Path Integrals, Spontaneous Localisation, and the Classical Limit

Bhavya Bhatt; Manish Ram Chander; Raj Patil; Ruchira Mishra; Shlok Nahar; Tejinder P. Singh

doi:10.1515/zna-2019-0251

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Path Integrals, Spontaneous Localisation, and the Classical Limit

Bhavya Bhatt , Manish Ram Chander , Raj Patil , Ruchira Mishra , Shlok Nahar und Tejinder P. Singh

Veröffentlicht/Copyright: 22. November 2019

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Aus der Zeitschrift Zeitschrift für Naturforschung A Band 75 Heft 2

Abstract

The measurement problem and the absence of macroscopic superposition are two foundational problems of quantum mechanics today. One possible solution is to consider the Ghirardi–Rimini–Weber (GRW) model of spontaneous localisation. Here, we describe how spontaneous localisation modifies the path integral formulation of density matrix evolution in quantum mechanics. We provide two new pedagogical derivations of the GRW propagator. We then show how the von Neumann equation and the Liouville equation for the density matrix arise in the quantum and classical limit, respectively, from the GRW path integral.

Keywords: Path Integrals; Quantum Theory; Spontaneous Localisation

1 Introduction

Nonrelativistic quantum mechanics is a general framework for all systems moving at speeds negligible in comparison to the speed of light. The theory is immensely successful for having predicted phenomena, which have been experimentally verified extensively in the past 100 years. However, there are some fundamental questions about the theory that still remain unanswered.

One of these is the measurement problem, which is essentially about how the collapse of a state occurs and why is the outcome given by the Born rule [1]. Yet another is the absence of macroscopic superpositions. Quantum mechanics predicts that any system can be in a superposition of “states,” but strangely, the effect is not easy to see at large length scales. Today, there is an ongoing effort to study macroscopic superpositions experimentally as well as theoretically [2].

Several proposals have been suggested to address these problems. However, the only experimentally verifiable modifications are the spontaneous collapse theories [3], [4], where measurements are not considered special acts and are instead built into the evolution of the state. These theories are being experimentally tested by measuring the excess energy produced due to spontaneous localisation [5], [6], [7]. Recently, an anomalous energy gain was detected using ultracold cantilevers [8] whose origin remains to be understood. We look at the simplest of these, the Ghirardi–Rimini–Weber (GRW) model of quantum mechanics, from the path integral perspective.

First derived by Pearle and Soucek [9] in an alternative way, the GRW propagator is a generalisation of the Feynman propagator and accounts for the pertinent phenomenological modifications. In this article, we present two pedagogical derivations of this propagator, which we believe would be new additions to the literature. In particular, as we will see, the correction to the standard propagator amounts to adding a damping term to it. This has possible repercussions for applications of path integral to quantum field theory. In addition, these methods can easily be extended to systems obeying the Lindblad equation, which is ubiquitous in the study of open quantum systems as the GRW master equation is in Lindblad form. Thus, the applicability of this article is broader than just the GRW model, and it could improve our understanding of systems obeying the laws of standard quantum mechanics as well. We hope that our derivations would serve as an instructive source for the interested reader, beyond being a useful addition to the growing literature of collapse models.

1.1 Introducing the Model

The idea of spontaneous localisation, and collapse models in general, has been extensively studied in recent years as a possible approach to solve the quantum measurement problem and explain the absence of macroscopic position superpositions. This was first proposed by Pearle in the 1970s [3] and subsequently by other authors in [10] and generalised to the case of identical particles in the CSL model [11]. The proposal is that every quantum object in nature undergoes spontaneous localisation to a region of size r_c, at random times given by a Poisson process with a mean collapse rate λ. Between every two collapses, the wave function obeys Schrödinger evolution. The collapse rate can be shown to be proportional to the number N of nucleons in the object, and we write λ=NλGRW, where λ_GRW is the collapse rate for a nucleon. Thus, λ_GRW and r_C are two new constants of nature, whose values must be fixed by experiment. Formally, the two postulates of the GRW model are stated as follows:

Postulate 1. Given the wave function ψ (x₁, x₂,…,x_N) of an N particle quantum system in the Hilbert space ℒ2(R3N), the n^th particle undergoes spontaneous localisation to a random position x as described by the following jump operator:

(1)ψt(x1,x2,…xN)→Ln(x)ψt(x1,x2,…xN)∥Ln(x)ψt(x1,x2,…xN)∥

The jump operator Ln(x) is a linear operator, which is defined to be the normalised Gaussian:

(2)Ln(x)=1(πrC2)3/4e−(q^n−x)2/2rC2

Here, q^n is the position operator for the n^th particle of the system, and the random variable x is the spatial position to which the jump occurs. r_C, which is the width of the Gaussian, is a new constant of nature.

