Path Integrals, Spontaneous Localisation, and the Classical Limit

Home , Measurement problem, Open quantum system

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1 Introduction 3 1.1 Introducingthemodel ...... 3

2 The GRW path integral and its derivation 5 2.1 Method-1 ...... 5 2.1.1 Introduction...... 5 2.1.2 Getting the Hamiltonian Form ...... 5 2.1.3 DerivationofthePathIntegral ...... 6 2.2 Method-2 ...... 8 2.2.1 Introduction...... 8 2.2.2 DerivationofthePathIntegral ...... 8

3 Classical and Quantum Limits of GRW path integral 11 3.1 QuantumLimit ...... 11 3.2 ClassicalLimit ...... 13 3.3 Absence of macroscopic position superpositions ...... 15

4 Discussion and Conclusion 16

2 1 Introduction

Non-Relativistic 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 hundred 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. QM 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 ultra-cold 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 generalization of the Feynman propagator and accounts for the pertinent phenomenological modifications. In this paper, 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 paper 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 localization, 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 rc, at random times given by a Poisson process with a mean collapse rate λ. Between every two collapses, the wave function obeys Schr¨odinger 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 λGRW is the

3 collapse rate for a nucleon. Thus, λGRW and rC 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, ..., xN) of an N particle quantum system in the Hilbert space L2(R3N ), the n-th particle undergoes spontaneous localization to a random position x as described by the following jump operator:

Ln(x)ψt(x1, x2,... xN ) ψt(x1, x2,... xN ) −→ (1) kLn(x)ψt(x1, x2,... xN )k

The jump operator Ln(x) is a linear operator which is deﬁned to be the normalised Gaussian: 1 ˆq x 2 2 x −( n− ) /2rC Ln( )= 2 3/4 e (2) (πrC )

Here, ˆqn 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. rC , 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: 2 pn(x) ≡ kLn(x)ψt(x1, x2,... xN )k (3) Also, it is assumed that the jumps are distributed in time as a Poissonian process with fre-

quency λGRW. This is the second new constant of nature, in the model.

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

3 2 3 2 2 pn(x)= d xn|Ln(x)φn(xn)| = d xn[l(x, xn)] |φn(xn)| (4) Z Z where l(x, xn) is the position representative of the operator Ln(x), a Gaussian localised at x. Because it is an operator on nth particle’s space, integrals over all other degrees of freedom are trivial. Further note that, d3x[l(x, x′)]2 =1 (5) Z 3 This ensures that pn(x)d x = 1. On the surface of the deﬁnition, this may not be obvious. The result follows similarly for the entangled states. R Postulate 2. In between any two successive jumps, the wave function evolves according to the Schr¨odinger equation.

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

d i ρ(t)= − [H, ρ(t)] − λ(ρ(t) − d3xL (x)ρL (x)) (6) dt ~ n n Z

4 This can be rewritten in Lindblad form as d i 1 ρ(t)= − [H, ρ(t)] + d3xλ(L (x)ρL (x) − {[L (x)]2, ρ(t)}) (7) dt ~ n n 2 n Z 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 generalizes 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 an 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 [11, 4] is

dρ i = − (Hρ − ρH) − λ ρ − d3rL ρL (8) dt ~ r r Z where H is the Hamiltonian for Schr¨odinger evolution of the system and

1 (ˆq − r)2 L = exp − (9) r N 2r2 C 5 is the collapse operator for the particle to localize around r. λ is the collapse rate, and rC is the length scale to which localization 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 Eq. (8) into an equation of the form d |ψi i = − H˜ |ψi (10) dt ~ we define |ψi as

|ψi = ρmn |mi⊗|ni (11) m,n X where ρmn = hm| ρ |ni are elements of the density matrix ρ from Eq. (8). Here the set of all |mi , |ni form an orthonormal basis in the single particle Hilbert space. We note that there is an isomorphism between |ψi as deﬁned here, and ρ. Thus, knowing the evolution of |ψi would give us all the information about how ρ whould evolve. Using Einstein’s summation convention, we rewrite Eq. (8) as, dρ i mn = − (H ρ − ρ H ) − λ ρ − d3rL ρ L (12) dt ~ ma an ma an mn rma ab rbn Z From Eq. (10) and Eq. (11), it follows that the equation dρ i mn = − H˜ ρ (13) dt ~ mabn ab must also hold. Comparing Eq. (12) and Eq. (13) we get

