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The Transactional Interpretation of Quantum Mechanics

by John G. Cramer

2.0 The Copenhagen Interpretation

As was implied in the introduction, we consider the theory of quantum mechanics to be divisible into a formalism and an interpretation. We will assume for the purposes of the present work that the formalism of quantum mechanics is correct and is well supported by experimental evidence. We will therefore focus on the interpretational part of the theory and in particular on the Copenhagen interpretation.

Despite an extensive literature which refers to, discusses, and criticizes the Copenhagen interpretation of quantum mechanics, nowhere does there seem to be any concise statement which defines the full Copenhagen interpretation. We require a definition of the CI for the discussion which follows, and so we have attempted to provide a definitive statement by summarizing the extensive discussions by Jammer (1966) and Audi (1973) in a few sentences, identifying what we consider to be the key concepts. We have been able to identify five principal elements:

These five elements comprise, for the purposes of the present discussion {footnote 10}, the Copenhagen interpretation.

2.0.1 The Functions of the Copenhagen Interpretation

Two distinct functions are performed by the CI (or for that matter, by any physical interpretation of a mathematical formalism). First, as many authors have emphasized, the interpretation must provide a connection between the mathematics of the formalism and the physical world. This connection makes it possible to test the formalism by confronting its predictions with experimental results. Without some interpretation of the symbols of the formalism in terms which can be related to experimental observables the formalism remains abstract mathematics without a physical context. It is perhaps in this sense that Bohr maintained (Popper, 1967) that the Copenhagen interpretation had been "proven by experiment".

However, there is another function of the interpretation which is sometimes overlooked. This function relates to the question of how the theory deals with unobserved objects (Reichenbach, 1944). While participating in a colloquium at Cambridge, von Weizsäcker (1971) denied that the CI asserted: "What cannot be observed does not exist". He suggested instead that the CI follows the principle: "What is observed certainly exists; about what is not observed we are still free to make suitable assumptions. We use that freedom to avoid paradoxes." This principle does not, of course, uniquely define the CI, but it does give an important criterion for developing a consistent interpretation of a formalism. The interpretation must not only relate the formalism to physical observables. It must also define the domain of applicability of the formalism and must interpret the non-observables in such a way as to avoid paradoxes and contradictions.

It may seem surprising that the interpretation of a physical theory can perform the function of avoiding "paradoxes", i.e., internal contradictions and conflicts with other established theories. It is therefore useful to consider some examples. Newton's second law, F=ma, is of no physical significance until the symbol F is identified as a vector representing force, a as a vector representing acceleration, and m as a scalar representing mass. Further, while F and a can have any (real) magnitude and direction, the formalism is interpreted as meaningful only when m>0. This is because zero and negative masses lead to unphysical (or paradoxical) results, e.g., infinite acceleration or acceleration in a direction opposite that of the force vector.

Or consider the Lorentz transformations of special relativity for the case v>c. Until fairly recently physicists had always applied to this case Interpretation A: "The transformations with v>c produce unphysical imaginary values for the transformed variables and are therefore meaningless." But recently an alternative has been suggested by Feinberg (1967, 1978) as Interpretation B: "The transformations in the v>c domain describe a new kind of particle called the tachyon which has the characteristic of imaginary mass, which always travels at velocity v>c, and which approaches the v=c limit asymptotically from above when it is given additional kinetic energy."

While the tachyons of Interpretation B are by no means an established physical phenomenon, this example illustrates how a change in interpretation can alter the meaning of a formalism, can extend the range of its application, and can deal with "paradoxical" or unphysical results", e.g., v>c and imaginary mass. A study of the debate over interpretation in the early history of quantum mechanics (Jammer, 1966) will show a similar process at work in early attempts to interpret the QM formalism. It is this process which produced the Copenhagen interpretation.