The probability density for the n^th particle to jump to the position x is assumed to be given by:

(3)pn(x) ≡ ∥Ln(x)ψt(x1,x2,…xN)∥2

Also, it is assumed that the jumps are distributed in time as a Poissonian process with frequency λ_GRW. This is the second new constant of nature, in the model.

For an unentangled wave function, we may write ψt(x1,x2,…xN)=∏nϕn(xn), where ϕn(xn) is the wave function for n^th particle. Therefore, we have

(4)pn(x)=∫d3xn|Ln(x)ϕn(xn)|2=∫d3xn[l(x,xn)]2|ϕn(xn)|2

where l(x,xn) is the position representative of the operator Ln(x), a Gaussian localised at x. Because it is an operator on n^th particle’s space, integrals over all other degrees of freedom are trivial. Further note that

(5)∫d3x[l(x,x′)]2=1

This ensures that ∫pn(x)d3x=1. On the surface of the definition, this may not be obvious. The result follows similarly for the entangled states.

Postulate 2. In between any two successive jumps, the wave function evolves according to the Schrödinger equation.

With these postulates, we can calculate the evolution of the density matrix that represents the state of the system as [4]

(6)ddtρ(t)=−iℏ[H,ρ(t)]−λ(ρ(t)−∫d3xLn(x)ρLn(x))

This can be rewritten in Lindblad form as

(7)ddtρ(t)=−iℏ[H,ρ(t)]+∫d3xλ(Ln(x)ρLn(x)−12{[Ln(x)]2,ρ(t)})

Thus, the GRW equation is an example of a Lindblad equation, and the following methods to derive the path integral can also be used to derive the path integral for any open quantum system satisfying the Lindblad master equation.

For the above model, the process of spontaneous localisation serves to provide an exponential damping of the exponential oscillations in the path integral amplitude. Inevitably, the damping is important for macroscopic systems, but insignificant for microscopic ones.

2 The GRW Path Integral and Its Derivation

The path integral formulation of quantum mechanics is a description of quantum theory that generalises the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. As mentioned in the Introduction, the GRW path integral has been previously derived by Pearle and Soucek [9]; here we give two alternative derivations of their result and then discuss the classical and quantum limits of the GRW path integral. (For further applications of path integrals to collapse models, see also [12], [13], [14]).

2.1 Method 1

2.1.1 Introduction

Standard techniques [15] can be used to derive the propagator starting from the Schrödinger equation. However, these techniques cannot directly be used for mixed states represented by density matrices. Hence, we first purify the state vector [16] so that it obeys Schrödinger-like evolution with an effective Hamiltonian. The methods followed in [15] can then be directly applied to this pure state ket. Such a method of purification of a density matrix to simplify its treatment can also be used when deriving the path integral of a more general class of open quantum system as described in the Introduction as this method does not depend on any property of the GRW equation other than the fact that it preserves trace.

2.1.2 Getting the Hamiltonian Form

The GRW master equation for a single particle [4], [11] is

(8)dρdt=−iℏ(Hρ−ρH)−λ(ρ−∫d3rLrρLr)

where H is the Hamiltonian for Schrödinger evolution of the system, and

(9)Lr=1𝒩exp(−(q^−r)22rC2)

is the collapse operator for the particle to localise around r. λ is the collapse rate, and r_C is the length scale to which localisation takes place, as defined in the Introduction. This master equation was first derived for the CSL model [11], where the authors noted that for the one particle case this equation is the same as for the GRW model, although this is not true in general.

In order to convert (8) into an equation of the form

(10)d|ψ⟩dt=−iℏH~|ψ⟩

we define |ψ⟩ as

(11)|ψ⟩=∑m,nρmn|m⟩⊗|n⟩

where ρmn=⟨m|ρ|n⟩ are elements of the density matrix ρ from (8). Here, the set of all |m⟩,|n⟩ forms an orthonormal basis in the single particle Hilbert space. We note that there is an isomorphism between |ψ⟩ as defined here and ρ. Thus, knowing the evolution of |ψ⟩ would give us all the information about how ρ would evolve. Using Einstein’s summation convention, we rewrite (8) as

(12)dρmndt=−iℏ(Hmaρan−ρmaHan)−λ(ρmn−∫d3rLrmaρabLrbn)

From (10) and (11), it follows that the equation

(13)dρmndt=−iℏH~mabnρab

must also hold. Comparing (12) and (13), we get

(14)H~=(H⊗𝕀−𝕀⊗HT)−(iℏλ𝕀⊗𝕀)+iℏλ∫d3rLr⊗LrT

So, |ψ(t)⟩ evolves as

(15)|ψ(t)⟩=exp(−iH~t/ℏ)|ψ(0)⟩

This gives us the evolution of ρ(t) via (11), and the above equation can be used to derive the propagator and the path integral.

