*The non-technical presentation of this material is available in my new book for the general reader, available here and as a Kindle version here.*

**The Possibilist Transactional Interpretation:**

**A Unified Account of Relativistic and Non-Relativistic Quantum Theory **

**that Solves the Problem of Measurement**

Ruth E. Kastner

17 February 2015

1 Introduction

The transactional interpretation of quantum mechanics (TI) was initially proposed by John G. Cramer in the 1980s. His most comprehensive exposition is found in Cramer (1986). This time-symmetric interpretation of quantum mechanics gives rise to a physical basis for the Born Rule for the probability of an event. The Born Rule specifies that the probability of an outcome is given by the square of the wave function corresponding to that outcome.

TI was inspired by the Wheeler-Feynman (WF) time-symmetric or ‘direct action’ theory of classical electrodynamics (Wheeler and Feynman 1945, 1949). In the WF theory, radiation is a time-symmetric process. A charge emits a field in the form of half-retarded, half-advanced solutions to the wave equation; the response of absorbers then gives rise to a radiative process that transfers energy from an emitter to an absorber. In this picture, the usual quantum state is called an ‘offer wave’ (OW) and the advanced response from the absorber is called the ‘confirmation wave’ (CW). In terms of state vectors, the OW is represented by a ket, |Y>, and the CW by a dual state vector or ‘brac,’ <F|.

The basic transactional picture has been extended by this author, in a ‘possibilist’ ontology, to the relativistic domain. This version of the interpretation is referred to as ‘Possibilist Transactional Interpretation’ or PTI (Kastner 2012). A theoretical basis can be found in the Davies application of the direct-action picture of fields to quantum electrodynamics (Davies 1971, 1972). However, PTI departs from the Davies treatment in two ways: (i) virtual particles are clearly distinguished from real particles (see also Kastner 2014a) and (ii) the coupling amplitude is identified as the amplitude for generation of an offer or confirmation wave, in a transactional account. These developments allow the interpretation to provide a smooth transition between the non-relativistic and the relativistic domains. The latter can be viewed as the ‘birthplace’ of offer waves, in a physically well-defined (although fundamentally stochastic) manner. These points will be discussed in Sections 3 and 4.

It should also be noted that the direct-action theory of fields provides an elegant and effective escape from Haag’s Theorem, a famously vexing result showing that the interaction picture of the standard quantized fields does not exist. This point is discussed in Kastner (2015).

- How TI explains von Neumann’s Measurement Process and the Born Rule

John von Neumann formulated in precise terms an account of measurement which, despite its practical utility, remains mysterious in any interpretation except TI. To review, von Neumann delineated two different processes that take place in quantum systems. The second of these, which he called ‘Process 2’, is the unitary evolution described by the Schrödinger Equation. The first of these, called ‘Process 1’, is the transition of a quantum system from a pure state to a mixed state upon measurement, i.e.:

The coefficients |c_{n}|^{2} are the probabilities given by the Born Rule for each of the outcomes *y*_{n}.

Von Neumann noted that this transformation is acausal , nonunitary, and irreversible, yet he was unable to explain it in physical terms. He himself spoke of this transition as dependent on an observing consciousness. However, we need not view the measurement process as observer-dependent. If we take into account the advanced responses of absorbers, then for an OW described by |Y>, we have for a collection of numbered absorbers:

In the above diagram, an initial offer wave from emitter **E** passes through some measuring apparatus that separates it into components <y_{n}|Y> |y _{n}>, each reaching a different absorber *n*. Each absorber responds with an advanced (adjoint) confirmation <y_{n}| <Y|y_{ n}>. In TI, these OW/CW encounters are called *incipient transactions*. They are described in probabilistic terms by the product of the OW and CW, which gives a weighted projection operator: <y_{n}|Y><Y|y_{n}> |y _{n}><y _{n }| = |c_{n}|^{2} |y _{n}><y _{n }|. If we add all the incipient transactions, we clearly have the density operator representation of ‘Process 1”.

Thus, by including the advanced responses of absorbers, we have a physical account of measurement as well as a natural explanation of the Born Rule and Von Neumann’s ‘Process 1’. The response of absorbers is what creates the irreversible act of measurement and breaks the linearity of the basic deterministic propagation of the quantum state. Since the conserved physical quantities can only be delivered to one absorber, there is an indeterministic collapse into one of the outcomes y_{k} with a probability given by the weight |c_{k}|^{2 }of the associated projection operator |y _{k}><y _{k }|. This is called an *actualized transaction*, and it consists in the delivery of energy, momentum, angular momentum, etc., to absorber *k*. That absorber that figuratively wins the incipient transaction ‘lottery’ is called the ‘receiving absorber’ in PTI. The process of collapse precipitated in this way by absorber response(s) can be understood as a form of spontaneous symmetry breaking (this is discussed in Chapter 4 of Kastner 2012a).

