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The Transactional Interpretation of Quantum Mechanics

The Transactional Interpretation of QuantumMechanics

Jimmy Aames

Osaka University Graduate School of Human [email protected]

Kochi University of TehchnologyUnified Quantum Device Laboratory Seminar

10/09/2018

[email protected]

Takuju Zen


Outline

1 Introduction

2 The Quantum Measurement ProblemTwo Kinds of Time Evolution in QMThe Inadequacy of the Standard Approach

3 The Wheeler-Feynman Absorber TheoryDerivation of the Electromagnetic Wave EquationsRetarded and Advanced Waves

4 Basic Framework of TIDerivation of Born’s RuleTI’s Solution to the Measurement ProblemApplication to Sequential Stern-Gerlach ExperimentTwo Kinds of Uncertainty

5 TI and Nonlocality


Introduction

Introduction

The Transactional Interpretation (TI) is a time-symmetricinterpretation of quantum mechanics developed by the physicistJohn G. Cramer in a series of papers in the 1980s:

(1) 1980. “Generalized absorber theory and theEinstein-Podolsky-Rosen paradox.” Physical Review D 22: 362–76.(2) 1983. “The arrow of electromagnetic time and the generalizedabsorber theory.” Foundations of Physics 13: 887–902.(3) 1986. “The transactional interpretation of quantummechanics.” Reviews of Modern Physics 58: 647–88.(4) 1988. “An overview of the transactional interpretation.”International Journal of Theoretical Physics 27: 227.


Introduction

Introduction

An alternative version of TI called the Possibilist TransactionalInterpretation (PTI) has also been developed by philosopher ofphysics Ruth E. Kastner. Today I’ll focus on Cramer’s original TI.

Figure: John G. Cramer(Professor Emeritus, University ofWashington)

Figure: Ruth E. Kastner(Research Associate, University ofMaryland)


The Quantum Measurement Problem


One of the main problems that TI was designed to solve is themeasurement problem in quantum mechanics. In a nutshell, themeasurement problem is the problem of when and why a quantumsystem “collapses” to a certain state upon measurement.

The standard formulation of QM adopts John von Neumann’sdistinction between two kinds of time evolution of quantumsystems. (The distinction is introduced by von Neumann in his1932 Mathematische Grundlagen der Quantenmechanik.)



Two Kinds of Time Evolution in QM


The first is what von Neumann called “Process 2.” This is thedeterministic and unitary time evolution of a quantum systemunder Schrodinger’s equation. This is the kind of time evolutionthat occurs when the system is not being measured.

The second is what von Neumann called “Process 1.” This is thenon-deterministic and non-unitary time evolution that occurs uponmeasurement of the quantum system.





Let’s suppose we perform a measurement of an observablerepresented by the operator A on a quantum system prepared inthe pure state |Ψ〉〈Ψ|. Then, Process 1 can be expressed as atransition from this pure state to a mixed state, as follows:

|Ψ〉〈Ψ| →∑i

‖〈Ψ|ai 〉‖2 |ai 〉〈ai |

where |ai 〉 is the eigenvector corresponding to the eigenvalue ai ofA. The value obtained in any single act of measurement will beone of the eigenvalues ai of A. But when a measurement of A isrepeated on many quantum systems prepared in the same purestate |Ψ〉〈Ψ|, the state |ai 〉〈ai | will be realized and the value aiobtained with probability ‖〈Ψ|ai 〉‖2.



The Inadequacy of the Standard Approach

The Inadequacy of the Standard Approach

In the standard formulation of QM, it is not clear when this kind oftransition from a pure state to a mixed state will occur. The notionof “measurement” is left vague—there is no criterion by which wecan judge whether a given physical process is a measurement ornot—and hence there is no way of determining theoretically whenthe “collapse” of the quantum state will take place.

