Mathematical Formalism for Nonlocal Spontaneous Collapse in Quantum Field Theory

Abstract

Previous work has shown that spontaneous collapse of Fock states of identical fermions can be modeled as arising from random Rabi oscillations between two states. In this paper, a mathematical formalism is presented to incorporate this into many-body quantum field theory. This formalism allows for nonlocal collapse in the context of a relativistic system. While there is no absolute time-ordering of events, this approach allows for a coherent narrative of the collapse process.

Appendices

Appendix 1

1.1 Nonlocality in the Many-Worlds Framework

While the nonlocality intrinsic in the Copenhagen interpretation is well known and discussed [27], it is not as widely appreciated that the many-worlds interpretation [5] also involves nonlocality.

Let us consider a standard EPR-type experiment [23], with two correlated photons sent in opposite directions, as shown in Fig. 1. The two-particle state emitted from the source in this experiment can be written as

$$\begin{aligned} |\psi \rangle = \frac{1}{\sqrt{2}}(|H\rangle |H\rangle + |V\rangle |V\rangle ) \end{aligned}$$

(30)

where $|H\rangle$ is the horizontally polarized state and $|V\rangle$ is the vertically polarized state. The first ket represents the state of the particle which is going to the left (toward detector 1), and the second ket represents the particle which is going to the right (toward detector 2). We can represent these in vector form as

$$\begin{aligned} |H\rangle = \left( \begin{array}{c} 1 \\ 0 \end{array}\right) ,\,\, |V\rangle = \left( \begin{array}{c} 0 \\ 1 \end{array}\right) \end{aligned}$$

(31)

The action of a polarizer at angle $\theta$ relative to the horizontal acting on these states is

$$\begin{aligned} \hat{P} = \left( \begin{array}{cc}\cos ^2\theta &{} \sin \theta \cos \theta \\ \cos \theta \sin \theta &{}\sin ^2\theta \end{array}\right) \end{aligned}$$

(32)

Acting on a single photon, this operator gives the action of Malus’s law for polarizers, namely, the probability of a photon passing through the polarizer is equal to $\cos ^2(\theta -\theta _1)$, where $\theta _1$ is the angle of the photon’s polarization relative to the horizontal. We can see this, for example, by starting with a photon in the state $|H\rangle$ and finding the final state,

$$\begin{aligned} |\psi '\rangle = \hat{P}|H\rangle = \cos \theta (\cos \theta |H\rangle + \sin \theta |V\rangle ) \end{aligned}$$

(33)

The probability of a photon passing through is

$$\begin{aligned} \langle \psi '|\psi '\rangle = \cos ^2\theta (\cos ^2\theta +\sin ^2\theta ) = \cos ^2\theta \end{aligned}$$

(34)

Here we have used the orthogonality of the two polarization states, namely $\langle V| H\rangle = 0$.

We now write the initial state of the system with explicit time and position dependence as

$$\begin{aligned} \frac{1}{\sqrt{2}} (| H(r_1,t_1) \rangle | H(r_2,t_1) \rangle + | V(r_1,t_1) \rangle | V(r_2,t_1) \rangle ) |E_1\rangle |E_2\rangle \end{aligned}$$

(35)

where $r_1$ represents the position of a moving wave packet on the left, and $r_2$ represents the position of a moving wave packet on the right. $|E_1\rangle$ and $|E_2\rangle$ are the many-body states of the detectors and environment prior to any interaction with the photons.

We assume that the photon wave packet on the left encounters detector 1 with its polarizer first. If this polarizer is set to pass horizontally polarized photons, then after its encounter, the state of the system, in the many-worlds approach, is

$$\begin{aligned} \frac{1}{\sqrt{2}}( | D (r_1=R_1,t_2) \rangle | H(r_2,t_2) \rangle |E_2\rangle + | N (r_1=R_1,t_2) \rangle | V(r_2,t_2) \rangle |E_2\rangle ) \end{aligned}$$

(36)

where D indicates a many-body wave function of the many particles in the detector and its environment that make up the detection event of a horizontally polarized photon, and N indicates no detection event, with only heat dissipated in the polarizer. (The slight separation of the polarizers and detectors will be treated as negligible compared to the distance between the detectors on opposite sides, so that these all are assigned the position $R_1$ on the right and $R_2$ on the left.)