2.1.3 Derivation of the Path Integral

The total time t = T can be divided into N intervals such that ϵ = T/N, and the finite time propagator in (15) can be written as

(16)U=[exp(−iϵℏ(H⊗𝕀−𝕀⊗HT)−λϵ(𝕀⊗𝕀−∫d3rLr⊗LrT))]N

As N → ∞ and ϵ → 0, we can make the approximation

(17)U≈[exp(−iϵℏ(H⊗𝕀−𝕀⊗HT))×exp(−λϵ(𝕀⊗𝕀−∫d3rLr⊗LrT))]N

Introducing resolution of the identity

(18)∫−∞∞dxkϵdykϵ|xkϵ⟩|ykϵ⟩⟨xkϵ|⟨ykϵ|

between every time step, we get N terms, each of the form

(19)⟨xkϵ,ykϵ|exp[−iϵℏ(H⊗𝕀−𝕀⊗HT)]×exp[−λϵ(𝕀⊗𝕀−∫d3rLr⊗LrT)]|x(k−1)ϵ,y(k−1)ϵ⟩

Evaluating one of these terms

(20)⟨xkϵ,ykϵ|exp[−iϵℏ(H⊗𝕀−𝕀⊗HT)]×exp[−λϵ(𝕀⊗𝕀−∫d3rLr⊗LrT)]|x(k−1)ϵ,y(k−1)ϵ⟩=⟨xkϵ,ykϵ|exp[−iϵℏ(H⊗𝕀−𝕀⊗HT)]|x(k−1)ϵ,y(k−1)ϵ⟩exp[−λϵ(1−exp−(x(k−1)ϵ−y(k−1)ϵ)24rC2)]

The first exponent is simply the Feynman propagator^[1] for Schrödinger evolution.^[2] We now assume that the system is nonrelativistic, and hence, Hamiltonian is quadratic in the momentum, and the potential is position dependent. Thus, taking all N terms, we get

(21)U(xNϵ,yNϵ,x0,y0)=m2πϵℏ×∫−∞∞∏n=1N−1m2πϵℏdxnϵdynϵ×exp[∑n=1N(im2ℏϵ((xnϵ−x(n−1)ϵ)2−(ynϵ−y(n−1)ϵ)2))−iϵℏ(V(x(n−1)ϵ)−V(y(n−1)ϵ))]×exp[∑k=1N−λϵ(1−exp−(x(k−1)ϵ−y(k−1)ϵ)24rC2)]

(22)=m2πϵℏ∫−∞∞∏n=1N−1m2πϵℏdxnϵdynϵ×exp[∑n=1Niℏ(S[xnϵ,x(n−1)ϵ]−S[ynϵ,y(n−1)ϵ])]×exp[∑k=1N−λϵ(1−exp−(x(k−1)ϵ−y(k−1)ϵ)24rC2)]

In the continuum limit with N → ∞ while still keeping Nϵ=T, the evolution of the density matrix element thus becomes

(23)ρ(xT,yT,T)=∫allpaths[Dxt][Dyt]×exp(iℏ(S[xt,T,t=0]−S[yt,T,t=0]))×exp[−λ∫0Tdt(1−exp−(xt−yt)24rC2)]ρ(x0,y0)dx0dy0

where

(24)∫[Dxt]=limN→∞(m2πℏϵ)1/2∫∏n=1N−1(m2πℏϵ)1/2dxn

This is the same result as derived in [9]. Here, x_t and y_t can be understood as individual paths that might be traversed. Thus, ∫allpaths[Dxt][Dyt] can be understood as an integral over all such paths. The exponential in the second line of the above equation serves as the GRW-induced regulator of the Feynman path integral and improves the understanding of the classical limit, as we will see in the following section.

2.2 Method 2

2.2.1 Introduction

In this case, we use a more physically motivated approach. We use the fact that after every time interval ϵ the wave function has a probability λϵ to collapse. Thus, by taking discrete time steps and using the above fact, we can derive the propagator.

