The Transactional Interpretation

In the 1980s, responding to what he conceived as deficiencies in the København interpretation of quantum mechanics, John G. Cramer devised the transactional interpretation to augment the older view. Before he could do that, though, he had to have a concise description of the København interpretation. By abstracting an extensive, unsummarized literature, he was able to list five basic principles that summed up the København interpretation for him.

The København Interpretation

We have those five principles as

**K1**: Heisenberg’s indeterminacy principle, which includes wave-particle
duality, the role of canonically conjugate variables as descriptions of physical
observables, and the impossibility of simultaneously measuring certain pairs of
those variables to arbitrary accuracy.

**K2**: Max Born’s statistical interpretation of the state function (ψ),
which calculates the probability density (ρ=ψ*ψ)
and asserts the predictivity of the formalism only for the average behavior of a
group of similar events.

**K3**: Niels Bohr’s complementarity principle, which includes the wholeness
or unity of a microscopic system and a macroscopic apparatus making measurements
of it, the complementary nature of the wave-particle duality, and the character
of the indeterminacy principle as an intrinsic property of nature rather than as
a peculiarity of the measurement process.

**K4**: Werner Heisenberg’s identification of the state function with
"knowledge of the system", which includes use of this concept to explain the
collapse of the state function and to eliminate simple non-locality problems.

**K5**: Heisenberg’s positivism, which focused interpretive discussions
exclusively on observables and refused to discuss "meaning" or "reality".

Elements K1 and K2 in the above set give us the content of what most people think of as quantum mechanics. Physicists use those elements in making calculations that they then compare with observation or experiment. The interpretation makes a connection between the mathematical theory and the physical world revealed by experiment. Without that connection we would have nothing more than a piece of abstract mathematical reasoning empty of all physical content.

Elements K3, K4, and K5 provide the means of keeping the quantum theory free of paradoxes (i.e. internal contradictions and conflicts with the other established theories). They do so by enabling us to deal with unobserved, even unobservable, entities and through them to resolve any paradoxes associated with the collapse of the state function and with non-locality. Unfortunately they have so far failed to resolve the non-locality paradoxes associated with Bell’s theorem. Something is clearly missing from the København interpretation.

Alone of all the theories of physics quantum mechanics presents us with a central entity that we cannot measure. In dynamics we know how to measure things like force and acceleration; we know how to measure the strengths of electric and magnetic fields; in thermodynamics we know how to measure temperature and heat; but in the quantum theory we have no way to measure the state function that we associate with a particle or system of particles. The state function seems to describe a ghost.

Although it has the mathematical form that describes a wave, standing and/or propagating, the state function is in no way analogous to a real wave, such as an electromagnetic wave. Instead, in accordance with element K4 above, the state function encodes knowledge that we can obtain from the particle when we make some measurement with it. When a measurement or other interaction changes a particle’s accidental properties the state function changes instantly everywhere. That latter fact appears to conflict with the requirements of Relativity, but the solution of Schrödinger’s Equation (for example) doesn’t represent a physical entity; it represents potential information that we can obtain about a physical entity and that, perhaps, can be changed at super-luminal speed.

The Transactional Interpretation

John G. Cramer devised the transactional interpretation to augment the København interpretation; in particular, to solve some problems that he discerned in element K4. He wanted to replace certain evasions involving subjectivity (the association of the state function with an observer’s subjective knowledge) and non-locality with "an objective and explicitly nonlocal description of quantum processes". He sought to develop a version of K4 that would attain four goals:

"1. It should permit the operation of the microcosm to be isolated from the macrocosm and particularly from intrinsically complicated macroscopic concepts, e.g., knowledge, intelligent observers, consciousness, irreversibility, and measurement;

"2. it should account for the nonlocal correlations of the Bell inequality tests in a way consistent with relativity and causality;

"3. it should account for the collapse of the state vector without subjective ‘collapse triggers’ (e.g. consciousness); and

"4. it should give added meaning to the state vector and provide insights into the problems of complexity, completeness, and predictivity."

In the transactional interpretation the state function represents a wave-like physical entity that actually exists in real space. The collapse of the state function in the formation of an emitter-absorber transaction involving an exchange of advanced and retarded waves, a transaction like the one first described by John Wheeler and Richard Feynman in the late 1940s. Non-local correlation effects of the kind associated with Bell’s theorem occur through the advanced waves, which propagate backward in time.

Although Wheeler and Feynman devised their theory to
describe electrodynamic processes, Cramer was able to adapt it readily to
quantum mechanics. Consider how the emitter-absorber transaction model treats
the exchange of a single photon between an emitter in the present and a single
absorber in the future. At a time T_{1} the emitter generates a
time-symmetric combination of a retarded wave (which propagates into the future)
and an advanced wave (which propagates into the past). The retarded wave
propagates away from the emitter until it encounters an absorber at a time T_{2}>T_{1}.
The absorber recoils from the contact, gaining energy and producing a new
retarded wave that exactly cancels the incident wave. The assumption that all
radiative processes are time-symmetric necessitates that the emission of a new
retarded wave be accompanied by the emission of a new, out-of-phase advanced
wave that propagates from the absorber to the emitter from T_{2} to T_{1}.
At the emitter the new advanced wave absorbs energy from the emitter and cancels
the original advanced wave. Thus an observer would see only a photon traveling
from the emitter to the absorber.