At a later time, the wave packet on the right encounters detector 2 with a horizontal polarizer, at which point the system wave function is

$$\begin{aligned} \frac{1}{\sqrt{2}}( | D (R_1,t_3) \rangle | D (R_2,t_3) \rangle + | N(R_1,t_3) \rangle | N(R_2,t_3) \rangle ) \end{aligned}$$

(37)

If we pick the world in which detection of a photon occurs on the left, the state is

$$\begin{aligned} | D (R_1,t_3) \rangle | D (R_2,t_3) \rangle \end{aligned}$$

(38)

while if we pick the world in which no detection occurs, the state is

$$\begin{aligned} | N (R_1,t_3) \rangle | N (R_2,t_3) \rangle \end{aligned}$$

(39)

Suppose now that at the last moment before the photon hits detector 1, a person there suddenly changed its polarizer position to 45$^\circ$. Then at time $t_2$ the state would be

$$\begin{aligned}{} & {} \frac{1}{\sqrt{2}}\left( \frac{1}{\sqrt{2}}( | D (R_1,t_2) \rangle + | N(R_1,t_2) \rangle ) | H(r_2,t_2) \rangle |E_2\rangle \right. \nonumber \\{} & {} \quad \left. +\frac{1}{\sqrt{2}}( | D (R_1,t_2) \rangle - | N (R_1,t_2) \rangle ) | V(r_2,t_2) \rangle |E_2\rangle \right) \end{aligned}$$

(40)

Then when the other wave packet encounters the horizontal polarizer and detector on the right at time $t_3$, the state would be

$$\begin{aligned}{} & {} \frac{1}{\sqrt{2}}\left( \frac{1}{\sqrt{2}}( | D(R_1,t_3) \rangle + | N (R_1,t_3) \rangle ) | D(R_2,t_3) \rangle \right. \nonumber \\{} & {} \quad \left. +\frac{1}{\sqrt{2}}( | D (R_1,t_3) \rangle - | N (R_1,t_3) \rangle ) | N(R_2,t_3) \rangle \right) \nonumber \\{} & {} \quad = \frac{1}{2}\left( | D (R_1,t_3) \rangle | D(R_2,t_3) \rangle + | N (R_1,t_3) \rangle | D(R_2,t_3) \rangle \right. \nonumber \\{} & {} \quad \left. + | D(R_1,t_3) \rangle | N(R_2,t_3) \rangle - | N (R_1,t_3) \rangle \rangle | N(R_2,t_3) \rangle \right) \end{aligned}$$

(41)

If we once again pick the world in which detection of a photon occurred on the left (which will be physically the same as the former world that we picked, since the system is rotationally symmetric), the state would then be

$$\begin{aligned} | D (R_1,t_3) \rangle \frac{1}{\sqrt{2}} \left( | D(R_2,t_3) \rangle + | N(R_2,t_3) \rangle \right) \end{aligned}$$

(42)

while if we pick the world in which no photon was detected on the left, the state would be

$$\begin{aligned} | N (R_1,t_3) \rangle \frac{1}{\sqrt{2}} \left( | D(R_2,t_3) \rangle - | N(R_2,t_3) \rangle \right) \end{aligned}$$

(43)

In each world of the person on the left, the physical state of the detector on the right and any persons observing it has been put into a superposition. Thus, the last-moment rotation of the polarizer by the person on the left has created a different physical state on the right. This is guaranteed no matter how little time elapses between $t_2$ and $t_3$, i.e., even if the detection events are spacelike separated. Since in the many-worlds framework, the wave function of the system is fully reified, this is a real nonlocal change of the wave function due to the action of rotating the polarizer at $R_1$.

This nonlocality of the wave function is well known to many-worlds advocates; e.g., David Wallace [6] quotes David Deutsch favorably as saying, “Quantum theory is a theory of local interactions and non-local states.” Wallace argues that the “worlds” experienced by people are still local, however, as people cannot have access to the wave function at a spacelike distance. However, much of the appeal of the many-worlds view is that that it reifies the quantum states; that is, it treats the full quantum field is a physical reality independent of whether anyone is looking at it. If this is the case, we must affirm that this physical entity has nonlocal effects.

On the other hand, Frank Tipler has presented an argument that the many-worlds hypothesis does not require nonlocality [41]. In that work, he assumed that a measurement apparatus can act to always give the definite state of particle, i.e.,

$$\begin{aligned} \hat{U}|\psi \rangle |M(0) \rangle = |\psi \rangle |M(\psi )\rangle \end{aligned}$$

(44)

where $\hat{U}$ is a unitary evolution operator giving the interaction with the measurement system, $|\psi \rangle$ is the state of the particle of interest, and $|M(0)\rangle$ and $|M(\psi )\rangle$ are the quantum states of the measurement apparatus before and after the measurement. Crucially, the detector state $|M(\psi )\rangle$ is uniquely identified with the state $|\psi \rangle$ that the particle had before the measurement.