2.2.2 Derivation of the Path Integral

Consider ρ(x0,y0,t=0) to be a density matrix at initial time t = 0. We intend to find ρ(xT,yT,T) at final time t = T. We divide the total time into smaller intervals such that ϵ=TN. So, we have

(25)ρ(xϵ,yϵ,ϵ)=A∫−∞∞∫−∞∞exp[iℏ(m2(xϵ−x0ϵ)2−V(xϵ+x02))ϵ]exp[−iℏ(m2(yϵ−y0ϵ)2−V(yϵ+y02))ϵ]ρ(x0,y0,t=0)dx0dy0

(26)=A∫−∞∞∫−∞∞𝒫1ρ(x0,y0,t=0)dx0dy0

where

(27)𝒫i=Kρ(xi,yi,ϵi;xi−1,yi−1,ϵi−1)=exp[iℏ(m2(xi−xi−1ϵ)2−V(xi+xi−12))ϵ]

(28)exp[−iℏ(m2(yi−yi−1ϵ)2−V(yi+yi−12))ϵ]

is density matrix propagator for infinitesimal time step from ϵ_{i − 1} to ϵi=ϵi−1+ϵ. The above expression for ρ(xϵ,yϵ,ϵ) represents standard Schrödinger evolution,^[3] where A is the appropriate normalisation constant to recover von Neumann equation. Now from (8), we know that at a given instance, say, t = ϵ, the probability of collapse is λϵ, while that of it evolving according to Schrödinger equation is 1−λϵ. Thus, the new density matrix after ϵ time becomes

(29)ρnew(xϵ,yϵ,ϵ)=(1−λϵ)ρ1+λϵ∫−∞∞Lr(xϵ)ρ1Lr(yϵ)dr

where ρ1=ρ(xϵ,yϵ,ϵ) and Lr(xϵ)=⟨xϵ|Lr|xϵ⟩ are as defined in (9). Here, as ρ₁ does not depend on r (it is a function of xϵ,yϵ, x₀, and y₀), we can evaluate the above integral by taking ρ₁ outside the integration. We get

(30)∫−∞∞Lr(xϵ)ρ1Lr(yϵ)dr=(∫−∞∞Lr(xϵ)Lr(yϵ)dr)ρ1

(31)=[∫−∞∞exp(−(xϵ−r)22rC2)exp(−(yϵ−r)22rC2)dr]ρ1

(32)=exp[−(xϵ−yϵ)24rC2]ρ1

Now, we can write^[4]

(33)ρnew(xϵ,yϵ,ϵ)=(1−λϵ)ρ1+λϵ[exp(−(xϵ−yϵ)24rC2)]ρ1

For simplicity, we write

Gi=exp[−(xiϵ−yiϵ)24rC2]

and thus

(34)ρnew(xϵ,yϵ,ϵ)=[(1−λϵ)+λϵG1]ρ1

We propagate again according to Schrödinger equation from time t = ϵ to time t = 2ϵ,

(35)ρ(x2ϵ,y2ϵ,2ϵ)=A∫−∞∞𝒫2[(1−λϵ)+λϵG1]ρ1dxϵdyϵ

Substituting ρ₁ according to (25) and writing new ρnew(x2ϵ,y2ϵ,2ϵ), we get

(36)ρnew(x2ϵ,y2ϵ,2ϵ)=A2((1−λϵ)+λϵG2)∫−∞∞∫−∞∞𝒫2((1−λϵ)+λϵG1)∫−∞∞∫−∞∞𝒫1ρ(x0,y0,t=0)dx0dy0dxϵdyϵ

Further, we can rearrange the terms as all the functions 𝒫1,𝒫2,G1, and G₂ are commutative operations, so this gives us

ρnew(x2ϵ,y2ϵ,2ϵ)=A2∫−∞∞∫−∞∞∫−∞∞∫−∞∞𝒫2𝒫1((1−λϵ)+λϵG2)((1−λϵ)+λϵG1)

(37)ρ(x0,y0,t=0)dx0dy0dxϵdyϵ

We repeat the above procedure N − 1 times. Taking continuum limit, N → ∞ gives us the final density matrix as

(38)ρ(xT,yT,T)=limN→∞AN−1∫⋯∫∏i=0N−1𝒫i∏i=0N−1((1−λϵ)+λϵGi)ρ(x0,y0,t=0)dx0dy0⋯dx(N−1)ϵdy(N−1)ϵ

We know that

(39)limN→∞∏i=1N𝒫i=limN→∞∏i=1Nexp{iℏ[m2(xiϵ−x(i−1)ϵϵ)2−V(xiϵ+x(i−1)ϵ2)]ϵ}exp{−iℏ[m2(yiϵ−y(i−1)ϵϵ)2−V(yiϵ+y(i−1)ϵ2)]ϵ}

(40)=exp{limN→∞∑i=1Niℏ[m2(xiϵ−x(i−1)ϵϵ)2−V(xiϵ+x(i−1)ϵ2)]ϵ}exp{limN→∞∑i=1N−iℏ[m2(yiϵ−y(i−1)ϵϵ)2−V(yiϵ+y(i−1)ϵ2)]ϵ}