˜ I I T ~ I I ~ 3 T H =(H ⊗ − ⊗ H ) − (i λ ⊗ )+ i λ d rLr ⊗ Lr (14) Z So, |ψ(t)i evolves as |ψ(t)i = exp −iHt/˜ ~ |ψ(0)i (15) This gives us the evolution of ρ(t) via Eq. (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 ǫ = T/N and the ﬁnite time propagator in Eq. (15) can be written as

N −iǫ I I T I I 3 T U = exp ~ H ⊗ − ⊗ H − λǫ ⊗ − d rLr ⊗ Lr (16) Z As N →∞ and ǫ → 0, we can make the approximation

N −iǫ I I T I I 3 T U ≈ exp ~ (H ⊗ − ⊗ H ) × exp −λǫ( ⊗ − d rLr ⊗ Lr ) (17) Z 6 Introducing resolution of the identity

∞ dxkǫdykǫ |xkǫi|ykǫihxkǫ|hykǫ| (18) Z−∞ between every time step we get N terms, each of the form

−iǫ I I T I I 3 T hxkǫ,ykǫ| exp ~ (H ⊗ − ⊗ H ) × exp −λǫ ⊗ − d rLr ⊗ Lr x(k−1)ǫ,y(k−1)ǫ Z (19)

Evaluating one of these terms

−iǫ I I T I I 3 T hxkǫ,ykǫ| exp ~ H ⊗ − ⊗ H × exp −λǫ ⊗ − d rLr ⊗ Lr x(k−1)ǫ,y(k−1)ǫ Z −iǫ − (x − y )2 = hx ,y | exp H ⊗ I − I ⊗ HT x ,y exp −λǫ 1 − exp (k−1)ǫ (k−1)ǫ kǫ kǫ ~ (k−1)ǫ (k−1)ǫ 4r2 C (20)

The ﬁrst exponent is simply the Feynman propogator1 for Schr¨odinger evolution.2 We now assume that the system is non relativistic and hence, Hamiltonian is quadratic in the momentum. and the potential is position dependent. Thus, taking all N terms we get

m ∞ N−1 m N im U(x ,y , x ,y )= × dx dy × exp ((x − x )2 − (y − y )2) Nǫ Nǫ 0 0 2πǫ~ 2πǫ~ nǫ nǫ 2~ǫ nǫ (n−1)ǫ nǫ (n−1)ǫ −∞ n=1 n=1 Z Y h X iǫ N −(x − y )2 − V (x ) − V (y ) × exp −λǫ 1 − exp (k−1)ǫ (k−1)ǫ ~ (n−1)ǫ (n−1)ǫ 2 " 4rC # i k=1 X (21) m ∞ N−1 m N i = dx dy × exp (S[x , x ] − S[y ,y ]) × 2πǫ~ 2πǫ~ nǫ nǫ ~ nǫ (n−1)ǫ nǫ (n−1)ǫ −∞ n=1 "n=1 # Z Y X N 2 −(x(k−1)ǫ − y(k−1)ǫ) exp −λǫ 1 − exp 2 (22) " 4rC # Xk=1 In the continuum limit with N →∞ while still keeping Nǫ = T , the evolution of the density matrix element thus becomes,

i ρ(xT ,yT , T )= [Dxt][Dyt] × exp ~(S[xt,T,t = 0] − S[yt,T,t = 0]) × Zall paths T −(x − y )2 exp −λ dt 1 − exp t t ρ(x ,y )dx dy (23) 4r2 0 0 0 0 Z0 C 1The propagator is a function that speciﬁes the probability amplitude for a particle to travel from one place to another in a given time, or to travel with a certain energy and momentum. 2Refer R. Shankar [15]; Eq. (21.1.15)

7 where N−1 m 1/2 m 1/2 [Dxt] = lim dxn (24) N→∞ 2π~ǫ 2π~ǫ n=1 Z Z Y This is the same result as derived in [9]. Here, xt and yt can be understood as individual paths that might be traversed. Thus, all paths[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 R GRW induced regulator of the Feynman path integral, and improves the understanding of the classical limit, as we will see in the next 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 ǫ the wave function has a probability λǫ 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 T ǫ = N . So, we have ∞ ∞ i m x − x x + x ρ(x ,y , ǫ)= A exp ( ǫ 0 )2 − V ( ǫ 0 ) ǫ ǫ ǫ ~ 2 ǫ 2 Z−∞ Z−∞ i m y − y y + y exp − ( ǫ 0 )2 − V ( ǫ 0 ) ǫ ρ(x ,y , t = 0)dx dy ~ 2 ǫ 2 0 0 0 0 (25) ∞ ∞ = A P1ρ(x0,y0, t = 0)dx0dy0 (26) Z−∞ Z−∞ where