In the present context it should be clear that elements CI1 and CI2 fulfill the function of relating the formalism to experiment, while elements CI3-5 perform the function of avoiding paradoxes, and particularly those associated with the collapse of the state vector and with nonlocality (see Sections 2.3 and 2.4). Moreover, it is only elements CI1 and CI2 which are employed by working physicists in using quantum mechanics. Indeed CI1 and CI2 are represented in many quantum mechanics textbooks as "the Copenhagen interpretation". Elements CI3-5 are held in reserve and usually employed only in pedagogical and philosophical discussions. Thus, Bohr's contention that the CI has been "proven by experiment" is perhaps correct as it applies to elements CI1-2 but not as it applies to CI3-5. Moreover, CI4 has, in effect, been tested by experiment (see Section 2.4) and found wanting, in that it has failed to neutralize the manifest nonlocality exhibited by carefully designed Bell Inequality experiments.

In the remainder of this chapter we will list the interpretational problems presented by the QM formalism and will examine these problems from the point of view of the CI, as defined above.

2.1 Identity: What is the State Vector?

In the formalism of quantum mechanics the possible states of a system are described by a state vector (SV), a function (usually complex) which depends on position, momentum, time, energy, spin and isospin variables, etc. The SV (which will be represented as |S> in the notation of Dirac) is the most general form of the quantum mechanical wave function (Psi). The central problem of the interpretation of the QM formalism is to explain the physical significance of the SV. This we will call the problem of identity.

The early semiclassical interpretations of de Broglie (1926, 1927a, 1927b) and of Schrödinger (1927c) attempted to make the obvious and straightforward analogy between the matter waves of quantum mechanics and the classical waves of Maxwellian electrodynamics. This approach asserts that the state vector of an electron, for example, is the QM equivalent of the electric field of an electromagnetic wave. Thus the SV of an electron would be considered to start at the point of emission and to physically travel through space as a wave. It would exhibit the properties of a particle only when (and if) it interacted with a scatterer or an absorber.

This apparently simple interpretation was found to lead to many conceptual problems. In particular, severe problems were found with the intrinsic nonlocality of such an interpretation (see Section 2.4). Heisenberg recognized these problems and argued strongly and successfully against the semiclassical interpretation {footnote 11}. He devised CI4 and CI5 specifically to avoid any association of nonlocal implications with the formalism.

The CI approaches the problem of identity through CI2 and CI4. The statistical interpretation and the probability law of CI2 give limited meaning to the SV by representing it as the vehicle for describing the probabilities of various possible outcomes in a quantum event. This provides the needed connection between quantum mechanical calculations and experimental observations. CI2 is, however, vague on the question of whether there is some unique SV which describes the present and evolving state of the system and on the question of whether the SV has a physical location in space as the semiclassical interpretation would imply.

CI4 is a more radical departure from the semiclassical interpretation in its description of the SV. According to CI4 the SV is not analogous to the electric field of a classical light wave or indeed to any other directly observable entity. Rather it is a mathematical representation of "our knowledge of the system" {footnote6}, (or more properly, that knowledge which is obtained by an ideal observer in an optimum experiment, the latter qualification covering the possibility that the actual experiment performed may be less than optimum due to noise, insensitivity, or other instrumental problems). The SV is approachable only through the results of a physical measurement. The observations from measurements, in an average and statistical way, determine the values of the absolute square of components of the SV. When a measurement is performed our knowledge of the system changes, and therefore the SV also changes. It instantaneously changes all of its components, even those which describe the quantum state in regions of space quite distant from the site of the measurement.

2.1.1 Action at a Distance?

The instantaneous "propagation" of this change gives the appearance of action-at-a-distance, but it is accommodated by CI4 by associating it with a change in knowledge. According to CI4 when the SV describing the state of a particle (perhaps an electron) has a non-zero value at some position in space at some particular time, this does not mean that the SV is physically present at that point but only that our knowledge (or lack of knowledge) of the system allows the particle the possibility of being present at that point at that instant. Therefore, in CI4 the wave function which the Schrödinger equation or its relativistic equivalent provides as a solution is not a physical entity, but rather an encoded mathematical message describing our knowledge of a physical entity.

