inverted scenario Γ CS µ λ CS m m θ. µ Taking this one step further is to ask whether collapse models could solely be responsible for the decaying part of the neutral mesons, i.e. the dynamics of the spontaneous localization induces the decay of the mass eigenstates. For that we set Γ exp µ Γ CS µ. Certainly Γ CS needs to be positive, i.e θ, }, to obey equations 9. Then we obtain

7 7 Γ exp s Γ exp s m exp s m s m s K-mesons D-mesons B d -mesons B s-mesons TABE I: Experimental values of the decay rates, the mass difference and the computed values of the absolute masses for the neutral mesons system. Γ CS Γ CS Γ CS + Γ CS θ ± m m m + m K-mesons: D-mesons: B d -mesons: B s -mesons: The experimental values for the experimentally measured decay constants right hand side of the above equation are taken from the particle data book 4. The method how to deduce from the experimental values measured the decay rates is described in appendix D since it differs slightly for each meson. The minus sign holds for the inverted scenario. Together with the experimentally obtained value of m : m m, this allows to compute the absolute values of the masses m / via Γ CS Γ CS Γ CS + Γ CS ± m m + m m + m + m ± m m m m + m m + m ± m + m + m m +. m 6 In the case we have m > m m > we observe that the right hand side becomes negative if we do not reverse the mass ratio. Thus the two involved masses cannot be both positive. This is because the collapse models relate the decay rates with the corresponding masses directly proportionally: the heavier the mass the bigger the decay rate, the smaller the lifetime. This is physically intuitive from the collapse model perspective since heavier masses should be affected stronger by the spontaneous factorization. The counter-intuitive effect for applying that to neutral mesons decay is that the more massive state should decay faster. In literature there can be found experiments 43, 44 for K-mesons assigned to measure the sign of m and, herewith, if the heavier mass connects also to the lower decay rate longer lifetime and vice versa. The results are a positive sign of m, i.e. the heavier mass decays slower. Note that not for all mesons the sign has been determined. In summary, for positive mass differences m > we cannot find positive masses. In the reversed scenario positive values for the absolute masses are obtained and listed in table I. The obtained values for the absolute masses are in the regime of the weak interaction due to our identification with the decay rates. They are functions of the two decay rates and the mass difference. et us remind the reader that the rest mass of mesons is by many units higher since here the strong interaction rules. Note that the numerical values are very sensitive to the errors and the method to determine the decay constants which are very different to the specific mesons and the experiments considered. We stick here to the values published by the particle data group in their summary and review papers 4. Now we can use these values of absolute masses to estimate λ CS by λ estimated CS : Γ exp µ m µ m θ m Γ m Γ 7 θ.

8 8 s - λ CS θ Adler GRW FIG. : Color online These plots show the values of θ versus the deduced collapse rate based on the input parameters Γ exp, Γexp, mexp for the different types of neutral mesons including experimental errors. As a reference mass the respective rest mass of the neutral mesons is assumed. The predicted values are plotted in Fig. for the different meson types. Interestingly, these values correspond to the ones assumed by Adler, except for the K-meson system which is closer to the one of GRW even weaker. Fixing the collapse rate to the one proposed by GRW requires that θ converges to, only in the Adler case values are allowed. Taking the scenario with reversed masses seriously we have also to consider the modified contribution to the interference term, i.e. λ m m CS m m m λ CS m Γ Γ λ CS λ estimated CS θ Γ 4 Γ 8 Γ Γ. This term is negligible for