Pi = Kρ(xi,yi, ǫi; xi−1,yi−1, ǫi−1) (27) i m x − x x + x = exp ( i i−1 )2 − V ( i i−1 ) ǫ ~ 2 ǫ 2 i m y − y y + y exp − ( i i−1 )2 − V ( i i−1 ) ǫ (28) ~ 2 ǫ 2 is density matrix propagator for inﬁnitesimal time step from ǫi−1 to ǫi = ǫi−1 + ǫ. The 3 above expression for ρ(xǫ,yǫ, ǫ) represents standard Schr¨odinger evolution where A is the appropriate normalization constant to recover Von-Neumann Equation. Now from Eq. (8) we know that at a given instant say, t = ǫ the probability of collapse is λǫ while that of it

3Refer R. Shankar [15] Eq. (8.5.4)

8 evolving according to Schr¨odinger’s equation is 1 − λǫ. Thus, the new density matrix after ǫ time becomes, ∞ ρnew(xǫ,yǫ, ǫ)=(1 − λǫ)ρ1 + λǫ Lr(xǫ)ρ1Lr(yǫ)dr (29) Z−∞

where ρ1 = ρ(xǫ,yǫ, ǫ) and Lr(xǫ)= hxǫ| Lr |xǫi are as deﬁned in Eq. (9). Here, since ρ1 does not depend on r (it is a function of xǫ,yǫ, x0 and y0), we can evaluate the above integral by taking ρ1 outside the integration. We get,

∞ ∞ Lr(xǫ)ρ1Lr(yǫ)dr = Lr(xǫ)Lr(yǫ)dr ρ1 (30) Z−∞ Z−∞ ∞ (x − r)2 (y − r)2 = exp − ǫ exp − ǫ dr ρ (31) 2r2 2r2 1 Z−∞ C C (x − y )2 = exp − ǫ ǫ ρ (32) 4r2 1 C Now, we can write 4 (x − y )2 ρ (x ,y , ǫ)=(1 − λǫ)ρ + λǫ exp − ǫ ǫ ρ (33) new ǫ ǫ 1 4r2 1 C For simplicity we write (x − y )2 G = exp − iǫ iǫ i 4r2 C and thus ρnew(xǫ,yǫ, ǫ) = [(1 − λǫ)+ λǫG1] ρ1 (34) We propagate again according to Schr¨odinger’s equation from time t = ǫ to time t =2ǫ, ∞ ρ(x2ǫ,y2ǫ, 2ǫ)= A P2 [(1 − λǫ)+ λǫG1] ρ1dxǫdyǫ (35) Z−∞

Substituting ρ1 according to the Eq. (25) and writing new ρnew(x2ǫ,y2ǫ, 2ǫ), we get ∞ ∞ 2 ρnew(x2ǫ,y2ǫ, 2ǫ)=A ((1 − λǫ)+ λǫG2) P2 ((1 − λǫ)+ λǫG1) −∞ −∞ ∞ ∞ Z Z P1ρ(x0,y0, t = 0)dx0dy0dxǫdyǫ (36) Z−∞ Z−∞ Further, we can rearrange the terms as all the functions P1, P2,G1 and G2 are commutative operations so this gives us, ∞ ∞ ∞ ∞ 2 ρnew(x2ǫ,y2ǫ, 2ǫ)=A P2P1 ((1 − λǫ)+ λǫG2)((1 − λǫ)+ λǫG1) Z−∞ Z−∞ Z−∞ Z−∞ ρ(x0,y0, t = 0)dx0dy0dxǫdyǫ (37)

4Note that now the function in eq.(28) is commutative as it does not contain any diﬀerential operations

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

N−1 N−1 N−1 ρ(xT ,yT , T ) = lim A ··· Pi ((1 − λǫ)+ λǫGi) N→∞ i=0 i=0 Z Z Y Y ρ(x0,y0, t = 0)dx0dy0 ··· dx(N−1)ǫdy(N−1)ǫ (38)