(41)=exp{iℏ∫0TL(xt)dt}exp{−iℏ∫0TL(yt)dt}

(42)=exp{iℏ(S(xt,T,t=0)−S(yt,T,t=0))}

where L(xt,T,t=0) is the Lagrangian and S(xt,T,t=0) the action thus obtained. Expanding the second product term gives us

(43)limN→∞∏i=1N((1−λϵ)+λϵGi)=limN→∞(1−λϵ)N∑k=0∞(∑i=1NλGiϵ)kk!(1−λϵ)k

(44)=exp(−λT)exp(limN→∞∑i=1NλGiϵ)

(45)=exp(−λT)exp(λ∫0TG(t)dt)

(46)=exp(−λT)exp[λ∫0Texp(−(xt−yt)24rC2)dt]

Substituting these two terms back in (38), we get an integral form solution of (8) given by

(47)ρ(xT,yT,T)=∫exp(iℏ(S[xt,T,t=0]−S[yt,T,t=0]))exp(−λ∫0T(1−exp{−(xt−yt)24rC2})dt)[Dxt][Dyt]ρ(x0,y0,t=0)dx0dy0

where the integral in the above equation is defined in (24). The above derived propagator is the same as what we got using the previous method given in (23).

3 Classical and Quantum Limits of GRW Path Integral

3.1 Quantum Limit

From (23) or (47), the path integral for the GRW model is written as

(48)ρ(xT,yT,T)=∫allpathsexp[iℏ(S(xt,T,t=0)−S(yt,T,t=0))]exp[−λ∫0T(1−e−(xt−yt)24rC2)dt]ρ(x0,y0,t=0)[Dxt][Dyt]dx0dy0

If we consider the limit λT→0, i.e. we look at the system at timescales (t = T) much smaller than the time period of collapse (τ=1/λ), then the nonoscillating part of the above given propagator could be approximated as

(49)exp[−λ∫0T(1−e−(xt−yt)24rC2)dt]≈1

This makes the propagator of GRW look exactly like that for normal quantum mechanics,

(50)ρ(xT,yT,T)=∫all pathsexp[iℏ(S(xt,T,t=0)−S(yt,T,t=0))]ρ(x0,y0,t=0)[Dxt][Dyt]dx0dy0

From here, the standard quantum mechanical result follows easily – we recall the calculation here, for sake of completeness. We can write the above equation for infinitesimal time interval ϵ as

(51)ρ(xϵ,yϵ,ϵ)=A∫exp[iℏ∫0ϵ(mx˙22−V(x))dt]exp[−iℏ∫0ϵ(my˙22−V(y))dt]ρ(x0,y0,t=0)dx0dy0

where A is as defined in the previous section. Using the following finite difference substitution

x˙→xϵ−x0ϵx→x0+xϵ2

and using the standard substitution of ηx=x0−xϵ and ηy=y0−yϵ and rearranging the terms, we have

(52)ρ(xϵ,yϵ,ϵ)=A∫∫eiℏmηx22ϵe−iℏmηy22ϵexp[iℏ(−V(x)+V(y))ϵ]ρ(xϵ+ηx,yϵ+ηy,t=0)dηxdηy

The exponentials oscillate very rapidly as ϵ could be made arbitrarily small. When such a rapidly oscillating function multiplies a smooth function, the integral vanishes for the most part due to the random phase of the exponential. Just as in the case of the path integration, the only substantial contribution comes from the region where the phase is stationary. The region of constructive interference is

(53)mη22ℏϵ⩽π

Now, Taylor expanding the terms in (52) up to the first order in ϵ, i.e. up to order η², we get

(54)ρ(xϵ,yϵ,ϵ)=A∫∫eiℏmηx22ϵe−iℏmηy22ϵ(1−iℏV(x)ϵ+iℏV(y)ϵ)ρ(xϵ+ηx,yϵ+ηy,t=0)dηxdηy=A∫∫eiℏmηx22ϵe−iℏmηy22ϵ(1−iℏV(x)ϵ+iℏV(y)ϵ)(ρ(xϵ,yϵ,t=0)+∂ρ∂y|(xϵ,yϵ,t=0)ηy+∂ρ∂x|(xϵ,yϵ,t=0)ηx+∂2ρ2∂y2|(xϵ,yϵ,t=0)ηy2+∂2ρ2∂x2|(xϵ,yϵ,t=0)ηx2)dηxdηy=A∫∫eiℏmηx22ϵe−iℏmηy22ϵ(ρ(xϵ,yϵ,t=0)−iℏV(x)ρ(xϵ,yϵ,t=0)ϵ+iℏV(y)ρ(xϵ,yϵ,t=0)ϵ+∂2ρ2∂y2|(xϵ,yϵ,t=0)ηy2+∂2ρ2∂x2|(xϵ,yϵ,t=0)ηx2+∂ρ∂y|(xϵ,yϵ,t=0)ηy+∂ρ∂x|(xϵ,yϵ,t=0)ηx)dηxdηy