Furthermore, it is not clear in the standard approach why the“collapse” occurs. Is it a purely physical process, or does itsomehow involve the presence of an observer? The question ofwhen and why von Neumann’s “Process 1” occurs is the quantummeasurement problem, and one of the tasks of any interpretationof QM is to provide an answer to this problem.


The Wheeler-Feynman Absorber Theory


Cramer’s TI is a generalization to quantum phenomena of theWheeler-Feynman Absorber Theory (WFAT), a time-symmetricapproach to classical electromagnetism developed by JohnArchibald Wheeler and Richard Feynman.

Maxwell’s equations give rise to two independent time solutions: awave that travels toward the future (called a retarded wave), and awave that travels toward the past (called an advanced wave). Let’ssee how these two kinds of waves follow from Maxwell’s equations.



Derivation of the Electromagnetic Wave Equations


Maxwell’s equations:

∇× E = −∂B∂t

(1)

∇× B =1

c2∂E

∂t+ µ0J (2)

∇ · E =ρ

ε0(3)

∇ · B = 0 (4)





Take the curl of both sides of Faraday’s law of induction (1):

∇× (∇× E) = −∇× ∂B

∂t(5)

Then differentiate both sides of Ampere-Maxwell’s law (2) by t:

∇× ∂B

∂t=

1

c2∂2E

∂t2+ µ0

∂J

∂t(6)

Substituing (5) into (6), we get

∇× (∇× E) +1

c2∂2E

∂t2+ µ0

∂J

∂t= 0





Using the vector identity ∇× (∇× A) = ∇(∇ · A)−∇2A, we get

∇(∇ · E)−∇2E +1

c2∂2E

∂t2+ µ0

∂J

∂t= 0

Using Gauss’ law (3), we have

(∇2 − 1

c2∂2

∂t2)E =

1

ε0∇ρ+ µ0

∂J

∂t(7)

Similarly, we obtain

(∇2 − 1

c2∂2

∂t2)B = −µ0∇× J (8)





In the case of a source-free space, eqs. (7) and (8) reduce to

(∇2 − 1

c2∂2

∂t2)E = 0 (9)

(∇2 − 1

c2∂2

∂t2)B = 0 (10)

These are the electromagnetic wave equations. Since thesedifferential equations are second-order in both time and space, theyhave two independent time solutions and two independent spacesolutions.



Retarded and Advanced Waves


For simplicity, let’s restrict ourselves to one dimension by requiringthat the wave described by eqs. (9) and (10) moves along the xaxis. Then two possible time solutions of eqs. (9) and (10) are

E±(x , t) = E0 sin(kx ± ωt) (11)

and

B±(x , t) = B0 sin(kx ± ωt) (12)

where k and ω are respectively the wave number and angularfrequency of the wave, and the alternating signs represent the twoindependent time solutions.





We can see that the wave corresponding to E− and B− propagatesinto the future (i.e., propagates spatially in the same direction asits temporal propagation), since the value of x for any point ofconstant phase must increase as t increases.

In the case of the wave corresponding to E+ and B+, the spatialindex x must decrease as t increases in order to keep the phaseconstant. Conversely, for the spatial index to increase, the temporalindex must decrease—the wave must propagate “into the past.”





If we suppose that the source is at the origin and is emitting in thepositive x direction, then the waves will exist only for x > 0. Inthis case the wave corresponding to E− and B− will exist onlywhen t > 0, while the wave corresponding to E+ and B+ will existonly when t < 0. Thus the former wave arrives at a point x in atime t after emission, while the latter wave arrives at x in a time tbefore emission. It is for this reason that the former wave is calleda retarded wave, while the latter is called an advanced wave.