The identification of the SV in this way raises a number of questions which surround the phrases "our knowledge" and "the system". The language begs the questions "Whose knowledge?" and "What is meant by 'the system'?". The notion that the solution of a simple second-order differential equation (particularly an equation which is only an operator relationship between mass, momentum, and energy) is somehow a mathematical representation of "knowledge" is a very curious and provocative one. The concept of knowledge implies an observer who is the recipient of that knowledge. And because the results of a given experiment often contain information about the state of the system only in a very indirect and highly encoded way, that knowledge may be accessible only to a conscious and intelligent observer. Therefore the observer implicit in the CI has degrees of freedom which are not any explicit part of the QM formalism and which are not characteristics required of observers used, for example, in the interpretation of special relativity.

Further, the concept of knowledge implies stored information, i.e., a memory to store the knowledge, a time sequence before and after the creation of the memory in the mind of the observer, and a flow of information representing a time dependent change in knowledge. Thus the CI implicitly associates with quantum events a time directionality which, while appropriate to macroscopic observers, is quite alien to and inconsistent with the even-handedness with which microphysics deals with the flow of time. Somehow the thermodynamic irreversibility of the macroscopic observer is intruding into the description of a fully reversible microscopic process.

Moreover, the assertion that knowledge is changed by measurement is not free of ambiguity. Measurements performed on any real physical system invariably contain an element of noise which partially obscures the knowledge obtained from the measurement. CI4 makes no provision for such noise, but treats all measurements in the same way, even when the actual signal-to-noise ratio would be such as to preclude any real gain in knowledge from the measurement. The "measurement" which changes the "knowledge" is not the real measurement actually performed, but an ideal measurement from which optimum information is assumed to have been extracted. Further, the "measurement" event is implicitly given a special status which distinguishes it from otherwise identical interaction events, presumably because the measurement interaction effects the knowledge of the observer while otherwise similar interactions do not.

2.1.2 Uniqueness?

The question of the uniqueness of the SV is not directly addressed by CI4. This leads to two possible ways of applying the CI4 "knowledge" interpretation when more than one observer make observations (perhaps simultaneously) on the same quantum mechanical system. These are: {CI4a} There is one unique SV which describes the overall state of knowledge of the system, and this SV is changed when any observer makes a measurement of the state of the system; or {CI4b} There are several non-unique SV's for a given system, each describing the knowledge of some particular observer of the system, and the SV for one such observer is different and distinguishable from the SV for any other observer of the system. In Section 2.4 we will see that each of these alternatives has its own problems.

The seemingly innocuous phrase "the system" has also been found to provide semantic difficulties. Attempts to formulate a quantum mechanical version of general relativity and to employ the CI for its interpretation have foundered in attempting to treat the universe as a whole as a quantum mechanical "system", in the sense of CI4. In such a system there are (presumably) no external observers, and no "knowledge of the system" which can be changed by experiments external to the system. Therefore, CI4 cannot be used for a SV describing the universe as a whole. This calls into question the whole concept.

Moreover, Wigner (1962) has demonstrated (see Sections 4.3 and A.3) that there are severe conceptual problems which arise when CI4 is applied to the SV of any system which includes a conscious observer within it, particularly when measurements on this system are performed by a second conscious observer external to the system. This has led him and others to conclude that the CI implicitly must give a special role to consciousness in the application of CI4.

It is our conclusion from the above considerations that the approach of CI4 to the problem of identity is a relatively superficial one. It has raised as many problems as it has solved and has led its practitioners into very deep philosophical waters. We suspect that the broad acceptance of the CI's identification of the state vector with knowledge is attributable more to the lack of a satisfactory alternative than to its compelling logic.

2.2 Complexity: Why is the State Vector a Complex Quantity?

One of the serious objections to Schrödinger's (1927c) early semi-classical interpretation of the SV, as recounted by Jammer (1974), is that the SV is a complex quantity. Complex functions are also found in classical physics but are invariably interpreted either (1) as an indication that the solution is unphysical, as in the case of the Lorentz transformations with v greater than c or (2) as a shorthand way of dealing with two independent and equally valid solutions of the equations, one real and one imaginary, as in the case of complex electrical impedance. In the latter case the complex algebra is essentially a mathematical device for avoiding trigonometry, and the physical variables of interest are ultimately extracted as the real (or imaginary) part of the complex variables. Never in classical physics is the full complex function "swallowed whole" as it is in quantum mechanics. This is the problem of complexity.