Evaluating the Gaussian integral and using A=−2ϵℏπim2ϵℏπim, we get

(55)ρ(xϵ,yϵ,ϵ)=ρ(xϵ,yϵ,t=0)−iℏV(x)ρ(xϵ,yϵ,t=0)ϵ+iℏV(y)ρ(xϵ,yϵ,t=0)ϵ+−iℏ2m∂2ρ∂y2|(xϵ,yϵ,t=0)ϵ+iℏ2m∂2ρ∂x2|(xϵ,yϵ,t=0)ϵ=ρ(xϵ,yϵ,t=0)−iℏ[H,ρ]ϵ

which describes how a density operator evolves in time:

(56)dρdt=−iℏ[H,ρ]

The above equation is the von Neumann equation, and it describes the statistical state of a system in quantum mechanics. We refer to the above equation as the statistical quantum limit of GRW model.

3.2 Classical Limit

The following analysis is previously done by Ajanapon [17] for the propagator of the density matrix in standard quantum mechanics. We here make use of the same analysis for the propagator of the GRW model. From (23) or (47), the path integral for GRW model could be written as

(57)ρ(xT,yT,T)=∫all pathsexp[iℏ(S(xt,T,t=0)−S(yt,T,t=0))]exp[−λ∫0T(1−e−(xt−yt)24rC2)dt]ρ(x0,y0,t=0)[Dxt][Dyt]dx0dy0

Now we consider the limit λT≫1, which could be interpreted as waiting for a sufficiently long time, or the collapse rate λ for the system is sufficiently large. Large λ implies large mass as the collapse rate is directly proportional to the number of entangled particles in the system. As a result, large λ implies large action. On the other hand, large time also results in large action. As a result, large masses and large times are both representatives of classical limit, which causes S to be large and thus implies the limit S ≫ ℏ.

When a rapidly oscillating function is multiplied with a smooth function, then the integral of their product could be approximated by the smooth function at the stationary point of the rapidly oscillating function. This is commonly called the stationary phase approximation. Here, xtcl and ytcl are the stationary paths for S(xt,T,t=0) and S(yt,T,t=0), respectively, in the limit S ≫ ℏ. Thus, the stationary phase approximation leads us to the following equation:

(58)ρ(xT,yT,T)=∫exp[iℏ(S(xtcl,T,t=0)−S(ytcl,T,t=0))]exp[−λ∫0T(1−e−(xtcl−ytcl)24rC2)dt]ρ(x0,y0,t=0)dx0dy0

For brevity, we here drop the notation for stationary paths and use xtcl=xt and ytcl=yt. The ρ(xT,yT,T) in the above expression represents diagonal as well as off-diagonal terms in position basis [9]. Now we look for the off-diagonal terms of the final ρ, which are specified by large (xt−yt). In the limit (xt−yt)≫rC, the nonoscillating part of the propagator could be approximated as

(59)exp[−λ∫0T(1−e−(xt−yt)24rC2)dt]≈exp[−λT]

This leads to damping of the off-diagonal terms of the density matrix. Thus, in the limit λT≫1, the integral can be considered to be vanished. This could also be interpreted as destruction of interference in the system as the off-diagonal terms are the primary representatives of interference. Now let us consider the diagonal terms of the final ρ, specified by (xt−yt)≈0. In the limit (xt−yt)≪rC, the nonoscillating part of the propagator could be approximated as

(60)exp[−λ∫0T(1−e−(xt−yt)24rC2)dt]≈1

Now, we consider an infinitesimal time step ϵ.

(61)S(xt,ϵ,t=0)−S(yt,ϵ,t=0)=m2ϵ2(xϵ−x0)2ϵ−12[V(xϵ)+V(x0)]ϵ−m2ϵ2(yϵ−y0)2ϵ+12[V(yϵ)+V(y0)]ϵ

(62)=mϵ[12(xϵ+yϵ)−12(x0+y0)][(xϵ−yϵ)−(x0−y0)]−ϵ2[V(xϵ)−V(yϵ)]−ϵ2[V(x0)−V(y0)]

Motivated by the above expression, we implement the following change of variables:

(63)q¯t=12(xt+yt)

(64)Δt=(xt−yt)

(65)U(q¯t,Δt)=V(q¯t+12Δt)−V(q¯t−12Δt)

