Can Wavefunction Collapse Conserve Energy? Shan Gao∗ April 4, 2012 Abstract The answer may be yes. A discrete model of energy-conserved wave- function collapse is proposed. It is shown that the model is consistent with existing experiments and our macroscopic experience. 1 Introduction In standard quantum mechanics, it is postulated that when the wave function of a quantum system is measured by a macroscopic device, it no longer follows the linear Schr¨odingerequation, but instantaneously collapses to one of the wave functions that correspond to definite measurement results. However, this col- lapse postulate is ad hoc, and the theory does not tell us why and how a definite measurement result appears (Bell 1990). A promising solution to this measure- ment problem is dynamical collapse theories, in which the collapse evolution is dynamical and integrated with the normal Schr¨odingerevolution into a unified dynamics (Ghirardi 2008). However, the existing dynamical collapse models are plagued by the serious problem of energy non-conservation (Pearle 2007, 2009). For instance, in the CSL (Continuous Spontaneous Localization) model, the collapse due to an external noise field narrows the wave function in position space, thereby producing an increase of energy. Although it is expected that the conservation laws may be satisfied when the contributions of the noise field to the conserved quantities are taken into account (Pearle 2004; Bassi, Ippoliti and Vacchini 2005), a complete solution has not yet been found, and it is still unknown whether such a solution indeed exists. In this paper, we will propose a discrete model of energy-conserved wave- function collapse, and show that the model is consistent with existing exper- iments and our macroscopic experience. It has been demonstrated that the energy-driven collapse models that conserve energy can't explain the emergence of definite measurement results (Pearle 2004). However, this important re- sult does not imply that all energy-conserved collapse models are inconsistent with experiments. A detailed analysis of this paper will demonstrate that some energy-conserved collapse models can be consistent with existing experiments ∗Unit for HPS and Centre for Time, University of Sydney, NSW 2006, Australia. E-mail: [email protected]. 1 and our macroscopic experience. The key is to assume that the energy uncer- tainty driving the collapse of the entangled state of a many-body system is not the uncertainty of the total energy of all sub-systems as the energy-driven col- lapse models assume, but the sum of the absolute energy uncertainty of every sub-system. 2 A discrete model of energy-conserved wave- function collapse Consider a multi-level system with a constant Hamiltonian. Its initial state is: m X j (0)i = ci(0) jEii; (1) i=1 where jEii is the energy eigenstate of the Hamiltonian of the system, Ei is the corresponding energy eigenvalue, and ci(0) satisfies the normalization relation Pm 2 i=1 jci(0)j = 1. It is assumed that this superposition of energy eigenstates collapses to one of the eigenstates after a discrete dynamical process, and the collapse evolution satisfies the conservation of energy at the ensemble level1. This collapse process is composed of a finite number of discrete tiny collapses. The properties of the tiny collapses is assumed as follows. At each discrete instant t = ntP (where tP is the discrete unit of time), the probability of the tiny collapse happening in 2 each energy branch jEii is Pi(t) ≡ jci(t)j , and this collapse slightly increases the probability of the energy branch and decreases the probabilities of all other branches pro rata. Then during a finite time interval much larger than tP , the probability of each energy branch will undergo a discrete and stochastic evolution. In the end, the probability of one branch will be close to one, and the probabilities of other branches will be close to zero. In other words, the initial superposition will randomly collapse to one of the energy branches in the superposition. Since it has been generally conjectured that the Planck scale is the minimum spacetime scale2, we will assume that the size of each discrete instant, tP , is the Planck time in our following analysis. Now we will give a concrete analysis of this dynamical collapse process. Since the linear Schr¨odingerevolution does not change the energy probability distribution, we may only consider the influence of dynamical collapse on the energy probability distribution. Suppose at a discrete instant t = ntP the tiny 1It can be proved