Figure: Minkowski diagram of four solutions to the electromagnetic waveequations (source: John G. Cramer, 1986, “The transactionalinterpretation of quantum mechanics,” p. 659)





We can calculate the energy flow produced by an advanced waveas follows. Let’s fix our axes in such a way that the wave istravelling in the positive x direction, the electric field points in thepositive y direction, and the magnetic field points in the positive zdirection. That is, E0 = yE0 and B0 = zB0, where x, y, and z areunit vectors along the Cartesian axes. From Faraday’s law ofinduction (1),

∇× E± = −∂B±∂t

(13)





We also have

∇× E± = ∇× yE0 sin(kx ± ωt) = zE0k cos(kx ± ωt) (14)

and

∂B±∂t

=∂

∂tzB0 sin(kx ± ωt) = ±zB0ω cos(kx ± ωt) (15)

From eqs. (13), (14), and (15), we get B0 = ∓E0k/ω = ∓E0/c .Therefore, the Poynting vector, which indicates the direction andintensity of the wave’s energy flow, is

S = (E± × B±)/µ0 = ∓(y × z)E02 sin2(kx ± ωt)/cµ0





Taking the average over a complete wavelength, this becomes

S = (E± × B±)/µ0 = ∓xE02/2cµ0

where we have used the fact that for arbitrary θ, the average of

sin2 θ over a complete wavelength is sin2 θ = (1− cos 2θ)/2 = 1/2.

The upper sign in the above equation corresponds to an advancedwave which propagates in the positive x direction but whose energyflows in the negative x direction. The advanced wave is thus anegative-energy solution to the electromagnetic wave equations.





Since advanced waves can be detected before they are emitted,they are usually discarded as meaningless solutions. In this sense,the conventional approach imposes an ad hoc direction of time.The Wheeler-Feynman theory, on the other hand, proposestime-symmetric boundary conditions that do not impose an ad hoctime direction.

Firstly, instead of discarding the advanced solutions, it assumesthat all radiating particles emit an electromagnetic wave consistingof a half-amplitude retarded wave and half-amplitude advancedwave that lie along the same worldline (but with opposite timedirections).





Secondly, it assumes that the process of emission is identical withthe process of absorption, and occurs in such a way that the waveproduced by an absorber is 180◦ out of phase with the wave that itreceives from the emitter.

Thirdly, it assumes that an advanced wave can be reinterpreted asa retarded wave by reversing the time direction—that is, we canview the same wave as either diverging from a source into theinfinite past, or as converging onto the source from the infinitepast (see figure on next slide).





Figure: Minkowski diagram of (a) retarded solution and (b) advancedsolution (source: Ruth E. Kastner, 2013, The TransactionalInterpretation of Quantum Mechanics, p. 49)





These conditions immediately lead to a conflict with observation.From the second condition, the emission of a retarded wave isalways accompanied by the emission of an advanced wave. Butsince the advanced wave carries negative energy of the samemagnitude as the retarded wave’s positive energy, the emitter doesnot experience any energy loss in the act of emission—the energyof the retarded and advanced waves cancel each other out. Thisclearly does not fit with observation.

The correspondence with observation is restored if we furtherassume that the emitted retarded wave is absorbed by anotherparticle (see figure on next slide).


Basic Framework of TI


Cramer’s TI is a generalization of the Wheeler-Feynman AbsorberTheory to quantum phenomena (hence in his earlier works it issimply called “Generalized Absorber Theory”). The generalizationis achieved by noting that the fundamental equations of relativisticquantum mechanics, the Klein-Gordon equation

[1

c

∂2

∂t2−∇2 + (mc2/~)2]ψ(x, t) = 0 (16)

and the Dirac equation

(i~γµ∂µ −mc)ψ(x, t) = 0, (17)

give rise to retarded and advanced solutions, just like Maxwell’sequation. Note that for m = 0, the Klein-Gordon equationeffectively reduces to the electromagnetic wave equation.




In TI, any process in which energy quanta are produced is called anemission, and anything that produces energy quanta is called anemitter. Likewise, any process in which energy quanta aredestroyed is called absorption, and anything that destroys energyquanta is called an absorber. Formally, emission and absorption arerepresented by the action of creation and annihilation operators,respectively.