In general, this is only possible if the measurement apparatus is set to detect exactly the state $|\psi \rangle$. For example, in the case of a photon hitting a polarizer and detector considered above, if the photon is polarized at $0^\circ$ and the polarizer is set at $0^\circ$, then the above process Eq. (44) will hold true. However, if the photon is polarized at $45^\circ$, then for the setting of the polarizer at $0^\circ$ it will not be true that the detector goes into a state of having definitely detected a photon with polarization at $45^\circ$. Instead, it will project the photon state into either a state with polarization at $0^\circ$ or $90^\circ$. In traditional quantum mechanics, one or the other of these states will occur with a probability given by the Born rule; in the many-worlds approach, the detector goes into a superposition of both possibilities. But this superposition is not the equivalent of having a single definite measurement of a photon with polarization at $45^\circ$; a person living inside one of these two superposed worlds will see only one or the other possibility. This can be seen an example of environmentally induced selection, or einselection, discussed by Zurek and coworkers [20]. The decoherence of the detector allows it to only be one or the other of detecting the polarization states $0^\circ$ or $90^\circ$; in the language of Dirac notation, the detection apparatus forces a preferred set of “basis states,” unlike the propagation of the photon through free space.

The nonlocality of quantum mechanics comes fundamentally from the fact that entangled states of spacelike separated wavepackets can be created. This is intrinsic to the mathematical structure and not removable by any of the interpretations of quantum mechanics that agree with experimental results.

Appendix 2

1.1 Nonlocal Collapse and the Transactional Interpretation

The scenario considered here has some similarities to the transactional interpretation of Cramer and Kastner (e.g., Refs. [37,38,39,40]). Those authors argue correctly that the microscopic equations of physics do not demand an arrow of time, and argue for a type of spontaneous collapse based on the interaction of an emitting atom and receiving atom aided by backwards-in-time, (“advanced”) waves from the receiver.

A full analysis of this view is beyond the scope of this article, but in this appendix, two claims made by these authors are analyzed. The first is that standard quantum field theory includes advanced waves, and the second is that standard quantum field theory already has non-unitary behavior built into it.

1.1.1 Are there Advanced Waves in Quantum Field Theory?

As discussed in many textbooks (e.g., Ref. [28], Chapter 8), the Green’s function for electron or photon propagation is written as

$$\begin{aligned} G_{\vec {k}}(t)\equiv & {} -i \langle\text{vac} | a^{ }_{\vec {k}}(t)a_{\vec {k}}^{\dagger }(0) | \text{vac}\rangle \varTheta (t)\nonumber \\= & {} -ie^{-i\omega _{\vec {k}} t}\varTheta (t) \end{aligned}$$

(45)

where $|\textrm{vac}\rangle$ is the vacuum state. This can be understood physically as the overlap amplitude for two processes: one which which starts with definite creation of an excitation in state $\vec {k}$ at time 0, allows the system to evolve to a later time t, and another process in which the vacuum evolves on its own until time t, and at that time an excitation is created in state $\vec {k}$. In probability language, it is the probability amplitude for a particle remaining in state $\vec {k}$ after a time t has elapsed.

The Green’s function for antiparticles (“holes” in condensed matter) is

$$\begin{aligned} G_{\vec {k}}(0)\equiv & {} i \langle \textrm{vac} | a^\dagger _{\vec {k}}(0) a^{ }_{\vec {k}}(t) | \textrm{vac}\rangle \varTheta (-t) \nonumber \\= & {} ie^{-i\omega _{\vec {k}} t}\varTheta (-t) \end{aligned}$$

(46)

This superficially looks like a backward-in-time traveling wave, because it asks the probability of first removing a particle, and then at a later time creating it. However, this makes sense as a forward-going process in the context of holes, because holes are absences of fermions below the Fermi level. In the standard theory, the energy states of a system are filled up (by Pauli exclusion) with fermions up to some cutoff energy level $E_F$, known as the Fermi level. A hole creation operator therefore corresponds to the removal of an electron in state $\vec {k}$, i.e., an electron destruction operator for a state $\vec {k}$ below the Fermi level. In the same way, a hole destruction operator corresponds to an electron creation operator for a state below the Fermi level. The Green’s function Eq. (46) therefore does not correspond to a wave actually traveling backwards in time; it corresponds to hole creation and destruction operators in the same order as in the case of the electron Green’s function. In the case of electrons in the vacuum of free space, the same applies to positrons with negative energy.