- The Possibilist Ontology and why it is necessary

If the system under consideration consists only of a single quantum, its associated state vector has only 3 spatial degrees of freedom. In that case, one can think of the wave function as inhabiting spacetime. However, any composite system of N quanta has 3N spatial degrees of freedom, and therefore cannot be considered a spacetime object. Any realist interpretation of the quantum state must take this fact into account. As a realist interpretation, the transactional picture describes real objects that inhabit a realm described not by 3+1 spacetime dimensions but by the 3N dimensions of the relevant Hilbert space (in addition to any spin degrees of freedom). If we think of the spacetime realm as the realm of concrete, actualized events, then the quantum entities described by state vectors must have a different ontological status. In PTI they are viewed as physical possibilities. The latter are precursors to any actualized spacetime event. In view of quantum indeterminism, they are necessary but not sufficient conditions for observable events. The necessary condition is the emission of an offer wave and confirming response(s) to that offer wave from one or more absorbers. This sets up a set of incipient transactions as described in the previous section. At that point we have non-unitary collapse, and only one of the set is actualized to become a spacetime interval with well-defined emission and absorption endpoints—that is the sufficient condition. Thus, not all OW/CW exchanges will result in actualized transactions corresponding to spacetime events.

- The relativistic domain as the birthplace of offer waves

Nonrelativistic quantum mechanics describes a constant number of particles emitted at some locus and absorbed at another. However, in the relativistic domain we are dealing with interactions among various coupling fields, and the number and types of quanta are generally in great flux. Such interactions are described in terms of scattering. The internal connections –i.e., virtual particles– are characterized at each scattering vertex by the coupling amplitude (in the case of QED, the elementary charge *e*) and are neither offers nor confirmations according to relativistic PTI. These are ‘internal lines’ in which the direction of propagation is undefined; i.e., there is no fact of the matter concerning which current ‘emitted’ and which ‘absorbed’ the virtual quantum. In that case, as shown by Davies (1971), the Feynman propagator *D*_{F} of standard QED can be replaced by the time-symmetric propagator in the direct-action theory.

According to PTI, the field coupling amplitudes, which are not present in the non-relativistic case, represent *the amplitude for an offer or confirmation to be generated*. This is a feature of the interpretation appearing only at the relativistic level, in which the number and type of particles can change. It is a natural step, in view of the fact that in standard QED the coupling amplitude is the amplitude for a real photon to be emitted. In PTI, a ‘real photon’ corresponds to an offer wave.[1] So we can think of virtual particles as necessary but not sufficient conditions for real particles, which correspond to offer waves in PTI. We see that the relativistic level brings with it a subtler form of the uncertainty associated with the nonrelativistic form of the theory, i.e., the amplitudes of quantum states themselves.

Thus, in PTI we have a clear distinction between virtual and real particles: virtual particles are not offer waves. Rather, they are *precursors* to offers or confirmations that do not rise to that level. In general, they are not on the mass shell. In the direct-action theory underlying the transactional picture, they are represented by time-symmetric propagators, not Fock space states. On the other hand, real particles correspond to offer waves, which are on the mass shell and are represented by Fock space states. In the direct action picture, offer waves are always responded to by confirmations, so the term ‘real photon’ (in the context of QED) can refer either to a Fock space state |*k*> (i.e., an OW) or to the projection operator |**k**><**k**| (where **k** is 3-momentum), depending on the context.

One of the criticisms of the original Cramer theory was that it took emitters and absorbers as primitive. PTI overcomes this limitation by providing well-defined physical conditions for the generation of OW and CW: both have their origins in the incessant virtual particle activity that is the coupling between fields, such as between the Dirac field and the electromagnetic field. When the interacting photons are on the mass shell and satisfy energy conservation, they are capable of achieving OW status, which in turn generates a CW response from eligible interacting fermionic currents. This is an inherently indeterministic process, since it is described by two factors of the coupling amplitude—i.e., the fine structure constant for QED (as well as the square of the relevant transition amplitude between the initial and final states, with satisfaction of energy conservation). The two factors of the coupling amplitude correspond to the two vertices that are involved in each scattering interaction.