Thus, (58) could be written as

(66)ρ(q¯ϵ,Δϵ,ϵ)=A∫exp[iℏ(mϵ(q¯ϵ−q¯0)(Δϵ−Δ0)−ϵ2U(q¯ϵ,Δϵ)−ϵ2U(q¯0,Δ0))]ρ(q¯0,Δ0,t=0)dx0dy0

As the state of a system is specified by position and momentum in classical mechanics, we take the Fourier transform of Δ as given by

(67)ρ(q¯t,pt,t)=A∫e(−iptΔt)ρ(q¯t,Δt,t)dΔt

Thus, (66) in terms of p_t could be written as

(68)ρ(q¯ϵ,pϵ,ϵ)=A∫exp[iℏ(Δ0p0−Δϵpϵ+mϵ(q¯ϵ−q¯0)(Δϵ−Δ0)−ϵ2U(q¯ϵ,Δϵ)−ϵ2U(q¯0,Δ0))]ρ(q¯0,p0,t=0)dΔ0dΔϵdx0dy0

The ρ(q¯t,pt,t) could be interpreted as the phase space representation of the diagonal terms of the density matrix in the limit S ≫ ℏ. As the Δϵ≪rC, U(q¯t,Δt) could be approximated by Taylor expanding and ignoring Δt2 and its higher orders

(69)U(q¯t,Δt)≈Δt∂V∂q(q¯t)

Equation (68) could be further simplified by using the above approximation,

(70)ρ(q¯ϵ,pϵ,ϵ)=1N′∫exp[iΔ0ℏ(p0−mϵ(q¯ϵ−q¯0)−ϵ2∂V∂q(q¯0))]exp[−iΔϵmℏϵ(q¯0−q¯ϵ+ϵmpϵ+ϵ22m∂V∂q(q¯ϵ))]ρ(q¯0,p0,t=0)dΔ0dΔϵdx0dy0

(71)=1N′′∫δ(p0−mϵ(q¯ϵ−q¯0)−ϵ2∂V∂q(q¯0))δ(q¯0−q¯ϵ+ϵmpϵ+ϵ22m∂V∂q(q¯ϵ))ρ(q¯0,p0,t=0)dx0dy0

(72)=1N′′′ρ(q¯ϵ−ϵmpϵ,pϵ+ϵ∂V∂q(q¯ϵ),t=0)

The above equation could also be written as follows by changing the variables of ρ,

(73)ρ(q¯0+ϵmp0,p0−ϵ∂V∂q(q¯0),t=ϵ)=1N′′′ρ(q¯0,p0,t=0)

Now, Taylor expanding the left-hand side around the point (q₀, p₀, t = 0) and equating orders of ϵ, we get, at zeroth order,

(74)N′′′=1

at first order,

(75)∂ρ∂t|(q¯0,p0,t=0)=−p0m∂ρ∂q¯|(q¯0,p0,t=0)+∂V∂q¯|(q¯0)∂ρ∂p|(q¯0,p0,t=0)

and dropping the subscript,

(76)∂ρ∂t=−{ρ,H}

where H=12mp2+V(q¯). We refer to (76) as being the statistical classical limit of GRW. The above limit does not depend on a specific form of the initial density matrix and hence is a phase space representation of a general density matrix following GRW evolution.

3.3 Absence of Macroscopic Position Superpositions

To summarise the discussion this far, we first developed a path integral formulation of the GRW model. We then showed that this gives us the correct quantum and classical limits. We shall now illustrate some important features of the classical limit through some examples. As we are taking the classical limit, we would consider large action and large number of nucleons (which implies large λ). Hence, the stationary phase approximation shown in (58) would be valid. If we consider the case of a free particle, the stationary paths would be straight lines with x˙(t)=constant.

Let us consider an initial condition that is formed by the superposition of two Gaussians separated by a macroscopic distance |a1−a2|≫rC. The resulting density matrix would be

ρ(x0,y0,t=0)=∑i,j=12Aije−(x0−ai)2r2e−(y0−aj)2r2

with r≪rC. Here, the coefficients A_ij can be chosen such that the density matrix is a valid one (i.e. it has unit trace, it is positive semidefinite, and it is Hermitian). Putting this into (58), we get

(77)ρ(xt,yt,T)=∫exp[−λ∫0T(1−e−(xtcl−ytcl)24rC2)dt]exp[iℏ(S(xtcl,T,=0)−S(ytcl,T,t=0))]∑i,j=12Aijexp[−(x0−ai)2r2]exp[−(y0−aj)2r2]dx0dy0