We know that

N N 2 i m xiǫ − x(i−1)ǫ xiǫ + x(i−1)ǫ lim Pi = lim exp − V ǫ N→∞ N→∞ ~ 2 ǫ 2 i=1 i=1 ( " # ) Y Y i m y − y 2 y + y exp − iǫ (i−1)ǫ − V iǫ (i−1)ǫ ǫ ~ 2 ǫ 2 ( " # ) (39) N i m x − x 2 x + x = exp lim iǫ (i−1)ǫ − V iǫ (i−1)ǫ ǫ N→∞ ~ 2 ǫ 2 ( i=1 " # ) X N i m y − y 2 y + y exp lim − iǫ (i−1)ǫ − V iǫ (i−1)ǫ ǫ N→∞ ~ 2 ǫ 2 ( i=1 " # ) X (40) i T i T = exp ~ L (xt) dt exp −~ L (yt) dt (41) Z0 Z0 i = exp ~(S(xt,T,t = 0) − S(yt,T,t = 0)) (42) where, L (xt,T,t = 0) is the Lagrangian and S(xt,T,t = 0) the action thus obtained. Ex- panding the second product term gives us,

N ∞ N k N ( i=1 λGiǫ) lim ((1 − λǫ)+ λǫGi) = lim (1 − λǫ) (43) N→∞ N→∞ k!(1 − λǫ)k i=1 k=0 P Y X N

= exp(−λT ) exp lim λGiǫ (44) N→∞ i=1 ! T X = exp(−λT ) exp λ G(t)dt (45) Z0 T (x − y )2 = exp(−λT ) exp λ exp − t t dt (46) 4r2 Z0 C

10 Substituting these two terms back in Eq. (38) we get an integral form solution of Eq. (8) given by i ρ(xT ,yT , T )= exp ~(S[xt,T,t = 0] − S[yt,T,t = 0]) Z T (x − y )2 exp −λ (1 − exp − t t )dt [Dx ][Dy ]ρ(x ,y , t = 0)dx dy 4r2 t t 0 0 0 0 Z0 C (47) where, the integral in the above equation is deﬁned in the equation (24). The above derived propagator is the same as what we got using the previous method given in equation (23).

3 Classical and Quantum Limits of GRW path integral

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

2 T −(xt−yt) i 4r2 ρ(xT ,yT , T )= exp ~ ( S(xt,T,t = 0) − S(yt,T,t =0) ) exp −λ 1 − e C dt Zall paths " Z0 ! # ρ(x0,y0, t = 0)[Dxt][Dyt]dx0dy0 (48) 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 non-oscillating part of the above given propagator could be approximated as,

2 T −(xt−yt) 4r2 exp −λ (1 − e C )dt ≈ 1 (49) " Z0 # This makes the propagator of GRW look exactly like that for normal quantum mechanics, i ρ(xT ,yT , T )= exp ~ ( S(xt,T,t = 0) − S(yt,T,t =0) ) ρ(x0,y0, t = 0)[Dxt][Dyt]dx0dy0 Zall paths (50) 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 ǫ as, i ǫ mx˙ 2 ρ(x ,y , ǫ)= A exp − V (x) dt ǫ ǫ ~ 2 Z Z0 i ǫ my˙2 exp − − V (y) dt ρ(x ,y , t = 0)dx dy (51) ~ 2 0 0 0 0 Z0 where A is as defined in the previous section. Using the following finite difference substitution x − x x˙ → ǫ 0 ǫ 11 x + x x → 0 ǫ 2 and using the standard substitution of ηx = x0 − xǫ and ηy = y0 − yǫ and rearranging the terms we have

2 mη2 i mηx −i y i ~ ~ ρ(xǫ,yǫ, ǫ)= A e 2ǫ e 2ǫ exp ~ (−V (x)+ V (y)) ǫ ρ(xǫ + ηx,yǫ + ηy, t = 0)dηxdηy Z Z (52) The exponentials oscillate very rapidly as ǫ 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, mη2 ≤ π (53) 2~ǫ Now, Taylor expanding the terms in equation (52) upto ﬁrst order in ǫ i.e. upto order η2 we get

2 mη2 i mηx −i y i i ~ ~ ρ(xǫ,yǫ, ǫ)= A e 2ǫ e 2ǫ 1 − ~V (x)ǫ + ~V (y)ǫ ρ(xǫ + ηx,yǫ + ηy, t = 0)dηxdηy Z Z 2 mη2 i mηx −i y i i ~ ~ = A e 2ǫ e 2ǫ 1 − ~V (x)ǫ + ~V (y)ǫ ρ(xǫ,yǫ, t = 0) Z Z ∂ρ ∂ρ + ηy + ηx ∂y (xǫ,yǫ,t=0) ∂x (xǫ,yǫ,t=0) 2 2 ∂ ρ 2 ∂ ρ 2 + 2 ηy + 2 ηx dηxdηy 2∂y (xǫ,yǫ,t=0) 2∂x (xǫ,yǫ,t=0) 2 mη2 i mηx −i y i ~ ~ = A e 2ǫ e 2ǫ ρ(xǫ,yǫ, t = 0) − ~V (x)ρ(xǫ,yǫ, t = 0)ǫ Z Z 2 2 i ∂ ρ 2 ∂ ρ 2 + V (y)ρ(xǫ,yǫ, t = 0)ǫ + 2 ηy + 2 ηx ~ 2∂y (xǫ,yǫ,t=0) 2∂x (xǫ,yǫ,t=0)