all types of neutral mesons due to the tiny difference between the two decay constants assuming that the other values are of order except for the K-meson system. In this case we have a very sensitive tradeoff between obtaining the experimental values of the decay constant and the damping of the interference term. The best limit on such a possible modification of the interference term comes from the entangled K-meson system, however, this is not directly comparable. In summary a full description of the decay and oscillation properties in the dynamics of neutral mesons can be obtained demanding certain properties of collapse models. d s C. What is the physical meaning of the choice of θ,? et us remind the reader that it comes from the action of the classical noise underlying any collapse model. One assumes that the stochastic noise average of two Wiener processes is given by a delta distribution, Ewt ws δt s. This in turn is the assumption of the white noise scenario, i.e. the assumption of a constant power spectral density. In our derivation we had to compute the following type of time integrals θt θ θ for t s ds δt s θ θ t θ for t s 9 with δtd t. Note that the dependence on θ occurs only in case one matches amplitudes of different orders within the expansion. Assuming the independence of the time direction δt δ t leads to θ. In this case the collapse quantities Λ QMUP, Γ CS become zero, respectively. No effect of the collapse field arises in the evolution of the mass eigenstates. Consequently, in this case also no dependence on absolute masses m, m is proposed in line with the standard quantum mechanical approach. A value θ can be interpreted as the freedom that the time evolved states in the expansions in the out bra and the in ket states do depend on the particular time ordering within the expansion. Only in this case the interaction with the classical noise field leads to contributions not solely affecting the interference term with respect to the chosen basis. Consequently, here is the point where the physics of the noise field strongly enters the discussion. In particular non-white noise fields will change the very dynamics of neutral mesons, that in turn will be testable. D. Mimicking the effect of the CS model on the meson dynamics by a decoherence model To better understand the physics proposed by collapse models for the meson dynamics, let us see which decoherence model within standard quantum mechanics would in principle lead to the same predictions. et us here also say some words about the nonhermitian amiltonian that is the the standard starting point in describing the meson phenomenology. Neutral meson systems violate the CP symmetry for the mass matrix and have a non-vanishing lifetime difference in the width matrix. This leads to an effective amiltonian which is even not a normal operator with incompatible non-commuting masses and widths. In the Wigner- Weisskopf approach, by diagonalizing the entire amiltonian, the in general non-orthogonal stationary states M, M are obtained. These states have complex eigenvalues whose real imaginary part does not coincide

9 9 with the eigenvalues of the mass width matrix. The mesonic systems can also be described as an open quantum mechanical system 9, 45, 46. In particular, the following Gorini-Kossakowski-indblad-Sudarshan master equation does the job 9 d ˆρt iĥ, ˆρt 3 dt f ˆ i ˆ i ˆρt + ˆρt ˆ ˆ i i ˆ i ˆρt ˆ i i where we define ˆρ to live on a ilbert-schmidt space with a direct product structure s d s corresponds to the surviving part and d of the decaying part of the system. In our case we need at least a four-dimensional space. The amiltonian Ĥ and all indblad operators ˆ are defined to act only onto the surviving part of the system, i.e. Ĥ ˆM, ˆi> ˆi, 3 whereas the zero indblad operator entangles the surviving part with the decaying part ˆ. 3 ˆ Given these definition the total density matrix ˆρt ˆρ ss t ˆρ sd t ˆρ sd t ˆρ ddt 33 is normalized for all times. The