In three spatial dimensions the emitter produces the retarded offer wave, which propagates to the absorber, causing the absorber to produce an advanced confirmation wave, which travels back to the emitter to complete the transaction. That process repeats until the net exchange of energy and other conserved quantities between the emitter and the absorber satisfies the quantum boundary conditions of the situation. Another way of seeing that process is to conceive it as the establishment of a four-dimensional standing wave between the emitter and the absorber. That spacetime standing wave is the transaction itself.

Relativistic Quantum Mechanics

Because the transactional interpretation requires the use of both advanced and retarded waves, it only works with Lorentz-invariant wave equations, such as Dirac’s equation or the Klein-Gordon equation. It cannot be used with the solutions of the Schrödinger equation because that equation is semi-classical and has only retarded solutions.

Another implication of relativistic quantum mechanics
concerns Heisenberg’s indeterminacy principle. In the usual form we have
ΔpΔq≥h,
which means that the less precisely we determine the location of a particle, the
more precisely we can measure its momentum. But the speed limit associated with
the propagation of light necessitates a time interval
Δt=Δq/c
to broaden the position localization of the particle enough to obtain a desired
precision on the measurement of Δp,
so we also have ΔpΔt
h/c. Likewise, the precision in which we can determine the particle’s momentum
has an upper limit in a Δp
that corresponds to the particle’s mass-energy, E=mc^{2}, so we have the
imprecision in the particle’s location as
Δq=h/mc,
the de Broglie wavelength associated with the particle. Whereas the original
form of the indeterminacy principle refers to space and time, the relativistic
version must acknowledge the existence of spacetime and the rules associated
with it.

Cramer’s Rules

In devising the transactional interpretation Cramer did not replace the København interpretation; rather, he revised it. Thus we have Cramer’s version of the five elements described above:

**T1**: "The uncertainty principle is as in K1. It is a consequence of the
fact that a transaction in going to completion can project out and localize only
one of a pair of conjugate variables from the offer wave."

**T2**: "The statistical interpretation is unchanged from K2. It is a
consequence of the fact that the ‘echo’ received by the emitter in initiating
the transaction follows the Born probability law ñ=ø*ø."

**T3**: "All physical processes have equal status. The observer, intelligent
or otherwise, has no special status. Measurement and measuring apparatus have no
special status, except that they happen to be processes which connect to
observers. The ‘wholeness’ of K3 exists, but is not related to any special
character of measurements but rather to the connection between emitter and
absorber through the transaction. The ‘complementarity’ concept of K3 likewise
exists, but like the uncertainty principle is just a manifestation of the
requirement that a given transaction going to completion can project out only
one of a pair of conjugate variables."

**T4**: "The fundamental quantum mechanical interaction is taken to be the
transaction, as defined in the preceding section ("The Transactional
Interpretation" above). The state vector of the quantum mechanical formalism is
a real physical wave with spatial extent and is identical with the initial
‘offer wave’ of the transaction. The particle (photon, electron, etc.) and the
collapsed state vector are identical with the completed transaction. The
transaction may involve a single emitter and absorber or multiple emitters and
absorbers, but is only complete when appropriate quantum boundary conditions are
satisfied at all loci of emission and absorption. Particles transferred have no
separate identity which is independent from the satisfaction of these boundary
conditions. The correspondence of the state vector with ‘knowledge of the
system’ of K4 is a fortuitous but deceptive consequence of the transaction, in
that such knowledge must follow and describe the transaction."

**T5**: "A distinction is made between observable and inferred quantities.
The former are firm predictions of the overall theory and may be subjected to
experimental verification. The latter, particularly those which are complex
quantities, are not verifiable and are useful only for interpretational and
pedagogical purposes. It is assumed that both kinds of quantities must obey the
conservation laws, macroscopic causality conditions, relativistic invariance,
etc. Resort to the positivism of K5 is unnecessary and undesirable."

Regarding Born’s Theorem

Element T2 tells us to keep the mathematical formalism of the standard quantum theory, but it gives us a slightly different interpretation of the state function. In Cramer’s interpretation the state function ψ represents the retarded offer wave while its complex conjugate ψ* represents the resulting advanced confirmation wave. The product ψ*ψ represents the offer-confirmation echo, in the form of a probability density, that the emitter receives from a particular direction, so the volume integral

∫_{Vol}ψ*ψdV

represents the sum of all such offer-confirmation echos from all possible locations in space. As usual we can calculate our expectation value for any variable x that an operator X extracts from the state function (Xψ=xψ) through the integral

<x>=∫_{Vol}ψ*XψdV

Thus we gain a kind of average that we can compare with the results of appropriate experiments.