∂ρ ∂ρ + ηy + ηx dηxdηy ∂y (xǫ,yǫ,t=0) ∂x (xǫ,yǫ,t=0)

(54)

−2ǫ~πi 2ǫ~πi Evaluating the Gaussian integral and using A = m m we get i q q i ρ(xǫ,yǫ, ǫ)= ρ(xǫ,yǫ, t = 0) − ~V (x)ρ(xǫ,yǫ, t = 0)ǫ + ~V (y)ρ(xǫ,yǫ, t = 0)ǫ −i~ ∂2ρ i~ ∂2ρ (55) + 2 ǫ + 2 ǫ 2m ∂y (xǫ,yǫ,t=0) 2m ∂x (xǫ,yǫ,t=0)

i = ρ(xǫ,yǫ, t = 0) − ~[H, ρ]ǫ which describes how a density operator evolves in time: dρ i = − [H, ρ] (56) dt ~ 12 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 equations (23) or (47), the path integral for GRW model could be written as,

2 T −(xt−yt) i 4r2 ρ(xT ,yT , T )= exp ~ ( S(xt,T,t = 0) − S(yt,T,t =0) ) exp −λ (1 − e C )dt Zall paths " Z0 # ρ(x0,y0, t = 0)[Dxt][Dyt]dx0dy0 (57)

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 since the collapse rate is directly proportional to 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 ≫ ~.

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. cl cl Here xt and yt are the stationary paths for S(xt,T,t = 0) and S(yt,T,t = 0) respectively in the limit S ≫ ~. Thus the stationary phase approximation leads us to the following equation,

− cl− cl 2 T (xt yt ) i r2 cl cl 4 C ρ(xT ,yT , T )= exp ~ S(xt ,T,t = 0) − S(yt ,T,t = 0) exp −λ (1 − e )dt Z " Z0 # ρ(x0,y0, t = 0)dx0dy0 (58)

cl cl For brevity, we here drop the notation for stationary paths and use xt = xt and yt = 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 non-oscillating part of the propagator could be approximated as,

2 T −(xt−yt) r2 exp −λ (1 − e 4 C )dt ≈ exp [−λT ] (59) " Z0 # 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 vanish. This could also be interpreted as

13 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 non-oscillating part of the propagator could be approximated as, 2 T −(xt−yt) r2 exp −λ (1 − e 4 C )dt ≈ 1 (60) " Z0 # Now, we consider an infinitesimal time step ǫ.

S(xt, ǫ, t = 0)−S(yt, ǫ, t = 0) m 1 m 1 = (x − x )2ǫ − [V (x )+ V (x )] ǫ − (y − y )2ǫ + [V (y )+ V (y )] ǫ 2ǫ2 ǫ 0 2 ǫ 0 2ǫ2 ǫ 0 2 ǫ 0 (61) m 1 1 = (x + y ) − (x + y ) [(x − y ) − (x − y )] ǫ 2 ǫ ǫ 2 0 0 ǫ ǫ 0 0 ǫ ǫ − [V (x ) − V (y )] − [V (x ) − V (y )] (62) 2 ǫ ǫ 2 0 0 Motivated by the above expression, we implement the following change of variables, 1 q¯ = (x + y ) (63) t 2 t t

∆t =(xt − yt) (64) 1 1 U(¯q , ∆ )= V (¯q + ∆ ) − V (¯q − ∆ ) (65) t t t 2 t t 2 t Thus the equation (58) could be written as, i m ǫ ǫ ρ(¯q , ∆ , ǫ)= A exp (¯q − q¯ )(∆ − ∆ ) − U(¯q , ∆ ) − U(¯q , ∆ ) ǫ ǫ ~ ǫ ǫ 0 ǫ 0 2 ǫ ǫ 2 0 0 Z ρ(¯q0, ∆0, t = 0)dx0dy0 (66) As the state of a system is speciﬁed by position and momentum in classical mechanics, we take the Fourier transform of ∆ as given by,