differential equation decouples for the parts of the system. ence, the solution of the survive-decay part ˆρ sd t has no physical significance and the time dependence of the decay-decay contribution ˆρ dd t depends solely on ˆρ ss t, i.e. ˆρ dd t ˆ ˆρ ss t dt ˆ. 34 For clarity, let us rewrite the relevant differential equation explicitly for hermitian indblad generators ˆρ ss t i Ĥ, ˆρ sst ˆ, ˆρ ss t} } ˆ i, ˆρ ss t} ˆ i ˆρ ss tˆ i 35 i> is given in the mass eigenstate basis by ˆ diag Γ + Γ CS, Γ + Γ CS }. Choosing ˆ γ m 4πr C d m M M + m m M M formally leads to the same probabilities, see Eq. 3. This has the following physical intuitive picture behind it: the state vector undergoes a random unitary transformation in the time dt Ûφ ψt e iφĝ ψt iφĝ φ Ĝ +... ψt 36 with a Gaussian probability distribution with a width proportional to dt, namely with probability pφdφ pφ πσ e φ σ, 37 where we choose explicitly the width σ γ dt. 4πr C d Since we assume small dt we can neglect safely the higher order terms and find for the density matrix at time t + dt ˆρt + dt + ˆρt σ + dφ pφ dφ pφ Ûφ ˆρt Û φ ˆρt φ Ĝ, ˆρt} Ĝ ˆρt Ĝ } Ĝ, ˆρt} Ĝ ρt Ĝ. 38 This differential equation is equivalent to the one in the indblad form with ˆ if we choose for Ĝ m i i m M i M i which is just the flavor part of our collapse operators. Even though we formally arrive at the same formulae, let us stress that in this case no real collapse is assumed, in particular the spatial part of the wave function played no role. Moreover, the dependence on the decay rate Γ CS i is not generated by the dynamics, but introduced by hand. owever, it explains why the interference term in the flavor oscillation probabilities depends on m, this is a general feature of any random unitary noise with a Gaussian distribution. V. SUMMARY AND OUTOOK In this paper we have focused on two popular dynamical reduction models, QMUP Quantum Mechanics with Universal Position ocalization and massproportional CS Continuous Spontaneous ocalization model, and analyzed their effect on the neutral meson system. The beauty of these models is that they solve the measurement problem by introducing a physical mechanism for the collapse. In particular they assume that with a certain rate every quantum system undergoes a spontaneous localization in space. Taking these models seriously they have to also affect systems at higher energies, in particular, neutral mesons. We have considered the two-state phenomenological amiltonian for a neutral meson system giving rise to flavor oscillations and assumed the collapse mechanism

10 implied by dynamical reduction models. Since the collapse models assume the collapse to the spatial part of the state, we had to choose proper collapse operators relating the flavor space where the oscillation takes place with the spatial space. To calculate the effect of the collapse we have considered the white noise of the collapse models as a small perturbation by utilizing the Dyson series. The transition probabilities were calculated up to fourth perturbative order. This allowed to distinguish between exponential behavior observed for the CS model and non-exponential behavior observed for the QMUP model, consequently giving insight into the physics of the noise field underlying the collapse mechanism. Due to necessariness in solving the stochastic nonlinear differential equations by perturbation series we have observed a dependence on the choice of the eaviside function θ at time point. This function turns up from integrating the correlations of two or more Wiener processes. Mathematically, the value is not well defined, it can be in the interval θ,. Only in the case of θ we find that the lifetime state evolves independently of the collapse in spatial part. Indeed, having a closer view at the computations we observe that the value of the eaviside function at time zero matters only if we evaluate amplitudes connecting different orders in the expansion. Consequently, the effect is due to