Born's (1926b) probability law (P=PsiPsi*) is the basis of the statistical interpretation which is embodied in CI2. Together with CI4 it provides a way of dealing with the problem of complexity. The SV is not directly observable and is not a real physical entity, and therefore its complex character is irrelevant. All physical observables depend on the absolute squares of the components of the SV, which are always real. CI4 interprets the SV as an encoded mathematical representation of "knowledge" removed from the domain of physical reality and thus makes its complex character more acceptable.

However, this solution of the problem raises some questions of its own. Why is the probability equal to the absolute square of SV elements, rather than to the absolute value, or to the real part [as Born (1926a) first suggested], or to the square of the real part, or some other similar quantity? Why, moreover, is this mathematical representation of "our knowledge of the system" characterized by complex quantities which are very remote from our knowledge? And in particular, why does the SV involve an overall complex phase which can never, by any conceivable experiment, become a part of "our knowledge"?

2.2.1 Complexity and Time

Some insight into these questions can be gained from the observation that the time reversal operator of Wigner (1950) is the operation of complex conjugation, i.e., reversing the sign of the imaginary part or the complex phase of the SV elements. Thus, the complex character of the SV is a manifestation of its time structure. The real part of the SV is time-reversal even, and the imaginary part is time-reversal odd. Moreover, a reversal of the complex phase of the SV reverses its time-sense and the signs of its energy and frequency observables. Thus, CI2, Born's probability law, implicitly tells us that the probability of a particular observation is obtained by taking the product of a component of the SV with its time-reverse. However, the CI provides us with no insight into why this should be the case. Why should probability be compounded of "knowledge" and the time-reverse of knowledge ("information loss"?)?

2.3 Collapse: How and Why Does the State Vector Abruptly Change?

The SV of a system before a measurement is performed is very different from the SV immediately after the measurement, even when the measurement is not the final state of the system but rather one of a series of sequential measurements or operations, e.g., transmission through a polarizing filter or Stern-Gerlach apparatus. Wigner (1962), following von Neumann (1932), has pointed out that there are two distinctly different kinds of changes which the SV undergoes: (1) the SV changes smoothly and continuously with time as the system evolves; and (2) the SV changes abruptly and discontinuously with time in accordance with the laws of probability when (and only when) a measurement is made on the system. He further observed that from the point of view of classical physics these changes seem to be inverted: one would expect classically that the laws of probability and uncertainty would assert themselves in the time evolution of a wave but not in the act of measurement.

A change in the SV of the second type described above is conventionally referred to as the "collapse of the state vector", and we will use this terminology {footnote 12}. It is an aspect of the formalism of quantum mechanics (von Neumann, 1932) rather than its interpretation, and it is the source of many of the most severe interpretational problems. As will be discussed in Section 4, gedanken experiments have been devised to demonstrate that, for example, the collapse can be precipitated by the absence of an interaction with experimental apparatus (Section 4.1), but on the other hand that the SV must remain uncollapsed after a photon has interacted with a pair of slits on the way to an experiment which may determine through which slit the photon has passed (Section 4.2).

Element CI4 deals with the problem of collapse by identifying the SV with "our knowledge of the system", so that measurements which alter such knowledge will produce an abrupt change of type (2) above in the SV as a direct consequence of this change in knowledge. Since the SV is not physically present at the locations in space where it has a non-zero value, an abrupt change in these values does not lead to any problems with propagation times or speed-of-light delays in information transfer. On the other hand, Schrödinger's (1927c) interpretation of the SV as a real semiclassical wave physically present in space has severe intrinsic problems with SV collapse.

However, the CI4 account of collapse is not without its own problems. Wigner (1962) has pointed out (see Section 4.3) the conceptual difficulties implicit in the CI description of collapse when the SV describes a system containing an intelligent observer. He and others have suggested that the process of collapse should involve a special role for consciousness (Wigner, 1962), for permanent recording of experimental results (Schrödinger, 1935), or for entry of the system into the domain of thermodynamic irreversibility (Heisenberg, 1960). In fact, most of the efforts to revise or replace the CI have focused on the problem of collapse, which remains the most puzzling and counter-intuitive aspect of the interpretation of quantum mechanics.

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