(−ipt∆t) ρ(¯qt,pt, t)= A e ρ(¯qt, ∆t, t)d∆t (67) Z Thus the equation (66) in terms of pt could be written as, i m ǫ ǫ ρ(¯q ,p , ǫ)=A exp ∆ p − ∆ p + (¯q − q¯ )(∆ − ∆ ) − U(¯q , ∆ ) − U(¯q , ∆ ) ǫ ǫ ~ 0 0 ǫ ǫ ǫ ǫ 0 ǫ 0 2 ǫ ǫ 2 0 0 Z ρ(¯q0,p0, t = 0)d∆0d∆ǫdx0dy0 (68)

The ρ(¯qt,pt, t) could be interpreted as the phase space representation of the diagonal terms of the density matrix in the limit S ≫ ~. As the ∆ǫ ≪ rC , U(¯qt, ∆t) could be approximated 2 by Taylor expanding and ignoring ∆t and its higher orders ∂V U(¯q , ∆ ) ≈ ∆ (¯q ) (69) t t t ∂q t

14 The equation (68) could be further simpliﬁed by using the above approximation, 1 i∆ m ǫ ∂V ρ(¯q ,p , ǫ)= exp 0 p − (¯q − q¯ ) − (¯q ) ǫ ǫ N ′ ~ 0 ǫ ǫ 0 2 ∂q 0 Z −i∆ m ǫ ǫ2 ∂V exp ǫ q¯ − q¯ + p + (¯q ) ρ(¯q ,p , t = 0)d∆ d∆ dx dy ~ǫ 0 ǫ m ǫ 2m ∂q ǫ 0 0 0 ǫ 0 0 (70) 1 m ǫ ∂V ǫ ǫ2 ∂V = δ p − (¯q − q¯ ) − (¯q ) δ q¯ − q¯ + p + (¯q ) N ′′ 0 ǫ ǫ 0 2 ∂q 0 0 ǫ m ǫ 2m ∂q ǫ Z ρ(¯q0,p0, t = 0)dx0dy0 (71) 1 ǫ ∂V = ρ(¯q − p ,p + ǫ (¯q ), t = 0) (72) N ′′′ ǫ m ǫ ǫ ∂q ǫ The above equation could also be written as follows by changing the variables of ρ, ǫ ∂V 1 ρ q¯ + p ,p − ǫ (¯q ), t = ǫ = ρ (¯q ,p , t =0) (73) 0 m 0 0 ∂q 0 N ′′′ 0 0

Now Taylor expanding the left hand side around the point (q0, p0, t=0) and equating orders of ǫ, we get, at zeroth order, N ′′′ =1 (74) at ﬁrst order, ∂ρ p ∂ρ ∂V ∂ρ = − 0 + (75) ∂t (¯q0,p0,t=0) m ∂q¯ (¯q0,p0,t=0) ∂q¯ (¯q0) ∂p (¯q0,p0,t=0) and dropping the subscript, ∂ρ = −{ρ, H} (76) ∂t 1 2 where H = 2m p +V (¯q). We refer to this equation (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. Since 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 Eq. (58) would be valid. If we consider the case of a free particle, the stationary paths would be straight lines withx ˙(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

2 − 2 (y −a )2 (x0 ai) 0 j − 2 − 2 ρ(x0,y0, t =0)= Aije r e r i,j=1 X 15 with r ≪ rC . Here, the coeﬃcients Aij can be chosen such that the density matrix is a valid one (i.e. it has unit trace, it is positive semi-deﬁnite, and it is Hermitian). Putting this into Eq. (58), we get

− cl− cl 2 T (xt yt ) r2 i 4 C cl cl ρ(xt,yt, T )= exp −λ (1 − e )dt exp ~ S(xt ,T, = 0) − S(yt ,T,t = 0) Z " Z0 # 2 (x − a )2 (y − a )2 A exp − 0 i exp − 0 j dx dy (77) ij r2 r2 0 0 i,j=1 X We can see that the terms of the initial density matrix

− 2 − 2 − 2 − 2 (x0 a1) (y0 a2) (x0 a2) (y0 a1) − 2 − 2 − 2 − 2 A12e r e r + A21e r e r

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

− cl− cl 2 T (xt yt ) 4r2 −λT exp −λ (1 − e C )dt ≈ e Z0 ! 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 rC . 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 Eqn. (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

16 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 paper, and investigate how it might incorporate spontaneous localisation in quantum ﬁeld theory.

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