the virtual propagation of the states and hence portrays a kind of consistency within the calculus. Since θ physically implies that we have a time symmetric case of the two Wiener processes in the two dual spaces ket/bra. This means that only in this case θ the norm of the state under investigation is conserved. Differently stated, the effect of the spontaneous collapse appears only in the interference terms of the flavor oscillation, not in the decaying parts. Interestingly, any value θ leads to a dependence on the absolute masses energies of the eigenstates of the time evolution, which does not show up in the standard quantum approach. In the CS model they would not be directly measurable since they would effectively contribute to the decay constants of the standard quantum approach. On the other hand, the effect of the QMUP model is in principle observable due to its nonexponential behavior. Since this deviation has not yet been observed, experiments provide upper bounds on the absolute masses of the lifetime states. In a further step we analyzed whether the spontaneous localization could be considered as the only source of the decay in the neutral meson dynamics. We related the measured decay constants with the absolute masses appearing due to the nonlinear and stochastic modifications. The first observation was that for this identification the sign of the mass difference matters, i.e. if the longer-lived state corresponds to the more massive state or the lighter one. Experiments for K-mesons favor the first relation. This is in strong contrast to the philosophy of collapse models where a more massive system should localize faster in order to solve the measurement problem. To obtain still positive absolute masses one needs to identify the strength of the generation of a say heavy mass eigenstate with the lower mass. Doing so, we can deduce the absolute masses. Via that the decay mechanism of neutral mesons can be fully described. In turn we can also use them to predict values of the collapse rate Fig.. This rate is computed solely by the input parameter of the mass difference and the two decay rates in dependence on the value for θ. The range is in the expected region proposed by Adler or Ghirardi-Rimini- Weber except for the K-meson system. In order to obtain a different insight into the physics behind the collapse models we also defined a master equation within standard quantum mechanics that leads to the same probabilities. ere we extended the ilbert space to include the decay components as developed in Ref. 9. Then a Gorini-Kossakowski-indblad- Sudarshan master equation with Gaussian noise proportional to the masses does the job. This illustrates the dependence of the damping on the squared mass difference since it is a general feature of systems undergoing a random Gaussian distributed unitary noise. The strongest bounds on deviations from the expected quantum mechanical behaviors come from experiments with entangled mesons; or may be the case if the tiny CP violating effects are taken into account. Therefore, it would be necessary to extend our computations to these cases. Moreover, our computations are performed for a white noise scenario. Colored noise should change the dynamics considerably and in turn allow to limit the collapse rate. Acknowledgment: The authors gratefully thank Sandro Donadi Università degli studi di Trieste for fruitful discussions and the Austrian Science Fund FWF-P6783. Appendix A: Computations for the QMUP model. Transition probabilities for the mass eigenstates We start the computations of the transition probabilities for the QMUP model with the -dimensional case. The five terms 5a 5b form the transition amplitude up to fourth order of the Dyson series which we calculate here. Inserting the definition 8b for the ˆN QMUP t operator -dimensional case and calculating the flavor part of matrix elements, we obtain the following expression for the components up to n-th order of the transition amplitudes T µν n p f, p i, α; t e imµt F n t i λ m n µ m p f ˆq n p i, α δ µν, A