that only when the collapse states are energy eigenstates of the total Hamiltonian for each identical system in an ensemble, can energy be conserved at the ensemble level for wavefunction collapse (see Pearle 2000 for a more detailed analysis). Note that for the linear Schr¨odingerevolution under an external potential, energy is conserved but momentum is not conserved even at the ensemble level, and thus it is not momentum conservation but energy conservation that is a more universal restriction for wavefunction collapse. Moreover, as we will see later, the conservation of energy may also hold true at the individual level for the collapse evolution of some special wave functions. 2Note that the existing arguments, which are based on some sort of combination of quantum theory and general relativity (see, e.g. Garay 1995 for a review), do not imply but only suggest that space and time are discrete. Moreover, the meanings and realization of discrete spacetime are also different in the existing models of quantum gravity. 2 collapse randomly happens in an energy branch jEii, and the probability of the branch Pi(t) changes to i Pi (t + tP ) = Pi(t) + ∆Pi; (2) where the superscript i denotes this tiny collapse event, and ∆Pi is a functional of Pi(t). Due to the conservation of probability, the probability of another 3 branch Pj(t)(j 6= i) correspondingly turns to be i Pj(t)∆Pi Pj (t + tP ) = Pj(t) − ; (3) 1 − Pi(t) where the superscript i still denotes this random event. The probability of this tiny collapse happening in the energy branch jEii at the instant is p(Ei; t) = Pi(t). Then we can work out the diagonal density matrix elements of the evo- lution4: m X j ρii(t + tP ) = p(Ej; t)Pi (t + tP ) j=1 X Pi(t)∆Pj(t) = Pi(t)[Pi(t) + ∆Pi] + Pj(t)[Pi(t) − ] 1 − Pj(t) j6=i X ∆Pj(t) = ρii(t) + Pi(t)[∆Pi − Pj(t) ]: (4) 1 − Pj(t) j6=i Here we shall introduce the first rule of dynamical collapse in our model, which says that the probability distribution of energy eigenvalues for an ensem- ble of identical systems is constant during the dynamical collapse process. It can be seen that this rule is entailed by the principle of energy conservation at the ensemble level. By this rule, we have ρii(t + tP ) = ρii(t) for any i. This leads to the following set of equations: X Pj(t)∆Pj(t) ∆P1(t) − = 0; 1 − Pj(t) j6=1 X Pj(t)∆Pj(t) ∆P2(t) − = 0; 1 − Pj(t) j6=2 ::: X Pj(t)∆Pj(t) ∆Pm(t) − = 0: (5) 1 − Pj(t) j6=m 3One can also obtain this result by first increasing the probability of one branch and Pi(t)+∆ then normalizing the probabilities of all branches. This means that Pi(t + tP ) = 1+∆ Pj (t) ∆ and Pj (t + tP ) = 1+∆ for any j 6= i. In this way, we have ∆Pi = 1+∆ (1 − Pi(t)) and ∆ ∆Pj = 1+∆ Pj (t) for any j 6= i. 4The density matrix describes the ensemble of states which arise from all possible random events. 3 By solving this equations set (e.g. by subtracting each other), we find the following relation for any i: ∆P i = k; (6) 1 − Pi(t) where k is an undetermined dimensionless quantity that relates to the state j (t)i. By using Eq. (6), we can further work out the non-diagonal density matrix elements of the evolution. But it is more convenient to calculate the following variant of non-diagonal density matrix elements: m X l l ρij(t + tP ) = p(El; t)Pi (t + tP )Pj (t + tP ) l=1 X = Pl(t)[Pi(t) − kPi(t)][Pj(t) − kPj(t)] l6=i;j +Pi(t)[Pi(t) + k(1 − Pi(t))][Pj(t) − kPj(t)] +Pj(t)[Pj(t) + k(1 − Pj(t))][Pi(t) − kPi(t)] 2 = (1 − k )ρij(t): (7) 1 Since the usual collapse time, τc, is defined by the relation ρij(τc) = 2 ρij(0), we may use a proper approximation, where k is assumed to be the same as its initial value during the time interval [0; τc], to simplify the calculation of the collapse time. Then we have: 2 n ρij(t) ≈ (1 − k ) ρij(0): (8) The corresponding collapse time is in the order of: 1 τ ≈ t ; (9) c k2 P In the following, we shall analyze the formula of k defined by Eq. (6). To begin with, the probability restricting condition 0 6 Pi(t) 6 1 for any i requires that 0 6 k 6 1. When k = 0, no collapse happens, and when k = 1, collapse happens instantaneously. Note that k cannot be smaller than zero, as this will lead to the negative value of Pi(t) in some cases. For instance, when k is jkj negative and Pi(t) < 1+jkj , Pi(t + tP ) = Pi(t) + k[1 − Pi(t)] will be negative and violate the probability restricting condition. That k is positive indicates that at each discrete instant only the probability of one branch increases and the probabilities of all other branches decrease, which is consistent with our previous assumption.