Details of the conditions for the emission of an offer wave instead of an internal process (i.e. a propagator linking two vertices) are given in Kastner (2014a), along with a discussion of how this picture can shed light on the computations necessary to obtain atomic decay rates. The latter are spontaneous emission processes in which virtual photon exchange is spontaneously elevated to a real photon offer and confirmation. Such elevations must of course satisfy energy conservation. That is the only way a would-be virtual photon becomes a real photon—only those photons satisfying the mass shell condition and the energy conservation requirements are eligible to be elevated in this way.

One way to visualize the new level of uncertainty presented in the relativistic domain is in terms of a coin flip. In the nonrelativistic theory, the coin flip has two outcomes: (1) an offer wave |Y> is emitted; (2) no offer wave is emitted. So we either have an offer wave or we don’t. Of course, if we do have an offer wave, it is characterized by its amplitudes <x|Y> for reaching various absorbers X. (If we have an offer wave, we also have confirming responses <x| to that offer, so the two always go hand-in-hand.) This is what made the notion of emitter and absorber in the original TI seem primitive or arbitrary: at what point do we call something an emitter or absorber? However, this problem is remedied at the relativistic level. At that level it is as if the coin becomes much thicker, and a third option becomes available: the coin lands on its side. This in-between, ‘sideways’ outcome corresponds to the virtual quantum. This result has a well-defined probability to yield an emitted offer and confirming response(s) (subject also to the satisfaction of conservation laws as noted above). That probability is the square of the coupling amplitude—i.e., the fine structure constant for QED. So we have two levels of probability amplitudes: (i) the usual, nonrelativistic amplitude of a quantum state <x|Y> corresponding to a particular outcome X; and (ii) the relativistic amplitude for a virtual quantum to become a real quantum representable by a quantum state |Y>.

- Spacetime as the growing set of actualized events

It is often claimed that relativity implies a block world – that is, an ever-present spacetime in which all past, present and future events exist in an equally robust sense. The main argument used in support of that claim is termed ‘chronogeometrical fatalism’ (cf. Stein 1991, pp. 148-9). However, that argument rests on certain assumptions, such as the ontological status of ‘lines of simultaneity’, and a substantival view of spacetime as a ‘container’ for events, that do not necessarily hold. (See Kastner 2012a, §8.1.4 for a rebuttal of chronogeometrical fatalism. See also Sorkin (2007) for a rebuttal to this common but erroneous assumption that a ‘block world’ is necessarily implied by relatibity.)

A different model, of a growing spacetime, is perfectly viable; one such model is the ‘causal set’ approach as proposed by Bombelli et al (1987) (see also Sorkin (2003) and Marolf and Sorkin (2006)). A causal set (causet) C is a locally finite partially ordered set of elements on which is defined a binary relation < . The following properties apply to the causet:

(i) transitivity: (∀x, y, z ∈ C)(x ≺ y ≺ z ⇒ x ≺ z)

(ii) irreflexivity: (∀x ∈ C)(x ~< x)

(iii) local finiteness: (∀x, z ∈ C) (cardinality {y ∈ C | x ≺ y ≺ z} < ∞)

Properties (i) and (ii) imply that the elements are acyclic, while (iii) dictates that the elements form a discrete rather than continuous set. The result is a well-defined causal order of distinct events that can be associated with a spacetime manifold possessing a temporal direction defined by that causal order. As new elements are introduced into the causet, its growth describes a growing, temporally unidirectional spacetime.

The actualized transactions of PTI can readily function as the dynamics underlying the process of ‘sprinkling’ new events into the causal set that is the spacetime manifold. An introduction to the basic concepts involved can be found in Kastner (2014b). The addition of new events to the spacetime causal set must be a Poissonian process in order to preserve relativistic covariance. Interestingly, the generation of offer waves is just such a process, since (as noted in the previous section) these are governed by Poissonian decay rates for the bound states, such as atoms, that give rise to those offer waves.

Offers are always responded to by confirmations, one of which will result in the actualization of a transaction that defines a spacetime interval in terms of its corresponding emission and absorption event. The absorption event of such an actualized transaction defines the present for that receiving absorber, while the emission event is actualized in the timelike (or lightlike) past relative to that absorption. The transferred quantum of energy constitutes a link between the two events that establishes their temporal relationship. It is this linking process between actualized events that establishes spacetime intervals and provides a clearly defined structure of the spacetime ‘fabric’, including its naturally oriented temporal direction towards the unactualized future. In this picture, the present is a strictly local property and cannot be extended along a ‘line of simultaneity’ into the spacelike ‘elsewhere’. Rather, events that are unrelated by the partial ordering of the transactional links have no temporal relationship.