Quantum states represented by kets |Ψ〉 in conventional QM arecalled offer waves in TI. These correspond to the retarded waves inthe Wheeler-Feynman theory. Quantum states represented by bras〈Φ| in conventional QM are called confirmation waves in TI. Thesecorrespond to the advanced waves in the Wheeler-Feynman theory.




Each absorber is assumed to correspond to a state Φ, and that it iscapable of absorbing (annihilating) only the correspondingcomponent of an offer wave that encounters it. Thus, when anoffer wave represented by the state |Ψ〉 reaches an absorbercorresponding to the state Φ, a component of the offer wave equalto the projection of |Ψ〉 onto |Φ〉, i.e. 〈Φ|Ψ〉 |Φ〉, will be absorbedby the absorber. This can be thought of as an “attenuation” of theoriginal offer wave from the perspective of the absorber. Next, inresponse to this absorption, the absorber emits a confirmation waverepresented by 〈Ψ|Φ〉〈Φ|, the adjoint of the absorbed componentof the offer wave, back to the emitter. This is in accord with thefact that the operation of time reversal in QM takes kets into bras.



Derivation of Born’s Rule


As a more general case, assume that there are multiple absorberscorresponding to states a1, a2, a3, etc. The component of the offerwave absorbed by the absorber corresponding to a1 is 〈a1|Ψ〉 |a1〉,the component of the offer wave absorbed by the absorbercorresponding to a2 is 〈a2|Ψ〉 |a2〉, and so on. Likewise, theconfirmation wave emitted by a1 is 〈Ψ|a1〉〈a1|, the confirmationwave emitted by a2 is 〈Ψ|a2〉〈a2|, and so on. If we take the outerproduct of the confirmation wave and the absorbed component ofthe offer wave, we get ‖〈ai |Ψ〉‖2 |ai 〉〈ai | for (i = 1, 2, ...). Thus,the amplitude of the product of the offer and confirmation waves isexactly the probability of measuring ai as given by Born’s rule.





The observable being measured is the one represented by theoperator

∑i ai |ai 〉〈ai |. Each ‖〈ai |Ψ〉‖2 |ai 〉〈ai | represents a

possible, mutually competing transaction, weighted by the value‖〈ai |Ψ〉‖2. These competing transactions are called incipienttransactions. In general, only one of these possible transactions isactualized and its corresponding eigenvalue measured. Theprobability of actualization is given by the weight ‖〈ai |Ψ〉‖2.





The emission-absorption process according to TI can be summedup as follows. An emitter sends out an offer wave or “probe wave”in various directions, seeking a transaction. An absorber, sensingthis offer wave, sends a confirmation wave back to the emitter,confirming the transaction and arranging for the transfer of energy.This is analogous to the way banks transfer money. Just as amoney transfer is not considered complete until the transaction isconfirmed and verified, so an energy transfer does not take placeuntil the offer for energy is confirmed by an absorber; hence thename “transactional interpretation.”



TI’s Solution to the Measurement Problem

TI’s Solution to the Measurement Problem

Recall that the measurement problem is the problem of when andwhy von Neumann’s Process 1

|Ψ〉〈Ψ| →∑i

‖〈Ψ|ai 〉‖2 |ai 〉〈ai |

takes place. Given the framework of TI developed so far, there isnothing mysterious about this transition. It is simply the processwherein absorbers capable of responding to an offer wave returnconfirmation waves to the emitter, establishing a set of incipienttransactions. TI thus explains the colllapse of a quantum state as apurely physical process of energy absorption, rather than as aprocess that somehow involves the observer.



Application to Sequential Stern-Gerlach Experiment


As an example of an application of TI, let’s consider a sequentialStern-Gerlach experiment. The following example is taken from J.J. Sakurai’s Modern Quantum Mechanics (2nd Edition), Chap. 1,Sec. 4 (pp. 32–33).