Regarding the State Function

In standard quantum mechanics the state function (or state vector) represents a particle and its transfers of energy and momentum. In transactional quantum mechanics the state function corresponds only to the offer wave emanating from the emitter: the particle itself is represented by the completed transaction. Because of those distinctions the transactional interpretation solves three problems that occur in standard quantum mechanics:

1. The state function represents a real physical wave that spreads out over space. Its non-local nature creates the possibility of action-at-a-distance interactions when it’s interpreted as constituting the complete quantum phenomenon. But interpreted as an offer wave that cannot by itself transfer momentum and energy, the state function no longer has any connection to action-at-a-distance.

2. In order to deal with Problem #1 the founders of standard quantum mechanics asserted that the state function occurs as wave packets, in which energy, momentum, and other properties are localized. But the mathematical description shows wave packets dispersing as they propagate, losing their localization. In the transactional model only the actual formation of a transaction transfers energy and momentum and, thus, localizes them. No wave packets are needed and they don’t appear in transactional quantum mechanics.

3. Because we represent the state function as a wave, either standing or propagating, we must describe it with complex numbers; in particular, complex exponentials. That formulation seems to confront us with the necessity of explaining what the imaginary parts of the state function represent. But only at the emitter and absorber loci (spacetime point-instants, usually marked by events) does a physical interaction occur in consequence of the completed transaction. At those loci we have a superposition of advanced and retarded waves of equal amplitude, so ψ+ψ*=2Re(ψ) and the imaginary part drops out of the collapsed state function. Where there was no interaction, no transfer of energy or momentum, the state function remains complex.

The collapse of the state function corresponds to the completion of a transaction, which emerges from the state function along the entire four-vector that extends between the emission locus and the absorption locus. The transaction thus sends out an influence that enforces the correlations of the quantum event, such as an event involving quantum entanglement, in a way that is explicitly non-local and atemporal.

As we expect, the state function brings to the absorber all possible outcomes of the transaction and the absorber selects one of those outcomes to emerge at the end of the transaction. The selection occurs at random, so we must still use Born’s probability law to calculate expectation values for the variables we want to observe. And, of course, if the transaction localizes one quantity of a canonically conjugate pair, it will also de-localize the other quantity of the pair as required by Heisenberg’s principle.

Causality

We understand the concepts of before and after, that the continuous elapse of time carries us inexorably from the past into the future; thus, we possess an inherent understanding of causality, the idea that cause always precedes effect. But the non-locality of and the use of advanced waves in the transactional model of quantum mechanics raise a question about the validity of causality in describing the quantum nature of Reality.

The emitter-absorber transaction certainly has the effect of enforcing non-local correlations, such as maintaining conservation of angular momentum in the orientation of the spins of widely separated particles used in experiments aimed at testing Bell’s theorem. But it cannot be used to provide non-local (i.e. faster-than-light) communication between observers, because communication requires that one observer be able to control the outcome of a measure but the transactions are still purely stochastic. Because the transaction forms atemporally along the entirety of the four-intervals lying between the emission locus and the absorption loci, both measurements at the absorbers participate equally and symmetrically in the formation of the transaction. That proposition stands true to Reality because the waves involved in the transaction move at the speed of light and, thus, are unaffected by the Lorentz Transformation. The form of the outcome and of its transactional description have the same form in all inertial reference frames, as they must do if Relativity and quantum mechanics are to be compatible with each other.

But looking at the basic transaction described above, we may gain the impression that the absorber causes the transaction and that the effect occurs at the emission locus, where the transaction is completed. But that is a false-to-Reality impression based on the omniscient perspective used in the transactional analysis. In Reality, because all of the waves extending pastward from the emission locus and futureward from the absorption locus all cancel each other, we can only observe what appears to us as a purely retarded wave propagating from the emitter to the absorber, maintaining the proper order of cause and effect.

But consider a wider scale. The transactional formulation of quantum mechanics, with its advanced and retarded waves, is perfectly symmetrical in time, but the history of the Universe is not. If we draw a line representing time, we know that we experience the elapse of time in the same direction in which space expands and that we cannot turn around and go the other way. The transactional interpretation shows us a reason for that fact: there exists a Prime Emitter at what we might call the Zero Locus and what we commonly call the Big Bang. The existence of that Prime Emitter gives us the fundamental boundary condition on all existence, thereby ensuring that the Universe is dominated by retarded waves rather than advanced waves. The relativistic nature of spacetime then ensures that nothing can reverse that evolution. The transactional interpretation of quantum mechanics thus explains what people usually call the Arrow of Time.

Finally we have one last task ahead of us. We need to find a logical path by which we can deduce the transactional version of quantum mechanics from a small set of axioms and/or theorems and thereby incorporate it properly into the Map of Physics. That will be the subject of another essay.

eabf