- Conclusion

This article has reviewed the essential concepts of the ‘Possibilist Transactional Interpretation’ (PTI) of quantum theory. It has been argued that PTI can provide a solution to the notorious problem of measurement, as well as providing a unified account of nonrelativistic and relativistic quantum theory. At the nonrelativistic level, we deal only with pre-existent offer waves |Y> and their confirmations from one or more absorbers indexed by X, which correspond to dual states <Y|x> <x|. The confirming response constitutes the onset of the measurement process, where the collapse to a particular outcome |x><x| concludes the measurement process. The collapse is not a process that occurs within spacetime, and that is why it has been so notoriously difficult to give a spacetime account of collapse (cf. Aharonov and Albert (1981) and Kastner (2012, §6.7). Rather, collapse corresponds to the creation of spacetime events from the quantum substratum. That substratum is composed of physical possibilities described by quantum states and of the virtual processes, described by time-symmetric propagators, which are the precursors to those states. The two events created via an actualized transaction are (i) the emission and (ii) the absorption of real energy. That transfer of real energy from the emitter to the receiving absorber defines a spacetime interval and a temporal direction, the emitter defining the past and the absorption defining the present for that absorber. Thus we gain deep physical meaning corresponding to the mathematical fact that energy is the generator of time translation.

At the relativistic level, we take into account virtual processes and their couplings with candidate emitters and absorbers. Such virtual processes are necessary but not sufficient conditions for the generation of offers and confirmations. The latter occur on a stochastic basis, where the probabilities for their occurrence correspond to decay rates. Thus, while the transactional picture of measurement involves objective uncertainty, that uncertainty is precisely quantifiable both at the nonrelativistic and relativistic levels..

References

Aharonov, Y. and Albert, D. (1981) . “Can we make sense out of the measurement process in relativistic quantum mechanics?” *Physical Review D 24*, 359- 370.

Bombelli, L., J. Lee, D. Meyer and R.D. Sorkin (1987). “Spacetime as a causal set”, *Phys. Rev. Lett. *59: 521-524.

Cramer J. G. (1986). “The Transactional Interpretation of Quantum Mechanics.” *Reviews of Modern Physics 58*, 647-688.

Davies, P. C. W. (1971).”Extension of Wheeler-Feynman Quantum Theory to the Relativistic Domain I. Scattering Processes,” *J. Phys. A: Gen. Phys. 6*, 836

Davies, P. C. W. (1972).”Extension of Wheeler-Feynman Quantum Theory to the Relativistic Domain II. Emission Processes,” *J. Phys. A: Gen. Phys. 5*, 1025-1036.

Kastner, R. E. (2012). *The Transactional Interpretation of Quantum Mechanics: The Reality of Possibility.* Cambridge University Press.

Kastner, R. E. (2014a). “On Real and Virtual Photons in the Davies Theory of Time-Symmetric Quantum Electrodynamics,” *Electronic Journal of Theoretical Physics* *11*, 75–86. Preprint version: http://arxiv.org/abs/1312.4007

Kastner, R. E. (2014b). “The Emergence of Spacetime: Transactions and Causal Sets,” to appear in Ignazio Licata, ed.*, The Algebraic Way. *Springer.

Kastner, R. E. (2015). “Haag’s Theorem as a Reason to Reconsider Direct-Action Theories,” to appear in *International Journal of Quantum Foundations*. Preprint version: http://arxiv.org/abs/1502.03814

Marolf, D. and R.D. Sorkin,”Geometry from order: causal sets ” in: *Einstein Online* **Vol.** **02** (2006), 1007.

Sorkin, R. D. (2003) “*Causal Sets: Discrete Gravity (Notes for the Valdivia Summer School)*,” In Proceedings of the Valdivia Summer School, edited by A. Gomberoff

Sorkin, R. D. (2007). “Relativity theory does not imply that the future already exists: a counterexample,” in Vesselin Petkov (editor), *Relativity and the Dimensionality of the World*. Springer. Preprint version: http://arxiv.org/abs/gr-qc/0703098

Wheeler, J.A. and R. P. Feynman, “Interaction with the Absorber as the Mechanism of Radiation,” *Reviews of Modern Physics*, 17, 157–161 (1945)

Wheeler, J.A. and R. P. Feynman, “Classical Electrodynamics in Terms of Direct Interparticle Action,” *Reviews of Modern Physics*, 21, 425–433 (1949).

[1] The term ‘real photon’ can also be applied to an actualized transaction, depending on the context. The term is really a conflation of two different physical situations in the standard approach which can be clearly disambiguated in the transactional approach.