11 where F t for all t, F n t... n dt n n j wt j. The transition probabilities up to second order in time t are The transition amplitudes derive to p f ˆq n p i, α απ e α p f p i ζn A with P Mµ M ν α; t P α; t + P α; t + P α; t, A3 ζ, ζ i α p f p i, ζ α αp f p i, ζ3 iα 3p f p i αp f p i 3, ζ4 α 3 6αp f p i + α p f p i 4. which decompose in terms of the transition amplitudes to P α; t π P α; t π P α; t π dp f E T µν p f, p i, α; tt µν p f, p i, α; t, A4a dp f E T µν p f, p i, α; tt µν p f, p i, α; t + T µν p f, p i, α; tt + T µν p f, p i, α; tt µν p f, p i, α; t, dp f E T µν p f, p i, α; tt 4 µν p f, p i, α; t + T 4 µν p f, p i, α; tt + T µν p f, p i, α; tt µν 3 p f, p i, α; t + T µν 3 p f, p i, α; tt µν p f, p i, α; t + T µν p f, p i, α; tt µν p f, p i, α; t. µν p f, p i, α; t A4b µν p f, p i, α; t A4c The first term gives P α; t απ dp f e αp f p i δ µν δ µν. π For the first order in time we need dp f E T µν p f, p i, α; tt µν p f, p i, α; t π + T µν p f, p i, α; tt µν p f, p i, α; t λm µ δ µν m α απ π dt Ewt wt A5 dp f αp f p i e αp f p i θ t, δ µν αλ m µ m where the computation of the two-point correlation function dt Ewt wt is explicitly derived in the Appendix C. The second term derives to dp f E T µν p f, p i, α; tt µν p f, p i, α; t π λm µ δ µν m α απ π δ µν α λm µ m dt Ewt wt t, dp f p f p i e αp f p i where the two-point correlation function t dt dt Ewt wt is derived in the Appendix C note the difference in the integration limits. Consequently, the transition probabilities in first order in time t result in P α α; t δ µν λm µ m θ t. A6

12 To obtain the solution in the second order in time t we have to compute the five components, T T 4, T 4 T, T T 3, T 3 T and T T, where we have for the first time to evaluate a four point function in the noise which is done in detail in Appendix C. We compute π dp f E T µν p f, p i, α; tt µν 4 p f, p i, α; t + T µν p f, p i, α; tt 4 µν p f, p i, α; t field, which are calculated in Appendix C. Summing up, we obtain the transition probabilities in second order time t P α; t 3α δ µν 4 3α δ µν 8 λ m 4 µ m 4 λ m 4 µ m 4 θθ + t θ t. A7 λ m 4 µ δ µν m 4 C α απ π 4, t dp f 3 6αp f p i + α p f p i 4 e αp f p i 3α δ µν and and π π λ m 4 µ m 4 C 4, t δ 3α λ m 4 µ µν m 4 dp f E T µν p f, p i, α; tt µν 3 p f, p i, α; t + T 3 µν p f, p i, α; tt µν p f, p i, α; t δ µν λ m 4 µ m 4 α3 απ π e αp f p i δ µν 3α δ µν 3α C 3, t θ t, dp f 3 αp f p i p f p i λ m 4 µ m 4 C 3, t λ m 4 µ m 4 θ t, dp f E T µν p f, p i, α; tt µν p f, p i, α; t λ m 4 µ δ µν m 4 C, t α απ dp f αp f p i e αp f p i π 3α δ µν 4 3α δ µν 4 λ m 4 µ m 4 C, t λ m 4 µ m 4 θ + t. where C 4, t, C 3, t and C, t correspond to the integrals of the 4-point correlation functions of the noise Finally, collecting all the terms A5 A7, we obtain the transition probabilities for mass eigenstates up to second order in time t P Mµ M ν α; t δ µν α + 3α 8 λ m 4 µ t m 4 θ. λm µ m θ t. d-dimensional case A8 In the case of d-dimensional space the components 5a 5b of transition amplitudes have to be generalized in the following way where T n µν p f, p i, α; t A9 e imµt F n p f, p i, α; t i λ m nδµν µ, m F p f, p i, α; t p f p i, α, F n p f, p i, α; t p f... n ˆq wt j p i, α. j n dt n ere one can think of basically two different ways the noise would act onto the system. Either a factorization in any of the possible dimensions happens and contributes to the first order in time, or a factorization of the wave function has to occur in all dimensions simultaneously. The second one seems to be less natural to assume. Since we assume white noise and an initial Gaussian wave function in all dimensions, however, integrals give the same value and the only difference is how often the integral occurs. Therefore, we stick to the first case. Explicitly, we find