Suppose we perform a series of selective measurements of the spinstates of a beam of silver atoms, using the sequentialStern-Gerlach apparatus arrangements shown in figs. (a) and (b).

Let’s first consider arrangement (a). The first filter A only letssome particular spin component |a′〉 of the silver atoms throughand blocks all others. Next, filter B selects the component |b′〉 andrejects all others, and finally, filter C selects the component |c ′〉and rejects all others. We want to find the probability of obtaining|c ′〉 when the beam coming out of the first filter is normalized tounity. Since the probabilities are multiplicative, we have∥∥⟨c ′∣∣b′⟩∥∥2∥∥⟨b′∣∣a′⟩∥∥2 (18)





Next, let’s consider the total probability for going through allpossible b′ routes. Operationally, this means that we first recordthe probability of obtaining |c ′〉 with all but the first b′ blocked,then we repeat the procedure with all but the second b′ blocked,and so on; then we sum up all the probabilities. The result is∑

b′

∥∥⟨c ′∣∣b′⟩∥∥2∥∥⟨b′∣∣a′⟩∥∥2 =∑b′

⟨c ′∣∣b′⟩ ⟨b′∣∣a′⟩ ⟨a′∣∣b′⟩ ⟨b′∣∣c ′⟩ (19)





Next let’s consider arrangement (b), where the filter B is absent ornon-operative. In this case, the probability of obtaining |c ′〉 is just‖〈c ′|a′〉‖2, which can also be written as follows:

∥∥⟨c ′∣∣a′⟩∥∥2 =

∥∥∥∥∥∑b′

⟨c ′∣∣b′⟩ ⟨b′∣∣a′⟩∥∥∥∥∥

2

=∑b′

∑b′′

⟨c ′∣∣b′⟩ ⟨b′∣∣a′⟩ ⟨a′∣∣b′′⟩ ⟨b′′∣∣c ′⟩ (20)

Thus, we see that the total probability for obtaining |c ′〉 is differentin arrangements (a) and (b).





In the case of arrangement (a), the probability is given as the sumof squares of the amplitudes, whereas in arrangement (b), theprobability is given as the square of the sum of the amplitudes,giving rise to interference terms that are absent in arrangement(a). This is remarkable, considering that in both cases the |a′〉beam coming out of filter A can be regarded as being made up ofthe B eigenkets: ∣∣a′⟩ =

∑b′

∣∣b′⟩ ⟨b′∣∣a′⟩ (21)





Here is Sakurai’s explanation: “The crucial point to be noted isthat the result coming out of the C filter depends on whether ornot B measurements have actually been carried out. In the firstcase, we experimentally ascertain which of the B eigenvalues areactually realized; in the second case, we merely imagine |a′〉 to bebuilt up of the various |b′〉’s in the sense of eq. (21). Put inanother way, actually recording the probabilities of going throughthe various b′ routes makes all the difference even though we sumover b′ afterwards. Here lies the heart of quantum mechanics” (p.33).





However, even though this kind of difference arises depending onwhether or not we actually carry out measurements, in thestandard interpretation of QM it is not clear what counts as a“measurement” and what does not. In TI, the difference isexplained as follows.

First, in arrangement (a), the beam |a′〉 coming out of filter A isan offer wave. Filter B absorbs a component of this offer waveequivalent to 〈b′|a′〉 |b′〉 and responds with the confirmation wave〈a′|b′〉〈b′|. The product of the amplitudes of the confirmationwave and absorbed component of the offer wave, ‖〈b′|a′〉‖2, givesthe probability for this transaction.





Next, upon receiving the offer wave |b′〉 emitted from filter B,filter C absorbs the component 〈c ′|b′〉 |c ′〉 and returns theconfirmation wave 〈c ′|b′〉〈c ′| to filter B. The probability of thistransaction is thus ‖〈c ′|b′〉‖2.