We can see that the terms of the initial density matrix

A12e−(x0−a1)2r2e−(y0−a2)2r2+A21e−(x0−a2)2r2e−(y0−a1)2r2

would have |xtcl−ytcl|≫rC for a large time. Hence, the final density matrix would have these terms damped exponentially as

exp(−λ∫0T(1−e−(xtcl−ytcl)24rC2)dt)≈e−λT

Additionally, in the remaining terms where both paths start in the same Gaussian, the paths must finally also remain within a distance, which is of the order r_C. Thus, the so-called off-diagonal terms are destroyed, while the approximately diagonal terms are preserved. Note that the system transforms from a state with the superposition of two Gaussians to a statistical ensemble of the two Gaussians with probabilities A₁₁ and A₂₂, respectively. Note also that this statistical ensemble is different from a superposition as this represents classical probabilities, which do not interfere. In this way, GRW destroys macroscopic superpositions.

4 Discussion and Conclusion

In our work, we have derived the GRW propagator in two new ways. As mentioned in the Introduction, the GRW propagator amounts to adding a damping term to the standard propagator that destroys macroscopic superpositions. We note that in this approach the transition from GRW to classical and quantum mechanics is quite naturally obtained. In order to see the transition to standard quantum mechanics, we took the limit λT≪1 of the path integral for the GRW model and were left with quantum mechanics for a density matrix, i.e. the von Neumann equation. In order to see the transition to classical mechanics, we took the limit λT≫1 in the path integral for the GRW model and were left with the classical Liouville equation.

Our study suggests methods for generalising spontaneous localisation to the relativistic case, via the path integral representation of quantum field theory. What we see in (23) is that spontaneous localisation is equivalent to modifying the standard path integral by a regulator. In relativistic quantum field theory, we replace space–time coordinates by quantum fields over space–time, so that the action function S(x,T) is replaced by a functional, S(ϕ(x,T)). We propose to introduce a regulator, analogous to the one introduced in the present article, and investigate how it might incorporate spontaneous localisation in quantum field theory.

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Received: 2019-07-31

Accepted: 2019-10-24

Published Online: 2019-11-22

Published in Print: 2020-02-25

Artikel in diesem Heft

Frontmatter
Atomic, Molecular & Chemical Physics
Numerical Investigation of the Cooling Temperature of the InGaP/InGaAs/Ge Subcells Under the Concentrated Illumination
Dynamical Systems & Nonlinear Phenomena
Ion-Acoustic Cnoidal Waves with the Density Effect of Spin-up and Spin-down Degenerate Electrons in a Dense Astrophysical Plasma
Hydrodynamics
Landau Quantised Modification of Rayleigh–Taylor Instability in Dense Plasmas
Interaction of a Singular Surface with a Characteristic Shock in a Relaxing Gas with Dust Particles
Quantum Theory
Path Integrals, Spontaneous Localisation, and the Classical Limit
Proposal for a New Quantum Theory of Gravity III: Equations for Quantum Gravity, and the Origin of Spontaneous Localisation
Quantum-Phase-Field: From de Broglie–Bohm Double-Solution Program to Doublon Networks
Solid State Physics & Materials Science
Photovoltaic Generator Based on Laser-Induced Reversible Aggregation of Magnetic Nanoparticles
Thermodynamics & Statistical Physics
Investigation of the Finite Size Properties of the Ising Model Under Various Boundary Conditions

https://doi.org/10.1515/zna-2019-0251

Schlagwörter für diesen Artikel

Path Integrals; Quantum Theory; Spontaneous Localisation

Artikel in diesem Heft

Frontmatter
Atomic, Molecular & Chemical Physics
Numerical Investigation of the Cooling Temperature of the InGaP/InGaAs/Ge Subcells Under the Concentrated Illumination
Dynamical Systems & Nonlinear Phenomena
Ion-Acoustic Cnoidal Waves with the Density Effect of Spin-up and Spin-down Degenerate Electrons in a Dense Astrophysical Plasma
Hydrodynamics
Landau Quantised Modification of Rayleigh–Taylor Instability in Dense Plasmas
Interaction of a Singular Surface with a Characteristic Shock in a Relaxing Gas with Dust Particles
Quantum Theory
Path Integrals, Spontaneous Localisation, and the Classical Limit
Proposal for a New Quantum Theory of Gravity III: Equations for Quantum Gravity, and the Origin of Spontaneous Localisation
Quantum-Phase-Field: From de Broglie–Bohm Double-Solution Program to Doublon Networks
Solid State Physics & Materials Science
Photovoltaic Generator Based on Laser-Induced Reversible Aggregation of Magnetic Nanoparticles
Thermodynamics & Statistical Physics
Investigation of the Finite Size Properties of the Ising Model Under Various Boundary Conditions