In the case of bosons, there is no Fermi level, so there is no switch to a destruction operator as the effective creation operator. Also, the boson operators do not pick up a − sign when they are commuted. The complementary Green’s function for bosons is then

$$\begin{aligned} G_{\vec {k}}(t)\equiv & {} -i \langle\text{vac} |(a_{\vec {k}}(0) a^{\dagger}_{{k}}(t)| \text{vac}\rangle \varTheta (-t). \end{aligned}$$

(47)

The boson Green’s function Eq. (45) can be switched to the frequency domain by the Fourier transform

$$\begin{aligned} G({\vec {k}},\omega )= & {} -i\int _{-\infty }^{\infty }dt \ e^{i\omega t}\ e^{-i \omega _{{k}} t}\varTheta (t) \nonumber \\= & {} \lim _{\epsilon \rightarrow 0} \ -i \int _{0}^{\infty } dt \ e^{i(\omega -\omega _{{k}}) t} e^{-\epsilon t}\nonumber \\= & {} \frac{1}{\omega -\omega _{{k}} + i\epsilon } \end{aligned}$$

(48)

and the complementary term corresponding has the transform

$$\begin{aligned} G(-{\vec {k}},\omega )= & {} - i\int _{-\infty }^{\infty }dt \ e^{i\omega t}\ e^{i \omega _{{k}} t}\varTheta (-t) \nonumber \\= & {} \lim _{\epsilon \rightarrow 0} \ - i \int _{-\infty }^{0} dt \ e^{i(\omega +\omega _{{k}}) t} e^{\epsilon t}\nonumber \\= & {} \frac{1}{-\omega -\omega _{{k}} + i\epsilon } \end{aligned}$$

(49)

This term accounts for the fact that a particle emitting a boson with momentum $\vec {k}$ and energy $\hbar \omega$ has the same effect as absorbing the same type of boson with momentum $-\vec {k}$ and energy $-\hbar \omega$. Figure 4 shows these two processes separately, which are typically accounted together as a single, effective interaction between the two electrons.

Fig. 4

Two processes for virtual phonon exchange between two electrons

What do we mean by a photon with negative frequency in this case? One interpretation is to treat this as an advanced wave traveling backwards in time. But if we remember the reason why we have two terms, it is because the phonon (and photon) waves are real-valued, and to have a Hermitian operator corresponding to a real amplitude, we need the sum of $a_{\vec {k}}^\dagger + a^{ }_{\vec {k}}$ to appear in every term of the Hamiltonian proportional to that amplitude.

The first Green’s function corresponds to the traveling wave $e^{i(\vec {k}\cdot x - \omega t)}$, while the second, complementary wave corresponds to $e^{i(-\vec {k}\cdot x + \omega t)} = e^{-i(\vec {k}\cdot x - \omega t)}$, which is just the complex conjugate of the first wave. Both of these propagate in the same direction, and the sum of the two is $\cos (\vec {k}\cdot x - \omega t)$. In other words, the field theory simply ensures that the interaction of the electrons responds to the real part of the phonon wave. The use of negative frequency is common in optics to account for the complex conjugate part that gives the real part of traveling waves.

We thus see that for both fermions and bosons, the Green’s functions that are often written as backwards-in-time-traveling waves are not really tachyons! They are simply bookkeeping conveniences in the theory.

Although we have done this calculation for phonons and electrons in a solid, for simplicity, the same argument applies to the case of photons in vacuum, worked out by P.C.W. Davies [42].

1.1.2 Is there Non-unitarity in Quantum Field Theory?

A unitary system cannot give non-unitary behavior; the mathematical approximations of the S-matrix expansion in quantum field theory do not change this.

Does the inclusion of the $i\epsilon$ term in the Green’s functions above mean that there is irreversible, non-unitary behavior intrinsic to quantum field theory? Nominally, this term corresponds to decay proportional to $e^{-\epsilon t}$, which is non-unitary. But as discussed in Ref. [28], Chapter 8, this imaginary term can be seen as arising from a small imaginary self-energy of the states of interest, which in turn corresponds to dissipation due to decoherence derived within the fully unitary quantum field theory. The introduction of the $i\epsilon$ term arises from the need for self-consistency when higher-order terms in the field theory are taken into account, and not as an ad hoc introduction of something non-unitary.

As noted by Davies [42], in an infinite system, unitary evolution gives irreversible behavior which looks like non-unitary behavior, because energy can flow outward forever without returning. In the discussion of Davies, this corresponds to outgoing photons that are never absorbed. This does not mean that there is a general non-unitarity of standard quantum mechanics, but rather that part of the system (at $t=\infty$) has been placed “off the books,” in the same way that an “environment” is often placed off the books in decoherence theory.