13 3 F p f, p i, α; t απ d/e α p f p i, F p f, p i, α; t i d/α t απ F p f, p i, α; t d/α t απ F 3 p f, p i, α; t i d/α απ p f p i wt e α p f p i, dt wt wt α p f p i wt p f p i wt e α p f p i, + p f p i wt wt wt 3 dt p f p i wt wt wt 3 + p f p i wt 3 wt wt α p f p i wt p f p i wt p f p i wt 3 e α p f p i, F 4 p f, p i, α; t d/α απ dt 3 dt 4 wt wt wt 3 wt 4 + wt wt 3 wt wt 4 + wt wt 4 wt wt 3 α p f p i wt p f p i wt wt 3 wt 4 α p f p i wt p f p i wt 3 wt wt 4 α p f p i wt p f p i wt 4 wt wt 3 α p f p i wt p f p i wt 3 wt wt 4 α p f p i wt p f p i wt 4 wt wt 3 α p f p i wt 3 p f p i wt 4 wt wt + α p f p i wt p f p i wt p f p i wt 3 p f p i wt 4 e α p f p i. and herewith the probabilities P α; t δ µν, P α α; t δ µν P α; t δ µν 3α 4 λm µ m θ t, λ m 4 µ m 4 θθ + t, which are identical to the ones of the -dimensional case and, consequently, lead to the same transition probabilities. 3. Transition probabilities for the flavor states Transition amplitude for a flavor state can be expanded in the following way: T M M / M p f, p i, α; t M / M, p f M t, p i, α α µ βν M ν, p f M µ t, p i, α µ,ν α µ βνt µν p f, p i, α; t, µ,ν

14 4 where µ, ν, and α α β, β ± plus sign refers to a meson, minus sign refers to an antimeson. In the same manner, transition probability for a flavor state can be defined as P M M / M α; t α µ βνα µ β ν A µ,ν,µ,ν π d dp f ET µν p f, p i, α; ttµ ν p f, p i, α; t α µ βνα µ β ν P µνµ ν α; t, µ,ν,µ,ν Furthermore, since each transition amplitude T µν p f, p i, α; t contains a Kronecker delta δ µν, as can be seen from A and A9, we can leave just one index in an amplitude and correspondingly two indexes in probabilities P µνµ ν α; t P M M / M α; t α µ βµα µ β µ P µµ α; t A µ,µ 4 P α; t ± P α; t ± P α; t + P α; t, Using the transition probabilities which were calculated above we obtain the terms for the transition probability, with same indexes P aa and different ones P ab P aa α; t α λm a m θ t A + 3α λ m 4 a t 8 m 4 θ, P ab α; t e ima m bt A3 α λ m a + m b θ m a m b t m + 3α λ 8 m 4 m 4 a + m 4 b θ m 3 am b + m a m 3 b θ + m am b θ + t }. Putting the terms together we finally obtain the transition probability for the flavor states P M M / M α; t ± λα m α 4 λm + m m θ t + 3α 6 m + m θ m m t m m m + m θ + m m θ + λ m 4 + m4 m 4 θ t λ α m 4 m 4 + m 4 θ t cos m m t }. A4 Taking the decay into account we obtain P M M / M α; t e Γt + e Γt λα m 4 e Γt + m e Γt θ t A5 m + 3 λ α 8 m 4 m 4 e Γt + m 4 e Γt θ t ± λα m + 3 λ α 8 m 4 m 4 + m 4 θ m m m + m θ + m m θ } cos m m t e Γ +Γ t. m + m θ m m t + t Appendix B: Computations for the CS model. Transition probabilities for mass eigenstates For the CS model we also have five terms which form the transition amplitude up to fourth order of the Dyson series. Putting the expressions for the ˆN I operators in we obtain: T n µν p f, p i, α; t e imµt i γ n K n µν p f, p i, α; t,

15 5 where K µν p f, p i, α; t M ν, p f M µ, p i, α, K n µν p f, p i, α; t M ν, p f k, n j... wt j, x j t n dt n dx... m k m ˆψk I t j, x j ˆψ I k t j, x j M µ, p i, α. dx n Accordingly, we will calculate the matrix elements in the same manner as done in 6. At first, we make an expansion of field operators into a superposition of plane waves ˆψ k I t, x d q ˆbq e iek q t q x, B where the energy of a meson of mass m k and momentum q is taken in non-relativistic limit, E q k q + m k m k. ere the system is assumed to be quantized in a box of size with using periodic boundary conditions. While calculating the transition amplitudes and probabilities we take the limit and perform an integration by momentum q dq. d π d Using the coordinate representation and calculating the matrix elements, we obtain components of the transition amplitudes in the following form K µν p f, p i ; t απ d/ e α p f p i δ µν, K µν p f, p i ; t m µ m απ d/ K µν p f, p i ; t m µ m π d dq απ d/ wt, x wt, x e ip f qx e iq pix e x K µν 3 p f, p i ; t m3 µ m 3 π d dx wt, x e ip f p ix e x α δ µν, d/ dq dq απ dt dt dx dx α δ µν, wt, x wt, x wt 3, x 3 e ip f q x e iq qx e iq pix3 e x K µν 4 p f, p i, α; t m4 µ m 4 π 3d d/ dq dq dq 3 απ dt 3 α δ µν, dx dx dx 3 3 dt 4 wt, x wt, x wt 3, x 3 wt 4, x 4 e ip f q x e iq qx e iq q3x3 e iq3 pix4 e x 4 α dx dx dx 3 dx 4 δ µν. The next step is to compute the transition probability which consists of three terms where each term corresponds to zeroth, first and second order by time P Mµ M ν t P t + P t + P t, B3