The reason we square the amplitudes before summing them inarrangement (a) is because there are confirmation waves comingout of filter B. In arrangement (b), on the other hand, we sum theamplitudes before squaring because there are no confirmationwaves coming out of any of the routes of filter B. Thus, in TI,squared amplitudes represent confirmed offer waves.



Two Kinds of Uncertainty


Note that just because an offer wave is confirmed does notnecessarily mean that the state corresponding to the confirmationwave is actually realized. The confirmation of an offer wave onlymeans that there is a determinate fact of the matter as to whetherthat state is actualized, or some other state of the same observableis actualized. In other words, through confirmation and theestablishment of a set of incipient transactions, we shift from asituation of quantum uncertainty (“the cat is in a superposition ofbeing alive and dead”) to a situation of classical uncertainty (“thecat is either alive or dead”).





We have seen that TI explains the transition from quantumuncertainty to classical uncertainty (von Neumann’s Process 1) interms of the physical process of absorption. The question thatarises, then, is: how is the classical uncertainty resolved? That is,how do we explain the process in which only one among thepossible competing transactions is actualized?

One possible approach would be to say that the classicaluncertainty is merely epistemic—the underlying process is actuallydeterministic, and the uncertainty merely reflects our ignorance ofthe underlying process.





This would imply a subjectivist view of probability, according towhich probability statements are really statements about the stateof knowledge of some agent. However, the subjectivist view ofprobability does not sit well with TI, since in TI Born’s probabilityrule is derived from considerations of the purely physical process ofabsorption, without any reference to knowledge, agents, etc.

Ruth Kastner gives an account of the process of actualization bydrawing an analogy with the phenomenon of spontaneoussymmetry breaking.





Figure: Example of spontaneous symmetry breaking: milk drop coronet(source: Harold E. Edgerton, Milk Drop Coronet, 1957)


TI and Nonlocality

TI and Nonlocality

We have seen how TI gives a solution to the measurementproblem, by interpreting it as a process of absroption. Anotherproblem that any interpretation of QM must answer is the problemof nonlocality. This is the problem of how we ought tounderstand the fact that QM violates Bell’s inequality (of whichthere are several variant formulations).

Bell’s inequality is a mathematical relationship that holds for anylocal realist theory. A local realist theory is a theory that satisfiesthe following two conditions:

Locality. The effect of a cause cannot travel faster than light.

Realism. Every observable has a determinate value, regardlessof whether it is measured or not.


TI and Nonlocality

TI and Nonlocality

Bell’s inequality can be derived from these two conditions.However, it can be proved that the predictions of QM areincompatible with Bell’s inequality (Bell’s Theorem). Hence weare faced with two competing view of reality: QM and localrealism.

Experiments conducted by Stuart Freedman and John Clauser in1972 and later refined by Alain Aspect and others in the 1980sgave results that agreed with the predictions of QM, thus providingstrong evidence against local realist theories.


TI and Nonlocality

TI and Nonlocality

Hence we must give up either the assumption of realism or locality(or both). Most physicists and philosophers of physics renouncerealism in order to save locality, since faster-than-light causationimplies a violation of the principle of causality, according towhich a cause must always precede its effect. This is becausesuperluminal influences are influences between events separated bya spacelike interval, and there is no absolute time order for eventsseparated by a spacelike interval—in some reference frames eventP precedes event Q, while in other reference frames Q precedes P.


TI and Nonlocality

TI and Nonlocality

TI, on the other hand, is a non-local theory, and hence iscompatible with realism. To see how TI is non-local, let’s considerFreedman and Clauser’s experimental test of Bell’s inequality(Phys. Rev. Lett. 28: 938–42). Excited calcium atoms undergo anatomic cascade and produce a pair of photons, which are assumedto be emitted back-to-back, in identical polarization states. Eachphoton passes through a linear polarization filter whose axes areset at arbitrary angles θ1 and θ2 which are varied independently,and are detected by photon detectors A and B (see figure on nextslide).