16 6 P p i, α; t π d P p i, α; t π d P p i, α; t π d dp f E T µν p f, p i, α; tt µν p f, p i, α; t, B4a dp f E T µν p f, p i, α; tt µν p f, p i, α; t + T µν p f, p i, α; tt µν p f, p i, α; t B4b + T µν p f, p i, α; tt µν p f, p i, α; t, dp f E T µν p f, p i, α; tt 4 µν p f, p i, α; t + T 4 µν p f, p i, α; tt + T µν p f, p i, α; tt µν 3 p f, p i, α; t + T µν 3 p f, p i, α; tt µν p f, p i, α; t + T µν p f, p i, α; tt µν p f, p i, α; t. µν p f, p i, α; t B4c First term is trivial and given by: P t απ π d dp f e αp f p i δ µν δ µν. B5 Second term consists of three components, T µν T µν, T µν T µν and T µν T µν, where the first two components result in π d δ µν γm µ m dp f E T µν p f, p i, α; tt µν p f, p i, α; t + T µν p f, p i, α; tt µν p f, p i, α; t e α p f p i απ /4 π απ /4 d dp f dq dt Ewt, x wt, x dx dx cos δ µν γ m µ d m 4πr C d π dp f dq dx dx e x x C + e ip f qx e iq pix e x α e α p f p i C δ µν γm µ 4πr C d m θ t., t p f qx + q p i x e x e ip f qx e iq pix α The third component equals to

17 7 π d dp f E T µν p f, p i, α; tt µν p f, p i, α; t γm µ d δ µν m π απ δ µν γm µ m dp f d 4πr C d π dp f απ δ µν γm µ 4πr C d m t. dx dx e ip f p ix x e x +x α dt Ewt, x wt, x dx dx e x x C e ip f p ix x e x +x α C, t Consequently: P t δ µν γm µ 4πr C d m θ t. B6 The computations of the integrals C, t and C, t which contain -point correlation functions of the noise field, can be found in the Appendix C. Second term consists of five components, T T 4, T 4 T, T T 3, T 3 T and T T, where the first two components result in π d δ µν γ m 4 µ m 4 dp f E T µν p f, p i, α; tt µν 4 p f, p i, α; t + T µν 4 p f, p i, α; tt µν p f, p i, α; t dp f dq dq dq 3 απ /4 π 4 απ /4 d + q q 3 x 3 + q 3 p i x 4 e x 4 α e α p f p i t dt dx dx dx 3 dx 4 cos p f q x + q q x 3 dt 4 Ewt, x wt, x wt 3, x 3 wt 4, x 4 γ m 4 µ d δ µν 4πrC d m 4 π 4 dp f dq dq dq 3 dx dx dx 3 dx 4 e ip f q x e iq qx e iq q3x3 e iq3 pix4 + e ip f q x e iq qx e iq q3x3 e e iq3 pix4 x 4 α e α p f p i e x x C e x 3 x 4 C U 4, x x3 t + e C e x x 4 C U 4, x x4 t + e C e x x 3 C U 4, 3 t γ m 4 µ d δ µν 4πrC d m 4 π 4 dp f dq dq dq 3 dx dx dx 3 dx 4 e ip f q x e iq qx e iq q3x3 e iq3 pix4 + e ip f q x e iq qx e iq q3x3 e iq3 pix4 e x 4 α e α p f p i e x x δ µν 4πr C d γ m 4 µ m 4 C e x 3 x 4 C θ t, t θ the second two components result in

18 8 π d δ µν γ m 4 µ m 4 dp f E T µν p f, p i, α; tt µν 3 p f, p i, α; t + T µν 3 p f, p i, α; tt µν p f, p i, α; t d dp f dq dq p f p i x 4 π 3 απ e x 3 +x 4 α dt dx dx dx 3 dx 4 cos p f q x + q q x + q p i x 3 dt 4 Ewt, x wt, x wt 3, x 3 wt 4, x 4 γ m 4 µ d δ µν 4πrC d m 4 π 3 dp f dq dq dx dx dx 3 dx 4 απ e ip f q x e iq qx e iq pix3 e ip f p ix 4 + e ip f q x e iq qx e iq pix3 e ip f p ix 4 e x 3 +x 4 α e x x C e x 3 x 4 C U 3, x x3 t + e C e x x 4 C U 3, x x4 t + e C e x x 3 C U 3, 3 t γ m 4 µ d δ µν 4πrC d m 4 π 3 dp f dq dq dx dx dx 3 dx 4 απ e ip f q x e iq qx e iq pix3 e ip f p ix 4 + e ip f q x e iq qx e iq pix3 e ip f p ix 4 e x 3 +x 4 α e x x C e x 3 x 4 C + e x x 4 γ m 4 µ δ µν 4πrC d m 4 θ t, C e x x 3 C θ t and the last component equals to π d δ µν γ m 4 µ m 4 dp f E T µν p f, p i, α; tt µν p f, p i, α; t d π 3 απ e ip f q x 3 e iq pix4 e x 4 α dp f dq dq γ m 4 µ d δ µν 4πrC d m 4 π 3 dp f dq dq απ e ip f q x e iq pix e x α e ip f q x 3 e iq pix4 e x 4 α dt 3 dx dx dx 3 dx 4 e ip f q x e iq pix e x dt 4 Ewt, x wt, x wt 3, x 3 wt 4, x 4 α dx dx dx 3 dx 4 e ip f q x e iq pix e x e x x C e x 3 x 4 C U, x x3 t + e C e x x 4 C U, x x4 t + e C e x x 3 C U, 3 t γ m 4 µ d δ µν 4πrC d m 4 π 3 dp f dq dq dx dx dx 3 dx 4 e ip f q x e iq pix e x απ e ip f q x e iq pix e x α e ip f q x 3 e iq pix4 e x 4 α e x x C e x 3 x 4 C x x 3 θ + e C e x x 4 C t γ m 4 µ δ µν 4πrC d m 4 θ + t. α α