TI and Nonlocality

TI and Nonlocality

Figure: Schematic representation of Freedman-Clauser experiment(source: John G. Cramer, 1986, “The Transactional Interpretation ofQuantum Mechanics”)

The experiment measures the correlation between the readings ofthe photon detectors as a function of the relative angle θ1 − θ2 ofthe polarization filters. The results are in agreement with thepredictions of QM and in conflict with Bell’s inequality. Thequestion is how the correlations are enforced when the photondetection events are separated by a spacelike interval.


TI and Nonlocality

TI and Nonlocality

Figure: Minkowski diagram of the TI analysis of the Freedman-Clauserexperiment (source: John G. Cramer, 1980, “Generalized absorber theoryand the Einstein-Podolsky-Rosen paradox,” p. 369)


TI and Nonlocality

TI and Nonlocality

Let’s consider the offer and confirmation waves to be four-vectorsthat provide lightlike spacetime connections between detectors D1and D2 and the source SO. Let the source be separated from thetwo detectors by radius vectors r1 and r2. The transit time for lightto traverse these distances are t1 and t2 respectively, and these arerelated to the distances by ct1 = r1 and ct2 = r2. The offer wavestravelling from the source to the detectors move along four-vectorswhich we will call R01 and R02, and the confirmation wavesreturning from the detectors to the source move along four-vectorswhich we will call A10 and A20.


TI and Nonlocality

TI and Nonlocality

Then the communication path from detector D1 to detector D2 isA10 + R02 (see figure below).

Figure: Minkowski diagram of the sums of lightlike four-vectors (source:John G. Cramer, 1980, “Generalized absorber theory and theEinstein-Podolsky-Rosen paradox,” p. 369)


TI and Nonlocality

TI and Nonlocality

Noting that A10 = −R01, we have

A10 = −R01 = −(r1, ict1) = −(r1, ir1) (22)

andR02 = (r2, ict2) = (r2, ir2). (23)

Their sum is therefore

A10 + R02 = ((r2 − r1), i(r2 − r1)) (24)


TI and Nonlocality

TI and Nonlocality

The squares of the space and time parts of this vector arerespectively

(r2 − r1)2 = r12 + r2

2 − 2(r1 · r2)

and

(r2 − r1)2 = r12 + r2

2 − 2r1r2.

But since r1r2 ≥ (r1 · r2), this four-vector will be spacelike unless r1and r2 are parallel, which is not the case in any experimental testof Bell’s inequality. TI thus provides a mechanism wherebydetectors can communicate over spacelike intervals and therebyproduce nonlocal correlations.


TI and Nonlocality

TI and Nonlocality

Does this mean that TI embraces the violation of the principle ofcausality? Cramer avoids this conclusion by distinguishing betweentwo forms of the principle:

Strong-causality principle. A cause must always precede allof its effects in any reference frame. Information, microscopicor macroscopic, can never be transmitted over a spacelikeinterval.

Weak-causality principle. A macroscopic cause must alwaysprecede its macroscopic effects in any reference frame.Macroscopic information can never be transmitted over aspacelike interval.


TI and Nonlocality

TI and Nonlocality

Cramer defines a macroscopic cause as “a cause initiated by anobserver,” a macroscopic effect as an “effect which would allow anobserver to receive information,” and macroscopic information as“information which would allow one observer to communicate withanother” (Cramer 1980: 368). Although it is not clear what hemeans by “observer,” he argues that TI violates the strongcausality principle but not the weak causality principle.


TI and Nonlocality

Recommended Reading

Figure: John G. Cramer (2016)The Quantum Handshake:Entanglement, Nonlocality andTransactions. Dordrecht: Springer.

Figure: Ruth E. Kastner (2017)The Transactional Interpretation ofQuantum Mechanics: The Reality ofPossibility. Cambridge: CambridgeUniversity Press.

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